Determining the Quark Mass with the Gradient Flow

Fabian Lange; Hiromasa Takaura; Robert Harlander

REVIEW 3 major objections 4 minor 1 cited by

Determining the Quark Mass with the Gradient Flow

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

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Pith's one-line read The paper establishes that the flowed quark condensate, computed at O(alpha_s) with full $m^2t$ dependence, supplies the perturbative input needed to extract heavy-quark MS-bar masses by matching a gauge-invariant ratio of flowed quark…

desk verdict New O(alpha_s) mass-dependent condensate terms with a promising but unproven expansion method; solid proceedings paper worth refereeing. read the letter →

arxiv 2411.13782 v1 pith:GWYHD3MA submitted 2024-11-21 hep-lat

Hiromasa Takaura , Robert Harlander , Fabian Lange This is my paper · ORCID

classification hep-lat MSC 81T2581T1381V05

keywords gradientflowquarkmassdeterminationcondensateheavyquarksperturbativeexpansionlatticeQCDLaplacetransformMS-barscheme

topics Yang-Mills Mass Gap

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 proposes determining heavy-quark masses from the gradient flow by comparing the lattice value of a ratio of flowed quark condensates to perturbation theory. The ratio is gauge invariant, involves only one-point functions, and is finite after renormalization, which should give clean control of perturbative errors and suppressed lattice noise. The missing ingredient was the quark-mass dependence of the flowed quark condensate; the paper computes it at $O(\alpha_s)$ in both the small-$m^2t$ and large-$m^2t$ limits, and also numerically over the full mass range. The next-to-leading small-mass term and all the $O(\alpha_s)$ terms of the large-mass expansion are new. If the result is right, lattice data on this ratio can be matched to determine MS-bar charm and bottom masses at next-to-leading order.

What carries the argument

The central mechanism is the Laplace-transform residue expansion for massive gradient-flow integrals. Given an integral $I(m^2,t)$, one forms $\tilde I(s,t)=\int_0^\infty d(m^2)\,(m^2)^{-s-1}I(m^2,t)$, inverts with a contour integral, and obtains the $m^2t\ll 1$ series as a residue sum over positive singularities and the $m^2t\gg 1$ series as a residue sum over negative singularities of $\tilde I(s,t)(m^2)^s$. This bypasses the failure of expansion by regions in the large-mass limit, where the flow-time integral covers hard and soft momentum regions simultaneously. The observable it is applied to is the ratio $R(t,m_1,m_2)=\langle\bar{\chi}_1\chi_1\rangle/\langle\bar{\chi}_2{\overleftrightarrow{D}}\chi_2\rangle$, which is UV finite because the only divergent part of the flowed fermion fields is the wave-function renormalization.

What would settle it

Evaluate one of the eight integrals in Eq. (3) at an intermediate value such as $m^2t=10$ with an independent high-precision numerical integrator and compare with the large-$m^2t$ expansion of Eq. (9) truncated at the $1/(m^2t)^2$ term; a disagreement larger than the $O(1/(m^2t)^3)$ remainder would falsify the residue-expansion method.

Watch

Extended reading notes

Core claim

At $O(\alpha_s)$, the flowed quark condensate $\langle[\bar{\chi}(t,x)\chi(t,x)]_R\rangle$ is given by Eq. (8) for $m^2t\ll 1$ and by Eq. (9) for $m^2t\gg 1$. In the small-mass expansion the coefficient of the $m^2t$ term at $O(\alpha_s)$ is new; in the large-mass expansion all $O(\alpha_s)$ terms are new. These analytic results are supplemented by a numerical evaluation with full $m^2t$ dependence, so the perturbative prediction for the ratio $R(t,m_1,m_2)$ can be evaluated for charm and bottom quarks across their physical flow-time windows. The calculation uses a Laplace transform in $m^2$ followed by a contour inversion in the transform variable; the small- and large-$m^2t$ series are obtained from residues at positive and negative singularities, respectively, avoiding the failure of expansion by regions when the flow-time integral spans hard and soft momenta.

Load-bearing premise

The calculation assumes that closing the inverse-Laplace contour and summing the residues at the singularities of $\tilde I(s,t)$ reproduces the full massive gradient-flow integral for small and large $m^2t$, a claim whose proof is deferred to a future publication and which is only checked by numerical self-consistency.

Editorial extensions

If this is right

Lattice groups can match $R(t,m,0)$ or $R(t,m,m)$ to the new $O(\alpha_s)$ expressions and extract MS-bar charm and bottom masses at next-to-leading order, once sufficiently precise flowed-condensate data exist.
The large-$m^2t$ expansion is valid in the physical bottom-quark window $1.0 \ll 8m_b^2t \ll 200$, where the previously known small-mass results do not apply.
The numerical full-mass-dependence result covers the intermediate $m^2t$ region, so a mass determination does not depend on which asymptotic expansion happens to converge.
Because the observable is gauge invariant, the leading nonperturbative corrections start at dimension-four condensates, giving a controlled perturbative uncertainty that gauge-dependent schemes such as RI-MOM lack.
The same framework should extend to light-quark masses, as the paper notes, since the ratio does not require the quark to be heavy.

Reading between the lines

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

The Laplace-transform residue technique should transfer directly to the denominator observable $\langle\bar{\chi}\overleftrightarrow{D}\chi\rangle$ and to $O(\alpha_s^2)$ calculations, which would raise the matching precision to next-to-next-to-leading order.
Because expansion by regions is known to fail in the large-$m^2t$ flow-time case, the same method may also unlock analytic expansions of other massive gradient-flow quantities, such as flowed observables involving massive quarks.
A stronger check than numerical self-consistency would be to apply the residue expansion to a simpler massive gradient-flow integral with a known closed form; passing that test would harden the evidence before the detailed proof appears.
If lattice data reach the required precision, this gauge-invariant one-point observable could serve as an independent cross-check of existing lattice averages for charm and bottom MS-bar masses, which rest on different renormalization schemes.

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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

Hi [Name],

Punchline: this proceedings paper contains a genuinely new perturbative result—the O(alpha_s) mass-dependent term for the flowed quark condensate, in both small- and large-m^2t expansions, plus a full numerical evaluation. If the coefficients are right, this is a useful building block for a proposed gauge-invariant, noise-suppressed heavy-quark mass scheme. The paper is honestly scoped: it calls the computation a first step and explicitly lists lattice feasibility and higher orders as open issues.

What's good: the authors derive the expansions and cross-check them against an independent numerical evaluation (ftint) in Fig. 1, with agreement in the charm and bottom windows. That is real internal evidence, not fitting. They also clearly mark which terms are new (Eq. 8's NLO small-mass term, all of Eq. 9's NLO terms) and which come from earlier work. The citation pattern is clean; the self-citations are to published LO results.

Now the soft spots, in proportion. The load-bearing expansion method—Laplace transform, contour closure, residue sums in Eqs. (4)–(7)—is asserted, with details deferred to 'a future publication.' The stress-test note is correct that meromorphy, arc vanishing, and pole completeness are not demonstrated here. That means the new numerical constants rest on an unproven method. The Fig. 1 check is meaningful but bounded: it is a joint test of the analytic and numerical pipelines, covers a limited flow-time window (especially narrow for charm), and carries no stated precision. A common-mode error could hide, though I think that is unlikely given the method's analogy to prior work and the standard structure of the integrals.

Minor: the abstract says 'two-loop level' while the body consistently says O(alpha_s); that wording should be fixed. The constants in Eqs. (8)–(9) lack precision statements, a common proceedings omission.

Verdict: solid, worth a serious referee. The referee's main request should be the expansion-method derivation, either in an appendix or by pointing to the promised follow-up. The paper is not a complete proof, but the result is new, useful, and probably correct. I'd take it for peer review and ask for the missing details.

Referee Report

3 major / 4 minor

Summary. This proceedings paper proposes a new method to determine heavy-quark MS-bar masses from lattice QCD by matching the ratio of flowed quark condensates, Eq. (1), to perturbation theory. The perturbative input is the O(alpha_s) mass-dependent flowed quark condensate, which the authors compute in two complementary ways: analytic expansions in m^2t and 1/(m^2t), Eqs. (8)-(9), and a full numerical evaluation using ftint. The paper claims that the next-to-leading small-m^2t O(alpha_s) term in Eq. (8) and all O(alpha_s) terms in Eq. (9) are new, and it shows results for charm and bottom quarks in Fig. 1.

Significance. If the results are correct, they provide a genuinely useful perturbative ingredient for a novel, gauge-invariant, one-point-function-based lattice determination of heavy-quark masses. The proposed Laplace-residue expansion technique for massive gradient-flow integrals is also of methodological interest, and the explicit numerical cross-check with ftint is a strength. The novelty claims for Eqs. (8)-(9) are concrete and falsifiable, and the paper is refreshingly direct about what remains to be done. However, the central technical tool is only asserted, not demonstrated, so the significance is conditional on the missing derivation.

major comments (3)

[Section 2, Eqs. (4)-(7)] The inverse-Laplace residue expansion is the load-bearing element of the paper, but its validity is not established. After Eq. (5), the text simply says the contour is closed and the result is given by residue sums, with 'Further details will be explained in a future publication.' For Eqs. (6)-(7) to hold, one must show that the Laplace-transformed integrals are meromorphic in the relevant half-plane, that the arc contributions at |s| -> infinity vanish, and that the catalog of singularities is complete for each of the eight scalar integrals. None of these conditions is demonstrated. Since every new O(alpha_s) coefficient in Eqs. (8)-(9) depends on this method, the central claim is not yet fully supported. Please include the derivation, or at least a rigorous statement of the analyticity assumptions and a proof sketch, in the manuscript.
[Section 2, Fig. 1 and surrounding text] The agreement between the expansions and the ftint evaluation shown in Fig. 1 is a useful internal cross-check, but it is not an independent validation of the residue method: ftint evaluates the same scalar integrals that the expansions are supposed to represent. The plotted flow-time range is also limited, and the large-m^2t expansion is not shown for the charm case. The text states that Fig. 1 provides a 'convincing verification' of the expansion method; this overstates what a self-consistency check in a finite window can establish. Please quantify the plotted range, the order of neglected terms, the numerical uncertainties of ftint, and, if possible, test the expansions against an independent method (for example, expansion by regions or direct high-precision integration) at selected values of m^2t.
[Section 2, Eqs. (8)-(9) and the paragraph following them] The novelty claims--'the next-to-leading term in m^2t at O(alpha_s) of Eq. (8) is new' and 'in Eq. (9), all the O(alpha_s) terms are new'--cannot be verified from the manuscript because no intermediate steps are shown. A reader cannot see which residues produce which coefficients, how the MS-bar renormalization of the flowed operator and mass is implemented beyond the quoted factors R_chi and Z_m^MS, or how the numerical values such as 0.759581 and 2.3334 are obtained. For a proceedings contribution this level of detail may be acceptable, but for a refereed journal article the derivation must be present or the claims must be explicitly labeled as relying on a forthcoming publication.

minor comments (4)

[Eq. (3)] The momentum-space integration measure is garbled in the displayed integrals; please typeset the d-dimensional measures ∫_p ∫_k clearly so that the reader can distinguish the loop integrations from the flow-time integrations.
[Section 1, flow-time window] The inequalities '0.1 ≪ 8m_c^2 t ≪ 20' and '1.0 ≪ 8m_b^2 t ≪ 200' are written without explicitly stating the unit convention; adding a sentence that t has mass dimension -2 and m^2t is dimensionless would help the reader.
[Fig. 1] The figure axes and legends are difficult to read in the current rendering, and it is not clear over which t-interval each expansion is expected to be accurate. Please improve the figure and add a statement of the approximate validity ranges of the small- and large-m^2t expansions.
[References and footnotes] The footnotes marked 'one.sup' and 'two.sup' in the text are not rendered as ordinary footnotes, and the reference to the latest arXiv version of Ref. [17] is made only in a footnote; please incorporate these corrections into the reference list and remove the placeholder superscripts.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new O(alpha_s) mass dependence is derived analytically and checked numerically, not fitted; existing self-citations are not load-bearing.

full rationale

The paper derives the flowed quark condensate at O(alpha_s) from eight scalar integrals. The new analytic expansions in Eqs. (8)-(9) follow from the Laplace-transform residue method (Eqs. (4)-(7)); the numerical coefficients are obtained from residue sums, not tuned to reproduce any target. The check in Fig. 1 compares these expansions with an independent numerical evaluation using ftint [20]; this is a consistency check, not a fit. Prior work by the same authors is cited for the leading-order terms ('All the O(alpha_s^0) terms were calculated in Ref. [17]', Section 2) and for the numerical tool ftint, but neither citation supplies the claimed new NLO content. The sentence 'This approach is analogous to the ideas found in Refs. [18, 19]' is an analogy, not an import of the result, and the paper explicitly defers the proof ('Further details will be explained in a future publication', Section 2). That deferral is a completeness/correctness risk: the meromorphy of the Laplace transform, arc-vanishing of the contour, and completeness of the catalogue of singularities are not demonstrated. However, an unproven derivation is not equivalent to its inputs; no parameter is fitted and renamed as a prediction, no definition is circular, and no uniqueness theorem is imported. The self-citations are therefore not load-bearing, and no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameter is fitted to data in this paper. The physically relevant inputs used for Figure 1 (mb(mb)=4.18 GeV, mc(mc)=1.27 GeV, alpha_s(MZ)=0.1179) are adopted from prior literature solely for illustration. The new analytic results depend on the gradient-flow perturbative framework (Eq. 2, from Refs. [6-8]), the finiteness argument for the ratio (Refs. [8,12]), the asserted Laplace and residue expansion method, and the perturbativity window for the flow time. The only potentially self-referential inputs are prior results of the same authors (Refs. [17] and [19]), used as published inputs rather than fitted quantities. No new particles, forces, dimensions, or conserved quantities are introduced; the ratio of Eq. (1) is a new composite observable, not an entity with independent falsifiable content beyond the perturbative computation.

assumptions (4)

domain assumption Flowed fermion and gauge fields satisfy the gradient-flow equations (Eq. 2) with boundary conditions at t=0 and gauge parameter alpha_0=1.
Framework taken from Refs. [6-8]; renormalization of the flowed fermion field Z_chi is taken from Ref. [8].
domain assumption The ratio R(t,m1,m2) of Eq. (1) is finite after wave-function renormalization, giving a well-defined continuum limit.
Quoted from Refs. [8,12]; used to justify the lattice matching premise.
domain assumption Perturbation theory is valid in the flow-time window of Section 1, with 8t << Lambda_QCD^-2 and the quoted mass constraints; nonperturbative effects start at dimension-four condensates.
Section 1; the basis for claiming control over perturbative errors in the proposed method.
ad hoc to paper The inverse Laplace transform of Eq. (5) can be evaluated by closing the contour and summing residues (Eqs. 6-7), with the flow-time integrals handled in both the small- and large-m^2t limits.
The paper's new method; justification deferred: 'Further details will be explained in a future publication' (Section 2).

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Cite this review

Pith. "Pith review of Determining the Quark Mass with the Gradient Flow." pith.science (2026). https://pith.science/paper/GWYHD3MA

@misc{pith2026241113782,
  author       = {Pith},
  title        = {Pith review of: Determining the Quark Mass with the Gradient Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWYHD3MA}},
  note         = {Machine review of arXiv:2411.13782}
}

abstract

We propose a new method to determine the quark mass by using bilinear operators of the flowed quark field defined within the gradient-flow formalism. This method enables the quark mass determination through a comparison of perturbative calculations with lattice data. The gauge-invariant nature of the observable should allow clear control over perturbative errors. At the same time, the gradient flow suppresses the noise in the lattice measurements of the observable, which simply consists of one-point functions. Concerning the perturbative input in this framework, we study the mass dependence of the flowed quark condensate $\langle \bar{\chi}(t,x) \chi(t,x) \rangle$ at the two-loop level. For this purpose, we develop a novel approach for expanding massive gradient-flow integrals in the limit of small and large $(m^2t)$. We also present a fully numerical result which includes the full mass dependence.

Figures

Figures reproduced from arXiv: 2411.13782 by the authors.

**Figure 1.** ˚(, , 0) as a function of the flow time for the bottom-quark case ( = ) (left) and the charm-quark case ( = ) (right). The gray line shows the O ( 0 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

A new approach to quark mass determination using the gradient flow
hep-lat 2025-06 conditional novelty 7.0 of 10

The MS quark mass can be extracted by matching lattice data to ratios of flowed quark bilinear VEVs, and this paper provides the required next-to-leading-order expressions with full mass dependence.

Reference graph

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

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