Rescaling the isospin triangle argument for constraining ${\varphi }_2$ ($\alpha$): consolidating Belle II and a potential path forward for LHCb

Abstract

A rescaling of the SU(2) isospin triangles constraining ${\varphi }_2$ ($\alpha$) 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 ${\varphi }_2$ constraint in the $B\to \rho \rho$ system at LHCb that is more independent of Belle II input is shown to become viable even without a measurement of $CP$ violation in $B^0\to {\rho }^{+}{\rho }^{-}$.

Comments: 10 pages, 7 figures, replaced with journal version. This article should be considered in conjunction with arXiv:2108.06182