{"id":"0ae5016b-ab0f-4ee3-b20e-52e267d1bf50","arxiv_id":"2411.18323","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Time-reversal-odd CP observables in baryon decays are proportional to the cosine of strong-phase differences, so they complement sine-suppressed direct CP asymmetries.","lead":"This paper derives general rules for CP-violating observables in baryon decays, showing that observables built from spin and momentum correlations can stay large even when strong phases are small. It provides ready-to-use angular distribution formulas for several bottom-baryon decay channels that LHCb, Belle II, and BESIII can use to search for baryonic matter-antimatter asymmetry.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. 4.2 asserts the cosine dependence for T-odd CP asymmetries without deriving it, and the assignment of definitions (2.25)/(2.26) to parity-odd/even operators conflicts with the explicit examples; the general proof needs completion.","rationale":"The paper's most important theoretical contribution is the exact statement that T-odd CP asymmetries are proportional to cos Δδ under conditions (i) and (ii). The proof in Sec. 4.2 shows only the imaginary-part structure of the expectation value; the decisive step to cosine dependence is asserted rather than derived. This is not merely a typographical issue: the text's assignment of definition (2.25) to the parity-odd case is contradicted by the paper's own explicit calculations, where β and sin φ_R (both T-odd and parity-odd) use the sum definition to obtain cos Δδ. A reader who follows Sec. 4.2 literally would derive a different dependence for β. The general theorem may well be true—the worked examples in Secs. 4.3 and 4.4 are convincing—but the manuscript does not yet contain a complete derivation. The reader's verdict correctly noted skipped algebra and reliance on prior work, but I regard the missing proof as load-bearing because the central claim is precisely this theorem. The paper is also honest about the condition (i)/(ii) limitation for four-body momentum triple products, which the reader identified; I agree that this limits practical reach, but it is a clearly stated caveat rather than a hidden flaw. The appropriate verdict is therefore CONDITIONAL: the scientific content is sound in the explicit examples, but the general proof and sign conventions must be completed and corrected before the central claim can be accepted as proven.","tokens_in":64969,"tokens_out":31559,"duration_ms":258619,"concrete_test":"Complete the general derivation: expand an arbitrary T-odd operator satisfying conditions (i) and (ii) into partial-wave interference terms, apply the CP transformation of Eq. (2.6) (S̄ = -S(ϕ→-ϕ), P̄ = +P(ϕ→-ϕ)), and compute the CP asymmetry for both sign choices in (2.25)/(2.26). Verify that, for each parity class, the definition that cancels weak-phase-independent terms yields only terms of the form cos(δ_i - δ_j) sin Δϕ and no terms of the form sin(δ_i - δ_j) cos Δϕ. As a specific cross-check, re-derive Eq. (4.107) for a_{T2}^{CP} using the general prescription and confirm that no sine-type strong-phase terms survive; this would test whether the asserted theorem holds beyond the worked examples.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim of Sec. 4.2 is that any T-odd CP asymmetry satisfying conditions (i) and (ii) is proportional to cos Δδ. The proof establishes only that ⟨O−⟩ ∝ Σ Im(h_{λ'}h_λ^*) (Eq. 4.24). The subsequent statement that 'one can demonstrate' the cosine dependence is not shown; this is the load-bearing step. The cosine dependence follows only after combining ⟨O−⟩ with its CP conjugate, which requires the CP transformation of the amplitudes. The text does not supply this general derivation, and its sign convention is internally inconsistent: Sec. 4.2 assigns the parity-odd case to definition (2.25) (the difference), but the explicit parity-odd examples β (Eq. 3.60) and sin φ_R (Eq. 4.54) both use the sum (2.26) to obtain cos Δδ. Following the text literally would produce sin Δδ cos Δϕ for β, contradicting Eq. (3.61). The proof can likely be completed by expanding into partial waves and using Eq. (2.6), but as written the general theorem is not established. The paper's own limitation in Sec. 4.4.2 (four-body momentum triple products do not satisfy conditions (i)(ii)) further narrows applicability, so the cosine-dependence result rests entirely on the unproven general step plus the worked examples.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":65264,"tokens_out":5249,"duration_ms":50519,"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":[{"comment":"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.","section":"Sec. 4.2, Eq. (4.24) and the following paragraph"},{"comment":"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.","section":"Sec. 2.3, Eqs. (2.25)-(2.26), and Secs. 3.5, 4.4.1"},{"comment":"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.","section":"Sec. 4.3.2, Eqs. (4.36)-(4.39)"},{"comment":"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.","section":"Sec. 4.4.2 and Abstract"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Sec. 3.5, Eqs. (3.60) and (3.85)"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Sec. 4.1, Eq. (4.5)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a broad review with an original central claim, but the novelty boundary with the authors' earlier works (refs. [56, 70, 107]) and with the work of Geng and Liu [115] should be made explicit. The main theorem is plausible and likely completeable, but the skipped steps in Secs. 4.2 and 4.3.2 and the sign-convention inconsistency in Secs. 2.3 and 3.5 are load-bearing. If the proof cannot be completed, the paper should be reframed as a pedagogical review with worked examples, and the claim of an exact proof should be softened. The scope is appropriate for a journal in this field, though the absence of numerical predictions means the acceptance case rests on the theoretical framework and the experimental outlook."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWhere 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":65827,"tokens_out":2421,"would_cite":false,"duration_ms":26597,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["CP violation","baryon decays","T-odd correlations","T-even correlations","triple products","asymmetry parameters","strong phase differences","b-baryons"],"falsifier":"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.","tokens_in":64719,"feed_emoji":"⚛️","tokens_out":10542,"duration_ms":87437,"temperature":0.7,"pith_summary":"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.","feed_headline":"T-odd CP asymmetries follow cosine, not sine, of the strong phase","feed_subtitle":"T-odd and T-even partners give cosine and sine strong-phase dependencies, so one survives small strong phases.","key_machinery":"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.","core_discovery":"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 $\\Lambda_b \\to p\\,a_1(1260)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the eigenstate analysis of T-odd operators and the sin phi / sin 2 phi correlations reused in the proof and in the angular distributions.","marker":"[115]"},{"why":"Provides the b-baryon T-odd correlation construction and triple-product asymmetries that the paper extends into complementary pairs.","marker":"[71]"},{"why":"Defines the CP, P and T transformation properties of triple-product correlations used to identify the T-odd observables.","marker":"[68]"},{"why":"Gives the angular distribution of B to V1 V2 with T-odd sin phi terms that the paper adapts to baryon quasi-two-body decays.","marker":"[109]"},{"why":"Provides the Jacob-Wick helicity formalism underlying all the angular distributions and the helicity-basis part of the proof.","marker":"[72]"},{"why":"Supplies prior arguments that T-odd correlation-induced CP asymmetries depend on cos Delta-delta, which the paper turns into an exact statement.","marker":"[107]"},{"why":"Supplies the asymmetry-parameter definitions for alpha, beta, gamma CP asymmetries and the observation that the beta-over-alpha ratio has cosine strong-phase dependence.","marker":"[111]"},{"why":"Gives the general amplitude forms for spin-1/2 and spin-3/2 baryon decays used to express the helicity amplitudes explicitly.","marker":"[131]"}],"fun_headline_variants":["T-odd CP asymmetry follows cosine of strong phase, not sine","Complementary baryon CP observables dodge small strong-phase suppression","Baryon T-odd CP asymmetry: cos strong phase, T-even: sin","Strong-phase complementarity rescues baryon CP violation from small phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["T-odd CP asymmetry follows cosine of strong phase, not sine","Complementary baryon CP observables dodge small strong-phase suppression","Baryon T-odd CP asymmetry: cos strong phase, T-even: sin","Strong-phase complementarity rescues baryon CP violation from small phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001418,"raw_usage":{"total_tokens":5831,"prompt_tokens":1160,"completion_tokens":4671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":776,"completion_tokens_details":{"reasoning_tokens":4592}},"tokens_in":776,"tokens_out":4671,"duration_ms":32140,"temperature":1.0,"reasoning_tokens":4592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:18:48.117032+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Searching for possible evidences of new physics in $B\\to V_1 V_2$","cited_arxiv_id":"2106.10628","evidence_quote":"Supplies the eigenstate analysis of T-odd operators and the sin phi / sin 2 phi correlations reused in the proof and in the angular distributions."},{"cited_title":"Time-reversal asymmetries and angular distributions in $\\Lambda_b \\to \\Lambda V$","cited_arxiv_id":"2109.09524","evidence_quote":"Provides the b-baryon T-odd correlation construction and triple-product asymmetries that the paper extends into complementary pairs."}],"review_version":1}