Unambiguous cancellation of divergences of $H \to \gamma \gamma$ process via one W loop in unitary gauge: Gauge invariance in Dyson scheme and physical boundary condition

Abstract

Following the thread of R. Gastmans, S. L. Wu and T. T. Wu, the calculation in the unitary gauge for the $H \to \gamma \gamma$ process via one W loop is repeated, but without the specific choice of the independent loop momentum for the Feynman diagrams. This is based on the original 'Dyson scheme' provided in Dyson's classical paper. I.e., the original integrations on all propagator momenta are kept, not expressed by the specified independent loop momentum. Correspondingly, the 4-dimension $\delta$ function at each vertex in which the 4-momentum conservation is embedded, is retained. Together with the Ward identity of the W-W-photon vertex, the 4-momentum conservation of each vertex guarantee the cancellation of all divergent integrals worse than logarithmic without any uncertainty or ambiguity, with any shift of integrated momentum eschewed. The calculation is in 4-dimension Minkowski phase space and without any help of regularization. The resulting integrals are to the most logarithmically divergent, hence is invariant for various setting of the independent loop momentum and any of its shift. At last the logarithmically divergent symmetric (tensor) integration is determined by the boundary condition at infinity in momentum space inherent of the free Feynman propagator, and the gauge (both $SU(2)\times U_Y(1)$ and $U_{em}(1)$) invariant finite result can be obtained without the introduction of the 'Dyson subtraction'. The physical boundary conditions at infinity of phase space can make sense in many problems.