The running coupling from the QCD Schr\"odinger functional -- a one-loop analysis

Rainer Sommer; Stefan Sint

REVIEW

The running coupling from the QCD Schr\"odinger functional -- a one-loop analysis

Stefan Sint, Rainer Sommer

High Energy Physics - Lattice · Submitted 9 Aug 1995

Pith's verdict: The paper that puts quarks into the Schrödinger functional coupling and hands you the one-loop matching to MSbar plus the boundary improvement coefficient — the numbers hold up.

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arXiv IDhep-lat/9508012 (version 1)

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CitationRainer Sommer, Stefan Sint. "The running coupling from the QCD Schr\"odinger functional -- a one-loop analysis." arXiv:hep-lat/9508012 (1995). https://pith.science/paper/ZZXTOQUL

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Editor's note

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This is the perturbative scaffolding for the whole ALPHA program, and it is done properly. The genuinely new content is the fermionic matching coefficient c_{1,1}(z) between the SF coupling and MSbar (eqs. 3.15–3.18, table 4), the θ = π/5 spatial boundary condition with its Dirac-gap justification, and the one-loop boundary coefficient c_t^{(1,1)} = 0.0191410(1) (eq. 4.5). The pure-gauge version was already known from Lüscher–Weisz–Wolff; this extends it to dynamical quarks, which is the step that made the later non-perturbative α_s determination possible. The θ = π/5 choice is a practical, well-motivated trick: table 1 shows the lowest eigenvalue of the continuum operator roughly doubling, and the lattice numbers in table 2 track it sensibly for both actions. That is the kind of detail that saves a simulation real time, and it is argued rather than asserted (abstract:s1). The computation is cross-checked by two independent routes — a variational plane-wave basis and a hypergeometric/recursion solution — and Appendix A gives the first-order D₅ recursion in enough detail to reimplement. The s_0 coefficient comes out at -0.00844344(2), matching the exact -1/(12π²) to about a part in 10⁶, and s_1 vanishes to 1e-4 for the tree-level improved action. Those are real structural checks, not decoration. On the soft spots: the 1/l extrapolation behind r_0(z) and r_1(z) is the right thing to worry about, but it is narrower than it looks. The s_0 agreement validates the leading log, and the 1/l² term is cancelled by numerical differentiation before the fit, with l up to 64 in REAL*16. What the quoted errors do not do is independently bound the truncation systematic — they are fit errors — so an unanticipated non-integer power or ln²l would slip through. Symanzik power counting says that will not happen, so I would call this a controlled assumption rather than an open hole. The §4.1–4.2 conjecture that no extra fermionic surface operators are needed is a genuine logical gap, but the authors flag it, check it against v̄, and it does not touch the matching coefficient. The heavy-quark omission bound (eq. 3.26) is one-loop only, and they say so. No circularity anywhere: the Chebyshev and large-z fits are representations of their own numerically evaluated function, not fits to data. This is for anyone doing lattice QCD with fermions in finite volume, and for anyone who wants to see how a matching calculation should be documented. Send it to referees; the questions worth pressing are the truncation systematic and the surface-operator conjecture, and both are answerable.

Referee report

Significance

The paper supplies exactly the perturbative input that the finite-volume non-perturbative programme needs: a one-loop conversion from a measurable Schrödinger-functional coupling to MSbar, including the quark-mass dependence that a realistic simulation with dynamical flavours will encounter. Several features deserve explicit credit. The quark determinant is evaluated by two independent routes, a second-order recursion and the first-order D₅ recursion of Appendix A, and the results agree. The leading logarithm s_0 is not merely assumed but fitted, giving −0.00844344(2) against the exact −1/(12π²), and is reported independent of z, θ and c_sw; for the tree-level improved action s_1 = 0 is recovered to 1e-4. The internal consistency of eqs. (4.3) and (4.5) is visible by inspection: −2(0.0191410) + 0.038282 = 0. The Λ_SW/Λ_W ratio of eq. (3.25) is checked in an independent SU(2) computation. The observation in §4.2 that the z/l artefact is removable by a redefinition of the bare coupling, and therefore is not a genuine lattice artefact, is a useful clarification for practitioners.

Major objections (2)

[§3.1, eqs. (3.9)–(3.11) and (3.15)] The headline numbers c_{1,1}(0) = 0.039863(2) and 0.022504(2) in eq. (3.15), and hence c_t^{(1,1)} in eq. (4.5), descend from r_0(0) and r_1(0), which are extracted from the asymptotic series (3.9) with s_0 held fixed at its exact value and the 1/l² term cancelled by numerical differentiation. The quoted two-digit precision therefore appears to be a fit error and does not by itself bound the truncation systematic. Because these are the paper's central deliverables, the reader should be shown the size of that systematic: report the spread of r_0(0) when the number of retained terms in (3.9) is varied, and when s_0 is left free rather than fixed, and state whether the resulting variation is below, at, or above the quoted (2). A short table or a sentence in §3.1 would settle the question.
[§3.3, eqs. (3.16)–(3.18) and Fig. 1] The mass dependence c_{1,1}(z) is represented by two different approximations: a Chebyshev fit on z ∈ [0,10] (eq. 3.16) and a large-z form with a logarithm plus three inverse powers fitted on z ∈ [6,25] (eq. 3.17). The two representations overlap only on [6,10], and Fig. 1 as well as the threshold discussion of §3.6 depend on which is used where. Please state the maximum discrepancy between the two representations on the overlap region, and specify which one is plotted in Fig. 1 and which one enters eq. (3.26). If the discrepancy exceeds the quoted 5×10⁻⁴–5×10⁻⁵ residuals, the interpolation, not the fit, is the limiting error.

Minor objections (8)

[§1 and §2.3, ref. [10]] The statement that the Schrödinger functional in QCD is renormalized by the usual QCD renormalizations plus a multiplicative renormalization of the quark boundary fields, and that this has been confirmed at one loop, rests on ref. [10], which is listed as 'to appear in Nucl. Phys. B'. The preprint number is given, so this is acceptable, but the editor may wish to confirm the final publication details before the paper goes to press.
[Eq. (3.15) and Table 3] Please state explicitly the sign convention and which entries of Table 3 enter eq. (3.15), so that the two quoted values of c_{1,1}(0) can be reproduced from the table and eq. (3.14) by inspection. These are the paper's most-quoted numbers and should be checkable without reconstructing the calculation.
[§2.7 and Tables 1–2] The choice θ = π/5 is justified by the statement that a maximal gap was observed around this value, but only θ = 0 and θ = π/5 appear in the tables. Please state the range of θ that was scanned, or add a short plot of the lowest eigenvalue versus θ, so that the word 'maximal' is supported.
[Eq. (3.16)] The Chebyshev argument is written as z/5 − 1, which maps z ∈ [0,10] onto [−1,1] but reads as though the shift were applied before the division. Writing it as (z − 5)/5 would remove the ambiguity.
[Eq. (3.9)] Since the whole extraction rests on the assumed form of the series, it would help to say explicitly that ln²l/l² and non-integer powers of 1/l are excluded by Symanzik power counting, rather than leaving this to be inferred from the citation of ref. [22].
[Table 2 caption] The caption says the eigenvalues are 'ordered according to their continuum limits', but the rows are not labelled by the quantum numbers that would let a reader match a row at L/a = 6 to the corresponding row at L/a = 24. Adding the colour component n_c and the node number to the table, or to the caption, would make the comparison possible.
[§4.2] The authors conjecture that the dimension-four boundary operators are already present with the correct tree-level weight, and note that two or more fermionic surface operators could in principle cancel in the observables studied. It would strengthen the paper to say what measurement would falsify the conjecture, for example a different observable or a different background field.
[§4.3 and Table 5] The statement that ε_{1,1} for the unimproved action 'practically takes a constant for all lattice sizes accessible to simulations' is drawn from L/a = 4 to 16, over which the entry changes from −0.00273 to −0.00244. Quantifying this as a fit to c_0 + c_1 a/L, and quoting c_1, would make the claim precise.

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The paper

The paper claims

A finite-volume definition of the running QCD coupling with quarks is matched to MSbar at one loop, and the boundary improvement coefficient needed for controlled continuum extrapolations is computed.

The reading

The paper's goal is to make the strong coupling computable from first principles on a lattice when quarks are present, not just in the pure-gauge theory. The authors define a running coupling in a finite box by measuring how the QCD effective action responds to a change in a color-electric boundary field; the box size $L$ sets the renormalization scale. They choose quark boundary conditions that are periodic up to a phase $theta$, finding that $theta=pi/5$ gives a larger lowest Dirac eigenvalue than $theta=0$. At one loop they compute the conversion factor to the $bar{mathrm{MS}}$ scheme, the quark-mass dependence of the $beta$-function, and the $O(a)$ boundary improvement coefficient. If this is right, Monte Carlo measurements of the finite-volume coupling can be converted to $alpha_{bar{mathrm{MS}}}$ with a controlled perturbative error and the main lattice artefacts can be removed or shown to be small; the authors note that the complete $O(a)$-improved action for the Schroedinger functional is not yet known beyond this one-loop, single-observable analysis.

What carries the argument

The load-bearing object is the Schroedinger functional effective action $Gamma[B]$ on a space-time cylinder whose spatial boundary values are fixed color-electric gauge fields. The running coupling is defined by differentiating $Gamma$ with respect to a parameter $eta$ that scales the boundary field, normalized by the classical action so that the coupling equals the bare coupling at tree level. Quark fields are periodic up to a phase $theta$; the choice $theta=pi/5$ is made because the massless Dirac operator in the classical background then has a larger lowest eigenvalue, which improves the condition number of the fermion matrix. The quark determinant is evaluated by a first-order recursion in each fixed spatial-momentum and color sector, and the continuum matching is extracted from the asymptotic $1/l$ expansion of the lattice coefficient. The integrated $beta$-function is packaged in the step scaling function $Sigma(s;u,z)$, which is the quantity a non-perturbative simulation would measure.

Extended reading notes

Core claim

The central claim is that the Schroedinger functional gives a non-perturbative definition of a running coupling in QCD with quarks, and that its one-loop relation to the $bar{mathrm{MS}}$ scheme is under control. For massless quarks the quark contribution to the matching coefficient is $c_{1,1}(0)=0.039863(2)$ for $theta=pi/5$ and $0.022504(2)$ for $theta=0$, both small and positive. The quark-mass dependence is parametrized by Chebyshev fits for $z=mL$ below about 10 and by a $ln z$ plus power-correction form for large $z$; the resulting threshold behaviour in the one-loop $beta$-function is broad rather than sharp. The paper also computes the boundary improvement coefficient $c_t^{(1,1)}=0.0191410(1)$ needed to cancel the $O(a)$ artefacts from the time boundaries. For the step scaling function, one massless flavour induces only sub-percent artefacts with the standard action, and these fall roughly like $(a/L)^2$ once the clover term and the boundary improvement are switched on.

Load-bearing premise

The one-loop continuum matching coefficient is extracted from a lattice series assumed to have the asymptotic form $p_{1,1}$ approaching $r_0+s_0ln l+(r_1+s_1ln l)/l$, with $s_0$ fixed to its exact value and only the first $1/l$ corrections retained; if that truncation or the numerical $1/l$ extrapolation is inadequate, the quoted $c_{1,1}$ values shift.

Editorial extensions

If this is right

Monte Carlo measurements of the finite-volume Schroedinger-functional coupling can be converted to $alpha_{bar{mathrm{MS}}}$ with a one-loop conversion factor.
Heavy quarks with $z=mL$ above about 2 can be omitted from the one-loop $beta$-function evolution with an error of about $0.003$ in $g^{-2}$, below the precision of the pure-gauge step-scaling result.
The boundary coefficient $c_t^{(1,1)}$ can be tuned to remove the linear-in-$a$ artefacts from the step scaling function at one loop.
With the clover term, lattice artefacts in the step scaling function fall roughly like $(a/L)^2$; without it they can look almost constant over the lattice sizes accessible to simulations.
The construction extends recursive finite-size scaling to QCD with dynamical quarks, which is the step needed for a first-principles determination of the strong coupling.

Reading between the lines

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

The $theta=pi/5$ choice is justified by the free Dirac spectrum; a direct Monte Carlo measurement of the low-lying spectrum at realistic coupling would test whether the gain survives gauge fluctuations.
Because the matching coefficients are small, the Schroedinger-functional coupling may be numerically close to $bar{mathrm{MS}}$ at accessible scales; whether that closeness persists at two loops is an open calculation.
The broad threshold behaviour and the $1/z$ tail suggest that omitting a heavy flavour is safe at the quoted precision, but a precision extraction near a flavour threshold may need the full mass dependence.
The improvement coefficient is computed at one loop; non-perturbative tuning at stronger coupling may be needed before continuum extrapolations of dynamical simulations are fully controlled.

Original abstract

Starting from the Schr\"odinger functional, we give a non-perturbative definition of the running coupling constant in QCD. The spatial boundary conditions for the quark fields are chosen such that the massless Dirac operator in the classical background field has a large smallest eigenvalue. At one-loop order of perturbation theory, we determine the matching coefficient to the $\MSbar$-scheme and discuss the quark mass effects in the $\beta$-function. To this order, we also compute the Symanzik improvement coefficient necessary to remove the $\Oa$ lattice artefacts originating from the boundaries. For reasonable lattice resolutions and the standard Wilson action, lattice artefacts are found to be only weakly dependent on the lattice spacing $a$, while they vanish quickly with the improved action of Sheikholeslami and Wohlert.

The compiled record

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

Starting from the Schr\"odinger functional, we give a non-perturbative definition of the running coupling constant in QCD. The spatial boundary conditions for the quark fields are chosen such that the massless Dirac operator in the classical background field has a large smallest eigenvalue. At one-loop order of perturbation theory, we determine the matching coefficient to the $\MSbar$-scheme and discuss the quark mass effects in the $\beta$-function. To this order, we also compute the Symanzik improvement coefficient necessary to remove the $\Oa$ lattice artefacts originating from the boundaries. For reasonable lattice resolutions and the standard Wilson action, lattice artefacts are found to be only weakly dependent on the lattice spacing $a$, while they vanish quickly with the improved action of Sheikholeslami and Wohlert.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Starting from the Schr\"odinger functional, we give a non-perturbative definition of the running coupling constant in QCD.	contribution 0.99	0.93	0.34	asserted 1.00
s1	The spatial boundary conditions for the quark fields are chosen such that the massless Dirac operator in the classical background field has a large smallest eigenvalue.	method 0.98	0.84	0.03	asserted 0.99
s2	At one-loop order of perturbation theory, we determine the matching coefficient to the $\MSbar$-scheme and discuss the quark mass effects in the $\beta$-function.	contribution 0.64	0.96	0.15	asserted 1.00
s3	To this order, we also compute the Symanzik improvement coefficient necessary to remove the $\Oa$ lattice artefacts originating from the boundaries.	contribution 0.69	0.95	0.09	asserted 1.00
s4	For reasonable lattice resolutions and the standard Wilson action, lattice artefacts are found to be only weakly dependent on the lattice spacing $a$, while they vanish quickly with the improved action of Sheikholeslami and Wohlert.	result 0.99	0.93	0.20	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from proof or derivation 0.92.

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relations to other papers

levels run

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References

Forward citations

Cited by 16 Pith papers

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Lattice QCD with nonperturbative step scaling from 0.2 to 70 GeV gives Lambda_MS-bar^(3) = 341(12) MeV and alpha_MS-bar^(5)(m_Z) = 0.11852(84).
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Lattice simulations of SU(2) gauge theory with two adjoint fermions locate an infrared fixed point at 1/g*^2 = 0.20(4)(3) with mass anomalous dimension gamma_m(g*) = 0.31(6), placing the theory inside the conformal window.
Mass anomalous dimension in SU(2) with two adjoint fermions
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The first Schroedinger-functional measurement of the running mass in SU(2) with two adjoint fermions bounds the chiral condensate anomalous dimension at a putative infrared fixed point to 0.05 < gamma < 0.56.
Zero of the discrete beta function in SU(3) lattice gauge theory with color sextet fermions
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The discrete beta function of SU(3) gauge theory with two sextet Wilson fermions crosses zero near g^2 = 2.0, suggesting an infrared fixed point at unexpectedly weak coupling, though the result rests on a single latti...

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Improved Wilson-clover lattice simulations show QCD-like running at four fermion flavours and a possible infrared fixed point at ten, while the six-flavour case remains undecided within statistical and cutoff errors.
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The authors derive and apply RG equations for the lattice spacing in 2+1 clover fermion QCD, finding only a small quark mass dependence and smoother lattice spacing ratios.
Non-perturbative renormalization and O$(a)$-improvement of the non-singlet vector current with $N_{\mathrm{f}}=2+1$ Wilson fermions and tree-level Symanzik improved gauge action
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The authors determine non-perturbatively the renormalization factor and O(a) improvement coefficients of the local and conserved non-singlet vector currents for Nf=2+1 Wilson fermions, leaving only fV undetermined.
High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks
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The chirally rotated Schrödinger functional yields non-perturbative axial and vector current renormalization constants for Nf=2 and Nf=3 CLS-matched lattice QCD, with errors up to an order of magnitude smaller than Wa...
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For two-flavour QCD the gradient-flow scales t0 and w0 and the topological susceptibility are extrapolated to the continuum, and the resulting scale ratios differ substantially from the pure-gauge (Nf=0) values.
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hep-lat 2012-05 conditional novelty 5.0 of 10

Two-flavor lattice QCD gives Lambda_MS = 310(20) MeV and m_s = 138(3)(1) MeV, with r_0 = 0.503(10) fm, using the kaon decay constant to set the scale.
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A lattice QCD simulation with two light and one strange quark yields alpha_s(M_Z) = 0.12047(81)(48)(+0/-173) and Lambda_MS^(5) = 239(10)(6)(+0/-22) MeV, using non-perturbative Schroedinger-functional running over five...
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How the literature uses it

44 arXiv papers cite this in 106 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 · 59 sentences without a confident label

Builds on it 2 papers

For this, we calculate the one loop running coupling in our family of SF schemes following closely the discussions in [ref] and adapting them to arbitrary $N$.

Schr\"odinger functional boundary conditions and improvement for N>3 · Introduction · 0.82
This is a generalisation of the program used earlier to compute the improved action for two fundamental representation fermions in SU(3) gauge theory [ref] .

Nonperturbative improvement of SU(2) lattice gauge theory with adjoint or fundamental flavors · Introduction · 0.98

Uses its method or tool 19 papers

The renormalized coupling $g^2(2L)$ at the scale $2L$ is corrected perturbatively in order to cancel the deviation of the PCAC mass from zero at the scale $L$ [ref] [equation] The PCAC mass at the scale $L$ has been tuned such that the deviation $Φ(0)u^2mL$ is smaller than the typical statistical error.

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Step scaling function · 0.98
Here $Σ^(k)(u,a/L)$, whose lattice artifacts are perturbatively removed at k-loop, is defined by [equation] where $δ_n$ $(n=1,2…)$ is given by [equation] and $δ_1(a/L)$ for the RG improved action is given in Ref. [ref] .

Universality of the Nf=2 Running Coupling · Result · 0.85
For the details of the numerical method used to determine coefficient $c_t^(1,1)$, we refer to the original literature where the method was developed for fundamental representation fermions [ref] .

Perturbative Improvement of the Schrodinger Functional for Lattice Strong Dynamics · Introduction · 0.97

Uses its data or values 5 papers

We translate to the $$ scheme using $Λ_^(2)=2.382035(3)Λ_SF^(2)$ [ref] as well as to physical units Λ_^(2) = 310(20) MeV , where =155 enters.

Non-perturbative computation of the strong coupling constant on the lattice · Scale setting for $=2$ · 0.93
By combining the previous results from Ref. [ref] , the values of $A_0$ can be obtained, and are shown in the second line (numbers with $∗$) in each $s$ in Table [ref] for $θ=0,π/5$.

Perturbative analysis of the Neuberger-Dirac operator in the Schr\"odinger functional · Coefficients of Symanzik's expansion · 0.92
[equation] where the right-hand side of eq.([ref]) is taken from [ref] .

Universality of the Nf=2 Running Coupling · Simulation details · 0.81

Compares with it 6 papers

The first coefficient $A_0$ is generally a function of the parameter $s$, and I get $A_0(s)|_s=0=0.012567(3)$, while by combining the results of Ref. [ref] , $A_0(s)|_s=0=0.012566$ can be deduced.

Universality check of the overlap fermions in the Schroedinger functional · Universality check in perturbation theory · 0.94
We show numerical results for the one-loop deviation in Tables [ref] and [ref] and the plots in Figure [ref], where we include those of the Wilson type fermions [ref] and overlap fermion [ref] for comparison. $L_s$-dependence of DWFs is small.

A formulation of domain-wall fermions in the Schr\"odinger functional · Lattice artifacts of the step scaling function to one-loop order · 0.98
We show numerical results of the one-loop deviation in Table [ref] and plots in Figure [ref], where we include those of the Wilson-Dirac and the clover action for comparison [ref] .

Perturbative analysis of the Neuberger-Dirac operator in the Schr\"odinger functional · Lattice artifacts of the step scaling function to one-loop order · 0.87

Cites it as background 6 papers

The renormalised coupling in the Schrödinger functional (SF) scheme has been defined in [ref] and its scale evolution has been studied in QCD with zero and two quark flavours [ref] .

The SF running coupling with four flavours of staggered quarks · Introduction · 0.87
In this section we briefly recall the ideas introduced in [ref] and collect the expressions necessary for the subsequent discussion.

Schr\"odinger functional boundary conditions and improvement for N>3 · Schrödinger functional · 0.91
[ref] for some important examples.]

Past, present, and future of precision determinations of the QCD coupling from lattice QCD · The high-energy regime of $=3$ QCD · 0.94

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