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

Abstract

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 ($\chi$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 $eB=0.2\mathrm{Ge}{\mathrm{V}}^2$, and the (renormalized) magnetization is in agreement up to $eB=0.3\mathrm{Ge}{\mathrm{V}}^2$, while three-flavor $\chi$PT, which gives a non-zero shift in the difference between the light quark condensates unlike two-flavor $\chi$PT, underestimates the difference compared to lattice QCD.

Comments: 10 pages, 3 figures