Hadronic corrections to $\mu$-$e$ scattering at NNLO with space-like data

TL;DR

This paper presents a novel method to compute hadronic NNLO corrections to muon-electron scattering using space-like data, avoiding reliance on time-like $R$ ratio data, and provides analytic kernels for these calculations.

Contribution

It introduces a hyperspherical integration approach to evaluate irreducible hadronic vacuum polarization diagrams directly from space-like data, enhancing precision in Standard Model predictions.

Findings

Analytic expressions for two-loop vertex and box correction kernels.

Numerical evaluation consistent with traditional dispersive methods.

Method reduces dependence on time-like $R$ ratio data.

Abstract

The Standard Model prediction for $\mu$-$e$ scattering at Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD contributions given by diagrams with a hadronic vacuum polarization insertion in the photon propagator. By taking advantage of the hyperspherical integration method, we show that the subset of hadronic NNLO corrections where the vacuum polarization appears inside a loop, the irreducible diagrams, can be calculated employing the hadronic vacuum polarization in the space-like region, without making use of the $R$ ratio and time-like data. We present the analytic expressions of the kernels necessary to evaluate numerically the two types of irreducible diagrams: the two-loop vertex and box corrections. As a cross check, we evaluate these corrections numerically and we compare them with the results given by the traditional dispersive approach and with analytic two-loop vertex results in QED.