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Properties of magnetized neutral mesons within a full RPA evaluation

We consider the two flavor Nambu--Jona-Lasinio model within the RPA framework to evaluate the masses of the $σ$ and $π^0$ mesons and the $π^0$ decay constant in the presence of a magnetic field at vanishing temperatures and baryonic densities. The present work extends other RPA applications by fully considering the external momenta, which enter the integrals representing the magnetized polarization tensor, so that more accurate results can be obtained in the evaluation of physical quantities containing pionic contributions. As we show, this technical improvement generates results which agree well with those produced by lattice simulations and chiral perturbation theory. Our method may also prove to be useful in future evaluations of quantities such as the shear viscosity and the equation of state of magnetized quark matter with mesonic contributions.