Scalar Charges in a Self-Dual Background

In this work, we present a $3+1d$ scalar QED model in a constant self-dual magnetic field configuration. We provide exact closed form analytic calculations of the partition function, the $\beta$-function, and the propagators. To our knowledge, we present the first closed-form finite expression for the matter field propagators in a self-dual background, which is made accessible by working entirely in the Landau level basis. This theory serves as a toy model for charged particles in parallel electric and magnetic fields, with natural extensions to studies of $3+1d$ chiral magnetic effects and pulsar physics.