Radiative E1 decays of X(3872)

Radiative E1 decay widths of $\mathrm{X}(3872)$ are calculated through the relativistic Salpeter method, with the assumption that $\mathrm{X}(3872)$ is the ${\chi }_{c1}$(2P) state, which is the radial excited state of ${\chi }_{c1}$(1P). We firstly calculated the E1 decay width of ${\chi }_{c1}$(1P), the result is in agreement with experimental data excellently, then we calculated the case of $\mathrm{X}(3872)$ with the assignment that it is ${\chi }_{c1}$(2P). Results are: $\mathrm{\Gamma }(\mathrm{X}(3872)\to \gamma \text{sl}J\mathrm{/}\psi )=33.0$ keV, ${\Gamma}({\rm X(3872)}\rightarrow \gamma \psi(2S))=146$keVand${\Gamma}({\rm X(3872)}\rightarrow \gamma \psi(3770))=7.09$keV\mathrm{.}Theratio${{\rm Br(X(3872)}\rightarrow\gamma\psi(2{\rm S}))}/{{\rm Br(X(3872)}\rightarrow \gamma {\sl J}/\psi)}=4.4$ agrees with experimental data by BaBar, but larger than the new up-bound reported by Belle recently. With the same method, we also predict the decay widths: ${\Gamma}(\chi_{b1}(1\rm P))\rightarrow \gamma \Upsilon(1\rm S))=30.0$keV,${\Gamma}(\chi_{b1}(2\rm P))\rightarrow \gamma \Upsilon(1\rm S))=5.65$keVand${\Gamma}(\chi_{b1}(2\rm P))\rightarrow \gamma \Upsilon(2S))=15.8$keV,andthefullwidths:${\Gamma}(\chi_{b1}(1\rm P))\sim 85.7$keV,${\Gamma}(\chi_{b1}(2\rm P))\sim 66.5$ keV.