Gauge invariance in the presence of a cutoff

TL;DR

This paper develops a method to construct electromagnetic currents in two-nucleon systems with a finite cutoff, preserving gauge invariance and the long-range exchange current, crucial for effective field theory applications.

Contribution

It introduces a gauging equations approach that maintains gauge invariance and the equivalence of T-matrix and five-point functions in cutoff and non-cutoff theories.

Findings

Ensures cutoff theory reproduces reference theory results.

Maintains long-range exchange currents with a cutoff.

Provides linear Ward-Takahashi identities in the cutoff framework.

Abstract

We use the method of gauging equations to construct the electromagnetic current operator for the two-nucleon system in a theory with a finite cutoff. The employed formulation ensures that the two-nucleon T-matrix and corresponding five-point function, in the cutoff theory, are identical to the ones formally defined by a reference theory without a cutoff. A feature of our approach is that it effectively introduces a cutoff into the reference theory in a way that maintains the long-range part of the exchange current operator; for applications to effective field theory (EFT), this property is usually sufficient to guarantee the predictive power of the resulting cutoff theory. In addition, our approach leads to Ward-Takahashi (WT) identities that are linear in the interactions. From the point of view of EFT's where such a WT identity is satisfied in the reference theory, this ensures that gauge invariance in the cutoff theory is maintained order by order in the expansion.