# Evaluation of CKM matrix elements from exclusive $P_{\ell 4}$ decays

## TL;DR

This paper analyzes exclusive $P_{ ext{ell}4}$ decays to refine the extraction of CKM matrix elements, accounting for pole contributions beyond the dominant amplitude, and applies findings to improve measurements of $|V_{ub}|$ and $|V_{cd}|$.

## Contribution

It identifies and evaluates the impact of pole contributions in $P_{ ext{ell}4}$ decays on CKM matrix element determinations, providing more accurate extraction methods.

## Key findings

- Pole contributions significantly affect CKM element extraction.
- Inclusion of pole effects increases the $|V_{ub}|$ value from $B\to\pi\pi\ell\nu$ data.
- Pole effects in $D$ decays influence $|V_{cd}|$ measurements.

## Abstract

We consider the exclusive $P_{\ell 4}$ decays, $P\to (P_1P_2)_V\ell \nu_{\ell}$, where the subindex $V$ means that the invariant mass of the pseudoscalar pair is taken within a small window around the mass of the vector meson $V$. Pole contributions beyond the dominant $P\to V(\to P_1P_2)\ell\nu_{\ell}$ amplitude of $P_{\ell 4}$ decays are identified, which, in turn, affects the determination of the CKM matrix elements $|V_{qq'}|$. We evaluate the effects of those contributions in the extraction of bottom and charm quark mixings. An application to $B\to (\pi\pi)_{\rho}\ell \nu_{\ell}$ data from Belle collaboration, shows an increase in the extracted value of $|V_{ub}|$ in better agreement with determinations based on $B\to \pi\ell\nu_{\ell}$ decays. The effect of the $\rho$ and $D^*$ pole contributions in the determination of $|V_{cd}|$ from the decay $D\to \pi\pi \ell^- \bar{\nu}_{\ell}$, has been also investigated.