Subprocesses $\rho (770,1450)\to K\bar{K}$ for the three-body hadronic $D$ meson decays

We construct the theoretical framework for quasi-two-body $D$ meson decays with the help of pion and kaon electromagnetic form factors, and with which we study the contributions of the subprocesses $\rho (770,1450)\to K\bar{K}$ for the three-body $D$ decays within the flavour $SU(3)$ symmetry. Because of the limitations imposed by phase space and strong coupling, the contributions for kaon pair from the virtual bound state $\rho (770)$ are channel-dependent and generally small for the concerned three-body $D$ decays. But some quasi-two-body processes could still be observed in the Dalitz plot analyses for related decays, such as $D^0\to K^{-}\rho (770{)}^{+}\to K^{-}{K}^{+}{K}_S^0$ and $D^+ \to K_S^0\rho(770)^+ \to K_S^0 K^+K_S^0$, they are predicted to have the branching fractions $\mathcal{B}=(0.82\pm 0.04)\times {10}^{-4}$ and $\mathcal{B}=0.47_{-0.03}^{+0.05}\times {10}^{-4}$, which are $(1.86\pm 0.16)\mathrm{\%}$ and $(1.84_{-0.16}^{+0.21})\mathrm{\%}$, respectively, of the total branching fractions for the corresponding three-body $D$ decays. We find in this work that the normal subprocesses like $\rho (1450{)}^{+}\to {\pi }^{+}{\pi }^0$ or $\rho(1450)^+\to K^+\bar{K}^0$, which are bound by the masses of decaying initial states, will provide virtual contributions in some special decays.