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Exact evaluation and extrapolation of the divergent expansion for the Heisenberg-Euler Lagrangian I: Alternating Case

Christian D. Tica, Eric A. Galapon

Abstract

We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral representations of the Heisenberg-Euler Lagrangian from QED for a constant magnetic field and magnetic-like self-dual background. We also devise a prescription based on the finite-part integration of a generalized Stieltjes integral to sum and extrapolate to the strong-field regime the alternating divergent weak-field expansions of the Heisenberg-Euler Lagrangians.

Exact evaluation and extrapolation of the divergent expansion for the Heisenberg-Euler Lagrangian I: Alternating Case

Abstract

We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral representations of the Heisenberg-Euler Lagrangian from QED for a constant magnetic field and magnetic-like self-dual background. We also devise a prescription based on the finite-part integration of a generalized Stieltjes integral to sum and extrapolate to the strong-field regime the alternating divergent weak-field expansions of the Heisenberg-Euler Lagrangians.
Paper Structure (12 sections, 2 theorems, 86 equations, 2 figures, 6 tables)

This paper contains 12 sections, 2 theorems, 86 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Hadamard's Finite Part
  3. Canonical Representation
  4. Contour Integral Representation
  5. Regularized Limit Representation
  6. Finite-Part Integration
  7. The Heisenberg-Euler Lagrangian
  8. Spin-0
  9. Self-dual electromagnetic background.
  10. Summation and extrapolation of the weak-field expansion for the Heisenberg-Euler Lagrangian
  11. Self-Dual Background
  12. Conclusion

Key Result

Lemma 2.1

Let the complex extension, $f(z)$, of $f(x)$, be analytic in the interval $[0,a]$. If $f(0)\neq 0$, then where $\log z$ is the complex logarithm whose branch cut is the positive real axis and $\mathrm{C}$ is the contour straddling the branch cut of $\log z$ starting from $a$ and ending at $a$ itself, as depicted in figure tear2. The contour $\mathrm{C}$ does not enclose any pole of $f(z)$.

Figures (2)

  • Figure 1: The contour $\mathrm{C}$ used in the representation \ref{['result1']} for the Hadamard's finite part. The contour $\mathrm{C}$ excludes any of the poles of $f(z)$. The upper limit $a$ can be infinite. $\epsilon$ is a small positive parameter.
  • Figure 2: The contour of integration. The upper limit $a$ can be infinite. The poles of the generalized Stieltjes integral at $z=\pm i/\sqrt{\beta}$ lie inside the contour $\mathrm{C}$. $\epsilon$ is a small positive parameter.

Theorems & Definitions (4)

  • Lemma 2.1
  • Definition 2.1
  • Theorem 4.1
  • proof