On the Charm Contribution to the Muon $g-2$ Light-by-Light

TL;DR

The paper tackles the charm-quark contribution to hadronic light-by-light scattering in the muon g-2. It uses perturbative QCD in the $ ext{MS}$-bar scheme, includes the NLO $\alpha_s$ correction, and converts from the pole mass to $m_c(m_c)$ to improve convergence and reduce renormalization-scale dependence; the key perturbative expressions for LO and NLO are provided in this scheme. The main result is $a_\mu^{\mathrm{HLbLc}}=3.65(25)\times 10^{-11}$, consistent with recent lattice determinations and achieved with improved scale stability. This work delivers a precise, perturbatively controlled charm HLbL benchmark that informs Standard Model predictions for $a_\mu$ and offers a cross-check against lattice QCD calculations.

Abstract

We combine existing perturbative results to show that a precise analytic determination of the charm-quark contribution to the hadronic light-by-light (HLbL) part of the muon anomalous magnetic moment is possible. Working in the $\overline{\mathrm{MS}}$ scheme, we include the NLO $\mathcal{O}(α_s)$ correction, which significantly reduces the residual renormalization-scale dependence and the perturbative uncertainty. Our final result is $a_μ^{\mathrm{HLbLc}}=3.65(25)\times 10^{-11}$, in good agreement with recent lattice determinations.

Figures (5)

• Figure 1: Types of diagrams that contribute. (a) Leading-order diagram, (b) crossed-gluon diagram, (c) diagram with mass renormalization.
• Figure 2: The contribution from the charm quark to $a_\mu^{\textrm{HLbL}}$. LO $1/m_c^{10}$ is the result at lowest order including higher orders in $m_\mu/m_c$Kuhn:2003pu. LO $1/m_c^2$ is the result of only the lowest order in $m_\mu/m_c$. LO+NLO is our full result with the central values of \ref{['eq:inputs']} as input, i.e. leading in $m_\mu/m_c$ but containing $\alpha_s$ corrections.
• Figure 3: The contribution from the charm quark to $a_\mu^{\textrm{HLbL}}$. The result is LO+NLO while varying the input value of the charm quark mass and $\alpha_s$ within the errors given in \ref{['eq:inputs']}. The last curve is done with the same input but 2-loop running of $m_c$ and $\alpha_s$.
• Figure 4: The charm quark contribution to the electron HLbL.
• Figure 5: The charm loop results for the tau HLbL. The X indicates the full LO result at $m_c=1.273$.