Closed analytic formula for the pair‑production rate in a harmonic Yukawa background

Derive a closed analytic expression for the imaginary part of the effective action (the pair‑creation probability density) of a scalar field Yukawa‑coupled to a time‑dependent background potential V(t)=½ V0 sin^2(ω t), as represented in Eq. (1.3.32) with improved Schwinger–DeWitt derivative corrections, in arbitrary spacetime dimension D.

Background

In Chapter 1 the authors develop a worldline formalism for scalar fields in Yukawa backgrounds, including resummations and derivative corrections, and analyze non‑quadratic time‑dependent potentials to study assisted pair creation. For the specific harmonic potential V(t)=½ V0 sin^2(ω t), they obtain an expression for the imaginary part of the effective action in Eq. (1.3.32) involving a series that can be truncated for numerical evaluation.

While the framework yields systematic approximations and numerical results, the authors were unable to produce a closed‑form analytic expression for this case. Obtaining such a formula would sharpen analytic control over assisted pair production in inhomogeneous scalar backgrounds and benchmark the resummed worldline approach.