IL NUOV0 CIMENT0 VoL. 101 B, N. 1 Gennaio 1988 Axially Symmetric Bi-Skyrmionic Static Solutions. :E. SOI~A(3E ~nd ~/~. TAI~LINI Istituto Nazionale di Fisica Nucleate - Largo E. Fermi 2, 1-50125 l~irenze, Italia (rieevuto il 26 Gennaio 1988) Summary. -- The spectrum of the minimal-energy Skyrme field configu- rations with barionic number B ~ 2 is studied as a function of the dis- stance 2a between the two centres of a bispherical chart, which are the points at which the field U takes the value -- 1. No a priori conditions have been imposed on the fields, assumed axially symmetric, but those coming from the request to recover the product of two static spherical skyrmions in the large separation limit. The solutions have been found by means of a direct variational approach on interpolating bidimensional spline. The resulting spectrum appears to be very flat till distances of the order of two fermi, where it shares a minimum followed by a fast growing for smaller distances. The possibility of recovering a system of two fermions at the lowest-order quantization is discussed. PACS. 02.40. - Geometry, differential geometry and topology. PACS. 02.60. - Numerical approximation and analysis. PACS. 21.30. - Nuclear forces. 1. - Introduction. In recent times a relevant part of the research in the Skyrme model has been devoted to the problem of the interaction of two skyrmions, with the main purpose of recovering with reasonable approximation the peculiar features of the nucleon-nucleon forces. An approach very often used consists in numerical computations modelled on the different versions of the ((product ansatz ~ (1), going from the original (1) Solitons in nuclear and elementary particle physics, in Proceedings of the Lewes Workshop, 198d, edited by A. CHODOS, E. HADJIMICttA]~L and C. Tz]~ (World Scien- tific, Singapore, 1984). For a recent review see I. ZAHE~) and G. E. B~owzq: Phys. Re/~., 142, 1 (I986). 85
Transcript
IL NUOV0 CIMENT0 VoL. 101 B, N. 1 Gennaio 1988
Axially Symmetric Bi-Skyrmionic Static Solutions.
:E. SOI~A(3E ~ n d ~/~. TAI~LINI
Istituto Nazionale di Fisica Nucleate - Largo E. Fermi 2, 1-50125 l~irenze, Italia
(rieevuto il 26 Gennaio 1988)
S u m m a r y . - - The spectrum of the minimal-energy Skyrme field configu- rations with barionic number B ~ 2 is s tudied as a function of the dis- stance 2a between the two centres of a bispherical chart, which are the points a t which the field U takes the value -- 1. No a priori conditions have been imposed on the fields, assumed axial ly symmetric, but those coming from the request to recover the product of two stat ic spherical skyrmions in the large separation limit. The solutions have been found by means of a direct var ia t ional approach on interpolat ing bidimensional spline. The resulting spectrum appears to be very flat t i l l distances of the order of two fermi, where i t shares a minimum followed by a fast growing for smaller distances. The possibil i ty of recovering a system of two fermions at the lowest-order quantization is discussed.
PACS. 02.40. - Geometry, differential geometry and topology. PACS. 02.60. - Numerical approximation and analysis. PACS. 21.30. - Nuclear forces.
1. - I n t r o d u c t i o n .
I n r e c e n t t i m e s a r e l e v a n t p a r t of t h e r e s e a r c h in t h e S k y r m e m o d e l h a s
b e e n d e v o t e d to t h e p r o b l e m of t h e i n t e r a c t i o n of two s k y r m i o n s , w i t h t h e
m a i n p u r p o s e of r e c o v e r i n g w i t h r e a s o n a b l e a p p r o x i m a t i o n t h e p e c u l i a r f e a t u r e s
of t h e n u c l e o n - n u c l e o n forces .
A n a p p r o a c h v e r y o f t e n u s e d cons i s t s in n u m e r i c a l c o m p u t a t i o n s m o d e l l e d
on t h e d i f f e ren t v e r s i o n s of t h e ( (p roduc t a n s a t z ~ (1), g o i n g f r o m t h e o r i g i n a l
(1) Solitons in nuclear and elementary particle physics, in Proceedings of the Lewes Workshop, 198d, edited by A. CHODOS, E. HADJIMICttA]~L and C. Tz]~ (World Scien- tific, Singapore, 1984). For a recent review see I. ZAHE~) and G. E. B~owzq: Phys. Re/~., 142, 1 (I986).
85
86 E. soEAc~ and ~. TA~LINI
proposal of Skyrme to the generalized formulat ion (~.5) in which the produc t is t aken between an ordinary skyrmion centred in rl and another , ro t a t ed in the SU(2) space, cent red in r~: the energy is thus dependent , besides the relat ive distance r---- Ir~-- r~[, on the Euler paramete rs (~, fl, y) of the SU(2) rotat ion. I t was found by numerical computat ions tha t the min imum of the r -dependent energies is reached for (~, fl, y) ~-- (0, z, 0). The predict ions of this model can be compared with the very recent numerical results based on the re laxat ion me thod (e). They repor t to have found the minimal B ~-- 2 non- concentr ic configuration of the Skyrme field b y imposing a priori only the discrete reflection symmet ry with respect to the z-axis, while the result ing minima possess the reflection symmetr ies with respect to the x- and y-axis, too. Obviously the last symmetr ies are not present in the configuration of ref. (2), which is the s tar t ing point of the re laxat ion i terations. ~o reove r , following the note added in proof of ref. (s) it seems t h a t the minimal-energy configuration shares a continuous axial symmetry , whose appearance mus t be
theoret ical ly explained. 1floweret, there is a physical difficulty intrinsic to this k ind of solutions
when we want to use the collect ive-co-ordinate me thod for quantizing the system. In fact , the goal of these researches should be a deutonlike or J~'-J~ system once the zero modes are quantized. But those classical configurations which have - -p rov ided some addit ional constraints on the Hami l ton ian select the physical states (a)-- the topology compatible (7,s) with a two-fermion- composed quan tum system describe, in fact, a unique extended object pos- sessing only those zero modes associated to the global isorotat ional symmet ry . Therefore~ these configurations share the energy spectrum of a s ingle-quantum rigid ro ta tor even for great r. One can get, of course, more physically satis- fying quan tum states by merging deeply the theory in the known hadronic phenomenology with the direct use of distinct quark and pionic states and the hybr id iza t ion with the ehiral bag model (9). Even the addit ion of a s ixth-order term, to account for the co-meson contributions, has been used (~0).
(3) A. JACKSON, A.D. JACKSON and V. PASQIIIEI~: 2Vucl. Phys. A, 432, 567 (1985). (3) E. BI~AATEN and L. CARSON: Phys. lCev. Lett., 56, 1897 (1986). (4) R. VINH MAy, M. LACO~BE, B. LOISEAC, W.N. CO~TINGI-IAI~ and P. LISBOA: Phys..Lett. B, 150, 259 (1985). (5) U.B. KAVLFUSS and U. G. MEISS~m~: Phys. Bey. C, 30, 2058 (1984) and Phys. Rev. D, 31, 3024 (1985). (6) J . J . M. V~I~BAAI~SCEOT, T. S. WALI~OCT, J. WAMBACI-I and H.W. WYLD: 2Yuel. Phys. A, 468, 520 (1987). (7) D. FINKELSTEIN and J. RUBISTEIN: J. Math. Phys. (s 9, 1762 (1968). (s) L.C. BIEDENEA~, E. SO~ACE and M. TAI~LINI: Topological Concepts in s Physics: the Deuteron as a Bi-Skyrmion, Symmetries in Science II , edited by B. G~VBER and R. LENCZEWS~:I (Plenum Press, New York, N.Y., 1986). (9) s HOSAKA and H. TOKI: Phys. •ett. B, 188, 301 (1987). (lo) j .M. EISENBE~G, A. E~ELL and R.R. SILBAR: Phys. Rev. C, 33, 1531 (1985);
All these efforts have produced m a n y interes t ing results bu t i t appears ve ry difficult to find the main propert ies of the Ao-A ~ interact ions all together . In par t icular the a t t rac t ive behaviour at medium distances described in the nuclear modellistic of the one-boson exchange by means of the fictitious a-meson has been af ter all elusive, a possible description having been found by insert ing in the computa t ion the isobar contr ibutions (11).
Another approach, quite s t ra ightforward and using the collective-co- ordinate method, was proposed by the authors with encouraging results (1~). We described the Skyrme field by means of bispherieal co-ordinates (U, 0, ~0) with polar distance 2a, very well suited to describe configurations with two centres, barionic number B = 2, and separat ing to the product of two skyr- mions for a - > oo. By restr ict ing the solutions to the form exp [iJ(u)~.~], we calculated numerical ly the minimal J(~) and its energy for each a: as long as the stat ic minima are concerned, there is no a t t rac t ion at all. But i t hap- pens tha t the energy is invar iant under separate independent isorotations of the two specular lobes composing the configuration so we have one zero mode for each lobe and the quant izat ion produces a system which separates con- t inuously in the Ao-A ~ Moreover, the semi-classical t t ami l ton ian thus pro- duced depends l inearly on the inverse of the momenta of inert ia which are strongly dependent on the distance a, thus producing an a t t rac t ive t e rm depending on the pion mass m. and the Skyrme paramete r e. By adjusting one or bo th these two parameters , it is easy to find a t t rac t ive central potentials of the s t rength one likes as a funct ion of a.
The aim of this paper is to explore the possibility of extending this approach to a larger class of solutions allowing the fields to depend also on the angle 0. This new dependence can a priori produce fields different f rom zero on the plane separat ing the two lobes (V = 0, 0 • 0). The complete isorotational s ymm e t r y for each lobe obta ined in the previous case is only par t ia l ly re- covered if the field results different f rom zero on such a plane. F ro m the axial symm e t r y of the problem the only nonzero fields on the separation plane can be the t angen t ones: we find definitely tha t the minimal-energy con- figuration exploits this possibility then allowing an independent isorotat ional s y m m e t r y for each lobe only along a given axis. For this reason i t is not possible to repeat step by step the quant izat ion procedure shown in our pre-
vious paper. In this paper we l imit ourselves to present the computat ional me thod and the resul t of the stat ic solution in the general case of axial sym- metry , showing the interest ing features of this configuration in comparison with the previous ones.
With in the scope of finding the axial stat ic solutions we have used a nu-
A. DE PACE, H. ~/[ULL~R and A. FAESSLER: Z. Phys., 325, 229 (1986). (11) A. Dr, PAC~, It. MULL]~R and A. FAESSL]~R: Nud. Phys. A, 457, 541 (1986). (13) E. S o~ c ~ and M. TA~LINI: .Phys. t~ev. D, 33, 253 (1986).
8 8 ]~. SORACE a n d ~ . TARLINi
merical computa t ional s t ra tegy in which the direct var ia t ional calculus and interpola t ion methods are bound together . In this way i t is possible to find the mass spectrum and the associated Skyrme fields f rom the asympto t ic distances, where the two skyrmions are free, t i l l t he ve ry short distances where the in terac t ion is jumping up. The energy densi ty is representable as a sum of squares so the funct ional is ve ry well suited for our direct var ia t ional ap- proach. The results are ra the r unexpected , for we find a near ly flat spect rum unt i l ve ry short distances where i t begins to increase quickly. W h a t could be interes t ing is t ha t we find in the region of medium distances a spectrum growing ve ry slowly in the energy followed for smaller distances b y a decrease showing a flat min imum around a ~ 1 fm; the value of the energy at this min imum is ve ry near to t ha t of the asymptot ic free configuration.
The intui t ive explanat ion is t ha t the two lobes of the field configuration around the two centres are identical by pa r i ty symmetry~ each of t h em pos- sessing a barionic number B ~ i and half of the to ta l energy. So, if the Skyrme spherical solution is the absolute min imum in the B = 1 sector, the minimal energy of each lobe could not be lower than it. A mathemat ica l problem re- mains open however, for, a l though i t is B ~ 1 in each lobe, the field mapping from each lobe in SU(2) cannot be classified in the absolute homotopy classes because the fields have no constant values on the boundary , as the t ransverse fields are not zero on the plane separating the two lobes. But~ if all the previous hypotheses are exact , it is clear t ha t the only wayout should be in the search of solutions with less symmet ry t han the problem allows, al though this choice would be ra the r surprising given the existence (la) of the lower positive bound E>(F~/e)3z~[Q[ (analogous to the one by BOGO~OL'~u (14)).
In the nex t section, we introduce the funct ional of the stat ic Skyrme en- ergy in its general form and in bispherical co-ordinates and we show the regular- i t y conditions, the B ~ 2 conditions and the boundary values one must choose to get the r ight asymptot ic limit. The symmetr ies of the field configurations are then discussed and used to res t r ic t the boundary values. In sect. 3 we
give a brief account of the numerical methods and sect. 4 and 5 are in tended for the presentat ion of the results and conclusion, respectively.
2. - The Skyrme functional energy.
We star t f rom the original Skyrme Lagrangian
(2.1) L = ~-~ Tr(~uU~U*) + ~ Tr ([Ut~U, Ut~ U] 2)
(18) T. H. R. SKYR~]~: Proc. ~. Soc. London, Set. A, 260, 127 (1961); 262, 237 (1961). (1~) E.B. BoGoMoT.'NYI: Soy. J. •ucl. ~hys., 24, 449 (1976).
(2.2) H~--~-~ dx a F~Tr(V~q- Y 2) q- Tr(C : q - D ~) ,
with
Vo=iUt~oU, V~iU~U, C=I[Vo, D = ~ I (V•
If we then introduce the usual representation of the SU(2) matrices, U can be expressed by means of exponentiated pionic fields F/~(x):
m.3) U = exp [i//o(x)To].
We get for the energy of the static configurations
(2.4)
1
where no = HdtH]. To gel finite-energy configurations one must have Vo --> 0 and V-~ 0 at the infinite boundary, in the static ease these conditions imply ]HI -+ 0 at spatial infinity. As is well known the finite-energy field configurat- ions (supposed continuous mappings) fall in the classes of the homotopy group H~ (S U(2)), classified by integer numbers B, which are the values of the topolo- gical charge (15)
(2.5) B - - 1 ~ 24~ s~,ddx ~ Tr [V~ V~ Vk].
For the sector B ~ 1 an absolute minimum of E has been found which appears also to be a minimum to H, giving a true solution of the field equation. This is the Skyrme (~ hedge hog )) (1~) model of the nucleon, having spherical sym- metry and boundary values for the scalar field /~(r) zero and ~ for r - ~ and r - ~ 0, respectively. However, a crucial point in the treatment of the functional E lies in the conditions one imposes on the fields: the uniformity of the fields at the physical infinity and the need to recover in some appropriately defined limits the Skyrme solution are examples of conditions the field must satisfy.
(15) C.J. ISHA~{: J. Phys. A, 10, 1397 (1977).
90 ~.. SOl, ACE and ~. TARLISII
In general, i t can be observed tha t one has to do with boundary values ra ther then wi th initial values. An impor tan t mathemat ica l question concerns then the existence and uniqueness of the ex t r emum of ]~ in terms of the fields Ha and na (and H, too) given the boundary conditions: we t r y to admi t (as in the Skyrme solution) tha t , if the funct ional with its boundary conditions is invar ian t under a group of t ransformations, t hen even the minimizing con- figuration field (F(r) in the Skyrme case) is invar iant under these transfor- mat ions (16).
As we are in teres ted in bicentric configurations which reduce to two free skyrmions when the distance between the two centres goes to infinity, we can express the funct ional E in terms of a bispherical char t (12) U, 0, ~o wi th the centres at + a and -- a on the z-axis. In general, w e then use in the expres- sion of E a vector field 1I (*l, O, qJ) -~ H~ + O0 + qb~, f rom which we can t r y to drop out a priori the dependence on ~0 owing to the sphericM s y m m e t r y of J~ in (2.4) and the axial s y m m e t r y of the boundary conditions imposed b y the free a -+ oo l imit describing two separate spherical skyrmions. At the end of ra the r long manipulat ions by vector calculus in these curvilinear co-ordinates
we get
f a a sin 0 B C
--co 0
with the three positive expressions A, B and C defined b y
2 C = ~ (2 ~ cos ~ q(aq'-- flO)2 + (aF cos q - - 2y~ sin q)~ +
+ (~F cos q - , ~ q ' sin q)"},
where
r H (2.10) ]H I = v / - ~ , q ---- aresin F rr--E, g ---- - - arcg ~ ,
(16) S. COL]~A~r in New Phenomena in Subnuclear Physics, Part A, edited by A. ZI- cI~ICni (Plenum Press, l~ew York, N. Y., 1977); A. KUNDO, YU P. ]~YBACOV and V. I. SAI~tYtlI~: Indian J. Pure Appl. Phys., 17, 673 (1979).
AXIALLY SYMMETRIC BI-SKYRMIONIC STATIC SOLUTIONS ~ l
and
(2.zl)
= cosh fl - - cos 0 ,
/ ~ = e o s h ~ c o s 0 - - 1 ,
= )~ @ sin 0 ,
fi = ~g ' - - sinh ~],
# y = cos g sinh ~ - - sin g sin 0 '
F = - - sin g sinh ~ - - cos g sin 0 '
the dot is the ~-der ivat ive and the p r ime the 0-derivative. Energy (2.6) is t hen a comple te ly general expression of the Skyrme stat ic
funct ional except for the imposed axia l s y m m e t r y on the fields. We wan t
now to s tudy wi th more precision the conditions d ic ta ted f rom topological
and a sympto t i c reasons; af terwards , we will discuss the conditions f rom the a d m i t t e d axial s y m m e t r y . As we have no ted in ref. (12) the surface ~ = 0 is
composed of the infinite spherical bounda ry plus the z = 0 plane. By the
definition of 0 - - t h e angle under which f rom the given point the vector con-
nect ing (0,0, z = - - a ) wi th ( 0 , 0 ~ z = @ a ) is s e e n - - w e have t h a t all the
infinite boundary~ disregarding the ~ dependence, corresponds to the point r / = O , 0 = 0 .
On the other hand, the set ~ = • 0% 0, ~ describes the two centres @ a and - - a . So the spat ia l infini ty bounda ry condit ions impose [r~(0, 0)1 = 0, thus
(s.ls) ~ ( o , o) - - o ( o , o) = ~ ( o , o) = o.
The ~ -~ • c~ l imits are equivalent to the a -~ ~ limits, in bo th we wan t
to recover the spherical Skyrme funct ional dens i ty for the energy. I t is im- p r o t a n t to notice t h a t in the spherical ease F(0) = z is a condit ion of regular i ty ; f rom the spherical s y m m e t r y the only possible definition for the field in the
origin is U = J = l . Thus we impose t h a t YI(• ~ , 0 ) = n • a B = 2 con-
figuration is chosen if n = 1 and n+ = - 1. To describe f rom the energy
(2.6) the Sky rme spherical dens i ty in the ~ -~ • c~ l imit , we mus t use
(2.~3) / / ( + ~ , o) = - = ,
I / ( - ~ , o) = = ,
o ( § ~ , o) = o ,
o ( - ~ , 0) = o,
~ ( + ~ , o) = o ,
~ ( - ~ , o) = o .
Let us consider now the axial s y m m e t r y ; we can repea t the same reasoning
f rom which F ( 0 ) ~ z for the spherical case to p rove t h a t the regular i ty of
92 ~. so~Ac~ and ~. TARLINI
the fields induces
(2.:[4:) O(v, o) = O(v, ~) = r o) = r ~t) = o ,
we recall t ha t {U, 0} u {~], z} represents the s y m m e t r y axis z. Our system does not have any continuous sy m m et ry left bu t is symmetr ic
by discrete t ransformations. F rom the fact t ha t the energy is symmetr ic in q - . - - q and the assumption of uniqueness of the solution we can assert t h a t the solution has q = 0; this implies ~b = 0. I t is easy to see t h a t ~ - * - U, H - - > - H and 0 --> 0 is an invariance of the energy and is coherent with the conditions (2.12)-(2.14), f rom which we can assume tha t for the solution
Let us explain now by a simple topological a rgument the s t a t emen t tha t we have a B : 2 configuration; f rom our boundary conditions !it is possible to deform continuously the field set t ing a configuration with H = H(U) and 0 ~ 0, r ~ 0, wi thout changing the homotopy class of the applications. For this t e rmina l stage the calculus of B is trivial, i t becomes (~2)
(2.16) B ~ ] f 8 sin 0 d0 d~ _ --2-~ ~ H(v) sin~H(U)sinh~v (eosh ~ - - cos 0) ~ -
-{-co
2 f sins H d H = ~ ~ d ~ . - - c o
Obviously this a rgument does not allow us to say any th ing about the sepa- ra te existence and values of B in U > 0 and U ~< 0, however; sy m m et ry consider- ations are in order even on this point. Our vectorial field solution is zero in the origin and has no orthogonal component in an y other point of the plane z : 0 (~ ~ 0, 0 =/= x), whereas the t ransverse pa r t can be different f rom zero.
Thus in looking at the expression of the energy • and the barionic number B
we find tha t the contr ibut ion to these integrals of the hemispace U ~> 0 is equal to the ~ < 0 one. When a -* oo we know tha t ~] ~ ~(r+), ~ ~ 0 and U ~ u(r-),
< 0, while 0 becomes the polar angle with respect to (0, 0, -~ a) or (0, 0, -- a). So we have (considering ~) two spherical charts centred in (0, 0, q- a). I f one
searches the field solution in this situation, one will find just the U produc t of two skyrmionie fields centred in (0, 0, • a). In the nex t section, we will expose the procedure we followed to minimize the funct ional ~ with B ~ 2, bu t before we want to solve a doubt about the concrete dependence on a of its min imum values. In fact , one could think, since the change of a is mere ly an admissible char t t ransformat ion, i t should not have any physical con- sequence. B u t this would be t rue only if all the boundary conditions remMn
AXIALLY Su BI-SI~LYI%IVs STATIC SOLUTIO:NS 9 3
intrisieally the same: this happens for the spatial infinity boundary but not for the n• z values of H, necessarily associated to the point of co-ordinates (0, 0, • a). 5Ioreover, it can be observed tha t in the expression of B actual ly enters the adimensional variable ~ ~- F= ea, so tha t all the results can be read as the minimizations of the energy funct ional in a fixed geometric framework with different values of / ~ e.
3. - The n u m e r i c a l m i n i m i z a t i o n o f the energy.
Even a fleeting glance at the functionals shows tha t the search for a nu- merical solution of the Euler-Lagrange system of part ial differential equation ~E(H, O) = 0 appears quite hopeless. This approach seemed to us v e ry dif- ficult to follow even if one limits oneself to the s tudy of solutions with the symmet ry previously discussed. We have thus t r ied for a direct var ia t ional approach on the energy funct ional (2.6). Our me thod is based on the ap- proximation of the fields to be var ied by means of in terpolat ing functions constructed so as to always satisfy the boundary constraints. The funct ional is then minimized with respect to the var ia t ion of the values of these functions in the interpolat ion points: in this way we mime strictu sensu the genuine calculus of variat ions.
At the beginning we used the interpolat ing Lagrange polynomials because of thei r successful use in the approximate solution of complicated linear operator eigenvalue problems (17) and we obta ined very encouraging results. To avoid the well-known problems related with polynomial interpolat ion, we switched to the interpolat ing cubic spline functions (is). These are cubic polynomials between pairs of knots (interpolating points); adjacent polynomials join con- t inuously with continuous first and second derivates. I f we call S(x) the spline and we indicate with (x~, z~) the interpolat ing points, we have S(xr : z~, this imposes n relations on the parameters of the cubic sptine curve, the cont inui ty with the first and second derivatives at each of n - 2 interior nodes x~ amo- unts to 3 ( n - 2) more conditions on S(x). The comparison between the to ta l 4 n - - 4 parameters of the q~-- 1 sections of cubic curves and the to ta l 4 n - - 6 conditions shows t ha t we need 2 more conditions to determine the spline. We use the conditions picked up in (is). The interpolat ion of a surface can be obta ined with a combinat ion of two one-dimensional splines s tar t ing from rectangular lat t ice of interpolat ing points (x~, y~) having interpolat ing values z~j. The procedure we adopted is to s tar t building the one-dimensional splines in the x-direction using the sets of points (x~, z~)~ ~ (x~, z~) get t ing j ~ splines,
(17) F. CALOG]~O and E. FR• 2v~uovo Cimento B, 89, 161 (1985). (is) G.E. FOP.S~H~, M. A. M ~ c O L ~ and C. B. MOL]~R: Computer Methods ]or Mathe- matical Computations (Prentice Hall Inc., Englewood Cliffs, N.J . , 1971).
9~ ]~. SORAC]~ allCl M. TA~LINI
each of t h e m defined b y the value of yj. The evaluat ion of the in terpola t ing
surface on the generic poin t {x, y} is given b y choosing the values of the pre- vious splines on {x, y~} and using these in terpolat ing points to build a y-directed
(cubic in the y-co-ordinate) spline. Hav i ng the coefficients of this spline we
can compute its values and the par t ia l der iva t ive in the y-direct ion a t the po in t {x, y}. To compute the x pa r t i a l der ivat ive, we can repea t the procedure
b y inver t ing the order: first the spline on the y-direct ion is ob ta ined a f t e r
the spline on the x-direction gives easily the x pa r t i a l der ivat ive. B y this
in terpola t ion mechan i sm we have a discret izat ion of a surface and i ts pa r t i a l der ivat ives ; a funct ional of a generic field of two var iables and its der ivat ives
is t r ans fo rmed into a funct ion of the ({~, y,.} ; z~) lat t ice in terpola t ing points and values. B y keeping {~, yj} fixed we can minimize the act ion funct ional
va ry ing the ~; values. Coming back to our physical p rob lem we have a rec tangular domain in
the ~ and 0 var iable G the bounda ry conditions are H(O, O) = 0, H ( • c~, O) = = ~=~, 0(~ b 0) = 0(7 , ~) = 0, 0 ( • 0% 0) = 0. Because of the s y m m e t r y wi th respec t to the z = 0 p lane we can consider ~ e [0, -]- c~] and double the ba-
rionie and act ion integrals (in the s ta t ic case the mass). To t r ea t a finite
domain we define the new var iable m = exp [-- 7/] which has the good p rope r ty t h a t for a -+ c~ (or ~ -+ + ~ ) i t is the dis tance divided by 2a f rom the point (0,0 , + a), i.e. i t is p ropor t iona l to the var iab le r of the spherical B = 1 skyrmion. I f we choose K equidis tant w~ in [0, 1] a n d / ~ equid is tan t 0j in [0, z]
we need 2K/? values for H(w~, 0;) and O(x~, 0j) bu t we are constra ined to impose H(xK, 0~) ----- 0, H(m~, 0~) = - - ~ ~nd O(w~, 0~) =- 0 ( ~ , 0L) --~ O(m~, 0j) = 0. The in terpola t ion da ta we va r i a t e to minimize the act ion are then _AT = ( K - 2).
�9 L + ( K - - 1 ) ( L - - S). Of course r the choice of equidis tant lat t ice is the s implest bu t not the bes t ;
for example , for a high va lue of a (near the region of noninteraet ion) we have
r ~ 2ax, f rom the free solution we know t h a t the field U is app rox ima te ly the un i t y for r>20/e~' , , for a-----1000/e2~= we have x ~ 0.01. This means
t h a t the points of the la t t ice f rom 0.01 and 1 are a lmost useless to de te rmine
the funct ion solution. To p reven t such a waste of in format ion we call Xm~ ~
the va lue on which the ~ var iable can be though t as zero and H(Xm~, 0~) ---- 0 ( ~ , 0~)= O ( x ~ , O ~ r ) = 9. We le% x va r i a t e together wi th the in-
te rpola t ing values (when different f rom zero for bounda ry conditions) to obta in
the m i n i m u m of the action, of course the K points in the x-direction are equi-
d i s t an t in [0, x ~ J , thus all of t h e m are in a region of physical significance.
The to t a l n u m b e r of var iables in the minimizing p rob lem is then N ---- ( 2 K - - 3)- �9 ( ~ - - 1).
Near the a sympto t i c region (big ~) the dependence on 0 is ve ry weak and
we need a small 15 to describe the solution, indeed we find a convergence of the min ima , ob ta ined increasing JL, for /~ = 3. The s tory is different for the
var iab le ~, on the cont rary , we need for K to be a t leas t 16 to have a good
answer when the two centres are ve ry far apar t . Obviously, the a s y m p t o t i c
non in te rac t ing region is defined b y f inding wha t va lue of ~ reproduces the
a sympto t i c spher ical mass well known b y previous computa t ions . I n the
in t e rac t ing region we find t h a t t he la t t ices g iving the bes t resul ts (for t ime
comput ing problems i t is imposs ib le to have a va lue of N ~ 1 0 0 ) are K~--10
and ~ 6 ( N ~ 8 5 ) and for ~ increas ing K ~ 1 3 and ~ 5 followed b y
K ~ 16 and L ---- 4 for ~ near t he a s y m p t o t i c region. I t is possible to follow
the decreasing of the mass b y improv ing the l a t t i ce a n d the final resu l t ( the
min ima l mass) is t e s t e d b y the convergence oR the values o b t a i n e d wi th dif-
fe rent la t t ices .
4. - The numer ica l results .
I n th is sect ion we wan t to summar ize t he large q u a n t i t y of meaningfu l
results ob ta ined using the prev ious ly descr ibed methods . Our first t a sk has
been to reproduce b y means of our d i rec t va r i a t i ona l i n t e rpo la t ing me thod
TABz]~ I. - The adimensional static mass values ( (e/E=)M) in the model without angular dependence are calculated as ]unctions o/ the parameter ~ using two di]]erent methods. The ]irst is the standard Merson method /or the integration o/ di]]erential equations, the second one is our interpolating minimization with K spline points.
TABLE II . - The values o/ the components H and 0 o/ the St~yrme /ield minimizing the energy means of these values one can recover the explicit form of the interpolat ing solution using his-
= 7, mass = Eu/e 72.930
Bispline solution of H(x, O)
l ~ k 1 2 3 4 5 6
1 - - z --1.9517 --0.9574 --0.4132 --0.1931 2 - - z --1.9753 --1.0275 --0.4870 --0.2535 3 - - z --2.0307 --1.1849 --0.6644 --0.3932 4 - - z --2.0826 --1.3224 --0.8347 --0.5430 5 - - z --2.1025 --1.3733 --0.8966 --0.6018
cannot , of course, p roduce damage in the in tegrals and this is p roved even
b y numer ica l results . Fo r a less t h a n 9 the code chooses Xm~ ~ ~-- 1, for g rea t
values of a the Xma ~ goes l ike 1/a which indica tes the a sympto t i c free regime.
As one can see f rom t ab l e I for low values of ~ we get ex t r eme ly good values
for the energy wi th few points . The field configurat ion we discussed in this
pape r is obvious ly more compl ica ted , as we have to in t e rpo la t e two funct ions
in two var iab les . Our first a s sumpt ion has been to use for the gl"aining of the
l a t t i ce in the ~7-direction the number of poin ts i nd ica t ed b y t ab le I. The gra in-
ing in the 0-direct ion has been de te rmined b y the cons t ra in t on the m a x i m u m
number of i n t e rpo la t i ng poin ts on which i t is possible to run the min imiza t ion
code. Moreover, i t is i n tu i t i ve t h a t for increas ing a the dependence on 0 is
weaker and weaker , and we need a decreasing number of poin ts in 0 to get
good m i n i m a in t h a t region in which, on the cont ra ry , we need more points
in ~/.
The p rob lem of the in t roduc t ion of the cut-off in x has been solved by
s imply defining an xm~ ~ i n d e p e n d e n t of 0 for the component H(x, 0), i.e. H(x, 0)~--0 for X>~Xmax, while such an op t imiza t ion of the in te rpo la t ion on
the effective domain is not t a k e n for O(x, 0). As for the one-dimensional case
x.n~. is assumed a min imiza t ion v~riablc.
I n fig. 1 we exh ib i t the spec t rum of the masses as a funct ion of g, for g f rom
7 - I1 Nuovo Cimento B.
98 ~. SORACE and ~. T~RL~NZ
73.8
73.6
73J*
732
s~c~t ic -n~ss vo~zes ( e /F~ H )
73.0 �9 �9 �9
O i o 0 !
' ' ' ,'s ' 2o= sep~re~tion l~o~r~rneter
Fig. 1. - The comparison between the mass spectra of the scalar model without angular dependence (studied in ref. (4)) and of the general field solution (described in this paper) is given in small energy scale. �9 with angular dependence, A without angu'ar dependence.
1 to 20; in fig. 2 the same spec t rum is shown in a larger scale and compared wi th the old one of the 0- independent configurations. ~Such care has been
devo ted to the control of the behav iour f rom a z 5 to a ~ 15, where there is the mos t in teres t ing s t ructure . A possible i m p r o v e m e n t in the es t imat ion
of the masses could be derived b y observing t h a t the closer one is to the t rue
m i n i m u m the more justified is the use of the corresponding configurations as the basis of a l inearized difference calculus for a be t t e r approx imat ion . B u t
the a p p r o x i m a t e par t ia l differenti~'l equat ion of the corrections would t hen be
l inear and the solutions separable in 0 and ~. F r o m our results i t can be observed tha t , unlike the s i tuat ion be tween
~ 1000 and a--~ 20 - -where xm~ x changes essential ly (mainly like l / a ) while
the others values change ve ry s l owly - - t he configurations for a < 20 are ve ry quickly changing wi th ~. These results indicate t h a t the quasi-flatness of the
spec t rum is not the consequence of a subs tan t ia l indifference of the fields to
the d is tance; on the contrary , i t results f rom the complex i ty of the configurat-
ions of the min im a which grows with 1/~. So the increasing of the repuls ive energy of the in te rac t ion for ~ -~ 0 is control led b y the major influence of the
t ransverse componen t and the ma jo r dependence of the fields on 0.
Fig. 2. - The comparison between the mass spectra of the scalar model without angular dependence (studied in ref. (4)) and of the general field solution (described in this paper) is given in large energy scale. �9 with angular dependence, �9 without angular dependence.
I t is i m p o r t a n t to note t h a t following this me thod the barionic n u m b e r is exact ly 2 a t every s tage of the computa t ions , so we do not have the p rob lem to recover the correct B as in the re laxat ion procedures (~).
I n tab le I I we describe the configuration corresponding a t the energy m i n i m u m in a giving Xma x and the la t t ice values of the fields; the values in every point can be recons t ruc ted b y using the splines.
5. - Conc lud ing remarks .
We proved in the previous pape r (1~) t h a t i t is possible to describe
a q u a n t u m centra l po ten t ia l be tween two spinning B z 1 skyrmions which gives rise to free nucleons a t large separat ion. An i m p o r t a n t point was the
a t t r ac t ive charac te r of such a potent ia l , a f te r quant iza t ion of the isorotat ional
collective co-ordinates, in a region of the order of some fermi. The a t t r ac t ive
contr ibut ion arose comple te ly f rom the in t roduct ion of the isorotat ional zero
modes as the s ta t ic solution in t h a t case produced energy growing a t every distance. Of course, th is fea ture rises some formal difficulties on the quantiza-
100 E. SORACE a n d M. T A R L I N I
t ion procedure as the classical spectrum does not show a min imum in the
distance. The goal of the present paper has been to investigate the possibility
t ha t the absence of the static min imum were a consequence of the restriction
to fields depending only on the co-ordinate ~. The answers of our numerical
calculations are satisfying in this respect, in fact a min imum appears in
the spect rum energy if the fields are assumed to depend also on 0. On the
other side, we face a general problem in the Skyrme model of nucleon in-
teraction, the lack of zero modes necessary to allow the separation of the
system in two quant ized free fermions. We find the ~ component of the fields
on the mirror s y m m e t r y plane (~/----0, 0 r 0) different from zero, then the
independent isorotational s y m m e t r y for the two interact ing lobes is part ial ly
lost. Thus one has to choose between two al ternat ives: either to impose a
priori a constra int fixing the fields on the separat ing plane equal to zero or
to focus himself to a phenomenology of the interact ion of such a system (1~)
neglecting the description of the separated free fermions. Our general idea is
to follow the first hypothesis, t ak ing into account, moreover, tha t in principle
the be t te r procedure for t reat ing this model would be to minimize not the
static t t ami l ton ian but the classical dynamical Hamil tonian with the collective
co-ordinate already in action, and to quantize the obtained s t ructure (~o).
However, even this method has big problems, the technical difficulties ~re
insurmountable and some theoretical aspects, e.g.~ stabili ty problems, arise
ye t at the B ---- 1 level opposing the phenomenological fit of the parameters (~1).
Moreover, we wan t to stress t ha t the numerical methods we have con- s t ructed have been successful in solving a problem otherwise ~ery complicated
giving the possibility to go fur ther in the program of analysing the nucleon-
nucleon interactions arising from this model.
(19) M. LACOMBE, B. LOISEAU, J.M. RICHARD, R. VINH MAY, J. CST~, P. PIR~S and R. DE TOURR]~L: Phys. Rev. C, 21, 861 (1980). (e0) R. RAJARAMAN: Solitons and Instantons (North-Holland Co., Amsterdam, 1982); R. RAJARAMAN and E.J . WEINB]~RG: Phys. Rev. D, 11, 2950 (1975). (21) R. RAJARAMAN, H. M. SOMMERMAN, J. WAMBACH and H. W. WYLD : Phys. Rev. D, 33, 287 (1986).
�9 R I A S S U N T 0
Lo spettro delle configurazioni del campo di Skyrme di energia minima con numero barionieo B = 2 viene studiato in funzione della distanza 2a tra i due centri di una carta bisferiea coincidenti con i punti in eui il eampo U vale -- 1. Per i eampi viene assunta la simmetria assiale e nessuna altra condizione a priori vien6 imposta tranne quelle derivanti dalla richiesta di ottenere il prodotto di due skyrmioni sferiei statici nel limite di grandi distanze tra i due centri. Le soluzioni sono state trovate utilizzando
u n m e t o d o var i~z iona le d i r e t t o app l i ca to a spl ine i n t e r p o l a n t i b id imens iona l i . I r isul- t a t i n u m e r i c i m o s t r a n o uno s pe t t r o p i u t t o s t o p i a t t o sino a d i s t anze de l l ' o rd ine di due fermi, dove esso s e m b r a p r e s e n t a r e u n min imo segui to da u n a r a p i d a eresei t~ pe r d i s tnnze pi~t piecole. Discubiamo qu ind i 1~ poss ibi l i t5 di r i cos t ru i r e u n s i s t ema di due fe rmion i nello s c h e m a semi-elassieo di qu~nt izzaz ione .
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