Roles of $a_0(980)$ and $a_0(1710)$ in Cabibbo-suppressed process $D^+\to π^0π^+η$

TL;DR

The paper investigates the Cabibbo-suppressed decay $D^+ \to \pi^0\pi^+\eta$, incorporating a dynamically generated $a_0(980)$ through the chiral unitary approach and including $\rho$ and $a_0(1710)$ intermediate states. It develops a hadronization and final-state-interaction framework, solving a coupled-channel Bethe-Salpeter equation to obtain $a_0(980)$ amplitudes and uses Breit–Wigner forms for $a_0(1710)$, all combined with interference phases to predict invariant-mass distributions. Fits to BESIII data show that $a_0(980)$ dynamics reproduce the $~1\ \mathrm{GeV}$ peaks in $π^0η$ and $π^+η$, while $a_0(1710)$ is essential to describe the $1.6\ \mathrm{GeV}$ enhancement; the results support a mixed interpretation of these scalars and motivate further high-precision measurements. The work provides a quantitative link between light scalar meson structure and charmed-hadron decays, with potential implications for understanding scalar meson spectroscopy.

Abstract

Motivated by the BESIII amplitude analysis of the single Cabibbo-suppressed process $D^+\to π^0π^+η$, we investigate this reaction by taking into account the contributions from the $a_0(980)$, $ρ$, 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 $π^0η$, $π^+η$, and $π^+π^0$ invariant mass distributions are in agreement with the BESIII measurements, especially the clear peaks around 1~GeV in the $π^0η$ and $π^+η$ 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 $π^{0/+}η$ 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 $a_0(1710)$.

Figures (5)

• Figure 1: Diagrammatic representation of the $D^+$ decay. (a) The internal emission of $D^+\to\pi^+d\bar{d}$ and hadronization of the $d\bar{d}$ through $\bar{q}q$ with vacuum quantum numbers; (b) The internal emission of $D^+\to\pi^0u\bar{d}$ and hadronization of the $u\bar{d}$ through $\bar{q}q$ with vacuum quantum numbers; (c) The external emission of $D^+\to\pi^+d\bar{d}$ and hadronization of the $d\bar{d}$ through $\bar{q}q$ with vacuum quantum numbers; (d) The external emission of $D^+\to K^+s\bar{d}$ and hadronization of the $s\bar{d}$ through $\bar{q}q$ with vacuum quantum numbers.
• Figure 2: Mechanisms of the $D^+\to\pi^+\pi^0\eta$: (a) tree diagram, (b) the final state interaction of $\pi^+\eta$, $K^+\bar{K}^0$, and (c) the final state interaction of $\pi^0\eta$, $K^0\bar{K}^0$, and $K^+K^-$.
• Figure 3: Process $D^+\to \rho^+ \eta \to \pi^0\pi^+\eta$ via the intermediate vector $\rho^+$.
• Figure 4: $\pi^0\eta$ (a), $\pi^+\eta$ (b) and $\pi^0\pi^+$ (c) invariant mass distributions for the process $D^+\to \pi^0\eta\pi^+$ considered the $a_0(980)$, $a_0(1710)$, $\rho^+$; $\pi^0\eta$ (d), $\pi^+\eta$ (e) and $\pi^0\pi^+$ (f) invariant mass distributions for the process $D^+\to \pi^0\eta\pi^+$ by turning off the contribution of $a_0(1710)$. The experimental data of BESIII is represented by the points with error bars BESIII:2024tpv.
• Figure 5: Dalitz plots of '$M^2_{\pi^0\eta}$' vs. '$M^2_{\pi^+\eta}$' for the process $D^+\to \pi^0\pi^+\eta$. The results of Fig. \ref{['Dalitz(1710)']} are obtained from our full model, and Fig. \ref{['Dalitz(rho)']} are obtained by turning off the contribution of the $a_0(1710)$.