Constructing vertices in QED
We study the Dyson Schwinger Equation for the fermion propagator in the quenched approximation. We construct a non-preservative fermion-boson vertex that ensures the fermion propagator satisfies the Ward-Takahashi identity, is multiplicatively renormalizable, agrees with the lowest order perturbation theory for weak couplings and has a critical coupling for dynamical mass generation that is strictly gauge independent. This is in marked contrast to the rainbow approximation in which the critical coupling changes by 50% just between the Landau and Feynman gauges. We also show how to construct a vertex which not only has the aforementioned properties but also agrees with the results obtained from the CJT effective potential for the critical exponent of the mass function. These vertices are expressed in terms of two functions which satisfy an integral and a derivative condition. By considering the perturbative expansion for the transverse vertex, we have performed numerical evaluation of the first of these functions which will hopefully guide their non-perturbative structure. The use of vertices satisfying these properties should lead to a more believable study of mass generation.