Normalization of partial wave CP asymmetries in three-body decays of heavy hadrons

TL;DR

This paper tackles the normalization problem of Partial-Wave CP Asymmetries (PWCPAs) in multi-body heavy-hadron decays by introducing quasi-normalized PWCPAs $A_{CP,l}=\eta_l \mathring{A}_{CP,l}$, where $\mathring{A}_{CP,l}$ is built from Legendre-weighted amplitudes. It defines experiment-friendly observables $\hat{A}_{CP,l}$ and relates them to the PWCPAs through a matrix $\Omega$, enabling practical determination of the normalization factors $\eta_l$. By proposing explicit choices for $\eta_l$ (including $\tilde{\eta}_l$ and $\eta_l^{(L)}$) based on correlation matrices and error considerations, the method yields PWCPAs with roughly equal uncertainties and bounded magnitudes. The framework is applied to the $B^\pm\rightarrow\pi^\pm\pi^+\pi^-$ channel using LHCb data near the $\rho^0(1450)$ region, demonstrating that the quasi-normalized PWCPAs align with experimental observables, remain robust against alternative definitions, and mitigate misleading interpretations due to normalization artifacts. The approach offers a principled, transferable tool for PWCPA analyses in complex three-body decays.

Abstract

CP violation in hadronic multi-body decays has been extensively studied, and the experimental breakthrough in the heayy baryon sector was made recently. Partial-Wave CP Asymmetries (PWCPAs) in multi-body decays of heavy hadrons, which although provide us with more interference information, suffer from the normalization problem, as is pointed out in this paper. We propose a novel solution to this problem. We introduce a set of extra factors to rescale the PWCPAs to the proper sizes. Instead of determining the set of factors according to the normalization requirement, we demand that all the PWCPAs have the same statistical errors. In this way, we obtain a set of quasi-normalized PWCPAs, in the sense that they are close to the ideal normalized ones. As an application, we perform an analysis of PWCPAs in the decay channel $B^\pm\toπ^+π^-π^\pm$. We focus on the phase space region where the invariant mass of the $π^+π^-$ pair varies around the vector resonance $ρ^0(1450)$. Based on the data of the LHCb collaboration, the interference patterns among the resonances $ρ^0(1450)$, $f_2(1270)$, and $f_0(1500)$ and their contributions to the quasi-normalized PWCPAs are analyzed. The analysis indicates that the quasi-normalized PWCPAs can avoid potential misleading or distorted results comparing with some other alternatively defined ones.

Figures (4)

• Figure 1: The fitting results of the parameters (shown in the upper right corner of each figure) from the event yields of $B^+\to\pi^+\pi^-\pi^+$ (a) and $B^-\to\pi^+\pi^-\pi^-$ (b) for $\cos\theta>0$ and $\cos\theta<0$.
• Figure 2: Predictions of the quasi-normalized PWCPAs $\left.A_{CP,l}\right|_{\eta_l=\eta_l^{(4)}}$ for $B^\pm\to \pi^+\pi^-\pi^\pm$ as a function of the invariant mass $m_{\pi\pi,\mathrm{low}}$. The solid, dotted, dashed, and dash-dotted lines correspond to $l=1$, 2, 3, and 4, respectively.
• Figure 3: Comparisons of various definitions of PWCPAs, where the solid, dotted, dashed , and dot-dashed lines correspond to $\left.A_{CP,l}\right|_{\eta_l=\eta_l^{(4)}}$, $\left.A_{CP,l}\right|_{\eta_l=\tilde{\eta}_l}$, $\hat{A}_{CP,l}$, and $\mathring{A}_{CP,l}$, respectively. (a), (b), (c), and (d) correspond to $l=1$, 2, 3, and 4, respectively. Note that $\left.A_{CP,l}\right|_{\eta_l=\eta_l^{(4)}}$ and $\left.A_{CP,l}\right|_{\eta_l=\tilde{\eta}_l}$ almost coincide with each other.
• Figure 4: The conventionally defined PWCPAs, $A_{CP,l}^{\text{conv}}$, as a function of $m_{\pi\pi,\text{low}}$, where the solid, dotted, dashed and dash-dotted lines correspond to $l=1$, 2, 3, and 4, respectively. We also present the results for another alternative PWCPAs, which are defined as $\tilde{A}_{CP,l}^{\text{conv}}\equiv \frac{w_l-\bar{w}_l}{|w_l|+|\bar{w}_l|}$, with $l$ corresponding to the same line types as those for $A_{CP,l}^{\text{conv}}$. For $l=1$ and 3, $\tilde{A}_{CP,l}^{\text{conv}}$ are exactly $-1$ in the main part of the region of $m_{\pi\pi,\text{low}}$; while for $l=2$ and 4, $\tilde{A}_{CP,l}^{\text{conv}}$ exactly coincides with ${A}_{CP,l}^{\text{conv}}$.