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

Abstract

It has been pointed out that the currently most precise determination of the weak phase ${\varphi }_2=\alpha$ 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.

Comments: to be published in Physics Letters B