Pith. sign in

REVIEW 3 major objections 4 minor 86 references

Hilbert Series of Pseudoscalar Mesons: Operators and Sum Rules

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that all leading-order B-meson pseudoscalar decay sum rules are the null vectors of one projection matrix built from a complete four-meson, two-derivative operator basis with CKM spurions.

desk verdict A genuinely new Hilbert-series application to B-meson flavor sum rules, but the central rank claim is asserted rather than shown; referee it, but require the explicit operator reduction. read the letter →

arxiv 2608.11118 v1 pith:2QTKEL4T submitted 2026-08-11 hep-ph hep-exhep-th

Joydeep Chakrabortty , Sabyasachi Chakraborty , Siddhartha Karmakar This is my paper · ORCID
classification hep-phhep-exhep-th PACS 11.30.Rd12.39.Fe13.25.Hw
keywords HilbertseriesoperatorbasisCKMspurionsBmesondecaysflavorsumruleschiralperturbationtheorySU(3)symmetryeffectivefield
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

The paper establishes that the symmetries of the light and heavy pseudoscalar mesons, together with the CKM matrix treated as a spurion, determine the full set of leading-order amplitude relations in $B$ decays to two and three pseudoscalars. Using the Hilbert series to enumerate non-redundant local operators with four meson fields and two derivatives, the authors show that physical decay amplitudes are linear combinations of a small number of universal Wilson coefficients, and that every amplitude sum rule is a null vector of the amplitude-projection matrix. For the 24 charmless two-body modes this matrix has rank five, so nineteen independent relations follow, including the familiar isospin and $U$-spin relations and seven new relations involving the $\eta_8$ octet state. The same construction yields eleven relations for $B\to DP$ with $b\to c\bar u q$, sixteen for the $b\to u\bar c q$ class, and twenty-eight for charmless three-body contact amplitudes. A sympathetic reader would care because the basis provides a model-independent parametrization in which Wilson coefficients play the role of reduced hadronic amplitudes.

What carries the argument

The machinery is the Hilbert series with Haar projection: a plethystic exponential encodes all field monomials, and character orthogonality projects out the invariants under $SU(3)_L\times SU(3)_R\times SU(2)_{L_H}\times SU(2)_{R_H}$, with a momentum generating function that removes integration-by-parts redundancies and derivative constraints. The CKM elements are promoted to non-dynamical spurion fields whose representations (Table 1) carry the flavor structure of the weak Hamiltonian. The load-bearing output is the projection matrix $M$ whose rows are physical decay amplitudes expanded in the operator basis; its rank deficit $N-\mathrm{rank}(M)$ counts the independent sum rules. A secondary mechanism is the unfolding of each invariant operator into a pair of mesonic currents joined by a mediator, which identifies the SM tree, penguin, exchange, and annihilation topology behind each Wilson coefficient.

What would settle it

Measure the reduced amplitudes of the twenty-four charmless $B\to P_1P_2$ modes, stripping the common CKM factors and the single momentum variable as in Table 4 and Appendix B.1, and test the null space of that matrix; if the data fail to lie in the five-dimensional operator row space beyond the expected $\pi^0$--$\eta_8$ mixing and $SU(3)$-breaking corrections, the D6 four-meson, two-derivative basis is not the leading-order parametrization.

Watch

Extended reading notes

Core claim

The central discovery is that, once the full non-redundant $D6$ operator basis is fixed, no separate topological-amplitude or $SU(3)_V$ reduced-matrix-element input is needed: the amplitude sum rules follow automatically as the null space of the matrix $M$ that projects the operators onto physical channels, with $N$ amplitudes and $r$ independent operators giving $N-\mathrm{rank}(M)$ relations. Concretely, Table 4 gives twenty-four charmless $B\to P_1P_2$ amplitudes and only five independent operator structures, so $\mathrm{rank}(M)=5$ and the nineteen relations in eqs. (3.15)--(3.33) emerge, reproducing the isospin relations (3.15)--(3.16), the $U$-spin relations (3.17)--(3.19), and the full-$SU(3)$ relations (3.20)--(3.33). Seven of these, eqs. (3.27)--(3.33), involve an $\eta_8$ in the final state and are new leading-order predictions, because $\eta_8$ is an isosinglet that is not a $U$-spin eigenstate. For three-body charmless modes the operator count again restricts the 32 amplitudes to rank four, yielding 28 relations that hold locally in phase space for the non-resonant contact amplitudes.

Load-bearing premise

The construction assumes that the leading low-energy weak decays are described by local contact operators made from four meson fields and two derivatives, with the quark-mixing matrix encoded as a fixed background structure; anything beyond that—higher derivatives, quark-mass insertions, or resonance cascades—must be suppressed at this order.

Editorial extensions

If this is right

  • The nineteen charmless $B\to P_1P_2$ relations include the known isospin and $U$-spin identities as particular null vectors, so those classic results are recovered from the operator basis without a separate reduced-amplitude decomposition.
  • Seven relations containing $\eta_8$ are new leading-order flavor predictions; after octet-singlet mixing they constrain $\eta$ and $\eta'$ modes and provide pure-octet reference amplitudes.
  • The twenty-eight charmless three-body relations hold locally in the Dalitz plot for non-resonant contact pieces, allowing information from well-measured $B_{u,d}\to K\pi\pi$ modes to be transferred to largely unmeasured $\bar B_s$ channels.
  • For $B\to DP$, the basis yields eleven independent relations in the $b\to c\bar u q$ sector and sixteen in the $b\to u\bar c q$ sector, extending the usual isospin triangles and $U$-spin equalities with full-$SU(3)$ octet identities.
  • The unfolding procedure assigns each operator to a definite weak topology, and marks operators requiring a flavor-octet mediator as having no SM tree-level origin, thereby isolating candidate new-physics contributions.

Reading between the lines

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

  • If the D6 contact basis dominates, the new $\eta_8$ relations translate into quantitative tests of $\eta$--$\eta'$ mixing: measurable violations in physical $\eta$ and $\eta'$ modes would pinpoint octet-singlet contamination or the first corrections to the leading-order basis.
  • The same null-space counting applies to other Goldstone sectors; for example, extending the basis to vector mesons or adding mass spurions should produce a nested hierarchy of sum rules whose breaking pattern could be fitted from data to determine which operator is first to contribute.
  • The comparison with conventional $SU(3)_V$ analyses suggests an algebraic criterion: whenever chiral power counting keeps fewer operators than the reduced-amplitude count, the EFT will predict extra leading-order relations, a fact that could be used to search for additional testable identities in other heavy-meson decay classes.
  • Reading the unfolding backwards, operators without SM tree-level origin identify specific new-physics mediators, and the measured quality of the D6 sum rules could bound their masses or couplings.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constructs a non-redundant operator basis for pseudoscalar mesons using the Hilbert series method, with the global flavor symmetry SU(3)_L × SU(3)_R × SU(2)_LH × SU(2)_RH, and promotes CKM matrix elements to spurion fields. The authors classify D6 operators containing four meson fields and two derivatives, and project them onto two- and three-body B-meson decay amplitudes. The central claim is that the 24 charmless B→P1P2 amplitudes are governed by a 24×5 amplitude matrix of rank 5, yielding 19 independent sum rules (eqs. (3.15)–(3.33)), which reproduce known isospin and U-spin relations and include seven new η8 octet relations. The framework is also applied to B→DP and three-body decays, and an 'unfolding' procedure is proposed to identify ultraviolet origins of the operators.

Significance. If the central rank assertions are correct, the paper provides a systematic and model-independent method to generate flavor sum rules for B-meson decays, with the operator basis serving as the reduced-amplitude basis. The use of Hilbert series to construct mesonic operator bases is a useful methodological advance, and the explicit catalogues of operators and amplitudes in the tables and appendices are extensive. The paper honestly acknowledges that the new η8 relations are leading-order statements and are subject to π0–η8 mixing and SU(3) breaking. The main result, however, depends crucially on an unshown reduction from 12 operators to 5 independent contributions; until that reduction is exhibited, the completeness of the 19 sum rules and the novelty of the η8 relations are not independently verifiable.

major comments (3)
  1. [Sec. 3.4 and Table 4] The paper asserts that 'among the 12 operators of Table 3 only 5 independent operators contribute' to B→P1P2, but it never shows the projection of O1a, O2a, O3a–d, or O4b onto the 24 modes. This reduction is load-bearing because the rank(M)=5 claim and the 19 sum rules in eqs. (3.15)–(3.33) are null vectors of that specific matrix. Since O2b contributes through the λb background of ΣH, the vanishing of O3a–d (which also contain ΣH†) is not obvious. If any omitted operator has a nonzero projection that is not a linear combination of the five retained columns, the null-space dimension drops below 19 and the claimed completeness fails. The authors should provide the explicit row-reduction (e.g., the zero columns, a reduction log, or a small script) so that the rank statement can be checked.
  2. [Sec. 3.4, eqs. (3.15)–(3.33)] The sum rules are presented as relations among physical amplitudes A(mode), but they are derived after factoring out 'the common CKM and the momentum dependence' from each row of Table 4. For rows with different CKM factors (e.g., V_dαV_bα vs. V_bαV_sα), such as in eq. (3.25), the relation is not directly a relation among physical amplitudes unless the CKM factors are first restored. The paper states this caveat in Sec. 4.2.1 for B→DP, but not in Sec. 3.4. The authors should clarify in Sec. 3.4 that eqs. (3.15)–(3.33) hold for the reduced amplitudes, and should either suppress the physical-amplitude notation or explicitly define the reduced amplitudes Ã(mode).
  3. [Sec. 3.4, eqs. (3.27)–(3.33)] The text calls the seven η8 relations 'new', but the same paragraph states that all relations (3.22)–(3.33) 'can be also derived from the topological amplitudes given in [61–65]'. These statements need to be reconciled: are the relations new because they are not explicitly written in the literature, or are they new consequences of the EFT operator basis? The distinction matters for the paper's novelty claim. Additionally, the leading-order nature and the corrections from π0–η8 mixing are acknowledged, but the reduced-amplitude caveat of the previous comment is especially relevant for these relations and should be discussed when assessing their phenomenological utility.
minor comments (4)
  1. [Sec. 3.4, Table 4] The notation A(mode) is used throughout eqs. (3.15)–(3.33) for quantities that are actually reduced amplitudes after CKM and momentum factoring; using a tilde (e.g., Ã) would avoid ambiguity.
  2. [Sec. 2.3] The notation 'bQi bQ†j' in the sentence defining the operator class is undefined; the spurion labels should be written out or defined before use.
  3. [Sec. 3.4, first paragraph] There is a typo in 'B → P Psum rules' (missing space); the authors should proofread for similar spacing and typesetting errors.
  4. [Sec. 6, Table 13] The 'Comment' column in Table 13 uses abbreviations such as PSM, P ASM, ASM without definition in the caption or text; these should be spelled out on first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sum rules are null vectors of an explicitly projected operator matrix with no fitted parameters, and the one author-overlap citation (Ref. [34]) is non-load-bearing.

full rationale

The paper's derivation chain is self-contained rather than circular. The Hilbert series machinery (Sec. 2) constructs invariant operators from declared field content and spurion representations; the projection onto 24 B→P1P2 channels is written out in Table 4 and Appendix B.1; and the nineteen sum rules are computed as the null space of that projection matrix (Sec. 3.4). No step fits a parameter to a subset of the amplitudes and then renames it a prediction, and no amplitude relation is inserted into the operator basis by construction. The Wilson coefficients enter as undetermined coefficients, so the null-space relations are genuine consequences of the stated operator basis. The known isospin and U-spin relations are reproduced, and the 'new' η8 relations are checked against the topological-amplitude framework of Refs. [61-65], providing an external consistency test. The only author-overlapping citation is Ref. [34], used for the CKM spurion matrices and the traced-form basis; those are standard SM current-matching results that are parameter-free and independently checkable, and they organize rather than force the sum-rule claim. There is a real verification gap: Sec. 3.4 states without proof that only five of the twelve Table 3 operators contribute, and the omitted operators, especially O3a-d, are not shown to have zero or linearly dependent B→P1P2 projections. That is an omitted derivation and a correctness risk, but it is not a circular step. No equation in the paper reduces by construction to its own input, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard group-integration technology (character orthogonality, Haar measures, plethystic exponentials) and on three physical-modeling choices: the nonlinear chiral realization of the light mesons, the assignment of heavy-light and heavy-heavy mesons to SU(2)_LH x SU(2)_RH representations, and the promotion of CKM elements to spurions with the charges in Table 1. No new particles are postulated; the heavy-quark condensate parameters lambda_b and lambda_c are the only hand-chosen constants, and they drop out of the null-space sum rules.

free parameters (1)
  • lambda_b and lambda_c = not fitted
    Introduced in Eq. (3.5) as real parameters proportional to heavy-quark condensates <bb> and <cc>; they appear in the projected amplitudes via Sigma_H background insertions but cancel in the null-space sum rules.
assumptions (5)
  • standard math Character orthogonality and Haar integration correctly enumerate all invariant polynomials under G3322.
    Used in Sec. 2.1 to project singlet operators from the plethystic exponential; this is the core counting assumption of the Hilbert series method.
  • domain assumption The chiral field Sigma realizes the spontaneously broken SU(3)_L x SU(3)_R in the nonlinear form Sigma = (f_pi/2) exp(2iM/f_pi).
    This is the standard chiral perturbation theory building block invoked in Sec. 3.1, and all light-meson expansions are based on it.
  • domain assumption The weak CKM matrix elements are promoted to spurion fields bQ_ij with the transformation properties of Table 1, taken from Ref [34].
    This is the mechanism by which the SM weak interactions are inserted into the flavor-covariant operators; if these representations are incomplete, the derived sum rules are incomplete.
  • ad hoc to paper The heavy-light and heavy-heavy mesons are assigned to SU(2)_LH x SU(2)_RH representations, a non-dynamical bookkeeping symmetry.
    The paper states in Sec. 3.1 that this is not a dynamical symmetry of QCD but a spurion symmetry used to track flavor flow; the doubling is a modeling choice specific to this construction.
  • domain assumption Leading contributions to two- and three-body B decays are captured by D6 operators with four meson fields and two derivatives; higher-derivative and mass-spurion operators are suppressed.
    This truncation is defined in Sec. 2.3 and defended only qualitatively in Sec. 7, with no estimate of the size of neglected corrections.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hilbert Series of Pseudoscalar Mesons: Operators and Sum Rules." pith.science (2026). https://pith.science/paper/2QTKEL4T

@misc{pith2026260811118,
  author       = {Pith},
  title        = {Pith review of: Hilbert Series of Pseudoscalar Mesons: Operators and Sum Rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QTKEL4T}},
  note         = {Machine review of arXiv:2608.11118}
}
abstract

We construct the complete non-redundant basis of operators built from pseudoscalar mesons using the Hilbert series method, subject to the global flavor symmetry $SU(3)_L \times SU(3)_R \times SU(2)_{L_H} \times SU(2)_{R_H}$, together with the underlying shift symmetry. The SM weak interactions are incorporated systematically by promoting the CKM matrix elements to spurion fields transforming under the global symmetry. The resulting operator basis provides a model-independent parametrization of two- and three-body weak decay amplitudes of heavy mesons, with the corresponding Wilson coefficients serving as reduced hadronic amplitudes. We demonstrate that all amplitude sum rules follow directly from the structure of the operator basis and propose an unfolding procedure that connects the effective operators to their possible ultraviolet origins. Our framework reproduces the known flavor relations while predicting additional identities arising from the symmetry-constrained operator basis.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

86 extracted references · 25 canonical work pages

  1. [1]

    Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327–340

    S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327–340

    1979

  2. [2]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop , Annals Phys. 158 (1984) 142

    1984

  3. [3]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark , Nucl. Phys. B 250 (1985) 465–516

    1985

  4. [4]

    Isgur and M

    N. Isgur and M. B. Wise, Weak Decays of Heavy Mesons in the Static Quark Approximation , Phys. Lett. B 232 (1989) 113–117

    1989

  5. [5]

    Eichten and B

    E. Eichten and B. R. Hill, An Effective Field Theory for the Calculation of Matrix Elements Involving Heavy Quarks, Phys. Lett. B 234 (1990) 511–516

    1990

  6. [6]

    Isgur and M

    N. Isgur and M. B. Wise, Weak transition form factors between heavy mesons , Phys. Lett. B 237 (1990) 527–530

    1990

  7. [7]

    Georgi, An Effective Field Theory for Heavy Quarks at Low-energies , Phys

    H. Georgi, An Effective Field Theory for Heavy Quarks at Low-energies , Phys. Lett. B 240 (1990) 447–450

    1990

  8. [8]

    Grinstein, The Static Quark Effective Theory , Nucl

    B. Grinstein, The Static Quark Effective Theory , Nucl. Phys. B 339 (1990) 253–268

    1990

Show all 86 references
  1. [9]

    Neubert, Heavy quark symmetry , Phys

    M. Neubert, Heavy quark symmetry , Phys. Rept. 245 (1994) 259–396, [ hep-ph/9306320]

    1994 arXiv

  2. [10]

    A. V. Manohar and M. B. Wise, Heavy quark physics , vol. 10. Cambridge University Press, 2000, 10.1017/9781009402125

    2000 doi

  3. [11]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays , Phys. Rev. D 63 (2001) 114020, [ hep-ph/0011336]

    2001 arXiv

  4. [12]

    C. W. Bauer, S. Fleming and M. E. Luke, Summing Sudakov logarithms in B → Xsγ in effective field theory., Phys. Rev. D 63 (2000) 014006, [ hep-ph/0005275]

    2000 arXiv

  5. [13]

    C. W. Bauer and I. W. Stewart, Invariant operators in collinear effective theory , Phys. Lett. B 516 (2001) 134–142, [ hep-ph/0107001]

    2001 arXiv

  6. [14]

    C. W. Bauer, D. Pirjol and I. W. Stewart, Soft collinear factorization in effective field theory , Phys. Rev. D 65 (2002) 054022, [ hep-ph/0109045]

    2002 arXiv

  7. [15]

    Beneke, A

    M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power , Nucl. Phys. B 643 (2002) 431–476, [ hep-ph/0206152]

    2002 arXiv

  8. [16]

    Gasser and H

    J. Gasser and H. Leutwyler, Low-Energy Expansion of Meson Form-Factors , Nucl. Phys. B 250 (1985) 517–538

    1985

  9. [17]

    Colangelo, J

    G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125–179, [hep-ph/0103088]

    2001 arXiv

  10. [18]

    Bernard, N

    V. Bernard, N. Kaiser and U. G. Meissner, πK scattering in chiral perturbation theory to one loop , Nucl. Phys. B 357 (1991) 129–152

    1991

  11. [19]

    Gasser and H

    J. Gasser and H. Leutwyler, Spontaneously broken symmetries: Effective lagrangians at finite volume , Nucl. Phys. B 307 (1988) 763–778

    1988

  12. [20]

    Colangelo, S

    G. Colangelo, S. Durr and C. Haefeli, Finite volume effects for meson masses and decay constants , Nucl. Phys. B 721 (2005) 136–174, [ hep-lat/0503014]

    2005 arXiv

  13. [21]

    Burdman, Z

    G. Burdman, Z. Ligeti, M. Neubert and Y. Nir, The Decay B → πℓν in heavy quark effective theory , Phys. Rev. D 49 (1994) 2331–2345, [ hep-ph/9309272]. – 52 –

    1994 arXiv

  14. [22]

    Charles, A

    J. Charles, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Heavy to light form-factors in the heavy mass to large energy limit of QCD , Phys. Rev. D 60 (1999) 014001, [ hep-ph/9812358]

    1999 arXiv

  15. [23]

    Grinstein and D

    B. Grinstein and D. Pirjol, Exclusive rare B → K ∗ℓ+ℓ− decays at low recoil: Controlling the long-distance effects, Phys. Rev. D 70 (2004) 114005, [ hep-ph/0404250]

    2004 arXiv

  16. [24]

    Caprini, L

    I. Caprini, L. Lellouch and M. Neubert, Dispersive bounds on the shape of B → D(∗)ℓν form-factors, Nucl. Phys. B 530 (1998) 153–181, [ hep-ph/9712417]

    1998 arXiv

  17. [25]

    Beneke, T

    M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive B → V l+l−, V γdecays, Nucl. Phys. B 612 (2001) 25–58, [ hep-ph/0106067]

    2001 arXiv

  18. [26]

    Beneke, T

    M. Beneke, T. Feldmann and D. Seidel, Exclusive radiative and electroweak b → d and b → s penguin decays at NLO , Eur. Phys. J. C 41 (2005) 173–188, [ hep-ph/0412400]

    2005 arXiv

  19. [27]

    C. W. Bauer, D. Pirjol and I. W. Stewart, Factorization and endpoint singularities in heavy to light decays, Phys. Rev. D 67 (2003) 071502, [ hep-ph/0211069]

    2003 arXiv

  20. [28]

    Becher and M

    T. Becher and M. D. Schwartz, A precise determination of αs from LEP thrust data using effective field theory, JHEP 07 (2008) 034, [ 0803.0342]

    2008 arXiv

  21. [29]

    Bisht, S

    D. Bisht, S. Chakraborty, S. Karmakar and A. Samanta, Soft Collinear Effective Field Theory for Heavy QCD Axions , (In preparation) (2026)

    2026

  22. [30]

    M. B. Wise, Chiral perturbation theory for hadrons containing a heavy quark , Phys. Rev. D 45 (1992) R2188

    1992

  23. [31]

    Burdman and J

    G. Burdman and J. F. Donoghue, Union of chiral and heavy quark symmetries , Phys. Lett. B 280 (1992) 287–291

    1992

  24. [32]

    Yan, H.-Y

    T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin and H.-L. Yu, Heavy quark symmetry and chiral dynamics, Phys. Rev. D 46 (1992) 1148–1164

    1992

  25. [33]

    Casalbuoni, A

    R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phenomenology of heavy meson chiral Lagrangians , Phys. Rept. 281 (1997) 145–238, [ hep-ph/9605342]

    1997 arXiv

  26. [34]

    Chakraborty, N

    S. Chakraborty, N. Mahajan and T. S. Roy, A Phenomenological Model of Mesons for Charged Current Weak Decays, 2605.13977

    arXiv

  27. [35]

    Hanany, E

    A. Hanany, E. E. Jenkins, A. V. Manohar and G. Torri, Hilbert Series for Flavor Invariants of the Standard Model, JHEP 03 (2011) 096, [ 1010.3161]

    2011 arXiv

  28. [36]

    Lehman and A

    L. Lehman and A. Martin, Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox, Phys. Rev. D 91 (2015) 105014, [ 1503.07537]

    2015 arXiv

  29. [37]

    Henning, X

    B. Henning, X. Lu, T. Melia and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT , JHEP 08 (2017) 016, [ 1512.03433]

    2017 arXiv

  30. [38]

    Henning, X

    B. Henning, X. Lu, T. Melia and H. Murayama, Hilbert series and operator bases with derivatives in effective field theories , Commun. Math. Phys. 347 (2016) 363–388, [ 1507.07240]

    2016 arXiv

  31. [39]

    L. Graf, B. Henning, X. Lu, T. Melia and H. Murayama, 2, 12, 117, 1959, 45171, 1170086, . . . : a Hilbert series for the QCD chiral Lagrangian , JHEP 01 (2021) 142, [ 2009.01239]

    2021 arXiv

  32. [40]

    Lagrangian

    U. Banerjee, J. Chakrabortty, S. Prakash and S. U. Rahaman, Characters and group invariant polynomials of (super)fields: road to “Lagrangian” , Eur. Phys. J. C 80 (2020) 938, [ 2004.12830]

    2020 arXiv

  33. [41]

    Chakrabortty, B

    J. Chakrabortty, B. Siqueira Eduardo, S. Karmakar and P. Schicho, Finite-temperature operator basis on R3 × S1 for SMEFT , 2605.02878. – 53 –

    arXiv

  34. [42]

    S. R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1. , Phys. Rev. 177 (1969) 2239–2247

    1969

  35. [43]

    C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247–2250

    1969

  36. [44]

    B. Feng, A. Hanany and Y.-H. He, Counting Gauge Invariants: The Plethystic Program , JHEP 03 (2007) 090, [ hep-th/0701063]

    2007 arXiv

  37. [45]

    Br¨ ocker and T

    T. Br¨ ocker and T. tom Dieck,Representations of Compact Lie Groups , vol. 98 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003

    2003

  38. [46]

    A. B. Balantekin and P. Cassak, Character expansions for the orthogonal and symplectic groups , J. Math. Phys. 43 (2002) 604–620, [ hep-th/0108130]

    2002 arXiv

  39. [47]

    R. J. Plymen, On the Weyl Character Formula for SU(N) , Int. J. Theor. Phys. 15 (1976) 201–206

    1976

  40. [48]

    Koike and I

    K. Koike and I. Terada, Littlewood’s formulas and their application to representations of classical Weyl groups, in Commutative Algebra and Combinatorics (Tokyo, 1985) , vol. 11 of Advanced Studies in Pure Mathematics, (Tokyo, Japan), pp. 147–160, Mathematical Society of Japan, 1987

    1985

  41. [49]

    D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups . AMS Chelsea Publishing Series. University Microfilms, Ann Arbor, Mich., 1977

    1977

  42. [50]

    Rossmann, Lie Groups: An Introduction Through Linear Groups , vol

    W. Rossmann, Lie Groups: An Introduction Through Linear Groups , vol. 5 of Oxford Graduate Texts in Mathematics. Oxford University Press, 2006

    2006

  43. [51]

    Lie algebras and their representations: Graduate course lecture notes

    M. Foster, “Lie algebras and their representations: Graduate course lecture notes.” http://mf23.web.rice.edu/, 2016

    2016

  44. [52]

    Slansky, Group theory for unified model building , Physics Reports 79 (1881) 1–128

    R. Slansky, Group theory for unified model building , Physics Reports 79 (1881) 1–128

  45. [53]

    Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building , 1511.08771

    N. Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building , 1511.08771

    arXiv

  46. [54]

    Hanany and R

    A. Hanany and R. Kalveks, Highest Weight Generating Functions for Hilbert Series , JHEP 10 (2014) 152, [1408.4690]

    2014 arXiv

  47. [55]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, QCD factorization for B → ππ decays: Strong phases and CP violation in the heavy quark limit , Phys. Rev. Lett. 83 (1999) 1914–1917, [hep-ph/9905312]

    1999 arXiv

  48. [56]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, QCD factorization in B → πK, ππ decays and extraction of Wolfenstein parameters , Nucl. Phys. B 606 (2001) 245–321, [ hep-ph/0104110]

    2001 arXiv

  49. [57]

    Wirbel, B

    M. Wirbel, B. Stech and M. Bauer, Exclusive Semileptonic Decays of Heavy Mesons , Z. Phys. C 29 (1985) 637

    1985

  50. [58]

    Bauer, B

    M. Bauer, B. Stech and M. Wirbel, Exclusive Nonleptonic Decays of D, D(s), and B Mesons , Z. Phys. C 34 (1987) 103

    1987

  51. [59]

    Buchalla, A

    G. Buchalla, A. J. Buras and M. E. Lautenbacher, Weak Decays beyond Leading Logarithms, Rev. Mod. Phys. 68 (1996) 1125–1144, [ hep-ph/9512380]

    1996 arXiv

  52. [60]

    Beneke, G

    M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, QCD factorization for exclusive, nonleptonic B meson decays: General arguments and the case of heavy-light final states , Nucl. Phys. B 591 (2000) 313–418, [hep-ph/0006124]. – 54 –

    2000 arXiv

  53. [61]

    Gronau and D

    M. Gronau and D. London, Isospin analysis of CP asymmetries in B decays, Phys. Rev. Lett. 65 (1990) 3381–3384

    1990

  54. [62]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Electroweak penguins and two-body B decays, Phys. Rev. D 52 (1995) 6374–6382, [ hep-ph/9504327]

    1995 arXiv

  55. [63]

    Fleischer, New strategies to extract β and γ from Bd → π+π− and Bs → K +K −, Phys

    R. Fleischer, New strategies to extract β and γ from Bd → π+π− and Bs → K +K −, Phys. Lett. B 459 (1999) 306, [ hep-ph/9903456]

    1999 arXiv

  56. [64]

    Gronau, U -spin symmetry in charmless B decays, Phys

    M. Gronau, U -spin symmetry in charmless B decays, Phys. Lett. B 492 (2000) 297–302, [hep-ph/0008292]

    2000 arXiv

  57. [65]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Decays of B mesons to two light pseudoscalars, Phys. Rev. D 50 (1994) 4529–4543, [ hep-ph/9404283]

    1994 arXiv

  58. [66]

    Feldmann, P

    T. Feldmann, P. Kroll and B. Stech, Mixing and decay constants of pseudoscalar mesons , Phys. Rev. D 58 (1998) 114006, [ hep-ph/9802409]

    1998 arXiv

  59. [67]

    Neubert and B

    M. Neubert and B. Stech, Non-leptonic weak decays of B mesons, Adv. Ser. Direct. High Energy Phys. 15 (1998) 294–344, [ hep-ph/9705292]

    1998 arXiv

  60. [68]

    Neubert and A

    M. Neubert and A. A. Petrov, Comments on color suppressed hadronic B decays , Phys. Lett. B 519 (2001) 50–56, [ hep-ph/0108103]

    2001 arXiv

  61. [69]

    Chiang and J

    C.-W. Chiang and J. L. Rosner, Final-state phases in B → Dπ, D∗π, and Dρ decays, Phys. Rev. D 67 (2003) 074013, [ hep-ph/0212274]

    2003 arXiv

  62. [70]

    Fleischer, N

    R. Fleischer, N. Serra and N. Tuning, Tests of Factorization and SU (3) Relations in B Decays into Heavy-Light Final States , Phys. Rev. D 83 (2011) 014017, [ 1012.2784]

    2011 arXiv

  63. [71]

    Altmannshofer et al., The Belle II Physics Book , PTEP 2019 (2019) 123C01, [1808.10567]

    Belle-II collaboration, W. Altmannshofer et al., The Belle II Physics Book , PTEP 2019 (2019) 123C01, [1808.10567]

    2019

  64. [72]

    Aaij et al., Measurement of CP violation in B0 → D+D− decays, Phys

    LHCb collaboration, R. Aaij et al., Measurement of CP violation in B0 → D+D− decays, Phys. Rev. Lett. 117 (2016) 261801, [ 1608.06620]

    2016 arXiv

  65. [73]

    Fayyazuddin, Final state phases in B → Dπ, ¯Dπ decays and CP-asymmetry , Phys. Rev. D 70 (2004) 114018, [hep-ph/0402189]

    2004 arXiv

  66. [74]

    Dunietz, Clean CKM information from Bd(t) → D∗±π∓, Phys

    I. Dunietz, Clean CKM information from Bd(t) → D∗±π∓, Phys. Lett. B 427 (1998) 179–182, [hep-ph/9712401]

    1998 arXiv

  67. [75]

    Fleischer, New strategies to obtain insights into CP violation through Bs → D± s K ∓, D∗± s K ∓,

    R. Fleischer, New strategies to obtain insights into CP violation through Bs → D± s K ∓, D∗± s K ∓, . . . and Bd → D±π∓, D∗±π∓, . . .decays, Nucl. Phys. B 671 (2003) 459, [ hep-ph/0304027]

    2003 arXiv

  68. [76]

    M. J. Savage and M. B. Wise, SU(3) Predictions for Nonleptonic B Meson Decays , Phys. Rev. D 39 (1989) 3346

    1989

  69. [77]

    Grinstein and R

    B. Grinstein and R. F. Lebed, SU(3) decomposition of two-body B decay amplitudes , Phys. Rev. D 53 (1996) 6344, [ hep-ph/9602218]

    1996 arXiv

  70. [78]

    A. S. Dighe, M. Gronau and J. L. Rosner, Amplitude relations for B decays involving eta and eta’ , Phys. Lett. B 367 (1996) 357, [ hep-ph/9509428]

    1996 arXiv

  71. [79]

    Bhattacharya, M

    B. Bhattacharya, M. Gronau, M. Imbeault, D. London and J. L. Rosner, Charmless B →PPP decays: The fully-symmetric final state , Phys. Rev. D 89 (2014) 074043, [ 1402.2909]

    2014 arXiv

  72. [80]

    Wang, Isospin sum rules for the nonleptonic B decays , Eur

    D. Wang, Isospin sum rules for the nonleptonic B decays , Eur. Phys. J. C 84 (2024) 949, [ 2405.20698]. – 55 –

    2024 arXiv

  73. [81]

    He, G.-N

    X.-G. He, G.-N. Li and D. Xu, SU(3) and isospin breaking effects on B → P P Pamplitudes, Phys. Rev. D 91 (2015) 014029, [ 1410.0476]

    2015 arXiv

  74. [82]

    Aaij et al., Measurement of CP violation in the phase space of B± → K ±π+π− and B± → K ±K +K − decays, Phys

    LHCb collaboration, R. Aaij et al., Measurement of CP violation in the phase space of B± → K ±π+π− and B± → K ±K +K − decays, Phys. Rev. Lett. 111 (2013) 101801, [ 1306.1246]

    2013 arXiv

  75. [83]

    Cheng and C.-K

    H.-Y. Cheng and C.-K. Chua, Charmless three-body decays of Bs mesons, Phys. Rev. D 89 (2014) 074025, [1401.5514]

    2014 arXiv

  76. [84]

    Chau, Quark Mixing in Weak Interactions , Phys

    L.-L. Chau, Quark Mixing in Weak Interactions , Phys. Rept. 95 (1983) 1–94

    1983

  77. [85]

    Chau and H.-Y

    L.-L. Chau and H.-Y. Cheng, Analysis of Exclusive Two-Body Decays of Charm Mesons Using the Quark Diagram Scheme , Phys. Rev. D 36 (1987) 137

    1987

  78. [86]

    Zeppenfeld, SU (3) Relations for B Meson Decays, Z

    D. Zeppenfeld, SU (3) Relations for B Meson Decays, Z. Phys. C 8 (1981) 77–81. – 56 –

    1981

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.