$\pi$-$\pi$ scattering lengths: electric corrections in the linear sigma model

TL;DR

This paper investigates how an external electric field influences $$-$$ scattering lengths within the linear sigma model, highlighting the significance of electric corrections and their contrast to magnetic effects.

Contribution

It provides the first explicit calculation of electric field effects on $$-$$ scattering lengths, including box diagrams previously neglected in literature.

Findings

Electric corrections oppose magnetic corrections.

Explicit box diagram calculations are performed.

Electric fields significantly modify scattering lengths.

Abstract

In the frame of the linear sigma model and working in the weak field approximation, we discuss the role played by an external electric field on the behavior of $\pi$-$\pi$ scattering lengths. For this purpose, we have considered all relevant one-loop diagrams in the $s$, $t$, and $u$ channels where we have Schwinger propagators for charged pions. An important novelty of our analysis, compared with previous existing discussions in the literature, is the explicit calculation of box diagrams which were previously not considered in discussions regarding magnetic corrections to $\pi$-$\pi$ scattering lengths. Our analysis shows that electric field corrections have an opposite effect with respect to the previously calculated magnetic corrections.