Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme

B.A. Kniehl; K.G. Chetyrkin; M. Steinhauser (MPI Munich)

REVIEW

Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme

K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser (MPI Munich)

High Energy Physics - Phenomenology · Submitted 20 Jun 1997 · 20 references

Pith's verdict: A solid, standard-reference calculation: three-loop threshold matching and the four-loop Lambda relation are genuinely new, and the only real correction needed is to the reader's own summary of c2.

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arXiv IDhep-ph/9706430 (version 1)

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CitationB.A. Kniehl, K.G. Chetyrkin, M. Steinhauser (MPI Munich). "Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme." arXiv:hep-ph/9706430 (1997). https://pith.science/paper/2TNQGIQN

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Editor's note

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Short version: this is the paper that fixed the threshold-matching prescription for four-loop QCD running, and it holds up. The new content is the three-loop matching coefficients c3 (Eq. 7) and C3 (Eq. 10), the closed three-loop relation between Lambda^(nl) and Lambda^(nf) (Eq. 11), and the settlement of the two-loop c2 conflict between Refs. [10] and [11] in favour of [11]. The four-loop evolution formula (Eq. 3) is just the integral of the beta function with beta_3 plugged in, and the paper says so; the value is in the matching, not the running. It also explicitly corrects PDG Eq. (9.7), and it is honest about why that equation was wrong: the old c2 from [10] plus a factor-of-4 slip in the beta_2 terms. That kind of bookkeeping is worth having in print. The checks are the right ones for this genre: UV cancellation, gauge-parameter independence, RG invariance, and c2 agreeing with the independent Larin–van Ritbergen–Vermaseren result. The 442-diagram reduction is not something a reader can verify from the text, but nothing in the paper asks you to take it on faith beyond the standard level for a three-loop diagrammatic calculation. One thing to flag: the reader's summary quotes c2 = 117/72. The paper says 11/72, and Eq. (10) confirms it — C2 = 11/72 - 4/9 = -7/24, exactly as printed. The stress-test caught this correctly; the reader's strongest_claim is simply misstated, and it does not touch the paper. I also don't buy the reader's 'weakest assumption' about the Ward identity in Eqs. (4)-(5). That is the standard Slavnov-Taylor relation for coupling renormalization, linear in the heavy-quark contribution, and it is not where the risk lives. The real residual risk is the unverified three-loop integrals, which is normal and low. Minor: no code or intermediate results, unremarkable for 1997. Who should read it: anyone doing four-loop alpha_s evolution, PDF fitting, or threshold crossing in precision collider work. It deserves a referee, though the report will mostly be a confirmation.

Referee report

Significance

If correct, this is a useful and durable result: it supplies the last ingredient needed for four-loop QCD evolution with consistent three-loop threshold matching, and it does so in closed analytic form rather than as a numerical fit. The paper earns credit for several things that are easy to understate. The coefficients are given analytically in both the MS-bar-mass and pole-mass conventions (Eqs. (6)–(10)), so users of either convention can apply them without redoing the conversion. The Λ-relation is accompanied by explicit substitution rules for the inverse and on-shell cases, which makes the result reproducible rather than merely quotable. The calculation is backed by stated internal checks (ultraviolet cancellation, gauge-parameter independence, renormalization-group invariance), and the two-loop coefficient is cross-checked against an independent published computation. The paper also performs the service of identifying a specific defect in a widely used PDG formula. The Ward identity in Eqs. (4)–(5) is the standard Slavnov-Taylor relation for coupling renormalization and is linear in the heavy-quark contribution, so it is not a weak point of the derivation.

Major objections (2)

[Eq. (11) and the paragraph following it] The O(1/l_h^2) term of Eq. (11) is advertised as a new result and is precisely the piece that the paper says disagrees with Eq. (9.7) of Ref. [1], yet the step from Eqs. (2), (3), and (6) to Eq. (11) is asserted (''we obtain'') without intermediate algebra. Because this term is load-bearing for the correction claim, please supply either the derivation in a short appendix or an explicit check that a reader can reproduce: for instance, that applying the stated inverse substitutions twice returns Eq. (11) identically, or that truncating Eq. (11) at O(1/l_h) reproduces the known two-loop Λ-relation. Either would let the reader verify the new term without redoing the full matching calculation.
[Introduction and the discussion around Eq. (11)] The paper states that settling the conflict between Refs. [10] and [11] on the two-loop matching condition is one of its purposes, and it reports agreement with Ref. [11] and disagreement with Ref. [10], but it never quotes the value of c2 obtained in Ref. [10]. Since the numerical difference is the whole substance of the dispute, please state the competing value explicitly (and, if possible, indicate where the discrepancy arises, e.g. in a particular diagram class or in the treatment of the light-quark vacuum polarization). The same applies to the remark that ''the terms involving β2 should be divided by 4'' in Eq. (9.7) of Ref. [1]: naming the affected terms would make the correction actionable rather than a pointer.

Minor objections (7)

[Eq. (8)] Please state explicitly whether the n_f appearing in the O(A^2) coefficient of Eq. (8) counts the heavy quark itself, i.e. whether it is n_f or n_l = n_f − 1, since the constant term and the n_f term are convention-dependent and a reader converting Eq. (6) to Eq. (9) needs to know which is meant.
[Fig. 3] The text quotes a four-loop versus three-loop difference in α_s^(5)(M_Z) of ''almost 0.001'', but the vertical scale of Fig. 3 spans 0.12 to 0.13, so a difference of that size cannot be read off the figure. A short table of α_s^(5)(M_Z) at μ^(5) = M_b for the one-, two-, three-, and four-loop cases would make the central numerical claim verifiable.
[Eqs. (6)–(10)] The coefficients are given only as stacked fractions, which makes comparison with Refs. [10] and [11] unnecessarily laborious. A small table listing the numerical values of c2, c3, C2, and C3 for n_f = 3, 4, 5, 6 would let readers check the two-loop agreement at a glance and would make the size of the three-loop correction immediately visible.
[Paragraph following Eq. (11)] The on-shell Λ-relation and both inverse relations are specified only through substitution rules (Λ ↔ Λ', β_N ↔ β'_N, c_N → −c_N, and so on). These rules are unambiguous, but an explicit equation for at least the on-shell case would reduce the chance of a user misapplying them.
[Table 1] Table 1 lists β_N^(n_f) only for n_f = 3 to 6. Since the matching formulae are given for general n_l, including the general-n_f expressions (or at least the n_f-dependence in a footnote) would make the table consistent with the rest of the paper.
[Fig. 1 caption] The one-loop curve (''coarsely dotted'') and the exact four-loop solution (''densely dotted'') may be difficult to distinguish in a black-and-white print. Distinct symbols or line styles would improve legibility.
[Paragraph describing the calculation, after Eq. (5)] The checks are named but not quantified. A sentence giving the residual gauge-parameter dependence and the magnitude of the uncancelled ultraviolet poles before renormalization would let the reader judge how stringent the checks actually were.

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The paper

The paper claims

The paper derives the three-loop matching conditions that let the strong coupling constant be run at four loops across quark-flavour thresholds in the MS-bar scheme, and in doing so settles a long-standing disagreement about the two-loop匹配

The reading

Quantum chromodynamics predicts how the strong coupling $a = alpha_s/pi$ changes with energy, but the standard running formula breaks down near a quark mass: heavy quarks do not automatically drop out of MS-bar quantities, so one must switch to an effective theory with fewer flavours and match the two couplings at the threshold. The authors compute that matching to three loops in closed analytic form, then combine it with the recently obtained four-loop $\beta$-function coefficient $beta_3$ to write a four-loop expression for $alpha_s^{(n_f)}(mu)$ and a closed relation between the asymptotic scale parameters $Lambda^{(n_f)}$ for neighbouring flavour numbers. Along the way they resolve a published contradiction: two earlier papers disagreed on the two-loop matching coefficient, and the authors' independent diagrammatic calculation gives $c_2 = 117/72$, agreeing with one and disagreeing with the other. The payoff is bookkeeping: if the coupling is evolved at $N+1$ loops, the thresholds must be matched at $N$ loops for the residual scale dependence to cancel at the stated order, so a four-loop evolution needs exactly this three-loop matching. The paper also shows numerically that the four-loop curve for $alpha_s^{(5)}(mu)$ is nearly flat under variation of the threshold scale, whereas lower orders drift.

What carries the argument

The load-bearing device is a Ward identity that fixes the coupling decoupling constant without computing a coupling-renormalization diagram directly. Writing $g_s' = zeta_g g_s$ for the effective-theory coupling, and $zeta_g^0 = zeta_g Z_g'/Z_g$, the authors use $zeta_g^0 = zeta-tilde_1^0 / (zeta-tilde_3^0 sqrt{zeta_3^0})$, where the three factors are the heavy-quark contributions to the gluon vacuum polarization, the ghost vacuum polarization, and the ghost-gluon vertex correction at vanishing momentum. This reduces the whole problem to 3+1+5 two-loop and 189+25+228 three-loop Feynman diagrams, generated with QGRAF and evaluated with MATAD. The checks are the standard ones: cancellation of ultraviolet poles, gauge-parameter independence, and renormalization-group invariance. The second piece of machinery is the integrated $\beta$ function, Eq. (2), whose expansion constant $C$ fixes the convention for $Lambda$; the conventional MS-bar choice $C = (b_1/beta_0) ln beta_0$ is what makes Eq. (3) and Eq. (11) well defined.

Extended reading notes

Core claim

The central claim is a set of analytic matching coefficients. Writing $a' = a^{(n_l)}(mu^{(n_f)})$ and $a = a^{(n_f)}(mu^{(n_f)})$, the ratio is given to third order in $a$ by Eq. (6), with the heavy-quark logarithm $ell_h = ln[(mu^{(n_f)})^2/mu_h^2]$ measured in the renormalization-group-invariant MS-bar mass $mu_h$, and with $c_2 = 117/72$ and $c_3 = -82043/27648, zeta(3) + 564731/124416 - 2633 n_l/31104$. An equivalent form in terms of the pole mass $M_h$ is given in Eq. (9) with $C_2 = -7/24$ and $C_3$ as in Eq. (10). Feeding these into the integrated $\beta$ function yields the four-loop running formula of Eq. (3) and, for the convenient threshold choice $mu^{(n_f)} = mu_h$, the closed three-loop relation between $Lambda^{(n_l)}$ and $Lambda^{(n_f)}$ in Eq. (11). The authors state that their $c_2$ agrees with Ref. [11] and disagrees with Ref. [10], and that Eq. (11) disagrees with the corresponding formula in the Particle Data Group compilation, partly because that formula used the Ref. [10] value and partly because its $beta_2$ terms should be divided by four.

Load-bearing premise

Everything rests on the Ward identity that expresses the coupling decoupling constant purely through the heavy-quark pieces of the gluon and ghost vacuum polarizations and the ghost-gluon vertex; if that factorization needs extra terms or operators at three loops, the coefficients $c_3$ and $C_3$ and the $Lambda$ relation all shift.

Editorial extensions

If this is right

Any prediction carried to next-to-next-to-next-to-leading order in QCD can now be quoted with a consistent four-loop evolution of $alpha_s$ and three-loop threshold matching, removing a systematic ambiguity at that order.
The two-loop matching coefficient is fixed at $c_2 = 117/72$, so the earlier conflicting value and the PDG formula built on it are superseded.
Crossing the bottom threshold, the relation between $alpha_s^{(4)}(M_tau)$ and $alpha_s^{(5)}(M_Z)$ becomes nearly independent of the arbitrary matching scale $mu^{(5)}$, and the central value of $alpha_s^{(5)}(M_Z)$ shifts downward by almost 0.001 relative to three-loop treatment.
The scale parameters $Lambda^{(n_f)}$ for different flavour numbers are now connected by a closed three-loop formula, so a determination of $Lambda$ in one effective theory can be translated to another without redoing the evolution.
The same machinery extends to the pole-mass version of the relation, giving an on-shell threshold prescription alongside the MS-bar one.

Reading between the lines

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

Because the matching coefficients enter the $Lambda$ relation multiplied by powers of $1/l_h$, the practical impact of the three-loop terms grows for heavy quarks whose mass sits not far above $Lambda$; charm-threshold applications are therefore where the new terms should be tested most sharply.
The residual scale dependence of the four-loop curve in Fig. 3 is a diagnostic of truncation error, not of correctness: a future five-loop beta coefficient would let one check whether the near-flatness persists or was partly accidental cancellation.
The same decoupling-constant Ward identity should apply to any MS-bar quantity built from the gluon and ghost sector, suggesting that the three-loop matching computed here could be reused as a building block for other threshold-crossing observables without recomputation.
Independent confirmation could come from a completely different route, such as a large-$n_f$ or lattice-style determination of the matching coefficient at the same order, since the present result rests on a single diagrammatic pipeline.

Original abstract

We present in analytic form the matching conditions for the strong coupling constant alpha_s^(n_f)(mu) at the flavour thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient beta_3 of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavours n_f.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present in analytic form the matching conditions for the strong coupling constant alpha_s^(n_f)(mu) at the flavour thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient beta_3 of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavours n_f.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present in analytic form the matching conditions for the strong coupling constant alpha_s^(n_f)(mu) at the flavour thresholds to three loops in the modified minimal-subtraction scheme.	contribution 0.65	0.92	0.34	asserted 1.00
s1	Taking into account the recently calculated coefficient beta_3 of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavours n_f.	result 0.69	0.94	0.70	asserted 1.00

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

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Figures

**Figure 1.** µ dependence of α (5) s (µ) calculated from α (5) s (MZ) = 0.118 using Eq. (3) at one (coarsely dotted), two (dashed), three (dot-dashed), and four (solid) loops. The densely dotted line represents the exact solution of Eq. (1) at four loops. MATAD [16], respectively. The cancellation of the ultraviolet singularities, the gaugeparameter independence, and the renormalization-group (RG) invariance serve as strong checks for our calculation. If we measure the matching scale µ (nf ) in units of the RG-invariant MS mass µh = mh(µh), our result for the ratio of a ′ = a (nl) (µ (nf ) ) to…

**Figure 2.** Typical three-loop diagrams pertinent to Π

**Figure 3.** µ (5) dependence of α (5) s (MZ) calculated from α (4) s (Mτ ) = 0.36 and Mb = 4.7 GeV using Eq. (3) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eq. (9) at the respective order. in Ref. [10], which differs from the value listed in Eq. (7). Furthermore, in the same equation, the terms involving β2 should be divided by 4. Equation (11) represents a closed three-loop formula for Λ(nl) in terms of Λ(nf ) and µh. For consistency, it should be used in connection with the four-loop expression (3) for α (nf ) s (µ) with the understandin

Figures reproduced from arXiv:hep-ph/9706430 by the authors.

References

Works cited (20)

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Revisiting the Global Electroweak Fit of the Standard Model and Beyond with Gfitter
hep-ph 2008-11 accept novelty 5.0 of 10

Combining electroweak precision data with direct Higgs searches gives M_H = 116.4 +18.3/-1.3 GeV and alpha_S(M_Z) = 0.1193 +0.0028/-0.0027, and excludes a charged Higgs below 240 GeV in the Type-II 2HDM.
The Determination of alpha_s from Tau Decays Revisited
hep-ph 2008-03 accept novelty 5.0 of 10

Using the newly computed fourth-order QCD coefficient and new BABAR data, the authors obtain alpha_s(m_tau) = 0.344 and, evolved to the Z mass, alpha_s(M_Z) = 0.1212, the most precise determination to date.
SOFTSUSY: a program for calculating supersymmetric spectra
hep-ph 2001-04 accept novelty 5.0 of 10

SOFTSUSY 3.3.3 is a documented, freely available C++ code that solves the MSSM renormalisation group equations iteratively and returns a full superparticle spectrum, with its approximations and failure flags stated ex...
Lattice-motivated QCD coupling and hadronic contribution to muon $g-2$
hep-ph 2020-09 conditional novelty 4.0 of 10

Within a lattice-motivated analytic QCD coupling, reproducing tau spectral sum rules and the muon g-2 hadronic contribution requires a positive D=4 condensate and restricts alpha_s(MZ) to 0.1171 to 0.1180.
Electroweak Precision Observables, New Physics and the Nature of a 126 GeV Higgs Boson
hep-ph 2013-06 conditional novelty 4.0 of 10

A global fit to electroweak precision data with a 126 GeV Higgs constrains the Higgs-to-gauge-boson coupling to within about 5% of the Standard Model value, strongly disfavoring generic composite Higgs models.
Review of lattice results concerning low energy particle physics
hep-lat 2010-11 accept novelty 4.0 of 10

The review sets quality criteria for lattice QCD calculations and uses them to produce recommended averages for light-quark masses, f+(0), fK/fpi, low-energy constants, and B_K.
A Higher Order Perturbative Parton Evolution Toolkit (HOPPET)
hep-ph 2008-04 accept novelty 4.0 of 10

HOPPET delivers NNLO DGLAP evolution of parton distributions in x-space at roughly 10^-4 relative accuracy and about 10 ms per new initial condition, benchmarked against the Mellin-space code QCD-Pegasus.
Review of lattice results concerning low energy particle physics
hep-lat 2013-10 accept novelty 3.0 of 10

FLAG's 2014 review fixes consensus lattice values for light-quark masses, f+(0), fK/fpi, BK, heavy-meson decay constants and alpha_s, and finds the first row of the CKM matrix unitary at the permille level.
The QCD Running Coupling
hep-ph 2016-04 accept novelty 2.0 of 10

A comprehensive review of the QCD running coupling across perturbative and nonperturbative regimes, closing with a proposal that effective charges defined directly from observables give the most unified definition of alpha_s.
Renormalisation
hep-ph 2025-11 unverdicted novelty 1.0 of 10

An introductory review of renormalisation procedures, the renormalisation group, and scale-setting optimisation in gauge theories such as QCD.
The Theory of Deeply Inelastic Scattering
hep-ph 2012-08 accept novelty 1.0 of 10

A comprehensive review of perturbative QCD calculations for unpolarized and polarized deeply inelastic scattering, covering coefficient functions, anomalous dimensions, heavy-flavor and target-mass corrections, and th...

How later papers used it

62 later papers cite it

Evidence: substantial. 61 of the citing papers share no author with it, and 35 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 76 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

12 later papers by other authors are listed below: 10 used its method or tool and 2 used its data or values.

Uses its method or tool 10 of 27 papers with this label

Obtained the 4-flavor coupling at the charm mass using this paper's 3-loop matching equation.

Determination of $\alpha_s$ from static QCD potential: OPE with renormalon subtraction and lattice QCD (2018) · 1.00 · paraphrased: the licence does not allow quoting
RGE-evolved a value to low energies using this paper's 3-loop threshold relations at the quark thresholds.

Lattice-motivated holomorphic nearly perturbative QCD (2016) · 0.99 · paraphrased: the licence does not allow quoting
Incorporated quark thresholds at the three-loop level using this paper's matching conditions.

Simple analytic QCD model with perturbative QCD behavior at high momenta (2010) · 0.99 · paraphrased: the licence does not allow quoting
Incorporated the quark threshold matching conditions at the three-loop level according to this paper.

Reconciling the analytic QCD with the ITEP operator product expansion philosophy (2010) · 0.98 · paraphrased: the licence does not allow quoting
RGE-evolved the coupling up to M_Z using this paper's three-loop matching conditions at the quark thresholds.

Operator Product Expansion with analytic QCD in tau decay physics (2012) · 0.98 · paraphrased: the licence does not allow quoting
Used this paper's matching between n_f and n_f+1 flavor couplings.

Nonsinglet parton distribution functions from the precise next-to-next-to-next-to leading order QCD fit (2009) · 0.86 · paraphrased: the licence does not allow quoting
Used this paper's matching between n_f and n_f+1 flavor couplings.

NNLO QCD contributions to the flavor non-singlet sector of ${\bf {F_2(x,Q^2)}}$ (2008) · 0.86 · paraphrased: the licence does not allow quoting
Obtained its N_f=3 values by following this paper's four-loop running procedure.

Toolkit for staggered $\Delta S=2$ matrix elements (2014) · 0.99 · paraphrased: the licence does not allow quoting
Estimated the light-quark Yukawa couplings using this paper's QCD four-loop RGEs and three-loop decoupling effects from heavy quarks.

Nucleon Decay as a Probe of Flavor Symmetry: The Case of Fake Unification (2024) · 0.99 · paraphrased: the licence does not allow quoting
Took the quark thresholds according to this paper's 3-loop matching conditions.

Perturbative QCD in acceptable schemes with holomorphic coupling (2014) · 0.96 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 7 papers with this label

Used this paper's expressions for k_j.

The bottom quark mass from the $\Upsilon(1S)$ system at NNNLO (2014) · 0.90 · paraphrased: the licence does not allow quoting
Used this paper's value of Lambda^(n_ℓ=3)=0.332 GeV.

Determination of $|V_{cb}|$ using N$^3$LO perturbative corrections to $\Gamma(B \to X_c \ell \nu)$ and 1S masses (2022) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

57 arXiv papers cite this in 71 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 · 31 sentences without a confident label

Builds on it 1 paper

To obtain this expression from Ref. [ref] , results in Refs. [ref] are needed to restore the $n_h$ dependence.

Relations between Heavy-light Meson and Quark Masses · Synthesis · 0.83

Uses its method or tool 25 papers

The couplings are then run to high energy using the three-loop renormalization group equations with four-loop running included for $α_s$ [ref] .

Scale Invariant Instantons and the Complete Lifetime of the Standard Model · Numerical Results · 0.97
For this purpose we used the matching formula near mass thresholds for the $MS$ scheme at three-loop order in Refs. [ref] .

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Our strategy · 0.99
The QCDNUM [ref] program is used to evolve these distributions to larger values of $Q^2$ in the -scheme at NNLO in perturbative QCD [ref] .

New constraints on the up-quark valence distribution in the proton · Analysis Procedure · 1.00

Uses its data or values 6 papers

Due to experimental uncertainty of the value $a(M_Z^2,) ≡a(M_Z^2)$, and because several of the results of our analysis depend crucially on the value of $a(M_Z^2)$, we will consider in the following the three values $a(M_Z^2) = 0.1185/π$ [ref] , $0.1181/π$ [ref] and $0.1189/π$, which has $N_f=5$ in all cases.[We recall that the world average given by Particle Data Group in 2014 is $a(M_Z^2)=(0.1185 ±0.0006)/π$ [ref] , and in 2016 is $a(M_Z^2)=(0.1181 ±0.0011)/π$ [ref] . ] We RGE-evolve this value down into the $N_f=3$ regime of positive $Q^2$, with four-loop $$ beta function [ref] , and take into account the corresponding three-loop quark threshold relations [ref] at $Q^2_thr. = (κm_q)^2$ ($q=b, c$) with $κ=2$ and with the $$ quark masses $m_q ≡m_q(m_q^2)$ set equal to $m_b=4.20$ GeV and $m_c=1.27$ GeV.

Nearly perturbative lattice-motivated QCD coupling with zero IR limit · The underlying pQCD coupling · 0.98
where the expressions for $k_j$ ($j=1,2,3$) are given in Ref. [ref] ($k_1 = -ℓ_h/6$, etc.), with the logarithm there being $ℓ_h = ln(ν_f^2/_c^2)$ and $N_ℓ=3$ (see also Appendix D of Ref. [ref] ).

The bottom quark mass from the $\Upsilon(1S)$ system at NNNLO · Relations between different couplings · 0.90
First, at the mass of the $Z$ boson, $m_Z = 91.2$ GeV with $α_s^(m_Z^2) = 0.1189$ [ref] , from Eq. ([ref]) we obtain [equation] for $N_f = 5$, while at the mass of the $τ$ lepton, $m_τ= 1.777$ GeV, [equation] where, to be specific, we have used $(m^2_τ) = 0.322$, the value obtained by running $(m_Z^2) = 0.1189$ down to the $N_f=3$ scale $m^2_τ$ using the standard 4-loop running [ref] .

The strong coupling and its running to four loops in a minimal MOM scheme · The minimal MOM scheme · 0.98

Compares with it 1 paper

In Ref. [ref] , the use of four-loop $$ $β$-function [ref] with the corresponding three-loop quark threshold matching [ref] was made (with $κ=2$).[ For this reason, for example, for $α_s(M_Z^2;)=0.1181$ the value of $Λ_L=0.1136$ GeV was obtained in [ref] , while here $Λ_L=0.1130$ GeV.]

Lattice-motivated QCD coupling and hadronic contribution to muon $g-2$ · Construction of the coupling $(Q^2)$ · 0.98

Cites it as background 4 papers

Finally, we remind that to derive constraints on the effective coefficients, the QCD running below and above the electroweak scale must be taken into account, which impacts scalar and tensor operators and is known up to four-loops [ref] .

Neutrinoless Double-Beta Decays from Operator Mixing · Low-energy EFT for $0νββ$ · 0.81
However, one should keep in mind that proper inclusion of three-loop non-QCD decoupling corrections is required to evaluate $α_s$ consistently at the electroweak scale in the full SM (for pure QCD results up to the four-loop level see Refs. [ref] and for the leading two-loop electroweak corrections see Ref. [ref] ).

Four-loop strong coupling beta-function in the Standard Model · Introduction · 0.86
(See e.g. [ref] for recent progress on gluon amplitudes.) In this paper, we will study the two-loop Higgs amplitudes in the Higgs effective theory [ref] with full QCD corrections.

Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD · Introductioin · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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