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

TL;DR

The paper develops a complete resurgence framework for the two-field Euler–Heisenberg Lagrangian in spinor and scalar QED, deriving explicit large-order asymptotics that reveal a richer Borel-plane structure with both real and imaginary axis singularities. By recasting the two-variable weak-field expansion into a single-variable series with a fixed field ratio and applying Padé–Borel and Padé–Conformal–Borel resummations, it demonstrates accurate reconstruction of both the real and imaginary parts of the Lagrangian from finite perturbative data, with spinor QED benefiting most from conformal improvements. The results interpolate smoothly between the single-field magnetic and electric limits and show that nonperturbative Schwinger pair production is encoded in perturbative coefficients, thereby completing the resurgence program for constant backgrounds. This analytic framework provides robust tools for exploring nonperturbative phenomena in strong-field QED and scalar QED and suggests natural extensions to higher loops and inhomogeneous backgrounds relevant to future high-intensity laser experiments.

Abstract

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 Padè-Borel and Padè-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. In the scalar case, it yields only marginal 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.

Figures (10)

• Figure 1: Contour integrals for Borel dispersion relations for the two-field scalar and spinor EHLs. The singularities on the Borel plane occur at $k\pi i$ and $k\pi/\kappa$ with $k \in \mathbb{Z}-\{0\}$. Notice that, unlike the single-field case, the Borel plane has singularities both on the real and imaginary axes, which leads to a contour $\mathcal{C}_{\infty}$ that has slots both on the real and imaginary axes instead of just the imaginary axis, as is the case in the single-field Borel plane.
• Figure 2: Values of the weak-field expansion coefficient for $n=50$ plotted as a function of $\kappa$ for the (a) spinor, and (b) scalar QED two-field EHL compared with the expected large-N expansion computed in Eqs. (\ref{['eq:large-N-behaviour-spinor']}) and (\ref{['eq:large-N-behaviour-scalar']}). Note that these are plots of absolute values.
• Figure 3: Comparison of numerically computed ratios (orange) and with Richardson extrapolation applied (blue) of consecutive weak-field coefficients of (a) spinor and (b) scalar two-field EHLs with the expected ratio (grey) computed from the leading order contributions in Eqs. (\ref{['eq:large-N-behaviour-spinor']}) and (\ref{['eq:large-N-behaviour-scalar']}) respectively. The value of $\kappa$ is fixed at $\kappa=2$.
• Figure 4: The location of the poles of the Padé sums for (a) $P^N_{N+1}\left[\widehat{\mathcal{S}}^{\text{sp}}_{\lambda,N^*} \right](\zeta,\kappa)$ and (b) $P^N_{N+1}\left[\widehat{\mathcal{S}}^{\text{sc}}_{\lambda,N^*} \right](\zeta,\kappa)$ for $\kappa=0.5$ and varying values of $N$.
• Figure 5: The Padé-Borel reconstruction of the one-loop two-field spinor and scalar EHL for different values of $N$ with $\kappa=0.5$. The parameter values used here are $\lambda=1,f=1,\varepsilon=0.1$. (a), (b) are plots of the real part of the Padé-Borel reconstructed spinor and scalar EHLs, respectively, compared with their special function expansions in Eqs. (\ref{['eq:ehl-numerical-real-part-qed']}) and (\ref{['eq:ehl-numerical-real-part-sqed']}). (c), (d) are the plots of the imaginary part of the Padé-Borel reconstructed spinor and scalar EHLs, respectively, compared to the instanton expansions in Eqs. (\ref{['eq:ehl-imaginary-instanton-expansion-qed']}) and (\ref{['eq:ehl-imaginary-instanton-expansion-sqed']}).