Unveiling the Strong Interaction origin of Baryon Masses with Lattice QCD

Bolun Hu; Haiyang Du; Keh-Fei Liu; Peng Sun; Xiangyu Jiang; Yi-Bo Yang

REVIEW 4 major objections 4 minor 2 cited by

Unveiling the Strong Interaction origin of Baryon Masses with Lattice QCD

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

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Pith's one-line read First-principles lattice QCD reproduces the lightest baryon masses to within 1% and decomposes them into a small Higgs-induced part and a flavor-blind gluon trace anomaly of about 0.8–1.2 GeV.

desk verdict A serious lattice calculation with a valuable mass table, but the headline claims outrun the error budget and the gluon-anomaly decomposition rests on an unquantified fit-form assumption. read the letter →

arxiv 2411.18402 v2 pith:NV6KKASB submitted 2024-11-27 hep-lat hep-ph

Bolun Hu , Haiyang Du , Xiangyu Jiang , Keh-Fei Liu , Peng Sun , Yi-Bo Yang This is my paper · ORCID

classification hep-lathep-ph MSC 81T2581V05 PACS 12.38.Gc12.15.-y

keywords latticeQCDbaryonmasstraceanomalyHiggsmechanismsigmatermsFeynman-Hellmantheoremcharmedbaryonsgap

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 claims that lattice QCD, with no adjustable parameters beyond the Standard Model, predicts the ground-state spin-1/2 and spin-3/2 baryon masses built from light, strange, and charm quarks to within 1% of experiment. Using the Feynman–Hellman theorem on a global fit of baryon masses, it splits each mass into quark sigma terms (the effective Higgs–quark couplings) and the remainder, the gluon trace anomaly. The central result is that the quark sigma terms receive flavor-dependent enhancements—4 to 8 times the bare light-quark mass, 2 to 3 for strange, 1.2 to 1.3 for charm—while the gluon trace anomaly is nearly flavor-insensitive and contributes roughly 0.8 to 1.2 GeV to every baryon. If correct, most of the mass of visible matter is not generated by the Higgs mechanism but by the QCD trace anomaly.

What carries the argument

The central object is the energy-momentum tensor trace decomposition (Eq. 2), which partitions hadron mass into quark $\sigma$ terms plus the gluon trace anomaly. The operational machinery is the Feynman–Hellman theorem applied to a global chiral perturbation theory fit: the nucleon and $\Delta$ masses are fitted with a SU(4|2) partially quenched heavy baryon chiral perturbation theory ansatz (Eq. 5), and other baryons with a linear or interpolation ansatz, all as functions of valence and sea pion masses, the $\eta_s$ mass, lattice spacing, and volume. Derivatives of this fitted mass surface with respect to quark masses give $\sigma_{q,H}$; subtracting $(1+\gamma_m)\sum_q\sigma_{q,H}$ from the physical mass yields the gluon trace anomaly. The machinery also includes tuned valence strange and charm quark masses to remove mismatch effects and a decoupling argument that neglects charm sea quarks.

What would settle it

Compute the scalar matrix elements $\langle \bar{q}q\rangle_H$ directly on the same ensembles instead of via the Feynman–Hellman derivative of the fit and show that the resulting $\sigma_{q,H}$ differ from the paper's values by more than the quoted uncertainties, or repeat the joint fit with a different chiral perturbation theory ansatz (for example, including higher-order terms or a different handling of the $m_\pi^3$ correction) and observe the gluon trace anomaly shifting outside 0.8–1.2 GeV.

Watch

Extended reading notes

Core claim

The central claim is that a baryon mass $m_H$ decomposes as $m_H = \sum_q \sigma_{q,H} + \gamma_m \sum_q \sigma_{q,H} + \frac{\beta(\alpha_s)}{2\alpha_s}\langle G^2\rangle_H$, where $\sigma_{q,H} \equiv m_q \langle \bar{q}q\rangle_H$ is the Higgs contribution of flavor $q$ and the last term is the gluon trace anomaly. From a joint fit of 14 gauge ensembles at five lattice spacings, the paper extracts the $\sigma$ terms by differentiating the fitted hadron masses with respect to valence and sea quark masses. It finds that the ratio $(1+\gamma_m)\sigma_{q,H}/(n_q m_q)$ is 4–8 for light quarks, 2–3 for strange quarks, and 1.2–1.3 for charm quarks at the $\overline{\mathrm{MS}}$ 2 GeV scale, and that the subtracted gluon trace anomaly $\langle H_a^g\rangle_H$ converges to 0.8–1.2 GeV across all baryons. The authors conclude that the gluon trace anomaly, not the Higgs mechanism, dominates the mass of visible matter.

Load-bearing premise

The entire mass decomposition rests on the assumption that the global fit ansatz (Eq. 5) correctly captures the quark-mass dependence of the baryon masses over the fitted range (sea pion masses 122–351 MeV), so the derivatives with respect to quark masses are trustworthy.

Editorial extensions

If this is right

Ground-state baryon masses across the light, strange, and doubly and triply charmed sectors are now predictable at the sub-percent level from first principles.
The Higgs-generated quark masses contribute to baryon mass with flavor-dependent enhancement factors (4–8 for light, 2–3 for strange, 1.2–1.3 for charm), meaning the effective Higgs–nucleon coupling is larger than the bare quark masses would suggest.
The gluon trace anomaly contributes a nearly constant 0.8–1.2 GeV to every baryon, making it the dominant component of the proton's mass and of visible matter in general.
The charm sea-quark contribution to baryon mass is suppressed by heavy-quark decoupling, with a leading-order estimate of about 74(15) MeV for the baryons studied.

Reading between the lines

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

If the gluon trace anomaly is as flavor-blind as claimed, the same decomposition should hold for excited baryons and for most mesons (with the pion as a special case); testing that would separate a universal mass-generation mechanism from hadron-specific structure.
The extracted light-quark sigma terms directly constrain dark-matter–nucleon scattering cross sections, and the reported enhancement factors of 4–8 imply stronger Higgs-mediated couplings than models built from bare quark masses would predict.
The near-constancy of the gluon trace anomaly across baryons suggests a connection to the Yang–Mills mass gap: a nonzero, hadron-independent scale in the trace of the energy-momentum tensor may be the same phenomenon that gives rise to the mass gap in pure gauge theory.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a real piece of work: 19 baryons and 8 charmed mesons in one framework, 14 ensembles, 23k measurements, careful continuum extrapolation, and an error budget that actually separates statistical, scale-setting, and fit-ansatz contributions. The mass predictions mostly land close to experiment and agree with the best prior lattice results. That part deserves credit and will be useful to people working on heavy-baryon spectroscopy.

What is genuinely new is the systematic decomposition of each hadron mass into quark sigma terms and a remainder gluon trace anomaly, with the observation that the gluon part hovers around 1 GeV across the whole spectrum. That is a plausible and interesting pattern. But it is not as solid as the abstract implies.

First, the "within 1%" claim is contradicted by the paper's own Table III: Delta is 1.273 vs 1.232 GeV (3.3% high) and Sigma* is 1.417 vs 1.384 GeV (2.4% high). That is not a nitpick; it is the central quantitative claim.

Second, the decomposition is only as good as the sigma-term extraction. The sigma terms come from Feynman-Hellman derivatives of global fits. For N and Delta they use a chiPT ansatz, but for every other baryon the quark-mass dependence is modeled with a linear interpolation in the valence masses. The quoted systematic "fit ansatz" error only covers additive vs multiplicative discretization errors, not the functional form of the quark-mass dependence. Since the gluon anomaly is defined as m_H - (1+gamma_m) sigma_H, any bias in sigma_H shifts the anomaly directly. For Omega_ccc with sigma_c around 3 GeV, a few percent functional-form bias is hundreds of MeV, which is the entire claimed 0.8-1.2 GeV window. So the flavor-insensitivity is not yet established.

Third, there is an internal inconsistency in the strangeness enhancement: the abstract says 2-3 for strange quarks, but Section 2 and Fig. 11 quote roughly 1.5. That needs fixing regardless of the physics.

The neglect of ISB and QED is explicitly stated as a deferred systematic; that is fine, but then "within 1%" should not be in the abstract without a caveat.

Bottom line: the mass predictions and the error-budget table are worth keeping, and the decomposition is worth discussing. But the paper needs a major revision of the claims, and ideally a direct check of at least one sigma term, before the anomaly result is advertised. I would send it to peer review with the expectation of heavy revision, not desk-reject it. For the lattice community, the mass table alone makes it a useful reference; for the decomposition, treat it as suggestive until the fit-form uncertainty is quantified.

Referee Report

4 major / 4 minor

Summary. The paper reports CLQCD lattice QCD calculations of the ground-state spin-1/2 and spin-3/2 baryon masses for light, strange, and charmed baryons using 14 ensembles with 2+1 flavors, and it uses these masses to propose a decomposition of the baryon mass into quark sigma terms and the gluon trace anomaly. The central phenomenological claims are that all predicted masses agree with experiment within 1% and that the decomposition yields flavor-dependent Higgs enhancement factors (4–8 for light, 2–3 for strange, 1.2–1.3 for charm quarks) plus a flavor-insensitive gluon trace anomaly of 0.8–1.2 GeV. The sigma terms are extracted from Feynman–Hellman derivatives of a joint mass fit, and the gluon trace anomaly is obtained as the remainder in Eq. (3).

Significance. If the mass decomposition and the quoted uncertainties were reliable, the paper would provide a unified lattice-QCD picture of baryon mass generation and would strengthen the case that the gluon trace anomaly dominates visible mass. The strengths of the paper include the use of a large number of gauge ensembles, the detailed error budget in Table III, cross-checks with the distillation method, and comparisons with earlier lattice calculations. However, as detailed below, the strongest claims are not currently supported by the paper's own tables and by the stated treatment of the fit ansatz.

major comments (4)

[Abstract; Table III] The abstract's claim of predicting ground-state spin-1/2 and spin-3/2 baryon masses 'within 1% of experimental values' is contradicted by the paper's own Table III for the Δ and Σ* rows: the central values are 1.2732 GeV and 1.4172 GeV, which are roughly 3.3% and 2.4% above the experimental Δ(1232) and Σ*(1385) masses. The claim should either be restricted to the states for which it actually holds, or the comparison should explicitly account for the resonance-pole definition and the finite decay widths of the experimental states.
[Supplement §3 and §5; Eq. (3)] The gluon trace anomaly in Eq. (3) is a remainder, ⟨H_g^a⟩ = m_H − (1+γ_m)σ_H, so any systematic error in the fitted sigma terms propagates directly into the central decomposition. For all baryons except N and Δ, the sigma terms are derivatives of a fit whose quark-mass dependence is not the SU(4|2) χPT form: the supplement states that χPT is not applicable for baryons with strange and charm valence quarks and uses a linear/interpolation ansatz. The 'fit ansatz' systematic quoted in Table III is defined as additive versus multiplicative discretization errors, so it does not cover the choice of the quark-mass functional form. For Ω_ccc, where σ_c ≈ 2.98 GeV, even a 5% functional-form bias changes the anomaly by roughly 150 MeV, comparable to the width of the advertised 0.8–1.2 GeV range. The authors should either provide a direct scalar-matrix-element check on these ensembles or quantify the functional-form uncertainty by varying the fit ansatz.
[Section 2; Fig. 11; Abstract] The abstract's strange-quark enhancement factor of 2–3 is inconsistent with Section 2, which states that the strange enhancement factor is approximately 1.5, and with the middle panel of Fig. 11. This internal inconsistency matters because the enhancement factors are the quantitative output of the sigma-term fits; the abstract needs to be revised to match the values actually obtained.
[Section 3, item 4] The paper defers isospin-breaking and QED corrections with the estimate that both effects are 'at the 1% level'. Since the headline claim is agreement with experimental masses at the 1% level, a 1% estimate is not a controlled systematic; for the Δ and Σ*, where the paper's own deviations are already 2–3%, the comparison with experiment is ambiguous. The manuscript should either include an estimate of these effects in the error budget or explicitly state that the mass predictions are for isospin-symmetric, QCD-only quantities.

minor comments (4)

[Throughout] There are several typographical errors, including 'valance' instead of 'valence' in figure labels and the Supplement, 'whrere' in Supplement §3, and 'the the' in Supplement §3.
[Fig. 1 caption] The caption says 'Baryon mass prediction' and 'experimental values .' with an irregular period; the figure would be clearer if the experimental references and the meaning of the shaded width bands were stated in a single convention.
[Supplement §5, Table III] The table would benefit from a row or column explicitly listing the experimental masses used for comparison, since several central values are compared to experiment but the experimental inputs are only shown graphically.
[Supplement §3, Eq. (5)] The notation 'm_val^π' and 'm_sea^π' is used before being defined; defining these quantities at the first occurrence in Eq. (5) would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: baryon masses are genuine lattice QCD results, and the gluon trace anomaly is extracted from the exact trace relation after independently evaluated sigma terms; self-citations are not load-bearing.

full rationale

The paper's primary claim, the 1% mass predictions of ground-state baryons, rests on lattice correlators, continuum/chiral extrapolations, and comparisons with experiment and independent lattice determinations (Fig. 1). No fitted parameter is renamed as a prediction: the physical masses are extrapolated outputs of the global fit, not inputs. The sigma terms are obtained from Feynman-Hellman derivatives of the fitted mass surface (Supplement Sec. 4), and the gluon trace anomaly is then computed from the exact QCD trace relation, Eq. (2)/(3), as m_H - (1+gamma_m) sigma_H. That is a derived decomposition, not a definition of the anomaly equal to an input. The flavor-insensitivity claim is an observation about the resulting residuals, not imposed by the fit ansatz; the anomalies for N and Delta, for example, differ by about 0.27 GeV, which is not a constant by construction. The SU(4|2) chiPT ansatz is standard and cited to Tiburzi as well as to the authors' earlier work; the linear interpolation for non-N/Delta baryons is an openly stated modeling choice. The main caveat, that the quoted 'fit ansatz' systematic (Supplement Sec. 5) covers only additive vs. multiplicative discretization errors and not the valence-quark functional form, is a systematic-uncertainty limitation rather than a circular step. Self-citations to CLQCD ensembles and earlier xQCD/CLQCD methodology are normal prior-work citations and do not carry the load of the present derivation. Overall, the derivation chain is self-contained enough that no equation reduces to its own input.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The analysis is a global fit with many coefficients. The central numbers are not parameter-free predictions: the sigma terms are derivatives of a fitted ansatz, and the gluon trace anomaly is a remainder after subtracting these terms. No new particles or forces are introduced.

free parameters (2)

Global fit coefficients C^tag_j, C_s, C_L, C_{a,2i} in Eq. (5) = Not quoted individually; fitted jointly
Fitted to baryon masses on 14 CLQCD ensembles; they encode all quark-mass, volume, and lattice-spacing dependence and directly determine the Feynman-Hellman sigma terms.
Valence quark mass correction coefficients d^s_l, d^s_s, d^c_l, d^c_s in Eq. (14) = d^c_l=-0.161(11), d^c_s=-0.388(44), d^s_l=-0.0267(30), d^s_s=-0.0090(10) GeV^-1
Fitted to the renormalized strange and charm quark masses; these enter the sea-quark corrections in Eqs. (10) and (12) and thus the gluon trace anomaly via the sum rule.

assumptions (6)

domain assumption The trace anomaly mass decomposition (Eq. 2) is valid for ground-state hadrons.
Standard QCD result (Collins, Duncan, Joglekar; Nielsen; Shifman, Vainshtein, Zakharov). The paper applies it to all baryons and mesons in Table III.
standard math The Feynman-Hellman theorem relates the quark mass derivative of the hadron mass to the scalar matrix element (Eq. 6).
Used throughout to extract sigma terms from the fitted mass function.
ad hoc to paper The SU(4|2) partially quenched heavy baryon chiPT ansatz (Eq. 5) correctly describes the nucleon mass for m_pi in 0.12-0.35 GeV, and similar truncated forms suffice for the other baryons.
The central extraction of sigma terms depends on this functional form; its validity is asserted, not derived, and the paper notes the m_pi^3 terms are omitted because chiPT is not applicable.
domain assumption Charm sea quark loops are negligible via the heavy quark decoupling relation m_Q<QbarQ> = -(alpha_s/12pi)F^2.
Stated after Fig. 2: 'The charm sea quark contribution to the baryon mass is neglected here... due to the heavy quark relation... leading to the decoupling theorem.'
domain assumption Isospin symmetry breaking and QED effects are at the 1% level and can be deferred.
Section 4: 'Naive power counting indicates that both effects are at the 1% level and then unlikely to alter the conclusions presented in this work.' This is a power-counting estimate, not a calculation.
domain assumption The anomalous dimension gamma_m=0.295 at MS 2 GeV from the PDG is accurate and scale-appropriate for the subtraction in Eq. (3).
Used to separate the quark trace anomaly from the gluon trace anomaly; a 10% uncertainty in gamma_m shifts the reported gluon anomaly by up to about 50 MeV.

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Cite this review

Pith. "Pith review of Unveiling the Strong Interaction origin of Baryon Masses with Lattice QCD." pith.science (2026). https://pith.science/paper/NV6KKASB

@misc{pith2026241118402,
  author       = {Pith},
  title        = {Pith review of: Unveiling the Strong Interaction origin of Baryon Masses with Lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NV6KKASB}},
  note         = {Machine review of arXiv:2411.18402}
}

abstract

Both the Higgs mechanism and strong interactions contribute to the masses of visible matter, yet how the six Higgs-generated quark masses and uniform strong interaction strength determine the hundreds of hadron masses remains unclear. Additionally, the role of massless, flavor-neutral gluons on hadron mass formation is central to the unresolved Millennium Prize problem on the mass gap in Yang-Mills theory. Addressing these questions requires advanced simulations on state-of-the-art supercomputers using Lattice Quantum Chromodynamics (QCD), which offers a rigorous, non-perturbative definition of QCD solvable numerically. Here we present first-principles lattice QCD calculations using comprehensive gauge ensembles that accurately predict ground state spin-1/2 and spin-3/2 baryon masses with light, strange, and charm quarks within 1\% of experimental values. At the $\overline{\mathrm{MS}}$ 2 GeV scale, our results unveil two fundamental mass generation mechanisms for those baryon masses in QCD: 1) the flavor-dependent enhancement of Higgs contributions, 4-8 for light, 2-3 for strange, and 1.2-1.3 for charm quarks; and 2) the flavor-insensitive contribution 0.8-1.2 GeV from gluon quantum anomaly. This breakthrough significantly advances our comprehension of strong interaction dynamics and the genesis of visible matter's mass.

Figures

Figures reproduced from arXiv: 2411.18402 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The effective mass obtained from the two-point correlation functions (2pt) of the nucleon in two representative [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Similar figures for the Ω baryon. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. Similar figures for the Ω [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: FIG. 6. The nucleon mass on 14 ensembles at 5 lattice spacings (colored data points and bands), and also that in the continuum [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. Fitting results for nucleon, Ω, and Ω [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Fitting results for the masses of all baryons, showing their dependence on lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Lattice spacing dependence of the renormalized strange quark mass at [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: FIG. 11. Ratio (1 + [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: FIG. 12. Results for quark sigma terms and trace anomaly contributions to charmed meson masses. The upper-left, upper-right, [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

117 extracted references · 13 canonical work pages · cited by 2 Pith papers

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Mismatch effects arising from the masses of the light and strange quarks: The ensembles [2, 3] use different 3 light quark masses ranging from 0.8 to 6.7 times the av- eraged mass of the up and down quarks. This allows us to interpolate the light quark mass to its physical value, using the SU(4 |2) partially quenched heavy baryon chi- ral perturbation the...
[2]

Impact of tuning the charm quark mass and con- sidering the missing charm quark loop: Similar to our approach with the strange quark, we adjust the valence charm quark mass to correspond with the physical mass of the Ds meson in each ensemble, and then eliminate the mismatch effects from the light and strange quark loops through a joint fit. As demonstrat...
[3]

Additionally, the a4 correction is crucial for capturing the a dependence of certain charmed baryons

Continuum and infinite volume extrapolation: We find that the linear a2 correction adequately describes the masses of baryons without the charm quark across differ- ent lattice spacings a. Additionally, the a4 correction is crucial for capturing the a dependence of certain charmed baryons. Consequently, we retain the a4 term in the con- tinuum extrapolati...
[4]

Therefore, we defer their investigation to future studies

Isospin symmetry breaking (ISB) and quantum elec- trodynamics (QED) effects: Naive power counting in- dicates that both effects are at the 1% level and then unlikely to alter the conclusions presented in this work. Therefore, we defer their investigation to future studies. More evidence validating above procedure are provided in the supplementary informat...
[5]

2 displays the light quark σ terms, representing the direct contributions of light quark masses to the baryon mass

Light quark sigma term σπH ≡ σuH + σdH : The upper left panel of Fig. 2 displays the light quark σ terms, representing the direct contributions of light quark masses to the baryon mass. Some of the enhancement originates solely from the light quark loop (a gluon splits into a quark and anti-quark pair which combines into an- other gluon at a later time), ...
[6]

Conse- quently, as observed in the upper-right panel of Fig

Strange quark sigma term σsH : When the quark mass reaches the magnitude of the strange quark mass 92.4(1.0) MeV at MS 2 GeV [28, 59–63, 66], the enhance- ments from both the quark loop and connected-sea mech- anisms diminish, as the gluon must provide significantly more energy for the related processes to occur. Conse- quently, as observed in the upper-r...
[7]

2 illustrates that the charm quark scenario is no- tably more consistent, with the valence σcH being di- rectly proportional to the charm quark by a factor of ap- proximately 1

Charm quark sigma term σcH : The lower-left panel of Fig. 2 illustrates that the charm quark scenario is no- tably more consistent, with the valence σcH being di- rectly proportional to the charm quark by a factor of ap- proximately 1. In comparison to the charm quark mass of 1.093(7) GeV at MS 2 GeV [28, 63, 67–71], the ra- tio may be slightly smaller th...
[8]

(3), the val- ues of ⟨Ha⟩H (represented by hollow triangles) for dif- ferent hadrons H are displayed in the lower-right panel of Fig

Trace anomaly contribution: By subtracting the to- tal σ term from the baryon mass using Eq. (3), the val- ues of ⟨Ha⟩H (represented by hollow triangles) for dif- ferent hadrons H are displayed in the lower-right panel of Fig. 2. It is evident that ⟨Ha⟩H increases with the number of heavy valence quarks, resulting in a twofold enhancement from the lightes...

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They were generated using a tadpole-improved tree-level Symanzik (TITLS) gauge action with a 2+1 flavor tadpole-improved tree-level Clover (TITLC) fermion action

Details of the ensembles The ensembles used in this work are summarized in Table I. They were generated using a tadpole-improved tree-level Symanzik (TITLS) gauge action with a 2+1 flavor tadpole-improved tree-level Clover (TITLC) fermion action. The TITLS gauge action, denote...
[110]

Here, ˜V = ˜L3 × ˜T is the dimensionless 4-D volume of the lattice, and we use ˜O for the dimensionless value of any quantity O

6 g2 0 u4 0 ≡ 10 g2 0 u4 0 , with c0 1 = − 1 12 , c1 = c0 1 u2 0 , and u0 = * ReTrP x,µ<ν P U µν(x) 6Nc ˜V +1/4 being the tadpole improvement factor. Here, ˜V = ˜L3 × ˜T is the dimensionless 4-D volume of the lattice, and we use ˜O for the dimensionless value of any quantity O...
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Note that some of the particles require linear combination of different permutations of three quarks

Interpolation fields, 2-point functions and ground state baryon mass extraction For the interpolation fields of the baryons, we useϵabcP + (q1 a)T Cγ 5q2 b q3 c and ϵabcP + (q1 a)T Cγ µq2 b q3 c for the spin- 1/2 and 3/2 particles, respectively. Note that some of the particles...
[112]

[99] is the pure QCD ηs mass which corresponds to the physical strange quark mass

Joint fit of the baryon mass We fit the nucleon mass using the the SU(4 |2) partially quenched heavy baryon chiral perturbative theory ( χPT) ansatz [8, 27], mH mval π , msea π , msea ηs , a,1/L =mphys H + X tag=val/pq,j=2,3 C tag j [ mtag π j − mphys π j ] + Cs (msea ηs )2 − ...

1966
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(6) 14 Here, mval/sea q denote the valence/sea quark masses, and mH (mval q , msea q ) is derived from joint fits of lattice data across ensembles with varying mval q and msea q

σqH extraction The Feynman-Hellman theorem establishes the relation ⟨¯qq⟩H = ∂mH ∂mq , which can be further decomposed into contributions from valence and sea quarks: ⟨¯qq⟩H = ⟨¯qq⟩val H + ⟨¯qq⟩sea H , ⟨¯qq⟩val H = ∂mH (mval q , msea q ) ∂mvalq , ⟨¯qq⟩sea H = ∂mH (mval q , mse...
[114]

Error budget of our predictions In table III, we summarized the final numerical results of the baryon masses, sigma terms and also trace anomaly contributions. 17 The statistical uncertainty is categorized by its sources: the statistical fluctuations in the dimensionless pion ...
[115]

The experimental Ds mass measurement
[116]

multiplicative discretization errors)

Choice of fit ansatz (additive vs. multiplicative discretization errors)
[117]

(10) and (12), which propagate to uncertainties of the sea quark mass contribution of sigma terms, and then also that of the gluon trace anomaly due to the sum rule

Sea quark dependence parameters ds(c) 1,2 in Eqs. (10) and (12), which propagate to uncertainties of the sea quark mass contribution of sigma terms, and then also that of the gluon trace anomaly due to the sum rule. The primary source of uncertainty is statistical, arising fro...

1970

Pith tools

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

[1] [1]

Mismatch effects arising from the masses of the light and strange quarks: The ensembles [2, 3] use different 3 light quark masses ranging from 0.8 to 6.7 times the av- eraged mass of the up and down quarks. This allows us to interpolate the light quark mass to its physical value, using the SU(4 |2) partially quenched heavy baryon chi- ral perturbation the...

[2] [2]

Impact of tuning the charm quark mass and con- sidering the missing charm quark loop: Similar to our approach with the strange quark, we adjust the valence charm quark mass to correspond with the physical mass of the Ds meson in each ensemble, and then eliminate the mismatch effects from the light and strange quark loops through a joint fit. As demonstrat...

[3] [3]

Additionally, the a4 correction is crucial for capturing the a dependence of certain charmed baryons

Continuum and infinite volume extrapolation: We find that the linear a2 correction adequately describes the masses of baryons without the charm quark across differ- ent lattice spacings a. Additionally, the a4 correction is crucial for capturing the a dependence of certain charmed baryons. Consequently, we retain the a4 term in the con- tinuum extrapolati...

[4] [4]

Therefore, we defer their investigation to future studies

Isospin symmetry breaking (ISB) and quantum elec- trodynamics (QED) effects: Naive power counting in- dicates that both effects are at the 1% level and then unlikely to alter the conclusions presented in this work. Therefore, we defer their investigation to future studies. More evidence validating above procedure are provided in the supplementary informat...

[5] [5]

2 displays the light quark σ terms, representing the direct contributions of light quark masses to the baryon mass

Light quark sigma term σπH ≡ σuH + σdH : The upper left panel of Fig. 2 displays the light quark σ terms, representing the direct contributions of light quark masses to the baryon mass. Some of the enhancement originates solely from the light quark loop (a gluon splits into a quark and anti-quark pair which combines into an- other gluon at a later time), ...

[6] [6]

Conse- quently, as observed in the upper-right panel of Fig

Strange quark sigma term σsH : When the quark mass reaches the magnitude of the strange quark mass 92.4(1.0) MeV at MS 2 GeV [28, 59–63, 66], the enhance- ments from both the quark loop and connected-sea mech- anisms diminish, as the gluon must provide significantly more energy for the related processes to occur. Conse- quently, as observed in the upper-r...

[7] [7]

2 illustrates that the charm quark scenario is no- tably more consistent, with the valence σcH being di- rectly proportional to the charm quark by a factor of ap- proximately 1

Charm quark sigma term σcH : The lower-left panel of Fig. 2 illustrates that the charm quark scenario is no- tably more consistent, with the valence σcH being di- rectly proportional to the charm quark by a factor of ap- proximately 1. In comparison to the charm quark mass of 1.093(7) GeV at MS 2 GeV [28, 63, 67–71], the ra- tio may be slightly smaller th...

[8] [8]

(3), the val- ues of ⟨Ha⟩H (represented by hollow triangles) for dif- ferent hadrons H are displayed in the lower-right panel of Fig

Trace anomaly contribution: By subtracting the to- tal σ term from the baryon mass using Eq. (3), the val- ues of ⟨Ha⟩H (represented by hollow triangles) for dif- ferent hadrons H are displayed in the lower-right panel of Fig. 2. It is evident that ⟨Ha⟩H increases with the number of heavy valence quarks, resulting in a twofold enhancement from the lightes...