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arxiv: 2605.06768 · v1 · submitted 2026-05-07 · ✦ hep-ph · hep-ex

Recognition: 2 theorem links

· Lean Theorem

Unbinned extraction of γ from Bto DK with normalizing flows

Yuval Grossman , Tony Menzo , Stefan Schacht , Chinhsan Sieng , Jure Zupan
Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:57 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords normalizing flowsCKM angle gammaB to DK decaysDalitz plotunbinned analysisCP violationamplitude modeling
0
0 comments X

The pith

Normalizing flows trained on D decays enable unbinned extraction of the CKM angle γ from B to DK data.

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

The paper develops an unbinned technique to extract the CKM angle γ from B± decays to a D meson that decays to KS π+ π- and a charged kaon. Normalizing flows are trained on D decay data to create a smooth model of the amplitude and strong phase variation across the Dalitz plot. This model is then applied to B decay observations to determine the magnitude ratio r_B, the strong phase δ_B, and γ. The approach is validated on Monte Carlo data, where it recovers the injected value of γ within uncertainties, and its accuracy improves as the size of the D decay training sample grows.

Core claim

By training normalizing flows on D decay data, the method learns a faithful continuous representation of the amplitude and strong phase variation over the D to KS π+ π- Dalitz plot. With this representation, the B decay data can be used to extract the parameters r_B, δ_B, and γ. The method was tested on Monte Carlo generated data and successfully recovers the injected value of γ within uncertainties.

What carries the argument

Normalizing flows that learn a continuous representation of the amplitude and strong phase variation over the D to KS π+ π- Dalitz plot from D decay data.

Load-bearing premise

The normalizing flows must accurately capture the true underlying amplitude and strong phase variation from the finite D decay training data without introducing biases.

What would settle it

Applying the trained flows to Monte Carlo B decay data with a known injected γ value and finding that the extracted γ lies outside the estimated uncertainties would falsify the claim that the learned representation is sufficient for accurate parameter extraction.

read the original abstract

We introduce an unbinned method for extracting the CKM angle $\gamma$ from the decay chain $B^\pm \to (D \to K_S \pi^+ \pi^-) K^\pm$ using normalizing flows (NFs). The NFs, trained on $D$ decay data, learn a faithful continuous representation of the amplitude and strong phase variation over the $D\to K_S\pi^+\pi^-$ Dalitz plot whose fidelity improves with increased data sample sizes. With this input, the $B$ decay data can be used to extract the parameters $r_B$, $\delta_B$, and $\gamma$. We test the method on Monte Carlo generated data, where it successfully recovers the injected value of $\gamma$ within uncertainties. The present implementation propagates statistical uncertainties from finite training data via an ensemble of independently trained flows, and does not attempt to capture the effects of systematic experimental errors. We explore two versions of the method that differ in how the trigonometric constraint on phase variation is encoded, and comment on the possible extension to Bayesian NFs, which would provide direct uncertainty estimates on the learned densities without requiring ensemble training.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an unbinned method to extract the CKM angle γ from B± → (D → KS π+ π−) K± decays. Normalizing flows are trained on D decay samples to learn a continuous representation of both the D decay amplitude magnitude and strong phase variation over the Dalitz plot. This representation is inserted into the B decay likelihood to fit simultaneously for r_B, δ_B, and γ. Two variants are considered that differ in the encoding of trigonometric constraints on the phase. The method is validated exclusively on Monte Carlo data, where the injected γ is recovered within uncertainties; statistical uncertainties from finite training samples are propagated via an ensemble of flows. Systematic experimental effects are not modeled.

Significance. If the central claim were valid, the approach would offer a notable advance by enabling a fully unbinned, data-driven extraction of γ that avoids Dalitz-plot binning and explicit isobar models for the D decay amplitude. The fidelity of the NF representation improving with larger D samples, the ensemble-based uncertainty propagation, and the suggestion of Bayesian NF extensions are constructive elements that could enhance precision in future γ measurements.

major comments (2)
  1. [Abstract] Abstract: The claim that NFs 'learn a faithful continuous representation of the amplitude and strong phase variation' cannot hold. D decay training data constrain only the density p(x) ∝ |A(x)|^2; any phase function φ(x) produces identical samples. The two versions differing in how the 'trigonometric constraint on phase variation is encoded' can at best impose smoothness or periodicity but supply no information selecting the physically correct φ(x) required to evaluate the interference term 2 r_B |A(x) A_bar(x)| cos(δ_B + φ(x) − γ) in the B decay likelihood.
  2. [Monte Carlo validation] Monte Carlo tests (described in the validation section): Recovery of the injected γ is consistent with the above limitation if the test likelihood implicitly re-uses the generator's true phase function when constructing the B decay amplitude. This does not demonstrate that the NF has extracted φ(x) from density samples alone. A non-circular test would require fitting with an independent phase model or real data.
minor comments (2)
  1. [Abstract] The abstract supplies no quantitative details on fit quality (e.g., pull distributions, bias, or coverage), making it difficult to assess the strength of the MC recovery.
  2. [Discussion] The manuscript explicitly states that systematic experimental effects are not modeled; this should be emphasized more clearly when discussing applicability to real data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the insightful comments provided. We address the two major comments point by point below. We believe our responses clarify the methodology and demonstrate that the approach remains valid, with some revisions to enhance precision in the description.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that NFs 'learn a faithful continuous representation of the amplitude and strong phase variation' cannot hold. D decay training data constrain only the density p(x) ∝ |A(x)|^2; any phase function φ(x) produces identical samples. The two versions differing in how the 'trigonometric constraint on phase variation is encoded' can at best impose smoothness or periodicity but supply no information selecting the physically correct φ(x) required to evaluate the interference term 2 r_B |A(x) A_bar(x)| cos(δ_B + φ(x) − γ) in the B decay likelihood.

    Authors: We agree that the training data from D decays provides information solely on the squared magnitude |A(x)|^2, and thus the normalizing flow models the continuous density corresponding to this magnitude. The strong phase variation is not extracted directly from the samples but is instead incorporated through the explicit encoding of trigonometric constraints in the two variants of the method. These encodings are designed to capture the expected phase behavior consistent with unitarity and other physical principles, allowing us to evaluate the interference terms in the B decay likelihood. The manuscript will be revised to make this distinction clearer in the abstract and method sections, avoiding any implication that the phase is learned solely from the density. revision: partial

  2. Referee: [Monte Carlo validation] Monte Carlo tests (described in the validation section): Recovery of the injected γ is consistent with the above limitation if the test likelihood implicitly re-uses the generator's true phase function when constructing the B decay amplitude. This does not demonstrate that the NF has extracted φ(x) from density samples alone. A non-circular test would require fitting with an independent phase model or real data.

    Authors: The Monte Carlo validation is intended to show that, when the phase constraints are set consistently with the generation model, the unbinned fit recovers the injected γ value within uncertainties, including propagation of training sample uncertainties via the flow ensemble. We acknowledge that this test reuses the phase information implicitly through the constraints and does not independently extract φ(x) from density alone. To strengthen the validation, we will add a test using an independent phase parameterization in the fit (different from the generator) and discuss the results. Application to real data will ultimately test the method without knowledge of the true phase. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; separate D-training supplies independent input to B fit

full rationale

The paper trains normalizing flows exclusively on D→KSπ+π− samples to obtain a continuous representation of amplitude and phase variation, then inserts that representation into a distinct likelihood for B-decay data to extract r_B, δ_B and γ. Monte Carlo tests use independently generated B samples. No equation or procedure reduces the extracted γ (or the phase function) to a quantity defined by the D-training itself; the two trigonometric-constraint encodings are explicit modeling choices rather than self-definitions. The derivation therefore remains self-contained against the stated MC benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the assumption that normalizing flows can capture the D decay amplitude and phase variation without introducing model-dependent bias beyond what is captured by the ensemble uncertainty.

axioms (1)
  • domain assumption Normalizing flows trained on D decay data can faithfully represent the amplitude and strong phase variation over the Dalitz plot
    This is the core premise that allows the B decay data to be used for parameter extraction.

pith-pipeline@v0.9.0 · 5515 in / 1253 out tokens · 42585 ms · 2026-05-11T00:57:04.470654+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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

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Reference graph

Works this paper leans on

45 extracted references · 30 canonical work pages

  1. [1]

    Zupan,Introduction to flavour physics,CERN Yellow Rep

    J. Zupan,Introduction to flavour physics,CERN Yellow Rep. School Proc.6(2019) 181–212, [1903.05062]

    work page arXiv 2019

  2. [2]

    Grossman and P

    Y. Grossman and P. Tanedo,Just a taste: lectures on flavor physics., inTheoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics, pp. 109–295, 2018.1711.03624

    work page arXiv 2018

  3. [3]

    A. B. Carter and A. I. Sanda,CP Violation in Cascade Decays of B Mesons,Phys. Rev. Lett. 45(1980) 952

    1980

  4. [4]

    A. B. Carter and A. I. Sanda,CP Violation in B Meson Decays,Phys. Rev. D23(1981) 1567

    1981

  5. [5]

    Gronau and D

    M. Gronau and D. London,How to determine all the angles of the unitarity triangle from B0 d →DK s andB 0 s →D ϕ,Phys. Lett. B253(1991) 483–488

    1991

  6. [6]

    Gronau and D

    M. Gronau and D. Wyler,On determining a weak phase from CP asymmetries in charged B decays,Phys. Lett. B265(1991) 172–176

    1991

  7. [7]

    Brod and J

    J. Brod and J. Zupan,The ultimate theoretical error onγfromB→DKdecays,JHEP01 (2014) 051, [1308.5663]

    work page arXiv 2014

  8. [8]

    Brod,Electroweak effects in the extraction of the CKM angleγfrom B→Dπdecays,Phys

    J. Brod,Electroweak effects in the extraction of the CKM angleγfrom B→Dπdecays,Phys. Lett. B743(2015) 56–60, [1412.3173]

    work page arXiv 2015

  9. [9]

    Bondar,Improved Gronau–Wyler method forϕ3 extraction, inBINP special analysis meeting on Dalitz analysis, Sep

    A. Bondar,Improved Gronau–Wyler method forϕ3 extraction, inBINP special analysis meeting on Dalitz analysis, Sep. 24–26, 9, 2002. unpublished

    2002

  10. [10]

    A. Giri, Y. Grossman, A. Soffer, and J. Zupan,Determining gamma usingB± →DK ± with multibodyDdecays,Phys. Rev. D68(2003) 054018, [hep-ph/0303187]. [11]BelleCollaboration, A. Poluektovet al.,Measurement of phi(3) with Dalitz plot analysis of B+- —>D**(*) K+- decay,Phys. Rev. D70(2004) 072003, [hep-ex/0406067]

    work page arXiv 2003

  11. [11]

    Bondar and A

    A. Bondar and A. Poluektov,Feasibility study of model-independent approach to phi(3) measurement using Dalitz plot analysis,Eur. Phys. J. C47(2006) 347–353, [hep-ph/0510246]. – 32 –

    work page arXiv 2006

  12. [12]

    Ceccucci, T

    A. Ceccucci, T. Gershon, M. Kenzie, Z. Ligeti, Y. Sakai, and K. Trabelsi,Origins of the method to determine the CKM angleγusingB ± →DK ±,D→K 0 Sπ+π− decays,2006.12404

    work page arXiv 2006

  13. [13]

    Bondar and A

    A. Bondar and A. Poluektov,On model-independent measurement of the angle phi(3) using Dalitz plot analysis, in4th International Workshop on the CKM Unitarity Triangle (CKM 2006), 3, 2007.hep-ph/0703267

    work page arXiv 2006

  14. [14]

    Bondar and A

    A. Bondar and A. Poluektov,The Use of quantum-correlated D0 decays for phi3 measurement, Eur. Phys. J. C55(2008) 51–56, [0801.0840]

    work page arXiv 2008

  15. [15]

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

    LHCbCollaboration, R. Aaijet al.,Simultaneous determination of CKM angle γ and charm mixing parameters,JHEP12(2021) 141, [2110.02350]. [17]LHCbCollaboration, R. Aaijet al.,Measurement of the CKM angleγinB ± →DK ± and B± →Dπ ± decays withD→K 0 Sh+h−,JHEP02(2021) 169, [2010.08483]

    work page arXiv 2021

  16. [16]

    Poluektov,Unbinned model-independent measurements with coherent admixtures of multibody neutralDmeson decays,Eur

    A. Poluektov,Unbinned model-independent measurements with coherent admixtures of multibody neutralDmeson decays,Eur. Phys. J. C78(2018), no. 2 121, [1712.08326]

    work page arXiv 2018

  17. [17]

    J. Lane, E. Gersabeck, and J. Rademacker,A novel unbinned model-independent method to measure the CKM angleγin B→DK decays with optimised precision,JHEP09(2023) 007, [2305.10787]

    work page arXiv 2023

  18. [18]

    J. V. Backus, M. Freytsis, Y. Grossman, S. Schacht, and J. Zupan,Toward extractingγfrom B→DKwithout binning,Eur. Phys. J. C83(2023) 877, [2211.05133]

    work page arXiv 2023

  19. [19]

    L. Dinh, D. Krueger, and Y. Bengio,Nice: Non-linear independent components estimation, 2015

    2015

  20. [20]

    Kobyzev, S

    I. Kobyzev, S. J. Prince, and M. A. Brubaker,Normalizing flows: An introduction and review of current methods,IEEE Transactions on Pattern Analysis and Machine Intelligence43 (Nov., 2021) 3964–3979

    2021

  21. [21]

    D. J. Rezende and S. Mohamed,Variational inference with normalizing flows, 2016

    2016

  22. [22]

    Banerjeeet al.(Heavy Flavor Averaging Group (HFLAV)), Phys

    Heavy Flavor A veraging Group (HFLA V)Collaboration, S. Banerjeeet al.,Averages of b-hadron, c-hadron, andτ-lepton properties as of 2023,Phys. Rev. D113(2026), no. 1 012008, [2411.18639]

    work page arXiv 2023

  23. [23]

    Aubertet al.,Amplitude analysis of the decay B± →π ±π±π∓, Phys

    HFLAV Collaboration, “Results on Time-Dependent CP Violation and Measurements Related to the Angles of the Unitarity Triangle.” https://hflav-eos.web.cern.ch/hflav-eos/triangle/summer2025/#gamma_comb, 2025. Summer 2025 averages and combinations (EPS 2025, Lepton Photon 2025, CKM2025, etc.). [26]BaBarCollaboration, B. Aubertet al.,An amplitude analysis of ...

    work page arXiv 2025

  24. [24]

    Backet al.,LAURA ++: A Dalitz plot fitter,Comput

    J. Backet al.,LAURA ++: A Dalitz plot fitter,Comput. Phys. Commun.231(2018) 198–242, [1711.09854]

    work page arXiv 2018

  25. [25]

    Grossman, A

    Y. Grossman, A. Soffer, and J. Zupan,The Effect ofD−¯Dmixing on the measurement ofγ inB→DKdecays,Phys. Rev. D72(2005) 031501, [hep-ph/0505270]

    work page arXiv 2005

  26. [26]

    Martone and J

    M. Martone and J. Zupan,B± →DK ± with direct CP violation in charm,Phys. Rev. D87 (2013), no. 3 034005, [1212.0165]. – 33 –

    work page arXiv 2013

  27. [27]

    Rama,Effect of D-Dbar mixing in the extraction of gamma with B- ->D0 K- and B- -> D0 pi- decays,Phys

    M. Rama,Effect of D-Dbar mixing in the extraction of gamma with B- ->D0 K- and B- -> D0 pi- decays,Phys. Rev. D89(2014), no. 1 014021, [1307.4384]. [32]CLEOCollaboration, H. Muramatsuet al.,Dalitz analysis of D0 —>K0(S) pi+ pi-,Phys. Rev. Lett.89(2002) 251802, [hep-ex/0207067]. [Erratum: Phys.Rev.Lett. 90, 059901 (2003)]. [33]BelleCollaboration, K. Abeet ...

    work page arXiv 2014

  28. [28]

    Gilman,Quantum correlation of neutral charmed mesons at BESIII, in32nd International Symposium on Lepton Photon Interactions at High Energies: Lepton-Photon 2025, 1, 2026

    A. Gilman,Quantum correlation of neutral charmed mesons at BESIII, in32nd International Symposium on Lepton Photon Interactions at High Energies: Lepton-Photon 2025, 1, 2026. 2601.16768

    work page arXiv 2025

  29. [29]

    Durkan, A

    C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios,Neural Spline Flows, inAdvances in Neural Information Processing Systems, vol. 32, Curran Associates, Inc., 2019.1906.04032

    work page arXiv 2019

  30. [30]

    Durkan, A

    C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios,nflows: normalizing flows in PyTorch, 2020

    2020

  31. [32]

    Astonet al.,A Study of K- pi+ Scattering in the Reaction K- p —>K- pi+ n at 11-GeV/c,Nucl

    D. Astonet al.,A Study of K- pi+ Scattering in the Reaction K- p —>K- pi+ n at 11-GeV/c,Nucl. Phys. B296(1988) 493–526

    1988

  32. [33]

    Gronau, Y

    M. Gronau, Y. Grossman, and J. L. Rosner,Measuring D0 - anti-D0 mixing and relative strong phases at a charm factory,Phys. Lett. B508(2001) 37–43, [hep-ph/0103110]

    work page arXiv 2001

  33. [34]

    James and M

    F. James and M. Roos,Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations,Comput. Phys. Commun.10(1975) 343–367. – 34 –

    1975

  34. [35]

    D. J. C. MacKay,Probable networks and plausible predictions – a review of practical Bayesian methods for supervised neural networks,Network: Computation in Neural Systems6(1995), no. 3 469–505

    1995

  35. [36]

    R. M. Neal,Bayesian Learning for Neural Networks. PhD thesis, University of Toronto, 1995

    1995

  36. [37]

    Gal,Uncertainty in Deep Learning

    Y. Gal,Uncertainty in Deep Learning. PhD thesis, University of Cambridge, 2016

    2016

  37. [38]

    Louizos and M

    C. Louizos and M. Welling,Multiplicative Normalizing Flows for Variational Bayesian Neural Networks, inInternational Conference on Machine Learning, 2017.1703.01961

    work page arXiv 2017

  38. [39]

    Bollweg, M

    S. Bollweg, M. Haußmann, G. Kasieczka, M. Luchmann, T. Plehn, and J. Thompson, Deep-Learning Jets with Uncertainties and More,SciPost Phys.8(2020), no. 1 006, [1904.10004]

    work page arXiv 2020

  39. [40]

    Ernst, L

    F. Ernst, L. Favaro, C. Krause, T. Plehn, and D. Shih,Normalizing flows for high-dimensional detector simulations,SciPost Phys.18(2025), no. 3 081, [2312.09290]

    work page arXiv 2025

  40. [41]

    Bierlich, P

    C. Bierlich, P. Ilten, T. Menzo, S. Mrenna, M. Szewc, M. K. Wilkinson, A. Youssef, and J. Zupan,Towards a data-driven model of hadronization using normalizing flows,SciPost Phys. 17(2024), no. 2 045, [2311.09296]

    work page arXiv 2024

  41. [42]

    Butteret al.,Iterative HOMER with uncertainties,SciPost Phys.20(2026), no

    A. Butteret al.,Iterative HOMER with uncertainties,SciPost Phys.20(2026), no. 2 042, [2509.03592]

    work page arXiv 2026

  42. [43]

    L. Dinh, J. Sohl-Dickstein, and S. Bengio,Density estimation using Real-NVP, in International Conference on Learning Representations, 2017.1605.08803

    work page arXiv 2017

  43. [44]

    J. A. Gregory and R. Delbourgo,Piecewise Rational Quadratic Interpolation to Monotonic Data,IMA J. Numer. Anal.2(1982), no. 2 123–130

    1982

  44. [45]

    Rahaman, A

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. A. Hamprecht, Y. Bengio, and A. Courville,On the Spectral Bias of Neural Networks, inInternational Conference on Machine Learning, 2019.1806.08734

    work page arXiv 2019

  45. [46]

    Tancik, P

    M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. T. Barron, and R. Ng,Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains, inAdvances in Neural Information Processing Systems, vol. 33, 2020.2006.10739. – 35 –

    work page arXiv 2020