Precise prediction of the decay rate for ${\eta }_b\to \gamma \gamma$ from lattice QCD

Abstract

We calculate the decay rate for ${\eta }_b\to \gamma \gamma$ in lattice QCD for the first time, providing a precise prediction for the Belle II experiment. Our calculation includes $u$, $d$, $s$ and $c$ quarks in the sea, using gluon field configurations generated by the MILC collaboration, at three values of the lattice spacing from $0.06\mathrm{fm}$ to $0.03\mathrm{fm}$. All quarks are treated in the Highly Improved Staggered Quark formalism, which enables us to reach the $b$ quark mass for our valence quarks on these fine lattices. We calculate quark-line connected correlation functions only. By working at additional heavy quark masses between those of $c$ and $b$ we map out the behaviour of the ratio $f_{{\eta }_h}/(M_{{\eta }_h}^2{F}_{{\eta }_h}(0,0))$, where $f$ is the decay constant, $M$, the mass and $F(0,0)$, the form factor for decay to two on-shell photons for the pseudoscalar heavyonium meson, ${\eta }_h$. This ratio takes the approximate value 0.5 in leading-order non-relativistic QCD (NRQCD) but we are able to give a much more accurate analysis than this. Focussing on the $b$ quark mass, we find a ratio of $0.467(11)$, giving $\Gamma ({\eta }_b\to \gamma \gamma )=0.559(32{)}_{\text{fit}}(1{)}_{\text{syst}}\mathrm{keV}$. Combined with a value for the branching fraction from NRQCD, our result can be used to determine the total width of the ${\eta }_b$ with a $6\%$ uncertainty.

Comments: Minor changes to the text. Additional checks of the fits in the Supplementary Materials. Version accepted by Physical Review D. 5 pages, 3 figures, 2 tables, plus Supplementary Materials: 6 pages, 9 figures, 5 tables