Impact of systematic and amplitude model correlations within and between systems of combined input: A case study with $\phi_2$ ($\alpha$)

TL;DR

This study investigates how systematic and amplitude model correlations affect the precision and bias in measuring the $CP$-violating phase $oldsymbol{\phi_2}$, emphasizing the importance of proper correlation treatment in combined decay analyses.

Contribution

It demonstrates the impact of ignoring correlations in combined measurements of $oldsymbol{\phi_2}$ and proposes methods to correctly account for them, reducing bias and improving uncertainty estimates.

Findings

Ignoring correlations can bias $oldsymbol{\phi_2}$ by about 1 degree.

Proper correlation treatment reduces bias and uncertainty.

Bias depends on the relative statistical strength of decay channels.

Abstract

The pursuit of experimental precision in the $CP$-violating weak phase $\phi_2$ ($\alpha$) is not without its challenges, in part due to the need to combine multiple physical observables from various related decay channels, and therein lies a fundamental issue. Similarities in analysis procedures give rise to systematic correlations between the measured inputs constraining $\phi_2$ that must be taken into account to avoid bias. Specifically, in the case of the irreducible model uncertainty accompanying analyses involving the $\rho$ meson, it is demonstrated that ignoring correlations derived from its pole parameters, or indeed even treating correlations individually contained within each decay channel, can ultimately lead to a bias in $\phi_2$ of $\mathcal{O}(1^\circ)$. Correct treatment on the other hand, markedly reduces wandering of its central value as a function of the model uncertainty strength with the added dividend of a further improved overall uncertainty. Bias in the combination of $B^0 \to (\rho \pi)^0$ and $B \to \rho \rho$ is also seen to depend on the statistical strength of the former in relation to that of the model uncertainty in the latter. This work can inspire other studies into the points at which systematic correlations beyond those determined in single measurements matter in combinations leading to other $CP$-violating weak phases such as $\phi_1$ ($\beta$), $\phi_3$ ($\gamma$) and $\phi_s$.