On the Surviving Pion-Skyrmion Yukawa Coupling

Abstract

Recently two papers have been written by Saito l ) and Holzwarth et al.,2) which aim at giving the decay width of a L1(1232) state in the Skyrmion model. Their aim is to remedy the unpleasant situation on the pion-Skyrmion Yukawa coupling:3),4) Since pion fields are introduced as quantum fluctuations around a rotating Skyrmion, the Yukawa coupling, which is linear in the fluctuating pion field, disappears from the resultant pion-Skyrmion Lagrangi~n owing to the stability condition of the Skyrmion. The Yukawa of a pion to the axial vector current of a nucleon has been long known to be of order me,5) where Ne is the numbr of colors: The GoldbergerTreiman relation for the nucleon is written as I4iifNN=mgA/F, where jNN is the pion-nucleon Yukawa constant, and gA(F,,) is the axial vector (pion decay) constant, which is known to be of order N e(N//2).5),6) Thus, jNN is of order me. The same Ne dependence is also seen in the Skyrmion model,6) but the pionSkyrmion coupling in this case is the measure of the classical and spacelike pion tail of the extended soliton, but not the measure of the emission or absorption of the quantum and timelike pions. In this paper we point out that the Yukawa will only survive at order of Ne -3/2 within the tree approximation. This is because a rotating Skyrmion field with collective coordinates is not an exact solution of the equations of motion, nor minimizes the rotating soliton masses. It is also discussed that the Yukawa couplings thus obtained could not be identified with the standard pion-baryon Yukawa couplings. According to Refs. 1) and 2), the pion-Skyrmion Lagrangian has a term linear in the time-derivative of the pion field ¢a(X, t) with a=l, 2 and 3, which is written as