Lattice QCD determination of the radiative decay rates $h_c\to {\eta }_c\gamma$ and $h_b\to {\eta }_b\gamma$

We present the results of our lattice QCD computation of the hadronic matrix elements relevant to the $h_c\to {\eta }_c\gamma$ and $h_b\to {\eta }_b\gamma$ decays by using the gauge configurations produced by the Extended Twisted Mass Collaboration with $N_f=2+1+1$ dynamical Wilson-Clover twisted mass fermions at five different lattice spacings with physical dynamical $u$ , $d$, $s$ and $c$ quark masses (except for the the coarsest lattice for which the lightest sea quark corresponds to a pion with $m_{\pi }\simeq 175\mathrm{M}\mathrm{e}\mathrm{V}$). While the hadronic matrix element for $h_c\to {\eta }_c\gamma$ is obtained directly, the one relevant to $h_b\to {\eta }_b\gamma$ is reached by working with heavy quark masses $m_H^{(n)}={\lambda }^{n-1}{m}_c$, with $\lambda \sim 1.24$ and $n=1,2,\dots ,6$, and then extrapolated to $m_b$ by several judicious ansätze. In the continuum limit we obtain $\mathrm{\Gamma }(h_c\to {\eta }_c\gamma )=0.604(24)\mathrm{M}\mathrm{e}\mathrm{V}$, which is by a factor of $2.3$ more accurate than the previous lattice estimates, and in good agreement with the experimental measurement. In the $b$-quark case we obtain $\mathrm{\Gamma }(h_b\to {\eta }_b\gamma )=46.0(4.8)\mathrm{k}\mathrm{e}\mathrm{V}$.