Five-body $D\to V$ Semileptonic Decays

Our main objective is to derive the decay rate for the semileptonic decays $D\to V\ell^+\nu_{\ell}\,(\ell=e,\mu)$, where $V$ represents a vector particle. In these decays, the vector particle $V$ decays into three pseudo-scalar particles. To accomplish this, we evaluate the phase-space factor for the five-body decay with a set of eight independent variables which uniquely define a point in the phase space. We further conduct a detailed investigation of the $D\to \omega\ell^+\nu_{\ell}$, where $\omega$ subsequently decays into $\pi^+\pi^-\pi^0$, within the Standard Model and in a general effective field theory description of the weak interactions at low energies. The outcomes of this study have potential applications in the measurement of $D\to \omega$ form factors. These measurements can be performed using data obtained from BESIII.