Nonlinear effects in Schwinger-Dyson Equation

We study nonlinear effects in the QED ladder Schwinger-Dyson(SD) equation. Without further approximations, we show that all nonlinear effects in the ladder SD equation can be included in the effective couplings and how a linear approximation works well. The analyses is generalized in the case of the improved ladder calculation with the Higashijima-Miransky approximation. The dynamical symmetry breaking (DSB) is widely studied by the Schwinger-Dyson (SD) equation. (See references [1] and [2], and the references therein. ) In QED, the lowest-order approximation is a ladder approximation. It was pointed out that there is DSB of the chiral symmetry in the QED ladder SD equation with a finite cut off. (See, e.g., [3, 4, 2]. ) In QCD, the calculation was modified by introducing a running coupling. The modified calculation is called an improved ladder approximation and gives a good results even quantitatively. Although the original SD equation is an integral equation, it is sometimes rewritten in the form of a nonlinear differential equation. (See, e.g., [4, 5, 1, 2]. ) Even in the most simple ladder approximation, the nonlinear equation can not be solved exactly and a linear approximation is widely used. (See, e.g., [6, 2]. ) The aim of this paper is to re-examine the nonlinear effects in the QED ladder SD equation and show how the linear approximation works well. After that, we generalize our analyses in the case of the improved ladder calculation with the HigashijimaMiransky approximation. [7, 8] In Landau gauge, the QED ladder Schwinger-Dyson equation is given by B(p) = 3e ∫ dq (2π)4 B(q) q2 +B2(q2) 1 (p− q)2 +m0, (1) where B(p) is a fermion self-energy with four momentum p and m0 is a bare mass of the fermion. We restrict our discussions on Euclidean region only. Putting p = me t 2 (m is a some energy scale) and rescaling x(t) = B(p)/p, after some algebra, we get a nonlinear second differential equation [4, 5, 1] ẍ+ 2ẋ+ 3 4 x+ λ x 1 + x2 = 0 (2) with two boundary conditions v1 ≡ e 3t 2 (ẋ+ 1 2 x) → 0 (t → −∞) (3) and v3 ≡ e t 2 (ẋ + 3 2 x) → m0 m (t → ∞). (4)