Phenomenological description of the $D^*_{s0}(2317)\to D_sπ^0$ decay

TL;DR

The paper develops a phenomenological, multi-channel description of isospin-violating D^*_{s0}(2317)^+ decays to D_s^+ pi^0 by combining an I=0 resonance with I=1 nonresonant backgrounds. It constructs unmixed I=0 and I=1 amplitudes, then treats their mixing with a two-channel propagator formalism, enforcing unitarity and respecting the Watson theorem for the final-state interaction. The analysis yields a resonance decay amplitude whose phase matches the nonresonant scattering phase, and provides width estimates in the 95–130 keV range for reasonable phase choices, in agreement with other theoretical approaches. The framework complements unitarized chiral perturbation theory and can be extended to include possible mixing with an isovector partner and to related decays of heavier strange-charm states.

Abstract

For coupled channels $D^0K^+$, $D^+K^0$, $D_s^+η$, and $D_s^+π^0$, the $S$-wave scattering amplitudes are constructed taking into account the mixing of the isoscalar resonance $D^*_{s0}(2317)^+$ with nonresonance amplitudes with isospin $I=1$. The phenomenological approach we use allows us to quite simply clear up the general structure of the $D^*_{s0}(2317)^+\to D_s^+π^0$ decay amplitude violating isospin. We show that the phase of this amplitude coincides with the phase of the nonresonanct $D_s^+π^0 $ scattering amplitude in agreement with the Watson theorem. Its modulus squared, as it should be, determines the width of the resonance peak in the $D_s^+π^0$ channel. Taking into account the $π^0-η$ mixing in internal lines up to the second order inclusively ensures that the unitarity condition is fulfilled. The presented analysis complements the description of the $D^*_{s0}(2317 )^+\to D_s^+π^0$ decay based on the coupled channel unitarized chiral perturbation theory. The numerical estimates obtained by us for the $D^*_{s0}(2317)^+\to D_s^+π^0$ decay width do not contradict those available in the literature.

Figures (7)

• Figure 1: (a) Decay of the $I=0$ state, $D^*_{s0}(2317 )^+$, into $D_s^+\pi^0$. (b) $D_s^+\pi^0$ scattering via the $D^*_{s0}(2317)^+$ intermediate state resulting from the $\pi^0-\eta$ mixing. Each black circle in this figure, as well as in Figs. 2, 3, and 6, denotes the $\pi^0-\eta$ mixing amplitude $\Pi_{\pi^0\eta}$
• Figure 2: Equations for propagators of the mixed $\eta$ and $\pi^0$ mesons and propagators of the $\pi^0 \leftrightarrow \eta$ transitions; the argument $q^2$, on which the propagators depend, is omitted in the figure (the notations are explained in detail in the text).
• Figure 3: According to Eq. (\ref{['Eq2']}) there are two equivalent ways of calculating the function $\ddot{G}_{44}(s)$ using the right-hand or left-hand side of the equality shown in the figure. In diagram (b), $\langle \pi^0\rangle_\eta$ denotes that the pion proragator $1/D_\pi(q^2)$ is taken at $q^2=m^2_\eta$.
• Figure 4: Coupling of the $c\bar{s}$ state with channels $D^0K^+$, $D^+K^0$, and $D_s^+\eta$.
• Figure 5: Example diagram of the quark rearrangement during scattering.