The ratio of decay widths of X(3872) to ${\psi }^{\prime }\gamma$ and $J/\psi \gamma$ as a test of the X(3872) dynamical structure

Abstract

Radiative decays of X(3872) with $J^{PC}=1^{++}$ are studied in the coupled-channel approach, where the $c\bar{c}$ states are described by relativistic string Hamiltonian, while for the decay channels $D{D}^{\ast }$ a string breaking mechanism is used. Within this method a sharp peak and correct mass shift of the $2{}^3{P}_1$ charmonium state just to the $D^0{D}^{\ast 0}$ threshold was already obtained for a prescribed channel coupling to the $D{D}^{\ast }$ decay channels. For the same value of coupling the normalized wave function (w.f.) of X(3872) acquires admixture of the $1{}^3{P}_1$ component with the w.f. fraction $c_1=0.153(\theta =8.8^{\circ }$), which increases the transition rate $\Gamma (X(3872)\to J/\psi \gamma )$ up to 50-70 keV, making the ratio $R=\frac{\mathcal{B}(X(3872)\to {\psi }^{\prime }\gamma )}{\mathcal{B}(X(3872)\to J/\psi \gamma )}=0.8\pm 0.20(th)$ significantly smaller, as compared to $R\simeq 5$ for X(3872) as a purely $2{}^3{P}_1$ state.

14 pages,2 Tables

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