External Gauge Invariance and Anomaly in BS Vertices and Boundstates

A systematic method is given for obtaining consistent approximations to theSchwinger-Dyson(SD) and Bethe-Salpeter(BS) equations which maintain theexternal gauge invariance. We show that for any order of approximation to theSD equation there is a corresponding approximation to the BS equations suchthat the solutions to those equations satisfy the Ward-Takahashi identities ofthe external gauge symmetry. This formulation also clarifies the way how we cancalculate the Green functions of current operators in a consistent manner withthe gauge invariance and the axial anomaly. We show which type of diagrams forthe $\pi^0\rightarrow\gamma\gamma$ amplitude using the pion BS amplitude giveresult consistent with the low-energy theorem. An interesting phenomenon isobserved in the ladder approximation that the low energy theorem is saturatedby the zeroth order terms in the external momenta of the pseudoscalar BSamplitude and the vector vertex functions.Comment: 30 pages, plain TeX (using `phyzzx' macro), 17 Postscript Figures are included as uuencoded files (need `epsf.tex'), KUNS-1236, HE(TH) 93/1