Lattice QCD determination of the radiative decay rates $h_c\to \hat{I}{\cdot }_c{\hat{I}}^3$ and $h_b\to \hat{I}{\cdot }_b{\hat{I}}^3$

Abstract

We present the results of our lattice QCD computation of the hadronic matrix elements relevant to the $h_c\to \hat{I}{\cdot }_c{\hat{I}}^3$ and $h_b\to \hat{I}{\cdot }_b{\hat{I}}^3$ 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 $m_{\ddot{I}}€\simeq 175\mathrm{MeV}$). While the hadronic matrix element for $h_c\to \hat{I}{\cdot }_c{\hat{I}}^3$ is obtained directly, the one relevant to $h_b\to \hat{I}{\cdot }_b{\hat{I}}^3$ is reached by working with heavy quark masses $m_H^{(n)}=\hat{I}{»}^{n-1}{m}_c$, with $\hat{I}»\sim 1.24$ and $n=1,2,\dots ,6$, and then extrapolated to $m_b$ by several judicious ansätze. In the continuum limit we obtain $\hat{I}“(h_c\to \hat{I}{\cdot }_c{\hat{I}}^3)=0.604(24)\mathrm{MeV}$, 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 $\hat{I}“(h_b\to \hat{I}{\cdot }_b{\hat{I}}^3)=46.0(4.8)\mathrm{keV}$.

25 pages, 16 figures, 5 tables