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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 tocombine multiple physical observables from various related decay channels, andtherein lies a fundamental issue. Similarities in analysis procedures give riseto systematic correlations between the measured inputs constraining $\phi_2$that must be taken into account to avoid bias. Specifically, in the case of theirreducible 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 decaychannel, 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 centralvalue as a function of the model uncertainty strength with the added dividendof 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 statisticalstrength of the former in relation to that of the model uncertainty in thelatter. This work can inspire other studies into the points at which systematiccorrelations beyond those determined in single measurements matter incombinations leading to other $CP$-violating weak phases such as $\phi_1$($\beta$), $\phi_3$ ($\gamma$) and $\phi_s$.

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