Controlling $ρ$ width effects for a precise value of $α$ in $B \to ρρ$

Abstract

It has been pointed out that the currently most precise determination of the weak phase $ϕ_2 = α$ of the Cabibbo-Kobayashi-Maskawa (CKM) matrix achieved in $B \to ρρ$ decays is susceptible to a small correction at a level of $(Γ_ρ/m_ρ)^2$ due to an $I=1$ amplitude caused by the $ρ$ width. Using Breit-Wigner distributions for the two pairs of pions forming $ρ$ mesons, we study the $I=1$ contribution to $B\to ρρ$ decay rates as function of the width and location of the $ρ$ band. We find that in the absence of a particular enhancement of the $I=1$ amplitude reducing a single band to a width $Γ_ρ$ at SuperKEKB leads to results which are completely insensitive to the $ρ$ 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 $ρ$ bands of width $Γ_ρ$. 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.