QCD $\theta$-vacuum in a Uniform Magnetic Field

We study the $\theta$-vacuum of QCD using two-flavor chiral perturbation theory ($\chi$PT) in the presence of a uniform, background magnetic field calculating the magnetic field-dependent free energy density, the topological density, the topological susceptibility and the fourth cumulant at one-loop order. We find that the topological susceptibility is enhanced by the magnetic field while the fourth topological cumulant is diminished at weak fields and enhanced at larger fields when $\theta=0$. However, in the QCD vacuum with $\theta\neq 0$, the topological susceptibility can be either monotonically enhanced or diminished relative to their $\theta$-vacuum values. The fourth cumulant also exhibits monotonic enhancement or suppression except for regions of $\theta$ near $0$ and $2\pi$, where it is both diminished and enhanced. Finally, the topological density is enhanced for all magnetic fields with its relative shift being identical to the relative shift of the up and down quark condensates in the $\theta$-vacuum.