Muon-Electron Scattering at Next-To-Next-To-Leading Order: The Hadronic Corrections

The standard model prediction for muon-electron scattering beyond leading order requires the inclusion of QCD contributions which cannot be computed perturbatively. At next-to- and next-to-next-to-leading order, they arise from one- and two-loop diagrams with hadronic vacuum polarization insertions in the photon propagator. We present their evaluation using the dispersive approach with hadronic $e^{+}{e}^{-}$ annihilation data and estimate their uncertainty. We find that these corrections are crucial for the analysis of future high-precision muon-electron scattering data, like those of the recently proposed MUonE experiment at CERN.

Figure 1

(a) Diagram contributing to the hadronic correction to $\mu e$ scattering at NLO. (b)–(e) Examples of diagrams contributing to the four classes of hadronic corrections at NNLO. Electrons, muons, and photons are depicted with thin, thick, and wavy lines, respectively. The gray blobs indicate hadronic vacuum polarization insertions.

Figure 2

$K_h^{\mathrm{NNLO}}(t_e)$ factor for a positive (upper panel) and negative (lower panel) muon beam of energy $E_{\mu }=150\mathrm{GeV}$. The total hadronic NNLO correction are depicted in black, while the contributions of class I (II–IV) are shown separately in red (blue).