Magnetic Corrections to ${\pi}$-${\pi}$ Scattering Lengths in the Linear Sigma Model

TL;DR

This paper investigates how magnetic fields influence ${}$-${}$ scattering lengths within the linear sigma model, revealing that magnetic corrections increase the $I=2$ channel but decrease the $I=0$ channel, contrasting with thermal effects.

Contribution

It provides the first detailed calculation of magnetic corrections to ${}$-${}$ scattering lengths in the linear sigma model, including all one-loop channels and small magnetic field effects.

Findings

Magnetic field increases the $I=2$ scattering length.

Magnetic field decreases the $I=0$ scattering length.

The $I=1$ channel remains unaffected.

Abstract

In this article we consider the magnetic corrections to ${\pi}$-${\pi}$ scattering lengths in the frame of the linear sigma model. For this we consider all the one loop corrections in the s, t and u channels, associate to the insertion of a Schwinger propagator for charged pions, working in the region of small values of the magnetic field. Our calculation relies on an appropriate expansion for the propagator. It turns out that the leading scattering length, $l = 0$ in the S-channel, increases for an increasing value of the magnetic field, in the isospin $I = 2$ case whereas the opposite effect is found for the $I = 0$ case. The isospin symmetry is valid because the insertion of the magnetic field occurs through the absolute value of the electric charges. The channel $I = 1$ does not receive any corrections. These results, for the channels $I = 0$ and $I = 2$ are opposite with respect to the thermal corrections found previously in the literature.