Radiative transitions of $χ_{_{cJ}}\toψγ$ and $χ_{_{bJ}}\toΥγ$

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 $χ_{_{cJ}}$$\rightarrow$$γψ$ and $χ_{_{bJ}}$$\rightarrow$$γΥ$ ($J=0,1,2$), 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 $χ_{_{cJ}}(2P)\toγψ(1S)$ ($J=0,1,2$) are $\{49.7\%,~30.9\%,~37.5\%\}$, much larger than the corresponding $\{17.8\%,~7.08\%,~12.9\%\}$ of $χ_{_{bJ}}(2P)\rightarrowγΥ(1S)$. The decay of $χ_{_{cJ}}(2P)\toγψ$ can be used to distinguish between $χ_{_{c0}}(3860)$ and $χ_{_{c0}}(3915)$, which particle is the charmonium $χ_{_{c0}}(2P)$. Although our result of $χ_{_{c1}}(3872)$$\rightarrow$$γψ(2S)$ is consistent with data, but the one of $χ_{_{c1}}(3872)$$\rightarrow$$γψ(1S)$ is much larger than data, so whether $χ_{_{c1}}(3872)$ is the conventional $χ_{_{c1}}(2P)$ remains an open question. The undiscovered $Υ(1D)$ and $Υ(2D)$ have large production rates in decays of $χ_{_{b0}}(2P)\rightarrowγΥ(1D)$ and $χ_{_{bJ}}(3P)\rightarrowγΥ(2D)$ ($J=0,1$), respectively. To search for $χ_{_{bJ}}(3P)$ $(J=0,1,2)$, the most competitive channels are the decays $χ_{_{bJ}}(3P)\rightarrowγΥ(3S)$. And the best way to find $χ_{_{b2}}(1F)$ is to search for the decay of $χ_{_{b2}}(1F)\rightarrowγΥ(1D)$.