QCD phase diagram in the $T-eB$ plane for varying pion mass

We study the effect of a varying pion mass on the quantum chromodynamics (QCD) phase diagram in the presence of an external magnetic field, aiming to understand it, for the first time, using Nambu\textendash Jona-Lasinio like effective models. We compare results from both its local and nonlocal versions. In both cases, we find that the inverse magnetic catalysis (IMC) near the crossover is eliminated with increasing pion mass, while the decreasing trend of crossover temperature with increasing magnetic field persists for pion mass values at least up to $440$ MeV. Thus, the models are capable of capturing qualitatively the results found by lattice QCD (LQCD) for heavy (unphysical) pions. The key feature in the models is the incorporation of the effect of a reduction in the coupling constant with increasing energy. Along with reproducing the IMC effect, it enables models to describe the effects of heavier current quark masses without introducing additional parameters. For the local NJL model, this agreement depends on how the parameters of the model are fit at the physical point. In this respect, the nonlocal version, which, due to its formulation, automatically exhibits the IMC effect around the crossover region, captures the physics more naturally. We further use the nonlocal framework to determine the pion mass beyond which the IMC effect around the transition region does not exist anymore.