Rescaling the isospin triangle argument for constraining $\phi_2$ ($\alpha$): consolidating Belle II and a potential path forward for LHCb

TL;DR

The paper proposes a rescaling method for isospin triangle analysis that uses relative branching fractions to better constrain the CKM angle $oldsymbol{ heta_2}$, reducing systematic uncertainties and enabling independent measurements at LHCb.

Contribution

It introduces a novel rescaling approach for isospin analysis that minimizes systematic uncertainties and facilitates independent $oldsymbol{ heta_2}$ constraints at LHCb without relying on Belle II data.

Findings

Rescaling reduces systematic uncertainties in $oldsymbol{ heta_2}$ measurements.

Method enables $oldsymbol{ heta_2}$ constraints at LHCb without $CP$ violation data.

Potential for more sustainable and independent isospin analyses.

Abstract

A rescaling of the SU(2) isospin triangles constraining $\phi_2$ ($\alpha$) 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 $\phi_2$ 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 $C\!P$ violation in $B^0 \to \rho^+\rho^-$.