The Yukawa potential under weak magnetic field

TL;DR

This paper investigates how weak magnetic fields modify the Yukawa potential between nucleons via pion exchange, revealing small but anisotropic corrections that depend on pion and quark masses.

Contribution

It provides analytical and semi-analytical expressions for magnetic field corrections to the Yukawa potential, considering pion propagator, form factor, and an effective gluon mass.

Findings

Corrections are about 2-5% of the Yukawa potential at 2 fm distance.

Anisotropic corrections are smaller than isotropic ones.

Magnetic field causes a significant splitting between neutral and charged pion contributions.

Abstract

Weak magnetic field induced corrections for the Yukawa potential due to one pion exchange between two constituent quarks (nucleons) are presented. For that, the constant magnetic field effect on the pion propagator and on the pion form factor are taken into account. An effective gluon propagator parameterized with an effective gluon mass ($M_g\sim 0.5$\,GeV) is considered. In the limit of magnetic field weak with respect to the constituent quark mass and pion mass, analytical and semi-analytical expressions can be obtained. Different types of contributions are found, isotropic or anisotropic, dependent on the pion mass and also on the constituent quark and effective gluon masses. Overall the corrections are of the order of $2\%$ to $5\%$ of the Yukawa potential at distances close to $2$fm, and they decrease slower than the Yukawa potential. The anistropic corrections are considerably smaller than the isotropic components. A sizable splitting between results due to magnetic field dependent neutral or charged pion mass is found.