Vacuum free energy, quark condensate shifts and magnetization in three-flavor chiral perturbation theory to $\mathcal{O}(p^6)$ in a uniform magnetic field

We study three-flavor QCD in a uniform magnetic field using chiral perturbation theory ($\chi$PT). We calculate the vacuum energy density, quark condensate shifts and the renormalized magnetization to $\mathcal{O}(p^6)$ in the low-energy expansion. We find that the calculation of the vacuum density is greatly simplified by cancellations among two-loop diagrams involving charged mesons. Comparing our results with recent $2+1$-flavor lattice QCD data, we find that the light quark condensate shift at $\mathcal{O}(p^6)$ is in better agreement than the shift at next-to-leading order. We also find that the renormalized magnetization, which is positive definite at next-to-leading order can be either positive or negative at $\mathcal{O}(p^{6})$ due to the uncertainties in the low-energy constants.