About consistence between pi N Delta spin-3/2 gauge couplings and electromagnetic gauge invariance

TL;DR

This paper investigates the compatibility of spin 3/2 gauge couplings for the Delta resonance with electromagnetic gauge invariance, revealing limitations in maintaining both symmetries simultaneously within effective field theory.

Contribution

It demonstrates that electromagnetic gauge invariance cannot be fully preserved with spin 3/2 gauge couplings at all orders, highlighting fundamental constraints in the theoretical framework.

Findings

Electromagnetic gauge invariance is only approximately maintained at a given order.

The Ward identity cannot be satisfied with a trimmed 3/2 propagator.

Radiative corrections reintroduce conventional interactions, breaking the gauge symmetry.

Abstract

We analyze the consistence between the recently proposed "spin 3/2 gauge" interaction for the Delta resonance with nucleons and pions, and the fundamental electromagnetic gauge invariance in any radiative amplitude. Chiral symmetric pion-derivative pi N Delta couplings can be substituted through a linear transformation to get Delta-derivative ones, which have the property of decoupling the 1/2 field components of the Delta propagator. Nevertheless, the electromagnetic gauge invariance introduced through minimal substitution in all derivatives, can only be fulfilled at a given order n without destroying the spin 3/2 one by dropping n+1 order terms within an effective field theory (EFT) framework with a defined power counting. In addition, we show that the Ward identity for the gamma Delta gamma vertex cannot be fulfilled with a trimmed 3/2 propagator, which should be necessary in order to keep the spin 3/2 gauge symmetry in the radiative case for the gamma Delta gamma amplitude. Finally, it is shown that radiative corrections of the spin 3/2 gauge strong vertexes at one loop, reintroduce the conventional interaction.