Up-down Asymmetries and Angular Distributions in $D\to K_1(\to K\pi \pi ){\ell }^{+}{\nu }_{\ell }$

Abstract

Using the helicity amplitude technique, we derive differential decay widths and angular distributions for the decay cascade $D\to K_1(1270,1400){\ell }^{+}{\nu }_{\ell }\to (K\pi \pi ){\ell }^{+}{\nu }_{\ell }(\ell =e,\mu )$, in which the electron and muon mass is explicitly included. Using a set of phenomenological results for $D\to K_1$ form factors, we calculate partial decay widths and branching fractions for $D^0\to K_1^{-}{\ell }^{+}{\nu }_{\ell }$ and $D^{+}\to K_1^0{\ell }^{+}{\nu }_{\ell }$, but find that results for $\mathcal{B}(D\to K_1(1270)e^{+}{\nu }_e)$ are larger than recent BESIII measurements by about a factor 1.5. We further demonstrate that the measurement of up-down asymmetry in $D\to K_1{e}^{+}{\nu }_e\to (K\pi \pi )e^{+}{\nu }_e$ and angular distributions in $D\to K_1{\ell }^{+}{\nu }_{\ell }\to (K\pi \pi ){\ell }^{+}{\nu }_{\ell }$ can help to determine the hadronic amplitude requested in $B\to K_1(\to K\pi \pi )\gamma$. Based on the Monte-Carlo simulation with the LHCb geometrical acceptance, we find that the angular distributions of MC events can be well described.

Comments: 13 pages, 6 figures