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Study of $a_0^0(980)$-$f_0(980)$ mixing from $a_0(1450) \to a_0^0(980) f_0(500) \to π^+ π^- f_0(500)$

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 $ξ_{af}$ measured recently at BESIII, we calculate the the branching ratio of the the isospin violation decay $J/ψ\rightarrowγη_c \rightarrow γπ^0 a_0^0(1450)\rightarrow γπ^0 a_0^0(980)f_0(500)\rightarrow γπ^0 f_0(980) f_0(500) \rightarrow γπ^0 π^+π^- π^+π^-$. The value of the branching ratio is found to be $O(10^{-6})$, which can be observed with $10^{10}$ $J/ψ$ events collected at BESIII. The narrow peak from the $f_0(980)$-$a_0^0(980)$ mixing in the $π^+π^-$ mass square spectrum can also be observed. In addition, we study the non-resonant decay $a_0^0(1450)\rightarrow f_0(980) π^+π^-(\text{non-resonant})$, which is dominated by the $a_0^0(980)$-$f_{0}(980)$ mixing. We find that the non-resonant decay $a_0^0(1450)\rightarrow f_0(980) π^+π^-$ and the decay $a_0^0(1450)\rightarrow f_0(980) f_0(500)$ can be combined to measure the mixing intensity $ξ_{af}$ in experiment. These decays are the perfect complement to the decay $χ_{c1}\rightarrow f_{0}(980)π^{0}\toπ^{+}π^{-}π^{0}$ which had been observed at BESIII, the observations of them will make the measurement of the mixing intensity $ξ_{af}$ more precisely.