Asymptotics of the Pseudoscalar Bethe-Salpeter Amplitude

Abstract

We examine the asymptotic behaviour of the Bethe-Salpeter amplitude for the pseudoscalar Nambu-Goldstone boson in QCD in the chirallimit. The Bethe-Salpeter equation in the improved ladder approximation is shown to produce the correct asymptotic behaviour of the amplitude as predicted by the OPE and renormalisation group in the exact theory. This provides a highly non-trivial check of the ladder 'approximation, which is important considering its widespread applica­ tion in numerical work. reproduces the correct asymptotic behaviour in the solution for the BS amplitude of the Nambu-Goldstone (NG) bound state. If we expand this amplitude in powers of q, where q is the eM momentum of the bound state, the lowest term (obtained by setting q=O) is known to be the solution of the SD equation: however the order q components of the BS amplitude are new amplitudes whose asymptotic behaviour has not yet been discussed. In the first part we use the operator product expansion in QeD to derive the asymptotic behaviour of these O(q) BS amplitudes. We show that the coefficients of the leading terms are uniquely determined in terms of the decay constant fn. We then examine the asymptotic behaviour of the amplitudes given by the solution of the BS equation in the Higashijima approximation and show that they correctly produce the behaviour predicted by the OPE results of the exact theory. The Higashijima approximation is a particular way of incorporating the running coupling constant so that the SD equation can be converted into a simple differential equation. Further the analysis of the BS equation is also made easy in this approxi­ t) Postdoctoral fellow of the J<lpan Society for the Promotion of Science.