ISSN: 0256-307X 中国物理快报 Chinese Physics Letters A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://www.iop.org/journals/cpl http://cpl.iphy.ac.cn CHINESE PHYSICAL SOCIETY CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201 Analytical Solution for the SU(2) Hedgehog Skyrmion and Static Properties of Nucleons * JIA Duo-Jie(_õ#)1**, WANG Xiao-Wei(王¡维), LIU Feng(4¹) Institute of Theoretical Physics, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070 (Received 5 May 2010) An analytical solution for symmetric Skyrmion is proposed for the SU(2) Skyrme model, which takes the form of the hybrid form of a kink-like solution, given by the instanton method. The static properties of nucleons is then computed within the framework of collective quantization of the Skyrme model, in 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 presented. PACS: 12. 38. −t, 11. 15. Tk, 12. 38. Aw DOI: 10.1088/0256-307X/27/12/121201 [1] [15] SK 2 The Skyrme model is an effective field theory of ity for (1) implies E ≥ 6휋 (f휋=e)jBj, where 2 R 3 ijk mesons and baryons in which baryons arise as topo- B ≡ (1=24휋 ) d x" T r(LiLjLk) is the topologi- logical soliton solutions, known as Skyrmions. The cal charge, known as the baryon number. Using the model is based on the pre-QCD nonlinear 휎 model of hedgehog ansatz, U(x) = cos(F ) − i(^x · 휎) sin(F ) (휎 the pion meson and was usually regarded to be consis- are the three Pauli matrices) with F ≡ F (r) depend- tent with the low-energy limit of large-N QCD.[2] For ing merely on the radial coordinate r, the static energy this reason, among others, it has been extensively re- for (1) becomes visited in recent years (see Refs. [3–9], and Refs. [10,11] (︁f )︁ Z h for review). Owing to the high nonlinearity, the solu- ESK = 2휋 휋 dx x2F 2 + 2 sin2(F )(1 + F 2) tion to the Skyrme model was mainly studied through e x x the numerical approach. It is worthwhile, however, sin4(F )i [12−14] + ; (2) to seek the analytic solutions of Skyrmions due x2 to its various applications in baryon phenomenology. One of the noticeable analytic methods for studying with x = ef휋r a dimensionless variable and Fx ≡ the Skyrmion solutions is the instanton approach pro- dF (x)=dx. The equation of motion of (2) is posed by Atiyah and Manton,[8] which approximates (︂ sin2 F )︂ 2 critical points of the Skyrme energy functional. 1 + 2 F + F In this Letter, we address the static solution of x2 xx x x hedgehog Skyrmion in the SU(2) Skyrme model with- sin(2F ) (︂ sin2 F )︂ out pion mass term and propose an analytical solution + F 2 − 1 − = 0; (3) x2 x x2 for the hedgehog Skyrmion by writing it as the hybrid form of a kink-like solution and the analytic solution where the boundary condition F (0) = 휋; F (1) = 0 [8] obtained by the instanton method. Two lowest-order will be imposed so that it corresponds to the physi- Padé approximations were used and the correspond- cal vacuum for (1): U = ±1. Equation (3) is usually ing solutions for Skyrmion profile are given explicitly solved numerically due to its high nonlinearity.[3−6] by using the downhill simplex method. The Skyrmion A kink-like analytic solution was given by[5] mass and static properties of nucleon as well as delta were computed and compared to the previous results. F1(x) = 4 arctan[exp(−x)]; (4) The SU(2) Skyrme action[1] without pion mass SK 2 term is given by with E = 1:24035(6휋 f휋=e), while an alternative Skyrmion profile, proposed based on the instanton Z [︂ f 2 1 ]︂ SSK = d3x − 휋 T r(L )2 + T r([L ;L ]2) ; method, takes the form[8] 4 휇 32e2 휇 휈 (1) h (︁ 휆 )︁−1=2i y F (x) = 휋 1 − 1 + ; (5) where L휇 = U @휇U, U(x; t) 2 SU(2) is the 2 x2 nonlinear realization of the chiral field describing 2 the 휎 field and 휋 mesons with the unitary con- with the corresponding energy 1:2432(6휋 f휋=e) for the y strain U U = 1, 2f휋 the pion decay constant, numeric factor 휆 = 2:109. The singularity at r = 0 and e a dimensionless constant characterizing non- is gauge dependent and can be gauged away without linear coupling. The Cauchy–Schwartz inequal- affecting the value for the Skyrme field. *Supported in part by the National Natural Science Foundation of China under Grant Nos 10547009 and 10965005, and the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCS), State Education Ministry. **Email: [email protected]; [email protected] ○c 2010 Chinese Physical Society and IOP Publishing Ltd 121201-1 CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201 Table 1. The Skyrmion energy from different theoretical calculations. Work [3][13][6][14][12] Hyb(2/2) Hyb(4/4) Num. SK E 36.5 36.47 36.484 36.474 36.47 36.4638 36.4638 36.4616 (2f휋 =e) M 2 1.233 1.2317 1.2322 1.23186 1.2317 1.23152 1.23146 1.23145 (6휋 f휋 =e) To find the more accurate analytic solution, we profile F (x). For small x ≪ 1 the solution F (x) to first improve the solution (4) into 4 arctan[exp(−cx)] Eq. (3) is given by with c a numeric factor and then take 휆 to be a x- F (x) = 휋 + Ax + Bx3 + Cx5 + ··· dependent function: 휆 ! 휆(x). Hence, we propose a 3 Skyrmion profile function in the hybrid form mixing = 휋 − 2:007528x + 0:358987x (4) and (5), − 0:146499x5 + ··· ; (10) (also see Ref. [12], where the variable x used is twice of Fw(x) = 4w arctan[exp(−cx)] x in this study) while the analytic solution (6), when h (︁ 휆(x))︁−1=2i + 휋(1 − w) 1 − 1 + ; (6) (8) and (9) is used, behaves like x2 3 5 Fw(x) = 휋−2:028368x+0:518855x −0:641539x +··· : with w 2 [0; 1] being a positive weight factor. In prin- (11) ciple, one can find the governing equation for the un- One can see that Eq. (11) agrees well with Eq. (10) up known 휆(x) by substituting Eq. (6) into Eq. (3) and to x6. For large x ! 1 the series solution for F (x) obtain a series solution of 휆(x) by solving the gov- can be obtained by solving Eq. (3) with x replaced by erning equation. Here, however, we choose the Padé 1=y and using the series expansion for small y. After approximation to parameterize 휆(x) re-changing y to x, one finds n 2 2:1596 0:222 116:0 1 + ax + ··· F (x) = 2 1 − 4 − 6 휆(x) = 휆0 ; (7) x x x 1 + bx2 + ··· 0:113 2:71 o + 8 + 10 + ··· : (12) since it has an equal potential as series in approximat- x x ing a continuous function. Note that we have already On the other hand, the solution Hyb(2/2), at large x, written 휆(x) as a function of x2 instead of x since so has the asymptotic form is F2(x) in Eq. (5). The simplest nontrivial case of the 2:0348n 0:91576 5:8524 9:6819 F (x) = 1 + − + above Padé approximation is the [2/2] approximant w x2 x2 x4 x6 3:1677 51:943 o 1 + ax2 + − + ··· ; (13) 휆(x) = 휆 : (8) x8 x10 0 1 + bx2 which agrees globally with Eq. (12) except for a small The minimization of the energy (2) with the trial bit difference. The detailed differences between (12) function (6) with respect to the variational parame- and (13) at large x can be due to the fact that the variationally-obtained solution (6) approximates the ters (a; b; c; w; 휆0) was carried out numerically for the [2/2] Padé approximant (8) using the downhill simplex Skyrmion profile globally and may produce small er- method (the Neilder-Mead algorithm). The result for rors in local region, for instance, Fw(50) = 8:142 × −4 −4 the optimized parameters is given by 10 while F (50) = 8:638 × 10 . a = 0:330218; b = 1:331975; c = 2:094056; 3.5 w = 0:286566; 휆0 = 7:323877; (9) 3.0 Hyb(2/2) SK 2 Hyb(4/4) with E = 1:23152(6휋 f휋=e). The solution (6), with 2.5 The solution (4) 휆(x) given by Eq. (7) and the parameters (9), is re- The solution (5) 2.0 ferred as the solution Hyb(2/2) in brief in this study Num. Ponciano and is plotted in Fig.1, compared to the solutions Yamashita et al (4) and (5), and the numerical solution (Num.) to 1.5 Eq. (3). We also include the analytic solutions given 1.0 in Ref. [14] and the solution in the form of the purely Chiral angle Padé approximant[13] for comparison. A quite well 0.5 agreement of our solution with the numerical solution 0.0 can be seen from this plot. We note that the inequal- 0 2 4 6 8 10 SK 2 ity E =(6휋 f휋=e) ≥ jBj = 1 is fulfilled for all of cited Radial distance results of the energy (see Table1).