Precise prediction of the decay rate for $η_b\to γγ$ from lattice QCD

TL;DR

This work presents the first lattice QCD prediction for the heavyonium two-photon decay width $\Gamma(\eta_b \rightarrow \gamma\gamma)$ by computing the on-shell form factor $F(0,0)$ and the heavy-quark decay-constant ratio $R_{\eta_h}$ across heavy masses, using $u$, $d$, $s$, $c$ sea quarks with HISQ on MILC configurations and multiple lattice spacings to enable a controlled continuum limit. The authors use connected heavy-quark correlators, RI-SMOM renormalization of the vector current, and a NRQCD-inspired extrapolation in $M_{\eta_h}$ to determine $R_{\eta_b}$ = $0.467(11)$ and $F(0,0)_{\eta_b}$ = $0.01754(50)\ \mathrm{GeV}^{-1}$, yielding $\Gamma(\eta_b \rightarrow \gamma\gamma) = 0.559(32)(1)\ \mathrm{keV}$. The result, aided by an external lattice determination of $f_{\eta_b}$, leads to a precise estimate of the total width via a NRQCD branching ratio, $\Gamma_{\text{tot}}(\eta_b)=13.81^{+17}_{-28}\ \mathrm{MeV}$, and demonstrates a small but nonzero deviation from the naive NRQCD limit $R_{\eta_h} \to 1/2$. The analysis provides a benchmark for Belle II and highlights the reliability of lattice QCD for heavyonium electromagnetic decays, including a clear mass dependence of $R_{\eta_h}$ across $M_{\eta_h}$.

Abstract

We calculate the decay rate for $η_b \to γγ$ 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 $f_{η_h}/(M_{η_h}^2F_{η_h}(0,0))$, 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, $η_h$. 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 $Γ(η_b \to γγ) = 0.559(32)_{\text{fit}}(1)_{\text{syst}} \: \mathrm{keV}$. Combined with a value for the branching fraction from NRQCD, our result can be used to determine the total width of the $η_b$ with a $6\%$ uncertainty.

Figures (3)

• Figure 1: Schematic diagram of the connected 3-point correlation function between $O_{\eta_h}$ and two vector currents, see text. The lines between the operators represent $h$ quark propagators. We do not include any quark-line disconnected correlation functions in our calculation.
• Figure 2: Lattice results (open red circles and square) and fit (red band) for the ratio $R_{\eta_h}$ (Eq. \ref{['eq:ratio']}) for the $\eta_b$. The black star and error bar denotes our result in the continuum limit.
• Figure 3: Lattice results (open red squares) for the ratio $R_{\eta_h}$ (Eq. \ref{['eq:ratio']}) for meson masses corresponding to those of $\eta_b$ and $\eta_c$ as well as two intermediate values of 4 GeV and 6.62 GeV, plotted against $1\,\mathrm{GeV}/M_{\eta_h}$. The $\eta_c$ result is derived from data in Colquhoun:2023zbc. The blue dash-dotted line shows the leading order NRQCD result of 0.5 and the purple dashed line and band is a simple fit to the lattice QCD results of NRQCD form adding in higher order corrections in $\alpha_s$ and powers of $1\,\mathrm{GeV}/M_{\eta_h}$ (see Eq. \ref{['eq:nrqcdfit']}).