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Renormalization and Running of Quark Mass and Field in the Regularization Invariant and MS-bar Schemes at Three and Four Loops

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K. G. Chetyrkin, A. Retey

High Energy Physics - Phenomenology · Submitted 13 Oct 1999 · 61 references

Pith's verdict: The NNNLO MS-bar→RI conversion factors are the real deliverable and they look solid; the 3 GeV advice is a heuristic the authors themselves flag as '(apparent)'.

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arXiv IDhep-ph/9910332 (version 2)
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CitationA. Retey, K. G. Chetyrkin. "Renormalization and Running of Quark Mass and Field in the Regularization Invariant and MS-bar Schemes at Three and Four Loops." arXiv:hep-ph/9910332 (1999). https://pith.science/paper/TX3IQ6EM
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The paper

The paper claims

The paper computes the three-loop conversion between MS-bar and RI quark masses and fields and argues that lattice matching should move from 2 GeV to about 3 GeV because the new third-order mass term is comparable to the second.

The reading

This paper wants to make the bridge between lattice QCD and the MS-bar quark masses precise enough to trust. Lattice calculations naturally produce masses in the RI scheme, while most phenomenological work uses MS-bar, so the conversion factor between them has to be computed in perturbation theory. The authors compute the three-loop quark propagator in the high-energy limit and extract the next-to-next-to-next-to-leading-order conversion factors for the quark mass and field. The practical finding is that at 2 GeV the new third-order mass-conversion term is about 5 percent, not much smaller than the 7.5 percent second-order term, so the perturbative series is not under control there. They recommend matching at roughly 3 GeV, where the known terms shrink, and they also extract four-loop RI anomalous dimensions as a by-product.

What carries the argument

The load-bearing object is the set of conversion equations that relate a general mass-independent scheme to MS-bar through the renormalized quark propagator. Writing the propagator as $S(q)=i/(m-/q-Sigma)$ with $Sigma=/q Sigma_V + m Sigma_S$, the RI renormalization condition sets the amputated Green function to its tree value in Landau gauge at $q^2=-mu^2$ and $m=0$. This gives $C^{RI}_2 = [1+Sigma_V + (1/2) partial Sigma_V/partial ell]^{-1}$ and $C^{RI}_m = [(1+Sigma_V + (1/2) partial Sigma_V/partial ell)/(1-Sigma_S)]$ evaluated at $q^2=-mu^2$, with $ell=log(-q^2/mu^2)$; the RI' variant drops the derivative term. The authors compute the needed massless three-loop $Sigma_V$ and $Sigma_S$ using QGRAF for diagram generation, LMP for small-mass expansion, and MINCER and MATAD for the resulting integrals. The same conversion functions then feed into $gamma'_m = gamma_m - beta, partial_{alpha_s} ln C_m$ and the analogous field equation, producing the four-loop RI anomalous dimensions. The paper's discussion of the absent $zeta(4)$ rests on defining scale- and scheme-invariant functions $Gamma^{RI}_{?}$ from the $q^2$ dependence of the conversion factor.

Extended reading notes

Core claim

The central claim is that the RI-to-MS-bar conversion for the quark mass and field is now known through order $O(alpha_s^3)$, and that this new term changes the practical advice for lattice quark-mass determinations. In Eqs. (34)-(37) the authors give explicit conversion functions $C^{RI}_m$, $C^{RI}_2$, $C^{RI'}_m$, and $C^{RI'}_2$ as polynomials in $alpha_s/(4pi)$ with coefficients depending on $n_f$. For $n_f=4$ and $alpha_s/pi=0.1$, the RI mass conversion reads $C^{RI}_m = 1.0 - 0.1333 - 0.0754 - 0.0495$ at $mu=2$ GeV: the third-order term is comparable to the second. At $mu=3$ GeV the same series becomes $1.0 - 0.111 - 0.0526 - 0.0289$, which is the basis for the recommendation to move the matching scale upward. The paper also reports that the massless approximation used for the RI conversion is poor for a charm quark near 2 GeV, because the mass-dependent MOM scheme differs from RI by terms containing $z ln z$ with $z=m^2/q^2$.

Load-bearing premise

The recommendation to move the matching scale to 3 GeV rests on the apparent shrinkage of the first three terms and assumes the uncomputed fourth-order coefficient will not reverse that pattern; the paper offers no bound or resummation for it.

Editorial extensions

If this is right

  • Lattice groups that convert RI masses to MS-bar at 2 GeV should treat the missing fourth-order term as a real truncation uncertainty rather than assuming the three-loop result is precision.
  • Moving the matching scale to about 3 GeV reduces the known mass-conversion series from $1.0-0.111-0.0526-0.0289$ at $alpha_s/pi=0.1$, giving a better-behaved starting point if the pattern continues.
  • The four-loop RI and RI' anomalous dimensions allow consistent running of RI masses and fields and a four-loop relation between the RI mass and the renormalization-group-invariant mass for $n_f=3,4,5$.
  • The absence of $zeta(4)$ in the RI and RI' anomalous dimensions, in contrast to their MS-bar counterparts, supports the interpretation that this constant is an artifact of the MS-bar scheme rather than a physical feature.
  • For the charmed quark, the RI massless limit is a poor approximation near 2 GeV because mass-dependent MOM corrections enter as $z ln z$; a mass-dependent matching is needed for charm.

Reading between the lines

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

  • A natural next step is to estimate the uncomputed four-loop coefficient with a Padé, Borel, or conformal-mapping analysis; if that estimate is small at 3 GeV, the paper's practical recommendation becomes much stronger, and if it is large, the recommendation loses its basis.
  • The scheme-invariant $Gamma^{RI}_{?}$ construction could be applied to other momentum-subtraction anomalous dimensions to test whether the disappearance of $zeta(4)$ is a general property of physical MOM schemes rather than a special feature of the quark mass and field.
  • A lattice systematic study that varies the matching scale from 2 to 3 GeV and propagates the conversion uncertainty into the final light-quark masses would show whether this truncation error dominates other lattice errors at current precision.
  • For charm, a practical extension is to compare RI, RI', and mass-dependent MOM conversions directly at the same scale, rather than treating RI as an approximation to MOM in the charm region.
Original abstract

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the MS-bar scheme with the corresponding quantities defined in any other scheme. By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the MS-bar quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (RI) scheme which is more suitable for lattice QCD calculations. The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV --- the typical normalization scale employed in lattice simulations. Thus, in order to get a precise prediction for the MS-bar masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. We also compute two more terms in the high-energy expansion of the MS-bar renormalized quark propagator. The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the MS-bar mass of the charmed quark. As a by-product of our calculations we determine the four-loop anomalous dimensions of the quark mass and field in the Regularization Invariant scheme. Finally, we discuss some physical reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,525 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.78 · compares with prior results 0.89 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the MS-bar scheme with the corresponding quantities defined in any other scheme. By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the MS-bar quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (RI) scheme which is more suitable for lattice QCD calculations. The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV --- the typical normalization scale employed in lattice simulations. Thus, in order to get a precise prediction for the MS-bar masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. We also compute two more terms in the high-energy expansion of the MS-bar renormalized quark propagator. The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the MS-bar mass of the charmed quark. As a by-product of our calculations we determine the four-loop anomalous dimensions of the quark mass and field in the Regularization Invariant scheme. Finally, we discuss some physical reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the MS-bar scheme with the corresponding quantities defined in any other scheme. contribution 0.93 0.95 0.15 asserted 1.00
s1 By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the MS-bar quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (RI) scheme which is more suitable for lattice QCD calculations. contribution 0.57 0.97 0.29 asserted 1.00
s2 The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV --- the typical normalization scale employed in lattice simulations. result 0.96 0.92 0.23 asserted 1.00
s3 Thus, in order to get a precise prediction for the MS-bar masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. interpretation 0.64 0.49 0.32 asserted 0.50
s4 We also compute two more terms in the high-energy expansion of the MS-bar renormalized quark propagator. contribution 0.85 0.96 0.03 asserted 0.99
s5 The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the MS-bar mass of the charmed quark. contribution 0.42 0.85 0.04 asserted 0.87
s6 As a by-product of our calculations we determine the four-loop anomalous dimensions of the quark mass and field in the Regularization Invariant scheme. result 0.91 0.97 0.12 asserted 1.00
s7 Finally, we discuss some physical reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions. interpretation 0.95 0.66 0.04 asserted 0.60

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.94.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The ratio of the RI and MOM scheme conversion functi
Figure 2. The ratio of the RI and MOM scheme conversion functi

Figures reproduced from arXiv:hep-ph/9910332 by the authors.

References
How later papers used it

68 later papers cite it

Evidence: substantial. 67 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 26 of the 111 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 5 of 18 papers with this label
How the literature uses it

64 arXiv papers cite this in 101 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 58 sentences without a confident label

Uses its method or tool 17 papers
  • We use $μ'=μ= 10$ GeV as the renormalisation scale ; $c_0Z_q^RI'(p^2,μ^2)$ is computed from the four loop perturbative running of $Z_q$ [ref] : c_0Z_q^RI'(p^2,μ^2) ≡Z_q^pert RI'(p^2, g^2_bare)/Z_q^pert RI'(μ^2, g^2_bare) ;

    Renormalisation of quark propagators from twisted-mass lattice QCD at $N_f$=2 · Running including $A^2$ corrections from OPE · 0.95

  • _c(2 )=1.14(3)(3) =1.14(4) → _c(_c)=1.28(4) , where the evolution to the more conventional scale given by $_c$ itself has been performed at N$^3$LO [ref] with $N_f=2$, consistently with our non-perturbative evaluation of the renormalization constant.

    Quark masses with Nf=2 twisted mass lattice QCD · Charm quark mass · 0.96

  • _c(2 ) = 1.14(3)(3) =1.14(4) → _c(_c) = 1.28(4) ,where the evolution to the more convential scale given by $_c$ itself has been performed at N$^3$LO [ref] with $N_f=2$, consistently with our non-perturbative evaluation of the renormalization constant.

    Average up/down, strange and charm quark masses with Nf=2 twisted mass lattice QCD · Charm quark mass · 0.95

Uses its data or values 11 papers
Compares with it 1 paper
  • We see that the lattice QCD data exhibit small discretization errors and agree in the continuum limit with the perturbative counterpart $Σ_P^pt(μ_A^2,μ_B^2)=1.058$, which is computed to N$^3$LO [ref] .

    Quark masses using twisted mass fermion gauge ensembles · Analysis method and safety checks against hadronic contaminations · 0.81

Cites it as background 6 papers

Discussion

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