$a_0(980)$ and $f_0(980)$ excitation in the $D^+ \to π^+ ηη$ decay

TL;DR

This work resolves the BESIII-like puzzle in $D^+ \to \pi^+ \eta\eta$ by showing that triangle-singularity and $f_0(1370)$ explanations are too small to account for the data, while a correlated $f_0(980)$ excitation decaying to $\eta\eta$—tied to $a_0(980)$ production—reproduces the observed enhancement in the high-$M_{\pi^+\eta}$ region and corrects the low-$M_{\eta\eta}$ distribution. The authors employ a hadronization framework with a $P$-matrix and final-state interactions in a chiral-unitary approach, incorporating both external and internal emission with a color-based suppression factor, and evaluate absolute rates for subleading mechanisms. Their quantitative analysis yields ${\rm BR}(D^+ \to \pi^+\eta\eta)$ in agreement with the measured ${\rm BR}=(3.67\pm0.12\pm0.06)\times10^{-3}$ when including the $f_0(980)$ contribution, while the triangle and $f_0(1370)$ channels remain at the $10^{-5}$–$10^{-4}$ level. The results support a molecular picture for the scalar resonances and highlight the importance of final-state interactions in charm decays for describing Dalitz-plot features.

Abstract

We have made a thorough study of the $D^+ \to π^+ ηη$ reaction, recently measured by the BESIII collaboration, which shows an abnormal strength at high invariant masses in the $πη$ mass distribution. We studied in detail the triangle mechanism and the $f_0(1370)$ excitation modes that have been suggested to explain this abnormal feature, and concluded that they are too small to have any important role in the solution to that problem. We have also studied other possible solutions evaluating the contribution of excitations of other $f_0$, $a_0$ and $f_2$ resonances and reached the same conclusion. Unexpectedly, the solution to the problem is found considering the $f_0(980)$ excitation, with the $f_0(980)$ decaying to two $η$, which is tied to the $a_0(980)$ production, and well under control. At the same time, the consideration of the $f_0(980)$ excitation solves another non reported problem, which is the $ηη$ mass distribution that comes when only the $a_0(980)$ resonance is allowed to be excited, which produces a large deficiency at low invariant masses compared with experiment.

Figures (16)

• Figure 1: Triangle mechanism suggested in Ref. BESIII:2025yag.
• Figure 2: Diagrams of external emission at the quark level: (a) Cabibbo suppressed $W u \bar{s}$ vertex, (b) Cabibbo suppressed $Wcd$ vertex.
• Figure 3: Diagrams of internal emission at the quark level: (a) Cabibbo suppressed $W u \bar{s}$ vertex, (b) Cabibbo suppressed $Wcd$ vertex.
• Figure 4: Hadronization in the diagrams of Fig. \ref{['fig:figs1frompaper']}.
• Figure 5: Hadronization in the diagrams of Fig. \ref{['fig:figs2frompaper']}.