Rescaling the isospin triangle argument for constraining $ϕ_2$ ($α$): consolidating Belle II and a potential path forward for LHCb

A rescaling of the SU(2) isospin triangles constraining $ϕ_2$ ($α$) that relies on measurements of the experimentally cleaner relative branching fractions, as opposed to those absolute, is proposed. Paving the way towards more systematically sustainable analysis, this method promises to eliminate a dominant systematic at Belle II amongst others, namely the uncertainty on the number of $B \bar B$ pairs in data. Furthermore, a $ϕ_2$ constraint in the $B \to ρρ$ system at LHCb that is more independent of Belle II input is shown to become viable even without a measurement of $C\!P$ violation in $B^0 \to ρ^+ρ^-$.

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