Two-loop self-dual Euler-Heisenberg Lagrangians (II): Imaginary part and Borel analysis

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We analyze the structure of the imaginary part of the two-loop Euler-Heisenberg QED effective Lagrangian for a constant self-dual background. The novel feature of the two-loop result, compared to one-loop, is that the prefactor of each exponential (instanton) term in the imaginary part has itself an asymptotic expansion. We also perform a high-precision test of Borel summation techniques applied to the weak-field expansion, and find that the Borel dispersion relations reproduce the full prefactor of the leading imaginary contribution.