Thermomagnetic effects on light pseudo-scalar meson masses within the SU(3) Nambu-Jona--Lasinio model

We calculate the screening masses of pseudoscalar mesons in a hot and strongly magnetized medium within the framework of the SU(3) Nambu-Jona--Lasinio model, using a magnetic field-independent regularization scheme. Inverse magnetic catalysis (IMC) is implemented through the use of a magnetic field-dependent coupling $G(B)$, fitted to reproduce lattice quantum chromodynamics (QCD) results for the pseudocritical chiral transition temperature $T_c^B$. For the external homogeneous magnetic field considered, neutral screening masses separate in two types: perpendicular and parallel to the direction of the field, while for charged mesons only parallel energies can be defined for each Landau level. We obtain $m_{\mathrm{scr},\perp} > m_{\mathrm{scr},\parallel}$, as expected from causality. Thermally, all screening energies are almost constant until some critical temperature, whose behavior is correlated with $T_c^B$. They rapidly increase around this value, keeping a steady enhancement afterward due to thermal excitation. Magnetically, neutral parallel masses are enhanced (suppressed) at high temperatures when considering $G(B)$ ($G$). Perpendicular ones display a non-monotonic magnetic behavior for $G$ (due to increasing $T_c^B$) when $T \lesssim 500$~MeV, but become magnetically enhanced when $T \gtrsim 500$~MeV. For $G(B)$ they always increase with $B$. Charged parallel energies are always magnetically enhanced, for both couplings. In the high-temperature limit, we show that both neutral and charged screening energies converge to $2\pi T$. At $B=0$ the model overestimates the remaining quark interaction in this regime. At $B\neq 0$ we find that, when IMC is accounted for, the interaction is suppressed as $B$ increases, a fact that appears to be at odds with currently available lattice QCD results.