Partial-Wave CP Asymmetries

Updated 20 November 2025

Partial-Wave CP Asymmetries are observables that isolate CP violation by decomposing multi-body decay amplitudes into distinct angular momentum components.
They are extracted using angular moment analyses such as Legendre polynomial and spherical harmonic expansions, providing model-independent and statistically robust measurements.
PWCPAs are applied across heavy baryon, meson, and leptonic decays, enhancing the sensitivity to CP violating phases and guiding new physics searches.

Partial-Wave CP Asymmetries (PWCPAs) are a class of observables that measure charge-parity (CP) violation in multi-body decays of hadrons, leptons, or heavy baryons by isolating the contributions associated with specific angular momentum components (partial waves) of the decay amplitude. Formulated to harness the interference patterns between different partial-wave amplitudes—where varying strong and weak phases can produce enhanced CP-violating effects—PWCPAs have become central to statistically robust, model-independent strategies for the search for CP violation in systems where composite or overlapping resonance contributions are prevalent.

1. Theoretical Foundation and General Definition

PWCPAs are rooted in the partial-wave decomposition of decay amplitudes for processes such as $H \to h_1 h_2 \dots h_n$ , where $H$ is a heavy hadron or lepton. The amplitude $A(\{p_i\})$ is expanded in a complete set of basis functions—typically spherical harmonics $Y_{\ell m}(\Omega)$ for angular momentum $\ell$ , projection $m$ , or Legendre polynomials $P_\ell(\cos\theta)$ for systems with azimuthal symmetry:

$A(\{p_i\}) = \sum_{\ell, m} A_{\ell m} Y_{\ell m}(\Omega)$

$\bar A(\{p_i\}) = \sum_{\ell, m} \bar{A}_{\ell m} Y_{\ell m}(\Omega)$

Here, $A_{\ell m}$ and $\bar A_{\ell m}$ are the partial-wave coefficients for the process and its CP-conjugate, respectively. The partial-wave CP asymmetry (PWCPA) for a given $(\ell, m)$ is defined as

$A_{CP}^{\ell m} \equiv \frac{|A_{\ell m}|^2 - |\bar A_{\ell m}|^2}{|A_{\ell m}|^2 + |\bar A_{\ell m}|^2}$

For practical applications in systems exhibiting azimuthal symmetry, an expansion in Legendre polynomials is used, reducing to

$|M|^2(\Omega) = \sum_{j=0}^J w^{(j)} P_j(\cos\theta), \qquad A_{CP}^{(j)} = \frac{w^{(j)} - \bar w^{(j)}}{w^{(j)} + \bar w^{(j)}}$

where $w^{(j)}$ are the event-weighted partial-wave intensities.

2. Extraction from Experiment and Methodological Aspects

PWCPAs are extracted by leveraging the orthogonality of the basis functions on the angular space. Strategies for experimental extraction include:

Fitting the full differential decay rate $d\Gamma/d\Omega$ to the chosen basis and reading off the intensities $W_{\ell m}$ .
Direct computation via event-weighted sums (e.g., $Y_{\ell m}$ - or $P_j$ -weighted) to obtain yields $N_{\ell m}, \bar{N}_{\ell m}$ or $N_j, \bar N_j$ for signal and CP-conjugate samples.

For the Legendre expansion, the event sum is:

$N_{j\,\mathrm{wgt}} = \sum_k P_j(\cos\theta_k), \quad A_{CP}^{(j), \mathrm{exp}} = \frac{N_{j\,\mathrm{wgt}} - \bar{N}_{j\,\mathrm{wgt}}}{N_{j\,\mathrm{wgt}} + \bar{N}_{j\,\mathrm{wgt}}}$

Event selection is performed in specified phase-space windows, often around resonant structures, and binning in invariant mass is used to paper differential PWCPAs and the underlying interference effects.

3. Model Independence and Interference-Driven Sensitivity

A critical property of PWCPAs is their independence from specific amplitude models; they merely require measurement of angular distributions and identification of relevant subsystems for partial-wave analysis. This renders the method robust to uncertainties in hadronic models for heavy baryons or multibody hadronic decays.

CP violation is manifest in PWCPAs whenever at least two interfering amplitudes carry different weak phases and different strong phases (rescattering phases). The difference $w^{(j)} - \bar w^{(j)}$ is proportional to $\sin(\Delta\phi_{\mathrm{weak}}) \sin(\Delta\delta_{\mathrm{strong}})$ , allowing sizable CP asymmetries even for small weak-phase differences if rescattering phases are large. Interference between different resonances (selectable via angular momentum selection rules) leads to enhanced sensitivity in certain $j$ channels.

4. Normalization Issues and Quasi-Normalized PWCPAs

The direct ratio definition of PWCPAs,

$A_{CP, \ell}^{\mathrm{conv}} = \frac{w_\ell-\bar w_\ell}{w_\ell + \bar w_\ell}$

is problematic for small $w_\ell$ (statistical instability, unbounded range). Alternative choices like

$\mathring A_{CP, \ell} = \frac{w_\ell-\bar w_\ell}{w_0 + \bar w_0}$

still suffer from inconsistent normalization, since physical phase-space weights vary drastically across $\ell$ .

To address this, a quasi-normalization scheme introduces rescaling factors $f_\ell$ for each partial wave such that all PWCPAs exhibit equal statistical uncertainties:

$A_{CP, \ell}^{\mathrm{quasi}} = f_\ell \frac{w_\ell - \bar w_\ell}{w_0 + \bar w_0}$

$f_\ell = \left( \left[ \omega_L^{-1} \rho (\omega_L^{-1})^T \right]_{\ell\ell} \right)^{-1/2}$

where $\omega_{\ell k}$ is the overlap matrix and $\rho$ the correlation matrix for the sign-weighted observables. This ensures all $A_{CP, \ell}^{\mathrm{quasi}}$ observables are bounded, statistically comparable, and interpretable across partial waves, circumventing the pathologies of earlier definitions (Qi et al., 16 Nov 2025).

5. Applications Across Systems

Heavy Baryon Decays: The PWCPA formalism provides a systematic strategy to search for CPV in multi-body decays such as $\Lambda_b^0\to p \pi^-\pi^+\pi^-$ , $\Lambda_b\to p K^- \pi^+\pi^-$ , and related channels. Illustration for $\Lambda_b^0\to p \pi^- \pi^+ \pi^-$ demonstrates enhanced asymmetries (e.g., $A_{CP}^{(2)}$ up to 20–30%) when exploiting interference between resonances with large strong-phase differences (Zhang et al., 2021).

Three-Body Meson Decays: In $B^{\pm} \to \pi^+ \pi^- \pi^\pm$ , PWCPAs resolve CPV patterns associated with interference among $\rho^0(1450)$ (P-wave), $f_2(1270)$ (D-wave), and $f_0(1500)$ (S-wave) resonances. Quasi-normalized PWCPAs in this channel remain well-behaved and reveal interference-driven structure not visible in conventional global CP asymmetries (Qi et al., 16 Nov 2025).

Four-Body and Multi-Body Systems: In $B\to \phi(\to K\bar K)K^*(\to K\pi)$ , the formalism accommodates interferences among S- and P-wave helicity amplitudes. PWCPAs and triple-product asymmetries (TPAs) constructed through angular variables further probe CPV and final-state interactions (Zhang et al., 2021). Similarly, in $D^0\to K^+K^-\pi^+\pi^-$ , PWCPAs are constructed from kinematic functions isolating specific paired partial-wave interferences, providing sensitivity at the $10^{-3}$ level (Collaboration et al., 2018).

Leptonic Hadronic Decays: For $\tau\to K_S\pi\nu_\tau$ , the partial-wave expansion in the K $\pi$ system permits definition of S- and P-wave CP asymmetries projected through Legendre moments. In the Standard Model, all direct CPV vanishes aside from the indirect component from $K^0$ – $\bar K^0$ mixing. Under model-independent effective operator analyses, strong external constraints from neutron EDM and $D^0$ – $\bar D^0$ mixing tightly limit possible new-physics contributions to angular PWCPAs (Chen et al., 30 Jun 2024).

PWCPAs are closely related to:

Regional CP Asymmetries (rCPAs): Finer binning in the Dalitz plot can define regional CP asymmetries, but these generally have inferior statistical properties compared to PWCPAs for low-statistics multi-body samples (Zhang et al., 2021).
Triple-Product Asymmetries (TPAs): T-odd observables, constructed from correlations like $\vec{p}_1\cdot (\vec{p}_2\times\vec{p}_3)$ , can be decomposed into partial-wave interferences, serving as complementary probes of CPV and strong phases (Zhang et al., 2021, Collaboration et al., 2018).

Resonance-dominated models underlie the interpretation of enhancements in localized Dalitz regions—e.g., in $B^\pm \to \pi^\pm(\pi^+\pi^-)_{m^2 < 0.4}$ , the observed large PWCPAs emerge from interference between $\rho(770)$ (P-wave) and $f_0(500)$ (S-wave), enforced by U-spin and CPT symmetry arguments (Bhattacharya et al., 2013).

7. Statistical and Practical Considerations

The PWCPA methodology, especially with quasi-normalization, is optimized for low-statistics environments and broad resonant or non-resonant phase-space regions. Summary characteristics:

Statistical precision: Quasi-normalized PWCPAs maintain equal uncertainties across partial waves, providing direct comparability.
Model-independence: No amplitude model input required; applicable to arbitrary-spin systems, multi-body decays, or systems with overlapping resonances.
Implementation: Amenable to analysis via event-weighting and angular-moment projections, readily generalized to three-, four-, or higher-body decays.
Physical interpretability: Enhanced sensitivity to CPV due to interference-driven amplification when strong phases are large, potentially reaching sizable values (e.g., 20–30%) in heavy-baryon channels.

Future directions include extending the PWCPA framework to high-statistics data sets, refining the treatment of multiple resonant structures, systematic studies of phase-space partitioning, and probing their sensitivity to new-physics sources in channels where Standard Model background is heavily suppressed (Qi et al., 16 Nov 2025, Zhang et al., 2021, Chen et al., 30 Jun 2024).