Resolving the ${\varphi }_2$ ($\alpha$) ambiguity in $B\to \rho \rho$

Abstract

I propose an alternative method for measuring the $CP$ violating phase ${\varphi }_2$ ($\alpha$) without ambiguity in an extended SU(2) isospin triangle analysis, which can ultimately be achieved by exploiting interference effects between $B^0\to {\rho }^0{\rho }^0$ and $B^0\to a_1^{\pm }{\pi }^{\mp }$ in a time-dependent flavour-tagged amplitude analysis. Under certain assumptions on the effective ${\varphi }_2$ in each channel, I demonstrate with an idealised amplitude model that potential deviations in the measured ${\varphi }_2$ due to penguin contamination in $B^0\to a_1^{\pm }{\pi }^{\mp }$ are sufficiently large within current experimental uncertainties that this programme could be executed with Run 3 data at LHCb and easily at Belle II.

Comments: 19 pages, 7 figures, replaced with journal version