Dimensional reduction and the generalized pion in a magnetic field within the NJL model

In this work, the mass of the neutral pion is investigated in the presence of background magnetic fields in the framework of the Nambu--Jona-Lasinio model. Taking into account the anisotropic four-fermion interactions, a tensor current arises in the magnetized QCD system, which forms an anomalous magnetic moment (AMM) coupling in the Dirac equation for the quarks. By solving the gap equations, we find that the sign of the dynamically generated AMM is opposite to the sign of the quark's charge and its magnitude is definitely smaller than the constituent mass. We construct two generalized Nambu-Goldstone pions, which emerge as combinations of the quantum fluctuations around the conventional scalar and the emergent tensor chiral condensates. We analytically demonstrate that the Goldstone nature has been spoiled by the dimensional reduction in the two-particle state and the corresponding decreasing mass of the lighter generalized pionic mode is a remnant of the infrared dynamics.