Triangle singularity in the $J/\psi \to \varphi {\pi }^{+}{a}_0^{-}({\pi }^{-}\eta )$, $\varphi {\pi }^{-}{a}_0^{+}({\pi }^{+}\eta )$ decays

We study the $J/\psi \to \varphi {\pi }^{+}{a}_0(980{)}^{-}(a_0^{-}\to {\pi }^{-}\eta )$ decay, evaluating the double mass distribution in terms of the ${\pi }^{-}\eta$ and ${\pi }^{+}{a}_0^{-}$ invariant masses. We show that the ${\pi }^{-}\eta$ mass distribution exhibits the typical cusp structure of the $a_0(980)$ seen in recent high statistics experiments, and the ${\pi }^{+}{a}_0^{-}$ spectrum shows clearly a peak around $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})=1420\mathrm{MeV}$, corresponding to a triangle singularity. When integrating over the two invariant masses we find a branching ratio for this decay of the order of ${10}^{-5}$, which is easily accessible in present laboratories. We also call attention to the fact that the signal obtained is compatible with a bump experimentally observed in the $\eta {\pi }^{+}{\pi }^{-}$ mass distribution in the $J/\psi \to \varphi \eta {\pi }^{+}{\pi }^{-}$ decay and encourage further analysis to extract from there the $\varphi {\pi }^{+}{a}_0^{-}$ and $\varphi {\pi }^{-}{a}_0^{+}$ decay modes.

Figure 1

Mechanisms in $J/\psi \to \eta {\pi }^0\varphi (\varphi \to K\overline{K})$: (a),(b) TS; (c),(d) the tree level.

Figure 2

Triangle diagrams for $J/\psi \to \varphi {\pi }^{+}{a}_0^{-}$ decay (a) and $J/\psi \to \varphi {\pi }^{-}{a}_0^{+}$ decay (b). (c) and (d) illustrate the processes of (a) and (b) respectively, with a clear depiction of the decay channel of $a_0^{-}$ and $a_0^{+}$. In (a), the momenta of the particles are shown, where $P=p_{J/\psi }-p_{\varphi }$.

Figure 3

Factor $A\equiv [-\frac{2}{\pi }{M}_{\mathrm{inv}}({\pi }^{-}\eta )\mathrm{Im}{t}_{K^{-}{K}^0,K^{-}{K}^0}(M_{\mathrm{inv}}({\pi }^{-}\eta ))]$ as a function of $M_{\mathrm{inv}}({\pi }^{-}\eta )$.

Figure 4

${\tilde{t}}_{\mathrm{TS}}^{\prime }$ given by Eq. (29) as a function of $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})$ when fixing $M_{\mathrm{inv}}({\pi }^{-}\eta )=m_{a_0}$.

Figure 5

$\frac{1}{{\mathrm{\Gamma }}_{J/\psi }}\frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{J/\psi \to \varphi {\pi }^{+}{a}_0(980{)}^{-}}}{\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{-}\eta )\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})}$ as a function of $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})$ when fixing $M_{\mathrm{inv}}({\pi }^{-}\eta )=m_{a_0}$.

Figure 6

${\tilde{t}}_{\mathrm{TS}}^{\prime }$ given by Eq. (29) as a function of $M_{\mathrm{inv}}({\pi }^{-}\eta )$ when fixing $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})=1416\mathrm{MeV}$.

Figure 7

$\frac{1}{{\mathrm{\Gamma }}_{J/\psi }}\frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{J/\psi \to \varphi {\pi }^{+}{a}_0(980{)}^{-}}}{\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{-}\eta )\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})}$ as a function of $M_{\mathrm{inv}}({\pi }^{-}\eta )$ when fixing $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})=1416\mathrm{MeV}$.

Figure 8

$\frac{1}{{\mathrm{\Gamma }}_{J/\psi }}\frac{d^2{\mathrm{\Gamma }}_{J/\psi \to \varphi {\pi }^{+}{a}_0(980{)}^{-}}}{\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{-}\eta )\mathrm{d}{M}_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})}$ as a function of $M_{\mathrm{inv}}({\pi }^{+}{a}_0^{-})$ when integrating over $M_{\mathrm{inv}}({\pi }^{-}\eta )$ in the ranges $m_{a_0}\pm 10\mathrm{MeV}$, $m_{a_0}\pm 20\mathrm{MeV}$, $m_{a_0}\pm 50\mathrm{MeV}$ and $m_{a_0}\pm 100\mathrm{MeV}$.