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Light mesons around critical end points in $T-μ_B-μ_I-eB$ space

Light mesons $(σ, π^0, π^\pm)$ are investigated in $T-μ_B-μ_I-eB$ space by using a two-flavor NJL model, which are related to the chiral symmetry restoration and pion superfluid phase transition. In $T-μ_B-eB$ space, during the chiral restoration process, the mass of pseudo-Goldstone mode $π^0$ keeps increasing, together with the sudden mass jump. At the critical end point region, $π^0$ meson has a very sharp but continuous mass increase, together with a sudden mass jump at the Mott transition, and in the first order chiral phase transition region nearby, we observe twice $π^0$ mass jumps, induced by the Mott transition and quark mass jump, respectively. The mass of Higgs mode $σ$ first decreases and then increases associated with the chiral symmetry restoration, and shows a jump at the first order chiral phase transition. We plot the chiral phase diagram in terms of the change of quark mass, the Mott transition of $π^0$ and the minimum mass of $σ$. Due to the explicit breaking of chiral symmetry in physical case, the chiral restoration phase boundaries in $T-μ_B$ plane from the order parameter and meson side are different from each other. In $T-μ_I$ plane, the competition between pion superfluid phase transition and chiral symmetry restoration under magnetic fields is studied in terms of the Goldstone mode $π^+$ and the pseudo-Goldstone mode $π^0$. The separation of the two phase boundaries is enhanced by the external magnetic field. Different from the twice mass jumps of $π^0$ in the first order chiral phase transition region, the $π^+$ meson displays several mass jumps in the chiral crossover region. At the critical end point, $π^+$ also shows very sharp but continuous mass changes, together with a mass jump at the Mott transition.