CP violation observables in baryon decays

Fu-Sheng Yu; Jian-Peng Wang; Qin Qin

REVIEW 4 major objections 4 minor 4 cited by

CP violation observables in baryon decays

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that CP asymmetries built from T-odd spin-momentum correlations in baryon decays depend on the cosine of the strong phase difference, while T-even ones depend on the sine, and gives the exact conditions for this…

desk verdict A useful but uneven baryon-CPV paper: the explicit angular templates are worth referee time, but the advertised general proof of cosine strong-phase dependence is asserted rather than shown and the sign conventions are inconsistent. read the letter →

arxiv 2411.18323 v1 pith:QLF43EA5 submitted 2024-11-27 hep-ph hep-ex

Jian-Peng Wang , Qin Qin , Fu-Sheng Yu This is my paper · ORCID

classification hep-phhep-ex

keywords CPviolationbaryondecaysT-oddcorrelationsT-eventripleproductsasymmetryparametersstrongphasedifferencesb-baryons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Baryon CP violation has not yet been confirmed, and the usual direct CP asymmetry is suppressed when the strong phase difference $\Delta\delta$ is small. This paper tries to establish that CP-violating observables built from time-reversal-odd (T-odd) spin-momentum correlations instead scale as $\cos \Delta\delta$, so they survive small strong phases, while their T-even partners scale as $\sin \Delta\delta$. It states and proves the two conditions under which this cosine law is exact, and it gives criteria for when a T-odd observable and a T-even observable built from the same interference term form a complementary pair. The paper then constructs explicit complementary observables for $b$-baryon decays such as $\Lambda_b \to N^*(3/2^\pm) V$ and $\Lambda_b \to p\,a_1(1260)$ using helicity angular distributions. A sympathetic reader would care because these observables offer a way to search for baryonic CP violation without depending on large strong phases.

What carries the argument

The central object is the T-odd correlation, a rotational invariant built from spin and momentum vectors, for example a triple product $(\vec{v}_1 \times \vec{v}_2)\cdot \vec{v}_3$, whose expectation value changes sign under time reversal and is even under CP when compared with its CP conjugate. The proof uses the anti-unitarity of time reversal to show that the matrix elements of any T-odd operator in a helicity basis are purely imaginary, so the asymmetry isolates the imaginary part of an interference term between partial waves or helicity amplitudes. The two conditions in Sec. 4.2 guarantee that this imaginary part carries $\cos \Delta\delta$, while the T-even observable that takes the real part of the same interference term carries $\sin \Delta\delta$. This real/imaginary split of a single interference term is the mechanism that makes the two CP asymmetries exactly complementary.

What would settle it

Measure a T-odd CP asymmetry such as $a(\sin \varphi_R)$ in $\Lambda_b \to N^*(3/2) \rho$ using the sign-weighted angular binning of Eq. (4.83), in a channel where the strong phase difference $\Delta\delta$ is independently known to be small; if the asymmetry tracks $\sin \Delta\delta$ instead of $\cos \Delta\delta$, the theorem is falsified.

Watch

Extended reading notes

Core claim

The central claim is a strong-phase theorem: any CP asymmetry defined from a T-odd correlation through the paper's equations (2.25) and (2.26) is proportional to $\cos \Delta\delta$ (multiplied by $\sin \Delta\phi$), provided two conditions hold: there is a unitary $U$ with $U\mathcal{T}|\psi_n\rangle = e^{i\alpha}|\psi_n\rangle$ for the chosen final-state basis, and $U Q_- U^\dagger = Q_-$ for the T-odd operator $Q_-$. The paper proves this from the anti-unitarity of time reversal, which forces the matrix elements of $Q_-$ to be purely imaginary. The companion T-even observable from the real part of the same interference term then depends on $\sin \Delta\delta$, so the pair is exactly complementary. The proof is carried out in both partial-wave and helicity languages, and the general criteria are applied to build concrete angular-distribution observables for $b$-baryon quasi-two-body and quasi-three-body decays, including the $\sin \varphi_R$, $\sin 2\varphi_R$, $\cos \varphi_R$ and $\cos 2\varphi_R$ asymmetries in $\Lambda_b \to N^*(3/2^\pm) V$ and the $T_2$, $T_2^P$, $B_2$, $B_2^P$ observables in $\Lambda_b \to p\,a_1(1260)$.

Load-bearing premise

The load-bearing premise is that the final-state basis and the T-odd operator can be arranged so that time reversal is undone by a single unitary transformation that leaves the operator's form intact, a condition the authors themselves say fails for pure momentum triple products in four-body decays, where the cosine dependence is therefore not guaranteed.

Editorial extensions

If this is right

A baryon decay with a small strong phase difference can still show observable CP violation through its T-odd correlation, because that asymmetry is not suppressed by $\sin \Delta\delta$.
Once a T-odd asymmetry satisfying the conditions is found, the same angular distribution contains a T-even partner from the same interference term, giving an internal cross-check of the CP violation.
In quasi-two-body channels such as $\Lambda_b \to N^*(3/2^\pm) V$, the terms $\sin \varphi_R$ and $\sin 2\varphi_R$ are T-odd while $\cos \varphi_R$ and $\cos 2\varphi_R$ are T-even, and the paper's criterion makes them complementary pairs.
The proposed redefinition in Eq. (3.85) keeps these CP asymmetries inside $[-1,1]$ and removes the artificial enhancement from small denominators, so experimental limits remain meaningful.
Experiments should bin with sign-weighted functions such as $\mathrm{Sign}(\sin 2\theta_L \sin 2\theta_R)\, \sin \varphi_R$ rather than simple triple-product cuts, because the $\sin \varphi_R$ term integrates to zero over naive phase-space bins.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to apply the same complementarity test to charmed-baryon decays, where strong phases are predicted to be very small; if the two conditions can be met there, T-odd observables would outperform direct CP asymmetries by a large factor.
The two conditions give a practical classifier: spin-correlation-based T-odd observables that satisfy them should be prioritized in experimental searches, while pure momentum triple products in four-body decays, which the paper notes do not satisfy them, should be treated with more caution.
The complementarity is stated at the level of interference terms and helicity amplitudes, so a similar real/imaginary pairing could be hunted in four-body $B$-meson decays with intermediate resonances, where triple-product asymmetries are already measured.
A concrete test of the paper's claim could be made with an amplitude-level Monte Carlo of $\Lambda_b \to N^* \rho$ using toy strong phases: fitting the $\sin \varphi_R$ and $\cos \varphi_R$ coefficients should return $\cos \Delta\delta$ and $\sin \Delta\delta$ with equal magnitudes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful, sometimes messy phenomenological paper on baryonic CP observables. The genuinely new pieces are the explicit angular-distribution templates for Λb → N*(3/2±)V, Λb → p a1, and Λb → Δ++π−π−, plus the bounded CP-asymmetry definitions in Eqs. (3.85)–(3.87). Those templates are concrete and likely to be used by LHCb-oriented colleagues. The core complementarity claim — T-odd CP asymmetries go as cos Δδ, T-even as sin Δδ — is not new in full generality (Donoghue-He-Pakvasa and the authors' own earlier preprint have it), but the paper does more: it states explicit criteria and carries the claim through many helicity amplitudes.

The soft spots are real. The general proof in Sec. 4.2 is the main one. It establishes ⟨O−⟩ ∝ Im(h_{λ'} h_λ^*) and then says "one can demonstrate" the cosine dependence. That step is exactly the claim needing proof, and it is not supplied in general. On top of that, the assignment of definitions (2.25) vs (2.26) to parity-odd/even cases is inconsistent with the worked β and sin φ_R examples; following the text literally gives the wrong phase dependence for β. The stress-test note lands. In practice, the paper's own examples carry the argument: the explicit helicity manipulations in Secs. 4.3–4.4 do produce cos Δδ for the sin φ and sin 2φ observables, so I do not think the theorem is false — it is unproven and mis-stated as written. That is fixable in revision, but the general claim should be softened or completed. Second, the manuscript is dense with typos, duplicated sentences, and notation glitches; the journal version needs a serious editing pass. Third, self-citation is heavy but not disqualifying: ref. [107] is the companion preprint and the overlap should be disclosed, not hidden.

Where I depart from the reader: I would not call the derivation "mostly self-contained" given the skipped algebra in (4.36)–(4.39) and the sketched criterion-1 proof. But the practical content is solid enough to merit a referee. The paper will be useful to experimentalists and to theorists building angular analyses; it does not need to be right in every formal statement to be worth engaging.

Recommendation: send to referees, with instruction that the general proof in Sec. 4.2 and the sign conventions in (2.25)/(2.26) be fixed or explicitly scoped to the worked examples.

Referee Report

4 major / 4 minor

Summary. This paper presents a systematic framework for CP-violating observables in baryon decays, concentrating on b-baryon two- and three-body decay modes. It reviews the Lee-Yang asymmetry parameters and their CP counterparts, develops the helicity formalism for angular distributions, and proposes that T-odd and T-even correlation observables form complementary pairs: the T-odd CP asymmetry is claimed to depend on the strong phase as cos Δδ, while the T-even counterpart depends on sin Δδ. The central theoretical assertion is formulated in Sec. 4.2 as a theorem subject to two conditions and is then applied to explicit quasi-two-body and quasi-three-body decay chains such as Λ_b → N*(3/2)V and Λ_b → p a_1. The paper also contains a pedagogical appendix on Jacob-Wick helicity techniques and extensive angular distributions intended for LHCb, Belle II, and BESIII analyses.

Significance. If the cosine-dependence theorem is fully established, the paper provides a valuable strategy for baryonic CP searches: it shows how to construct observables that are not suppressed by small strong phases. The explicit helicity-amplitude expansions in Sec. 4.4, the angular distributions in Eqs. (4.46), (4.89), (4.115), and the appendix are detailed and potentially useful for future experimental analyses. The paper is self-contained in the sense that the main relations are derived from the amplitude decomposition of Eq. (2.6) rather than imported from an external calculation, and no numerical fits are introduced. However, the significance depends on completing the proof of the central theorem and resolving the sign-convention inconsistencies; as written, the advertised exact proof is partly asserted, and several explicit examples use a convention that differs from the one stated in the theorem.

major comments (4)

[Sec. 4.2, Eq. (4.24) and the following paragraph] The central theorem states that a T-odd CP asymmetry satisfying conditions (i) and (ii) is proportional to cos Δδ. The derivation as written stops at ⟨O−⟩ ∝ Σ Im(h_{λ'}h_λ^*) in Eq. (4.24); the sentence ‘one can demonstrate’ that the CP asymmetry defined via (2.25) or (2.26) is proportional to cos Δδ is not a proof. This step is load-bearing: one must combine ⟨O−⟩ with the CP-conjugate expectation and use the explicit CP transformation of the partial-wave amplitudes, e.g. Eq. (2.6), to isolate sin Δφ and cos Δδ. Please supply the full derivation, or alternatively state the theorem as two lemmas with proofs and restrict the claim to those cases.
[Sec. 2.3, Eqs. (2.25)-(2.26), and Secs. 3.5, 4.4.1] The assignment of definitions (2.25) and (2.26) to parity-even and parity-odd T-odd operators is inconsistent with the worked examples. The text says that the operator O− exhibits parity-even and parity-odd properties in a_CP,1 and a_CP,2 respectively, but the explicitly P-odd T-odd parameter β is defined with the sum convention in Eq. (3.60), aβ_CP ∝ Im(S*P)+Im(̅S*̅P), and the same plus sign is used for the P-odd sin φ_R observable in Eq. (4.54). If Eq. (2.25) were applied literally to a P-odd operator, the result would not agree with Eq. (3.61). Fix a single convention, introduce a sign variable that distinguishes the P-even and P-odd cases, and rederive Eqs. (3.60)-(3.61), (4.54)-(4.55), and (4.58)-(4.59) consistently.
[Sec. 4.3.2, Eqs. (4.36)-(4.39)] The proof of Criterion 2 relies on the cancellation of weak-phase-independent terms when passing from Eq. (4.36) to Eq. (4.37). The text states that this follows from relations similar to Eq. (4.35), but the cancellation is not shown explicitly. This is precisely the mechanism that isolates the sin Δφ factor and the sine/cosine strong-phase factors, so the derivation should be completed. In addition, the pair in Eq. (4.40) is asserted to be ‘also completely complementary’ without a derivation; either provide the calculation or state explicitly that the proof is identical to the parity-even case.
[Sec. 4.4.2 and Abstract] The paper correctly notes that pure momentum triple products in four-body decays do not simultaneously satisfy conditions (i) and (ii), so the general cosine theorem does not apply there. Nevertheless, the abstract and Sec. 1.2 present the cosine dependence as a general property of T-odd correlations. Please state the restriction explicitly in the abstract and in the introduction, and clarify which of the proposed observables in Sec. 4.4 satisfy the theorem rather than merely sharing a formal similarity with β-like observables.

minor comments (4)

[Throughout] There are numerous typos and grammatical issues, e.g. ‘establised’ in the abstract, ‘oftem’ in Sec. 2.2, ‘hat are’ in Sec. 5, ‘assin’ in Sec. 2.1, and inconsistent notation between ‘φ_R’ and ‘ϕ_R’ in Sec. 4.4 and Fig. 7. A careful proofreading pass is needed.
[Sec. 3.5, Eqs. (3.60) and (3.85)] The definitions in Eq. (3.60) are written with ‘∝’ only, and the denominators are not specified precisely; the later redefinition in Sec. 3.5 is useful but the paper should state that the phenomenological formulas in Secs. 3.5 and 4.4 use the older convention. This would help readers avoid assigning the wrong normalization to aβ_CP and aO_CP.
[References] The reference list contains duplicates: [57] and [103] are the same Belle paper, and [70] and [107] are also the same work. The list should be pruned and cross-referenced carefully.
[Sec. 4.1, Eq. (4.5)] The transformation properties in Eq. (4.5) mix the momentum vectors of the decay and of the CP-conjugate process without defining the action of C, P, and T on each momentum in a common convention. Clarifying this would remove possible ambiguity in the subsequent derivation of aψ_CP.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cosine-dependence claim is derived from the amplitude parameterization and CP-conjugation relations, not from the conclusion itself.

full rationale

The central derivation chain is self-contained. The input is the tree+penguin decomposition with weak and strong phases (Eq. (2.6)), together with the CP-conjugate amplitudes. The general expectation value of a T-odd operator is reduced to Im(h_{lambda'} h_lambda^*) in Eq. (4.24) using the anti-unitary property of time reversal, rotational invariance and hermiticity. The subsequent claim that the CP asymmetry is proportional to cos Delta-delta is not a fit or an input; it is the result of combining this imaginary interference term with the CP-conjugate amplitudes under the appropriate sign in definitions (2.25)/(2.26). Explicit algebra carrying out this step appears in Sec. 4.3.2 (Eqs. (4.37)-(4.39)) and Sec. 4.4.1 (Eqs. (4.72)-(4.73)), where the tree/penguin decomposition is inserted and the sin Delta-phi factor and cos strong-phase factors are exhibited. No parameter is fitted to data and then renamed a prediction; no uniqueness theorem is imported from the authors' earlier work; the self-citations (e.g., refs. [56,59,66,70,92,100,107]) occur in review/context passages and the relevant formulas are re-derived rather than assumed. Two caveats are noted but they are correctness issues, not circularity: (i) Sec. 4.2 states 'One can demonstrate' the cosine dependence for the general theorem without displaying the demonstration, and (ii) the assignment of the parity-odd case to definition (2.25) conflicts with the explicit beta and sin phi_R examples, which use the sum form (2.26). The paper itself also restricts the theorem in Sec. 4.4.2, where pure four-body momentum triple products do not satisfy conditions (i)-(ii). These points weaken the proof's exposition but do not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard quantum field theory (CPT, helicity formalism, parity conservation in strong decays) and on two domain assumptions: the tree+penguin amplitude decomposition, and the two conditions in Sec. 4.2 that are required for the cosine theorem.

assumptions (5)

domain assumption Tree-plus-penguin decomposition of amplitudes with one weak-phase difference and process-dependent strong phases
Used in Eqs. (2.6), (3.61), (4.25), (4.35) and throughout to derive the sine/cosine dependence. It is standard phenomenology but not derived in the paper.
ad hoc to paper Conditions (i) and (ii) of Sec. 4.2: unitary U with U T |ψn⟩ = e^{iα}|ψn⟩, and U Q- U† = Q-
These conditions are required for the proof that T-odd CP asymmetries vary as cos Δδ. They are process-dependent and the authors show they fail for momentum triple products in four-body decays.
standard math Rotational invariance, hermiticity, and SO(3) scalar property of T-odd operators
Used in deriving Eq. (4.24) and the angular-distribution formulas.
domain assumption Parity conservation in strong decays (e.g., |h_{λ}| = |h_{-λ}|)
Used in simplifying angular distributions in Sec. 4.4 and Appendix A.
domain assumption CPT conservation and the interpretation of differences of conjugate-process asymmetries as CP violation
Basis for definitions (2.24)-(2.26).

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Cite this review

Pith. "Pith review of CP violation observables in baryon decays." pith.science (2026). https://pith.science/paper/QLF43EA5

@misc{pith2026241118323,
  author       = {Pith},
  title        = {Pith review of: CP violation observables in baryon decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLF43EA5}},
  note         = {Machine review of arXiv:2411.18323}
}

abstract

The era of baryon physics is on the horizon with the accumulation of increasing data by collaborations such as LHCb, Belle II, and BESIII. Despite the wealth of data, one of the critical issues in flavor physics, namely CP violation in baryon decays, still awaits experimental confirmation. It is evident that the development of formulas and phenomenological analyses will play a pivotal role in advancing theoretical investigations and experimental measurements in this domain. In this work, we discuss the fundamental definitions and properties of polarization, asymmetry parameters and their associated CP violating observables that may arise in baryon decays. Additionally, we provide a detailed analysis of b-baryon quasi-two and -three body decays based on angular distributions. We highlight the significance of CP asymmetries defined by spin and momentum correlations, as non-trivial spin distributions are expected in baryon decays due to their non-zero polarization in productions and decays. Of particular importance is a complementary relation can be establised for two different types of observables, that are $\mathcal{T}$-even and -odd CP asymmetries respectively. The complementarity indicates that one observable exhibits the sine dependence on strong phase $\Delta\delta$, while another exhibits as cosine. This gives us an distinctive opportunity to avoid the potential large suppression to CP asymmetries from small phase $\Delta\delta$. Two points (\textbf{i}) the exact proof of strong phase dependence behaviour, and (\textbf{ii}) the criterion for complementary observables are clarified. Based on these arguments from theoretical aside, some brief discussions and suggestions on experimental measurements are offered in the final discussions. Lastly, we present a pedagogical introduction to angular distribution techniques based on helicity descriptions for both two and three body decays.

Figures

Figures reproduced from arXiv: 2411.18323 by the authors.

**Figure 2.** (a) A general case of a polarized hyperon decay with [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** The kinematics of three body decay can be represented by a closed triangle in the frame of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The depicted figures of angular distributions of [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: The kinematics of three body decay modes of Λ [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: Decay planes in the rest frame of mother particle. [PITH_FULL_IMAGE:figures/full_fig_p044_6.png]

**Figure 7.** Figure 7: The quadrant diagram of (sin ϕR,sin 2ϕR). Asymmetry parameters A(sin φR) and A(sin 2φR) are therefore extracted by following two observation A(sin 2φR) = Γ(sin 2φR > 0) − Γ(sin 2φR < 0) Γ(sin 2φR > 0) + Γ(sin 2φR < 0) (4.82) A(sin φR) = Γ[Sign(sin 2θL sin 2θR) sin φR >…

**Figure 8.** Figure 8: The signs of sin 2θL sin 2θR. From the angular distributions of Λb → N∗ρ or N∗K∗ in Eq.(4.46), it can be seen that the term of sin φR vanishes with the integration of θL and θR. That might be a reason why the current measurements of triple-product asymmetries in Λb dec…

**Figure 9.** Figure 9: The kinematic diagram for the scattering process [PITH_FULL_IMAGE:figures/full_fig_p053_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Revisiting $\bar B^0 \rightarrow \Lambda_c^+ \bar p$ decay with higher twist corrections
hep-ph 2026-07 accept novelty 6.0 of 10

Higher-twist LCDAs plus W-exchange produce destructive interference that lowers the PQCD branching fraction of B-bar0 to Lambda_c+ p-bar to ~1.6e-5, matching experiment, while predicting the suppressed mode at order 10^{-8}.
$CP$ asymmetries in the $\Lambda_c^+\to pK^0_S$ and $\Xi^+_c\to \Sigma^+K^0_S$ decays
hep-ph 2025-09 conditional novelty 5.0 of 10

The Xi_c+ to Sigma+ K0_S decay can show an interference-induced CP asymmetry up to about 10^-3, several times larger than in D meson decays.
Charmful two-body $\Omega_b$ decays in the light-front quark model
hep-ph 2024-12 conditional novelty 5.0 of 10

Predicts branching fractions for charmful Omega_b decays up to 10^-4, with external W-emission channels as the most stable targets for LHCb.
CP violation studies at Super Tau-Charm Facility
hep-ex 2025-02 conditional novelty 4.0 of 10

A future Super Tau-Charm Facility is projected to test CP violation in hyperon, tau, and charm decays at 1e-4 to 1e-3 sensitivities and improve the kaon CPT mass-difference limit tenfold.

Reference graph

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