Analytical Solution for the SU(2) Hedgehog
Skyrmion and Static
Properties of Nucleons
An analytical solution for symmetric Skyrmion was proposed for the SU(2) Skyrme model, which take the form of the hybrid form of a kink-like solution and that given by the instanton method. The static properties of nucleons was then computed within the framework 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:12.38.-t, 11.15.Tk, 12.38.Aw
The Skyrme model Skyrme () is an effective field theory of mesons and baryons in which baryons arise as topological soliton solutions, known as Skyrmions. The model is based on the pre-QCD nonlinear model of the pion meson and was usually regarded to be consistent with the low-energy limit of large-N QCDWitten (). For this reason, among others, it has been extensively revisited in recent years ANW (); JR (); Sutcliffe (); Battye97 (); Atiyah89 (); Battye () (see, Zahed (); Topsoliton (), for a review). Owing to the high nonlinearity, the solution to the Skyrme model was mainly studied through the numerical approach. It is worthwhile, however, to seek the analytic solutions JMP (); Ponciano (); Yamashita () of Skyrmions due to its various applications in baryon phenomenology. One of noticeable analytic method for studying the Skyrmion solutions is the instanton approach proposed by Atiyah and MantonAtiyah89 () which approximates critical points of the Skyrme energy functional.
In this Letter, we address the static solution of hedgehog Skyrmion in the Skyrme model without pion mass term and propose an analytical solution for the hedgehog Skyrmion by writing it as the hybrid form of a kink-like solution and the analytic solution obtained by the instanton method Atiyah89 (). Two lowest order of Padé approximations was used and the corresponding solutions for Skyrmion profile are given explicitly by using the downhill simplex method. The Skyrmion mass and static properties of nucleon as well as delta was computed and compared to the previous results.
2 Analytic solution to Skyrme model
The Skyrme action Skyrme () without pion mass term is
in which , is the nonlinear realization of the chiral field describing the field and mesons with the unitary constrain , the pion decay constant, and a dimensionless constant characterizing nonlinear coupling. The Cauchy-Schwartz inequality for (1) impliesFaddeev () , where is the topological charge, known as baryon number. Using the hedgehog ansatz, ( are the three Pauli matrices) with depending merely on the radial coordinate , the static energy for (1) becomes
with a dimensionless variable and . The equation of motion of (2) is
where the boundary condition will be imposed so that it corresponds to the physical vacuum for (1):. The equation (3) is usually solved numerically due to its high nonlinearity ANW (); JR (); Battye97 (); Sutcliffe ().
A kink-like analytic solution was given by Sutcliffe ()
with , while an alternative Skyrmion profile, proposed based on the instanton method, takes the formAtiyah89 ()
with corresponding energy for the numeric factor . The singularity at is gauge dependent and can be gauged away without affecting the value for the Skyrme field.
To find the more accurate analytic solution, we first improve the solution (4) into with a numeric factor and then take to be a -dependent function: . Hence, we propose a Skyrmion profile function in the hybrid form mixing (4) and (5)
with being a positive weight factor. In principle, one can find the governing equation for the unknown by substituting (6) into (3) and obtain a series solution of by solving the governing equation. Here, however, we choose the Padé approximation to parameterize
since it has as equal potential as series in approximating a continuous function. Note that we have already written as function of instead of since so is in (5). The simplest nontrivial case of the above Padé approximation is the [2/2] approximant
The minimization of the energy (2) with the trial function (6) with respect to the variational parameters () was carried out numerically for the [2/2] Padé approximant (8) using the downhill simplex method (the Neilder-Mead algorithm). The result for the optimized parameters is given by
with . The solution (6), with given by (7) and the parameters (9), is referred as solution Hyb(2/2) for short in this paper and is plotted in Fig.1, compared to the solutions (4) and (5), and the numerical solution (Num.) to the equation (3). We also include the analytic solutions given by Yamashita () and the solution in the form of the purely Padé approximant Ponciano () for comparison. A quite well agreement of our solution with the numerical solution can be seen from this plot. We note that the inequality is fulfilled for all of cited results of the energy(see Table I).
One can see that (11) agrees well with (10) up to . For large the series solution for can be obtained by solving (3) with replaced by and using the series expansion for small . After re-changing to , one finds
On the other hand, the solution Hyb(2/2), at large , has the asymptotic form
which agrees globally with (12) except for a small bit differences. The detailed differences between (12) and (13) at large can be due to the fact that the variationally-obtained solution (6) approximates the Skyrmion profile globally and may produce small errors in local region, for instance, while .
The disagreement can be improved by employing the Padé approximant of higher order than (8), for example, the [4/4] approximant
The solution (6) with specified by (14) and (15) will be referred as the Hyb(4/4) in this paper and is also plotted in Fig.1. The Fig.2 shows the profiles of at large for Hyb(2/2) as well as Hyb(4/4), and the numeric solution. The asymptotic expansion of the solution Hyb(4/4) shows that for small the profile becomes
Here, a better value is obtained for the latter asymptotic profile in contrast with the solution Hyb(2/2). The computed Skyrmion energies (2), measured in the unit of , are listed in Table I, including the corresponding results obtained by the numeric solution and obtained in the relevant references as indicated.
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 Lagrangian using the collective variablesANW (). Here, we will use the solution Hyb(2/2) and Hyb(4/4) to compute the static properties of nucleons and nucleon-isobar() within the framework of the bosonic quantization of Skyrme model.
with the action for the static hedgehog configuration, , and
which is independent of and . The Hamiltonian associated to (16), when quantized via the quantization procedure in terms of collective coordinates, yields an eigenvalue , with being the soliton energy of the Skyrmion. This yields the masses of the nucleon and -isobar
The isoscalor root mean square(r.m.s) radius and isoscalor magnetic r.m.s radius are given by
respectively. Combining with the masses of nucleon and the delta, one can evaluate the magnetic moments for proton and neutron via the following formula
where plus and minus correspond to proton and neutron, respectively. The calculated results for these quantities using two solution schemes (Hyb.(2/2) and Hyb.(4/4)) are shown explicitly in Table II, compared to the experimental values as well as that computed by the numeric solution for . The corresponding results from other predictions are also shown in this table. Here in Table II, we use the experimental values , for fixing and through (18), in contrast with the input , used by Ref.ANW () and Ref.Yamashita ().
To check the solution further, we also list, in the Table II, the axial coupling constant and the -coupling, which are given by
respectively. Here, the numeric factor is
4 Concluding remarks
We show that the hybrid form of a kink-like solution and that given by the instanton method are suited to approximate the exact solution for the hedgehog Skyrmion, when combining with Padé approximation. The resulted analytic solution (6) has two remarkable features: (1) it is simple in the sense that it is globally given in whole region; (2) it well approaches the asymptotic behavior of the exact solution. We note that the further generalization of (6), made by approximating in (6) via Padé approximation, does not exhibit remarkable improvement, particularly in the asymptotic behavior of the chiral angle at infinity. We expect that our solution can be useful in the dynamics study of the Skyrmion evolution and interactions.
D. J thanks C. Liu and ChuengRyong Ji for discussions. This work is supported in part by the National Natural Science Foundation of China (No.10547009) and (No.10965005), and the Knowledge and S&T Innovation Engineering Project of NWNU (No. NWNU-KJCXGC-03-41)
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