Hadronic light-by-light corrections to the muon g-2: the pion-pole contribution

TL;DR

This work targets a precise SM prediction for the muon g-2 by focusing on the hadronic light-by-light scattering, specifically the pion-pole contribution. It combines a large-$N_c$ QCD-inspired pi0 transition form factor with short-distance constraints and performs analytic angular integration via Gegenbauer polynomials, reducing the problem to a numerically tractable two-dimensional integral. The authors find $a_{\\mu}^{\\hbox{LbyL;\\pi^0}} = +5.8(1.0) \times 10^{-10}$ (LMD+V) and $+5.6 \times 10^{-10}$ (VMD) with a sign opposite to some prior results, and, including $\eta$ and $\eta'$ poles, obtain $a_{\\mu}^{\\hbox{LbyL;PS}} = +8.3(1.2) \times 10^{-10}$, which reduces the SM–experimental discrepancy. The methodology—analytic angular integration plus a 2D momentum integral across multiple form-factor models—provides a robust framework for refining hadronic LbyL contributions and guides future full four-point-function evaluations.

Abstract

The correction to the muon anomalous magnetic moment from the pion-pole contribution to the hadronic light-by-light scattering is considered using a description of the pi0 - gamma* - gamma* transition form factor based on the large-Nc and short-distance properties of QCD. The resulting two-loop integrals are treated by first performing the angular integration analytically, using the method of Gegenbauer polynomials, followed by a numerical evaluation of the remaining two-dimensional integration over the moduli of the Euclidean loop momenta. The value obtained, a_{mu}(LbyL;pi0) = +5.8 (1.0) x 10^{-10}, disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads a_{mu}(LbyL;pi0)_{VMD} = +5.6 x 10^{-10}, and differs only by its overall sign from the value obtained by previous authors. Inclusion of the eta and eta-prime poles gives a total value a_{mu}(LbyL;PS) = +8.3 (1.2) x 10^{-10} for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of a_{mu} and its theoretical counterpart in the standard model.

Figures (3)

• Figure 1: The three topologies involving hadronic contributions to the anomalous magnetic moment of the muon: from left to right, vacuum polarization insertion in the vertex, light-by-light scattering, and two-loop electroweak contributions. The cross indicates an insertion of the electromagnetic current, the shaded areas correspond to hadronic subgraphs, while the lines exchanged in the rightmost graph correspond to a photon and a neutral gauge boson.
• Figure 2: The pion-pole contributions to light-by-light scattering. The shaded blobs represent the form factor ${\cal F}_{\pi^0\gamma^*\gamma^*}$. The first and second graphs give rise to identical contributions, involving the function $T_1(q_1,q_2;p)$ in Eq. (\ref{['eq:a_pion_2']}), whereas the third graph gives the contribution involving $T_2(q_1,q_2;p)$.
• Figure 3: The weight functions of Eqs. (\ref{['eq:wf1']})--(\ref{['eq:wg2M']}). Note the different ranges of $Q_i$ in the subplots. The functions $w_{f_1}$ and $w_{g_1}$ are positive definite and peaked in the region $Q_1\sim Q_2\sim 0.5$ GeV. Note, however, the tail in $w_{f_1}$ in the $Q_1$ direction for $Q_2 \sim 0.2~\hbox{GeV}$. The functions $w_{g_2}(M_\pi,Q_1,Q_2)$ and $w_{g_2}(M_V,Q_1,Q_2)$ take both signs, but their magnitudes remain small as compared to $w_{f_1}(Q_1,Q_2)$ and $w_{g_1}(M_V,Q_1,Q_2)$. We have used $M_V=M_{\rho}=769$ MeV.