Roles of $a_0(980)$ and $a_0(1710)$ in Cabibbo-suppressed process $D^+\to \pi^0\pi^+\eta $
Motivated by the BESIII amplitude analysis of the single Cabibbo-suppressed process $D^+\to \pi^0\pi^+\eta$, we investigate this reaction by taking into account the contributions from the $a_0(980)$, $\rho$, and $a_0(1710)$, where the scalar meson $a_0(980)$ could be dynamically generated from the $S$-wave pseudoscalar meson-pseudoscalar meson interaction within the chiral unitary approach. Our theoretical predictions for the $\pi^0\eta$, $\pi^+\eta$, and $\pi^+\pi^0$ invariant mass distributions are in agreement with the BESIII measurements, especially the clear peaks around 1~GeV in the $\pi^0\eta$ and $\pi^+\eta$ invariant mass distributions could be associated with the dynamically generated state $a_0(980)$. Furthermore, we demonstrate that the intermediate $a_0(1710)$ is also necessary to describe the enhancement structure around $1.6\sim 1.7$~GeV in the $\pi^{0/+}\eta$ invariant mass distribution. More precise experimental measurements of this process could provide deeper insights into the nature of the scalar mesons $a_0(980)$ and $a_0(1710)$.