Controlling ρ width effects for a precise value of α in B → ρρ

It has been pointed out that the currently most precise determination of the weak phase ϕ2=α of the Cabibbo-Kobayashi-Maskawa (CKM) matrix achieved in B→ρρ decays is susceptible to a small correction at a level of (Γρ/mρ)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→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 Γρ 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 Γρ. 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.