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Computation of the strong coupling in QCD with two dynamical flavours

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Michele Della Morte, Roberto Frezzotti, Jochen Heitger, Juri Rolf, Rainer Sommer, Ulli Wolff (ALPHA Collaboration)

High Energy Physics - Lattice · Submitted 19 Nov 2004 · 63 references

Pith's verdict: The two-flavour running and -ln(ΛLmax)=1.09(7) are solid and worth citing; the MeV number is an acknowledged placeholder, and the only real soft spot is that the 0.07 error doesn't visibly absorb the spread among the three extrapolation procedures.

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arXiv IDhep-lat/0411025 (version 1)
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CitationJochen Heitger, Juri Rolf, Michele Della Morte, Rainer Sommer, Roberto Frezzotti, Ulli Wolff (ALPHA Collaboration). "Computation of the strong coupling in QCD with two dynamical flavours." arXiv:hep-lat/0411025 (2004). https://pith.science/paper/RKORYDBQ
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The paper

The paper claims

The running of the strong coupling is computed non-perturbatively in two-flavour QCD, and the result is matched to three-loop perturbation theory to extract the QCD Lambda parameter.

The reading

The paper asks a question that lattice QCD had previously answered only in the quenched approximation: how does the strong coupling $alpha_s$ actually run, as a matter of non-perturbative fact, across the two orders of magnitude in energy that separate hadronic scales from the regime where perturbation theory is trusted? The authors simulate QCD with two flavours of dynamical $O(a)$ improved Wilson fermions in a finite box with Schrödinger functional boundary conditions, tune the bare parameters so that the renormalized coupling takes a prescribed value $u$ at box size $L$, then double the box at the same lattice spacing and measure the new coupling. Repeating this at several lattice resolutions and extrapolating to zero lattice spacing gives the continuum step scaling function $sigma(u)$, which is an integrated $beta$-function. Iterating $sigma$ upward from a hadronic-scale coupling $u_{max}=5.5$ through seven or eight factors of two, and there matching onto the three-loop perturbative $beta$-function, yields $-ln(Lambda L_{max}) = 1.09(7)$. Combined with large-volume measurements of $r_0/a$ this becomes $Lambda^{(2)}_{MS} r_0 = 0.62(4)(4)$, or 245(16)(16) MeV if $r_0 = 0.5$ fm.

What carries the argument

The load-bearing object is the step scaling function $sigma(u) = lim_{a/L to 0} Sigma(u,a/L)$, where $Sigma(u,a/L) = bar g^2(2L)$ evaluated at $bar g^2(L) = u$ and zero quark mass. It is the discrete, finite-volume version of the renormalization group flow: one lattice simulation at coupling $u$ and resolution $L/a$ gives the coupling at twice the box size, and iterating the measured $sigma$ generates the running without ever needing a hierarchy of scales on a single lattice. Two pieces of machinery make it usable. First, the perturbative cutoff effects $delta_1(a/L) u + delta_2(a/L) u^2$ are computed and divided out, $Sigma^{(2)} = Sigma / (1 + delta_1 u + delta_2 u^2)$, which removes the leading lattice artifacts analytically before extrapolation; the paper notes that with the two-loop boundary improvement coefficient $c_t$ the residual $delta_2$ is tiny, of order $10^{-4}$, whereas with the one-loop $c_t$ it is of order $a/L$. Second, the exact renormalization group invariant $Lambda$ of Eq. (2.9) converts a coupling at a single high scale, together with the three-loop $beta$-function coefficient $b_2$ in the Schrödinger functional scheme, into $Lambda L_{max}$; the one-loop relation $Lambda_{MS} = 2.382035(3) Lambda_{SF}$ then carries the answer to the MS-bar scheme.

Extended reading notes

Core claim

The central claim is that the energy dependence of the strong coupling in two-flavour QCD is now known non-perturbatively over a scale separation of a factor 100 to 250, with a controlled continuum extrapolation, and that this non-perturbative running agrees with three-loop perturbation theory for $alpha$ below about 0.2 but departs from it rapidly above $alpha$ around 0.2 to 0.3. The chain runs as follows: the lattice step scaling function $Sigma(u,a/L)$ is measured for eight values of the renormalized coupling $u$ at lattice resolutions $L/a = 4,5,6,8$, the known perturbative cutoff effects are divided out, and the resulting $Sigma^{(2)}$ is extrapolated to $a/L to 0$ under three different fit procedures whose spread sets the systematic error. The continuum $sigma(u)$ is then interpolated by a polynomial with its first coefficients fixed by two-loop perturbation theory, and the recursion $sigma(bar g^2(L/2)) = bar g^2(L)$ is solved numerically starting from $u_{max} = 5.5$. The authors report $-ln(Lambda L_{max}) = 1.09(7)$ at that starting coupling, a seven percent determination, and note that had they used perturbation theory alone to evolve the coupling over the same range the $Lambda$ parameter would have been overestimated by up to 14 percent. Converting to physical units through $r_0$ gives $Lambda^{(2)}_{MS} r_0 = 0.62(4)(4)$ and 245(16)(16) MeV, values that sit close to the quenched result 0.60(5) and to four- and five-flavour extractions from high-energy data.

Load-bearing premise

Turning the dimensionless result into MeV assumes the hadronic radius $r_0$ equals 0.5 fm, a phenomenological estimate the paper takes from the literature rather than computes, and it also assumes that $r_0/a$ extrapolates linearly in the bare quark mass down to zero, with a systematic error set to half the distance between the last data point and the extrapolated value.

Editorial extensions

If this is right

  • The $beta$-function of QCD is now known non-perturbatively for two flavours from $alpha$ around 0.3 down to $alpha$ around 0.1, so the accuracy of three-loop perturbation theory in evolving the coupling can be quantified rather than assumed; the paper puts the induced error on $alpha_{MS}$ near the physical value at the Z mass at about 3 percent.
  • The bare parameters tabulated in the appendices give the lattice action's tuning points at each coupling, which makes possible the non-perturbative renormalization of quark masses and composite operators in the same two-flavour setup.
  • The comparison of $Lambda r_0$ across $N_f = 0, 2, 4, 5$ becomes a test of QCD itself: the two-flavour lattice number 0.62(4)(4) is compatible with the quenched 0.60(5) and with high-energy extractions, so the flavour dependence of $Lambda$ is small over that range once $r_0$ is held fixed.
  • The measured departure of the non-perturbative $beta$-function from its three-loop truncation at $alpha$ above roughly 0.2 sets a ceiling on where perturbative evolution of the coupling may be trusted, independent of any particular observable.
  • The machinery fixes a template: any gauge theory with a manageable finite-volume coupling can have its $Lambda$ parameter computed this way, with the continuum extrapolation replaced by a controlled fit to the known cutoff structure.

Reading between the lines

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

  • The 14 percent overestimate that would have resulted from purely perturbative evolution is the paper's most portable number: it says that for a coupling defined with a good perturbative expansion, the error of truncating at three loops over two decades of energy is a few percent in $alpha$ but a much larger fraction in $Lambda$, which is the quantity most high-energy fits actually quote.
  • The stated next step, a non-perturbative determination of $L_{max} F_pi$ or $r_0 F_pi$, would replace the phenomenological $r_0 = 0.5$ fm with a computed number; if that computation lands near 0.5 fm the 245 MeV stands, and if it lands near 0.45 or 0.55 fm the central value moves by roughly 10 percent, which is larger than the quoted error bar.
  • The metastability analysis in Sec. 3.3 is a caution for the whole programme: at $bar g^2$ around 3.3 the authors find secondary minima of the action populated at the 0.3 percent level and handle them by modified sampling plus exact reweighting. At stronger couplings, where the step scaling function is steepest, such effects grow, so any extension of the recursion to larger $u$ will need the same m
  • Because the continuum extrapolation is consistent with a constant in $a/L$ at the achieved precision, the result is limited by statistics and by the $r_0$ conversion rather than by lattice artifacts; a moderate increase in configuration count at $L/a = 8$ and a non-perturbative $r_0$ in physical units would tighten the answer more than pushing to finer lattices would.
Original abstract

We present a non-perturbative computation of the running of the coupling alpha_s in QCD with two flavours of dynamical fermions in the Schroedinger functional scheme. We improve our previous results by a reliable continuum extrapolation. The Lambda-parameter characterizing the high-energy running is related to the value of the coupling at low energy in the continuum limit. An estimate of Lambda*r_0 is given using large-volume data with lattice spacings a from 0.07 fm to 0.1 fm. It translates into Lambda_{MSbar}^{(2)}=245(16)(16) MeV [assuming r_0=0.5 fm]. The last step still has to be improved to reduce the uncertainty.

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, 4,861 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.97 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We present a non-perturbative computation of the running of the coupling alpha_s in QCD with two flavours of dynamical fermions in the Schroedinger functional scheme. We improve our previous results by a reliable continuum extrapolation. The Lambda-parameter characterizing the high-energy running is related to the value of the coupling at low energy in the continuum limit. An estimate of Lambda*r_0 is given using large-volume data with lattice spacings a from 0.07 fm to 0.1 fm. It translates into Lambda_{MSbar}^{(2)}=245(16)(16) MeV [assuming r_0=0.5 fm]. The last step still has to be improved to reduce the uncertainty.

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 present a non-perturbative computation of the running of the coupling alpha_s in QCD with two flavours of dynamical fermions in the Schroedinger functional scheme. contribution 1.00 0.96 0.13 asserted 1.00
s1 We improve our previous results by a reliable continuum extrapolation. contribution 0.95 0.92 0.05 asserted 1.00
s2 The Lambda-parameter characterizing the high-energy running is related to the value of the coupling at low energy in the continuum limit. background 0.81 0.16 0.08 asserted 1.00
s3 An estimate of Lambda*r_0 is given using large-volume data with lattice spacings a from 0.07 fm to 0.1 fm. method 0.80 0.93 0.11 asserted 0.99
s4 It translates into Lambda_{MSbar}^{(2)}=245(16)(16) MeV [assuming r_0=0.5 fm]. result 1.00 0.90 0.74 asserted 0.97
s5 The last step still has to be improved to reduce the uncertainty. limitation 0.50 0.25 0.04 asserted 0.89

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
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. Monte Carlo histories of S cool g and ∂Scool g /∂η from two independent replica (solid and dotted lines) in a simulation of a 164 lattice at ¯g 2 ≈ 3.3. to [16], adding to the HMC effective action a term γ ∂Sg ∂η
Figure 2. Continuum extrapolation of the step scaling function. 18
Figure 3. Step scaling function σ(u). The dashed lines show the perturbative results from the integration of the 2-loop and 3-loop β-function, respectively. expansion has broken down. In fact, already in section 3.3 we have discussed the indications that fluctuations around a secondary minimum of the action are important at this value of the coupling. Such a mechanism may represent one possible source of non-perturbative effects.6 We adopt the parametrized form of the step scaling function to compute the Λ-parameter. To this end we start a recursion with a maximal coupling umax = g¯ 2 (Lmax)
Figure 4. Running of the strong coupling in the Schr¨odinger functional scheme
Figure 5. Non-perturbative β-function in the Schr¨odinger functional scheme. 21
Figure 6. v¯ as a function of the strong coupling α. The dashed lines show the 2-loop perturbation theory while the solid lines are non-perturbative fits for α < 0.3 including an additional α 2 -term. these large-volume Nf = 2 simulations of QCD have been extended to smaller values of the lattice spacing, namely 5.2 ≤ β ≤ 5.4 [57]. In order to obtain Λr0 and to study its dependence on the lattice spacing, we here use results for r0/a of [57] at β = 5.2, 5.29, 5.4 and compare also to the numbers resulting from r0/a of [44]. First, we obtain the renormalized coupling on lattices with extent L/a…
Figure 7. Dependence of r0 on the bare quark mass mq = (1/κ − 1/κc)/2. Both quantities are rescaled (made dimensionless) by the extrapolated value of r0, denoted by r0(0). The uncertainty of this rescaling is not propagated into the errors. For κc, we take the values of [57], listed in table 7. r0/a do agree with those quoted in [56,44,57], where somewhat different ans¨atze – and in [44] also different data – have been used. Third, combining eqs. (4.4) and (2.11) with r0/a and Lmax/a of table 8, one arrives at the columns ΛMS r0 in that table. Here we have added errors in quadrature, except…

Figures reproduced from arXiv:hep-lat/0411025 by the authors.

References
How the literature uses it

52 arXiv papers cite this in 95 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 · 37 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 14 papers
  • Following the calculation procedure and the analysis in Ref. [ref] , we have calculated both $g̅^2_SF(L,a/L)$ and $g̅^2_SF(2L,a/L)$ at the same $β$, in order to obtain the lattice SSF as [equation] In order to make the continuum extrapolation as $lim_a/L→0Σ(u,a/L)=σ(u)$, we repeat this procedure by changing $L/a$ and $β$ while keeping $u=g̅^2_SF(L,a/L)$ fixed.

    Universality of the Nf=2 Running Coupling · Simulation details · 0.99

  • In this paper we adopt the same set up (scheme) as the Alpha collaboration [ref] for the boundary link eqn:boundary [equation] The parameter $η$ is used to define the renormalized coupling constant from the derivative of the effective action and is set to zero in the action after taking derivative with respect to it.

    Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Schrödinger functional formalism and action · 0.98

  • Small mismatches of the simulation results w.r.t. the target values, i.e. $(L_1)=4.484$ and $L_1(L_1)=0$, may be corrected by the non-perturbative $β$–function and the mass derivative of the coupling [ref] .

    Towards a non-perturbative matching of HQET and QCD with dynamical light quarks · Parameters for HQET simulations in $L_1^3×T$ with $T=L_1,L_1/2$ · 0.85

Uses its data or values 10 papers
Compares with it 12 papers
Cites it as background 11 papers

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