Amplitude Analysis of $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ and First Observation of the $W$-Annihilation Dominant Decays $D_s^{+}\to a_0(980{)}^{+}{\pi }^0$ and $D_s^{+}\to a_0(980{)}^0{\pi }^{+}$

We present the first amplitude analysis of the decay $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$. We use an $e^{+}{e}^{-}$ collision data sample corresponding to an integrated luminosity of $3.19{\mathrm{fb}}^{-1}$ collected with the BESIII detector at a center-of-mass energy of 4.178 GeV. We observe for the first time the $W$-annihilation dominant decays $D_s^{+}\to a_0(980{)}^{+}{\pi }^0$ and $D_s^{+}\to a_0(980{)}^0{\pi }^{+}$. We measure the absolute branching fraction $\mathcal{B}(D_s^{+}\to a_0(980{)}^{+(0)}{\pi }^{0^{(}+)},a_0(980{)}^{+(0)}\to {\pi }^{+(0)}\eta )=(1.46\pm 0.15_{\mathrm{stat}}\pm 0.23_{\mathrm{sys}})\%$, which is larger than the branching fractions of other measured pure $W$-annihilation decays by at least one order of magnitude. In addition, we measure the branching fraction of $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ with significantly improved precision.

Figure 1

$D_s^{+}\to a_0(980{)}^{+(0)}{\pi }^{0(+)}$ WA-topology diagrams, where the gluon lines can be connected with the quark lines in all possible cases and the contributions from FSI are included.

Figure 2

(a) Dalitz plot of $M_{{\pi }^{+}\eta }^2$ vs $M_{{\pi }^0\eta }^2$ for data, the projections of the fit on (b) $M_{{\pi }^{-}{\pi }^0}$, (c) $M_{{\pi }^{+}\eta }$, and (d) $M_{{\pi }^0\eta }$, and the projections on (e) $M_{{\pi }^{+}\eta }$ and (f) $M_{{\pi }^0\eta }$ after requiring $M_{{\pi }^{+}{\pi }^0}>1.0\mathrm{GeV}/c^2$. In (b)–(f), the dots with error bars and the solid line are data and the total fit, respectively; the dashed, dotted, and long-dashed lines are the contributions from $D_s^{+}\to {\rho }^{+}\eta$, $D_s^{+}\to ({\pi }^{+}{\pi }^0{)}_V\eta$, and $D_s^{+}\to a_0(980)\pi$, respectively. The (red) hatched histograms are the simulated background.

Figure 3

Fits to (a)–(g) the $M_{\mathrm{tag}}$ distributions of seven tag modes (indicated in each sub-figure) and (h) the $M_{\mathrm{sig}}$ distribution of signal candidates. The dots with error bars are data. The (blue) solid lines are the total fit. The (red) dashed and the (green) long-dashed lines are signal and background, respectively. In (a)–(g), the $D_s^{-}$ signal regions are between the arrows.