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arxiv: 2510.13969 · v1 · pith:PAJQLKUEnew · submitted 2025-10-15 · ✦ hep-ph

Isospin-based EWP-tree Relations

Bhubanjyoti Bhattacharya , Marianne Bouchard , Alexandre Jean , David London , Ipsita Ray This is my paper

Pith reviewed 2026-05-25 07:38 UTC · model grok-4.3

classification ✦ hep-ph
keywords B decaysisospin symmetryelectroweak penguinstree operatorsB to pi K puzzleCP violationSU(2) symmetryDelta S=1 decays
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The pith

Isospin symmetry alone produces EWP-tree relations in B decays that differ from SU(3) versions and raise the B to pi K puzzle to 4-5 sigma.

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

This paper shows that relations between electroweak penguin and tree operators in charmless B meson decays hold under isospin symmetry SU(2) without requiring the stronger SU(3) flavor symmetry. In decays where strangeness does not change, the relations are similar to the SU(3) ones and permit EWP contributions to be included when determining the CP-violating phase alpha from B to pi pi decays. In decays with Delta S equals 1, the isospin relations differ from the SU(3) relations. When applied to the B to pi K puzzle, these new relations produce a discrepancy with the Standard Model at the level of 4 to 5 sigma, which is larger than found in earlier analyses. The result indicates that any analysis of a set of decays linked by isospin must employ the corresponding SU(2) relations.

Core claim

Even assuming only SU(2)_I isospin symmetry, EWP-tree relations exist among the amplitudes. For Delta S=0 decays these relations are similar to the SU(3)_F relations and can be used to account for EWP contributions in the extraction of the CP phase alpha from B to pi pi decays. For Delta S=1 decays the SU(2)_I relations are different from the SU(3)_F ones; their application to the B to pi K puzzle yields a 4-5 sigma discrepancy with the Standard Model. The authors conclude that analyses of isospin-related decay sets require the use of SU(2)_I EWP-tree relations.

What carries the argument

The isospin SU(2)_I relations connecting reduced matrix elements of electroweak penguin operators to those of tree operators in B to light pseudoscalar meson decays.

If this is right

  • The relations permit inclusion of EWP effects in alpha extraction from Delta S=0 decays such as B to pi pi.
  • Application to Delta S=1 decays like B to pi K increases the SM discrepancy to 4-5 sigma.
  • Global analyses of B to PP decays must select the symmetry relations matching the isospin or SU(3) assumptions of the decay set.

Where Pith is reading between the lines

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

  • Previous global fits relying on SU(3) relations may have understated the tension in the B to pi K system.
  • More precise data on B to pi K observables could test whether the larger discrepancy persists or indicates new physics.
  • Similar isospin-based relations might be derivable for other decay classes related by SU(2) but not full flavor symmetry.

Load-bearing premise

The derived isospin relations among reduced matrix elements can be applied to the complete set of B to pi K amplitudes without additional corrections from SU(3) breaking that would change the size of the discrepancy.

What would settle it

A new global fit to B to pi K data using the SU(2)_I relations that finds the discrepancy reduced below 3 sigma, or direct evidence that the amplitude relations do not hold as predicted.

read the original abstract

In 1998, it was shown that, if flavor SU(3) symmetry [SU(3)$_F$] is assumed in charmless $B \to PP$ decays ($P$ is a light pseudoscalar meson), some reduced matrix elements involving electroweak penguin (EWP) operators are related to those involving tree operators. Similarly, EWP diagrams are related to tree diagrams. These SU(3)$_F$ EWP-tree relations were recently used in global analyses of $B \to PP$ decays. They have also been used over the years in analyses of the $B \to \pi K$ puzzle, even though the $B \to \pi K$ amplitudes are related by isospin symmetry [SU(2)$_I$], and not the full SU(3)$_F$. In this paper, we show that, even if only SU(2)$_I$ is assumed, there are still EWP-tree relations. In $\Delta S=0$ decays, these relations are similar to those of SU(3)$_F$, and can be used to take into account the EWP contributions in the extraction of the CP phase $\alpha$ from $B \to \pi\pi$ decays. In $\Delta S=1$ decays, the SU(2)$_I$ EWP-tree relations are quite different from those of SU(3)$_F$; when these are used to analyze the $B \to \pi K$ puzzle, one now finds a 4-5$\sigma$ discrepancy with the SM, much larger than what was previously found. We argue that, if one analyzes a set of hadronic $B$ decays whose amplitudes are related by isospin, one must use the SU(2)$_I$ EWP-tree relations for that set of decays in the analysis.

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 / 1 minor

Summary. The manuscript derives EWP-tree relations for charmless B→PP decays assuming only isospin SU(2)_I symmetry. It shows that these relations are similar to the SU(3)_F versions in ΔS=0 decays (useful for extracting α from B→ππ) but differ in ΔS=1 decays; when the SU(2)_I relations are substituted into analyses of the B→πK puzzle, a 4-5σ discrepancy with the SM emerges, larger than in prior SU(3)_F-based fits. The paper concludes that analyses of amplitude sets related by isospin must employ the corresponding SU(2)_I relations.

Significance. If the derivation is correct and the relations can be applied directly to existing B→πK parameterizations without additional corrections, the result would indicate that previous global fits using SU(3)_F relations for isospin-related decays have underestimated the tension with the SM. It provides a concrete symmetry argument for re-examining how EWP contributions are incorporated in B-decay analyses and could affect interpretations of the B→πK puzzle.

major comments (2)
  1. [Abstract] Abstract: the claim that the SU(2)_I EWP-tree relations produce a 4-5σ discrepancy when used for the B→πK puzzle assumes that these relations can be inserted into the same amplitude parameterization and data fits as prior SU(3)_F analyses without modification; the manuscript does not show the explicit substitution or the resulting fit parameters that yield the increased significance.
  2. [Abstract] Abstract (final paragraph): the assertion that the relations hold under SU(2)_I alone and can be directly applied to the full set of B→πK amplitudes does not address whether SU(3)-breaking effects (from mass differences or wave-function mixing in ΔS=1 decays) alter the effective EWP/tree ratios or the extracted discrepancy; this assumption is load-bearing for the 4-5σ claim.
minor comments (1)
  1. Notation for symmetry groups (SU(3)$_F$, SU(2)$_I$) is clear in the abstract but should be defined on first use in the main text with explicit statements of the assumed breaking patterns.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the SU(2)_I EWP-tree relations produce a 4-5σ discrepancy when used for the B→πK puzzle assumes that these relations can be inserted into the same amplitude parameterization and data fits as prior SU(3)_F analyses without modification; the manuscript does not show the explicit substitution or the resulting fit parameters that yield the increased significance.

    Authors: The SU(2)_I relations are obtained by applying the isospin transformation properties directly to the EWP and tree operators within the ΔS=1 sector, allowing them to be substituted into the standard isospin amplitude relations for B→πK in place of the previous SU(3)_F versions. While the manuscript emphasizes the derivation and the qualitative increase in tension, we agree that an explicit substitution into the amplitude parameterization together with the updated numerical significance would strengthen the presentation of the 4-5σ claim. This will be added in the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract (final paragraph): the assertion that the relations hold under SU(2)_I alone and can be directly applied to the full set of B→πK amplitudes does not address whether SU(3)-breaking effects (from mass differences or wave-function mixing in ΔS=1 decays) alter the effective EWP/tree ratios or the extracted discrepancy; this assumption is load-bearing for the 4-5σ claim.

    Authors: The derivation uses only the SU(2)_I transformation properties of the operators and makes no reference to SU(3)_F; therefore SU(3)-breaking corrections lie outside the assumed symmetry and do not modify the relations themselves. Any additional breaking effects would constitute model-dependent corrections beyond the symmetry analysis, but the central result is that the isospin relations differ from those obtained under SU(3)_F. We will add a clarifying sentence in the abstract and introduction to emphasize this distinction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; relations derived from SU(2)_I symmetry assumptions applied to data

full rationale

The paper derives EWP-tree relations directly from isospin symmetry (SU(2)_I) applied to reduced matrix elements in B decays, without fitting to the target B→πK discrepancy data. These relations are then substituted into amplitude parameterizations to compare against experiment, yielding a 4-5σ result. No step reduces by construction to a fit, self-definition, or self-citation chain; the central claim is a symmetry-based derivation whose output (discrepancy size) is falsifiable against external data. Self-citations to prior SU(3)_F work are not load-bearing for the new SU(2)_I results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption of exact SU(2)_I symmetry relating the relevant amplitudes; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption SU(2)_I isospin symmetry holds for the B→PP amplitudes under consideration
    Invoked throughout the abstract as the sole symmetry used to derive the EWP-tree relations.

pith-pipeline@v0.9.0 · 5887 in / 1409 out tokens · 20491 ms · 2026-05-25T07:38:54.257390+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case

    hep-ph 2026-04 unverdicted novelty 5.0

    A data-driven SU(3)-breaking analysis of B to PP decays yields QCD-factorization amplitudes that resemble dynamical predictions and require no enhanced annihilation terms.

  2. CP asymmetries in charged meson decay to two pions

    hep-ph 2026-05 unverdicted novelty 4.0

    CP asymmetries for B+ to pi+ pi0, D+ to pi+ pi0, and K+ to pi+ pi0 are estimated in the Standard Model at roughly 3 times 10 to the -3, 10 to the -5, and 10 to the -6 using a unified formalism for isospin violation.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 2 Pith papers · 24 internal anchors

  1. [1]

    Gronau and D

    M. Gronau and D. London,Isospin analysis of CP asymmetries in B decays,Phys. Rev. Lett. 65(1990) 3381

    work page 1990

  2. [2]

    Berthiaume, B

    R. Berthiaume, B. Bhattacharya, R. Boumris, A. Jean, S. Kumbhakar and D. London, Anomalies in Hadronic B Decays,Phys. Rev. Lett.133(2024) 211802 [2311.18011]

    work page arXiv 2024

  3. [3]

    Bhattacharya, M

    B. Bhattacharya, M. Bouchard, L. Hudy, A. Jean, D. London and C. MacKenzie,Anomalies in HadronicBDecays: an Update,Physical Review D(2025) [2505.11492]

    work page arXiv 2025

  4. [4]

    Weak Decays Beyond Leading Logarithms

    G. Buchalla, A.J. Buras and M.E. Lautenbacher,Weak decays beyond leading logarithms, Rev. Mod. Phys.68(1996) 1125 [hep-ph/9512380]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  5. [5]

    Decays of $B$ Mesons to Two Light Pseudoscalars

    M. Gronau, O.F. Hernandez, D. London and J.L. Rosner,Decays of B mesons to two light pseudoscalars,Phys. Rev. D50(1994) 4529 [hep-ph/9404283]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  6. [6]

    ELECTROWEAK PENGUINS AND TWO-BODY B DECAYS

    M. Gronau, O.F. Hernandez, D. London and J.L. Rosner,Electroweak penguins and two-body B decays,Phys. Rev. D52(1995) 6374 [hep-ph/9504327]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  7. [7]

    Model-independent electroweak penguins in B decays to two pseudoscalars

    M. Gronau, D. Pirjol and T.-M. Yan,Model independent electroweak penguins in B decays to two pseudoscalars,Phys. Rev. D60(1999) 034021 [hep-ph/9810482]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  8. [8]

    New Bound on gamma from B^+- -> pi K Decays

    M. Neubert and J.L. Rosner,New bound on gamma from B+- —>pi K decays,Phys. Lett. B441(1998) 403 [hep-ph/9808493]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  9. [9]

    Determination of the Weak Phase gamma from Rate Measurements in B^+- -> pi K, pi pi Decays

    M. Neubert and J.L. Rosner,Determination of the weak phase gamma from rate measurements in B+- —>pi K, pi pi decays,Phys. Rev. Lett.81(1998) 5076 [hep-ph/9809311]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  10. [10]

    The B -> pi K Puzzle and its Relation to Rare B and K Decays

    A.J. Buras, R. Fleischer, S. Recksiegel and F. Schwab,The B —>pi K puzzle and its relation to rare B and K decays,Eur. Phys. J. C32(2003) 45 [hep-ph/0309012]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  11. [11]

    B -> pi pi, New Physics in B -> pi K and Implications for Rare K and B Decays

    A.J. Buras, R. Fleischer, S. Recksiegel and F. Schwab,B —>pi pi, new physics in B —>pi K and implications for rare K and B decays,Phys. Rev. Lett.92(2004) 101804 [hep-ph/0312259]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  12. [12]

    Anatomy of Prominent B and K Decays and Signatures of CP-Violating New Physics in the Electroweak Penguin Sector

    A.J. Buras, R. Fleischer, S. Recksiegel and F. Schwab,Anatomy of prominent B and K decays and signatures of CP violating new physics in the electroweak penguin sector,Nucl. Phys. B697(2004) 133 [hep-ph/0402112]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  13. [13]

    S. Baek, P. Hamel, D. London, A. Datta and D.A. Suprun,The B —>pi K puzzle and new physics,Phys. Rev. D71(2005) 057502 [hep-ph/0412086]. – 26 –

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  14. [14]

    Is There Still a B -> pi K Puzzle?

    S. Baek and D. London,Is There Still a B —>pi K Puzzle?,Phys. Lett. B653(2007) 249 [hep-ph/0701181]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  15. [15]

    The B -> pi K Puzzle: 2009 Update

    S. Baek, C.-W. Chiang and D. London,The B —>pi K Puzzle: 2009 Update,Phys. Lett. B 675(2009) 59 [0903.3086]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  16. [16]

    The $B \to \pi K$ Puzzle Revisited

    N.B. Beaudry, A. Datta, D. London, A. Rashed and J.-S. Roux,TheB→πKpuzzle revisited,JHEP01(2018) 074 [1709.07142]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Bhattacharya, A

    B. Bhattacharya, A. Datta, D. Marfatia, S. Nandi and J. Waite,Axion-like particles resolve theB→πKandg−2anomalies,Phys. Rev. D104(2021) L051701 [2104.03947]

    work page arXiv 2021

  18. [18]

    Datta, J

    A. Datta, J. Kumar, S. Kumbhakar and D. London,Uniting low-energy semileptonic and hadronic anomalies within SMEFT,JHEP12(2024) 175 [2408.03380]

    work page arXiv 2024

  19. [19]

    A precise sum rule among four B -> Kpi CP asymmetries

    M. Gronau,A Precise sum rule among four B —>K pi CP asymmetries,Phys. Lett. B627 (2005) 82 [hep-ph/0508047]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  20. [20]

    Benchmarks for the New-Physics Search through CP Violation in B^0->pi^0 K_S

    R. Fleischer, S. Jager, D. Pirjol and J. Zupan,Benchmarks for the New-Physics Search through CP Violation in B0 —>pi0 K(S),Phys. Rev. D78(2008) 111501 [0806.2900]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  21. [21]

    Exploring $B\to\pi\pi, \pi K$ Decays at the High-Precision Frontier

    R. Fleischer, R. Jaarsma, E. Malami and K.K. Vos,ExploringB→ππ, πKdecays at the high-precision frontier,Eur. Phys. J. C78(2018) 943 [1806.08783]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    A global $SU(3)/U(3)$ flavor symmetry analysis for $B\to PP$ with $\eta-\eta'$ Mixing

    Y.-K. Hsiao, C.-F. Chang and X.-G. He,A globalSU(3)/U(3)flavor symmetry analysis for B→P Pwithη−η ′ Mixing,Phys. Rev. D93(2016) 114002 [1512.09223]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  23. [23]

    Huber and G

    T. Huber and G. Tetlalmatzi-Xolocotzi,Estimating QCD-factorization amplitudes through SU(3) symmetry inB→P Pdecays,Eur. Phys. J. C82(2022) 210 [2111.06418]

    work page arXiv 2022

  24. [24]

    Burgos Marcos, M

    M. Burgos Marcos, M. Reboud and K.K. Vos,DetailedSU(3)Flavour Symmetry Analysis of Charmless Two-BodyB-Meson Decays Including Factorizable Corrections,2504.05209

    work page arXiv

  25. [25]

    A General Analysis of $\gamma$ Determinations from $B\to\pi K$ Decays

    A.J. Buras and R. Fleischer,A General analysis of gamma determinations from B —>pi K decays,Eur. Phys. J. C11(1999) 93 [hep-ph/9810260]. [26]Belle-IIcollaboration,Measurement of branching fractions and direct CP asymmetries for B→Kπand B→ππdecays at Belle II,Phys. Rev. D109(2024) 012001 [2310.06381]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  26. [26]

    Isospin analysis of charmless B-meson decays

    J. Charles, O. Deschamps, S. Descotes-Genon and V. Niess,Isospin analysis of charmless B-meson decays,Eur. Phys. J. C77(2017) 574 [1705.02981]. [28]CLEOcollaboration,Observation of exclusive two-body B decays to kaons and pions,Phys. Rev. Lett.80(1998) 3456 [hep-ex/9711010]. [29]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110(2...

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  27. [27]

    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

    work page 1975

  28. [28]

    QCD Factorization in B -> pi K, pi pi Decays and Extraction of Wolfenstein Parameters

    M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda,QCD factorization inB→πK, ππ decays and extraction of Wolfenstein parameters,Nucl. Phys. B606(2001) 245 [hep-ph/0104110]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  29. [29]

    NNLO Vertex Corrections in charmless hadronic B decays: Imaginary part

    G. Bell,NNLO vertex corrections in charmless hadronic B decays: Imaginary part,Nucl. Phys. B795(2008) 1 [0705.3127]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  30. [30]

    NNLO vertex corrections in charmless hadronic B decays: Real part

    G. Bell,NNLO vertex corrections in charmless hadronic B decays: Real part,Nucl. Phys. B 822(2009) 172 [0902.1915]. – 27 –

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  31. [31]

    NNLO vertex corrections to non-leptonic B decays: Tree amplitudes

    M. Beneke, T. Huber and X.-Q. Li,NNLO vertex corrections to non-leptonic B decays: Tree amplitudes,Nucl. Phys. B832(2010) 109 [0911.3655]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  32. [32]

    G. Bell, M. Beneke, T. Huber and X.-Q. Li,Two-loop current–current operator contribution to the non-leptonic QCD penguin amplitude,Phys. Lett. B750(2015) 348 [1507.03700]. – 28 –

    work page internal anchor Pith review Pith/arXiv arXiv 2015