Analytical Solution for the SU(2) Hedgehog Skyrmion and Static Properties of Nucleons
aa r X i v : . [ h e p - ph ] D ec Analytical Solution for the SU(2) HedgehogSkyrmion and Static Properties of Nucleons
Duo-Jie Jia ∗ , Xiao-Wei Wang, Feng Liu Institute of Theoretical Physics, College of Physics and ElectronicEngineering, Northwest Normal University, Lanzhou 730070, P.R. China
Abstract
An analytical solution for symmetric Skyrmion was proposed for the SU(2) Skyrme model, which take the form of the hybrid formof a kink-like solution and that given by the instanton method. The static properties of nucleons was then computed within theframework of collective quantization of the Skyrme model, with a good agreement with that given by the exact numeric solution.The comparisons with the previous results as well as the experimental values are also given.
Keywords:
Skyrme Model, Soliton, Nucleons
PACS:
1. Introduction
The Skyrme model [1] is an e ff ective field theory of mesonsand baryons in which baryons arise as topological soliton so-lutions, known as Skyrmions. The model is based on the pre-QCD nonlinear σ model of the pion meson and was usuallyregarded to be consistent with the low-energy limit of large-N QCD[2]. For this reason, among others, it has been exten-sively revisited in recent years [3, 4, 5, 6, 8, 9] (see, [10, 11],for a review). Owing to the high nonlinearity, the solution tothe Skyrme model was mainly studied through the numericalapproach. It is worthwhile, however, to seek the analytic solu-tions [12, 13, 14] of Skyrmions due to its various applicationsin baryon phenomenology. One of noticeable analytic methodfor studying the Skyrmion solutions is the instanton approachproposed by Atiyah and Manton[8] which approximates criti-cal points of the Skyrme energy functional.In this Letter, we address the static solution of hedgehogSkyrmion in the S U (2) Skyrme model without pion mass termand propose an analytical solution for the hedgehog Skyrmionby writing it as the hybrid form of a kink-like solution and theanalytic solution obtained by the instanton method [8]. Twolowest order of Pad´e approximations was used and the corre-sponding solutions for Skyrmion profile are given explicitly byusing the downhill simplex method. The Skyrmion mass andstatic properties of nucleon as well as delta was computed andcompared to the previous results.
2. Analytic solution to Skyrme model
The
S U (2) Skyrme action [1] without pion mass term isgiven by ∗ Corresponding author
Email address: [email protected] (Duo-Jie Jia ∗ ) S S K = Z d x " − f π T r ( L µ ) + e T r ([ L µ , L ν ] ) (1)in which L µ = U † ∂ µ U , U ( x , t ) ∈ S U (2) is the nonlinear real-ization of the chiral field describing the σ field and π mesonswith the unitary constrain U † U =
1, 2 f π the pion decay con-stant, and e a dimensionless constant characterizing nonlinearcoupling. The Cauchy-Schwartz inequality for (1) implies[15] E S K ≥ π ( f π / e ) | B | , where B ≡ (1 / π ) R d x ε i jk T r ( L i L j L k )is the topological charge, known as baryon number. Using thehedgehog ansatz, U ( x ) = cos( F ) − i ( ˆ x · ~σ ) sin( F ) ( ~σ are the threePauli matrices) with F ≡ F ( r ) depending merely on the radialcoordinate r , the static energy for (1) becomes E S K = π ( f π e ) Z dx " x F x + ( F )(1 + F x ) + sin ( F ) x (2)with x = e f π r a dimensionless variable and F x ≡ dF ( x ) / dx .The equation of motion of (2) is + Fx ! F xx + x F x + sin(2 F ) x F x − − sin Fx ! = , (3)where the boundary condition F (0) = π, F ( ∞ ) = U = ±
1. The equation (3) is usually solved numericallydue to its high nonlinearity [3, 4, 6, 5].A kink-like analytic solution was given by [5] F ( x ) = − x )] , (4)with E S K = . π f π / e ), while an alternative Skyrmionprofile, proposed based on the instanton method, takes theform[8] F ( x ) = π (cid:20) − (1 + λ x ) − / (cid:21) , (5) Preprint submitted to Phys lett B November 7, 2018 ith corresponding energy 1 . π f π / e ) for the numericfactor λ = . r = ff ecting the value for theSkyrme field.To find the more accurate analytic solution, we first improvethe solution (4) into 4 arctan[exp( − cx )] with c a numeric fac-tor and then take λ to be a x -dependent function: λ → λ ( x ).Hence, we propose a Skyrmion profile function in the hybridform mixing (4) and (5) F w ( x ) = w arctan[exp( − cx )] + π (1 − w ) " − (1 + λ ( x ) x ) − / (6)with w ∈ [0 ,
1] being a positive weight factor. In principle,one can find the governing equation for the unknown λ ( x ) bysubstituting (6) into (3) and obtain a series solution of λ ( x ) bysolving the governing equation. Here, however, we choose thePad´e approximation to parameterize λ ( x ) λ ( x ) = λ + ax + · · · + bx + · · · , (7)since it has as equal potential as series in approximating a con-tinuous function. Note that we have already written λ ( x ) asfunction of x instead of x since so is F ( x ) in (5). The simplestnontrivial case of the above Pad´e approximation is the [2 /
2] ap-proximant λ ( x ) = λ + ax + bx . (8)The minimization of the energy (2) with the trial function(6) with respect to the variational parameters ( a , b , c , w , λ ) wascarried out numerically for the [2 /
2] Pad´e approximant (8) us-ing the downhill simplex method (the Neilder-Mead algorithm).The result for the optimized parameters is given by a = . , b = . , c = . , w = . , λ = . . (9)with E S K = . π f π / e ). The solution (6), with λ ( x )given by (7) and the parameters (9), is referred as solutionHyb(2 /
2) for short in this paper and is plotted in Fig.1, com-pared to the solutions (4) and (5), and the numerical solution(Num.) to the equation (3). We also include the analytic solu-tions given by [14] and the solution in the form of the purelyPad´e approximant [13] for comparison. A quite well agreementof our solution with the numerical solution can be seen fromthis plot. We note that the inequality E S K / (6 π f π / e ) ≥ | B | = F ( x ). For small x ≪ F ( x ) to the equation (3) is given by F ( x ) = π + Ax + Bx + Cx + · · · = π − . x + . x − . x + · · · , (10)(see also [12], where the variable x used is twice of x in thispaper) while the analytic solution (6), when (8) and (9) is used, behaves like F w ( x ) = π − . x + . x − . x + · · · . (11)One can see that (11) agrees well with (10) up to x . For large x → ∞ the series solution for F ( x ) can be obtained by solving(3) with x replaced by 1 / y and using the series expansion forsmall y . After re-changing y to x , one finds F ( x ) = . x ( − . x − . x + . x + . x + · · · ) . (12)On the other hand, the solution Hyb(2 / x , has theasymptotic form F w ( x ) = . x ( + . x − . x + . x + . x − . x + · · · ) , (13)which agrees globally with (12) except for a small bit di ff er-ences. The detailed di ff erences between (12) and (13) at large x can be due to the fact that the variationally-obtained solution (6)approximates the Skyrmion profile globally and may producesmall errors in local region, for instance, F w (50) = . × − while F (50) = . × − .The disagreement can be improved by employing the Pad´eapproximant of higher order than (8), for example, the [4 / λ ( x ) = λ + ax + a x + bx + b x . (14)The minimization of (2) using (14), as done for the [2 /
2] ap-proximant, yields the numerically optimal parameters, a = . , b = . , c = . , a = . , b = . , w = . , λ = . , (15)The solution (6) with λ ( x ) specified by (14) and (15) will be re-ferred as the Hyb(4 /
4) in this paper and is also plotted in Fig.1.The Fig.2 shows the profiles of F ( x ) at large x for Hyb(2 /
2) aswell as Hyb(4 / /
4) shows that for small x ≪ F w ( x ) = π − . x + . x − . x + · · · . while for x → ∞ F w ( x ) = . x ( − . x − . x − . x + . x − . x + · · · ) , Here, a better value F w (50) = . × − is obtained for thelatter asymptotic profile in contrast with the solution Hyb(2 / f π / e , are listed in Table I, including the corresponding resultsobtained by the numeric solution and obtained in the relevantreferences as indicated.2n solving (3) numerically, we employ the nonlinear shootalgorithm for the boundary values at x = .
001 and x = F ( x ).
3. The static properties of nucleons at low energy
The static properties of nucleons can be extracted by semi-classically quantizing the spinning modes of Skyrme La-grangian using the collective variables[3]. Here, we will usethe solution Hyb(2 /
2) and Hyb(4 /
4) to compute the static prop-erties of nucleons and nucleon-isobar( ∆ ) within the frameworkof the bosonic quantization of Skyrme model.Following Adkin et al.[3], one can choose S U (2)-variable A ( t ) as the collective variables, and substitute U = A ( t ) U ( x ) A ( t ) † into (1). In the adiabatic limit, one has S S K = S S K + I Λ Z dtT r [ ∂ A ∂ t ∂ A † ∂ t ] , (16)with S S K the action for the static hedgehog configuration, I = π/ (3 e f π ), and Λ = Z ∞ x dx sin F [1 + F x + sin F / x ] . (17)which is independent of f π and e . The Hamiltonian associ-ated to (16), when quantized via the quantization procedurein terms of collective coordinates, yields an eigenvalue h H i = M + J ( J + / (2 I Λ ), with M = E S K being the soliton energyof the Skyrmion. This yields the masses of the nucleon and ∆ -isobar M N = M + I Λ , M ∆ = M + I Λ . (18)By adjusting f π and e to fit the nucleon and delta masses in (18),one can fix the model parameters f π and e using the calculated M and Λ through (2) and (17).The isoscalor root mean square(r.m.s) radius and isoscalormagnetic r.m.s radius are given by e f π h r i / I = = {− π x sin FF x } / e f π h r i / M , I = = R ∞ x sin FF x dx R ∞ x sin FF x dx / (19)respectively. Combining with the masses of nucleon and thedelta, one can evaluate the magnetic moments for proton andneutron via the following formula µ p , n = µ I = p , n + µ I = p , n = h r i I = M N ( M ∆ − M N ) ± M N M ∆ − M N ) , (20)where plus and minus correspond to proton and neutron, respec-tively. The calculated results for these quantities using two so-lution schemes (Hyb.(2 /
2) and Hyb.(4 / F ( x ). The correspondingresults from other predictions are also shown in this table. Here in Table II, we use the experimental values M N = . MeV , M ∆ = MeV for fixing e and f π through (18), in contrastwith the input M N = MeV , M ∆ = MeV used by Ref.[3]and Ref.[14].To check the solution further, we also list, in the Table II, theaxial coupling constant and the π NN -coupling, which are givenby g A = − π e G , g π NN = M N f π g A (21)respectively. Here, the numeric factor G is G = Z ∞ dxx " F x + sin 2 Fx + sin 2 Fx ( F x ) + Fx F x + sin Fx sin 2 F . (22)
4. Concluding remarks
We show that the hybrid form of a kink-like solution andthat given by the instanton method are suited to approximatethe exact solution for the hedgehog Skyrmion, when combiningwith Pad´e approximation. The resulted analytic solution (6)has two remarkable features: (1) it is simple in the sense thatit is globally given in whole region; (2) it well approaches theasymptotic behavior of the exact solution. We note that thefurther generalization of (6), made by approximating c in (6) viaPad´e approximation, does not exhibit remarkable improvement,particularly in the asymptotic behavior of the chiral angle F ( x )at infinity. We expect that our solution can be useful in thedynamics study of the Skyrmion evolution and interactions. Acknowledgements
D. J thanks C. Liu and ChuengRyong Ji for discussions.This work is supported in part by the National Natural Sci-ence Foundation of China (No.10547009) and (No.10965005),and the Knowledge and S&T Innovation Engineering Project ofNWNU (No. NWNU-KJCXGC-03-41)
References [1] T.H.R. Skyrme, Nucl. Phys. 31(1961) 556.[2] E. Witten, Nucl. Phys. B 223(1983) 422.[3] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228(1983) 552.[4] A. Jackson, A.D. Jackson et al., Nucl. Phys. A 432(1985) 567.[5] P.M. Sutcli ff e, Phys. Lett. B 292(1992) 104.[6] R. Battye and P.M. Sutcli ff e, Phys. Rev. Lett.79(1997) 363.[7] M.F. Atiyah and N.S. Manton, Commun. Math. Phys.152(1993) 391.[8] M.F. Atiyah and N.S. Manton, Phys. Lett. B 222(1989) 438.[9] R. Battye and P.M. Sutcli ff e, Phys. Rev. C 73(2006) 055205.[10] I. Zahed and G. Brown, Phys. Rept. 142(1986) 1.[11] N.S. Manton and P.M. Sutcli ff e, Topological Solitons,Cambridge Univ.Press, Cambridge, 2004.[12] J. Ananias et al., J. Math. Phys. 32, 7(1991) 1949.[13] J.A. Ponciano et al., Phys. Rev. C 64(2001) 045205.[14] J. Yamashita and M. Hirayama, Phys. Lett. B 642(2006) 160.[15] L.D. Faddeev, Lett. Math. Phys. 1(1976) 289. /
2) Hyb(4 /
4) Num. E SK (2 f π / e ) . .
47 36 .
484 36 .
474 36 .
47 36 . . . M (6 π f π / e ) .
233 1 . . . . . . . /
2) Hyb.(4 /
4) Num. Expt. M / (2 f π / e ) 36 . . . . . − f π ( MeV ) 129 130 128 .
730 129 .
453 129 .
260 186 e .
45 5 .
48 5 . . . − Λ . . . . . −h r i / I = ( f m ) 0 .
59 0 .
586 0 . . . . h r i / M , I = ( f m ) 0 .
92 0 .
920 0 . . . . µ p . − . . . . µ n − . − − . − . − . − . | µ p /µ n | . − . . . . g A . − . . . . g π NN . − . . . . The solution (4)
The solution (5)
Hyb(2/2) T h e c h i r a l a ng l e F ( x ) The radial distance x ( = ef r ) Fig.1.
Hyb(4/4)
Ponciano et al
Yamashita et al Num . -9 -9 -8 -8 -8 -8 T h e c h i r a l a ng l e F ( x ) The radial distance x ( =ef r ) Fig.2.
Hyb(2/2)
Hyb(4/4) Num. r X i v : . [ h e p - ph ] D ec Analytical Solution for the SU(2) HedgehogSkyrmion and Static Properties of Nucleons
Duo-Jie Jia ∗ , Xiao-Wei Wang, Feng Liu Institute of Theoretical Physics, College of Physics and ElectronicEngineering, Northwest Normal University, Lanzhou 730070, P.R. China
Abstract
An analytical solution for symmetric Skyrmion was proposed for the SU(2) Skyrme model, which take the form of the hybrid formof a kink-like solution and that given by the instanton method. The static properties of nucleons was then computed within theframework of collective quantization of the Skyrme model, with a good agreement with that given by the exact numeric solution.The comparisons with the previous results as well as the experimental values are also given.
Keywords:
Skyrme Model, Soliton, Nucleons
PACS:
1. Introduction
The Skyrme model [1] is an e ff ective field theory of mesonsand baryons in which baryons arise as topological soliton so-lutions, known as Skyrmions. The model is based on the pre-QCD nonlinear σ model of the pion meson and was usuallyregarded to be consistent with the low-energy limit of large-N QCD[2]. For this reason, among others, it has been exten-sively revisited in recent years [3, 4, 5, 6, 8, 9] (see, [10, 11],for a review). Owing to the high nonlinearity, the solution tothe Skyrme model was mainly studied through the numericalapproach. It is worthwhile, however, to seek the analytic solu-tions [12, 13, 14] of Skyrmions due to its various applicationsin baryon phenomenology. One of noticeable analytic methodfor studying the Skyrmion solutions is the instanton approachproposed by Atiyah and Manton[8] which approximates criti-cal points of the Skyrme energy functional.In this Letter, we address the static solution of hedgehogSkyrmion in the S U (2) Skyrme model without pion mass termand propose an analytical solution for the hedgehog Skyrmionby writing it as the hybrid form of a kink-like solution and theanalytic solution obtained by the instanton method [8]. Twolowest order of Pad´e approximations was used and the corre-sponding solutions for Skyrmion profile are given explicitly byusing the downhill simplex method. The Skyrmion mass andstatic properties of nucleon as well as delta was computed andcompared to the previous results.
2. Analytic solution to Skyrme model
The
S U (2) Skyrme action [1] without pion mass term isgiven by ∗ Corresponding author
Email address: [email protected] (Duo-Jie Jia ∗ ) S S K = Z d x " − f π T r ( L µ ) + e T r ([ L µ , L ν ] ) (1)in which L µ = U † ∂ µ U , U ( x , t ) ∈ S U (2) is the nonlinear real-ization of the chiral field describing the σ field and π mesonswith the unitary constrain U † U =
1, 2 f π the pion decay con-stant, and e a dimensionless constant characterizing nonlinearcoupling. The Cauchy-Schwartz inequality for (1) implies[15] E S K ≥ π ( f π / e ) | B | , where B ≡ (1 / π ) R d x ε i jk T r ( L i L j L k )is the topological charge, known as baryon number. Using thehedgehog ansatz, U ( x ) = cos( F ) − i ( ˆ x · ~σ ) sin( F ) ( ~σ are the threePauli matrices) with F ≡ F ( r ) depending merely on the radialcoordinate r , the static energy for (1) becomes E S K = π ( f π e ) Z dx " x F x + ( F )(1 + F x ) + sin ( F ) x (2)with x = e f π r a dimensionless variable and F x ≡ dF ( x ) / dx .The equation of motion of (2) is + Fx ! F xx + x F x + sin(2 F ) x F x − − sin Fx ! = , (3)where the boundary condition F (0) = π, F ( ∞ ) = U = ±
1. The equation (3) is usually solved numericallydue to its high nonlinearity [3, 4, 6, 5].A kink-like analytic solution was given by [5] F ( x ) = − x )] , (4)with E S K = . π f π / e ), while an alternative Skyrmionprofile, proposed based on the instanton method, takes theform[8] F ( x ) = π (cid:20) − (1 + λ x ) − / (cid:21) , (5) Preprint submitted to Phys lett B November 7, 2018 ith corresponding energy 1 . π f π / e ) for the numericfactor λ = . r = ff ecting the value for theSkyrme field.To find the more accurate analytic solution, we first improvethe solution (4) into 4 arctan[exp( − cx )] with c a numeric fac-tor and then take λ to be a x -dependent function: λ → λ ( x ).Hence, we propose a Skyrmion profile function in the hybridform mixing (4) and (5) F w ( x ) = w arctan[exp( − cx )] + π (1 − w ) " − (1 + λ ( x ) x ) − / (6)with w ∈ [0 ,
1] being a positive weight factor. In principle,one can find the governing equation for the unknown λ ( x ) bysubstituting (6) into (3) and obtain a series solution of λ ( x ) bysolving the governing equation. Here, however, we choose thePad´e approximation to parameterize λ ( x ) λ ( x ) = λ + ax + · · · + bx + · · · , (7)since it has as equal potential as series in approximating a con-tinuous function. Note that we have already written λ ( x ) asfunction of x instead of x since so is F ( x ) in (5). The simplestnontrivial case of the above Pad´e approximation is the [2 /
2] ap-proximant λ ( x ) = λ + ax + bx . (8)The minimization of the energy (2) with the trial function(6) with respect to the variational parameters ( a , b , c , w , λ ) wascarried out numerically for the [2 /
2] Pad´e approximant (8) us-ing the downhill simplex method (the Neilder-Mead algorithm).The result for the optimized parameters is given by a = . , b = . , c = . , w = . , λ = . . (9)with E S K = . π f π / e ). The solution (6), with λ ( x )given by (7) and the parameters (9), is referred as solutionHyb(2 /
2) for short in this paper and is plotted in Fig.1, com-pared to the solutions (4) and (5), and the numerical solution(Num.) to the equation (3). We also include the analytic solu-tions given by [14] and the solution in the form of the purelyPad´e approximant [13] for comparison. A quite well agreementof our solution with the numerical solution can be seen fromthis plot. We note that the inequality E S K / (6 π f π / e ) ≥ | B | = F ( x ). For small x ≪ F ( x ) to the equation (3) is given by F ( x ) = π + Ax + Bx + Cx + · · · = π − . x + . x − . x + · · · , (10)(see also [12], where the variable x used is twice of x in thispaper) while the analytic solution (6), when (8) and (9) is used, behaves like F w ( x ) = π − . x + . x − . x + · · · . (11)One can see that (11) agrees well with (10) up to x . For large x → ∞ the series solution for F ( x ) can be obtained by solving(3) with x replaced by 1 / y and using the series expansion forsmall y . After re-changing y to x , one finds F ( x ) = . x ( − . x − . x + . x + . x + · · · ) . (12)On the other hand, the solution Hyb(2 / x , has theasymptotic form F w ( x ) = . x ( + . x − . x + . x + . x − . x + · · · ) , (13)which agrees globally with (12) except for a small bit di ff er-ences. The detailed di ff erences between (12) and (13) at large x can be due to the fact that the variationally-obtained solution (6)approximates the Skyrmion profile globally and may producesmall errors in local region, for instance, F w (50) = . × − while F (50) = . × − .The disagreement can be improved by employing the Pad´eapproximant of higher order than (8), for example, the [4 / λ ( x ) = λ + ax + a x + bx + b x . (14)The minimization of (2) using (14), as done for the [2 /
2] ap-proximant, yields the numerically optimal parameters, a = . , b = . , c = . , a = . , b = . , w = . , λ = . , (15)The solution (6) with λ ( x ) specified by (14) and (15) will be re-ferred as the Hyb(4 /
4) in this paper and is also plotted in Fig.1.The Fig.2 shows the profiles of F ( x ) at large x for Hyb(2 /
2) aswell as Hyb(4 / /
4) shows that for small x ≪ F w ( x ) = π − . x + . x − . x + · · · . while for x → ∞ F w ( x ) = . x ( − . x − . x − . x + . x − . x + · · · ) , Here, a better value F w (50) = . × − is obtained for thelatter asymptotic profile in contrast with the solution Hyb(2 / f π / e , are listed in Table I, including the corresponding resultsobtained by the numeric solution and obtained in the relevantreferences as indicated.2n solving (3) numerically, we employ the nonlinear shootalgorithm for the boundary values at x = .
001 and x = F ( x ).
3. The static properties of nucleons at low energy
The static properties of nucleons can be extracted by semi-classically quantizing the spinning modes of Skyrme La-grangian using the collective variables[3]. Here, we will usethe solution Hyb(2 /
2) and Hyb(4 /
4) to compute the static prop-erties of nucleons and nucleon-isobar( ∆ ) within the frameworkof the bosonic quantization of Skyrme model.Following Adkin et al.[3], one can choose S U (2)-variable A ( t ) as the collective variables, and substitute U = A ( t ) U ( x ) A ( t ) † into (1). In the adiabatic limit, one has S S K = S S K + I Λ Z dtT r [ ∂ A ∂ t ∂ A † ∂ t ] , (16)with S S K the action for the static hedgehog configuration, I = π/ (3 e f π ), and Λ = Z ∞ x dx sin F [1 + F x + sin F / x ] . (17)which is independent of f π and e . The Hamiltonian associ-ated to (16), when quantized via the quantization procedurein terms of collective coordinates, yields an eigenvalue h H i = M + J ( J + / (2 I Λ ), with M = E S K being the soliton energyof the Skyrmion. This yields the masses of the nucleon and ∆ -isobar M N = M + I Λ , M ∆ = M + I Λ . (18)By adjusting f π and e to fit the nucleon and delta masses in (18),one can fix the model parameters f π and e using the calculated M and Λ through (2) and (17).The isoscalor root mean square(r.m.s) radius and isoscalormagnetic r.m.s radius are given by e f π h r i / I = = {− π x sin FF x } / e f π h r i / M , I = = R ∞ x sin FF x dx R ∞ x sin FF x dx / (19)respectively. Combining with the masses of nucleon and thedelta, one can evaluate the magnetic moments for proton andneutron via the following formula µ p , n = µ I = p , n + µ I = p , n = h r i I = M N ( M ∆ − M N ) ± M N M ∆ − M N ) , (20)where plus and minus correspond to proton and neutron, respec-tively. The calculated results for these quantities using two so-lution schemes (Hyb.(2 /
2) and Hyb.(4 / F ( x ). The correspondingresults from other predictions are also shown in this table. Here in Table II, we use the experimental values M N = . MeV , M ∆ = MeV for fixing e and f π through (18), in contrastwith the input M N = MeV , M ∆ = MeV used by Ref.[3]and Ref.[14].To check the solution further, we also list, in the Table II, theaxial coupling constant and the π NN -coupling, which are givenby g A = − π e G , g π NN = M N f π g A (21)respectively. Here, the numeric factor G is G = Z ∞ dxx " F x + sin 2 Fx + sin 2 Fx ( F x ) + Fx F x + sin Fx sin 2 F . (22)
4. Concluding remarks
We show that the hybrid form of a kink-like solution andthat given by the instanton method are suited to approximatethe exact solution for the hedgehog Skyrmion, when combiningwith Pad´e approximation. The resulted analytic solution (6)has two remarkable features: (1) it is simple in the sense thatit is globally given in whole region; (2) it well approaches theasymptotic behavior of the exact solution. We note that thefurther generalization of (6), made by approximating c in (6) viaPad´e approximation, does not exhibit remarkable improvement,particularly in the asymptotic behavior of the chiral angle F ( x )at infinity. We expect that our solution can be useful in thedynamics study of the Skyrmion evolution and interactions. Acknowledgements
D. J thanks C. Liu and ChuengRyong Ji for discussions.This work is supported in part by the National Natural Sci-ence Foundation of China (No.10547009) and (No.10965005),and the Knowledge and S&T Innovation Engineering Project ofNWNU (No. NWNU-KJCXGC-03-41)
References [1] T.H.R. Skyrme, Nucl. Phys. 31(1961) 556.[2] E. Witten, Nucl. Phys. B 223(1983) 422.[3] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228(1983) 552.[4] A. Jackson, A.D. Jackson et al., Nucl. Phys. A 432(1985) 567.[5] P.M. Sutcli ff e, Phys. Lett. B 292(1992) 104.[6] R. Battye and P.M. Sutcli ff e, Phys. Rev. Lett.79(1997) 363.[7] M.F. Atiyah and N.S. Manton, Commun. Math. Phys.152(1993) 391.[8] M.F. Atiyah and N.S. Manton, Phys. Lett. B 222(1989) 438.[9] R. Battye and P.M. Sutcli ff e, Phys. Rev. C 73(2006) 055205.[10] I. Zahed and G. Brown, Phys. Rept. 142(1986) 1.[11] N.S. Manton and P.M. Sutcli ff e, Topological Solitons,Cambridge Univ.Press, Cambridge, 2004.[12] J. Ananias et al., J. Math. Phys. 32, 7(1991) 1949.[13] J.A. Ponciano et al., Phys. Rev. C 64(2001) 045205.[14] J. Yamashita and M. Hirayama, Phys. Lett. B 642(2006) 160.[15] L.D. Faddeev, Lett. Math. Phys. 1(1976) 289. /
2) Hyb(4 /
4) Num. E SK (2 f π / e ) . .
47 36 .
484 36 .
474 36 .
47 36 . . . M (6 π f π / e ) .
233 1 . . . . . . . /
2) Hyb.(4 /
4) Num. Expt. M / (2 f π / e ) 36 . . . . . − f π ( MeV ) 129 130 128 .
730 129 .
453 129 .
260 186 e .
45 5 .
48 5 . . . − Λ . . . . . −h r i / I = ( f m ) 0 .
59 0 .
586 0 . . . . h r i / M , I = ( f m ) 0 .
92 0 .
920 0 . . . . µ p . − . . . . µ n − . − − . − . − . − . | µ p /µ n | . − . . . . g A . − . . . . g π NN . − . . . . The solution (4)
The solution (5)
Hyb(2/2) T h e c h i r a l a ng l e F ( x ) The radial distance x ( = ef r ) Fig.1.
Hyb(4/4)
Ponciano et al
Yamashita et al Num . -9 -9 -8 -8 -8 -8 T h e c h i r a l a ng l e F ( x ) The radial distance x ( =ef r ) Fig.2.
Hyb(2/2)
Hyb(4/4) Num. r X i v : . [ h e p - ph ] D ec Analytical Solution for the SU(2) HedgehogSkyrmion and Static Properties of Nucleons
Duo-Jie Jia ∗ , Xiao-Wei Wang, Feng Liu Institute of Theoretical Physics, College of Physics and ElectronicEngineering, Northwest Normal University, Lanzhou 730070, P.R. China
Abstract
An analytical solution for symmetric Skyrmion was proposed for the SU(2) Skyrme model, which take the form of the hybrid formof a kink-like solution and that given by the instanton method. The static properties of nucleons was then computed within theframework of collective quantization of the Skyrme model, with a good agreement with that given by the exact numeric solution.The comparisons with the previous results as well as the experimental values are also given.
Keywords:
Skyrme Model, Soliton, Nucleons
PACS:
1. Introduction
The Skyrme model [1] is an e ff ective field theory of mesonsand baryons in which baryons arise as topological soliton so-lutions, known as Skyrmions. The model is based on the pre-QCD nonlinear σ model of the pion meson and was usuallyregarded to be consistent with the low-energy limit of large-N QCD[2]. For this reason, among others, it has been exten-sively revisited in recent years [3, 4, 5, 6, 8, 9] (see, [10, 11],for a review). Owing to the high nonlinearity, the solution tothe Skyrme model was mainly studied through the numericalapproach. It is worthwhile, however, to seek the analytic solu-tions [12, 13, 14] of Skyrmions due to its various applicationsin baryon phenomenology. One of noticeable analytic methodfor studying the Skyrmion solutions is the instanton approachproposed by Atiyah and Manton[8] which approximates criti-cal points of the Skyrme energy functional.In this Letter, we address the static solution of hedgehogSkyrmion in the S U (2) Skyrme model without pion mass termand propose an analytical solution for the hedgehog Skyrmionby writing it as the hybrid form of a kink-like solution and theanalytic solution obtained by the instanton method [8]. Twolowest order of Pad´e approximations was used and the corre-sponding solutions for Skyrmion profile are given explicitly byusing the downhill simplex method. The Skyrmion mass andstatic properties of nucleon as well as delta was computed andcompared to the previous results.
2. Analytic solution to Skyrme model
The
S U (2) Skyrme action [1] without pion mass term isgiven by ∗ Corresponding author
Email address: [email protected] (Duo-Jie Jia ∗ ) S S K = Z d x [ − f π T r ( L µ ) + e T r ([ L µ , L ν ] )] (1)in which L µ = U † ∂ µ U , U ( x , t ) ∈ S U (2) is the nonlinear real-ization of the chiral field describing the σ field and π mesonswith the unitary constrain U † U =
1, 2 f π the pion decay con-stant, and e a dimensionless constant characterizing nonlinearcoupling. The Cauchy-Schwartz inequality for (1), implies[15] E S K ≥ π ( f π / e ) | B | , where B ≡ (1 / π ) R d x ε i jk T r ( L i L j L k )is the topological charge, known as baryon number. Using thehedgehog ansatz, U ( x ) = cos( F ) − i ( ˆ x · ~σ ) sin( F ) ( ~σ are the threePauli matrices) with F ≡ F ( r ) depending merely on the radialcoordinate r , the static energy for (1) becomes E S K = π ( f π e ) Z dx " x F x + ( F )(1 + F x ) + sin ( F ) x (2)with x = e f π r a dimensionless variable and F x ≡ dF ( x ) / dx .The equation of motion of (2) is + Fx ! F xx + x F x + sin(2 F ) x F x − − sin Fx ! = , (3)where the boundary condition F (0) = π, F ( ∞ ) = U = ±
1. The equation (3) is usually solved numerically due to itshighly-nonlinearity, as done in most calculations[3, 4, 6, 5].A kink-like analytic solution was given by [5] F ( x ) = − x )] , (4)with E S K = . π f π / e ), while an alternative Skyrmionprofile, proposed based on the instanton method, takes theform[8] F ( x ) = π (cid:20) − (1 + λ x ) − / (cid:21) , (5) Preprint submitted to Phys lett B November 7, 2018 ith corresponding energy 1 . π f π / e ) for the numericfactor λ = . r = ff ecting the value for theSkyrme field.To find the more accurate analytic solution, we first improvethe solution (4) into 4 arctan[exp( − cx )] with c a numeric fac-tor and then take λ to be a x -dependent function: λ → λ ( x ).Hence, we propose a Skyrmion profile function in the hybridform mixing (4) and (5) F w ( x ) = w arctan[exp( − cx )] + π (1 − w ) " − (1 + λ ( x ) x ) − / (6)with w ∈ [0 ,
1] being a positive weight factor. In principle,one can find the governing equation for the unknown λ ( x ) bysubstituting (6) into (3) and obtain a series solution of λ ( x ) bysolving the governing equation. Here, however, we choose thePad´e approximation to parameterize λ ( x ) λ ( x ) = λ + ax + · · · + bx + · · · , (7)since it has as equal potential as series in approximating a con-tinuous function. Note that we have already written λ ( x ) asfunction of x instead of x since so is F ( x ) in (5). The simplestnontrivial case of the above Pad´e approximation is the [2 /
2] ap-proximant λ ( x ) = λ + ax + bx . (8)The minimization of the energy (2) with the trial function(6) with respect to the variational parameters ( a , b , c , w , λ ) wascarried out for the [2 /
2] Pad´e approximant (8) using the down-hill simplex method (the Neilder-Mead algorithm). The resultfor the numerically optimized parameters is given by a = . , b = . , c = . , w = . , λ = . . (9)with E S K = . π f π / e ). The solution (6), with λ ( x )given by (7) and the parameters (9), is referred as solutionHyb(2 /
2) for short in this paper and is plotted in Fig.1, com-pared to the solutions (4) and (5), and the numerical solution(Num.) to the equation (3). We also include the analytic solu-tions given by [14] and the solution in the form of the purelyPad´e approximant [13] for comparison. A quite well agreementof our solution with the numerical solution can be seen fromthis plot. We note that the inequality E S K / (6 π f π / e ) ≥ | B | = F ( x ). For small x ≪ F ( x ) to the equation (3) can be given bysolving the equation via series expansion, which is F ( x ) = π + Ax + Bx + Cx + · · · = π − . x + . x − . x + · · · , (10)(see also [12], where the variable x used is twice of x in thispaper) while the analytic solution (6), when (8) is used, behaves like F w ( x ) = π − . x + . x − . x + · · · . (11)One can see that (11) agrees well with (10) up to x . For large x → ∞ the series solution for F ( x ) can be obtained by solving(3) with x replaced by 1 / y and using the series expansion forsmall y . After re-changing y to x , one finds F ( x ) = . x ( − . x − . x + . x + . x + · · · ) . (12)On the other hand, the solution Hyb(2 / x , has theasymptotic form F w ( x ) = . x ( + . x − . x + . x + . x − . x + · · · ) , (13)which agrees globally with (12) except for a small bit di ff er-ences. The detailed di ff erences between (12) and (13) at large x can be due to the fact that the variationally-obtained solution (6)approximates the Skyrmion profile globally and may producesmall errors in local region, for instance, F w (50) = . × − while F (50) = . × − .The disagreement can be improved by employing the Pad´eapproximant of higher order than (8), for example, the [4 / λ ( x ) = λ + ax + a x + bx + b x . (14)The minimization of (2) using (14), as done for the [2 /
2] ap-proximant, yields the numerically optimal parameters, a = . , b = . , c = . , a = . , b = . , w = . , λ = . , (15)The solution (6) with λ ( x ) specified by (14) and (15) will be re-ferred as the Hyb(4 /
4) in this paper and is also plotted in Fig.1.The Fig.2 shows the profiles of F ( x ) at large x for Hyb(2 /
2) aswell as Hyb(4 / /
4) shows that for small x ≪ x → ∞ F w ( x ) = π − . x + . x − . x + · · · . while for x → ∞ F w ( x ) = . x ( − . x − . x − . x + . x − . x + · · · ) , Here, a better value F w (50) = . × − is obtained for thelatter asymptotic profile in contrast with the solution Hyb(2 / f π / e , are listed in Table I, including the corresponding resultsobtained by the numeric solution and obtained in the relevantreferences as indicated.2n solving (3) numerically, we employ the nonlinear shootalgorithm for the boundary values at x = .
001 and x = F ( x ).
3. The static properties of nucleons at low energy
The static properties of nucleons can be extracted by semi-classically quantizing the spinning modes of Skyrme La-grangian using the collective variables[3]. Here, we will usethe solution Hyb(2 /
2) and Hyb(4 /
4) to compute the static prop-erties of nucleons and nucleon-isobar( ∆ ) within the frameworkof the bosonic quantization of Skyrme model.Following Adkin et al.[3], one can choose S U (2)-variable A ( t ) as the collective variables, and substitute U = A ( t ) U ( x ) A ( t ) † into (1). In the adiabatic limit, one has S S K = S S K + I Λ Z dtT r [ ∂ A ∂ t ∂ A † ∂ t ] , (16)with S S K the action for the static hedgehog configuration, I = π/ (3 e f π ), and Λ = Z ∞ x dx sin F [1 + F x + sin F / x ] . (17)which is independent of f π and e . The Hamiltonian associ-ated to (16), when quantized via the quantization procedurein terms of collective coordinates, yields an eigenvalue h H i = M + J ( J + / (2 I Λ ), with M = E S K being the soliton energyof the Skyrmion. This yields the masses of the nucleon and ∆ -isobar M N = M + I Λ , M ∆ = M + I Λ . (18)By adjusting f π and e to fit the nucleon and delta masses in (18),one can fix the model parameters f π and e using the calculated M and Λ through (2) and (17).The isoscalor root mean square(r.m.s) radius and isoscalormagnetic r.m.s radius are given by e f π h r i / I = = {− π x sin FF x } / e f π h r i / M , I = = R ∞ x sin FF x dx R ∞ x sin FF x dx / (19)respectively. Combining with the masses of nucleon and thedelta, one can evaluate the magnetic moments for proton andneutron via the following formula µ p , n = µ I = p , n + µ I = p , n = h r i I = M N ( M ∆ − M N ) ± M N M ∆ − M N ) , (20)where plus and minus correspond to proton and neutron, respec-tively. The calculated results for these quantities using two so-lution schemes (Hyb.(2 /
2) and Hyb.(4 / F ( x ). The correspondingresults from other predictions are also shown in this table. Here in Table II, we use the experimental values M N = . MeV , M ∆ = MeV for fixing e and f π through (18), in contrastwith the input M N = MeV , M ∆ = MeV used by Ref.[3]and Ref.[14].To checkthe solution further, we also list in the table, theaxial coupling constant and the π NN -coupling, which are givenby g A = − π e G , g π NN = M N f π g A (21)respectively. Here, the numeric factor G is G = Z ∞ dxx " F x + sin 2 Fx + sin 2 Fx ( F x ) + Fx F x + sin Fx sin 2 F . (22)
4. Concluding remarks
We show that the hybrid form of a kink-like solution andthat given by the instanton method are suited to approximatethe exact solution for the hedgehog Skyrmion, when combiningwith Pad´e approximation. The resulted analytic solution (6)has two remarkable features: (1) it is simple in the sense thatit is globally given in whole region; (2) it well approaches theasymptotic behavior of the exact solution. We note that thefurther generalization of (6), made by approximating c in (6) viaPad´e approximation, does not exhibit remarkable improvement,particularly in the asymptotic behavior of the chiral angle F ( x )at infinity. We expect that our solution can be useful in thedynamics study of the Skyrmion evolution and interactions. Acknowledgements
D. J thanks C. Liu and ChuengRyong Ji for discussions.This work is supported in part by the National Natural Sci-ence Foundation of China (No.10547009) and (No.10965005),and the Knowledge and S&T Innovation Engineering Project ofNWNU (No. NWNU-KJCXGC-03-41)
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2) Hyb(4 /
4) Num. E SK (2 f π / e ) . .
47 36 .
484 36 .
474 36 .
47 36 . . . M (6 π f π / e ) .
233 1 . . . . . . . /
2) Hyb.(4 /
4) Num. Expt. M / (2 f π / e ) 36 . . . . . − f π ( MeV ) 129 130 128 .
730 129 .
453 129 .
260 186 e .
45 5 .
48 5 . . . − Λ . . . . . −h r i / I = ( f m ) 0 .
59 0 .
586 0 . . . . h r i / M , I = ( f m ) 0 .
92 0 .
920 0 . . . . µ p . − . . . . µ n − . − − . − . − . − . | µ p /µ n | . − . . . . g A . − . . . . g π NN . − . . . . The solution (4)
The solution (5)
Hyb(2/2) T h e c h i r a l a ng l e F ( x ) The radial distance x ( = ef r ) Fig.1.
Hyb(4/4)
Ponciano et al
Yamashita et al Num . -9 -9 -8 -8 -8 -8 T h e c h i r a l a ng l e F ( x ) The radial distance x ( =ef r ) Fig.2.
Hyb(2/2)