Ward-Takahashi identity in light-front formalism for a bound state of fermions
Deepesh Bhamre, J. P. B. C. de Melo
TL;DR
This work tackles the Ward-Takahashi identity in light-front quantum field theory for a spin-0 bound state of fermions. It first confirms the identity covariantly at one loop, $q_{\mu}J^{\mu}(P,P') = \Sigma(P) - \Sigma(P')$, and then carries out a detailed LF analysis, integrating over the energy component and decomposing fermion propagators into on-shell and off-shell parts. The key finding is that a faithful LF proof requires careful treatment of end-point singularities (zero modes) and the inclusion of the pair production diagram alongside the standard triangle diagram; without these, covariance and gauge invariance are not fully restored. The results reinforce the consistency of LF Hamiltonian perturbation theory with gauge symmetry and provide a concrete procedure to connect covariant and LF formulations in bound-state problems.
Abstract
We investigate the Ward-Takahashi identity at one-loop in the light-front (LF) formalism for a bound state of fermions. We consider a spinless bound state made up of two fermions in which the Ward-Takahashi identity is satisfied in the covariant formulation. Considering the same system in the light-front formalism, we investigate the proof of Ward-Takahashi identity by integrating the light-front energy component through the identification of the relevant ranges of the longitudinal LF momentum. We elucidate that the pair production diagram plays a crucial role in establishing the Ward-Takahashi identity. We also point out the necessity of taking into account the corresponding zero modes for truly establishing the Ward-Takahashi identity in the LF formalism.