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Quark condensates and magnetization in chiral perturbation theory in a uniform magnetic field

We reconsider the problem of calculating the vacuum free energy (density) of QCD and the shift of the quark condensates in the presence of a uniform background magnetic field using two-and-three-flavor chiral perturbation theory ($χ$PT). Using the free energy, we calculate the degenerate, light quark condensates in the two-flavor case and the up, down and strange quark condensates in the three-flavor case. We also use the vacuum free energy to calculate the (renormalized) magnetization of the QCD vacuum, which shows that it is paramagnetic. We find that the three-flavor light-quark condensates and (renormalized) magnetization are improvements on the two-flavor results. We also find that the average light quark condensate is in agreement with the lattice up to $eB=0.2 {\rm\ GeV^{2}}$, and the (renormalized) magnetization is in agreement up to $eB=0.3 {\rm\ GeV^{2}}$, while three-flavor $χ$PT, which gives a non-zero shift in the difference between the light quark condensates unlike two-flavor $χ$PT, underestimates the difference compared to lattice QCD.