Vacuum free energy, quark condensate shifts and magnetization in three-flavor chiral perturbation theory to O(p6) in a uniform magnetic field

We study three-flavor QCD in a uniform magnetic field using chiral perturbation theory (χPT). We construct the vacuum free energy density, quark condensate shifts induced by the magnetic field and the renormalized magnetization to O(p6) in the chiral expansion. We find that the calculation of the free energy is greatly simplified by cancellations among two-loop diagrams involving charged mesons. In comparing our results with recent 2+1-flavor lattice QCD data, we find that the light quark condensate shift at O(p6) is in better agreement than the shift at O(p4). We also find that the renormalized magnetization, due to its small-ness, possesses large uncertainties at O(p6) due to the uncertainties in the low-energy constants.