Pith. sign in

REVIEW 4 minor 55 references

A new approach to quark mass determination using the gradient flow

T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Flowed quark-bilinear VEV ratios give a new, gauge-invariant route to quark masses, with the two-loop perturbative input now complete.

desk verdict A solid two-loop gradient-flow calculation with a genuinely new Laplace-transform expansion technique, but the heavy-quark mass extraction claim outruns what the paper actually establishes. read the letter →

arxiv 2506.09537 v2 pith:U52WQJPM submitted 2025-06-11 hep-lat hep-ph

Hiromasa Takaura , Robert V. Harlander , Fabian Lange This is my paper
classification hep-lathep-ph
keywords gradientflowquarkmassdeterminationflowedbilinearoperatorstwo-loopperturbationtheoryLaplacetransformexpansionlatticeQCDheavymassesMSbarscheme
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

This paper proposes to determine quark masses by matching lattice data to perturbation theory for ratios of vacuum expectation values of flowed quark bilinear operators, whose ultraviolet divergences cancel in ratios. The concrete deliverable is the next-to-leading-order (two-loop, $\mathcal O(\alpha_s)$) evaluation of these VEVs with full dependence on the dimensionless combination $m^2t$, supplied as small- and large-$m^2t$ expansions and as numerical grids. Because practical flow times for charm and bottom quarks force $m^2t$ well above one, the previously known small-$m^2t$ results were insufficient; these new results cover the needed range. If the proposal works, it gives a renormalization-group-invariant way to extract the $\overline{\rm MS}$ quark mass with systematics different from existing lattice methods.

What carries the argument

The central object is a new expansion technique based on the Laplace transform in the variable $z=m^2t$: writing each loop integral as $t^{-\alpha/2}\hat I(z)$, the transform $\tilde I(v)=\int_0^\infty dz\, z^{-v-1}\hat I(z)$ has singularities whose residues give the small-$z$ expansion when closing the contour to the right and the large-$z$ expansion when closing to the left. The same $\tilde I(v)$ therefore yields both asymptotic limits symmetrically. Integrals that resist this treatment are handled by differential equations with $t$-flow, and a companion numerical program provides the full mass dependence on a grid in $m^2t$.

What would settle it

A lattice measurement of $r_b(m)=R(t,m)/R(t,0)$ for bottom quarks at several flow times with $m^2t\simeq 6$--$100$ that disagrees with the NLO prediction by more than the quoted scale-variation band would show the non-perturbative corrections are not negligible; alternatively, computing the $\mathcal O(\alpha_s^2)$ terms and finding shifts much larger than the NLO band would undermine the precision estimate.

Watch

Extended reading notes

Core claim

The paper's central claim is that the ratios $r_a(m)=S(t,m)/R(t,m)$, $r_b(m)=R(t,m)/R(t,m=0)$, and $r_c(m)=m\,\mathrm{d}/\mathrm{d}m\,(S/R)$ are finite, renormalization-group invariant, and computable both on the lattice and in perturbation theory, so that matching the two determines the $\overline{\rm MS}$ quark mass. To enable this, the paper evaluates $S(t)$ and $R(t)$ at next-to-leading order with exact mass dependence. The results reproduce the known small-$m^2t$ expansions as checks, agree with independent numerical integration, and extend into the large-$m^2t$ region needed for heavy quarks; for the bottom quark the scale-variation estimate at this order suggests a 1--2\% mass determination.

Load-bearing premise

That the unknown non-perturbative corrections to the flowed-bilinear ratios are negligible in the $m^2t\gg1$ window needed for charm and bottom quarks, a regime the paper explicitly says is little understood.

Editorial extensions

If this is right

  • The two-loop mass-dependent results for $S$ and $R$ complete the perturbative input needed to turn the flowed-bilinear ratios of eq. (3.10) into a working lattice method.
  • Charm- and bottom-quark extractions can now be attempted in the physically relevant window $m^2t \gg 1$, where previous expansions in small $m^2t$ were insufficient.
  • At the achieved order, scale variation suggests roughly 1--2\% expected precision for the bottom-quark mass, with better precision for strange and charm at small flow times.
  • For light quarks, the derivative ratio $r_c$ removes the leading $m$-independent non-perturbative term $\propto \Lambda_{\rm QCD}^3$, making it the recommended observable, while $1-r_b$ also suppresses non-perturbative effects.
  • The large-$m^2t$ expansions are asymptotic with zero radius of convergence, so the series must be truncated near its optimal order, e.g. $k_*\sim 2m^2t-2$ for the one-loop $S$ series.

Reading between the lines

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

  • The light-quark branch could be tested immediately if existing lattice simulations of flowed bilinears for $u,d,s$ quarks are reanalyzed through $r_c$; no new simulation would be needed for a first check.
  • The Laplace-transform expansion is not tied to bilinears: the same symmetric small/large-$m^2t$ treatment could be applied to other flow-time observables, such as the energy-momentum tensor or hadronic vacuum polarization, whenever two external scales compete.
  • If non-perturbative corrections at $m^2t\gg1$ turn out to be sizable, the heavy-quark branch could still be salvaged by modelling them with an operator-product expansion or by confining the extraction to the overlap region where the small-time expansion converges.
  • An $\mathcal O(\alpha_s^2)$ extension would sharpen the uncertainty estimate substantially, but requires flavor-resolved short-flow-time coefficients and the complete two-loop $\langle\bar\psi\psi\rangle$ VEV, which the paper notes are not yet available.
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

0 major / 4 minor

Summary. This paper proposes a new method for determining quark masses by matching lattice measurements of ratios of flowed quark-bilinear vacuum expectation values, S(t)=<χ̄χ> and R(t)=<χ̄D↔χ>, with their perturbative evaluations. The central technical deliverable is the next-to-leading order (two-loop) computation of S and R with full mass dependence, presented as small- and large-m^2t expansions (App. D) and as numerical grids obtained from the ftint code. The expansions are derived using a newly developed Laplace-transform technique. The paper also provides estimates of the expected precision of a mass extraction at this perturbative order and a discussion of non-perturbative corrections, explicitly stating that non-perturbative effects for m^2t ≫ 1 are unknown.

Significance. If correct, this work supplies the missing perturbative ingredient for a new, gauge-invariant scheme of quark mass determination in the gradient-flow formalism. The calculations pass several strong internal checks: the small-m^2t limits reproduce known results from the literature, the results satisfy the renormalization group equations (3.6), and the asymptotic expansions agree with the numerical integration to high precision (Fig. 1). The paper makes its results reproducible by providing Mathematica-readable expansions and numerical grids in ancillary files, and it uses the publicly available ftint/pySecDec packages. The proposed observables r_a, r_b, r_c are finite, renormalization-group invariant, and require no gauge-variant operators, which is a conceptual advantage over RI-MOM-type schemes. The main limitation, acknowledged in Sec. 4, is that for m^2t ≫ 1 the non-perturbative corrections are unquantified; this makes the heavy-quark branch of the proposal conditional but does not affect the validity of the perturbative calculation itself.

minor comments (4)
  1. [Sec. 3.2 and Fig. 5] The estimate of 1–2% expected accuracy for the bottom quark mass is derived from perturbative scale variation only; since the paper itself states in Sec. 4 that non-perturbative corrections for m^2t ≫ 1 are unknown, the abstract and the caption of Fig. 5 should explicitly state that this precision estimate neglects non-perturbative effects, to prevent over-reading.
  2. [Appendix A, J19] The displayed integrand for J19 contains an apparent typo: the term "((p^2+m^2)^2)^2" should presumably be "(p^2+m^2)^2". Please verify and correct.
  3. [Appendix C.2] The phrase after eq. (C.5) contains a typo: "Fyenmal" should be "Feynman".
  4. [Sec. 3.1, Fig. 1] The agreement between the asymptotic expansions and the numerical full-mass results is shown only graphically; consider adding a brief statement of the maximum deviation in the plotted range to quantify this internal consistency check.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the perturbative calculation is self-contained and cross-checked independently.

full rationale

The paper's central deliverable is the NLO (two-loop) perturbative evaluation of the flowed quark-bilinear VEVs S(t) and R(t) with full mass dependence. This calculation is performed directly: the integrals are reduced to scalar master integrals (eq. 2.9), evaluated by a new Laplace-transform expansion method (sec. 2.3), by differential equations (sec. 2.4), and by independent numerical integration with ftint/pySecDec (sec. 2.5). The small- and large-m^2t expansions are compared against the fully numerical results in fig. 1, and the small-m^2t coefficients are cross-checked against previously published operator-matching results (sec. 2.6). None of these steps define an output in terms of the quantity they claim to predict. The proposed quark-mass determination (sec. 3.2) is a matching prescription r(m)=r_exp that uses the computed perturbative functions as independent theoretical input; no parameter is fitted to lattice data within this paper, and the quoted 1-2% precision estimate is explicitly labelled a crude scale-variation forecast, not an extracted mass. The admitted limitation that non-perturbative corrections for m^2t >> 1 are unknown (sec. 4) is a correctness/validity risk for the heavy-quark branch, not a circularity: it does not make the perturbative derivation equivalent to its inputs. Some small-m^2t series coefficients are imported from prior papers by the same authors (refs. [4,11,37,38]), but those are independently published, externally checkable computations, and the same coefficients are also reproduced by the paper's own numerical evaluation; hence they are not load-bearing self-citation in a circular sense. Overall, the paper's derivation chain is self-contained and non-circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central calculations rely on standard QCD, dimensional regularization, and the gradient-flow formalism; they introduce no new physical entities. The renormalization-scale prescription used for numerical uncertainty estimates is a hand-chosen input, not fitted. The main unproven background assumptions are the completeness of the UV-divergence cancellation in ratios, the validity of the perturbative expansion in the quoted flow-time windows, and the suppression of unknown non-perturbative effects.

free parameters (1)
  • central renormalization scale mu_int(t) = sqrt(mu_t^2 + m^2(m)), mu_t = exp(-gamma_E/2)/sqrt(2t)
    Chosen by hand in Sec. 3.2 (eq. 3.16) to interpolate between small and large t; not fitted to data but affects the numerical uncertainty bands and precision estimates. The authors note it is not claimed optimal.
assumptions (4)
  • domain assumption The UV divergences of S(t) and R(t) are fully absorbed by a flowed-quark wave-function renormalization factor Z_chi, so ratios have a finite continuum limit.
    Invoked in Sec. 1 via refs. [3,4]; it is load-bearing for the lattice match but not proven in this paper.
  • domain assumption The perturbative expansion in alpha_s around the gradient-flow solution is valid in the chosen flow-time windows and for the considered quark masses.
    Needed to turn App. D results into numerical mass estimates; the paper discusses perturbative validity windows in Sec. 3.1 but cannot prove absence of non-perturbative corrections.
  • standard math For m^2t >> 1 the asymptotic large-flow-time series can be truncated optimally at k* ~ 2m^2t - 2 to give a reliable approximation.
    Sec. 3.1 notes the series has zero radius of convergence and motivates optimal truncation; this is a standard asymptotic-series assumption.
  • domain assumption The small-flow-time operator product expansion of eqs. (2.45)-(2.46) can be used to extract leading small-m^2t terms at O(alpha_s).
    Sec. 2.6 uses this known expansion and refs. [36,37] to confirm coefficients; it is an established framework but is an input to the calculation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A new approach to quark mass determination using the gradient flow." pith.science (2026). https://pith.science/paper/U52WQJPM

@misc{pith2026250609537,
  author       = {Pith},
  title        = {Pith review of: A new approach to quark mass determination using the gradient flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U52WQJPM}},
  note         = {Machine review of arXiv:2506.09537}
}
abstract

We propose a new method to determine quark masses using ratios of the vacuum-expectation values (VEVs) of flowed quark bilinear operators. They can be expressed as functions of the flow time $t$ and the ${\overline {\rm MS}}$ quark mass $\overline{m}$, which can then be determined by matching with the corresponding lattice results. Motivated by this, we evaluate these VEVs perturbatively through next-to-leading order in the strong coupling. We provide the results as expansions in the limits of small and large $\overline{m}^2 t$. To this end, we develop a new expansion technique based on the Laplace transform. Additionally, we present numerical results with the exact mass dependence over a wide range of $\overline{m}^2t$. We discuss the expected perturbative precision for the mass determination based on our next-to-leading order perturbative calculations, and possible non-perturbative corrections.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 17 canonical work pages

  1. [1]

    Narayanan and H

    R. Narayanan and H. Neuberger,Infinite N phase transitions in continuum Wilson loop operators, JHEP 03 (2006) 064 [hep-th/0601210]

    arXiv 2006

  2. [2]

    Lüscher,Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [1006.4518]

    M. Lüscher,Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [1006.4518]

    arXiv 2010

  3. [3]

    Lüscher,Chiral symmetry and the Yang–Mills gradient flow, JHEP 04 (2013) 123 [1302.5246]

    M. Lüscher,Chiral symmetry and the Yang–Mills gradient flow, JHEP 04 (2013) 123 [1302.5246]

    arXiv 2013

  4. [4]

    Makino and H

    H. Makino and H. Suzuki,Lattice energy–momentum tensor from the Yang–Mills gradient flow—inclusion of fermion fields, PTEP 2014 (2014) 063B02 [1403.4772]

    arXiv 2014

  5. [5]

    Fla vour Lattice A veraging Group (FLAG)collaboration, FLAG Review 2024, 2411.04268

    arXiv 2024

  6. [6]

    Martinelli, C

    G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa and A. Vladikas,A general method for non-perturbative renormalization of lattice operators, Nucl. Phys. B 445 (1995) 81 [hep-lat/9411010]

    arXiv 1995

  7. [7]

    Aoki et al.,Nonperturbative renormalization of quark bilinear operators andBK using domain wall fermions, Phys

    Y. Aoki et al.,Nonperturbative renormalization of quark bilinear operators andBK using domain wall fermions, Phys. Rev. D78 (2008) 054510 [0712.1061]

    arXiv 2008

  8. [8]

    Sturm, Y

    C. Sturm, Y. Aoki, N.H. Christ, T. Izubuchi, C.T.C. Sachrajda and A. Soni, Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D80 (2009) 014501 [0901.2599]

    arXiv 2009

Show all 55 references
  1. [9]

    HPQCD collaboration, High-precision charm-quark mass and QCD coupling from current-current correlators in lattice and continuum QCD, Phys. Rev. D78 (2008) 054513 [0805.2999]

    2008 arXiv

  2. [10]

    TUMQCD collaboration, Relations between heavy-light meson and quark masses, Phys. Rev. D 97 (2018) 034503 [1712.04983]

    2018 arXiv

  3. [11]

    Artz, R.V

    J. Artz, R.V. Harlander, F. Lange, T. Neumann and M. Prausa,Results and techniques for higher order calculations within the gradient-flow formalism, JHEP 06 (2019) 121 [1905.00882]

    2019 arXiv

  4. [12]

    Lange,Applications of the perturbative gradient flow at higher orders in quantum chromodynamics, Ph.D

    F. Lange,Applications of the perturbative gradient flow at higher orders in quantum chromodynamics, Ph.D. thesis, RWTH Aachen University, 2021. 10.18154/RWTH-2021-07652

    2021 doi

  5. [13]

    Takaura, R.V

    H. Takaura, R.V. Harlander and F. Lange,Determining the Quark Mass with the Gradient Flow, PoS LA TTICE2024(2025) 288 [2411.13782]

    2025 arXiv

  6. [14]

    Harlander, T

    R.V. Harlander, T. Nellopoulos, A. Olsson and M. Wesle,ftint: Calculating gradient-flow integrals with pySecDec, Comput. Phys. Commun.306 (2025) 109384 [2407.16529]

    2025 arXiv

  7. [15]

    Beneke and V.A

    M. Beneke and V.A. Smirnov,Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391]. – 29 –

    1998 arXiv

  8. [16]

    Harlander,The gradient flow at higher orders in perturbation theory, PoS LA TTICE2021(2022) 489 [2111.14376]

    R. Harlander,The gradient flow at higher orders in perturbation theory, PoS LA TTICE2021(2022) 489 [2111.14376]

    2022 arXiv

  9. [17]

    Beneke, P

    M. Beneke, P. Hager and A.F. Sanfilippo,Cosmological correlators in masslessϕ4-theory and the method of regions, JHEP 04 (2024) 006 [2312.06766]

    2024 arXiv

  10. [18]

    Lüscher and P

    M. Lüscher and P. Weisz,Perturbative analysis of the gradient flow in non-abelian gauge theories, JHEP 02 (2011) 051 [1101.0963]

    2011 arXiv

  11. [19]

    Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions, Phys

    F.V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B100 (1981) 65

    1981

  12. [20]

    Chetyrkin and F.V

    K.G. Chetyrkin and F.V. Tkachov,Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159

    1981

  13. [21]

    Neubert,Scale setting in QCD and the momentum flow in Feynman diagrams, Phys

    M. Neubert,Scale setting in QCD and the momentum flow in Feynman diagrams, Phys. Rev. D 51 (1995) 5924 [hep-ph/9412265]

    1995 arXiv

  14. [22]

    Kitano and H

    R. Kitano and H. Takaura,Quantum electrodynamics on the lattice and numerical perturbative computation of g−2, PTEP 2023 (2023) 103B02 [2210.05569]

    2023 arXiv

  15. [23]

    Smirnov,Analytical result for dimensionally regularized massless on-shell double box, Phys

    V.A. Smirnov,Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B460 (1999) 397 [hep-ph/9905323]

    1999 arXiv

  16. [24]

    Tausk,Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys

    J.B. Tausk,Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506]

    1999 arXiv

  17. [25]

    Mishima,High-energy expansion of two-loop massive four-point diagrams, JHEP 02 (2019) 080 [1812.04373]

    G. Mishima,High-energy expansion of two-loop massive four-point diagrams, JHEP 02 (2019) 080 [1812.04373]

    2019 arXiv

  18. [26]

    Zhang,Massive two-loop four-point Feynman integrals at high energies with AsyInt, JHEP 09 (2024) 069 [2407.12107]

    H. Zhang,Massive two-loop four-point Feynman integrals at high energies with AsyInt, JHEP 09 (2024) 069 [2407.12107]

    2024 arXiv

  19. [27]

    Huber and D

    T. Huber and D. Maître,HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun.175 (2006) 122 [hep-ph/0507094]

    2006 arXiv

  20. [28]

    Huber and D

    T. Huber and D. Maître,HypExp 2, Expanding hypergeometric functions about half-integer parameters, Comput. Phys. Commun.178 (2008) 755 [0708.2443]

    2008 arXiv

  21. [29]

    Mathematica, Version 13.2

    Wolfram Research, Inc., “Mathematica, Version 13.2.”

  22. [30]

    Binoth and G

    T. Binoth and G. Heinrich,An automatized algorithm to compute infrared divergent multi-loop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013]

    2000 arXiv

  23. [31]

    Binoth and G

    T. Binoth and G. Heinrich,Numerical evaluation of multi-loop integrals by sector decomposition, Nucl. Phys. B 680 (2004) 375 [hep-ph/0305234]

    2004 arXiv

  24. [32]

    Borowka, G

    S. Borowka, G. Heinrich, S.P. Jones, M. Kerner, J. Schlenk and T. Zirke,SecDec-3.0: Numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun.196 (2015) 470 [1502.06595]

    2015 arXiv

  25. [33]

    Borowka, G

    S. Borowka, G. Heinrich, S. Jahn, S.P. Jones, M. Kerner, J. Schlenk et al.,pySecDec: A toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun.222 (2018) 313 [1703.09692]

    2018 arXiv

  26. [34]

    Borowka, G

    S. Borowka, G. Heinrich, S. Jahn, S.P. Jones, M. Kerner and J. Schlenk,A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec, Comput. Phys. Commun.240 (2019) 120 [1811.11720]. – 30 –

    2019 arXiv

  27. [35]

    Heinrich, S.P

    G. Heinrich, S.P. Jones, M. Kerner, V. Magerya, A. Olsson and J. Schlenk,Numerical scattering amplitudes with pySecDec, Comput. Phys. Commun.295 (2024) 108956 [2305.19768]

    2024 arXiv

  28. [36]

    Lüscher,Future applications of the Yang-Mills gradient flow in lattice QCD, PoS LA TTICE2013(2014) 016 [1308.5598]

    M. Lüscher,Future applications of the Yang-Mills gradient flow in lattice QCD, PoS LA TTICE2013(2014) 016 [1308.5598]

    2014 arXiv

  29. [37]

    Harlander, F

    R.V. Harlander, F. Lange and T. Neumann,Hadronic vacuum polarization using gradient flow, JHEP 08 (2020) 109 [2007.01057]

    2020 arXiv

  30. [38]

    Borgulat, R.V

    J. Borgulat, R.V. Harlander, J.T. Kohnen and F. Lange,Short-flow-time expansion of quark bilinears through next-to-next-to-leading order QCD, JHEP 05 (2024) 179 [2311.16799]

    2024 arXiv

  31. [39]

    Hieda and H

    K. Hieda and H. Suzuki,Small flow-time representation of fermion bilinear operators, Mod. Phys. Lett. A31 (2016) 1650214 [1606.04193]

    2016 arXiv

  32. [40]

    Mereghetti, C.J

    E. Mereghetti, C.J. Monahan, M.D. Rizik, A. Shindler and P. Stoffer,One-loop matching for quark dipole operators in a gradient-flow scheme, JHEP 04 (2022) 050 [2111.11449]

    2022 arXiv

  33. [41]

    Borgulat, R

    J. Borgulat, R. Harlander, M.D. Rizik and A. Shindler,Two-loop matching of the chromo-magnetic dipole operator with the gradient flow, PoS LA TTICE2022(2023) 313 [2212.09824]

    2023 arXiv

  34. [42]

    Broadhurst,Chiral symmetry breaking and perturbative QCD, Phys

    D.J. Broadhurst,Chiral symmetry breaking and perturbative QCD, Phys. Lett. B101 (1981) 423

    1981

  35. [43]

    Spiridonov and K.G

    V.P. Spiridonov and K.G. Chetyrkin,Nonleading mass corrections and renormalization of the operators m ¯ψψ and G2 µν, Sov. J. Nucl. Phys.47 (1988) 522

    1988

  36. [44]

    Harlander,Quarkmasseneffekte in der Quantenchromodynamik und asymptotische Entwicklung von Feynman-Integralen, Ph.D

    R. Harlander,Quarkmasseneffekte in der Quantenchromodynamik und asymptotische Entwicklung von Feynman-Integralen, Ph.D. thesis, Universität (TH) Karlsruhe, 1998

    1998

  37. [45]

    Harlander, Y

    R.V. Harlander, Y. Kluth and F. Lange,The two-loop energy–momentum tensor within the gradient-flow formalism, Eur. Phys. J.C78 (2018) 944 [1808.09837]

    2018 arXiv

  38. [46]

    Braaten, S

    E. Braaten, S. Narison and A. Pich,QCD analysis of the tau hadronic width, Nucl. Phys. B 373 (1992) 581

    1992

  39. [47]

    Chetyrkin and J.H

    K.G. Chetyrkin and J.H. Kühn,Quartic mass corrections toRhad, Nucl. Phys. B 432 (1994) 337 [hep-ph/9406299]

    1994 arXiv

  40. [48]

    Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D110 (2024) 030001

    2024

  41. [49]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,Five-Loop Running of the QCD Coupling Constant, Phys. Rev. Lett.118 (2017) 082002 [1606.08659]

    2017 arXiv

  42. [50]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt,The five-loop beta function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [1701.01404]

    2017 arXiv

  43. [51]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard and Y. Schröder,The five-loop beta function for a general gauge group and anomalous dimensions beyond Feynman gauge, JHEP 10 (2017) 166 [1709.07718]

    2017 arXiv

  44. [52]

    Chetyrkin, B.A

    K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser,Strong Coupling Constant with Flavor Thresholds at Four Loops in the Modified Minimal-Subtraction Scheme, Phys. Rev. Lett.79 (1997) 2184 [hep-ph/9706430]

    1997 arXiv

  45. [53]

    Vermaseren, S.A

    J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen,The 4-loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B405 (1997) 327 [hep-ph/9703284]. – 31 –

    1997 arXiv

  46. [54]

    WHOT-QCD collaboration, Exploring Nf = 2+1 QCD thermodynamics from the gradient flow, Phys. Rev. D96 (2017) 014509 [1609.01417]

    2017 arXiv

  47. [55]

    WHOT-QCD collaboration, Nf = 2+1 QCD thermodynamics with gradient flow using two-loop matching coefficients, Phys. Rev. D102 (2020) 014510 [2005.00251]. – 32 –

    2020 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.