Departamento de Fısica Universidade Federal da Paraıba 58051-970 Joao Pessoa PB Brazil
Correspondence should be addressed to M A Marques mammatheusgmailcom
Received 15 January 2019 Accepted 18 March 2019 Published 3 April 2019
Academic Editor Diego Saez-Chillon Gomez
Copyright copy 2019 M A Marques This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
We investigate the presence of vortex solutions in potentials without vacuumstateThe study is conducted consideringMaxwell andChern-Simons dynamics Also we use a first-order formalism that helps us to find the solutions and their respective electromagneticfields and energy densities As a bonus we get to calculate the energy without knowing the explicit solutions Even though thesolutions present a large ldquotailrdquo which goes far away from the origin the magnetic flux remains a well defined topological invariant
1 Introduction
In high energy physics topological structures appear in adiversity of contexts and have been vastly studied over theyears [1 2] In spatial dimensions lower than three the mostknown ones are kinks and vortices which are static solutionsof the equations of motion
The simplest structures are kinks which appear in (1 1)spacetime dimensions by the action of scalar fields [3] Kinksconnect the minima of the potential and have a topologicalcharacter that ensures its stability However it was shown in[4] that topological defects may arise in potentials without avacuum state whose minima are located at infinity Regard-ing the kink in the vacuumless system it is asymptoticallydivergent and has infinite amplitude Nevertheless it is stableand can be associated with a topological charge by usinga special definition for the topological current [5] Overthe years many papers have studied vacuumless topologicaldefects in a diversity of contexts in high energy physics [6ndash14]
Potentials with extrema at infinity similar to the ones weare going to study here although inverted also appear inclassical mechanics [15] In this scenario if the energy is smallenough themotion is bounded As the energy gets higher theboundary values become far form each other until the limitwhere they are infinitely separated This limit distinguishesbounded and unbounded motion so for sufficiently highvalues of the energy the motion becomes unbounded A
similar situation happens in the interaction of a body withthe gravitational potential 119881 prop minus1119903 which vanishes only at119903 997888rarr infin when one calculates the escape velocity the zeroenergy of the system describes the limit between boundedand unbounded motion In high energy physics vacuumlesspotentials arise in the massless limit of supersymmetric QCDdue to nonperturbative effects [16] They also appear in thecosmological context where their energy densities could actas a cosmological constant that decreases slower than thedensities of matter and radiation [17 18]
By working in (2 1) spacetime dimensions one canfind vortices The first relativistic model that supports theseobjetcs was studied in [19 20] with the action of a complexscalar field coupled to a gauge field under the symmetry119880(1)inMaxwell dynamicsThese structures are electrically neutraland engender a quantized flux which is conserved and worksas a topological invariant Their equations of motion are ofsecond order with couplings between the fields thus they arehard to be solved To simplify the problem theBPS formalismwas developed in [21 22] for this model which allowed forthe presence of first order equations and the energy withoutknowing the explicit form of the solutions
Models with the gauge field governed by the Maxwelldynamics however are not the only ones which supportvortices solutions One can also investigate these structureswith the dynamics of the gauge field governed by the Chern-Simons term [23ndash25] In this case the vortex presents aquantized flux which also is topological invariant and a
HindawiAdvances in High Energy PhysicsVolume 2019 Article ID 9406585 16 pageshttpsdoiorg10115520199406585
2 Advances in High Energy Physics
quantized electric charge The first studies of vortices inChern-Simons dynamics are [26ndash28] for more on this see[29]
The importance of vortices in high energy physics andin other areas of physics can be found in [1 2 30] Forinstance they may appear during the cosmic evolution ofour Universe [1] and in models that includes the so-calledhidden sector which is of interest in dark matter [31ndash34] byenlarging the symmetry to119880(1)times119880(1) see [35ndash37] Followingthis direction of enlarged symmetries they are also presentin119880(1) times 119878119874(3)models with the addition of extra degrees offreedom to the vortex via the inclusion of a triplet scalar fieldand in119880(1)times1198852models with the inclusion of a neutral scalarfield that acts as a source to the internal structure of the vortex[38] Other motivations come from the context of condensedmatter where they may emerge in superconductors and inmagnetic materials as magnetic domains [39] They mayalso appear in dipolar Bose-Einstein condensates where theatoms interact as dipole-dipole which leads to the presenceof non standard vortex structures [40ndash42]
Topological structures may be studied with generalizedmodels [43 44] Vortices in particular firstly appeared innoncanonical models in [45 46] Since then several worksarised with other motivations In the context of inflationfor instance a model with a modified kinetic term wasintroduced in [47] In this scenario these models presentdistinct features from the standard case they may not needa potential to drive the inflation Moreover generalizedmodels were used in [48 49] as a tentative to explainwhy the universe is accelerated at a late stage of its evolu-tion
Noncanonical models considering defect structures wereseverely investigated over the years [50ndash63] Among themany investigations a first-order formalism was developedfor some classes of noncanonical models in [45 46 60 6465] However only in [66] it was completely developed forany generalized model An interesting fact is that compactstructures which were firstly presented in [67] are possibleto appear as Maxwell and Chern-Simons vortices only if gen-eralized models are considered see [68 69] Noncanonicalmodels also allow for the presence of vortices that sharethe same field configuration and energy density known astwinlike models [70]
This work deals with a class of generalized Maxwell andChern-Simons models that support vortex solutions in vac-uumless systems In Section 2 we investigate the propertiesof vortices with Maxwell dynamics including its first-orderformalism and introduce two new models one of themwith analytical results In Section 3 we conduct a similarinvestigation however in the Chern-Simons scenario alsoconsidering its first-order formalism and we introduce twonew models Finally in Section 4 we present our endingcomments and conclusions
2 Maxwell-Higgs Models
We deal with an action in (2 1) flat spacetime dimensionsfor a complex scalar field and a gauge field governed bythe Maxwell dynamics We follow the lines of [66] and
write 119878 = int1198893119909L with the Lagrangian density givenby
L = minus14119865120583]119865120583] + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (1)
In the above equation 120593 denotes the complex scalar field119860120583 is the gauge field 119865120583] = 120597120583119860] minus 120597]119860120583 represents theelectromagnetic strength tensor119863120583 = 120597120583+119894119890119860120583 stands for thecovariant derivative 119890 is the electric charge and 119881(|120593|) is thepotential which is supposed to present symmetry breakingThe function 119870(|120593|) is dimensionless and in principlearbitrary Nevertheless it has to admit solutions with finiteenergy It is straightforward to show that 119870(|120593|) = 1 gives thestandard case considered in [19] One may vary the actionwith respect to the fields 120593 and 119860120583 to get the equations ofmotion
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (2a)
120597120583119865120583] = 119869] (2b)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) and 119881|120593| =119889119881119889|120593| Invariance under spacetime translations 119909120583 997888rarr119909120583 + 119886120583 with 119886120583 constant leads to the energy momentumtensor
In order to investigate vortex solutions in the model weconsider static configurations As a consequence the ] = 0component of (2b) becomes an identity under the choice1198600 = 0 This makes the electric field vanish so the vortex iselectrically uncharged Since we are dealing with two spatialdimensions we define the magnetic field as 119861 = minus11986512 Inthis case the surviving components of the energymomentumtensor (3) are
11987911 = 11986122 + 119870 (10038161003816100381610038161205931003816100381610038161003816) (2 1003816100381610038161003816119863112059310038161003816100381610038162 minus 100381610038161003816100381611986311989412059310038161003816100381610038162) minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (4c)
11987922 = 11986122 + 119870 (10038161003816100381610038161205931003816100381610038161003816) (2 1003816100381610038161003816119863212059310038161003816100381610038162 minus 100381610038161003816100381611986311989412059310038161003816100381610038162) minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (4d)
The energy density is 120588 = 11987900 and the components 119879119894119895 definethe stress tensor We then take the usual ansatz for vortexsolutions
119860 119894 = minus120598119894119895 1199091198951198901199032 [119886 (119903) minus 119899] (5b)
Advances in High Energy Physics 3
where 119903 and 120579 are the polar coordinates and 119899 = plusmn1 plusmn2 is the vorticity The functions 119892(119903) and 119886(119903) must obey theboundary conditions
119892 (0) = 0119886 (0) = 119899
lim119903997888rarrinfin
119892 (119903) = Vlim119903997888rarrinfin
119886 (119903) = 0(6)
In the above equations V is a parameter that is involved in thesymmetry breaking of the potential Considering the ansatz(5a) and (5b) the magnetic field becomes
By integrating it all over the space one can show that the fluxis given by
Φ = int1198892119909119861= 2120587119899119890
(8)
Therefore the magnetic flux is conserved and quantized bythe vorticity 119899 As one knows it is possible to introduce theconserved topological current
in which the component 1198950119879 = 119861 plays the role of a topologicalcharge density By integrating this one can see that the flux(8) plays an important role in the theory since it gives thetopological charge of the system
The equations ofmotion (2a) and (2b) with the ansatz (5a)and (5b) become
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (11b)
11987911 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 cos2120579 minus 1) + 119886211989221199032 (2 sin2120579 minus 1))minus 119881 (119892)
(11c)
11987922 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 sin2120579 minus 1) + 119886211989221199032 (2 cos2120579 minus 1))minus 119881 (119892)
(11d)
As was shown in [66] the stability against contractionsand dilatations in the solutions requires the stressless condi-tion By setting 119879119894119895 = 0 we get the first-order equations
1198921015840 = plusmn119886119892119903and minus 1198861015840119890119903 = plusmnradic2119881 (119892)
(12)
The pair of equations for the upper signs are related tothe lower signs ones by the change 119886(119903) 997888rarr minus119886(119903) Theseequations are compatible with the equations of motion(10a) and (10b) if the potential and the function 119870(|120593|) areconstrained by
For 119870(119892) = 1 we have 119881(|120593|) = 1198902(V2 minus |120593|2)22 which is thestandard case firstly studied in [19]This constraint shows thatgeneralized models are required to study different potentialsand their correspondent vortexlike solutions from the ones ofthe standard caseThe first-order equations (12) also gives riseto the possibility of introducing an auxiliary function119882(119886 119892)in the form
Thus the energy of the stressless solutions may be calculatedwithout knowing their explicit form Below by properlychoosing 119870(|120593|) and 119881(|120593|) that satisfy the constraint in (13)we show new models that engender a set of minima of thepotential at infinity Thus we have V = infin in (6) In order toprepare the model for numerical investigation we work withdimensionless fields and consider unit vorticity 119899 = 1 whichrequires the upper signs in the first-order equations (12)
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05
025
0
05
025
0
0 15 3
K
V
0 1 2
Figure 1 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (17a) and (17b)
21 First Model The first example is given by the pair offunctions
119881 (10038161003816100381610038161205931003816100381610038161003816) = 12 (1 minus tanh (12 100381610038161003816100381612059310038161003816100381610038162))2 (17b)
The above potential does not present a vacuum state that isthe reason we call it vacuumless potential However since119881(infin) = 0 we see the set of mimima of the potencial islocated at infinity which allows it to support vortex solutionsIts maximum is at |120593119898| = 0 with 119881(|120593119898|) = 12 In Figure 1we plot the above functions We see that 119870(|120593|) which is thefunction that controls the kinetic term of the model behavessimilarly to the potential 119881(|120593|) having a maximum in theorigin and its set of minima at infinity
For this model the first-order equations (12) become
1198921015840 = 119886119892119903and 1198861015840119903 = minus(1 minus tanh(11989222 ))
(18)
Near the origin we can study the behavior of the solutions bytaking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to firstorder in 1198860(119903) and 1198920(119903) By substituting them in the aboveequations we get that
It is worth commenting that in this case since the set ofminima of the potential are at infinity we see from theboundary conditions (6) that 119892(119903) is asymptotically divergent
and has infinite amplitude ie 119892(119903 997888rarr infin) 997888rarr infinNevertheless even though 119892(119903) goes to infinity 119886(119903) stillvanishes at infinity similarly to what happens in the standardcase
Although (18) are of first order their nonlinearities makethe job of finding analytical solutions very hard Unfortu-nately we have not been able to find them for these equationsTherefore we must solve them by using numerical methodsIn Figure 2 we plot the solutions of the above equations Nearthe origin we see that the functions vary as expected from(19) As 119903 increases they tend to their boundary values veryslow whichmakes the tail of the solutions be present far awayfrom the originThis behavior is exactly the opposite from theone that appears in models which support compact vorticesin which the solutions attain their boundary values at a finite119903 [68]
Before going further we calculate the function 119882(119886 119892)given by (14)
By using (16) it is straightforward to show that the solutionsof (18) have energy 119864 = 2120587 The magnetic field is given by (7)and the energy density can be calculated from (11a) whichbecomes
We then use our numerical solutions and plot the magneticfield and the energy density in Figure 3 One can see the largetail that the solutions have far away from the origin is less
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1
05
00 100 200
r
a
1
08
060 05 1
3
15
0
g
0 100 200
r
1
05
00 05 1
Figure 2 The solutions 119886(119903) (left) and 119892(119903) (right) of (18) The insets show the behavior of the functions near the origin in the interval119903 isin [0 127]
1
05
0
B
0 3 6r
3
15
0
0 15 3r
Figure 3The magnetic field (left) and the energy density (right) for the solutions of (18)
evident in the magnetic field and in the energy density Bynumerical integration one can show that the magnetic fluxis well defined and given by Φ asymp 2120587 as expected from (8)Since the flux gives the topological charge associated withthe vortex this well defined behavior is different from theone for kinks in vacuumless systems which require a specialdefinition of topological current to get a topological characterwell defined [5] The numerical integration of the energydensity all over the space gives energy 119864 asymp 2120587 whichmatchesthe value obtained with the using of the function119882(119886 119892) in(20)
22 Second Model Our second model arises from the pair offunctions
119870 (10038161003816100381610038161205931003816100381610038161003816) = (2 minus (4 minus 31198782)1198622) 1198782 + (2 minus 31198782) 1198781198622 100381610038161003816100381612059310038161003816100381610038164 (22a)
119881 (10038161003816100381610038161205931003816100381610038161003816) = (1 minus 119878119862)2 11987842 100381610038161003816100381612059310038161003816100381610038164 (22b)
in which we have used the notation 119878 = sech(|120593|) and119862 = |120593| csch(|120593|) Given the above expressions one may
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
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06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
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0 100 200r
a
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08
060 075 15
3
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g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
2 Advances in High Energy Physics
quantized electric charge The first studies of vortices inChern-Simons dynamics are [26ndash28] for more on this see[29]
The importance of vortices in high energy physics andin other areas of physics can be found in [1 2 30] Forinstance they may appear during the cosmic evolution ofour Universe [1] and in models that includes the so-calledhidden sector which is of interest in dark matter [31ndash34] byenlarging the symmetry to119880(1)times119880(1) see [35ndash37] Followingthis direction of enlarged symmetries they are also presentin119880(1) times 119878119874(3)models with the addition of extra degrees offreedom to the vortex via the inclusion of a triplet scalar fieldand in119880(1)times1198852models with the inclusion of a neutral scalarfield that acts as a source to the internal structure of the vortex[38] Other motivations come from the context of condensedmatter where they may emerge in superconductors and inmagnetic materials as magnetic domains [39] They mayalso appear in dipolar Bose-Einstein condensates where theatoms interact as dipole-dipole which leads to the presenceof non standard vortex structures [40ndash42]
Topological structures may be studied with generalizedmodels [43 44] Vortices in particular firstly appeared innoncanonical models in [45 46] Since then several worksarised with other motivations In the context of inflationfor instance a model with a modified kinetic term wasintroduced in [47] In this scenario these models presentdistinct features from the standard case they may not needa potential to drive the inflation Moreover generalizedmodels were used in [48 49] as a tentative to explainwhy the universe is accelerated at a late stage of its evolu-tion
Noncanonical models considering defect structures wereseverely investigated over the years [50ndash63] Among themany investigations a first-order formalism was developedfor some classes of noncanonical models in [45 46 60 6465] However only in [66] it was completely developed forany generalized model An interesting fact is that compactstructures which were firstly presented in [67] are possibleto appear as Maxwell and Chern-Simons vortices only if gen-eralized models are considered see [68 69] Noncanonicalmodels also allow for the presence of vortices that sharethe same field configuration and energy density known astwinlike models [70]
This work deals with a class of generalized Maxwell andChern-Simons models that support vortex solutions in vac-uumless systems In Section 2 we investigate the propertiesof vortices with Maxwell dynamics including its first-orderformalism and introduce two new models one of themwith analytical results In Section 3 we conduct a similarinvestigation however in the Chern-Simons scenario alsoconsidering its first-order formalism and we introduce twonew models Finally in Section 4 we present our endingcomments and conclusions
2 Maxwell-Higgs Models
We deal with an action in (2 1) flat spacetime dimensionsfor a complex scalar field and a gauge field governed bythe Maxwell dynamics We follow the lines of [66] and
write 119878 = int1198893119909L with the Lagrangian density givenby
L = minus14119865120583]119865120583] + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (1)
In the above equation 120593 denotes the complex scalar field119860120583 is the gauge field 119865120583] = 120597120583119860] minus 120597]119860120583 represents theelectromagnetic strength tensor119863120583 = 120597120583+119894119890119860120583 stands for thecovariant derivative 119890 is the electric charge and 119881(|120593|) is thepotential which is supposed to present symmetry breakingThe function 119870(|120593|) is dimensionless and in principlearbitrary Nevertheless it has to admit solutions with finiteenergy It is straightforward to show that 119870(|120593|) = 1 gives thestandard case considered in [19] One may vary the actionwith respect to the fields 120593 and 119860120583 to get the equations ofmotion
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (2a)
120597120583119865120583] = 119869] (2b)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) and 119881|120593| =119889119881119889|120593| Invariance under spacetime translations 119909120583 997888rarr119909120583 + 119886120583 with 119886120583 constant leads to the energy momentumtensor
In order to investigate vortex solutions in the model weconsider static configurations As a consequence the ] = 0component of (2b) becomes an identity under the choice1198600 = 0 This makes the electric field vanish so the vortex iselectrically uncharged Since we are dealing with two spatialdimensions we define the magnetic field as 119861 = minus11986512 Inthis case the surviving components of the energymomentumtensor (3) are
11987911 = 11986122 + 119870 (10038161003816100381610038161205931003816100381610038161003816) (2 1003816100381610038161003816119863112059310038161003816100381610038162 minus 100381610038161003816100381611986311989412059310038161003816100381610038162) minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (4c)
11987922 = 11986122 + 119870 (10038161003816100381610038161205931003816100381610038161003816) (2 1003816100381610038161003816119863212059310038161003816100381610038162 minus 100381610038161003816100381611986311989412059310038161003816100381610038162) minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (4d)
The energy density is 120588 = 11987900 and the components 119879119894119895 definethe stress tensor We then take the usual ansatz for vortexsolutions
119860 119894 = minus120598119894119895 1199091198951198901199032 [119886 (119903) minus 119899] (5b)
Advances in High Energy Physics 3
where 119903 and 120579 are the polar coordinates and 119899 = plusmn1 plusmn2 is the vorticity The functions 119892(119903) and 119886(119903) must obey theboundary conditions
119892 (0) = 0119886 (0) = 119899
lim119903997888rarrinfin
119892 (119903) = Vlim119903997888rarrinfin
119886 (119903) = 0(6)
In the above equations V is a parameter that is involved in thesymmetry breaking of the potential Considering the ansatz(5a) and (5b) the magnetic field becomes
By integrating it all over the space one can show that the fluxis given by
Φ = int1198892119909119861= 2120587119899119890
(8)
Therefore the magnetic flux is conserved and quantized bythe vorticity 119899 As one knows it is possible to introduce theconserved topological current
in which the component 1198950119879 = 119861 plays the role of a topologicalcharge density By integrating this one can see that the flux(8) plays an important role in the theory since it gives thetopological charge of the system
The equations ofmotion (2a) and (2b) with the ansatz (5a)and (5b) become
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (11b)
11987911 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 cos2120579 minus 1) + 119886211989221199032 (2 sin2120579 minus 1))minus 119881 (119892)
(11c)
11987922 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 sin2120579 minus 1) + 119886211989221199032 (2 cos2120579 minus 1))minus 119881 (119892)
(11d)
As was shown in [66] the stability against contractionsand dilatations in the solutions requires the stressless condi-tion By setting 119879119894119895 = 0 we get the first-order equations
1198921015840 = plusmn119886119892119903and minus 1198861015840119890119903 = plusmnradic2119881 (119892)
(12)
The pair of equations for the upper signs are related tothe lower signs ones by the change 119886(119903) 997888rarr minus119886(119903) Theseequations are compatible with the equations of motion(10a) and (10b) if the potential and the function 119870(|120593|) areconstrained by
For 119870(119892) = 1 we have 119881(|120593|) = 1198902(V2 minus |120593|2)22 which is thestandard case firstly studied in [19]This constraint shows thatgeneralized models are required to study different potentialsand their correspondent vortexlike solutions from the ones ofthe standard caseThe first-order equations (12) also gives riseto the possibility of introducing an auxiliary function119882(119886 119892)in the form
Thus the energy of the stressless solutions may be calculatedwithout knowing their explicit form Below by properlychoosing 119870(|120593|) and 119881(|120593|) that satisfy the constraint in (13)we show new models that engender a set of minima of thepotential at infinity Thus we have V = infin in (6) In order toprepare the model for numerical investigation we work withdimensionless fields and consider unit vorticity 119899 = 1 whichrequires the upper signs in the first-order equations (12)
4 Advances in High Energy Physics
05
025
0
05
025
0
0 15 3
K
V
0 1 2
Figure 1 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (17a) and (17b)
21 First Model The first example is given by the pair offunctions
119881 (10038161003816100381610038161205931003816100381610038161003816) = 12 (1 minus tanh (12 100381610038161003816100381612059310038161003816100381610038162))2 (17b)
The above potential does not present a vacuum state that isthe reason we call it vacuumless potential However since119881(infin) = 0 we see the set of mimima of the potencial islocated at infinity which allows it to support vortex solutionsIts maximum is at |120593119898| = 0 with 119881(|120593119898|) = 12 In Figure 1we plot the above functions We see that 119870(|120593|) which is thefunction that controls the kinetic term of the model behavessimilarly to the potential 119881(|120593|) having a maximum in theorigin and its set of minima at infinity
For this model the first-order equations (12) become
1198921015840 = 119886119892119903and 1198861015840119903 = minus(1 minus tanh(11989222 ))
(18)
Near the origin we can study the behavior of the solutions bytaking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to firstorder in 1198860(119903) and 1198920(119903) By substituting them in the aboveequations we get that
It is worth commenting that in this case since the set ofminima of the potential are at infinity we see from theboundary conditions (6) that 119892(119903) is asymptotically divergent
and has infinite amplitude ie 119892(119903 997888rarr infin) 997888rarr infinNevertheless even though 119892(119903) goes to infinity 119886(119903) stillvanishes at infinity similarly to what happens in the standardcase
Although (18) are of first order their nonlinearities makethe job of finding analytical solutions very hard Unfortu-nately we have not been able to find them for these equationsTherefore we must solve them by using numerical methodsIn Figure 2 we plot the solutions of the above equations Nearthe origin we see that the functions vary as expected from(19) As 119903 increases they tend to their boundary values veryslow whichmakes the tail of the solutions be present far awayfrom the originThis behavior is exactly the opposite from theone that appears in models which support compact vorticesin which the solutions attain their boundary values at a finite119903 [68]
Before going further we calculate the function 119882(119886 119892)given by (14)
By using (16) it is straightforward to show that the solutionsof (18) have energy 119864 = 2120587 The magnetic field is given by (7)and the energy density can be calculated from (11a) whichbecomes
We then use our numerical solutions and plot the magneticfield and the energy density in Figure 3 One can see the largetail that the solutions have far away from the origin is less
Advances in High Energy Physics 5
1
05
00 100 200
r
a
1
08
060 05 1
3
15
0
g
0 100 200
r
1
05
00 05 1
Figure 2 The solutions 119886(119903) (left) and 119892(119903) (right) of (18) The insets show the behavior of the functions near the origin in the interval119903 isin [0 127]
1
05
0
B
0 3 6r
3
15
0
0 15 3r
Figure 3The magnetic field (left) and the energy density (right) for the solutions of (18)
evident in the magnetic field and in the energy density Bynumerical integration one can show that the magnetic fluxis well defined and given by Φ asymp 2120587 as expected from (8)Since the flux gives the topological charge associated withthe vortex this well defined behavior is different from theone for kinks in vacuumless systems which require a specialdefinition of topological current to get a topological characterwell defined [5] The numerical integration of the energydensity all over the space gives energy 119864 asymp 2120587 whichmatchesthe value obtained with the using of the function119882(119886 119892) in(20)
22 Second Model Our second model arises from the pair offunctions
119870 (10038161003816100381610038161205931003816100381610038161003816) = (2 minus (4 minus 31198782)1198622) 1198782 + (2 minus 31198782) 1198781198622 100381610038161003816100381612059310038161003816100381610038164 (22a)
119881 (10038161003816100381610038161205931003816100381610038161003816) = (1 minus 119878119862)2 11987842 100381610038161003816100381612059310038161003816100381610038164 (22b)
in which we have used the notation 119878 = sech(|120593|) and119862 = |120593| csch(|120593|) Given the above expressions one may
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
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Advances in High Energy Physics 3
where 119903 and 120579 are the polar coordinates and 119899 = plusmn1 plusmn2 is the vorticity The functions 119892(119903) and 119886(119903) must obey theboundary conditions
119892 (0) = 0119886 (0) = 119899
lim119903997888rarrinfin
119892 (119903) = Vlim119903997888rarrinfin
119886 (119903) = 0(6)
In the above equations V is a parameter that is involved in thesymmetry breaking of the potential Considering the ansatz(5a) and (5b) the magnetic field becomes
By integrating it all over the space one can show that the fluxis given by
Φ = int1198892119909119861= 2120587119899119890
(8)
Therefore the magnetic flux is conserved and quantized bythe vorticity 119899 As one knows it is possible to introduce theconserved topological current
in which the component 1198950119879 = 119861 plays the role of a topologicalcharge density By integrating this one can see that the flux(8) plays an important role in the theory since it gives thetopological charge of the system
The equations ofmotion (2a) and (2b) with the ansatz (5a)and (5b) become
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (11b)
11987911 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 cos2120579 minus 1) + 119886211989221199032 (2 sin2120579 minus 1))minus 119881 (119892)
(11c)
11987922 = 11988610158402211989021199032 + 119870 (119892)sdot (11989210158402 (2 sin2120579 minus 1) + 119886211989221199032 (2 cos2120579 minus 1))minus 119881 (119892)
(11d)
As was shown in [66] the stability against contractionsand dilatations in the solutions requires the stressless condi-tion By setting 119879119894119895 = 0 we get the first-order equations
1198921015840 = plusmn119886119892119903and minus 1198861015840119890119903 = plusmnradic2119881 (119892)
(12)
The pair of equations for the upper signs are related tothe lower signs ones by the change 119886(119903) 997888rarr minus119886(119903) Theseequations are compatible with the equations of motion(10a) and (10b) if the potential and the function 119870(|120593|) areconstrained by
For 119870(119892) = 1 we have 119881(|120593|) = 1198902(V2 minus |120593|2)22 which is thestandard case firstly studied in [19]This constraint shows thatgeneralized models are required to study different potentialsand their correspondent vortexlike solutions from the ones ofthe standard caseThe first-order equations (12) also gives riseto the possibility of introducing an auxiliary function119882(119886 119892)in the form
Thus the energy of the stressless solutions may be calculatedwithout knowing their explicit form Below by properlychoosing 119870(|120593|) and 119881(|120593|) that satisfy the constraint in (13)we show new models that engender a set of minima of thepotential at infinity Thus we have V = infin in (6) In order toprepare the model for numerical investigation we work withdimensionless fields and consider unit vorticity 119899 = 1 whichrequires the upper signs in the first-order equations (12)
4 Advances in High Energy Physics
05
025
0
05
025
0
0 15 3
K
V
0 1 2
Figure 1 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (17a) and (17b)
21 First Model The first example is given by the pair offunctions
119881 (10038161003816100381610038161205931003816100381610038161003816) = 12 (1 minus tanh (12 100381610038161003816100381612059310038161003816100381610038162))2 (17b)
The above potential does not present a vacuum state that isthe reason we call it vacuumless potential However since119881(infin) = 0 we see the set of mimima of the potencial islocated at infinity which allows it to support vortex solutionsIts maximum is at |120593119898| = 0 with 119881(|120593119898|) = 12 In Figure 1we plot the above functions We see that 119870(|120593|) which is thefunction that controls the kinetic term of the model behavessimilarly to the potential 119881(|120593|) having a maximum in theorigin and its set of minima at infinity
For this model the first-order equations (12) become
1198921015840 = 119886119892119903and 1198861015840119903 = minus(1 minus tanh(11989222 ))
(18)
Near the origin we can study the behavior of the solutions bytaking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to firstorder in 1198860(119903) and 1198920(119903) By substituting them in the aboveequations we get that
It is worth commenting that in this case since the set ofminima of the potential are at infinity we see from theboundary conditions (6) that 119892(119903) is asymptotically divergent
and has infinite amplitude ie 119892(119903 997888rarr infin) 997888rarr infinNevertheless even though 119892(119903) goes to infinity 119886(119903) stillvanishes at infinity similarly to what happens in the standardcase
Although (18) are of first order their nonlinearities makethe job of finding analytical solutions very hard Unfortu-nately we have not been able to find them for these equationsTherefore we must solve them by using numerical methodsIn Figure 2 we plot the solutions of the above equations Nearthe origin we see that the functions vary as expected from(19) As 119903 increases they tend to their boundary values veryslow whichmakes the tail of the solutions be present far awayfrom the originThis behavior is exactly the opposite from theone that appears in models which support compact vorticesin which the solutions attain their boundary values at a finite119903 [68]
Before going further we calculate the function 119882(119886 119892)given by (14)
By using (16) it is straightforward to show that the solutionsof (18) have energy 119864 = 2120587 The magnetic field is given by (7)and the energy density can be calculated from (11a) whichbecomes
We then use our numerical solutions and plot the magneticfield and the energy density in Figure 3 One can see the largetail that the solutions have far away from the origin is less
Advances in High Energy Physics 5
1
05
00 100 200
r
a
1
08
060 05 1
3
15
0
g
0 100 200
r
1
05
00 05 1
Figure 2 The solutions 119886(119903) (left) and 119892(119903) (right) of (18) The insets show the behavior of the functions near the origin in the interval119903 isin [0 127]
1
05
0
B
0 3 6r
3
15
0
0 15 3r
Figure 3The magnetic field (left) and the energy density (right) for the solutions of (18)
evident in the magnetic field and in the energy density Bynumerical integration one can show that the magnetic fluxis well defined and given by Φ asymp 2120587 as expected from (8)Since the flux gives the topological charge associated withthe vortex this well defined behavior is different from theone for kinks in vacuumless systems which require a specialdefinition of topological current to get a topological characterwell defined [5] The numerical integration of the energydensity all over the space gives energy 119864 asymp 2120587 whichmatchesthe value obtained with the using of the function119882(119886 119892) in(20)
22 Second Model Our second model arises from the pair offunctions
119870 (10038161003816100381610038161205931003816100381610038161003816) = (2 minus (4 minus 31198782)1198622) 1198782 + (2 minus 31198782) 1198781198622 100381610038161003816100381612059310038161003816100381610038164 (22a)
119881 (10038161003816100381610038161205931003816100381610038161003816) = (1 minus 119878119862)2 11987842 100381610038161003816100381612059310038161003816100381610038164 (22b)
in which we have used the notation 119878 = sech(|120593|) and119862 = |120593| csch(|120593|) Given the above expressions one may
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
119881 (10038161003816100381610038161205931003816100381610038161003816) = 12 (1 minus tanh (12 100381610038161003816100381612059310038161003816100381610038162))2 (17b)
The above potential does not present a vacuum state that isthe reason we call it vacuumless potential However since119881(infin) = 0 we see the set of mimima of the potencial islocated at infinity which allows it to support vortex solutionsIts maximum is at |120593119898| = 0 with 119881(|120593119898|) = 12 In Figure 1we plot the above functions We see that 119870(|120593|) which is thefunction that controls the kinetic term of the model behavessimilarly to the potential 119881(|120593|) having a maximum in theorigin and its set of minima at infinity
For this model the first-order equations (12) become
1198921015840 = 119886119892119903and 1198861015840119903 = minus(1 minus tanh(11989222 ))
(18)
Near the origin we can study the behavior of the solutions bytaking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to firstorder in 1198860(119903) and 1198920(119903) By substituting them in the aboveequations we get that
It is worth commenting that in this case since the set ofminima of the potential are at infinity we see from theboundary conditions (6) that 119892(119903) is asymptotically divergent
and has infinite amplitude ie 119892(119903 997888rarr infin) 997888rarr infinNevertheless even though 119892(119903) goes to infinity 119886(119903) stillvanishes at infinity similarly to what happens in the standardcase
Although (18) are of first order their nonlinearities makethe job of finding analytical solutions very hard Unfortu-nately we have not been able to find them for these equationsTherefore we must solve them by using numerical methodsIn Figure 2 we plot the solutions of the above equations Nearthe origin we see that the functions vary as expected from(19) As 119903 increases they tend to their boundary values veryslow whichmakes the tail of the solutions be present far awayfrom the originThis behavior is exactly the opposite from theone that appears in models which support compact vorticesin which the solutions attain their boundary values at a finite119903 [68]
Before going further we calculate the function 119882(119886 119892)given by (14)
By using (16) it is straightforward to show that the solutionsof (18) have energy 119864 = 2120587 The magnetic field is given by (7)and the energy density can be calculated from (11a) whichbecomes
We then use our numerical solutions and plot the magneticfield and the energy density in Figure 3 One can see the largetail that the solutions have far away from the origin is less
Advances in High Energy Physics 5
1
05
00 100 200
r
a
1
08
060 05 1
3
15
0
g
0 100 200
r
1
05
00 05 1
Figure 2 The solutions 119886(119903) (left) and 119892(119903) (right) of (18) The insets show the behavior of the functions near the origin in the interval119903 isin [0 127]
1
05
0
B
0 3 6r
3
15
0
0 15 3r
Figure 3The magnetic field (left) and the energy density (right) for the solutions of (18)
evident in the magnetic field and in the energy density Bynumerical integration one can show that the magnetic fluxis well defined and given by Φ asymp 2120587 as expected from (8)Since the flux gives the topological charge associated withthe vortex this well defined behavior is different from theone for kinks in vacuumless systems which require a specialdefinition of topological current to get a topological characterwell defined [5] The numerical integration of the energydensity all over the space gives energy 119864 asymp 2120587 whichmatchesthe value obtained with the using of the function119882(119886 119892) in(20)
22 Second Model Our second model arises from the pair offunctions
119870 (10038161003816100381610038161205931003816100381610038161003816) = (2 minus (4 minus 31198782)1198622) 1198782 + (2 minus 31198782) 1198781198622 100381610038161003816100381612059310038161003816100381610038164 (22a)
119881 (10038161003816100381610038161205931003816100381610038161003816) = (1 minus 119878119862)2 11987842 100381610038161003816100381612059310038161003816100381610038164 (22b)
in which we have used the notation 119878 = sech(|120593|) and119862 = |120593| csch(|120593|) Given the above expressions one may
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
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Submit your manuscripts atwwwhindawicom
Advances in High Energy Physics 5
1
05
00 100 200
r
a
1
08
060 05 1
3
15
0
g
0 100 200
r
1
05
00 05 1
Figure 2 The solutions 119886(119903) (left) and 119892(119903) (right) of (18) The insets show the behavior of the functions near the origin in the interval119903 isin [0 127]
1
05
0
B
0 3 6r
3
15
0
0 15 3r
Figure 3The magnetic field (left) and the energy density (right) for the solutions of (18)
evident in the magnetic field and in the energy density Bynumerical integration one can show that the magnetic fluxis well defined and given by Φ asymp 2120587 as expected from (8)Since the flux gives the topological charge associated withthe vortex this well defined behavior is different from theone for kinks in vacuumless systems which require a specialdefinition of topological current to get a topological characterwell defined [5] The numerical integration of the energydensity all over the space gives energy 119864 asymp 2120587 whichmatchesthe value obtained with the using of the function119882(119886 119892) in(20)
22 Second Model Our second model arises from the pair offunctions
119870 (10038161003816100381610038161205931003816100381610038161003816) = (2 minus (4 minus 31198782)1198622) 1198782 + (2 minus 31198782) 1198781198622 100381610038161003816100381612059310038161003816100381610038164 (22a)
119881 (10038161003816100381610038161205931003816100381610038161003816) = (1 minus 119878119862)2 11987842 100381610038161003816100381612059310038161003816100381610038164 (22b)
in which we have used the notation 119878 = sech(|120593|) and119862 = |120593| csch(|120593|) Given the above expressions one may
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
6 Advances in High Energy Physics
V
1
05
0
K
0 1 2
0 1 2
02
01
0
Figure 4 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (22a) and (22b)
wonder if these functions are finite in the origin It is worthto investigate their behavior for |120593| asymp 0 which is given by
119870|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 4445 minus 1676945 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23a)
119881|120593|asymp0 (10038161003816100381610038161205931003816100381610038161003816) = 29 minus 88135 100381610038161003816100381612059310038161003816100381610038162 + O (10038161003816100381610038161003816120593410038161003816100381610038161003816) (23b)
Then they are regular at |120593119898| = 0 which is a pointof maximum with 119881(|120593119898|) = 29 As in the previousmodel this potential also is vacuumless In Figure 4 we plotthese functions Notice that 119870(|120593|) behaves similarly to thepotential
In this case the first-order equations (12) take the form
1198921015840 = 119886119892119903 (24a)
1198861015840119903 = minus(1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (24b)
The behavior near the origin can be studied by considering119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) and going up to first orderin 1198860(119903) and 1198920(119903) Plugging them in the above equations weget the same behavior of (19) The above equations admit thesolutions
Therefore as expected 119892(119903) goes to infinity and 119886(119903) vanishesvery slowly as 119903 increases Then as in the previous model thetail of the solutions is present even for large distances fromthe origin This behavior is shown in Figure 5 in which weplot these solutions
In this case119882(119886 119892) given by (14) takes the form
119882(119886 119892) = minus119886 (1 minus 119892 sech (119892) csch (119892)) sech2 (119892)1198922 (26)
Then from (16) the solutions (25a) and (25b) have energy119864 = 41205873 Since we have the analytical solutions in this casewe can calculate the magnetic field from (7) and the energydensity from (11a) to get
In Figure 6 we plot the magnetic field and the energy densityA direct integration of the magnetic field (27a) gives exactlythe flux in (8) The energy obtained by an integration of theenergy density (27b) gives the same value obtained by the useof the auxiliary function 119882(119886 119892) in (26) that is 119864 = 41205873As in the previous model the long tail of the solutions doesnot seem tomodify the flux of the vortex which remains as in(8)Then the topological current (9) is a definition that leadsto a well behaved topological charge
3 Chern-Simons-Higgs Models
In order to investigate the presence of vortices with theChern-Simons dynamics we consider the action 119878 = int1198893119909L
Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
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[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
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a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
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Advances in High Energy Physics 7
1
05
00 100 200
r
a
1
08
060 1 2
3
15
0
g
0 100 200
r
15
075
00 1 2
Figure 5The solutions 119886(119903) (left) and 119892(119903) (right) as in (25a) and (25b) The insets show the behavior of the functions near the origin in theinterval 119903 isin [0 216]
08
04
0
B
0 3 6
r
2
1
00 1 2
r
Figure 6 The magnetic field in (27a) (left) and the energy density as in (27b) (right)
for a complex scalar field and a gauge field Here we study theclass of generalized models presented in [64]
L = 1205814120598120572120573120574119860120572119865120573120574 + 119870 (10038161003816100381610038161205931003816100381610038161003816) 119863120583120593119863120583120593 minus 119881 (10038161003816100381610038161205931003816100381610038161003816) (28)
In the above expression 120593 119860120583 119890 119863120583 = 120597120583 + 119894119890119860120583119865120583] = 120597120583119860] minus 120597]119860120583 and 119881(|120593|) have the same meaning ofthe previous section Here 120581 is a constant Regarding thedimensionless function 119870(|120593|) it is in principle arbritraryThe only restriction for it is to provide solutions with finiteenergy The standard case is given by 119870(|120593|) = 1 and was
studied in [27] Here we consider 119860120583 = (1198600 997888rarr119860) Thus theelectric and magnetic fields are
with the dot meaning the temporal derivative and (119864119909 119864119910) equiv119864119894 where 119894 = 1 2 The equations of motion for the scalar andgauge fields read
119863120583 (119870119863120583120593) = 1205932 10038161003816100381610038161205931003816100381610038161003816 (119870|120593|119863120583120593119863120583120593 minus 119881|120593|) (30a)
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
where the current is 119869120583 = 119894119890119870(|120593|)(120593119863120583120593 minus 120593119863120583120593) Since theChern-Simons term in the Lagrangian density (28) is metric-free it does not contribute to the energy momentum tensorwhich has the form
We now consider static solutions and the same ansatz of (5a)and (5b) with the boundary conditions (6) This makes theelectric and magnetic fields in (29) have the form
The magnetic flux can by calculated and it is given by (8)which shows that it is quantized and conservedTherefore theMaxwell andChern-Simons vortices share the samemagneticflux Furthermore we can also consider the topologicalcurrent as in (9) to show that the topological charge is givenby the magnetic flux We must be careful though with thetemporal component of the gauge field 1198600 In this case theGaussrsquo law that appears in (30b) for 120582 = 0 is not solved for1198600 = 0 Moreover 1198600 is not an independent function onecan show that it is given by
Since the electric field does not vanish Chern-Simons vor-tices engender electric charge given by
119876 = int11988921199091198690= minus120581Φ
(34)
Therefore given the quantized magnetic flux (8) the electriccharge is also quantized by the vorticity 119899 The equations ofmotion (30a) and (30b) with the ansatz (5a) and (5b) and1198600 = 1198600(119903) are given by
1119903 (1199031198701198921015840)1015840 + 119870119892(119890211986020 minus 11988621199032 )+ 12 ((1198902119892211986020 minus 11989210158402 minus 119886211989221199032 )119870119892 minus 119881119892) = 0
11987901 = minus2119870 (119892) 11989011988611989221198600 sin 120579119903 (36b)
11987902 = 2L11988311989011988611989221198600 cos 120579119903 (36c)
11987912 = 119870 (119892)(11989210158402 minus 119886211989221199032 ) sin (2120579) (36d)
11987911 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 cos2120579 minus 1)+ 119886211989221199032 (2 sin2120579 minus 1)) minus 119881 (119892)
(36e)
11987922 = 119870 (119892)(1198902119892211986020 + 11989210158402 (2 sin2120579 minus 1)+ 119886211989221199032 (2 cos2120579 minus 1)) minus 119881 (119892)
(36f)
The equations of motion (35a) (35b) and (35c) are coupleddifferential equations of second order To simplify the prob-lem and get first-order equations we follow [66] and take thestressless condition 119879119894119895 = 0 This leads to
For 119870(|120593|) = 1 we have the potential given by 119881(|120593|) =1198904|120593|2(1minus|120593|2)21205812 which was studied in [27]The first-orderequations allowus to introduce an auxiliary function119882(119886 119892)given by
119882(119886 119892) = minus 1205811198861198902119892radic119881(119892)119870 (119892) (40)
Advances in High Energy Physics 9
06
03
00 15 3
016
008
0
0 1 2
K
V
Figure 7 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (43a) and (43b)
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
This formalism allows us to calculate the energy of thestressless solutions without knowing their explicit form Asdone in the latter section for simplicity we neglect theparameters and work with unit vorticity 119899 = 1 Next wepresent models in the above class that admit vortices inpotentials with minima located at infinity ie V 997888rarr infin inthe boundary conditions (6)
31 First Model To start the investigation with the Chern-Simons dynamics we consider the same 119870(|120593|) of (17a) and(17b) but with other potential in order to satisfy the constraint(39) We then take
These functions are plotted in Figure 7The potential presentsa minimum at |120593| = 0 and a set of minima at |120593| 997888rarrinfin Its maximum is located at |120593119898| asymp 079 such that119881(|120593119898|) asymp 014 Furthermore even though the function119870(|120593|) is the same of (17a) and (17b) in Maxwell dynamics
we see its corresponding potential has a completely differentbehavior near the origin in the Chern-Simons dynamics witha minimum instead of a maximum at |120593| = 0
The first-order equations (38) in this case read
1198921015840 = 119886119892119903 (44)
1198861015840119903 = minus1198922sech2 (11989222 )(1 minus tanh(11989222 )) (45)
We have not been able to find analytical solutions for themHowever the behavior of the solutions near the origin may bestudied by taking 119886(119903) = 1 minus 1198860(119903) and 119892(119903) = 1198920(119903) similarlyto the previous sections By substituting them in the aboveequations we get that
This helps as a guide in the numerical calculations InFigure 8 we plot the solutions In fact we see the behavior ofthe functions near the origin as given above These solutionsbehave similarly to the ones in Maxwell dynamics 119892(119903) goesto infinity as 119903 increases but 119886(119903) tends to zero very slowlypresenting a tail that goes far away from the origin Thisfeature is the opposite of the one found for compact Chern-Simons vortices in [69]
We now turn our attention to the auxiliar function119882(119886 119892) from (40) It is given by
This is exactly the same function that appears in (20) Byusing (42) we get that the energy of the stressless solutions is
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
10 Advances in High Energy Physics
1
05
0
0 100 200r
a
1
08
060 075 15
3
15
0
g
0 100 200
r
1
05
00 075 15
Figure 8The functions 119886(119903) (left) and 119892(119903) (right) solutions of (44)The insets show the behavior near the origin in the interval 119903 isin [0 157]
119864 = 2120587 To calculate the electric field intensity and themagnetic field one has to use the numerical solutions of (44)in (32) The energy density must be calculated in a similarmanner by using the expression given below which comesfrom (36a)
In Figure 9 we plot the electric field the magnetic fieldthe temporal component of the gauge field from (33) andthe energy density As in the previous models a numericalintegration of the magnetic field and energy density gives thefluxΦ asymp 2120587 and energy 120588 asymp 2120587 Thus the tail of the solutionsdoes not seem to contribute to change the topological chargesince it is given by the flux Therefore in the Chern-Simonsscenario vortices in vacuumless systems have the topologicalcurrent (9) well defined that does not require any specialdefinitions as done in [5] for kinks
32 Second Model We now present a new model given bythe functions
Differently of the previous model the minima of both119870(|120593|)and the potential are located at |120593| = 0 and |120593| 997888rarr infin Thepotential presents a maximum at |120593119898| asymp 07500 such that119881(|120593119898|) asymp 00055 These features can be seen in Figure 10 inwhich we have plotted 119870(|120593|) and the potential
To calculate our solutions we consider the first-orderequations (38) to get
We have not been able to find the analytical solutions of theabove equations Nevertheless it is worth to estimate theirbehavior near the origin by taking 119886(119903) = 1 minus 1198860(119903) and119892(119903) = 1198920(119903) similarly to what was done before for the lattermodels This approach leads to
In Figure 11 we plot the solutions of (50) Notice that 119886(119903) isalmost constant near the originThis is due to the formof (52)As in the previous models 119892(119903) tends to infinity as 119903 becomeslarger and larger Also we see 119886(119903) tends to vanish very slowwhen 119903 997888rarr infin also presenting a tail which extends far awayfrom the origin
In this case the function119882(119886 119892) in (40) becomes
119882(119886 119892) = 1198863 (1 minus tanh3 (119892)) (53)
Therefore by using (42) we conclude that the energy is 119864 =21205873 To calculate the intensity of the electric and magnetic
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in High Energy Physics 11
08
04
0
E
0 4 8r
05
025
0
B
0 4 8r
1
05
0
A0
0 4 8r
1
05
0
0 2 4r
Figure 9 The electric field (upper left) the magnetic field (upper right) the temporal gauge field component (bottom left) and the energydensity (bottom right) for the solutions of (44)
fields one has to use the numerical solutions into (32) Thesameoccurs to evaluate the energy density which comes from(36a) that leads to
2119892+ 118119892 sech2 (119892) tanh2 (119892) (1 minus tanh3 (119892))2
(54)
In Figure 12 we plot the electric and magnetic fields thetemporal gauge component (33) and the above energy
density As for all of our previous models the topologicalcharge given by the flux remains unchanged from (8) havingthe value Φ asymp 2120587 obtained from a numerical integrationThe energy can be obtained numerically and it is given by119864 asymp 21205873 the same value obtained from the function119882(119886 119892)of (53) Also we see the energy density in this model presentsa valley deeper than in the previous one
4 Conclusions
In this work we have investigated vortices in vacuumlesssystems with Maxwell and Chern-Simons dynamics In bothscenarios we have studied the properties of the generalizedmodels in the classes (1) and (28) and following [66] we
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
12 Advances in High Energy Physics
02
01
0
K
0 15 3
V
0006
0003
00 15 3
Figure 10 The function119870(|120593|) (left) and the potential 119881(|120593|) (right) given by (49a) and (49b)
1
05
00 100 200
r
a
1
09
080 15 3
4
2
0
g
0 100 200
r
1
05
00 15 3
Figure 11The functions 119886(119903) (left) and119892(119903) (right) solutions of (50)The insets show the behavior near the origin in the interval 119903 isin [0 327]
have used a first-order formalism that allows calculatingthe energy without knowing the explicit form of the solu-tions
The behaviors of the potentials are different at |120593| = 0depending on the scenario in theMaxwell case they are non-vanishing whilst in the Chern-Simons models they are zeroThe hole around the origin in the potentials for the Chern-Simons dynamics makes the magnetic field vanish at 119903 = 0Regardless of the differences in the behavior of the magneticfield the magnetic flux is always quantized by the vorticity 119899Furthermore even though we have worked only with 119899 = 1
for simplicity in our examples it is worth commenting thatwe have checked the energy is also quantized by the vorticity119899
An interesting result is that the vortex solutions in vacu-umless systems present a large tail that extends far away fromthe origin The scalar field is asymptotically divergent andhas infinite amplitude Then the solutions lose the localityHowever the electric field if it exists the magnetic field andthe energy density are localized This avoids the possibilityof having infinite energies and fluxes The flux is well definedand still works as a topological invariant Unlike the kinks we
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in High Energy Physics 13
01
005
00 10 20
r
E
006
003
0
008
004
0
B
0 10 20r
10 200
r5 100
r
0
02A0
04
Figure 12The electric field (upper left) themagnetic field (upper right) the temporal gauge component (bottom left) and the energy density(bottom right) for the solutions of (50)
concluded that vortices in vacuumless systems do not requireany special definition of the topological current to study itstopological character
We then discovered vortices with a new behavior whosesolutions present a long tail We hope these results encouragenew research in the area stimulating the study of newmodelsin this and other contexts One can follow the direction of[14] and study the demeanor of fermions in the backgroundof these vortex structures Also the collective behavior ofthese vortices seems of interest since it may give rise to non-standard interactions due to the particular aforementionedfeatures of the solutions Furthermore following the linesof [6] one also can study the gravitational field of these
vortices Another perspective is to investigate these structuresin models with enlarged symmetries [35ndash38 71ndash73] whichmay make them appear in the hidden sector for instanceFinally one may try to extend the current investigation toother topological structures such as monopoles [74 75] andnontopological structures such as lumps [76ndash78] andQ-balls[70 79] Someof these issues are under consideration andwillbe reported in the near future
Data Availability
The data used to support the findings of this study areincluded within the article
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
14 Advances in High Energy Physics
Conflicts of Interest
The author declares that there are no conflicts of interest
Acknowledgments
We would like to thank Dionisio Bazeia and RobertoMenezes for the discussions that have contributed to thiswork We would also like to acknowledge the BrazilianagencyCNPq research project 1555512018-3 for the financialsupport
References
[1] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge Monographs on MathematicalPhysics Cambridge University Press Cambridge UK 2007
[2] N Manton and P Sutcliffe Topological Solitons CambridgeUniversity Press Cambridge UK 2007
[3] T Vachaspati Kinks and Domain Walls An Introduction toClassical and Quantum Solitons Cambridge University PressCambridge UK 2007
[4] I Cho and A Vilenkin ldquoVacuum defects without a vacuumrdquoPhysical Review D vol 59 Article ID 021701 1999
[5] D Bazeia ldquoTopological solitons in a vacuumless systemrdquoPhysical Review D vol 60 Article ID 067705 1999
[6] I Cho and A Vilenkin ldquoGravitational field of vacuumlessdefectsrdquo Physical Review D vol 59 Article ID 063510 1999
[7] D Bazeia F A Brito and J R S Nascimento ldquoSupergravitybrane worlds and tachyon potentialsrdquo Physical Review D vol68 Article ID 085007 2003
[8] A de Souza Dutra and A C Amaro de Faria ldquoVacuumless kinksystems from vacuum systems An examplerdquo Physical Review Dvol 72 Article ID 087701 2005
[9] D Bazeia F A Brito and L Losano ldquoScalar fields bent branesand RG flowrdquo Journal of High Energy Physics vol 0611 p 0642006
[10] D Bazeia F A Brito and F G Costa ldquoFirst-order frameworkand domain-wallbrane-cosmology correspondencerdquo PhysicsLetters B vol 661 p 179 2008
[11] G P de Brito and A de Souza Dutra ldquoMultikink solutions anddeformed defectsrdquo Annals of Physics vol 351 p 620 2014
[12] F C Simas A R Gomes and K Z Nobrega ldquoDegenerate vacuato vacuumless model and kink-antikink collisionsrdquo PhysicsLetters B Particle Physics Nuclear Physics and Cosmology vol775 pp 290ndash296 2017
[13] D Bazeia andD CMoreira ldquoFrom sine-Gordon to vacuumlesssystems in flat and curved spacetimesrdquo The European PhysicalJournal C vol 77 p 884 2017
[14] D Bazeia AMohammadi and D CMoreira ldquoFermion boundstates in geometrically deformed backgroundsrdquoChinese PhysicsC vol 43 Article ID 013101 2019
[15] A M Perelomov Integrable Systems of Classical Mechanics andLie Algebras vol I Birkhauser Basel Basel Switzerland 1990
[16] I AffleckMDine andN Seiberg ldquoDynamical supersymmetrybreaking in supersymmetric QCDrdquo Nuclear Physics B vol 241p 493 1984
[17] P J E Peebles and B Ratra ldquoCosmology with a time-variablecosmological rsquoconstantrsquordquo The Astrophysical Journal Letters vol325 p L17 1988
[18] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy componentwith general equation of staterdquoPhysical Review Letters vol 80 Article ID 1582 1998
[19] H B Nielsen and P Olesen ldquoVortex-line models for dualstringsrdquo Nuclear Physics B vol 61 pp 45ndash61 1973
[20] H J de Vega and F A Schaposnik ldquoClassical vortex solution ofthe Abelian Higgs modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 14 no 4 pp 1100ndash1106 1976
[21] E Bogomolrsquonyi ldquoThe stability of classical solutionsrdquo SovietJournal of Nuclear Physics vol 24 no 4 pp 449ndash454 1976
[22] M Prasad and C Sommerfield ldquoExact classical solution forthe rsquot hooft monopole and the julia-zee dyonrdquo Physical ReviewLetters vol 35 p 760 1975
[23] S-S Chern and J Simons ldquoCharacteristic forms and geometricinvariantsrdquo Annals of Mathematics vol 99 p 48 1974
[24] S Deser R Jackiw and S Templeton ldquoTopologically massivegauge theoriesrdquo Annals of Physics vol 140 no 2 pp 372ndash4111982
[25] S Deser R Jackiw and S Templeton ldquoThree-dimensionalmassive gauge theoriesrdquo Physical Review Letters vol 48 p 9751982
[26] J Hong Y Kim and P Y Pac ldquoMultivortex solutions of theAbelian Chern-Simons-Higgs theoryrdquo Physical Review Lettersvol 64 p 2230 1990
[27] R Jackiw and E J Weinberg ldquoSelf-dual Chern-Simons vor-ticesrdquo Physical Review Letters vol 64 p 2234 1990
[28] R Jackiw K Lee and E J Weinberg ldquoSelf-dual Chern-Simonssolitonsrdquo Physical Review D vol 42 p 3488 1990
[29] G Dunne Self-dual Chern-Simons Theories Springer-Verlag1995
[30] E Fradkin Field Theories of Condensed Matter Physics Cam-bridge University Press 2013
[31] A J Long J M Hyde and T Vachaspati ldquoCosmic strings inhidden sectors 1 radiation of standardmodel particlesrdquo Journalof Cosmology and Astroparticle Physics vol 09 p 030 2014
[32] A J Long and T Vachaspati ldquoCosmic strings in hiddensectors 2 cosmological and astrophysical signaturesrdquo Journalof Cosmology and Astroparticle Physics vol 12 p 040 2014
[33] A E Nelson and J Scholtz ldquoDark light dark matter and themisalignment mechanismrdquo Physical Review D vol 84 ArticleID 103501 2011
[34] P Arias D Cadamuro M Goodsell et al ldquoWISPy cold darkmatterrdquo Journal of Cosmology and Astroparticle Physics vol 06p 013 2012
[35] P Arias and F A Schaposnik ldquoVortex solutions of an AbelianHiggs model with visible and hidden sectorsrdquo Journal of HighEnergy Physics vol 1412 p 011 2014
[36] P Arias E Ireson C Nunez and F Schaposnik ldquoN=2 SUSYAbelian Higgs model with hidden sector and BPS equationsrdquoJournal of High Energy Physics vol 1502 p 156 2015
[37] D Bazeia L Losano M AMarques and R Menezes ldquoVorticesin a generalized Maxwell-Higgs model with visible and hiddensectorsrdquo httpsarxivorgabs180507369
[38] D Bazeia M A Marques and R Menezes ldquoMaxwell-Higgsvortices with internal structurerdquo Physics Letters B vol 780 p485 2018
[39] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer-Verlag 1998
[40] L E Sadler J M Higbie S R Leslie M Vengalattore andD M Stamper-Kurn ldquoSpontaneous symmetry breaking in
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
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International Journal of
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Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in High Energy Physics 15
a quenched ferromagnetic spinor Bose-Einstein condensaterdquoNature vol 443 p 312 2006
[41] M Vengalattore S R Leslie J Guzman and D M Stamper-Kurn ldquoSpontaneously modulated spin textures in a dipolarspinor bose-einstein condensaterdquo Physical Review Letters vol100 Article ID 170403 2008
[42] M O Borgh J Lovegrove and J Ruostekoski ldquoInternal struc-ture and stability of vortices in a dipolar spinor bose-einsteincondensaterdquo Physical Review A vol 95 Article ID 053601 2017
[43] E Babichev ldquoGlobal topological k-defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 74Article ID 085004 2006
[44] E Babichev ldquoGauge k-vorticesrdquo Physical Review D CoveringParticles Fields Gravitation and Cosmology vol 77 Article ID065021 2008
[45] J Lee and S Nam ldquoBogomolrsquonyi equations of Chern-SimonsHiggs theory from a generalized abelian Higgs modelrdquo PhysicsLetters B vol 261 no 4 pp 437ndash442 1991
[46] M Neubert ldquoSymmetry-breaking corrections to meson decayconstants in the heavy-quark effective theoryrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 46 p 18791992
[47] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999
[48] C Armendariz-Picon V Mukhanov and P J SteinhardldquoDynamical solution to the problem of a small cosmologicalconstant and late-time cosmic accelerationrdquo Physical ReviewLetters vol 85 p 4438 2000
[49] C Armendariz-Picon V Mukhanov and P J SteinbardtldquoEssentials of k-essencerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 63 Article ID 103510 2001
[50] X-H Jin X-Z Li and D-J Liu ldquoA gravitating global k-monopolerdquo Classical and Quantum Gravity vol 24 no 11 pp2773ndash2780 2007
[51] D Bazeia L Losano R Menezes and J C R E OliveiraldquoGeneralized global defect solutionsrdquo The European PhysicalJournal C vol 51 no 4 pp 953ndash962 2007
[52] S Sarangi ldquoDBI global stringsrdquo Journal of High Energy Physicsvol 018 p 0807 2008
[53] D Bazeia L Losano and R Menezes ldquoFirst-order frameworkand generalized global defect solutionsrdquo Physics Letters B vol668 no 3 pp 246ndash252 2008
[54] C Adam P Klimas J Sanchez-Guillen and A WereszczynskildquoCompact gaugeK vorticesrdquo Journal of Physics A MathematicalandTheoretical vol 42 Article ID 135401 2009
[55] D Bazeia A R Gomes L Losano and R MenezesldquoBraneworldmodels of scalar fieldswith generalized dynamicsrdquoPhysics Letters B vol 671 p 402 2009
[56] D Bazeia E da Hora C dos Santos and R Menezes ldquoBPSsolutions to a generalizedMaxwellndashHiggsmodelrdquoTheEuropeanPhysical Journal C vol 71 p 1833 2011
[57] R Casana MM Ferreira Jr and E da Hora ldquoGeneralized BPSmagnetic monopolesrdquo Physical Review D Covering ParticlesFields Gravitation and Cosmology vol 86 Article ID 0850342012
[58] R Casana E da Hora D Rubiera-Garcia and C dos SantosldquoTopological vortices in generalized BornndashInfeldndashHiggs elec-trodynamicsrdquo The European Physical Journal C vol 75 p 3802015
[59] H S Ramadhan ldquoMeasurement of spin correlations in ttproduction using the matrix element method in the muon+jetsfinal state in pp collisions at radic119904 = 8TeVrdquo Physics Letters B vol758 pp 321ndash346 2016
[60] A N Atmaja H S Ramadhan and E da Hora ldquoMoreon Bogomolrsquonyi equations of three-dimensional generalizedMaxwell-Higgs model using on-shell methodrdquo Journal of HighEnergy Physics vol 1602 p 117 2016
[61] R Casana A Cavalcante and E da Hora ldquoSelf-dual configu-rations in Abelian Higgs models with k-generalized gauge fielddynamicsrdquo Journal of High Energy Physics vol 1612 p 51 2016
[62] R Casana M L Dias and E da Hora ldquoTopological first-ordervortices in a gauged CP(2) modelrdquo Physics Letters B vol 768pp 254ndash259 2017
[63] D Bazeia M A Marques and R Menezes ldquoGeneralized born-infeldndashlike models for kinks and branesrdquo EPL (EurophysicsLetters) vol 118 p 11001 2017
[64] D Bazeia E da Hora C dos Santos and R Menezes ldquoGen-eralized self-dual Chern-Simons vorticesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 81 Article ID125014 2010
[65] A N Atmaja ldquoA method for BPS equations of vorticesrdquo PhysicsLetters B vol 768 pp 351ndash358 2017
[66] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoFirst order formalism for generalized vorticesrdquoNuclear PhysicsB vol 934 pp 212ndash239 2018
[67] P Rosenau and J M Hyman ldquoCompactons Solitons with finitewavelengthrdquo Physical Review Letters vol 70 p 564 1993
[68] D Bazeia L LosanoMAMarques RMenezes and I ZafalanldquoCompact vorticesrdquoThe European Physical Journal C vol 77 p63 2017
[69] DBazeia L LosanoMAMarques andRMenezes ldquoCompactchern-simons vorticesrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 772 pp 253ndash257 2017
[70] D Bazeia M A Marques and R Menezes ldquoTwinlike modelsfor kinks vortices and monopolesrdquo Physical Review D Parti-cles Fields Gravitation and Cosmology vol 96 no 2 Article ID025010 2017
[71] M Shifman ldquoSimple models with non-Abelian moduli ontopological defectsrdquo Physical Review D vol 87 Article ID025025 2013
[72] A Peterson M Shifman and G Tallarita ldquoLow energydynamics of U(1) vortices in systems with cholesteric vacuumstructurerdquoAnnals of Physics vol 353 p 48 2014
[73] A Peterson M Shifman and G Tallarita ldquoSpin vortices inthe AbelianndashHiggs model with cholesteric vacuum structurerdquoAnnals of Physics vol 363 p 515 2015
[74] G rsquot Hooft ldquoMagnetic monopoles in unified gauge theoriesrdquoNuclear Physics B vol 79 no 2 pp 276ndash284 1974
[75] D Bazeia M A Marques and R Menezes ldquoMagneticmonopoleswith internal structurerdquoPhysical ReviewD CoveringParticles Fields Gravitation And Cosmology vol 97 Article ID105024 2018
[76] A T Avelar D Bazeia L Losano and R Menezes ldquoNew lump-like structures in scalar-field modelsrdquo The European PhysicalJournal C vol 55 no 1 pp 133ndash143 2008
[77] A T Avelar D Bazeia W B Cardoso and L Losano ldquoLump-like structures in scalar-fieldmodels in 1+1 dimensionsrdquo PhysicsLetters A vol 374 pp 222ndash227 2009
16 Advances in High Energy Physics
[78] D Bazeia M A Marques and R Menezes ldquoCompact lumpsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 111 no 6 p 61002 2015
[79] S R Coleman ldquoQ-ballsrdquo Nuclear Physics B vol 262 pp 263ndash283 1985
![Monique(C.Aller(Georgia(Southern(University)(lana/DUST2016/...–23– Zn Si Mn Fe Cr Ti ) un Solar&Level&(0%&dust)&-2.5 -2 -1.5 -1 -0.5 0 0.5 Metallicity [X/H]-1.5-1-0.5 0 Depletion](https://static.documents.pub/doc/80x56/5ffc3ad1c8cbce3a3a2244af/moniquecallergeorgiasouthernuniversity-lanadust2016-a23a-zn-si-mn.jpg)
