Light-front Ward-Takahashi Identity for Two-Fermion Systems

TL;DR

This paper develops a light-front framework for two-fermion systems that preserves gauge invariance via a Ward-Takahashi identity, enabling consistent calculations of electromagnetic currents in bound and scattering states.

Contribution

It introduces a three-dimensional light-front current operator satisfying a Ward-Takahashi identity, derived from a quasi-potential reduction of the four-dimensional current, applicable to bound and scattering states.

Findings

The light-front current operator satisfies the Ward-Takahashi identity at any order.

Explicit construction verified in the Yukawa model with ladder approximation.

Instantaneous terms and pair contributions are crucial for current conservation.

Abstract

We propose a three-dimensional electromagnetic current operator within light-front dynamics that satisfies a light-front Ward-Takahashi identity for two-fermion systems. The light-front current operator is obtained by a quasi-potential reduction of the four-dimensional current operator and acts on the light-front valence component of bound or scattering states. A relation between the light-front valence wave function and the four-dimensional Bethe-Salpeter amplitude both for bound or scattering states is also derived, such that the matrix elements of the four-dimensional current operator can be fully recovered from the corresponding light-front ones. The light-front current operator can be perturbatively calculated through a quasi-potential expansion, and the divergence of the proposed current satisfies a Ward-Takahashi identity at any given order of the expansion. In the quasi-potential expansion the instantaneous terms of the fermion propagator are accounted for by the effective interaction and two-body currents. We exemplify our theoretical construction in the Yukawa model in the ladder approximation, investigating in detail the current operator at the lowest nontrivial order of the quasi-potential expansion of the Bethe-Salpeter equation. The explicit realization of the light-front form of the Ward-Takahashi identity is verified. We also show the relevance of instantaneous terms and of the pair contribution to the two-body current and the Ward-Takahashi identity.