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Extra Dimensional Models Extra Dimensional Models with Magnetic Fluxes with Magnetic Fluxes Tatsuo KobayashiTatsuo Kobayashi1.1. IntroductionIntroduction2. Magnetized extra dimensions2. Magnetized extra dimensions3. N-point couplings and flavor symmetries3. N-point couplings and flavor symmetries44 .. ModelsModels5. Summary5. Summary based on based on Abe, T.K., Ohki, arXiv: 0806.4748 Abe, T.K., Ohki, arXiv: 0806.4748 Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.2631, Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.2631,
0907.5274,0907.5274, Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.0395 Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.0395 T.K., Maruyama, Murata, Ohki, Sakai, 1002.2828T.K., Maruyama, Murata, Ohki, Sakai, 1002.2828
1 1 IntroductionIntroduction1-1. Introduction to higher D theory1-1. Introduction to higher D theory
Extra dimensional field theories, Extra dimensional field theories,
in particular in particular
string-derived extra dimensional field string-derived extra dimensional field theories, theories,
play important roles in particle physicsplay important roles in particle physics
as well as cosmology .as well as cosmology .
Extra dimensionsExtra dimensions 4 + n dimensions4 + n dimensions
4D ⇒4D ⇒ our 4D space-timeour 4D space-time
nD nD ⇒ ⇒ compact spacecompact space Examples of compact spaceExamples of compact space
torus, orbifold, CY, etc. torus, orbifold, CY, etc.
Field theory in higher Field theory in higher dimensionsdimensions
10D 10D ⇒ ⇒ 4D our space-time + 6D space4D our space-time + 6D space
10D vector10D vector
4D vector + 4D scalars 4D vector + 4D scalars
SO(10) spinor ⇒SO(10) spinor ⇒ SO(4) spinor SO(4) spinor
x SO(6) spinorx SO(6) spinor
internal quantum internal quantum number number
mM AAA ,
Field theory in higher Field theory in higher dimensionsdimensions
Mode expansionsMode expansions
KK decomposition KK decomposition
0)(
0)( 6410
mm
mm
DDi
AA
Zero-modesZero-modesZero-mode equationZero-mode equation
⇒⇒ non-trival zero-mode profile non-trival zero-mode profile
the number of zero-modesthe number of zero-modes
0 mmDi
4D effective theory4D effective theoryHigher dimensional Lagrangian (e.g. 10D)Higher dimensional Lagrangian (e.g. 10D)
integrate the compact space ⇒integrate the compact space ⇒ 4D 4D theorytheory
Coupling is obtained by the overlap Coupling is obtained by the overlap integral of wavefunctionsintegral of wavefunctions
)()()(6 yyyydgY
),(),(),( 6410 yxyxAyxyxddgL
)()()(44 xxxxdYL
Couplings in 4DCouplings in 4D Zero-mode profiles are quasi-localized Zero-mode profiles are quasi-localized
far away from each otherfar away from each other in compact in compact spacespace
⇒ ⇒ suppressed couplingssuppressed couplings
1-2. Introduction to magnetized 1-2. Introduction to magnetized torustorusChiral theoryChiral theory
When we start with extra dimensional field When we start with extra dimensional field theories, theories,
how to realize chiral theories is one of how to realize chiral theories is one of important issues from the viewpoint of important issues from the viewpoint of particle physics. particle physics.
Zero-modes between chiral and anti-Zero-modes between chiral and anti-chiral chiral
fields are different from each other fields are different from each other
on certain backgrounds, e.g. CY.on certain backgrounds, e.g. CY.
0 mmDi
Torus with magnetic flux Torus with magnetic flux
The limited number of solutions with The limited number of solutions with
non-trivial backgrounds are known.non-trivial backgrounds are known.
Torus background with magnetic flux Torus background with magnetic flux
is one of interesting backgrounds, is one of interesting backgrounds,
where one can solve zero-mode where one can solve zero-mode
Dirac equation.Dirac equation.
0 mmDi
Magnetic fluxMagnetic fluxIndeed, several studies have been done Indeed, several studies have been done
in both extra dimensional field theories in both extra dimensional field theories
and string theories with magnetic flux and string theories with magnetic flux
background.background.
In particular, magnetized D-brane models In particular, magnetized D-brane models
are T-duals of intersecting D-brane models.are T-duals of intersecting D-brane models.
Several interesting models have been Several interesting models have been
constructed in intersecting D-brane models, constructed in intersecting D-brane models,
that is, that is, the starting theory is U(N) SYM.the starting theory is U(N) SYM.
Magnetized extra dimensionsMagnetized extra dimensions
The (generation) number of zero-modes The (generation) number of zero-modes
is determined by the size of magnetic is determined by the size of magnetic flux. flux.
Zero-mode profiles are quasi-localized.Zero-mode profiles are quasi-localized.
=> several interesting => several interesting phenomenologyphenomenology
Phenomenology of magnetized Phenomenology of magnetized brane modelsbrane models
It is important to study phenomenological It is important to study phenomenological
aspects of magnetized brane models such as aspects of magnetized brane models such as
massless spectra from several gauge groups, massless spectra from several gauge groups,
U(N), SO(N), E6, E7, E8, ...U(N), SO(N), E6, E7, E8, ...
Yukawa couplings and higher order n-pointYukawa couplings and higher order n-point
couplings in 4D effective theory, couplings in 4D effective theory,
their symmetries like flavor symmetries, their symmetries like flavor symmetries,
Kahler metric, etc.Kahler metric, etc.
It is also important to extend such studies It is also important to extend such studies
on torus background to other backgrounds on torus background to other backgrounds
with magnetic fluxes, e.g. orbifold with magnetic fluxes, e.g. orbifold backgrounds.backgrounds.
2. Extra dimensions with magnetic 2. Extra dimensions with magnetic fluxes: basic toolsfluxes: basic tools
2-1. Magnetized torus model2-1. Magnetized torus model
We start with N=1 super Yang-Mills theory We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. in D = 4+2n dimensions. For example, 10D super YM theory For example, 10D super YM theory consists of gauge bosons (10D vector)consists of gauge bosons (10D vector) and adjoint fermions (10D spinor).and adjoint fermions (10D spinor).We consider 2n-dimensional torus We consider 2n-dimensional torus
compactification compactification with magnetic flux background.with magnetic flux background.
Higher Dimensional SYM theory with flux Cremades, Ibanez, Cremades, Ibanez, Marchesano, Marchesano, ‘‘0404
The wave functionsThe wave functions eigenstates of correspondinginternal Dirac/Laplace operator.
4D Effective theory <= dimensional reduction
Higher Dimensional SYM theory with flux
AbelianAbelian gauge field on magnetized torusgauge field on magnetized torus
Constant magnetic flux
The boundary conditions on torus (transformation under torus translations)
gauge fields of background
Dirac equation on 2D torus
with twisted boundary conditions (Q=1)
is the two component spinor.
|M| independent zero mode solutions in Dirac equation.
(Theta function)
Dirac equation and chiral fermion
Properties of theta functions
:Normalizable mode
:Non-normalizable mode
By introducing magnetic flux, we can obtain chiral theory.
chiral fermion
Wave functions
Wave function profile on toroidal background
For the case of M=3
Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.
Fermions in bifundamentals
The gaugino fields
Breaking the gauge group
bi-fundamental matter fields
gaugino of unbroken gauge
(Abelian flux case )
Bi-fundamentalBi-fundamentalGaugino fields in off-diagonal entries Gaugino fields in off-diagonal entries
correspond to bi-fundamental matter correspond to bi-fundamental matter fields fields
and the difference M= m-mand the difference M= m-m’’ of magnetic of magnetic
fluxes appears in their Dirac equation.fluxes appears in their Dirac equation.
F F
Zero-modes Dirac equations
Total number of zero-modes of
:Normalizable mode
:Non-Normalizable mode
No effect due to magnetic flux for adjoint matter fields,
4D chiral theory 4D chiral theory 10D spinor 10D spinor light-cone 8s light-cone 8s even number of minus signseven number of minus signs 11stst ⇒ ⇒ 4D, the other ⇒4D, the other ⇒ 6D space6D space If all of appear If all of appear in 4D theory, that is non-chiral theory.in 4D theory, that is non-chiral theory. If for all torus, If for all torus, only only
appear for 4D helicity fixed.appear for 4D helicity fixed. ⇒ ⇒ 4D chiral theory 4D chiral theory
),(
, , baab NN
U(8) SYM theory on T6U(8) SYM theory on T6
Pati-Salam group up to U(1) factorsPati-Salam group up to U(1) factors
Three families of matter fields Three families of matter fields
with many Higgs fieldswith many Higgs fields
3
2
1
3
2
1
0
0
2
N
N
N
zz
m
m
m
iF
2 ,2 ,4 321 NNNRL UUU )2()2()4(
other tori for the 1)()(
first for the 3)()(
1321
21321
mmmm
Tmmmm
2,1,41,2,4
2-2. Wilson lines 2-2. Wilson lines Cremades, Ibanez, Cremades, Ibanez,
Marchesano, Marchesano, ’’04, 04, Abe, Choi, T.K. Ohki, Abe, Choi, T.K. Ohki, ‘‘0909
torus without magnetic fluxtorus without magnetic flux constant Ai constant Ai mass shift mass shift every modes massiveevery modes massive magnetic fluxmagnetic flux
the number of zero-modes is the same.the number of zero-modes is the same. the profile: f(y) the profile: f(y) f(y +a/M) f(y +a/M) with proper b.c.with proper b.c.
0 )(2
0 )(2
aMy
aMy
U(1)a*U(1)b theory U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0magnetic flux, Fa=2πM, Fb=0
Wilson line, Aa=0, Ab=CWilson line, Aa=0, Ab=C
matter fermions with U(1) charges, matter fermions with U(1) charges, (Qa,Qb)(Qa,Qb)
chiral spectrum, chiral spectrum,
for Qa=0, massive due to nonvanishing for Qa=0, massive due to nonvanishing WLWL
when MQa >0, the number of zero-modeswhen MQa >0, the number of zero-modes
is MQa.is MQa.
zero-mode profile is shifted depending zero-mode profile is shifted depending
on Qb, on Qb,
))/(( )( ab MQCQzfzf
2.3 Orbifold with magnetic 2.3 Orbifold with magnetic fluxflux
Abe, T.K., Ohki, Abe, T.K., Ohki, ‘‘0808
The number of even and odd zero-modesThe number of even and odd zero-modes
We can also embed Z2 into the gauge We can also embed Z2 into the gauge space.space.
=> various models, various flavor => various models, various flavor structuresstructures
Zero-modesZero-modes on orbifoldon orbifold
Adjoint matter fields are projected by Adjoint matter fields are projected by
orbifold projection.orbifold projection.
We have degree of freedom to We have degree of freedom to
introduce localized modes on fixed introduce localized modes on fixed points points
like quarks/leptons and higgs fields.like quarks/leptons and higgs fields.
3.3. N-point couplings N-point couplings and flavor symmetries and flavor symmetries
The N-point couplings are obtained by The N-point couplings are obtained by overlap integral of their zero-mode w.f.overlap integral of their zero-mode w.f.’’s.s.
)()()(2 zzzzdgY kP
jN
iM
Zero-modes Zero-modes Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404
Zero-mode w.f. = gaussian x theta-Zero-mode w.f. = gaussian x theta-functionfunction
up to normalization factor up to normalization factor
),(0
/)]Im(exp[)( iMMz
MjzMziNz M
jM
,)()()(1
NM
m
MmjiNMijm
jN
iM zyzz
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
MjNM ,,1 factor,ion normalizat:
3-point couplings3-point couplings Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404
The 3-point couplings are obtained by The 3-point couplings are obtained by
overlap integral of three zero-mode w.f.overlap integral of three zero-mode w.f.’’s.s.
up to normalization factor up to normalization factor
*2 )()()( zzzzdY k
NMjN
iMijk
NM
mijmkmMjiijk yY
1,
ikkM
iM zzzd *2 )()(
Selection rule Selection rule
Each zero-mode has a Zg charge, Each zero-mode has a Zg charge,
which is conserved in 3-point couplings.which is conserved in 3-point couplings.
up to normalization factor up to normalization factor
)(, NMkmMjikmMji
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
),gcd( when mod NMggkji
4-point couplings4-point couplings Abe, Choi, T.K., Ohki, Abe, Choi, T.K., Ohki, ‘‘0909 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f.overlap integral of four zero-mode w.f.’’s.s. splitsplit
insert a complete setinsert a complete set
up to normalization factor up to normalization factor for K=M+Nfor K=M+N
*2 )()()()( zzzzzdY l
PNMkP
jN
iMijkl
modes all
*)'()()'( zzzz n
KnK
*22 )'()'()'()()(' zzzzzzzzdd l
PNMkP
jN
iM
lsksijs
lijk yyY
4-point couplings: another 4-point couplings: another splittingsplitting
i k i ki k i k
t t
j s l j lj s l j l
*22 )'()'()'()()(' zzzzzzzzdd l
PNMjN
kP
iM
ltjtikt
lijk yyY
ltjtikt
lijk yyY lsksij
slijk yyY
N-point couplingsN-point couplings Abe, Choi, T.K., Ohki, Abe, Choi, T.K., Ohki, ‘‘09 09
We can extend this analysis to generic n-point We can extend this analysis to generic n-point
couplings.couplings. N-point couplings = products of 3-point N-point couplings = products of 3-point
couplingscouplings = products of theta-functions= products of theta-functions
This behavior is non-trivial. (ItThis behavior is non-trivial. (It ’’s like CFT.) s like CFT.) Such a behavior wouldSuch a behavior would be satisfied be satisfied not for generic w.f.not for generic w.f.’’s, but for specific w.f.s, but for specific w.f.’’s.s. However, this behavior could be expected However, this behavior could be expected from T-duality between magnetized from T-duality between magnetized and intersecting D-brane models.and intersecting D-brane models.
T-dualityT-duality The 3-point couplings coincide between The 3-point couplings coincide between magnetized and intersecting D-brane models. magnetized and intersecting D-brane models. explicit calculationexplicit calculation Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404
Such correspondence can be extended to Such correspondence can be extended to 4-point and higher order couplings because of 4-point and higher order couplings because of CFT-like behaviors, e.g., CFT-like behaviors, e.g.,
Abe, Choi, T.K., Ohki, Abe, Choi, T.K., Ohki, ‘‘09 09
lsksijs
lijk yyY
Heterotic orbifold modelsHeterotic orbifold models
Our results would be useful to n-point couplings Our results would be useful to n-point couplings of twsited sectors in heterotic orbifold models.of twsited sectors in heterotic orbifold models.
Twisted strings on fixed points might correspond Twisted strings on fixed points might correspond to quasi-localized modes with magnetic flux, to quasi-localized modes with magnetic flux, zero modes profile = gaussian x theta-function zero modes profile = gaussian x theta-function
amplitude string closedamplitude stringopen 2
orbifold heterotic
in coupling
brane ngintersecti
in couplings2
Non-Abelian discrete flavor Non-Abelian discrete flavor symmetrysymmetry
The coupling selection rule is controlled by The coupling selection rule is controlled by
Zg charges.Zg charges.
For M=g,For M=g, 1 2 1 2 g g
Effective field theory also has a cyclic permutation Effective field theory also has a cyclic permutation symmetry of g zero-modes. symmetry of g zero-modes.
These lead to non-Abelian flavor symmetires These lead to non-Abelian flavor symmetires
such as D4 and Δ(27)such as D4 and Δ(27) Cf. heterotic orbifolds, Cf. heterotic orbifolds, T.K. Raby, Zhang, T.K. Raby, Zhang, ’’0404
T.K. Nilles, Ploger, Raby, T.K. Nilles, Ploger, Raby, Ratz, Ratz, ‘‘0606
3. Models3. Models We can construct several models by using We can construct several models by using the above model building tools. the above model building tools. What is the starting theory ?What is the starting theory ? 10D SYM or 6D SYM (+ hyper multiplets), 10D SYM or 6D SYM (+ hyper multiplets),
gauge groups, U(N), SO(N), E6, E7,E8,...gauge groups, U(N), SO(N), E6, E7,E8,... What is the gauge background ?What is the gauge background ? the form of magnetic fluxes, Wilson lines.the form of magnetic fluxes, Wilson lines. What is the geometrical background ?What is the geometrical background ? torus, orbifold, etc.torus, orbifold, etc.
E6 SYM theory on T6E6 SYM theory on T6 Choi, et. al. Choi, et. al. ‘‘0909
We introduce magnetix flux along U(1) direction, We introduce magnetix flux along U(1) direction,
which breaks E6 -> SO(10)*U(1)which breaks E6 -> SO(10)*U(1)
Three families of chiral matter fields 16Three families of chiral matter fields 16
We introduce Wilson lines breaking We introduce Wilson lines breaking
SO(10) -> SM group.SO(10) -> SM group.
Three families of quarks and leptons matter fields Three families of quarks and leptons matter fields
with no Higgs fieldswith no Higgs fields
1100 161614578
1 ,1 ,3 321 mmm
Splitting zero-mode profilesSplitting zero-mode profilesWilson lines do not change the Wilson lines do not change the
(generation) number of zero-modes, (generation) number of zero-modes, but change localization point.but change localization point.
1616
QQ ………… LL
E8 SYM theory on T6E8 SYM theory on T6 T.K., et. al. 1002:2828T.K., et. al. 1002:2828 E8 -> SU(3)*SU(2)*U(1)E8 -> SU(3)*SU(2)*U(1)YY*U(1)*U(1)11*U(1)*U(1)22*U(1)*U(1)33*U(1)*U(1)44
We introduce magnetic fluxes and Wilson lines We introduce magnetic fluxes and Wilson lines
along five U(1)along five U(1)’’s.s.
E8 248 adjoint rep. include various matter fields.E8 248 adjoint rep. include various matter fields.
248 = (3,2)248 = (3,2)(1,1,0,1,-1) (1,1,0,1,-1) + (3,2) + (3,2)(1,0,1,1,-1)(1,0,1,1,-1) + (3,2) + (3,2)(1,-1,-1,1,-1)(1,-1,-1,1,-1) + (3,2)+ (3,2)(1,1,0,1,-1)(1,1,0,1,-1) + + (3,2)(3,2)(1,0,1,1,-1) (1,0,1,1,-1) + ...... + ......
We have studied systematically the possibilitiesWe have studied systematically the possibilities
for realizing the MSSM.for realizing the MSSM.
Results: E8 SYM theory on T6Results: E8 SYM theory on T6 1. 1. We can get exactly 3 generations of We can get exactly 3 generations of quarks and leptons, quarks and leptons, but there are tachyonic modes and no top but there are tachyonic modes and no top
Yukawa. Yukawa. 2. 2. We allow MSSM + vector-like generations We allow MSSM + vector-like generations and require the top Yukawa couplings and and require the top Yukawa couplings and no exotic fieldsno exotic fields as well as no vector-like quark doublets.as well as no vector-like quark doublets. three classesthree classes
Results: E8 SYM theory on T6Results: E8 SYM theory on T6 A. A.
B. B.
C.C.
charges U(1)Yno with singlets
])1,1()1,1[(15])2,1()2,1[(12
])1,3()1,3[(21])1,3()1,3[(10
])1,1()2,1()1,3()1,3()2,3[(3
6633
2244
63241
charges U(1)Yno with singlets
])1,1()1,1[()156(])2,1()2,1[()44(
])1,3()1,3[()36(])1,3()1,3[()96(
])1,1()2,1()1,3()1,3()2,3[(3
662
332
222
442
63241
nnnn
nnnn
charges U(1)Yno with singlets
])1,1()1,1[(180])2,1()2,1[(45
])1,3()1,3[(18])1,3()1,3[(66
])1,1()2,1()1,3()1,3()2,3[(3
6633
2244
63241
Results: E8 SYM theory on T6Results: E8 SYM theory on T6 There are many vector-like generations. There are many vector-like generations.
They have 3- and 4-point couplings with They have 3- and 4-point couplings with
singlets (with no hypercharges).singlets (with no hypercharges).
VEVs of singlets VEVs of singlets mass terms of vector-like mass terms of vector-like
generations generations
Light modes depend on such mass terms. Light modes depend on such mass terms.
That is, low-energy phenomenologies That is, low-energy phenomenologies
depends on VEVs of singlets (moduli).depends on VEVs of singlets (moduli).
Results: E8 SYM theory on T6Results: E8 SYM theory on T6 It would be interesting to reduce the number It would be interesting to reduce the number
of vector-like generations. of vector-like generations.
orbifolding and/or non-Abelian Wilson lines orbifolding and/or non-Abelian Wilson lines
SummarySummaryWe have studiedWe have studied phenomenological aspects phenomenological aspects of magnetized brane models.of magnetized brane models.
Model building from U(N), E6, E7, E8Model building from U(N), E6, E7, E8
N-point couplings are comupted.N-point couplings are comupted. 4D effective field theory has non-Abelian 4D effective field theory has non-Abelian
flavor flavor symmetries, e.g. D4, Δ(27).symmetries, e.g. D4, Δ(27). Orbifold background with magnetic flux is Orbifold background with magnetic flux is also important. also important.