The $\eta \rightarrow \gamma^* \gamma^*$ transition form factor and the hadronic light-by-light $\eta$-pole contribution to the muon $g-2$ from lattice QCD

TL;DR

This paper uses lattice QCD to calculate the $ o ext{transition form factor}$ for the $ ext{eta meson}$, providing key inputs for the muon $g-2$ anomaly and offering first-principles insights into hadronic light-by-light scattering.

Contribution

It presents the first lattice QCD calculation of the double-virtual $ ext{eta}$ transition form factor at physical quark masses, relevant for muon $g-2$ calculations.

Findings

Partial decay width $ ext{Gamma}( ext{eta} o ext{gamma} ext{gamma})$ measured as 323(85)(22) eV.

Slope parameter $b_ ext{eta}$ determined as 1.19(36)(16) GeV$^{-2}$.

Eta-pole contribution to muon $g-2$ estimated as 13.2(5.2)(1.3) $ imes 10^{-11}$.

Abstract

We calculate the double-virtual $\eta \rightarrow \gamma^* \gamma^*$ transition form factor $\mathcal{F}_{\eta \to \gamma^* \gamma^*}(q_1^2,q_2^2)$ from first principles using a lattice QCD simulation with $N_f=2+1+1$ quark flavors at the physical pion mass and at one lattice spacing and volume. The kinematic range covered by our calculation is complementary to the one accessible from experiment and is relevant for the $\eta$-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment $a_\mu = (g-2)/2$ of the muon. From the form factor calculation we extract the partial decay width $\Gamma(\eta \rightarrow \gamma \gamma) = 323(85)_\text{stat}(22)_\text{syst}$ eV and the slope parameter $b_\eta=1.19(36)_\text{stat}(16)_\text{syst}$ GeV${}^{-2}$. For the $\eta$-pole contribution to $a_\mu$ we obtain $a_\mu^{\eta-\text{pole}} = 13.2(5.2)_\text{stat}(1.3)_\text{syst} \cdot 10^{-11}$.