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On validity of perturbative quantization of the breathing mode in the Skyrme model

We present a detailed discussion of the breathing mode quantization in the Skyrme model and demonstrate that the chiral angle of the hedgehog soliton is strongly affected by the breathing motion.

this Brief Report we investigate validity of this assumption and show that the breathing motion affects strongly the soliton chiral angle behavior.
We are starting from the standard Lagrangian of the model where L µ = U + ∂U ∂x µ ; U = U(t, x) is the SU(2) chiral field matrix; F π and m π stand for the pion decay constant and pion mass, respectively; e S is the dimensionless Skyrme constant.
The chiral angle θ 0 (r) is obtained from the variational principle supplemented by the boundary conditions θ 0 (r)|r →∞ = 0.
Eq. (3) is equivalent to the following differential equation Now let us consider the breathing and rotating hedgehog ansatz [2] where A ∈ SU(2), A = a 0 (t)+i τ a(t), a 2 0 + a 2 = 1. The quantities a p = (a 0 , a), p = 0, 1, 2, 3 and the homogeneous scale transformation parameter λ(t) are the quantum collective coordinates which describe rotation and vibration of the soliton, respectively. In terms of these collective coordinates the Lagrangian of the dynamical system is written as [2]: where and (Integrals with the subscripts 2, 4 and π are the contributions of the kinetic term, the Skyrme term and the symmetry breaking term, respectively.) In the perturbative quantization approach [2,3] the integrals (12)-(18) are estimated by means of substitution of the static solution θ 0 (r) instead of θ(r).
To take into account the influence of the breathing mode nonperturbatively, one has to determine the chiral angle θ(r) from the variational principle averaged over the quantum state |nj [5]: where n and j denote quantum numbers of the rotational and breathing excitations, respectively. Using the following notations one can rewrite (19) in the form 1 θ(r)|r →∞ = 0.
The main difference between the static case and the quantum breathing one consists in the factors of the second order derivative terms in Eqs. (6) and (26): contrary to (6), in Eq.(26) this factor contains the term proportional tor 4 . The latter arises from the breathing motion kinetic energy. This difference changes the asymptotic behavior of the chiral angle drastically. Let us compare the asymptotic solutions of the two equations at larger. First, Eq. (6) is asymptotically reduced tõ where µ 2 π = m 2 π e 2 S F 2 π and θ ∞ 0 (r) = lim r→∞ θ 0 (r) ≪ 1. The general solution of Eq.(29) is In order to satisfy the boundary condition (5) the coefficient C 1 in (30) must be zero which can be realized by appropriate choice of θ ′ (r)|r =0 . Next, the asymptotic form of Eq. (26) is , and the general solution is expressed as [6] θ ∞ (r) = where First of all, we have to note that (contrary to the static case) the coefficient C 1 in Eq.(32) cannot be chosen equal zero by appropriate choice of θ ′ (r)|r =0 due to the following reason. The factor at second order derivative of Eq.(26) is positive for smallr and becomes negative for larger, so there exists at least one pointr 1 , where g(r 1 ) = 0. This point is a singular point of Eq.(26) and in order to get regular solution the following condition must be fulfilled  (27) does not satisfy C 1 = 0. It is easily seen that for the case κ 1 ≥ 0 (which corresponds to ν ≥ 0 ) the condition (28) is not fulfilled unless C 1 = 0. At ν < 0 the condition (28) can be fulfilled for any C 1 and C 2 , but there remain some problems about the functionals Q 2 , V π and I 2 . If ν ≤ − 9 4 they are divergent for arbitrary nontrivial values of C 1 and C 2 . When − 9 4 < ν < 0, C 1 has to be zero in order to get finite values of these functionals. So there is no solution for the case of ν ≥ 0 (such situation arises, for instance, in the chiral limit (m π = 0) ), when ν < 0 the existence of solution is not ruled out, but the integrals Q 2 , V π and I 2 should be divergent.
It was shown that the nonperturbative quantization of the breathing and rotating modes gives stable soliton solutions in the nonlinear σ-model without the Skyrme term [6]. Their energy and mean square radii are finite in spite of divergency of the functionals Q 2 , V π and I 2 . One can expect that in the present model solutions with finite energy and size could also exist. But, in any case, the slow asymptotic decreasing (at − 9 4 ≤ ν < 0) and oscillating behavior (at ν < − 9 4 ) of the chiral angle at larger contradicts the Yukawa law. This problem inherent in the rotating soliton (γ 2 , γ 4 = 0) as well as in nonrotating one (γ 2 , γ 4 = 0) what means that the source of these difficulties is in the assumption about homogeneous global breathing which does not describe the soliton external part correctly.
Our main result implies that the nonperturbative quantization of the homogeneous global breathing of the skyrmion gives the solutions different drastically from the static ones. In this case the using of the perturbative approach is problematic. One would also face the similar problems considering the homogeneous global breathing quantization of multiskyrmions, solitons in modified Skyrme model and the global nonspherically-symmetrical time dependent deformation of skyrmions [7].