Condition on the Symmetry-breaking Solution of the Schwinger-dyson Equation

We derive a condition for a non-trivial solution of the Schwinger-Dyson equation to be accompanied by a Goldstone bound state. It implies that, for quenched planar QED, although chiral symmetry breaking occurs when there is a cutoff, the continuum limit fails to exist. 1 I. Introduction Dynamical chiral symmetry breaking 1 (DCSB) has been extremely useful in a wide variety of topics in physics, from the success of low energy theorems in current algebra, to the construction of composite models of the Higgs boson. There are two aspects in the mechanism of DCSB, the mass generation of fermions and the existence of the Goldstone boson. 2 More concretely, chiral symmetry breaking occurs when the self-consistent Schwinger-Dyson (SD) equation for the fermion mass develops a non-trivial solution as the coupling constant reaches a critical value. 3 In addition, for the same coupling constant, there must also exist a bound state solution of the Bethe-Salpeter (BS) equation, with vanishing 4-momenta, corresponding to the massless pseudoscalar Goldstone boson. When both of these conditions are met, the self-energy solution shall be called a symmetry-breaking solution. We should also mention that T. Maskawa and H. Nakajima 4 had proposed an equivalent condition for the existence of the symmetry-breaking solution. Their condition is that the self-energy solution has to satisfy the Ward-Takahashi identity. For definiteness, we will concentrate on QED in the massless, quenched planar approximation 5 , which is a well-known model in the study of DCSB. It is simple enough for a detailed study on its properties, yet it is also rich in structure which suggests general physical features in other models. Thus, with the incorporation of running coupling constant, it can be easily extended to QCD in the large N approximation. 6 In addition, results gleaned from studies of this model have been used in the construction of realistic models in the context of technicolor 7 , t ¯ t condensation 8 , etc. In the study of the quenched planar QED, an important question concerns the cutoff. Thus, Maskawa and Nakajima established the existence of a critical coupling constant α c in the solution of the SD equation with a cutoff, Λ. When α > α c , a nontrivial solution of the SD equation emerges which signals the advent of chiral symmetry breaking. A natural question is whether there exists a limit, α c (∞) = lim Λ→∞ α c (Λ), …