Combined study of the isospin-violating decay $D^{*}_{s}\to D_{s} \pi^0$ and radiative decay $D^*_s\to D_s\gamma$ with intermediate meson loops

We carry out a combined study of the isospin-violating decay $D_{s}^{*} \to D_{s} \pi^{0}$ and radiative decay $D^*_s\to D_s\gamma$ in an effective Lagrangian approach by taking into account the corrections from the one-loop transitions. By distinguishing the transition mechanisms of the long-distance interactions through the intermediate meson loops from the short-distance interactions through the $\eta-\pi^{0}$ mixing at the tree level the isospin-violating decay $D_{s}^{*} \to D_{s} \pi^{0}$ can be well constrained. In our approach the higher order corrections to the isospin-violating effects can involve the intermediate $D^{(*)}$ and $K^{(*)}$ scatterings. We find that the contributions from the destructive interference of intermediate meson loops via $D^{(*)0}(c\bar{u}){K}^{(*)+}(u\bar{s})$ and $D^{(*)+}(c\bar{d}){K}^{(*)0}(d\bar{s})$ rescatterings are significant. Within the commonly accepted ultra-violet (UV) cutoff range we obtain the partial decay width $\Gamma[D_{s}^{*} \to D_{s} \pi^{0}] = 9.92^{+0.76}_{-0.66}\,\mathrm{eV}$. This approach allows us to describe the $D_s^*$ radiative decay in the same framework via the vector meson dominance (VMD) model. We demonstrate that both the tree-level and one-loop transitions can be self-consistently determined if we adopt the experimental data for the branching ratio fraction of $D_{s}^{*} \to D_{s} \pi^{0}$ to $D^*_s\to D_s\gamma$. It then leads to a reliable estimate of the total decay width of $D_s^*$, i.e. $\Gamma_{\text{total}}(D_s^{*+})=170^{+ 13}_{-12}\,\mathrm{eV}$.