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

In this work, we show that detecting the polarization information of ${\chi }_{cJ}\to \varphi \varphi (J=0,1,2)$ 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 $|F_{1,1}^{(0)}/F_{0,0}^{(0)}|\approx 0.359$, $|F_{1,0}^{(1)}/F_{0,1}^{(1)}|=1$, $|F_{1,1}^{(1)}/F_{0,1}^{(1)}|=0$, $|F_{1,0}^{(2)}|/|F_{0,0}^{(2)}|=|F_{0,1}^{(2)}|/|F_{0,0}^{(2)}|\approx 1.285$, $|F_{1,-1}^{(2)}|/|F_{0,0}^{(2)}|=|F_{-1,1}^{(2)}|/|F_{0,0}^{(2)}|\approx 5.110$, 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 use the Monte Carlo events to show that the moments $⟨t_{ij}⟩$ 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 ${\chi }_{cJ}\to \varphi \varphi$ decay in the future, and the results are important to understand the decay mechanism underlying the ${\chi }_{cJ}$ decays.

Figure 1

Helicity system and angles definition for the $\psi (2S)\to \gamma {\chi }_{cJ}$, ${\chi }_{cJ}\to \varphi \varphi$, $\varphi \to K^{+}{K}^{-}$ decay.

Figure 2

The Feynman diagrams depicting the ${\chi }_{c0}\to \varphi \varphi$ decay via $D$ meson loop.

Figure 3

The Feynman diagrams depicting the ${\chi }_{c1}\to \varphi \varphi$ decay via $D$ meson loop.

Figure 4

The Feynman diagrams depicting the ${\chi }_{c2}\to \varphi \varphi$ decay via $D$ meson loop.

Figure 5

Distribution of $⟨{\sin }^2{\theta }_2{\sin }^2{\theta }_3⟩$ moment versus $\cos ({\varphi }_2+{\varphi }_3)$ for the ${\chi }_{c0}\to \varphi \varphi \to 2(K^{+}{K}^{-})$. Histogram is filled with the MC events, and the curve shows the distribution of $1+2{\cos }^2({\varphi }_2+{\varphi }_3)$.

Figure 6

Angular distribution of $\varphi$ meson in ${\chi }_{c1}$ decays. Histogram is filled with the MC events, and the curve shows the distribution of $1-\frac{1}{3}{\cos }^2{\theta }_1$.

Figure 7

Moment distribution of $⟨t_{55}⟩$ for ${\chi }_{c1}$ decays. Histogram is filled with the MC events, and the curve shows the distribution of $1-\frac{1}{2}{\cos }^2{\theta }_1$.

Figure 8

Angular distribution of the $\varphi$ meson for ${\chi }_{c2}$ decays. Histogram is filled with the MC events, and the curve shows the distribution of $1+0.736{\cos }^2{\theta }_1$.

Figure 9

Angular distribution of the $\varphi$ meson for ${\chi }_{c2}$ decays. Histogram is filled with the MC events, and the curve shows the distribution of $1+1.26{\cos }^2{\theta }_1$.

Figure 10

Distribution of the $⟨t_{76}⟩$ moment for ${\chi }_{c2}$ decays. Histogram is filled with the MC events, and the curve shows the distribution of $x\sqrt{1-x^2}$ with $x=\cos {\theta }_1$.