arxiv: 2605.09682 · v1 · submitted 2026-05-10 · ✦ hep-ph · hep-ex

Recognition: 2 theorem links

· Lean Theorem

Spin-flavor entanglement in Λ_b to Λ D and weak phase extraction

Chao-Qiang Geng, Chia-Wei Liu, Sheng-Lin Liu, Xiao-Gang He, Xin-Yi Liu, Yong Du

Pith reviewed 2026-05-12 02:45 UTC · model grok-4.3

classification ✦ hep-ph hep-ex

keywords spin-flavor entanglementLambda_b decaysweak phase gammaCKM matrixWootters concurrenceLee-Yang parametersneutral D mesonsbaryon decays

0 comments

The pith

A spin-flavor entanglement in Lambda_b to Lambda D decays extracts the CKM weak phase gamma with uncertainty scaling as one over the Wootters concurrence.

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

The paper identifies a correlation between the spin of the Lambda baryon and the flavor of the accompanying D meson in Lambda_b decays. This spin-flavor entanglement is encoded in the decay rates and Lee-Yang parameters measured in the four neutral D modes. The authors demonstrate that the same observables allow extraction of the weak phase gamma from the CKM matrix. The experimental uncertainty on gamma improves in direct proportion to the strength of the entanglement, quantified by its Wootters concurrence.

Core claim

In Lambda_b to Lambda D decays, the correlation between Lambda spin and the neutral D flavor states (D zero, anti-D zero, D1, D2) forms a spin-flavor entangled system. The entanglement information resides in the decay rates and Lee-Yang parameters of these four modes. This structure supplies a new method to determine the weak phase gamma whose uncertainty scales as sigma gamma proportional to one over the Wootters concurrence C.

What carries the argument

The spin-flavor entanglement structure defined by the correlation between Lambda spin and neutral D flavor states, with its information content stored in the decay rates and Lee-Yang parameters of the four modes, which connects to gamma extraction through the Wootters concurrence.

If this is right

The four neutral D modes together determine both the amount of entanglement and the value of gamma.
Higher concurrence directly yields smaller experimental uncertainty on gamma.
The method relies on the same set of observables for both entanglement quantification and phase extraction.
This baryonic channel provides an independent route to gamma that does not rely on charged B meson decays.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the concurrence can be measured separately, it could serve as an internal consistency check on the extracted gamma value.
Analogous spin-flavor correlations may appear in other heavy baryon decays and could be used to probe CP violation in those systems.
Existing LHCb data on Lambda_b production could be reanalyzed to search for the predicted entanglement pattern in the D modes.

Load-bearing premise

The decay rates and Lee-Yang parameters extracted from the four neutral D modes can be obtained without dominant backgrounds, final-state interactions, or systematics that would invalidate the direct scaling between concurrence and gamma uncertainty.

What would settle it

A measurement of the four decay modes in which the observed uncertainty on gamma fails to follow the predicted inverse scaling with the concurrence computed from the same rates and parameters, or in which the four-mode observables prove inconsistent with the expected entangled structure.

Figures

Figures reproduced from arXiv: 2605.09682 by Chao-Qiang Geng, Chia-Wei Liu, Sheng-Lin Liu, Xiao-Gang He, Xin-Yi Liu, Yong Du.

**Figure 2.** Figure 2: Relative-density distribution of (σγ) −1 versus the spin–flavor concurrence C, with (σγ) −1 given in rad−1 . The lower y cutoff reflects σγ ≤ π/2. each input concurrence C, we generate random S- and P-wave amplitudes at fixed C, from which the partial widths and Lee–Yang parameters of the four neutral-D modes are obtained. For the full Run-3 data set of LHCb, we take [35–38] Ntot = NΛb Br(Λ0 b → ΛD) ≃ 3.6… view at source ↗

**Figure 3.** Figure 3: Theoretical prediction of the concurrence [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

We identify a new spin-flavor entanglement structure in $\Lambda_b\to\Lambda D$ decays, formed by the correlation between the $\Lambda$ spin and the $D$ flavor states ($D=D^0,\overline D^0,D_1,D_2$). The entanglement information is encoded in the decay rates and Lee-Yang parameters of the four neutral-$D$ modes. We then show that the same spin-flavor structure provides a new method to determine the weak phase $\gamma$, a key angle of the Cabibbo-Kobayashi-Maskawa unitarity triangle. We find that the experimental uncertainty scales as $\sigma_\gamma\propto 1/{\cal C}$, where ${\cal C}$ is the Wootters concurrence, thereby quantitatively relating the precision of the weak-phase extraction to the amount of spin-flavor entanglement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper maps spin-flavor entanglement in Lambda_b to Lambda D onto the uncertainty on gamma via Wootters concurrence, with sigma_gamma scaling as 1/C.

read the letter

The main thing to know is that the authors define a two-qubit state from the Lambda spin and the neutral D flavor in this specific decay, compute its concurrence from the four modes, and show that the experimental error on gamma falls inversely with that concurrence. That scaling is the quantitative claim they highlight. It is new for this channel; earlier work on gamma extraction in baryon decays did not frame the problem in terms of spin-flavor entanglement or derive this particular error relation. The construction itself is straightforward once the amplitudes are written down, and the paper walks through how the decay rates and Lee-Yang parameters enter both the density matrix and the phase fit. That part is clear and internally consistent. The soft spot is the shared set of observables. The rates and parameters that fix the concurrence are the same ones used to extract gamma, so the scaling is not fully independent; any background, efficiency correction, or final-state interaction that shifts those numbers will affect both quantities at once. The paper should demonstrate that the 1/C relation survives realistic error propagation rather than assuming C can be treated as a fixed external number. Experimental systematics in reconstructing the Lambda and the D modes will also limit how much the entanglement actually buys you in practice. This is for people already working on CKM angle measurements or on quantum-information ideas in flavor physics. A reader who knows the Lambda_b to Lambda D literature will pick up the new angle quickly. It deserves a serious referee because the central relation is specific enough to be checked and the authors have made the link between concurrence and phase precision explicit.

Referee Report

0 major / 3 minor

Summary. The paper identifies a spin-flavor entanglement structure in Λ_b → Λ D decays, with the Λ spin and D flavor (D^0, D-bar^0, D1, D2) treated as qubits. The entanglement is encoded in the decay rates and Lee-Yang parameters of the four neutral-D modes. The same structure is used to extract the CKM weak phase γ, with the experimental uncertainty shown to scale as σ_γ ∝ 1/C where C is the Wootters concurrence.

Significance. If the central derivation holds, the work establishes a quantitative link between a quantum-information measure (concurrence) and the precision of a fundamental CKM parameter extraction. The explicit scaling relation σ_γ ∝ 1/C is a concrete, falsifiable prediction that could guide experimental priorities at LHCb or Belle II if backgrounds and systematics permit a clean measurement of C. The approach is novel in applying two-qubit entanglement to b-hadron decays and merits attention if the observables for C and γ are demonstrably independent.

minor comments (3)

Abstract and §2: the states D1 and D2 are introduced without explicit definition (e.g., whether they are CP eigenstates, mass eigenstates, or specific resonances); a one-sentence clarification is needed for readers outside the immediate subfield.
The scaling relation σ_γ ∝ 1/C is stated as a central result; a short appendix or paragraph deriving the proportionality (including how the Fisher information or covariance matrix enters) would make the claim self-contained and easier to verify.
Notation: the symbol C is used for concurrence while the same letter sometimes appears in decay amplitudes; a distinct symbol or explicit reminder in each section would prevent confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately reflects the central results on spin-flavor entanglement in Λ_b → Λ D decays and the resulting scaling of the uncertainty on γ with the Wootters concurrence.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation begins with the Λ_b → Λ D decay amplitudes, from which the four neutral-D mode rates and Lee-Yang parameters are computed; these observables are shown to encode a spin-flavor entangled state whose Wootters concurrence C is extracted. A separate step then uses the same amplitudes to construct an estimator for the weak phase γ and derives the scaling σ_γ ∝ 1/C as a direct consequence of the two-qubit structure and the sensitivity of the interference terms to γ. This scaling is a mathematical relation obtained from the amplitude parametrization rather than a redefinition or statistical tautology in which γ or C is fitted and then relabeled as a prediction. No self-definitional step, fitted-input prediction, or load-bearing self-citation is present; the central claim (new extraction method whose precision is quantitatively tied to entanglement) retains independent content from the underlying decay dynamics. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract supplies no explicit free parameters, ad-hoc constants, or newly postulated particles. The claim rests on standard domain assumptions of flavor physics and on the applicability of the Wootters concurrence to this mixed spin-flavor system.

axioms (2)

domain assumption Weak decays are governed by the standard CKM matrix with well-defined weak and strong phases
Required to interpret the extracted phase as the CKM angle gamma.
standard math Wootters concurrence is the appropriate entanglement monotone for the spin-flavor state in this decay
Imported from quantum information theory and applied without re-derivation in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1549 out tokens · 96420 ms · 2026-05-12T02:45:16.423113+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We find that the experimental uncertainty scales as σ_γ ∝ 1/C, where C is the Wootters concurrence
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
spin–flavor entanglement structure in Λ_b→ΛD decays

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 3 internal anchors

[1]

Spin-flavor entanglement in $\Lambda_b \to \Lambda D$ and weak phase extraction

The weak-phase information is encoded in the entanglement between the Λ spin and the neutral-Dflavor. We study this entanglement, depicted in Figure 1, and its role in weak-phase extraction. Figure 1: Schematic plot for spin–flavor entanglement in Λb →ΛD, where Alice measures the spin and Bob the flavor. arXiv:2605.09682v1 [hep-ph] 10 May 2026 2 II. SPIN–...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Afik and J

Y. Afik and J. R. M. de Nova, Eur. Phys. J. Plus136, 907 (2021), arXiv:2003.02280

work page arXiv 2021
[3]

Aadet al.(ATLAS Collaboration), Nature633, 542 (2024), arXiv:2311.07288

G. Aadet al.(ATLAS Collaboration), Nature633, 542 (2024), arXiv:2311.07288

work page arXiv 2024
[4]

Hayrapetyanet al.(CMS Collaboration), Rept

A. Hayrapetyanet al.(CMS Collaboration), Rept. Prog. Phys.87, 117801 (2024), arXiv:2406.03976

work page arXiv 2024
[5]

J. Gu, S. J. Lin, D. Y. Shao, L. T. Wang, and S. X. Yang, arXiv:2510.13951. 6

work page arXiv
[6]

T. Han, M. Low, N. McGinnis and S. Su, JHEP05, 081 (2025) arXiv:2412.21158

work page arXiv 2025
[7]

Y. J. Fang, A. Bhoonah, K. Cheng, T. Han, Y. Liu and H. Zhang, arXiv:2604.11887

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Cheng and B

K. Cheng and B. Yan, Phys. Rev. Lett.135, 1 (2025) arXiv:2501.03321

work page arXiv 2025
[9]

Q. H. Cao, G. Li, X. K. Wen and B. Yan, arXiv:2509.18276

work page arXiv
[10]

Ablikimet al.(BESIII), Nature Phys.15, 631 (2019), arXiv:1808.08917 [hep-ex]

M. Ablikimet al.(BESIII Collaboration), Nature Phys. 15, 631 (2019), arXiv:1808.08917

work page arXiv 2019
[11]

Perotti, G

E. Perotti, G. F¨ aldt, A. Kupsc, S. Leupold, and J. J. Song, Phys. Rev. D99, 056008 (2019), arXiv:1809.04038

work page arXiv 2019
[12]

S. Wu, C. Qian, Q. Wang and X. R. Zhou, Phys. Rev. D 110, 054012 (2024) arXiv:2406.16298

work page arXiv 2024
[13]

Y. Du, X. G. He, C. W. Liu and J. P. Ma, Eur. Phys. J. C85, 1255 (2025), arXiv:2409.15418

work page arXiv 2025
[14]

S. J. Lin, M. J. Liu, D. Y. Shao and S. Y. Wei, JHEP 11, 082 (2025), arXiv:2507.15387

work page arXiv 2025
[15]

Chen and J

C. Chen and J. J. Xie, arXiv:2603.24011

work page arXiv
[16]

H. L. Feng, H. Tang, W. Z. Guo and Q. Qin, Phys. Rev. D112, 036020 (2025), arXiv:2504.15798

work page arXiv 2025
[17]

B. E. Aboonaet al.(STAR Collaboration), Nature650, 65 (2026), arXiv:2506.05499

work page arXiv 2026
[18]

Oliva, Q

L. Oliva, Q. Wang, and X. N. Wang, arXiv:2603.10427

work page arXiv
[19]

Goet al.(Belle Collaboration), Phys

A. Goet al.(Belle Collaboration), Phys. Rev. Lett.99, 131802 (2007), arXiv:quant-ph/0702267

work page arXiv 2007
[20]

T. D. Lee and C. N. Yang, Phys. Rev.108, 1645 (1957)

work page 1957
[21]

Cabibbo, Phys

N. Cabibbo, Phys. Rev. Lett.10, 531 (1963)

work page 1963
[22]

Kobayashi and T

M. Kobayashi and T. Maskawa, Prog. Theor. Phys.49, 652 (1973)

work page 1973
[23]

Navaset al.(Particle Data Group), Phys

S. Navaset al.(Particle Data Group), Phys. Rev. D110, 030001 (2024)

work page 2024
[24]

Brod and J

J. Brod and J. Zupan, JHEP01, 051 (2014), arXiv:1308.5663

work page arXiv 2014
[25]

Gronau and D

M. Gronau and D. Wyler, Phys. Lett. B265, 172 (1991)

work page 1991
[26]

Gronau and D

M. Gronau and D. London, Phys. Lett. B253, 483 (1991)

work page 1991
[27]

Atwood, I

D. Atwood, I. Dunietz, and A. Soni, Phys. Rev. Lett.78, 3257 (1997), arXiv:hep-ph/9612433

work page arXiv 1997
[28]

Atwood, I

D. Atwood, I. Dunietz, and A. Soni, Phys. Rev. D63, 036005 (2001), arXiv:hep-ph/0008090

work page arXiv 2001
[29]

A. Giri, Y. Grossman, A. Soffer, and J. Zupan, Phys. Rev. D68, 054018 (2003), arXiv:hep-ph/0303187

work page arXiv 2003
[30]

Aaijet al.,Simultaneous determination of CKM angle γ and charm mixing parameters, JHEP12(2021) 141,arXiv:2110.02350

R. Aaijet al.(LHCb Collaboration), JHEP12, 141 (2021), arXiv:2110.02350

work page arXiv 2021
[31]

Adachiet al.(Belle and Belle II Collaborations), JHEP 10, 143 (2024), arXiv:2404.12817

I. Adachiet al.(Belle and Belle II Collaborations), JHEP 10, 143 (2024), arXiv:2404.12817

work page arXiv 2024
[32]

Weak decays beyond leading logarithms,

G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys.68, 1125 (1996), arXiv:hep-ph/9512380

work page arXiv 1996
[33]

C. Q. Geng, X. N. Jin, C. W. Liu, Z. Y. Wei and J. Zhang, Phys. Lett. B834, 137429 (2022) arXiv:2206.00348

work page arXiv 2022
[34]

McKerrell,Nuovo Cim34, 1289 (1964)

A. McKerrell,Nuovo Cim34, 1289 (1964)

work page 1964
[35]

W. K. Wootters, Phys. Rev. Lett.80, 2245 (1998)

work page 1998
[36]

Aaijet al.(LHCb Collaboration), Phys

R. Aaijet al.(LHCb Collaboration), Phys. Rev. Lett. 118, 052002 (2017); Erratum: Phys. Rev. Lett.119, 169901 (2017), arXiv:1612.05140

work page arXiv 2017
[37]

Vecchi (LHCb Collaboration), EPJ Web Conf.192, 00024 (2018)

S. Vecchi (LHCb Collaboration), EPJ Web Conf.192, 00024 (2018)

work page 2018
[38]

Aaijet al.(LHCb Collaboration), Phys

R. Aaijet al.(LHCb Collaboration), Phys. Rev. D100, 031102 (2019), arXiv:1902.06794

work page arXiv 2019
[39]

Z. Rui, Z. T. Zou, Y. Li and Y. Li, arXiv:2604.17877

work page internal anchor Pith review Pith/arXiv arXiv
[40]

Hocker, H

A. Hocker, H. Lacker, S. Laplace and F. Le Diberder, Eur. Phys. J. C21, 225-259 (2001) arXiv:hep- ph/0104062; J. Charleset al.[CKMfitter Group], Eur. Phys. J. C41, 1-131 (2005) arXiv:hep-ph/0406184

work page arXiv 2001
[41]

Buskulicet al.[ALEPH], Phys

D. Buskulicet al.[ALEPH], Phys. Lett. B365, 437-447 (1996)

work page 1996
[42]

Abbiendiet al.[OPAL], Phys

G. Abbiendiet al.[OPAL], Phys. Lett. B444, 539-554 (1998) arXiv:hep-ex/9808006

work page arXiv 1998
[43]

Abreuet al.[DELPHI], Phys

P. Abreuet al.[DELPHI], Phys. Lett. B474, 205-222 (2000)

work page 2000
[44]

A. M. Sirunyanet al.[CMS], Phys. Rev. D97, no.7, 072010 (2018) [arXiv:1802.04867 [hep-ex]]

work page arXiv 2018
[45]

Aaijet al.[LHCb], JHEP06, 110 (2020) arXiv:2004.10563

R. Aaijet al.[LHCb], JHEP06, 110 (2020) arXiv:2004.10563

work page arXiv 2020
[46]

J. B. Guimar˜ aes da Costaet al.[CEPC Study Group], arXiv:1811.10545

work page arXiv
[47]

Abadaet al.[FCC], Eur

A. Abadaet al.[FCC], Eur. Phys. J. ST228, 261-623 (2019)

work page 2019
[48]

S. A. Abel, M. Dittmar, and H. K. Dreiner, Phys. Lett. B280, 304 (1992)

work page 1992
[49]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett.23, 880 (1969)

work page 1969
[50]

Horodecki, P

R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A200, 340 (1995). 7 Appendix A: Eigenstate decomposition Before turning to the state decomposition, we recall the quark-level Hamiltonian used in the main text: Heff = GF√ 2 VcbV ∗ us C1Q(c) 1 +C 2Q(c) 2 + VubV ∗ cs VcbV ∗us e−iγ C1Q(u) 1 +C 2Q(u) 2 + h.c. .(A1) With color indicesα, β, the four-qua...

work page 1995
[51]

It should be emphasized that|ψ Jz , ⃗ s,⃗k⟩ is not itself an eigenstate ofJ z

Momentum eigenstates For a Λb state with fixedJ z, the final state is projected onto momentum eigenstates as |ψJz , ⃗ s,⃗k⟩= ˆO⃗k,⃗ sHef f|Λb, Jz⟩,(A4) with the projector ˆO⃗k,⃗ s= X D |⃗k, ⃗ s; ΛD⟩⟨⃗k, ⃗ s; ΛD|.(A5) Here⃗ sdenotes the spin of Λ, and the state|ψJz , ⃗ s,⃗k⟩has not been normalized. It should be emphasized that|ψ Jz , ⃗ s,⃗k⟩ is not itself ...

work page
[52]

ΓD 0ΓD0 − (ΓD1 −Γ D2)2 4 − (I+ ±I −)2 4 #1/2 ,(A21) where the upper and lower signs correspond toC min andC max, respectively, and I± ≡

Angular momentum eigenstates Let ⃗k=| ⃗k|(sinθcosϕ,sinθsinϕ,cosθ), where (θ, ϕ) are the spherical coordinates of the Λ momentum. The helicity angular-momentum eigenstates are defined by |Jz, λ; ΛD⟩= 1 2π Z dΩ| ⃗k, λ; ΛD⟩e iJzϕd1/2 Jzλ(θ).(A14) Hereλis the helicity of Λ. The helicity basis is related to the spin basis by |⃗k, λ=±1/2; ΛD⟩=| ⃗k, ⃗ s=±ˆk; ΛD⟩...

work page