Remarks on strong phase shifts in weak nonleptonic baryon decays

Hai-Bo Li; Hong-Jian Wang; Jusak Tandean; Pei-Rong Li; Xiao-Rui Lyu

REVIEW 2 major objections 3 minor 25 references

Remarks on strong phase shifts in weak nonleptonic baryon decays

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proposes a single sign-parameterized formula, Eq. (10), that reproduces the strong phase shift $\delta_P-\delta_S$ under all three conventions used in the literature.

desk verdict Useful convention-compilation note whose central equation appears malformed as printed; fix the typo and it earns publication. read the letter →

arxiv 2412.02170 v2 pith:VEF6WURY submitted 2024-12-03 hep-ph hep-ex

Hong-Jian Wang , Pei-Rong Li , Xiao-Rui Lyu , Jusak Tandean , Hai-Bo Li This is my paper · ORCID

classification hep-phhep-ex

keywords strongphaseshiftnonleptonicbaryondecaysCPviolationhyperoncharmedpolarizationparametersarctanambiguitydecayasymmetry

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

This paper tries to establish that the strong phase shift $\delta_P-\delta_S$ in two-body nonleptonic baryon decays, a quantity that directly controls the size of CP-violating asymmetries, can be evaluated by a single unified formula regardless of which of three parameterization conventions a measurement uses. The three conventions are complex amplitudes with magnitudes $|S|$ and $|P|$, real amplitudes $e_S$ and $e_P$ that may be negative, and amplitudes written in terms of a mixing angle $\zeta$. The paper proposes Eq. (10), a modified arctangent formula containing a sign constant, and shows that it reproduces all three conventions. Applied to published $\Lambda$, $\Xi$, and $\Lambda_c$ data, the formula shows that opposite sign choices shift the reported phase shift by $\pi$ while leaving $\tan(\delta_P-\delta_S)$ unchanged. This gives a convention-independent way to combine phase-shift measurements from different experiments and to input them into future baryon CP-violation searches.

What carries the argument

The carrying object is the sign-parameterized arctangent formula, Eq. (10): $$\delta_P-\delta_S = 2\arctan\left(\frac{\$\beta$\,\mathrm{sign}}{\sqrt{\$alpha^{2}$+\$beta^{2}$}+\$\alpha$\,\mathrm{sign}}\right).$$ The sign constant encodes the convention-specific sign of the product of the S- and P-wave amplitudes, or the sign of $\sin(2\zeta)$, which is often not stated in experimental papers. The formula works because $\alpha$ is proportional to the cosine of the phase shift and $\beta$ to its sine, and the denominator $\sqrt{\alpha^2+\beta^2}+\alpha\,\mathrm{sign}$ selects the correct branch of the arctangent, returning an angle in $[-\pi,\pi]$ rather than only in the principal branch $[-\pi/2,\pi/2]$. One formula thus replaces the case-by-case branch corrections previously applied convention by convention.

What would settle it

Find a published baryon-decay analysis whose S- and P-wave amplitudes are defined with a phase convention that cannot be reduced to Eqs. (2), (6), or (7), or show that with a known sign the phase shift from Eq. (10) disagrees with the paper's reported value by anything other than an integer multiple of $\pi$; either result would disprove the claimed unification.

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Extended reading notes

Core claim

The central claim is that the strong phase shift $\delta_P-\delta_S$ in any two-body nonleptonic baryon decay can be extracted from the measured $\alpha$ and $\beta$ by $$\delta_P-\delta_S = 2\arctan\left(\frac{\$\beta$\,\mathrm{sign}}{\sqrt{\$alpha^{2}$+\$beta^{2}$}+\$\alpha$\,\mathrm{sign}}\right),$$ where 'sign' is a constant equal to either $+1$ or $-1$. For the first convention, $\mathrm{sign}=+1$ reduces the formula to the expression recently proposed in Ref. [10]; for the second convention, sign is the sign of the product $e_S e_P$ of the real amplitudes; for the third, it is the sign of $\sin(2\zeta)$. The paper shows numerically, across the experiments collected in its table, that switching sign changes $\delta_P-\delta_S$ by $\pi$ but leaves $\tan(\delta_P-\delta_S)$ exactly unchanged, so the tangent is convention-independent while the angle itself carries a $\pi$ ambiguity. It further points out that in one published $\Xi^-\to\Lambda\pi^-$ analysis the reported $(\alpha,\delta_P-\delta_S)$ pair is internally inconsistent once this branch structure is respected.

Load-bearing premise

The unification assumes that every publication's convention is captured by one of the three parameterizations in Eqs. (2), (6), and (7), and that the relevant sign can in principle be identified; a fourth convention or a sign ambiguity that is not binary would break the formula.

Editorial extensions

If this is right

The same measured pair $(\alpha,\beta)$ yields two possible phase shifts differing by $\pi$, and $\tan(\delta_P-\delta_S)$ is identical for both, so any global average that uses only the tangent is convention-independent.
Published $\Lambda$, $\Xi$, and $\Lambda_c^+$ data can all be re-expressed in Eq. (10), with the sign usually recoverable from the original papers, as the table demonstrates.
The E756 result on $\Xi^-\to\Lambda\pi^-$, which reported $\delta_P-\delta_S=0.06\pm0.09$ with $\alpha<0$ under the first convention, is internally inconsistent; applying Eq. (10) gives $-3.08\pm0.09$ rad for the same sign choice.
BESIII's two solutions for $\Lambda_c^+\to\Xi^0K^+$, $-1.55\pm0.25$ rad and $1.59\pm0.25$ rad, correspond exactly to the two sign choices, so the measured large phase shift persists under the unified form.
Future experimental papers on baryon weak decays should quote the sign or explicitly state the amplitude convention, so that phase-shift values can be combined without bookkeeping errors.

Reading between the lines

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

If the same sign-parameterized formula is applied to future measurements in $\Lambda_b$ and $\Xi_b$ decays, the convention ambiguity should appear in the same two-branch form, because the standard S- and P-wave partial-wave decomposition has the same structure for any $1/2^+\to 1/2^+ + 0^-$ decay.
The sign that resolves each measurement's branch can be compared with the sign of the product of S- and P-wave amplitudes predicted by a theoretical amplitude scheme, giving a consistency test that is independent of absolute phase conventions.
A practical rule suggested by the paper, but not stated by it, is that data tables should report $\alpha$, $\beta$, and the sign rather than a phase-shift value alone, since the latter is not convention-free.

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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

The one thing to know: this is a short conventions/compilation note, not a physics advance. The proposed unified formula is a one-line modification of Eq. (4) from Ref. [10], and the numerical table is a recompilation of published measurements. That is fine—it is a genuinely useful service—but set expectations accordingly.

What it does well: it collects the three common parameterizations of S and P amplitudes, shows how they differ by a sign, and provides a single formula with a 'sign' parameter. The table is handy: for each decay it lists α, β, ϕ, the original phase shift, and the value from the unified formula for both signs. It also correctly flags that E756's published δP−δS = 0.06±0.09 is inconsistent with its α<0 under convention (2); that is a real observation worth having. The π-shift property of the opposite sign choice is clearly explained, as is the fact that tan(δP−δS) is unaffected.

The soft spot is load-bearing and simple: the version of Eq. (10) I have in front of me reads 'δP−δS = 2 arctan β× signp α2 +β2 +α× sign' — that is, the square root in the denominator is missing (and the fraction bar is gone). Taken literally, the formula is wrong, and a reader cannot reproduce any row of the table. The table only works with the intended form '2 arctan[β·sign/(√(α²+β²)+α·sign)]'. The intended formula is standard and correct; the printed one is not. This must be fixed in the manuscript, and a referee should verify that the final typeset equation matches. The paper also asserts, without proof, that the three conventions in Eqs. (2),(6),(7) exhaust the literature; that is a minor gap, not a fatal one, but a sentence acknowledging possible other conventions would help.

Overall: the paper is a useful reference for anyone combining baryon-decay measurements or planning CP-violation scans at BESIII/LHCb. It deserves a serious referee and publication after the equation typo is fixed. I would not describe it as a major contribution to the theory of CP violation; it is a tool, and a mostly good one.

Referee Report

2 major / 3 minor

Summary. This manuscript reviews the conventions used in the literature for the strong phase shift δP−δS in two-body nonleptonic baryon decays, proposes Eq. (10) as a unified parameterization that accommodates three common amplitude parameterizations, and presents a table of phase shifts computed from published α, β, and ϕ values. The paper also discusses the impact of sign ambiguities on global averages and on CP-violation studies in baryon decays.

Significance. If the proposed formula and table are correct, the paper provides a compact and useful service to the baryon-CP community: it collects experimental results from different experiments, exposes the sign ambiguity, and offers a common language for comparing phase shifts. The underlying identity is standard, but the compilation and the explicit attention to the sign convention are useful. However, the central equation is not correctly displayed in the manuscript as provided, and the definition of the sign parameter is incomplete; both issues must be fixed before the paper can serve as a reliable reference.

major comments (2)

[Eqs. (4) and (10)] The displayed form of Eq. (10), and likewise Eq. (4), is incomplete: it reads as 'δP−δS = 2 arctan β× signp α2 +β2 +α× sign', with no square root and no fraction bar. Taken literally, this expression is not the claimed phase-shift formula and cannot reproduce the values in Table 1. For example, in the E756 row (α=−0.458, β=−0.03, sign=+1) the literal expression gives about 0.24 rad, whereas the table lists −3.08±0.09 rad; the intended identity with √(α²+β²) in the denominator gives −3.08 rad. Because Eq. (10) is the central result, please correct the typesetting of Eqs. (4) and (10) and re-verify the table entries with the intended formula.
[Definition of 'sign' in Eq. (10)] The constant 'sign' is introduced without an explicit mapping to the three conventions. For convention 2, α2 and β2 in Eq. (8) are proportional to eS eP, and for convention 3, α3 and β3 in Eq. (9) are proportional to sin(2ζ); in convention 1 the coefficient is positive. The paper should state that 'sign' equals the sign of the common coefficient, namely sign(2|S||P|/(|S|²+|P|²))=+1, sign(2eS eP/(|eS|²+|eP|²)), or sign(sin 2ζ), respectively. Without this identification, a reader cannot determine which value of sign to use for a given published analysis, and the table's 'unknown' entries are ambiguous between a convention-dependent and experimental source.

minor comments (3)

[Eq. (4)] The text attributes Eq. (4) to Ref. [10] as 'the new formula', but the formula as shown is garbled; the same typographical correction needed for Eq. (10) applies here.
[Table 1] The column heading 'Value of sign' should be clarified to indicate whether the sign is inferred from the amplitude convention or explicitly reported by the experiment; for rows with no entry the paper should state explicitly that the sign is unknown.
[General] A one-line derivation of Eq. (10) using the identity tan(Δ/2)=sinΔ/(1+cosΔ) would help readers see why the formula is exact and why the sign choice changes δP−δS by π while leaving tan(δP−δS) invariant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the unified phase-shift formula is a rearrangement of the standard α-β definitions, applied to external data.

full rationale

The paper makes no predictive claim that reduces to its own inputs. Equation (10) is an algebraic rearrangement of the defining relations α = 2|S||P| cos(δP−δS)/(|S|²+|P|²) and β = 2|S||P| sin(δP−δS)/(|S|²+|P|²), using tan(Δ/2) = sinΔ/(1+cosΔ); it is not fitted to data, and the 'sign' parameter is an explicitly declared convention choice rather than a fitted constant. The numerical table applies Eq. (10) to independently published α, β, and phase-shift values, and the agreement with previously reported phase shifts is a consistency check, not a derivation of those values. Citations to prior work (e.g., Refs. [10], [11], [13]) provide external experimental results or earlier formulas; none of the load-bearing steps is justified solely by a publication by the present authors. The paper also openly flags the practical limitation that the sign in some experimental conventions is unknown, which is an acknowledgement of ambiguity rather than a circular maneuver. No self-definitional step, fitted-input-as-prediction, or imported uniqueness theorem appears. A separate typographical presentation issue in Eq. (10) (missing radical or fraction bar in the extracted text) is a correctness concern, not a circularity concern.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper contributes a convention parameter (sign) and otherwise relies on standard partial-wave analysis and published experimental results. No new physical entities are introduced.

free parameters (1)

sign = +1 or -1 (chosen by convention, not fitted)
Introduced in Eq. (10) to absorb the ambiguity between parameterization conventions; its value is not determined by data in this paper but must be specified by each experiment.

assumptions (3)

domain assumption Lee-Yang partial-wave analysis: a spin-1/2 baryon decaying to spin-1/2 + 0− is fully described by S- and P-wave amplitudes.
Invoked in the preamble when defining α, β, γ in Eq. (1).
domain assumption The published experimental values of α, β, and ϕ in Table 1 are correct and mutually consistent.
The paper's recomputed phase shifts inherit the accuracy of the input measurements.
domain assumption CP conservation is assumed in the extraction of strong phases from data.
The phase shift is interpreted as the final-state strong interaction phase only if CP-violating weak phases are neglected.

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Cite this review

Pith. "Pith review of Remarks on strong phase shifts in weak nonleptonic baryon decays." pith.science (2026). https://pith.science/paper/VEF6WURY

@misc{pith2026241202170,
  author       = {Pith},
  title        = {Pith review of: Remarks on strong phase shifts in weak nonleptonic baryon decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEF6WURY}},
  note         = {Machine review of arXiv:2412.02170}
}

abstract

A sizable strong-interaction phase shift in weak two-body nonleptonic baryon decay would enhance the possibility of discovering charge-conjugation parity ($CP$) violation in the baryon sector, which might help in the quest for understanding the matter-antimatter asymmetry in the universe. Over the past 60 years, empirical analyses involving different types of instruments, including fixed-target experiments and $e^+e^-$ colliders, have indicated that the phase shifts in nonleptonic hyperon decays are relatively small, below order ten degrees in size. A large phase shift, however, has been observed by BESIII in the decay of a charmed baryon into a hyperon and kaon, $\Lambda_c^+\to \Xi^0K^+$. In various experimental and theoretical studies on hyperon, charmed-baryon, and bottomed-baryon decays, different conventions have been adopted for defining the strong phases. It is important to be aware of this situation when obtaining global averages from different measurements and applying the results to future investigations on $CP$ violation among baryons. This paper gives an overview of the conventions employed in the literature for the strong phases and suggests a unified parameterization form applicable to the different alternatives. Numerical results under the unified parameterization form are also provided, which can serve as useful inputs to further pursuits of baryon $CP$ violation.

Figures

Figures reproduced from arXiv: 2412.02170 by the authors.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [10]

Analysis of hadronic weak decays of charmed baryons in the topological diagrammatic approach, Phys Rev D 2024;109:114027

Zhong HL, Xu FR, Cheng HY . Analysis of hadronic weak decays of charmed baryons in the topological diagrammatic approach, Phys Rev D 2024;109:114027

work page 2024

[2] [1]

Sakharov AD, et al, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh Eksp Teor Fiz 1967;5:32-35

work page 1967

[3] [2]

Observation of direct CP vio- lation in KS,L→ππ decays, Phys Rev Lett 1999;83:22-27

KTeV Collaboration, Alavi-Harati A, et al. Observation of direct CP vio- lation in KS,L→ππ decays, Phys Rev Lett 1999;83:22-27

work page 1999

[4] [3]

Observation of CP violation in charm decays, Phys Rev Lett 2019;122:211803

LHCb Collaboration, Aaij R, et al. Observation of CP violation in charm decays, Phys Rev Lett 2019;122:211803

work page 2019

[5] [4]

Observation of several sources of CP violation in B+→π+π+π− decays, Phys Rev Lett 2020;124:031801

LHCb Collaboration, Aaij R, et al. Observation of several sources of CP violation in B+→π+π+π− decays, Phys Rev Lett 2020;124:031801

work page 2020

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LHCb Collaboration, Aaij R, et al. Measurement of matter-antimatter dif- ferences in beauty baryon decays, Nature Phys 2017;13:391-396

work page 2017

[7] [6]

Study of Λ0 b and Ξ0 b decays to Λh+h′− and evidence for CP violation in Λ0 b→ ΛK+K− decays, arXiv: 2411.15441, 2024

LHCb Collaboration, Aaij R, et al. Study of Λ0 b and Ξ0 b decays to Λh+h′− and evidence for CP violation in Λ0 b→ ΛK+K− decays, arXiv: 2411.15441, 2024

work page arXiv 2024

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Sch ¨onning K, Batozskaya V , Adlarson P, et al. Production and decay of polarized hyperon-antihyperon pairs, Chin Phys C 2023;47:052002

work page 2023