Comment on "QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case''

Bhubanjyoti Bhattacharya; David London

arxiv: 2606.02538 · v1 · pith:FTI5PUC4new · submitted 2026-06-01 · ✦ hep-ph · hep-ex

Comment on "QCD-factorization amplitudes from flavour symmetries: beyond the SU(3) symmetric case''

Bhubanjyoti Bhattacharya , David London This is my paper

Pith reviewed 2026-06-28 13:35 UTC · model grok-4.3

classification ✦ hep-ph hep-ex

keywords B to PP decaysSU(3) flavour symmetryEWP-tree relationstopological diagramsQCD factorizationweak effective Hamiltonianflavour symmetrieselectroweak penguins

0 comments

The pith

ETRs are exact group-theory consequences of unbroken SU(3) when small Wilson coefficients c7,8 are neglected.

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

This comment shows that electroweak penguin-tree relations hold automatically as a pure group-theory result for B to PP amplitudes whenever SU(3) flavour symmetry remains unbroken and the small Wilson coefficients c7 and c8 are set aside. The authors note that this exactness requires no hadronic matrix-element calculations and directly reduces the number of free parameters in amplitude fits. They contrast this with a recent analysis that rejects the relations and obtains a good fit, while fits that retain the relations yield a poor fit, and they identify several weaknesses in the opposing formalism that mixes topological diagrams with QCD factorization.

Core claim

If SU(3) is unbroken and the small Wilson coefficients c7,8 in the weak effective Hamiltonian are neglected, ETRs follow automatically and are exact. That is, this is a group theory result -- no hadronic calculations are involved.

What carries the argument

EWP-tree relations (ETRs), which relate different topological diagrams or reduced matrix elements via unbroken SU(3) applied to the weak effective Hamiltonian after dropping c7,8.

If this is right

Fits to B to PP data that enforce the ETRs are reliable and the resulting poor fit quality reflects the data rather than an invalid assumption.
Analyses that omit the ETRs introduce unnecessary free parameters and risk unreliable conclusions.
The asserted invalidity of the ETRs in the compared paper is incorrect.
The formalism that combines topological diagrams with QCD factorization contains several weaknesses that undermine its use for these amplitudes.

Where Pith is reading between the lines

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

The exactness of the relations in the stated limit supplies a clean benchmark against which SU(3)-breaking effects can later be quantified.
The same group-theory logic may apply to other classes of B decays or to reduced matrix elements in different symmetry limits.
Retaining the ETRs would allow future global fits to test whether the current data tension persists once experimental uncertainties shrink.

Load-bearing premise

Neglecting the small Wilson coefficients c7,8 is justified when claiming the relations are exact, and the unbroken-SU(3) limit is the relevant regime for the B to PP amplitudes.

What would settle it

An explicit evaluation of the relevant matrix elements in the exact SU(3) limit with c7 and c8 set to zero that produces amplitudes violating the stated ETRs would falsify the claim.

read the original abstract

Recently, a fit to $B \to PP$ decays ($P \in \{\pi, K, \eta, \eta'\}$) was performed (arXiv:2604.19612, "QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case''}) using a formalism that combines topological diagrams with QCD factorization, and a good fit was found. We also recently performed such a fit, under the assumption that the $B \to PP$ amplitudes are related by flavour SU(3) symmetry, but we found a very poor fit. The two results therefore disagree with one another. The source of this disagreement is that we applied EWP-tree relations (ETRs). These were derived $\sim 30$ years ago, and relate different topological diagrams or reduced matrix elements, thus reducing the number of unknown parameters in the fit. In their paper, it is asserted that ETRs are invalid, so that analyses that use them are unreliable. We are writing this Comment to explain why this assertion is incorrect. The key point is that ETRs are mathematically rigorous, group theoretically. If SU(3) is unbroken, and the small Wilson coefficients $c_{7,8}$ in the weak effective Hamiltonian are neglected, ETRs follow automatically and are exact. That is, this is a group theory result -- no hadronic calculations are involved. In this Comment, we also point out several weaknesses of their formalism.

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

This comment correctly notes that ETRs are exact group-theory results under unbroken SU(3) with c7,8 dropped, but adds little beyond restating that point and flagging the fit disagreement.

read the letter

The central claim here is that ETRs follow automatically from SU(3) symmetry and the weak Hamiltonian structure once the small c7,8 coefficients are neglected. This is a group-theory statement with no hadronic modeling involved, and the comment is right to say the other paper's assertion that the relations are invalid does not hold under those conditions.

The paper does a clean job of laying out the assumptions and showing why the relations reduce parameter count without extra input. It also flags the practical disagreement in fit quality between the two analyses, which is the real issue worth resolving.

The soft spots are that the argument restates results from decades ago with no new derivation, so the novelty is limited. The text mentions weaknesses in the other formalism but does not detail them enough to let a reader judge their impact on the fit. The unbroken-SU(3) limit is the right regime for the claim, yet real decays involve breaking, and the comment could note how that affects the comparison.

This is for specialists running SU(3)-based fits to B to PP decays. A reader tracking how parameter counting and symmetry relations are applied will get direct value from the clarification. The thinking is coherent on its own terms and engages the literature properly.

I would bring it to a reading group focused on flavor physics. I would not cite it in my own work in the next year. It deserves peer review so the community can sort out whether the fit difference traces to the ETRs or to other modeling choices.

Referee Report

0 major / 2 minor

Summary. This Comment paper asserts that the electroweak penguin-tree relations (ETRs) applied in SU(3)-based fits to B→PP decays are exact group-theoretic consequences of unbroken SU(3) flavor symmetry once the small Wilson coefficients c_{7,8} are neglected in the weak effective Hamiltonian. It explains the disagreement with the fit of arXiv:2604.19612 (which reports a good fit without ETRs) by noting that the authors' own fit was poor when ETRs were imposed, and states that the other paper's claim of ETR invalidity is therefore incorrect. The manuscript further indicates that it identifies several weaknesses in the opposing formalism.

Significance. If the central claim holds, the paper usefully reaffirms that ETRs are parameter-free results of established SU(3) representation theory with no hadronic input required under the stated conditions. This strengthens the case for consistent use of symmetry relations in amplitude fits and clarifies why omitting them can produce discrepant numerical results. The explicit statement of the unbroken-SU(3) plus c_{7,8}=0 premises is a strength.

minor comments (2)

The abstract states that 'several weaknesses' of the arXiv:2604.19612 formalism are pointed out, but the provided text does not enumerate them; if the full manuscript contains a dedicated section, ensure each weakness is tied to a specific equation or assumption in the target paper.
The original derivation of the ETRs is referenced only as '~30 years ago'; adding a citation to the foundational group-theory papers would improve traceability without altering the group-theory argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee correctly summarizes our central claim that the ETRs are exact consequences of unbroken SU(3) flavor symmetry when the small Wilson coefficients c_{7,8} are neglected, with no hadronic input required. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's core argument asserts that ETRs are exact group-theory consequences of unbroken SU(3) plus neglect of c7,8 operators, citing derivations from ~30 years ago with no hadronic input required. This rests on long-established external SU(3) representation theory rather than any internal fit, self-definition, or load-bearing self-citation. The provided text states the assumptions explicitly and presents the result as independent of the hadronic modeling under discussion, with no equations or steps that reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption of unbroken SU(3) symmetry and the neglect of small Wilson coefficients; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption SU(3) flavor symmetry is unbroken
Invoked to assert that ETRs follow automatically as a group-theory result.
domain assumption Neglect of Wilson coefficients c7,8 is valid for exactness of the relations
Stated explicitly as the condition under which ETRs hold without hadronic input.

pith-pipeline@v0.9.1-grok · 5804 in / 1335 out tokens · 26035 ms · 2026-06-28T13:35:08.526628+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 9 linked inside Pith

[1]

the naive EWP–tree relations are badly broken

The claim that “the naive EWP–tree relations are badly broken” is incorrect. If SU(3) is unbroken and the Wilson coefficientsc 7,8 are neglected, the ETRs can be derived straightforwardly and are exact. That is, they are rigorous mathematical results, based only on group theory
[2]

The authors argue that, because the ETRs are badly broken, it is necessary to keep all 10 RMEs as independent in the TDA analysis

When one uses the ETRs, the number of independent SU(3) RMEs is reduced from 10 to 7. The authors argue that, because the ETRs are badly broken, it is necessary to keep all 10 RMEs as independent in the TDA analysis. The problem is that Ref. [9], cited in their paper, does not agree with this. Ref. [9] works in the TDA formalism, derives ETRs in terms of ...
[3]

Furthermore, when nonfactorizable corrections are included in the calculation, the ETRs are badly broken

The authors state that ETRs can be derived within QCDF, but this requires some very non-trivial assumptions. Furthermore, when nonfactorizable corrections are included in the calculation, the ETRs are badly broken. They conclude that there is a problem with the ETRs. This is incorrect – the problem is not with the ETRs, but with the QCDF calculation. Firs...
[4]

beyond theSU(3)-symmetric case

The authors say that their formalism goes “beyond theSU(3)-symmetric case.” In our opinion, their formalism cannot be characterized in this way. In order to properly analyze SU(3) breaking, it is necessary to begin with an SU(3)-symmetric theory (including the ETRs) and then systematically add SU(3)-breaking effects. This is not what they have done. At th...
[5]

But the fact is that this prediction is aconsequenceof the ETRs, those same ETRs that they claim are unreliable

The authors stress the importance of theB→Kπsum rule, which says that ∆ SR = 0 is predicted in the SM. But the fact is that this prediction is aconsequenceof the ETRs, those same ETRs that they claim are unreliable. ACKNOWLEDGMENTS We thank Marianne Bouchard, Alexandre Jean, and Ipsita Ray for discussions. This work was financially supported by the Nation...
[6]

Neubert and J

M. Neubert and J. L. Rosner, Phys. Lett. B441, 403 (1998), arXiv:hep-ph/9808493

Pith/arXiv arXiv 1998
[7]

Gronau, O

M. Gronau, O. F. Hernandez, D. London, and J. L. Rosner, Phys. Rev. D50, 4529 (1994), arXiv:hep-ph/9404283

Pith/arXiv arXiv 1994
[8]

Gronau, D

M. Gronau, D. Pirjol, and T.-M. Yan, Phys. Rev. D60, 034021 (1999), [Erratum: Phys.Rev.D 69, 119901 (2004)], arXiv:hep-ph/9810482

Pith/arXiv arXiv 1999
[9]

Berthiaume, B

R. Berthiaume, B. Bhattacharya, R. Boumris, A. Jean, S. Kumbhakar, and D. London, Phys. Rev. Lett.133, 211802 (2024), arXiv:2311.18011 [hep-ph]

arXiv 2024
[10]

Bhattacharya, M

B. Bhattacharya, M. Bouchard, L. Hudy, A. Jean, D. London, and C. MacKenzie, Phys. Rev. D112, 056014 (2025), arXiv:2505.11492 [hep-ph]

arXiv 2025
[11]

Bhattacharya, M

B. Bhattacharya, M. Bouchard, A. Jean, D. London, and I. Ray, JHEP05, 195, arXiv:2510.13969 [hep-ph]

Pith/arXiv arXiv
[12]

Grossman, Z

Y. Grossman, Z. Ligeti, and Y. Nir, (2026), arXiv:2605.04137 [hep-ph]

Pith/arXiv arXiv 2026
[13]

W.-S. Fang, T. Huber, X.-Q. Li, E. Malami, and G. Tetlalmatzi-Xolocotzi, (2026), arXiv:2604.19612 [hep-ph]

Pith/arXiv arXiv 2026
[14]

Y.-J. Shi, W. Wang, and J. Xu, Eur. Phys. J. C85, 1283 (2025), arXiv:2505.21283 [hep-ph]

arXiv 2025
[15]

He and W

X.-G. He and W. Wang, Chin. Phys. C42, 103108 (2018), arXiv:1803.04227 [hep-ph]

Pith/arXiv arXiv 2018
[16]

Beneke, Theory of (quasi-) two-body non-leptonic B decays,https://indico.mitp

M. Beneke, Theory of (quasi-) two-body non-leptonic B decays,https://indico.mitp. uni-mainz.de/event/177/contributions/989/attachments/771/825/mainz_beneke. pdf
[17]

Gronau, Phys

M. Gronau, Phys. Lett. B727, 136 (2013), arXiv:1308.3448 [hep-ph]. 7

Pith/arXiv arXiv 2013
[18]

Gronau, Phys

M. Gronau, Phys. Lett. B627, 82 (2005), arXiv:hep-ph/0508047. 8

Pith/arXiv arXiv 2005

[1] [1]

the naive EWP–tree relations are badly broken

The claim that “the naive EWP–tree relations are badly broken” is incorrect. If SU(3) is unbroken and the Wilson coefficientsc 7,8 are neglected, the ETRs can be derived straightforwardly and are exact. That is, they are rigorous mathematical results, based only on group theory

[2] [2]

The authors argue that, because the ETRs are badly broken, it is necessary to keep all 10 RMEs as independent in the TDA analysis

When one uses the ETRs, the number of independent SU(3) RMEs is reduced from 10 to 7. The authors argue that, because the ETRs are badly broken, it is necessary to keep all 10 RMEs as independent in the TDA analysis. The problem is that Ref. [9], cited in their paper, does not agree with this. Ref. [9] works in the TDA formalism, derives ETRs in terms of ...

[3] [3]

Furthermore, when nonfactorizable corrections are included in the calculation, the ETRs are badly broken

The authors state that ETRs can be derived within QCDF, but this requires some very non-trivial assumptions. Furthermore, when nonfactorizable corrections are included in the calculation, the ETRs are badly broken. They conclude that there is a problem with the ETRs. This is incorrect – the problem is not with the ETRs, but with the QCDF calculation. Firs...

[4] [4]

beyond theSU(3)-symmetric case

The authors say that their formalism goes “beyond theSU(3)-symmetric case.” In our opinion, their formalism cannot be characterized in this way. In order to properly analyze SU(3) breaking, it is necessary to begin with an SU(3)-symmetric theory (including the ETRs) and then systematically add SU(3)-breaking effects. This is not what they have done. At th...

[5] [5]

But the fact is that this prediction is aconsequenceof the ETRs, those same ETRs that they claim are unreliable

The authors stress the importance of theB→Kπsum rule, which says that ∆ SR = 0 is predicted in the SM. But the fact is that this prediction is aconsequenceof the ETRs, those same ETRs that they claim are unreliable. ACKNOWLEDGMENTS We thank Marianne Bouchard, Alexandre Jean, and Ipsita Ray for discussions. This work was financially supported by the Nation...

[6] [6]

Neubert and J

M. Neubert and J. L. Rosner, Phys. Lett. B441, 403 (1998), arXiv:hep-ph/9808493

Pith/arXiv arXiv 1998

[7] [7]

Gronau, O

M. Gronau, O. F. Hernandez, D. London, and J. L. Rosner, Phys. Rev. D50, 4529 (1994), arXiv:hep-ph/9404283

Pith/arXiv arXiv 1994

[8] [8]

Gronau, D

M. Gronau, D. Pirjol, and T.-M. Yan, Phys. Rev. D60, 034021 (1999), [Erratum: Phys.Rev.D 69, 119901 (2004)], arXiv:hep-ph/9810482

Pith/arXiv arXiv 1999

[9] [9]

Berthiaume, B

R. Berthiaume, B. Bhattacharya, R. Boumris, A. Jean, S. Kumbhakar, and D. London, Phys. Rev. Lett.133, 211802 (2024), arXiv:2311.18011 [hep-ph]

arXiv 2024

[10] [10]

Bhattacharya, M

B. Bhattacharya, M. Bouchard, L. Hudy, A. Jean, D. London, and C. MacKenzie, Phys. Rev. D112, 056014 (2025), arXiv:2505.11492 [hep-ph]

arXiv 2025

[11] [11]

Bhattacharya, M

B. Bhattacharya, M. Bouchard, A. Jean, D. London, and I. Ray, JHEP05, 195, arXiv:2510.13969 [hep-ph]

Pith/arXiv arXiv

[12] [12]

Grossman, Z

Y. Grossman, Z. Ligeti, and Y. Nir, (2026), arXiv:2605.04137 [hep-ph]

Pith/arXiv arXiv 2026

[13] [13]

W.-S. Fang, T. Huber, X.-Q. Li, E. Malami, and G. Tetlalmatzi-Xolocotzi, (2026), arXiv:2604.19612 [hep-ph]

Pith/arXiv arXiv 2026

[14] [14]

Y.-J. Shi, W. Wang, and J. Xu, Eur. Phys. J. C85, 1283 (2025), arXiv:2505.21283 [hep-ph]

arXiv 2025

[15] [15]

He and W

X.-G. He and W. Wang, Chin. Phys. C42, 103108 (2018), arXiv:1803.04227 [hep-ph]

Pith/arXiv arXiv 2018

[16] [16]

Beneke, Theory of (quasi-) two-body non-leptonic B decays,https://indico.mitp

M. Beneke, Theory of (quasi-) two-body non-leptonic B decays,https://indico.mitp. uni-mainz.de/event/177/contributions/989/attachments/771/825/mainz_beneke. pdf

[17] [17]

Gronau, Phys

M. Gronau, Phys. Lett. B727, 136 (2013), arXiv:1308.3448 [hep-ph]. 7

Pith/arXiv arXiv 2013

[18] [18]

Gronau, Phys

M. Gronau, Phys. Lett. B627, 82 (2005), arXiv:hep-ph/0508047. 8

Pith/arXiv arXiv 2005