Detecting the polarization in ${\chi }_{cJ}\to \varphi \varphi$ decays to probe hadronic loop effect

Abstract

In this work, we propose that detecting the polarization information of ${\chi }_{cJ}\to \varphi \varphi (J=0,1,2)$ can be as good test of the role of hadronic loop effect on these decays. Our results shows that the obtained ratios of helicity amplitudes are quite stable, which are $|F_{1,1}^{(0)}/F_{0,0}^{(0)}|\approx 0.359$, $|F_{1,0}^{(1)}/F_{0,1}^{(1)}|=1$, $|F_{1,1}^{(1)}/F_{0,1}^{(1)}|=0$, $|F_{1,0}^{(2)}|/|F_{0,0}^{(2)}|=|F_{0,1}^{(2)}|/|F_{0,0}^{(2)}|\approx 1.285,|F_{1,-1}^{(2)}|/|F_{0,0}^{(2)}|=|F_{-1,1}^{(2)}|/|F_{0,0}^{(2)}|\approx 5.110$ and $|F_{-1,-1}^{(2)}|/|F_{0,0}^{(2)}|=|F_{1,1}^{(2)}|/|F_{0,0}^{(2)}|\approx 0.465$. By adopting these predicted ratios, we further generate Monte-Carlo events of moments $⟨t_{ij}⟩$, which are directly related to the determination of helicity amplitudes. We suggest further experiments like BESIII and Belle II to perform an analysis on the polarizations of the ${\chi }_{cJ}\to \varphi \varphi$ process in the future, which is important to understand the underlying decay mechanism existing in ${\chi }_{cJ}$ decays.