Unbinned extraction of $\gamma$ from $B\to DK$ with normalizing flows

TL;DR

This paper presents a novel unbinned approach using normalizing flows to extract the CKM angle gamma from B meson decays, improving fidelity with larger data samples.

Contribution

It introduces a new method employing normalizing flows trained on D decay data to accurately extract gamma without binning.

Findings

Successfully recovers injected gamma values in Monte Carlo tests.

Propagates statistical uncertainties via ensemble of trained flows.

Explores phase constraint encoding and Bayesian NF extensions.

Abstract

We introduce an unbinned method for extracting the CKM angle $\gamma$ from the decay chain $B^\pm \to (D \to K_S \pi^+ \pi^-) K^\pm$ using normalizing flows (NFs). The NFs, trained on $D$ decay data, learn a faithful continuous representation of the amplitude and strong phase variation over the $D\to K_S\pi^+\pi^-$ Dalitz plot whose fidelity improves with increased data sample sizes. With this input, the $B$ decay data can be used to extract the parameters $r_B$, $\delta_B$, and $\gamma$. We test the method on Monte Carlo generated data, where it successfully recovers the injected value of $\gamma$ within uncertainties. The present implementation propagates statistical uncertainties from finite training data via an ensemble of independently trained flows, and does not attempt to capture the effects of systematic experimental errors. We explore two versions of the method that differ in how the trigonometric constraint on phase variation is encoded, and comment on the possible extension to Bayesian NFs, which would provide direct uncertainty estimates on the learned densities without requiring ensemble training.