Resonant $a_0(980)$ state in triangle rescattering $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ decays

We study the $D_s^{+}\to {\pi }^{+}(a_0(980{)}^0\to ){\pi }^0\eta$, ${\pi }^0(a_0(980{)}^{+}\to ){\pi }^{+}\eta$ decays, which have been recently measured by the BESIII collaboration. We propose that $D_s^+\to \pi^{+(0)}(a_0(980)^{0(+)}\to)\pi^{0(+)}\eta$ receives the contributions from the triangle rescattering processes, where $M^0$ and ${\rho }^{+}$ in $D_s^+\to M^0 \rho^+$,byexchanging$\pi^{0(+)}$,areformedas$a_0(980)^{0(+)}$and$\pi^{+(0)}$,respectively,with$M^0=(\eta,\eta’)$. Accordingly, we calculate that ${\cal B}(D_s^+\to a_0(980)^{0(+)}\pi^{+(0)})=(1.7\pm 0.2\pm 0.1)\times 10^{-2}$and${\cal B}(D_s^+\to \pi^{+(0)}(a_0(980)^{0(+)}\to)\pi^{0(+)}\eta)=(1.4\pm 0.1\pm 0.1)\times 10^{-2}$, being consistent with the data.