Partial wave effects in the heavy quarkonium radiative electromagnetic decays

Abstract

In a previous paper \cite{Bc}, it was pointed out that the wave functions of all particles are not pure waves, besides the main partial waves, they all contain {other partial waves}. It is very interesting to know what role these different partial waves play in particle transitions. Therefore, by using the Bethe-Salpeter equation method, we study the radiative electromagnetic decays $\ddot{I}ˆ\to {\hat{I}}^3\ddot{I}{‡}_{{}_{cJ}}$ and $\hat{I}\yen \to {\hat{I}}^3\ddot{I}{‡}_{{}_{bJ}}$ ($J=0,1,2$). We find that for the $S$ and $P$ wave dominated states, like the $\ddot{I}ˆ(2S)$, $\hat{I}\yen (2S)$, $\ddot{I}{‡}_{{}_{cJ}}(1P)$, and $\ddot{I}{‡}_{{}_{bJ}}(1P)$ etc., the dominant $S$ and $P$ waves provide main and nonrelativistic contrition to the decays; other partial waves mainly contribute to the relativistic correction. For the states like the $\ddot{I}ˆ(1D)$, $\hat{I}\yen (2D)$, $\ddot{I}{‡}_{c2}(1F)$, and $\ddot{I}{‡}_{b2}(1F)$ etc., they are the $S-P-D$ mixing state dominated by $D$ wave or the $P-D-F$ mixing state dominated by $F$ wave. Large decay widths are found in the transitions $\ddot{I}ˆ(2D)\to \ddot{I}{‡}_{c2}(1F)$, $\hat{I}\yen (1D)\to \ddot{I}{‡}_{bJ}(1P)$, and $\hat{I}\yen (2D)\to \ddot{I}{‡}_{bJ}(2P)$ etc., which may be helpful to study the missing states $\ddot{I}{‡}_{c2}(1F)$, $\hat{I}\yen (1D)$, and $\hat{I}\yen (2D)$.

31 pages, 19 tables