Up-down Asymmetries and Angular Distributions in $D\to K_{1}(\to K\pi\pi)\ell^+\nu_{\ell}$

TL;DR

This paper derives angular distributions and asymmetries in $D$ meson decays to $K_1$ resonances, providing theoretical predictions and simulations that can aid in understanding hadronic amplitudes relevant for $B$ decays.

Contribution

It introduces a detailed helicity amplitude analysis of $D o K_1 o K\pi\pi$ decays, including explicit lepton mass effects, and connects these to measurements of hadronic amplitudes in $B$ decays.

Findings

Calculated partial decay widths and branching fractions for $D$ decays.

Identified discrepancies between theoretical predictions and BESIII measurements.

Demonstrated the feasibility of extracting hadronic amplitudes from angular distributions.

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 ${\cal B}(D\to K_1(1270)e^+\nu_{e})$ 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.