One-loop contributions to decays $e_b\to e_a\gamma$ and $(g-2{)}_{e_a}$ anomalies, and Ward identity

In this paper, we will present analytic formulas to express one-loop contributions to lepton flavor violating decays $e_b\to e_a\gamma$, which are also relevant to the anomalous dipole magnetic moments of charged leptons $e_a$. These formulas were computed in the unitary gauge, using the well-known Passarino-Veltman notations. We also show that our results are consistent with those calculated previously in the 't Hooft-Veltman gauge, or in the limit of zero lepton masses. At the one-loop level, we show that the appearance of fermion-scalar-vector type diagrams in the unitary gauge will violate the Ward Identity relating to an external photon. As a result, the validation of the Ward Identity guarantees that the photon always couples with two identical particles in an arbitrary triple coupling vertex containing a photon.