Lepton anomalous magnetic moments: Theory

TL;DR

This work surveys the theory of lepton anomalous magnetic moments, detailing the division into QED, hadronic, and electroweak contributions and the methods used to compute them. It emphasizes two complementary approaches for hadronic effects: data-driven dispersive analyses of $e^+e^-\to\text{hadrons}$ cross sections and lattice QCD evaluations of the hadronic vacuum polarization and light-by-light scattering, including the time-momentum representation and window observables. The muon remains the most sensitive probe for potential new physics, with recent White Papers reporting SM predictions that are compatible with experiment but highlighting tensions between lattice and data-driven results for the leading-order hadronic vacuum polarization. The findings motivate future measurements and calculations (e.g., MUonE, E34) to sharpen the SM prediction and maximize sensitivity to Beyond-Standard-Model scenarios. The chapter also contextualizes the electron and tau cases, where QED dominates, and hadronic and EW effects play different roles due to mass scales.

Abstract

The anomalous magnetic moment of a lepton encodes the fraction of the lepton's interaction strength with an external magnetic field, which is generated by quantum corrections. Lepton anomalous magnetic moments are sensitive probes of fundamental interactions and play a pivotal role in the quest for "new physics" that may be able to explain the shortcomings of the Standard Model. This chapter introduces the basic concepts and describes the calculation of the individual contributions arising from electromagnetism, the strong and the weak interactions.

Figures (17)

• Figure 1: Diagrams representing the leading-order electromagnetic and weak contributions to the muon anomalous magnetic moment (diagrams (a)--(c)), as well as the hadronic vacuum polarisation (diagram (d)) and hadronic light-by-light scattering (diagram (e)) contributions that arise from the strong interaction.
• Figure 2: Feynman rules for Quantum Electrodynamics in momentum space. The lepton is characterised by its mass $m$ and electric charge $e$. Setting the gauge parameter $\xi$ to $\xi=1$ corresponds to Feynman gauge, the choice $\xi=0$ is called Landau gauge.
• Figure 3: The vertex correction in QED at one loop.
• Figure 4: Fourth-order vertex diagrams. There are 7 diagrams in total (note that the time-reversed versions of (a) and (c) are not shown). Diagram (e) gives rise to mass-dependent contributions whenever the mass of the lepton in the loop does not coincide with that of the extermal lepton. Photons are denoted by wavy lines, lepton loops by straight lines. Figure taken from Aoyama:2012qma.
• Figure 8: The contour $C$ of the integration path in Eq. (\ref{['eq:Cauchy']}). The vacuum polarisation function is analytic except for a cut along the real axis represented by the thick blue line. The threshold value $s_0$ is identified with the squared pion mass.