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

Abstract

We derived in arXiv:2206.04188 arXiv:2211.15712 a compact expression for the $N$-th rank QED polarization tensor $Π_{μ_1\cdots μ_N}(k_1,\cdots,k_N)$ in a $(0+1)$-dimensional worldline framework. This fully off-shell object, a function of $N$ external photon four-momenta, is a key ingredient in high-order computations of cusp anomalous dimensions and lepton anomalous magnetic moments. We demonstrate here that $Π_{μ_1\cdots μ_N}$ 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 functions and (propertime 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 $4$-th and $6$-th 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 $e^{N-1}/\sqrt{N}$ terms as opposed to the $e^{N-1} N!/\sqrt{N}$ terms in conventional perturbation theory. We reexpress worldline parameter integrals that define the $4$-th 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.

Figures (2)

• Figure 1: Light-by-light worldline diagrams with $N=4$ photons entering the three loop (a), and four-loop (b), $N=4$ calculation of the anomalous dimension in the RGE of a cusped Wilson line $\Gamma_\text{cusp}$. Notice that each photon vertex is located at a worldline parameter $\tau_i$ that will be integrated over $\tau_i=[0,1]$. These worldline diagrams encode all Feynman diagram topologies contributing to $\Gamma_\text{cusp}$.
• Figure 2: Worldline diagrams for the QED vacuum polarization tensors at lowest order for (a) $N=2$, (b) $N=4$, and (c) general $N$ photons. Diagrams with odd $N$ insertions vanish by Furry's theorem. Each photon vertex, located at the worldline parameter $\tau_i$, will be integrated over $\tau_i=[0,1]$, with both numerator and denominator expressions depending on all boson and fermion worldline Green functions $G^{B,F}(\tau_i,\tau_j)$ in the problem. In this manner, the worldline diagrams encode all $(N-1)!$ possible ordering permutations of Feynman diagrams in conventional perturbation theory.