The $c\bar{c}$ interaction above threshold and the radiative decay $X(3872)\to J/\psi \gamma$

Abstract

Radiative decays of $X(3872)$ are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels $D{\bar{D}}^{\ast }$ are described with the string breaking mechanism. In SCA the transition rate ${\tilde{\Gamma }}_2=\Gamma (2{}^3{P}_1\to \psi \gamma )=71.8$~keV and large ${\tilde{\Gamma }}_1=\Gamma (2{}^3{P}_1\to J/\psi \gamma )=85.4$~keV are obtained, giving for their ratio the value $\widetilde{R_{\psi \gamma }}=\frac{{\tilde{\Gamma }}_2}{{\tilde{\Gamma }}_1}=0.84$. In the CC approach three factors are shown to be equally important. First, the admixture of the $1{}^3{P}_1$ component in the normalized wave function of $X(3872)$ due to the CC effects. Its weight $c_{\mathrm{X}}(E_{\mathrm{R}})=0.200\pm 0.015$ is calculated. Secondly, the use of the multipole function $g(r)$ instead of $r$ in the overlap integrals, determining the partial widths. Thirdly, the choice of the gluon-exchange interaction for $X(3872)$, as well as for other states above threshold. If for $X(3872)$ the gluon-exchange potential is taken the same as for low-lying charmonium states, then in the CC approach ${\Gamma }_1=\Gamma (X(3872)\to J/\psi \gamma )\sim 3$~keV is very small, giving the large ratio $R_{\psi \gamma }=\frac{\mathcal{B}(X(3872)\to \psi (2S)\gamma )}{\mathcal{B}(X(3872)\to J/\psi \gamma )}\gg 1.0$. Arguments are presented why the gluon-exchange interaction may be suppressed for $X(3872)$ and in this case ${\Gamma }_1=42.7$~keV, ${\Gamma }_2=70.5$~keV, and $R_{\psi \gamma }=1.65$ are predicted for the minimal value $c_{\mathrm{X}}(\min )=0.185$, while for the maximal value $c_{\mathrm{X}}=0.215$ we obtained ${\Gamma }_1=30.8$~keV, ${\Gamma }_2=73.2$~keV, and $R_{\psi \gamma }=2.38$, which agrees with the LHCb data.

Comments: 12 pages, no figures