Resurgence in the two-field scalar and spinor Quantum Electrodynamics Euler-Heisenberg Lagrangian

We present the first systematic resurgent analysis of the Euler-Heisenberg Lagrangian in spinor and scalar quantum electrodynamics for the most general constant background field configuration. In contrast to the extensively studied single-field cases, the two-field case exhibits unique asymptotic structures, leading to a substantially richer pattern of singularities in the Borel plane. Explicit large-order asymptotic formulas for the weak-field coefficients in both spinor and scalar quantum electrodynamics are derived. These reveal a nontrivial interplay between alternating and non-alternating factorial growth, governed by distinct structures associated with electric and magnetic contributions, and smoothly interpolating between the known single-field limits. Using Borel dispersion techniques, we demonstrate that the complete instanton structure underlying Schwinger pair production in two-field backgrounds is encoded in the divergent perturbative coefficients. We then construct resurgent approximants using Pad\'{e}-Borel and Pad\'{e}-Conformal-Borel resummation schemes adapted to the two-field case. For the spinor case, conformal improvement results in a significant enhancement in reconstructing both the real and imaginary parts of the effective Lagrangian across a wide range of field ratios, accurately capturing the subtle sign-changing features in the strong-field regime while in the scalar case, it yields minor improvement. Detailed comparisons with exact special-function representations demonstrate the reliability of reconstructions from a modest number of weak-field coefficients. This work establishes a natural completion of the resurgence programme for constant electromagnetic backgrounds, providing a robust analytic framework for exploring nonperturbative physics and strong-field phenomena in spinor and scalar quantum electrodynamics, from finite perturbative data.