Controlling $\rho$ width effects for a precise value of $\alpha$ in $B \to \rho \rho$

TL;DR

This paper investigates how the $ ho$ meson width affects the precision of measuring the CKM angle $eta$ in $B o ho ho$ decays, proposing methods to control and subtract the $I=1$ amplitude contribution for improved accuracy.

Contribution

It introduces a detailed analysis of the $I=1$ amplitude effects due to $ ho$ width and suggests experimental strategies to mitigate their impact on $eta$ measurement.

Findings

In the absence of $I=1$ enhancement, results are insensitive to $ ho$ width.

Enhanced $I=1$ amplitude can be isolated using separated $ ho$ bands.

Subtracting $I=1$ contribution improves isospin analysis precision.

Abstract

It has been pointed out that the currently most precise determination of the weak phase $\phi_2 = \alpha$ of the Cabibbo-Kobayashi-Maskawa (CKM) matrix achieved in $B \to \rho\rho$ decays is susceptible to a small correction at a level of $(\Gamma_\rho/m_\rho)^2$ due to an $I=1$ amplitude caused by the $\rho$ width. Using Breit-Wigner distributions for the two pairs of pions forming $\rho$ mesons, we study the $I=1$ contribution to $B\to \rho\rho$ decay rates as function of the width and location of the $\rho$ band. We find that in the absence of a particular enhancement of the $I=1$ amplitude reducing a single band to a width $\Gamma_\rho$ at SuperKEKB leads to results which are completely insensitive to the $\rho$ width. If the $I=1$ amplitude is dynamically enhanced relative to the $I=0,2$ amplitude one could subject its contribution to a "magnifying glass" measurement using two separated $\rho$ bands of width $\Gamma_\rho$. Subtraction of the $I=1$ contribution from the measured decay rate would lead to a very precise determination of the $I=0,2$ amplitude needed for performing the isospin analysis.