$CP$ violation in singly Cabibbo suppressed $D\to \pi a_0(980)$ decays

The singly Cabibbo suppressed (SCS) decays $D\to \pi a_0$, with $a_0\equiv a_0(980)$, have been measured with the branching-fraction ratios $r_{\mathrm{e}\mathrm{x}}^{+\mathrm{/}-}\equiv \mathcal{B}(D^0\to {\pi }^{-}{a}_0^{+})\mathrm{/}\mathcal{B}(D^0\to {\pi }^{+}{a}_0^{-})=7.5_{-0.8}^{+2.5}\pm 1.7$ and $r_{\mathrm{e}\mathrm{x}}^{+\mathrm{/}0}\equiv \mathcal{B}(D^{+}\to {\pi }^0{a}_0^{+})\mathrm{/}\mathcal{B}(D^{+}\to {\pi }^{+}{a}_0^0)=2.6\pm 0.6\pm 0.3$, deviating significantly from the short-distance expectations $(r^{+\mathrm{/}-},r^{+\mathrm{/}0})\simeq (0.07,0.2)$. This discrepancy indicates the necessity of long-distance rescattering effects. In particular, the process $D\to K^{\ast }K\to a_0\pi$ generates ${\mathcal{M}}_s$ comparable in magnitude to ${\mathcal{M}}_d$ in the amplitude $\mathcal{M}={\lambda }_d{\mathcal{M}}_d+{\lambda }_s{\mathcal{M}}_s$, with ${\lambda }_q\equiv V_{cq}^{\ast }{V}_{uq}$, accompanied by nontrivial strong phases essential for $CP$ violation. Consequently, the direct $CP$ asymmetries naturally arise at the $\mathcal{O}({10}^{-3})$ level, for example, ${\mathcal{A}}_{CP}(D^0\to {\pi }^{-}{a}_0^{+})=(-0.7\pm 0.1\pm 0.1\pm 0.1)\times {10}^{-3}$, ${\mathcal{A}}_{CP}(D^{+}\to {\pi }^0{a}_0^{+})=(-1.4\pm 0.1\pm 0.1\pm 0.1)\times {10}^{-3}$, and ${\mathcal{A}}_{CP}(D^0\to {\pi }^{+}{a}_0^{-})=(-2.1\pm 0.9\pm 1.1\pm 0.4)\times {10}^{-3}$. These results establish SCS $D\to \pi a_0$ decays as a new avenue for probing $CP$ violation.