Yukawa potential under weak magnetic field

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 parametrized with an effective gluon mass (Mg similar to 0.5 GeV) is considered. In the limit of magnetic field that is weak with respect to the constituent quark mass and pion mass, analytical and semianalytical 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 anisotropic corrections are considerably smaller than the isotropic components. A sizable splitting between results due to the magnetic field dependent neutral or charged pion mass is found.