Full analysis of CP violation induced by the decay angular correlations in four-body cascade decays of heavy hadrons

Technical Summary: Full Analysis of CP Violation Induced by Decay Angular Correlations in Four-Body Cascade Decays of Heavy Hadrons

Problem Statement
While Charge-Parity (CP) violation (CPV) has been extensively observed in pure meson decays, its manifestation in processes involving baryons has remained elusive until very recently. The LHCb collaboration recently confirmed CP violation in the four-body baryon-to-baryon decay $\Lambda_b^0 \to pK^-\pi^+\pi^-$. However, the detailed dynamics driving this asymmetry remain unclear. Furthermore, CP violation in meson-to-baryon-anti-baryon-pair transitions (e.g., $B^0 \to p\bar{p}K^+\pi^-$) has not yet been observed, despite theoretical expectations that the complex phase space of such decays offers enhanced opportunities for CPV studies compared to simpler transitions. Existing literature lacks a general and complete formalism for analyzing CP violation corresponding to decay angular correlations in these four-body cascade decays, particularly those involving baryons.

Methodology
The authors propose a formalism for the full analysis of decay angular correlations in four-body cascade decays of heavy hadrons ($H_Q \to a(\to 12)b(\to 34)$). The approach utilizes the generalized Cabibbo-Maksymowicz parametrization of the phase space, describing the decay using two invariant masses ($m_{12}, m_{34}$) and five kinematic angles ($\theta_{HQ}, \theta_a, \theta_b, \phi_a, \phi_b$).

Key elements of the methodology include:

• Amplitude Decomposition: The squared decay amplitude is expressed as a sum over dynamical factors ($\gamma$) and kinematic factors ($\Psi$ and $\Phi$). These factors depend on Wigner-$d$ matrices and the relative azimuthal angle $\phi = \phi_a - \phi_b$.
• Angular Correlation Constraints: The authors derive constraints on the angular quantum numbers ($j_a, j_b$) based on triangular inequalities and parity symmetry. This determines the number of independent angular correlations ($C_{j_a j_b} = 2\min(j_a, j_b) + 1$).
• CPV Observables: Two sets of CP-violating observables are constructed:
  1. Asymmetries based on the real and imaginary parts of the interference terms ($A_{CP}^{\Re(\gamma)}$ and $A_{CP}^{\Im(\gamma)}$).
  2. Asymmetries based on the sign of kinematic functions ($A_{CP}^{Y}$), defined by integrating over regions where specific angular correlations are positive or negative.
• Interference Retrieval: The formalism accounts for interferences between different intermediate resonances (e.g., $K^*(892)$ and scalar contributions). To retrieve interference terms that might be integrated out by standard binning, the authors introduce a sign factor ($\text{sgn}(m_{34}^2 - m_{res}^2)$) to define additional observables ($\tilde{A}_{CP}^{Y}$).

Application and Results
The formalism is applied to the decay channel $B^0 \to p\bar{p}K^+\pi^-$, utilizing event yields inversely extracted from published LHCb data (Ref. [33]). The analysis focuses on the low $K^+\pi^-$ invariant mass region (dominated by $K^*(892)$) and the low $p\bar{p}$ region (showing threshold enhancement).

• Angular Correlation Identification: The authors identify that Forward-Backward Asymmetries (FBAs) in the $p\bar{p}$ and $K^+\pi^-$ systems are driven by angular correlations $\Psi_{10}^0$ and $\Psi_{01}^0$, respectively. A Left-Right Asymmetry (LRA) in $\cos \phi$ is induced by $\Phi_{11}^1$.
• CP Asymmetry Calculation: Based on the extracted event yields and assuming statistical errors, the authors calculate CP asymmetries for 19 of the 26 possible angular correlations.
• Key Findings: The analysis yields non-zero CP asymmetries of approximately 10% for specific correlations, notably:
  • $\tilde{A}_{CP}^{\Psi_{01}^0} \approx 10.7\%$
  • $\tilde{A}_{CP}^{\Psi_{11}^0} \approx 10.0\%$
  • $A_{CP}^{\Psi_{12}^1} \approx 9.2\%$
    These values are significantly larger than the CP asymmetries observed in the $\Lambda_b^0 \to pK^-\pi^+\pi^-$ channel. The non-zero result for $\tilde{A}_{CP}^{\Psi_{11}^0}$ suggests non-zero helicity amplitudes for the $\pm 1$ states of the $K^*(892)$, while the result for $A_{CP}^{\Psi_{12}^1}$ indicates the importance of both $0^\pm$ and $1^\mp$ contributions in the $p\bar{p}$ threshold enhancement.

Significance and Claims
The paper claims that full decay angular correlation analysis offers distinct advantages over existing methods (regional CP asymmetry analysis, amplitude analysis, and energy tests) for studying CPV in multi-body decays. As summarized in the authors' comparison (Table II), this method is:

1. Model-independent: It does not require prior knowledge of specific resonances to begin the analysis.
2. Efficient: It utilizes the entire phase space, resulting in smaller statistical uncertainties compared to regional analyses.
3. Dynamically Transparent: It directly links CPV observables to the underlying interference dynamics (specific spin-parity combinations).
4. Comprehensive: It is the only method identified that possesses all four features (statistical significance, dynamical inference, model independence, and efficiency).

Conclusion and Caveats
The authors emphasize that while the calculated 10% CP asymmetries are large and statistically significant, they rely on assumptions regarding the extraction of event yields and the universality of a scaling factor. Therefore, the paper does not claim to have definitively discovered CP violation in $B^0 \to p\bar{p}K^+\pi^-$. Instead, it presents these results as a "persuasive suggestion" for experimental colleagues to perform a full angular correlation analysis on this channel (and others) to eliminate assumption dependencies and confirm the presence of CP violation. The authors argue that such a full analysis is necessary to understand the dynamics behind CPV, even in channels where it has already been observed, such as $\Lambda_b^0 \to pK^-\pi^+\pi^-$.