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

Abstract

We calculate the double-virtual $\eta \to {\gamma }^{\ast }{\gamma }^{\ast }$ transition form factor ${\mathcal{F}}_{\eta \to {\gamma }^{\ast }{\gamma }^{\ast }}(q_1^2,q_2^2)$ 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 $\Gamma (\eta \to \gamma \gamma )=323(85{)}_{\text{stat}}(22{)}_{\text{syst}}$ 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 $a_{\mu }^{\eta -\text{pole}}=13.2(5.2{)}_{\text{stat}}(1.3{)}_{\text{syst}}\cdot 10^{-11}$.

arXiv comment12 pages, 8 figures; updated to published version

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• An optimized basis for hadronic light-by-light scattering
• Precision evaluation of the $\eta$- and ${\eta }^{\prime }$-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon
• Pion Transition Form Factor from Twisted-Mass Lattice QCD and the Hadronic Light-by-Light ${\pi }^0$-pole Contribution to the Muon $g-2$
• Hadronic Light by Light Corrections to the Muon Anomalous Magnetic Moment
• Self-consistent analysis for the ${\eta }_c\to \gamma \gamma$ process
• Hadronic contributions to $a_{\mu }$ within Resonance Chiral Theory