Quasi-Normalized PWCPAs in Heavy-Hadron Decays

• Quasi-Normalized PWCPAs are CP asymmetry observables rescaled by factors ηₗ to ensure bounded values and comparable statistical errors across partial waves.
• They resolve normalization ambiguities in traditional CP measurements by employing a systematic rescaling of Legendre moments in multi-body decays.
• Applied to decays like B± → π⁺π⁻π±, this approach enhances experimental reliability by eliminating unphysical artifacts from denominator zeros.

Quasi-normalized partial-wave CP asymmetries (PWCPAs) address persistent challenges in quantifying CP violation (CPV) across partial waves in multi-body decays of heavy hadrons. Standard definitions of PWCPAs suffer from normalization ambiguities that impair their physical interpretability and statistical reliability, particularly in the context of three- and four-body hadronic decays. The quasi-normalization framework introduces a systematic rescaling protocol that ensures boundedness, comparability, and robust statistical precision for PWCPA observables, significantly advancing experimental and amplitude analysis methodologies in heavy-flavor physics (Qi et al., 16 Nov 2025).

1. Definition and Motivation

In a three-body decay $M \to M_1 M_2 M_3$, the squared amplitude $|\mathcal{M}(s_{12}, s_{23})|^2$ is typically expanded as a Legendre series in the helicity angle $\theta$ of the $(12)$ pair:

$$|\mathcal{M}(s_{12}, s_{23})|^2 = \sum_{\ell} w_\ell P_\ell(\cos\theta),$$

where $w_\ell$ are Legendre moments and $P_\ell$ are Legendre polynomials. Conventional PWCPAs are naively defined as

$$A^{\rm conv}_{CP,\ell} = \frac{w_\ell - \bar{w}_\ell}{w_\ell + \bar{w}_\ell},$$

with $w_\ell$ and $\bar{w}_\ell$ for $B^+$ and $B^-$ (or particle/antiparticle) decays. This standard form is not well-defined: $w_\ell$ may take negative or zero values, rendering $A^{\rm conv}_{CP,\ell}$ unbounded and non-comparable between different $\ell$. Even the alternative

$$\mathring{A}_{CP,\ell} = \frac{w_\ell - \bar{w}_\ell}{w_0 + \bar{w}_0}$$

fails to guarantee a $( -1, +1 )$ range because $w_\ell$ can be negative and is not a positive-definite weight. The root issue lies in the properties of the Legendre moments themselves.

2. Quasi-Normalization Ansatz: Scaling Factors $\eta_\ell$

To circumvent these issues, Qi et al. introduce quasi-normalized PWCPAs by rescaling the difference of moments with a positive normalization constant $\eta_\ell$:

$$A_{CP,\ell} \equiv \eta_\ell \frac{w_\ell - \bar{w}_\ell}{w_0 + \bar{w}_0}$$

This can be equivalently expressed as

$$A_{CP,\ell} = \frac{\eta_\ell (2\ell + 1) \int [|\mathcal{M}|^2 - |\bar{\mathcal{M}}|^2] P_\ell(\cos\theta) d\tilde{c}_\theta}{\int [|\mathcal{M}|^2 + |\bar{\mathcal{M}}|^2] d\tilde{c}_\theta},$$

with $d\tilde{c}_\theta \equiv \Delta_{12} d(\cos\theta)$. The normalization constants $\eta_\ell$ are chosen such that the resulting asymmetries are both theoretically meaningful and experimentally robust, achieving two principal objectives: a well-defined $( -1, +1 )$ range and uniform, comparable statistical errors across $\ell$.

Two primary prescriptions for $\eta_\ell$ are provided:

$\ell$	$\eta_\ell$ (rough, $\tilde{\eta}_\ell$)	$\eta_\ell^{(L)}$ (equal-error, $L=4$)
0	1.000	1.000
1	0.500	0.542
2	0.385	0.385
3	0.325	0.331
4	0.287	0.283

• Rough estimate ($\tilde{\eta}_\ell$): $$\tilde{\eta}_\ell \equiv \omega_{\ell\ell} = \frac{1}{2} \int_{-1}^{1} |P_\ell(x)| dx$$
• Equal-error ($\eta_\ell^{(L)}$): Choose $\eta_\ell$ so that $\sigma(A_{CP,\ell})$ matches the benchmark error of the sign-weighted asymmetry, exploiting $\omega_{k\ell}$ and sign-correlation matrices $\rho_{k\ell}$, and computing $$\eta_\ell^{(L)} = \left[ (\omega_L^{-1}\rho (\omega_L^{-1})^T)_{\ell\ell} \right]^{-1/2}$$ for expansion truncation $L$.

Either normalization yields similar quasi-normalization in practice.

3. Explicit Construction and Experimental Prescription

The quasi-normalized PWCPA in each bin of $s_{12}$ is constructed as

$$A^{\rm QNPW}_{CP,\ell}(s_{12}) = \eta_\ell \frac{w_\ell(s_{12}) - \bar{w}_\ell(s_{12})}{w_0(s_{12}) + \bar{w}_0(s_{12})}$$

for $\ell = 1,2,\ldots,L$, where $L$ is the maximum partial wave considered (e.g., $L=4$ in regions dominated by three interfering resonances).

Experimental Recipe:

1. In each small bin of $s_{12}=m^2_{12}$, histogram $B^\pm$ event counts versus $\cos\theta$.
2. Fit the distribution in each bin to $\sum_{\ell=0}^L w_\ell P_\ell(\cos\theta)$, obtaining $w_\ell^+$ and $w_\ell^-$.
3. Select $L$ (e.g., $L=4$ near $\rho(1450)$) and choose $\eta_\ell$ from the Table above.
4. Compute $$A_{CP,\ell} = \eta_\ell (w_\ell^+ - w_\ell^-)/(w_0^+ + w_0^-)$$ for each bin.
5. Plot $A_{CP,\ell}$ versus $m_{12}$. Genuine CPV appears as nonzero asymmetries within $[-1, +1]$, with artifacts due to denominator zeros eliminated.

This procedure directly generalizes to any heavy-hadron three- or four-body decay admitting a partial-wave expansion in some helicity angle (Qi et al., 16 Nov 2025).

4. Application to $B^\pm \to \pi^+\pi^-\pi^\pm$ near $\rho^0(1450)$

Qi et al. apply the quasi-normalized method to $B^\pm\to\pi^+\pi^-\pi^\pm$ decays, focusing on $$1.262\,\mathrm{GeV} < m(\pi\pi)_{\rm low} < 1.676\,\mathrm{GeV}$$ dominated by

• $\rho^0(1450)$ ($J^{PC}=1^{--}$, P-wave, $\ell=1$),
• $f_2(1270)$ ($2^{++}$, D-wave, $\ell=2$),
• $f_0(1500)$ ($0^{++}$, S-wave, $\ell=0$).

Resulting Legendre moments up to $\ell=4$ encode both diagonal and interference terms: $$\begin{align*} w_0 &= |M_{f_0}|^2 + (1/5)|M_{f_2}|^2 + (1/3)|M_\rho|^2, \ w_1 &= 2\,\mathrm{Re}[M_\rho M_{f_0}^*] + (4/5)\,\mathrm{Re}[M_\rho M_{f_2}^*], \ w_2 &= (2/3)|M_\rho|^2 + (2/7)|M_{f_2}|^2 + 2\,\mathrm{Re}[M_{f_0} M_{f_2}^*], \ w_3 &= (6/5)\,\mathrm{Re}[M_\rho M_{f_2}^*], \ w_4 &= (18/35)|M_{f_2}|^2. \end{align*}$$ Amplitudes $M_R^\pm(m)$ are parameterized as $$c_R^\pm BW_R(m) e^{i\delta_R^\pm } P_{\ell_R}(\cos\theta )$$, with $c, \delta$ fitted for both $B^+$ and $B^-$. Constructed quasi-normalized asymmetries for $\ell=1,2,3,4$ are observed to remain within $[-0.2,+0.2]$ across the relevant mass region, with smooth variation and no unphysical excursions. Conventional alternatives ($A_{CP,\ell}^{\rm conv}$, $\mathring{A}_{CP,\ell}$) exhibit pathologies—unbounded spikes near denominator zeros—which are not present in the quasi-normalized method.

5. Comparative Analysis of Normalization Schemes

Qi et al. provide direct side-by-side numerical and graphical comparison among four asymmetry definitions:

• $A_{CP,\ell}$ with $\eta_\ell = \eta_\ell^{(L)}$,
• $A_{CP,\ell}$ with $\eta_\ell = \tilde{\eta}_\ell$,
• $\hat{A}_{CP,\ell}$ (sign-weighted, bounded $\pm 1$),
• $\mathring{A}_{CP,\ell}$ (alternative unscaled).

Both quasi-normalized constructions (either $\eta_\ell^{(L)}$ or $\tilde{\eta}_\ell$) closely track the ideal sign-weighted asymmetry, while $\mathring{A}_{CP,\ell}$ deviates and fails to maintain proper scaling. This demonstrates that quasi-normalized PWCPAs simultaneously:

• are bounded in $[-1, +1]$ (or very close),
• preserve interference information (by construction from Legendre moments),
• and exhibit uniform, comparable statistical errors (with errors $\simeq 1/\sqrt{N + \bar{N}}$).

6. Implications for Amplitude Analysis and Generalization

The quasi-normalized PWCPA framework enables theoretically sound and statistically robust searches for CPV in individual partial waves, particularly relevant for current and future amplitude analyses in three- or four-body decays of heavy baryons and mesons. Unlike previous definitions that can yield unphysical or misleading signals, the quasi-normalized approach eliminates artifacts from denominator zeros and ensures reliable interpretation. A plausible implication is that amplitude analyses targeting resonance interference—such as those in $B$ and $\Lambda_b$ decays—will benefit from systematic adoption of quasi-normalized PWCPAs for both discovery and precision phases of CPV studies (Qi et al., 16 Nov 2025). This methodology is directly extendable wherever a partial-wave expansion in an appropriate helicity angle can be performed.