Radiative transitions of ${\chi }_{{}_{cJ}}\to \psi \gamma$ and ${\chi }_{{}_{bJ}}\to {\rm Y}\gamma$

Abstract

In the framework of instantaneous Bethe-Salpeter equation, according to the $J^{PC}$ of quarkonia, we find that their wave functions all contain multiple partial waves, rather than pure waves. In the radiative electromagnetic transitions ${\chi }_{{}_{cJ}}$$\to$$\gamma \psi$ and ${\chi }_{{}_{bJ}}$$\to$$\gamma {\rm Y}$ ($J=0,1,2$), the main wave of quarkonium gives the non-relativistic contribution, while other waves provide the relativistic corrections. Our results indicate that the relativistic effect of charmonium, especially highly excited states, is significant. Such as the relativistic effects of ${\chi }_{{}_{cJ}}(2P)\to \gamma \psi (1S)$ ($J=0,1,2$) are $\{49.7\%,30.9\%,37.5\%\}$, much larger than the corresponding $\{17.8\%,7.08\%,12.9\%\}$ of ${\chi }_{{}_{bJ}}(2P)\to \gamma {\rm Y}(1S)$. The decay of ${\chi }_{{}_{cJ}}(2P)\to \gamma \psi$ can be used to distinguish between ${\chi }_{{}_{c0}}(3860)$ and ${\chi }_{{}_{c0}}(3915)$, which particle is the charmonium ${\chi }_{{}_{c0}}(2P)$. Although our result of ${\chi }_{{}_{c1}}(3872)$$\to$$\gamma \psi (2S)$ is consistent with data, but the one of ${\chi }_{{}_{c1}}(3872)$$\to$$\gamma \psi (1S)$ is much larger than data, so whether ${\chi }_{{}_{c1}}(3872)$ is the conventional ${\chi }_{{}_{c1}}(2P)$ remains an open question. The undiscovered ${\rm Y}(1D)$ and ${\rm Y}(2D)$ have large production rates in decays of ${\chi }_{{}_{b0}}(2P)\to \gamma {\rm Y}(1D)$ and ${\chi }_{{}_{bJ}}(3P)\to \gamma {\rm Y}(2D)$ ($J=0,1$), respectively. To search for ${\chi }_{{}_{bJ}}(3P)$ $(J=0,1,2)$, the most competitive channels are the decays ${\chi }_{{}_{bJ}}(3P)\to \gamma {\rm Y}(3S)$. And the best way to find ${\chi }_{{}_{b2}}(1F)$ is to search for the decay of ${\chi }_{{}_{b2}}(1F)\to \gamma {\rm Y}(1D)$.

Comments: 31 pages, 4 figures