On the Feynman Rules of Massive Gauge Theory in Physical Gauges

For a massive gauge theory with Higgs mechanism in a physical gauge, the longitudinal polarization of gauge bosons can be naturally identified as mixture of the goldstone component and a remnant gauge component that vanishes at the limit of zero mass, making the goldstone equivalence manifest. In light of this observation, we re-examine the Feynman rules of massive gauge theory by treating gauge fields and their corresponding goldstone fields as single objects, writing them uniformly as 5-component "vector" fields. The gauge group is taken to be $SU(2)_L$ to preserve custodial symmetry. We find the derivation of gauge-goldstone propagators becomes rather trivial by noticing there is a remarkable parallel between massless gauge theory and massive gauge theory in this notation. We also derive the Feynman rules of all vertices, finding the vertex for self-interactions of vector (gauge-goldstone) bosons are especially simplified. We then demonstrate that the new form of the longitudinal polarization vector and the standard form give the same results for all the 3-point on-shell amplitudes. This on-shell matching confirms similar results obtained with on-shell approach for massive scattering amplitudes by Arkani-Hamed et.al. Finally we calculate some $1\rightarrow 2$ collinear splitting amplitudes by making use of the new Feynman rules and the on-shell match condition.