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

TL;DR

This paper proposes using polarization measurements in $oldsymbol{ ext{chi}_{cJ} o oldsymbol{ extphi} oldsymbol{ extphi}}$ decays to test the influence of hadronic loop effects, providing stable amplitude ratios and encouraging future experimental analysis.

Contribution

It introduces a method to analyze polarization in $oldsymbol{ ext{chi}_{cJ} o extphi extphi}$ decays to probe hadronic loop effects, with specific amplitude ratios and Monte Carlo simulations.

Findings

Stable ratios of helicity amplitudes were identified.

Monte Carlo events were generated based on predicted ratios.

The study advocates for experimental polarization analysis in future experiments.

Abstract

In this work, we propose that detecting the polarization information of $\chi_{cJ}\to \phi\phi~(J=0,1,2)$ 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 $\chi_{cJ}\to \phi\phi$ process in the future, which is important to understand the underlying decay mechanism existing in $\chi_{cJ}$ decays.