Study of $a_0^0(980)$-$f_0(980)$ mixing from $a_0(1450)\to a_0^0(980)f_0(500)\to {\pi }^{+}{\pi }^{-}{f}_0(500)$

Abstract

The $a_0^0(980)-f_0(980)$ mixing is one of the most potential tools to learn about the nature of $a_0^0(980)$ and $f_0(980)$. Using the $f_0(980)$-$a_0^0(980)$ mixing intensity ${\xi }_{af}$ measured recently at BESIII, we calculate the the branching ratio of the the isospin violation decay $J/\psi \to \gamma {\eta }_c\to \gamma {\pi }^0{a}_0^0(1450)\to \gamma {\pi }^0{a}_0^0(980)f_0(500)\to \gamma {\pi }^0{f}_0(980)f_0(500)\to \gamma {\pi }^0{\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$. The value of the branching ratio is found to be $O(10^{-6})$, which can be observed with $10^{10}$ $J/\psi$ events collected at BESIII. The narrow peak from the $f_0(980)$-$a_0^0(980)$ mixing in the ${\pi }^{+}{\pi }^{-}$ mass square spectrum can also be observed. In addition, we study the non-resonant decay $a_0^0(1450)\to f_0(980){\pi }^{+}{\pi }^{-}(\text{non-resonant})$, which is dominated by the $a_0^0(980)$-$f_0(980)$ mixing. We find that the non-resonant decay $a_0^0(1450)\to f_0(980){\pi }^{+}{\pi }^{-}$ and the decay $a_0^0(1450)\to f_0(980)f_0(500)$ can be combined to measure the mixing intensity ${\xi }_{af}$ in experiment. These decays are the perfect complement to the decay ${\chi }_{c1}\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ which had been observed at BESIII, the observations of them will make the measurement of the mixing intensity ${\xi }_{af}$ more precisely.