Detecting the polarization in $chi_{cJ} to phi phi $ decays to probe hadronic loop effect

In this work, we propose that detecting the polarization information of $chi_{cJ}to phiphi~(J=0,1,2)$ 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 $|F_{1,1}^{(0)}/F_{0,0}^{(0)}| approx 0.359$, $|F_{1,0}^{(1)}/F_{0,1}^{(1)}| =1$, $|F_{1,1}^{(1)}/F_{0,1}^{(1)}| =0$, $|F^{(2)}_{1,0}|/|F^{(2)}_{0,0}|=|F^{(2)}_{0,1}|/|F^{(2)}_{0,0}| approx 1.285,~|F^{(2)}_{1,-1}|/|F^{(2)}_{0,0}|=|F^{(2)}_{-1,1}|/|F^{(2)}_{0,0}| approx 5.110$ and $|F^{(2)}_{-1,-1}|/|F^{(2)}_{0,0}|=|F^{(2)}_{1,1}|/|F^{(2)}_{0,0}| approx 0.465$. By adopting these predicted ratios, we further generate Monte-Carlo events of moments $langle t_{ij} rangle$, 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 $chi_{cJ}to phiphi$ process in the future, which is important to understand the underlying decay mechanism existing in $chi_{cJ}$ decays.