Divergence Cancellations in Spontaneously Broken Gauge Theories

Abstract

An extremely simple method is presented to calculate unambiguously higher-order corrections in the unitary ($U$) gauge of theories with spontaneously broken gauge symmetries. Manipulating Feynman integrals in coordinate space, the spurious nonrenormalizable infinities of this gauge are isolated in the form of (contracted) Feynman graphs. Without making reference to any specific global regularization scheme, the complete cancellation of these graphs is demonstrated in the cases of fermion-fermion scattering to fourth order in the Abelian model considered by Appelquist and Quinn and for the similar neutrino scattering in Weinberg's ${\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_L\times Y$ model. The reason for such complete cancellation is seen to be a consequence of the algebraic structure of the equal-time commutators among currents, their divergences, and various fields. This structure, of course, is dictated by the original gauge symmetry. As a check on our methods, the weak muon anomaly in Weinberg's model is calculated, and agreement is found with the (gauge-invariant) results of other authors.