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Theoretical analysis of direct $CP$ violation and differential decay width in $D^\pm\to π^\pm π^+π^-$ in phase space around the resonances $ρ^0(770)$ and $f_0(500)$

We perform a theoretical study on direct $CP$ violation in $D^\pm\to π^\pm π^+π^-$ in phase space around the intermediate states $ρ^0(770)$ and $f_0(500)$. The possible interference between the amplitudes corresponding to the two resonances is taken into account, and the relative strong phase of the two amplitudes is treated as a free parameter. Our analysis shows that by properly chosen the strong phase, both the $CP$ violation strength and differential decay width accommodate to the experimental results.