EXTERNAL GAUGE-INVARIANCE AND ANOMALY IN BS VERTICES AND BOUNDSTATES

A systematic method is given for obtaining consistent approximations t o the Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations which mai ntain the external gauge invariance. We show that for any order of app roximation to the SD equation there is a corresponding approximation t o the BS equations such that the solutions to those equations satisfy the Ward-Takahashi identities of the external gauge symmetry. This for mulation also clarifies the way how we can calculate the Green functio ns of current operators in a consistent manner with the gauge invarian ce and the axial anomaly. We show which type of diagrams for the pi0 - -> gammagamma amplitude using the pion BS amplitude give result consis tent with the low-energy theorem. An interesting phenomenon is observe d in the ladder approximation that the low-energy theorem is saturated by the zeroth order terms in the external momenta of the pseudoscalar BS amplitude and the vector vertex functions.