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REVIEW 2 major objections 4 minor 48 references

Patterns of flavour symmetry breaking in hadron matrix elements involving u, d and s quarks

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Along a fixed-average-quark-mass line, SU(3) flavour breaking in octet baryon matrix elements is governed by a small set of coefficients, with the last constraints ending at quadratic order in the quark-mass difference.

desk verdict The group-theoretic expansion for flavour breaking in octet baryon matrix elements is sound and useful; the numerical demonstration is illustrative, not a convergence test. read the letter →

arxiv 1909.02521 v2 pith:RQA5WEGN submitted 2019-09-05 hep-lat hep-phnucl-th

J.M. Bickerton , R. Horsley , Y. Nakamura , H. Perlt , D. Pleiter , P.E.L. Rakow , G. Schierholz , H. Stüben
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R.D. Young J.M. Zanotti
This is my paper · ORCID
classification hep-lathep-phnucl-th PACS 11.30.Hv12.38.Gc
keywords flavoursymmetrybreakingSU(3)octetbaryonshadronmatrixelementsquarkmassexpansionlatticeQCDformfactorsquark-linedisconnecteddiagrams
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 strong interaction is flavour-blind, so once the $u$, $d$ and $s$ quark masses are set equal, the only way SU(3) flavour symmetry can break in hadron matrix elements is through the quark-mass differences. This paper extends a known expansion programme from hadron masses to octet baryon matrix elements, expanding about the SU(3)-symmetric point along a trajectory of constant average quark mass, $\bar m$. It establishes exactly which expansion coefficients are independent: at linear order the first-class amplitudes are generated by the two couplings $f,d$ plus five octet coefficients, at quadratic order 11 coefficients describe 12 amplitudes so one 64-plet combination must vanish, and at cubic order a 12th coefficient appears so no further constraint exists. For second-class currents, the constrained pattern ends one order earlier, at linear order. The result is a complete group-theoretic map of flavour breaking that can guide lattice extrapolations and, ultimately, quantities such as hyperon semileptonic decay form factors.

What carries the argument

The load-bearing objects are the SU(3) flavour tensors $T_{ijk}$ in the triple product $8 \otimes 8 \otimes 8$: three-index arrays that couple octet hadrons $i,k$ to an octet current $j$. In the $n_f=2+1$ isospin limit, only the $I=0$, $Y=0$ central components survive, leaving two singlet tensors, eight octet tensors, six 27-plet tensors and one 64-plet tensor—17 in all—classified as first- or second-class and as $f$-like or $d$-like. These tensors convert the quark-mass polynomial classification into the explicit coefficient tables, and the counting of independent tensor entries against the 12 physical amplitudes yields the constraint structure, including the 64-plet identity $Q_{64}=O(\delta m_l^3)$.

What would settle it

Compute the 12 first-class octet amplitudes on four or more lattice ensembles with the average quark mass held fixed and pion masses spanning from about 465 MeV down toward the physical value, then fit them with the paper's coefficient tables; if the 64-plet combination $Q_{64}$ of eq. (62) is found nonzero at $O(\delta m_l^2)$, or if the d-fan requires more than three independent slopes at $O(\delta m_l)$, the claimed constraint structure fails.

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Extended reading notes

Core claim

The paper's central claim is that the flavour-symmetry-breaking pattern of octet baryon matrix elements is group-theoretically constrained at leading and next-to-leading order, and not beyond. Working in the $n_f = 2+1$ isospin limit on the trajectory $\bar m = \text{const.}$, the 12 first-class amplitudes are expressed in terms of the symmetric-point couplings $f$ and $d$, five octet coefficients at $O(\delta m_l)$, and 11 independent coefficients at $O(\delta m_l^2)$. Since there are 12 amplitudes, the single 64-plet combination $Q_{64}$ of eq. (62) must vanish at $O(\delta m_l^2)$; at $O(\delta m_l^3)$ the 64-plet contributes a 12th coefficient, so no further constraint exists. For second-class currents, five amplitudes meet three coefficients at $O(\delta m_l)$ and five coefficients at $O(\delta m_l^2)$, so the constrained pattern ends there. The same tensor analysis shows that at leading order only $r_1$ carries a quark-line-disconnected contribution, and that improvement coefficients for the clover vector current are absorbed into the expansion coefficients.

Load-bearing premise

The argument assumes hadronic matrix elements are analytic in the quark masses from the SU(3)-symmetric point down to the physical point, so that the truncated Taylor series in $\delta m_l$ remains accurate; the numerical demonstration covers only three pion masses, the lightest at 310 MeV, fitted linearly.

Editorial extensions

If this is right

  • All first-class octet baryon matrix elements can be parametrised through order $\delta m_l^2$ by a small set of coefficients; the fan-plot relations mean that many measured splittings must be described by only a few independent slopes.
  • The 64-plet combination $Q_{64}$ must vanish at $O(\delta m_l^2)$, a testable lattice prediction and a bridge to the one-loop chiral perturbation theory expectation.
  • For second-class currents such as the vector $F_3$ form factor, the constrained expansion ends at $O(\delta m_l)$, so no new symmetry relations appear at higher orders.
  • At leading order only the coefficient $r_1$ receives a quark-line-disconnected contribution, so the $f$-fan and related combinations are insensitive to the difficult disconnected diagrams at leading order.
  • Flavour-singlet-like combinations $X_D$ and $X_F$ contain no linear term, so constant fits extrapolate them to the physical point; the conserved-vector-current condition fixes the renormalisation constant and two improvement coefficients.

Reading between the lines

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

  • The same counting logic could be applied to the decuplet, to octet-decuplet transitions, and to meson octet matrix elements, where analogous flavour tensors would give their own fan relations and constraint counts.
  • If the expansions remain accurate down to the physical point, the flat $X$ functions provide a practical lattice procedure—constant fits for flavour-singlet-like combinations—that could also serve for scale setting or renormalisation checks without chiral perturbation theory.
  • A partially quenched implementation, with valence and sea quark masses different, could use the same coefficient tables to constrain the expansion parameters over a wider mass range than the unitary line permits, and would sharpen the test of analyticity.
  • Because the constrained structure closes at quadratic order, future lattice data showing significant curvature in the $X$ functions would indicate a breakdown of the Taylor expansion itself rather than missing SU(3) representations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper develops a systematic SU(3)-flavour-breaking expansion for octet-baryon matrix elements of generalized currents, expanding about the SU(3)-symmetric point along a trajectory of constant singlet quark mass. The core of the paper is a group-theoretic decomposition of the 8⊗8⊗8 tensor product, leading to complete coefficient tables (Tables 6 and 7 and the appendices) for the twelve independent first-class and second-class amplitudes up to third order in δm_l. The central formal results are: at O(δm_l^2) the twelve first-class amplitudes depend on eleven parameters, giving one constraint, identified as the 64-plet combination Q64 in Eq. (62); at O(δm_l^3) the parameter count equals the amplitude count, so no further constraint remains; for second-class currents the constraints end at O(δm_l^2). The paper also maps the expansion coefficients onto quark-line-connected and disconnected diagram contributions, discusses renormalization and O(a) improvement for the vector current, and illustrates the formalism with three lattice ensembles, including fan plots and extractions of Z_V, b_V, and f_V^con.

Significance. If the formal counting is correct, this is a significant and useful systematization of SU(3)-breaking in baryon matrix elements. It generalizes the usual f/d parametrization, provides testable linear relations among amplitudes, and yields a clean prediction, Eq. (62), that the 64-plet combination vanishes up to O(δm_l^2). The paper's strengths are the complete and explicit coefficient tables, the transparent group-theoretic derivation, the separation of connected and disconnected contributions, and the demonstration that improvement coefficients simply modify the expansion parameters. The numerical section is a nice illustration, but it is not a precision calculation: the three-ensemble linear extrapolation leaves the convergence of the δm_l expansion below M_pi=310 MeV untested, which should be stated clearly.

major comments (2)
  1. [Section 6.2] The abstract's claim that 'considering higher orders would give no further constraints' is not fully established by the counting argument given in this section. Equal numbers of parameters and amplitudes at O(δm_l^3) do not by themselves exclude a linear constraint if the twelve coefficient vectors are linearly dependent; one must show that the twelve tensor structures from the 1, 8, 27 and 64 representations are linearly independent when restricted to the twelve standard amplitudes. For orders beyond O(δm_l^3), the text should also explain why the same complete set of tensor structures reappears at every higher order, or otherwise justify the 'and higher orders' part of the claim.
  2. [Section 13 and Eqs. (116)-(118)] The numerical extrapolation rests on an unverified analyticity and truncation assumption. Only three ensembles at M_pi = 465, 360 and 310 MeV are used, and the physical point is reached by a linear extrapolation in δm_l over a range in which there are no data. If O(δm_l^2) terms or chiral-logarithmic behaviour are significant between 310 MeV and the physical point, the extracted expansion coefficients and the derived values of Z_V, b_V and f_V^con in Eqs. (116)-(118) would be biased. Please either provide a quantitative estimate of the omitted higher-order systematic uncertainty or explicitly present the numerical section as an illustration of the formalism, with a clear caveat about this limitation.
minor comments (4)
  1. [Eqs. (97) and (101)] The Ξ0 entries in these equations appear to use incorrect normalizations. For Vπ0, the Ξ0 matrix element should be 1/√2 (1−0), not 1/√6 (1−0); for Vη, the Ξ0 matrix element should be 1/√6 (1+0−4), not 1/√2 (1+0−4). The conclusions of Section 11.2 are unaffected once these normalizations are corrected, but the equations as printed are internally inconsistent.
  2. [Section 6.2] The statement that at O(δm_l^2) for second-class currents there are 'additional 2 parameters' is inconsistent with Eq. (61), which exhibits four new coefficients (t2^x, u1^x, x1, y1). The conclusion that no new constraints appear at this order survives, but the counting statement should be corrected.
  3. [Section 6.2] The sentence 'for second-class operators there is no point in going higher than linear in the quark mass' is confusing because Eq. (61) gives the quadratic expansion and the constraints end at O(δm_l^2). Please rephrase, e.g. 'no point in going beyond O(δm_l^2)'.
  4. [Section 13.1] The constant fits to the X quantities would be easier to interpret if the χ²/dof values or a comparable goodness-of-fit statement were given, especially since the constancy of X_D and X_F is used to justify the linear truncation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constraint counting and the CVC-based ratio checks are derived from SU(3) group theory and an exact lattice symmetry, while the numerical fan-line 'predictions' use parameters fitted only to diagonal combinations.

full rationale

The central formal claim, that the 12 octet-baryon amplitudes have 11 free parameters at O(delta_m_l^2) and therefore satisfy one constraint (the 64-plet combination Q64 = O(delta_m_l^3) in eq. (62)), is obtained from the explicit 8 x 8 x 8 decomposition and the coefficient tables in sections 5-6, not from any fitting. The counting is self-contained: the tensors are constructed by imposing the isospin/hypercharge conditions (eq. (50)) and the Casimir classification is quoted from standard representation theory, so the constraint is a genuine consequence of the stated SU(3) symmetry assumption rather than a restatement of the input. The numerical 'predictions' are also genuine over-constrained checks: the d-fan and f-fan parameters r1, r3, s1, s2 and f, d are fitted only to the diagonal combinations D1, D2, D4 and F1, F2, F3, after which the off-diagonal hyperon lines D6, F4 and F5 are evaluated without being fitted. Similarly, the ratio s2/s1 = 2*sqrt(3) follows from eqs. (97)-(100), which use the exactly conserved vector current (CVC) quark-counting values, and is then compared with the lattice-extracted slopes; this is an external exact-symmetry input, not a parameter fitted to the data being predicted. The paper does rely on the authors' previous work [1,2,21] for the constant-mbar trajectory, the classification of mass polynomials, and the value of delta_m_l* at the physical point, but those uses are methodological or parametric inputs, and the matrix-element expansion and its constraint structure are re-derived here with complete coefficient tables, so the self-citations are not load-bearing for the claimed new result. The principal limitation, the assumed analyticity and convergence of the truncated delta_m_l expansion from the 465-MeV symmetric point down to the physical point, is an assumption-risk rather than a circularity, so it does not raise the circularity score.

Assumptions & free parameters 12 free parameters · 5 assumptions · 0 invented entities

The framework introduces no new particles, forces or symmetries. It relies on the standard assumption that hadronic quantities are analytic in quark masses and on the group-theoretic decomposition of 8x8x8. All expansion coefficients are, by construction, unknown parameters of the effective expansion; in the demonstration they are fitted to lattice data, with the improvement coefficients additionally constrained by CVC and the Ademollo-Gatto theorem.

free parameters (12)
  • f = 0.814(1) at Q2=0 (from XF1_F)
    Leading SU(3) coupling for the vector F1 form factor; determined from CVC normalisation in section 13.3.
  • d = 0 at Q2=0 (consistent with CVC)
    Leading SU(3) coupling; eq. (98) gives d=0 for F1 at Q2=0, and it remains small for Q2>0.
  • r1^con = e.g., -3.65(8) used in eq. (118)
    LO slope parameter in the d-fan; fitted per Q2 bin to the connected diagonal matrix elements (section 13.2).
  • r3 = 0.06(2) at Q2=0
    LO slope parameter in the d-fan; fitted to the D-fan lines (section 13.2, Fig. 9).
  • s1 = -0.479(22) at Q2=0
    LO slope parameter in the f-fan; fitted to the F-fan lines (section 13.2, Fig. 9).
  • s2 = -1.643(44) at Q2=0
    LO slope parameter in the f-fan; fitted to the F-fan lines (section 13.2, Fig. 9).
  • a0, a1, a2 (singlet expansion) = not determined numerically (disconnected parts neglected)
    Coefficients of the flavour-singlet operator expansion, eq. (39); for the connected sector they are expressed in terms of f, d and the r, s slopes via eq. (81), while the disconnected combinations remain free but are not used in the demonstration.
  • kappa0 = 0.120900
    The SU(3) symmetric hopping parameter that defines the constant-mbar trajectory; taken from [2], not fitted in this paper.
  • delta m*_l = -0.01103
    The physical-point value of the mass splitting parameter, taken from the earlier determination in [21]; used to evaluate the form factors at the physical point.
  • ZV (hat) = 0.869(1)
    Vector current renormalisation constant determined from CVC via eq. (99); constrained by the theory rather than a free fit.
  • bV (hat) = 1.174(21)
    Improvement coefficient obtained from eq. (100) and the Q2=0 values of s1, s2; constrained by CVC and the Ademollo-Gatto theorem.
  • fV^con (hat) = 0.041(4)
    Improvement coefficient obtained from eq. (102) and r1^con; constrained by CVC.
assumptions (5)
  • domain assumption Hadronic matrix elements are analytic in the quark masses around the SU(3) symmetric point, so the Taylor expansion in delta m_l exists and is convergent down to the physical point.
    Invoked in section 3.2 when writing eq. (19) and throughout the numerical extrapolation; not proven in the paper.
  • standard math In the nf=2+1 isospin limit, only I=0, Y=0 flavour tensors contribute to the expansion.
    Consequence of isospin and hypercharge conservation, stated in section 5.2; it eliminates the 10, 10, 35 and 35 representations.
  • standard math The Ademollo-Gatto theorem holds, so the O(delta m_l) terms in the F1 form factor vanish at Q2=0 for B' != B.
    Used in section 11.2.3 to fix the improvement coefficients; cited to [28,29] rather than re-derived.
  • domain assumption The specific form of O(a) improvement coefficients for the vector current (bV, fV, dV, etc.) from [12] applies here.
    Used in eqs. (88)-(95) to absorb improvement terms into the expansion coefficients.
  • domain assumption The three lattice ensembles at pion masses 465, 360 and 310 MeV lie within the region where the linear expansion is adequate.
    The numerical fits in section 13 assume linearity over this quark-mass range; no test of curvature beyond the X-plot consistency is provided.

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Pith. "Pith review of Patterns of flavour symmetry breaking in hadron matrix elements involving u, d and s quarks." pith.science (2026). https://pith.science/paper/RQA5WEGN

@misc{pith2026190902521,
  author       = {Pith},
  title        = {Pith review of: Patterns of flavour symmetry breaking in hadron matrix elements involving u, d and s quarks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQA5WEGN}},
  note         = {Machine review of arXiv:1909.02521}
}
read the original abstract

By considering a flavour expansion about the SU(3)-flavour symmetric point, we investigate how flavour-blindness constrains octet baryon matrix elements after SU(3) is broken by the mass difference between quarks. Similarly to hadron masses we find the expansions to be constrained along a mass trajectory where the singlet quark mass is held constant, which provides invaluable insight into the mechanism of flavour symmetry breaking and proves beneficial for extrapolations to the physical point. Expansions are given up to third order in the expansion parameters. Considering higher orders would give no further constraints on the expansion parameters. The relation of the expansion coefficients to the quark-line-connected and quark-line disconnected terms in the 3-point correlation functions is also given. As we consider Wilson clover-like fermions, the addition of improvement coefficients is also discussed and shown to be included in the formalism developed here. As an example of the method we investigate this numerically via a lattice calculation of the flavour-conserving matrix elements of the vector first class form factors.

Figures

Figures reproduced from arXiv: 1909.02521 by the authors.

Figure 1
Figure 1. Left panel: The baryon octet. Right panel: The meson octet. all possible Dirac gamma matrix structure1 . While of intrinsic interest in itself, an obvious application of this formalism is the determination of semileptonic decay form factors and the associated CKM matrix element, |Vus|. In general disentangling quark mass and momentum de￾pendencies is helpful for determining generalised form factors of baryons, as de… view at source ↗
Figure 2
Figure 2. I3, Y plots for some of the SU(3) multiplets which appear in the decom￾position of 8 ⊗ 8 ⊗ 8. The left-hand plot illustrates the octet, 27-plet and 64-plet rep￾resentations (clockwise). The right-hand plot shows the 10 and 35-plets (left to right). The number of spots in the central location gives the number of flavour-conserving operators in each multiplet. 5.3 The SU(3) symmetry-breaking expansions 5.3.1 Basis Bec… view at source ↗
Figure 3
Figure 3. The three point quark correlation function for a baryon. The cross rep￾resents the current insertion. Left panel: the quark-line-connected piece; right panel: the quark-line-disconnected piece. panel of [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: X F1 con D and X F1 F for F1 at Q2 = 0, top panel and for Q2 = 0.49 GeV2 , lower panel. The lower filled circles in each plot are X F1 con D , the upper filled triangles are X F1 F . The dashed lines are constant fits and the stars represent the physical point. 46 [PI…
Figure 5
Figure 5. Figure 5: X F2 con D and X F2 F for F2 at Q2 = 0.25 GeV2 , top panel and for Q2 = 0.49 GeV2 , lower panel. The same notation as for [PITH_FULL_IMAGE:figures/full_fig_p047_5.png]
Figure 6
Figure 6. Figure 6: Top panel: X F1 F (filled circles) and X F1 con D (filled triangles) versus Q2 . Lower panel: Similarly for F2. 49 [PITH_FULL_IMAGE:figures/full_fig_p049_6.png]
Figure 7
Figure 7. Figure 7: Top panel: D˜F1 i ≡ D F1 i /XF1 F for i = 1 (filled circles), 2 (filled squares) and 4 (filled triangles) for Q2 = 0.49 GeV2 . The three fits are from eq. (68), the line for i = 6 is also shown. The vertical dotted line represents the physical point. Lower panel: F˜F1 …
Figure 8
Figure 8. Figure 8: Top panel: D˜F2 i for i = 1 (filled circles), 2 (filled squares) and 4 (filled triangles) for Q2 = 0.49 GeV2 . The three fits are from eq. (68) normalised by X F2 D , also shown is the i = 6 line. The vertical dotted line represents the physical point. Lower panel: F˜F…
Figure 9
Figure 9. Figure 9: Top panel: r con 1 (filled circles), r3 (filled triangles), s1 (filled squares) and s2 (filled diamonds) expansion coefficients for the vector F con 1 form factor as a function of Q2 . Lower panel: Similarly for the F2 form factor. 52 [PITH_FULL_IMAGE:figures/full_fig…
Figure 10
Figure 10. Figure 10: XF (Q2 )/XF (0) (filled circles) and ˜d(Q2 ) (filled triangles) for F con 1 against Q2 . The interpolation formulae used are given in eq. (121). It is interesting to determine the various contributions to the form factors from the expansion coefficients. For illustrat…
Figure 11
Figure 11. Figure 11: r˜ con 0 1 (filled circles), ˜s 0 2 (filled diamonds), ˜s 0 1 (filled squares) and ˜r 0 3 (filled triangles) against Q2 together with interpolation formulae also given by eq. (121). the form AQ2 1 + BQ2 + C(Q2 ) 2 . (121) From [PITH_FULL_IMAGE:figures/full_fig_p055_11.png]
Figure 12
Figure 12. Figure 12: F con R 1 for the proton (filled circles) and Ξ0 (filled triangles) at the physical point. The dashed line is XF (Q2 )/XF (0). The dashed-dotted lines are the complete leading terms, for the proton: XF (Q2 , m¯ )/XF (0, m¯ )(1 + 2/ √ 3 ˜d(Q2 , m¯ )) and for Ξ0 : XF (Q…

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Works this paper leans on

48 extracted references · 23 canonical work pages

  1. [1]

    Tuning the strange quark mass in lattice simulations

    W. Bietenholz, V. Bornyakov, N. Cundy, M. G¨ ockeler, R. Horsley, A. D. Kennedy, W. G. Lockhart, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Sch¨ afer, G. Schierholz, A. Schiller, H. St¨ uben and J. M. Zanotti [QCDSF–UKQCD Collaboration], Phys. Lett. B690 (2010) 436, [arXiv:1003.1114 [hep-lat] ]

    work page Pith review arXiv 2010

  2. [2]

    Bietenholz, V

    W. Bietenholz, V. Bornyakov, M. G¨ ockeler, R. Horsley, W. G. Lockhart, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, T. Streuer, H. St¨ uben, F. Winter and J. M. Zanotti [QCDSF–UKQCD Collab- oration], Phys. Rev. D84 (2011) 054509, [ arXiv:1102.5300 [hep-lat] ]

    arXiv 2011

  3. [3]

    Horsley, Y

    R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, R. Stokes, H. St¨ uben, R. D. Young and J. M. Zanotti J. Phys. G43 (2016) 10LT02, [arXiv:1508.06401 [hep-lat] ]. 66

    arXiv 2016

  4. [4]

    Horsley, Y

    R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, R. Stokes, H. St¨ uben, R. D. Young and J. M. Zanotti [QCDSF- UKQCD Collaboration], JHEP 1604 (2016) 093, [ arXiv:1509.00799 [hep-lat]]

    arXiv 2016

  5. [5]

    Isospin splittings in the decuplet baryon spectrum from dynamical QCD+QED

    R. Horsley, Z. Koumi, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben, R. D. Young and J. M. Zanotti [CSSM and QCDSF and UKQCD Collaborations], J. Phys. G46 (2019) 115004, [arXiv:1904.02304 [hep-lat] ]

    work page Pith review arXiv 2019

  6. [6]

    E. E. Jenkins, A. V. Manohar, J. W. Negele and A. Walker-Loud, Phys. Rev. D81 (2010) 014502, [ arXiv:0907.0529 [hep-lat] ]

    work page Pith review arXiv 2010

  7. [7]

    Gell-Mann, Phys

    M. Gell-Mann, Phys. Rev. 125 (1962) 1067

    work page 1962

  8. [8]

    Okubo, Prog

    S. Okubo, Prog. Theor. Phys. 27 (1962) 949

    work page 1962

Show all 48 references
  1. [9]

    A. N. Cooke, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schier- holz and J. M. Zanotti PoS LATTICE 2012 (2012) 116, arXiv:1212.2564 [hep-lat]

    2012 arXiv

  2. [10]

    QCDSF Collaboration, in preparation

  3. [11]

    Accardi et al

    A. Accardi et al. , Eur. Phys. J. A52 (2016) 268, [ arXiv:1212.1701 [nucl-ex]]

    2016 arXiv

  4. [12]

    Bhattacharya, R

    T. Bhattacharya, R. Gupta, W. Lee, S. R. Sharpe and J. M. S. Wu Phys. Rev. D73 (2006) 034504 [ arXiv:hep-lat/0511014]

    2006 arXiv

  5. [13]

    Cabibbo, E

    N. Cabibbo, E. C. Swallow and R. Winston, Ann. Rev. Nucl. Part. Sci. 53 (2003) 39, [ arXiv:hep-ph/0307298]

    2003 arXiv

  6. [14]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. 112 (1958) 1375

    1958

  7. [15]

    Georgi, Weak Interactions and Modern Particle Theory , Ben- jamin/Cummings (1984)

    H. Georgi, Weak Interactions and Modern Particle Theory , Ben- jamin/Cummings (1984)

    1984

  8. [16]

    V. G. Bornyakov, R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben, and J. M. Zanotti [QCDSF– UKQCD Collaboration], Phys. Lett. B767 (2017) 366, [arXiv:1612.04798 [hep-lat]]

    2017 arXiv

  9. [17]

    Gasiorowicz, Elementary Particle Physics , John Wiley & Sons, New York, 1966

    S. Gasiorowicz, Elementary Particle Physics , John Wiley & Sons, New York, 1966

    1966

  10. [18]

    M. J. Savage and J. Walden, Phys. Rev. D55 (1997) 5376, [arXiv:hep-ph/9611210]. 67

    1997 arXiv

  11. [19]

    Walker-Loud, Nucl

    A. Walker-Loud, Nucl. Phys. A747 (2005) 476, [arXiv:hep-lat/0405007]

    2005 arXiv

  12. [20]

    Pfeifer, The Lie Algebras su(N), An Introduction , Birkh¨ auser, Basel, 2003

    W. Pfeifer, The Lie Algebras su(N), An Introduction , Birkh¨ auser, Basel, 2003

    2003

  13. [21]

    Horsley, J

    R. Horsley, J. Najjar, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben and J. M. Zanotti [QCDSF–UKQCD Collaborations], Phys. Rev. D91 (2015) 074512, [ arXiv:1411.7665 [hep-lat]]

    2015 arXiv

  14. [22]

    Greiner and B

    W. Greiner and B. M¨ uller,Quantum Mechanics: Symmetries , Springer, Hei- delberg, 1994

    1994

  15. [23]

    Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019)

    2019

  16. [24]

    V. G. Bornyakov, R. Horsley, R. Hudspith, Y. Nakamura, H. Perlt, D. Pleiter, P .E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben and J. M. Zanotti [QCDSF-UKQCD Collaboration], arXiv:1508.05916 [hep-lat]

    arXiv

  17. [25]

    Horsley, Y

    R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben, R. D. Young and J. M. Zan- otti [QCDSF-UKQCD Collaborations], PoS(LATTICE2018) (2018) 119, arXiv:1901.04792 [hep-lat]

    2018 arXiv

  18. [26]

    Gerardin, T

    A. Gerardin, T. Harris and H. B. Meyer, Phys. Rev. D99 (2019) 014519, [arXiv:1811.08209 [hep-lat] ]

    2019 arXiv

  19. [27]

    Bakeyev, M

    T. Bakeyev, M. G¨ ockeler, R. Horsley, D. Pleiter, P. E. L. Rakow, G. Schier- holz and H. St¨ uben [QCDSF-UKQCD Collaboration], Phys. Lett. B580 (2004) 197, [ arXiv:hep-lat/0305014]

    2004 arXiv

  20. [28]

    Ademollo and R

    M. Ademollo and R. Gatto, Phys. Rev. Lett. 13 (1964) 264

    1964

  21. [29]

    Anderson and M

    J. Anderson and M. A. Luty, Phys. Rev. D47 (1993) 4975, [arXiv:hep-ph/9301219]

    1993 arXiv

  22. [30]

    Capitani, M

    S. Capitani, M. G¨ ockeler, R. Horsley, B. Klaus, H. Oelrich, H. Perlt, D. Pet- ters, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller and P. Stephenson, Nucl. Phys. Proc. Suppl. 73 (1999) 294, arXiv:hep-lat/9809172

    1999 arXiv

  23. [31]

    G¨ ockeler, T

    M. G¨ ockeler, T. R. Hemmert, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Sch¨ afer and G. Schierholz [QCDSF Collaboration],Phys. Rev. D71 (2005) 034508, [arXiv:hep-lat/0303019]

    2005 arXiv

  24. [32]

    C. Best, M. G¨ ockeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, A. Sch¨ afer, G. Schierholz, A. Schiller and S. Schramm,Phys. Rev. D56 (1997) 2743, [arXiv:hep-lat/9703014]. 68

    1997 arXiv

  25. [33]

    P. E. Shanahan, A. W. Thomas, R. D. Young, J. M. Zanotti, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schierholz and H. St¨ uben [CSSM and QCDSF/UKQCD Collaborations], Phys. Rev. D89 (2014) 074511, [arXiv:1401.5862 [hep-lat] ]

    2014 arXiv

  26. [34]

    P. E. Shanahan, A. W. Thomas, R. D. Young, J. M. Zanotti, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schierholz and H. St¨ uben [CSSM and QCDSF/UKQCD Collaborations], Phys. Rev. D90 (2014) 034502, [arXiv:1403.1965 [hep-lat] ]

    2014 arXiv

  27. [35]

    Cundy, M

    N. Cundy, M. G¨ ockeler, R. Horsley, T. Kaltenbrunner, A. D. Kennedy, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Sch¨ afer, G. Schierholz, A. Schiller, H. St¨ uben, and J. M. Zanotti [QCDSF–UKQCD Collaboration], Phys. Rev. D 79 (2009) 094507, [ arXiv:0901.3302 [hep-lat] ]

    2009 arXiv

  28. [36]

    Cundy, M

    N. Cundy, M. G¨ ockeler, R. Horsley, T. Kaltenbrunner, A. D. Kennedy, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Sch¨ afer, G. Schierholz, A. Schiller, H. St¨ uben and J. M. Zanotti [QCDSF-UKQCD Collaboration], PoS LATTICE 2008 (2008) 132, arXiv:0811.2355 [hep-lat]

    2008 arXiv

  29. [37]

    Constantinou, R

    M. Constantinou, R. Horsley, H. Panagopoulos, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller and J. M. Zanotti, Phys. Rev. D91 (2015) 014502, [arXiv:1408.6047 [hep-lat] ]

    2015 arXiv

  30. [38]

    Horsley, J

    R. Horsley, J. Najjar, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schier- holz and J. M. Zanotti [QCDSF–UKQCD Collaborations], Phys. Rev. D86 (2012) 114511, [ arXiv:1206.3156 [hep-lat] ]

    2012 arXiv

  31. [39]

    Cabibbo, E

    N. Cabibbo, E. C. Swallow and R. Winston, Phys. Rev. Lett. 92 (2004) 251803, [arXiv:hep-ph/0307214]

    2004 arXiv

  32. [40]

    Mateu and A

    V. Mateu and A. Pich, JHEP 0510 (2005) 041, [ arXiv:hep-ph/0509045]

    2005 arXiv

  33. [41]

    Guadagnoli, V

    D. Guadagnoli, V. Lubicz and M. Papinutto Nucl. Phys. B761 (2007) 63, [arXiv:hep-ph/0606181 [hep-lat] ]

    2007 arXiv

  34. [42]

    Sasaki and T

    S. Sasaki and T. Yamazaki, Phys. Rev. D79 (2009) 074508, [arXiv:0811.1406 [hep-lat] ]

    2009 arXiv

  35. [43]

    Sasaki, Phys

    S. Sasaki, Phys. Rev. D86 (2012) 114502, [ arXiv:1209.6115 [hep-lat] ]

    2012 arXiv

  36. [44]

    Sasaki, Phys

    S. Sasaki, Phys. Rev. D96 (2017) 074509, [arXiv:1708.04008 [hep-lat] ]

    2017 arXiv

  37. [45]

    Horsley, Z

    R. Horsley, Z. Koumi, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, H. St¨ uben, R. D. Young and J. M. Zanotti [QCDSF-UKQCD Collaborations], EPJ Web Conf. 175 (2018) 06017, arXiv:1711.02485 [hep-lat]. 69

    2018 arXiv

  38. [46]

    T. R. Haar, Y. Nakamura and H. St¨ uben,EPJ Web Conf. 175 (2018) 14011, arXiv:1711.03836 [hep-lat]

    2018 arXiv

  39. [47]

    R. G. Edwards and B. Jo´ o, Nucl. Phys. Proc. Suppl. 140 (2005) 832, arXiv:hep-lat/0409003

    2005 arXiv

  40. [48]

    G¨ ockeler, Ph

    M. G¨ ockeler, Ph. H¨ agler, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, A. Sch¨ afer, G. Schierholz, H. St¨ uben and J. M. Zan- otti [QCDSF/UKQCD Collaboration], PoS LATTICE 2010 (2010) 163, arXiv:1102.3407 [hep-lat] . 70

    2010 arXiv

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