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Efficient computation of the N-th rank QED polarization tensor: Universal worldline structure of form factors

X. Feal1,4,*, A. Tarasov2,3,†, and R. Venugopalan3,4,5,‡

  • *Contact author: xgarciafe@bnl.gov
  • †Contact author: ataraso@ncsu.edu
  • ‡Contact author: rajuv@bnl.gov
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Phys. Rev. D 114, 016019 – Published 20 July, 2026

DOI: https://doi.org/10.1103/s6zt-srrr

Abstract

We derived in [Phys. Rev. D 106, 056009 (2022), Phys. Rev. D 107, 096021 (2023)] a compact expression for the Nth rank QED polarization tensor Πμ1⋯μN(k1,…,kN) in a (0+1)-dimensional worldline framework. This fully off-shell object, a function of N external photon 4-momenta, is a key ingredient in high-order computations of cusp anomalous dimensions and lepton anomalous magnetic moments. We demonstrate here that Πμ1⋯μN can further be expressed simply in terms of a small number of independent “head” form factors [each representing (N−1)!/2 Feynman diagrams] which have a universal structure in terms of sums over fermion Green’s functions and (proper time derivative of) their boson worldline superpartners. This worldline representation bypasses explicit Wick contractions and avoids tensor reductions to scalar loop integrals à la Passarino and Veltman, order by order in perturbation theory. We give explicit expressions for the fourth and sixth rank head form factors and provide a computer script generalizing these results to arbitrary N external photons. The multiplicity of heads, and their growth with N, can be understood in terms of orbits of the permutation group. We employ the Burnside-Cauchy-Frobenius lemma to show that it scales as eN−1/N terms as opposed to the eN−1N!/N terms in conventional perturbation theory. We reexpress worldline parameter integrals that define the fourth rank heads as Feynman parameter integrals to reproduce the seminal results by Karplus and Neuman for the on-shell light-by-light amplitude and extend these to the fully off-shell case in massless QED employing a tailored integration-by-parts procedure. In a follow-up paper, we will discuss the direct computation of worldline integrals, potentially providing a further N! advantage relative to Feynman diagram computations at high orders in perturbation theory.

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Phys. Rev. D 114, 016019– Published 20 July, 2026
  • Received 13 April 2026
  • Accepted 22 May 2026
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