Detecting the polarization in χc J→ϕ ϕ decays to probe hadronic loop effect

In this work, we show that detecting the polarization information of χ_{c J}→ϕ ϕ (J =0 ,1 ,2 ) could be a practical way to test the hadronic loop effect in these decays. Our results show that the predicted ratios of helicity amplitudes are less dependent on the parameters involved in the calculation. The ratios are determined to be |F₁^{,1 (0 )}/F₀^{,0 (0 )}|≈0.359 , |F₁^{,0 (1 )}/F₀^{,1 (1 )}|=1 , |F₁^{,1 (1 )}/F₀^{,1 (1 )}|=0 , |F₁^{,0 (2 )}|/|F₀^{,0 (2 )}|=|F₀^{,1 (2 )}|/|F₀^{,0 (2 )}|≈1.285 , |F₁^{,-1 (2 )}|/|F₀^{,0 (2 )}|=|F_-1^{,1 (2 )}|/|F₀^{,0 (2 )}|≈5.110 , and |F_-1^{,-1 (2 )}|/|F₀^{,0 (2 )}|=|F₁^{,1 (2 )}|/|F₀^{,0 (2 )}|≈0.465 . By adopting these predicted ratios, we use the Monte Carlo events to show that the moments ⟨t_{i j}⟩ can be used as observables to reveal the polarization transfer in these decays, and their distributions are directly related to the determination of helicity amplitudes. We suggest experiments like at the BESIII and the Belle II performing a polarization analysis on the χ_{c J}→ϕ ϕ decay in the future, and the results are important to understand the decay mechanism underlying the χ_{c J} decays.