Lattice determinations of the strong coupling
LLattice determinations of the strong coupling
Luigi Del Debbio a , Alberto Ramos b,1 a School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK b School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
Abstract
Lattice QCD has reached a mature status. State of the art lattice computations include u, d, s (and even the c ) sea quark effects, together with an estimate of electromagnetic andisospin breaking corrections for hadronic observables. This precise and first principlesdescription of the standard model at low energies allows the determination of multiplequantities that are essential inputs for phenomenology and not accessible to perturbationtheory.One of the fundamental parameters that are determined from simulations of latticeQCD is the strong coupling constant, which plays a central role in the quest for precisionat the LHC. Lattice calculations currently provide its best determinations, and will play acentral role in future phenomenological studies. For this reason we believe that it is timelyto provide a pedagogical introduction to the lattice determinations of the strong coupling.Rather than analysing individual studies, the emphasis will be on the methodologies andthe systematic errors that arise in these determinations. We hope that these noteswill help lattice practitioners, and QCD phenomenologists at large, by providing a self-contained introduction to the methodology and the possible sources of systematic error.The limiting factors in the determination of the strong coupling turn out to be differ-ent from the ones that limit other lattice precision observables. We hope to collect enoughinformation here to allow the reader to appreciate the challenges that arise in order toimprove further our knowledge of a quantity that is crucial for LHC phenomenology. Keywords:
QCD, renormalization, strong coupling, Lattice field theory.
Preprint:
IFIC/20-56
Email addresses: [email protected] (Luigi Del Debbio), [email protected] (Alberto Ramos) Present address: Instituto de F´ısica Corpuscular (IFIC), CSIC-Universitat de Valencia 46071 -Valencia, SPAIN
Preprint submitted to Physics Reports January 14, 2021 a r X i v : . [ h e p - l a t ] J a n ontents α MS ( M Z ) . . . . . . . . . . . . 13 α MS ( M Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Systematics in the extraction of α MS . . . . . . . . . . . . . . . . . . . . . 18 Present and future of lattice determinations of α s α s . . . . . . . . . . . . . . . . . . 887.4.1 The pure gauge theory as a perfect laboratory . . . . . . . . . . . 91 A.1 Topology freezing and large autocorrelation times . . . . . . 94A.2 Signal to noise problem . . . . . . . . . . . . . . . . . . . . 96
B Scale variation estimation of truncation errors 98
B.1 The case of the Schr¨odinger Funcional coupling in full detail 99B.2 The static potential . . . . . . . . . . . . . . . . . . . . . . 101B.3 HQ correlators . . . . . . . . . . . . . . . . . . . . . . . . . 102B.4 Wilson loops . . . . . . . . . . . . . . . . . . . . . . . . . . 102
References 102
1. Introduction
Nowadays lattice QCD is a mature field. Several low energy tests of the strong interac-tions involve careful lattice QCD computations, which in the last 10 years have acquiredthe status of precision physics. Lattice determinations of the strong coupling constantare amongst these precise measurements; they actually represent the most precise resultsavailable in the literature for this fundamental parameter of the Standard Model. Thissituation will continue to improve as the combined result of the increase in computerpower and the ingenuity of new methodologies. The next few years will probably seeseveral new and very precise lattice determinations of the strong coupling. With theworld average of α s already dominated by lattice determinations this work seems fullyjustified.The reader might not feel comfortable with this situation. But one of the points thatwe want to emphasize in this review is that this is not necessarily a bad situation. Thereason is that there are several methods within lattice QCD to extract the strong cou-pling, and they are affected by systematic effects in different ways. These methods differamong themselves at least as much as the different extractions from phenomenologicaldata. Determinations of the strong coupling from lattice QCD are, as a matter of fact,a vast subject. The main objective of this review is to present the different techniquesto determine the strong coupling on the lattice. We do not aim to review the individual3apers, but instead to present the different methods and their general characteristics,advantages and limitations. Along the way we hope to clarify what each method needsin order to improve their current results substantially. Hopefully this review shouldprovide enough information so that the non-experts in the field can understand and becritical when reading the specialized lattice literature. In this respect, this work is acomplement, and not an alternative, to the excellent FLAG review [1], where a detaileddescription of each published work can be found, together with quality criteria on theresults. For instance, we will not aim to provide here a best value for the strong coupling,for which we refer to FLAG, but we will try to insist on the systematic errors in latticedeterminations.The systematic errors that affect most lattice QCD calculations are quite differentfrom those that impact on the determinations of the strong coupling. We live in anera where state of the art lattice QCD computations include electromagnetic and charmeffects with the aim of reaching a sub-percent precision in many observables. But theseeffects represent a very small contribution to the uncertainty in the strong coupling,well below our current precision. Instead, when it comes to the determinations of thestrong coupling, the limiting factor in lattice analyses is in fact very similar to those thatlimit the precision of many phenomenological studies: the use of perturbation theory atrelatively low scales makes it difficult to estimate the uncertainty associated with thetruncation of the perturbative series.Let us now summarize the material in the review. In section 2 we introduce someelementary facts about the strong interactions. We focus on the Λ parameter, whoseknowledge is equivalent to the direct determination of the value of the strong coupling.In this respect, it is important to realise that, in the absence of quark masses, thedetermination of the coupling in QCD is equivalent to setting the scale of theory byspecifying the value of just one dimensionful quantity. This point will be discussed indetail below. Section 3 focus on how the strong coupling is determined. We will seethat in fact lattice methods share the same basic strategy as other phenomenologicaldeterminations, and face the same challenges. In section 4 we provide an introduction tolattice field theory. Special emphasis is put on the topics that enter the determination ofthe strong coupling: continuum extrapolations, scaling violations and scale setting aresome of the topics that are explained in detail. Contrary to other lattice computations,that are intrinsically low-energy computations, the determination of the strong couplingrequires to make contact with the Standard Model at the electroweak scale. In section 5we focus on the effects that the heavy charm and bottom quarks have in this peculiarsituation. Section 6 introduces the different techniques used on the lattice to determinethe strong coupling. The focus will be on how they address the systematic uncertaintiesinherent in these computations. Finally section 7 will discuss the present status andour anticipation for the future of lattice determinations of the strong coupling, with anemphasis on the role of the different methods.The authors have their own (possibly sometimes divergent) opinions about the topicscovered in this review. Of course we find our position well founded and we are happyto defend it, but in a review work like this it is important to keep in mind that therecan be some controversies. This can only be positive in a field that is still an active areaof research. We have tried to explain our point of view as clearly as possible, and whennecessary, we have explicitly stated that we are exposing our own point of view. We hopethat this work becomes a useful reference even for those who disagree with some of our4pinions.
2. The standard model at low energies
The standard model (SM) of particle physics classifies all known fundamental particlesand describes their interaction via three of the four fundamental forces. Its predictionsagree with experiments with an astonishing precision. The gravitational force is the onlyinteraction that is unaccounted for by the SM. Of the three fundamental interactionsin the SM, the weak and electromagnetic interactions are two aspects of a unified elec-troweak force. At temperatures below the electroweak scale ( ∼
100 GeV) two differentinteractions (weak and electromagnetic) emerge. The weak interactions become relevantonly at very short distances, less than the diameter of a proton, due to the massivenature of the weak W ± , Z bosons that mediate the interaction. On the other hand theelectromagnetic force, being mediated by massless photons, is a long range interaction.It describes the interactions between electrically charged particles, and is responsible ofmany phenomena in different areas, from optics, to radiation or the structure of theatoms. From the theoretical point of view quantum electrodynamics (QED) is a relativis-tic quantum field theory amenable to precise computations by using perturbation theory,due to the smallness of the coupling between charged particles ( α EM ∼ / Quantum Chromodynamics (QCD), called confinement.The force between color charged particles remains constant and different from zero atlarge separations between the charges. Pulling two quarks apart requires an increasingamount of energy, until eventually new pairs of quarks are created. Particles chargedunder the strong interactions are therefore confined in color “neutral” hadrons (like theproton). On the other hand, the strong interactions become asymptotically weaker atvery short distances, a phenomenon called asymptotic freedom [4, 5]. Quarks behave asalmost free particles at distances much smaller than the size of a proton.This qualitative picture of the strong interactions explains many experimental phe-nomena, from scaling in deep inelastic scattering (DIS) experiments to the fact that nota single free quark has ever been observed. Because of asymptotic freedom perturbativepredictions of QCD can be compared with high-energy experiments. Quantitative com-parisons for processes like vector boson production, event shape observables at the LargeElectron-Positron collider (LEP) or scaling violations in DIS – just to name a few – re-main the most stringent tests of QCD as the theory of the strong interactions, althoughnone of them reaches the precision of the tests of QED. Low-energy predictions for thestrong interactions are more elusive; as the coupling increases, computations based onperturbation theory are no longer adequate. Accurate predictions in this regime requirea non-perturbative formulation of the theory, and have become possible only recently5hanks to large scale lattice QCD simulations.There is currently clear evidence supporting the idea that QED and QCD is allthat is needed to explain most experimental results of particle physics at scales belowthe electroweak scale with very high accuracy. From photoproduction in proton-protoncollision, to the mass of the proton or the energy binding of the atomic nucleus or theformation of the atom.But the attentive reader should have noted that at the core of this picture for thestrong interactions (free quarks at “high” energies and confinement at “low” energies)lies a fundamental question to be asked: high and low energies compared with what?how does a scale arise in QCD?
The strong interactions are described by a relativistic quantum field theory. It describesthe interactions between color charged particles: the 6 quark flavors and the gluons. It isa non-abelian gauge theory with symmetry group SU (3). Matter content and symmetriesis all that is needed to write down the action of QCD, that reads S [ A ] = (cid:90) d x (cid:40) − g Tr ( F µν F µν ) + (cid:88) i =1 ¯ ψ i ( γ µ D µ + m i ) ψ i (cid:41) , (1)where D µ = ∂ µ + A µ , m i is the bare mass of quark flavor i , and g is the bare gaugecoupling. The field strength is defined by F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] (2)It is worth noting that quark masses are the only dimensionful parameters of the QCDaction, since the gauge coupling g is dimensionless in 4 dimensions. At the classical levelquark masses are the only source of breaking of scale invariance.QCD predictions are made by computing expectations values of fields in the Euclideantheory as path integral averages with partition function Z = (cid:90) D A e − S [ A ] . (3)All physical information is then extracted from these correlators. The path integralwritten above is, naively, ill defined. A simple perturbative calculation for instance showsthat the path integral is plagued by ultraviolet (UV) divergences, i.e. divergences thatarise when summing over the high-energy modes in the theory. Expectation values can bemade finite by modifying the theory at short distances. There are several possibilities forsuch a regularization of the theory, the most natural consists in defining the theory on afour dimensional Euclidean lattice with spacing a . When performing Fourier transformsin a discretized spacetime, momenta are limited to the first Brillouin zone, which impliesthat the inverse lattice spacing provides a UV cutoff. There are other possibilities to make We are going to work in 4-dimensional Euclidean space. The gauge field A µ ( x ) lives in the Liealgebra su (3), and therefore, for matter in the fundamental representation of the gauge group, it is ananti-hermitian traceless 3 × P ( Q ), measured at a typical scale Q , computed from some expectation value in theregularized theory, will depend not only on Q and the particular values of the gaugecoupling and quark masses ( g, m i ), but also on the short distance scale (denoted a ) atwhich QCD is modified. Denoting the mass dimension of P by d P , we have: a d P P ( Q ) = P ( aQ, g, am i ) . (4)Note that the quantity on the left-hand side of Eq. (4) is the dimensionless product a d P P ( Q ), and that accordingly the function P only depends on dimensionless quantities.The problem is how to make any solid prediction when the arbitrary value of the shortdistance a appears in all determinations of physical quantities. The answer comes underthe name of renormalization . Even if determinations in the regularized theory dependon the particular choice of ultraviolet cutoff ( a ), the physics at large distances comparedwith the cutoff (the regime aQ (cid:28)
1) is universal if it is parametrized in terms of the renormalized coupling (¯ g ( µ )) and renormalized quark masses ( ¯ m i ( µ )). The renormaliza-tion scale µ is an arbitrary scale that is introduced in the renormalization procedure. Amore precise relation would then read P ( aQ, g, am i ) = ¯ P ( Q/µ, ¯ g ( µ ) , ¯ m i ( µ ) /µ ) + O (( aQ ) p , ( aµ ) p , ( am ) p ) + . . . . (5)Note that the arbitrary scale a does not show up in the first term on the right-hand side.Moreover in the limit where the short-distance scale a is much smaller than the physi-cal ( Q ) and renormalization ( µ ) scales a precise prediction for any physical observableemerges P ( Q ) M d P = 1( aM ) d P ¯ P ( Q/µ, ¯ g ( µ ) , ¯ m i ( µ ) /µ ) . (6)In the equation above we have expressed P ( Q ) in units of some physical mass scale M ,which in turn can be obtained from a lattice simulation in units of the cutoff a – this thequantity in the denominator in the RHS of the expression above.The renormalized quantities ¯ g ( µ ) , ¯ m i ( µ ) are functions of the quark masses and cou-pling constant of the finite theory (the bare parameters g, m i ), the cut-off a and therenormalization scale µ . The physics content of this renormalization process is that atlow energies the theory is sensitive to the particular choice of cutoff only via the relationbetween bare and renormalized parameters. This relation is not observable and remainsan arbitrary choice needed in order to make physical predictions. The set of prescriptionsthat are necessary to fully specify the relation between bare and renormalized quantitiesis called a renormalization scheme .Note that in the renormalization procedure, we have introduced a new scale µ . Thisis not an accident, and is unavoidable, independently of the chosen regularization and/orrenormalization schemes. The renormalization scale µ is arbitrary and physical quantitiesmust be independent on µ . This requirement can be expressed as a set of mathematicalconditions, which go under the name of Callan-Symanzik [6, 7] equations: µ dd µ ¯ P ( Q/µ, ¯ g ( µ ) , ¯ m i ( µ ) /µ ) = 0 . (7)7hese equations can be used to determine how the renormalized coupling ¯ g ( µ ) and therenormalized quark masses ¯ m i ( µ ) change (“run”) with the renormalization scale. Oneof the main characters of this review is the β -function, which dictates the dependence ofthe renormalized coupling on the renormalization scale [4, 5] µ dd µ ¯ g ( µ ) = β (¯ g ) . (8)This renormalization group (RG) equation is a first order equation, and thereforeits solution depends on exactly one integration constant. Moreover the solution to thisequation has to respect the correct boundary condition given by the asymptotic behaviorof the β -function determined in perturbation theory β (¯ g ) ¯ g → ∼ − ¯ g ∞ (cid:88) k =0 b k ¯ g k , (9)where b = π ) (cid:16) − N f (cid:17) , (10a) b = π ) (cid:16) − N f (cid:17) , (10b)and N f is the number of fermions in the fundamental representation ( i.e. quarks). Notethat for ¯ g → β -function is negative (at least for N f < i.e. the decrease of the coupling with increasing energy.It is instructive to discuss in some detail the integration of the RG equation, Eq. (8).We can readily see that dµµ = d ¯ gβ (¯ g ) = ⇒ log (cid:18) µ µ (cid:19) = (cid:90) ¯ g ¯ g dxβ ( x ) , (11)where ¯ g = ¯ g ( µ ), ¯ g = ¯ g ( µ ). The logarithmic divergence on the left-hand side ofEq. (11) when µ (resp. µ ) tends to infinity is reflected in the divergence of the integralon the right-hand side when ¯ g (resp. ¯ g ) tend to zero. The asymptotic behaviour of theintegrand is 1 β ( x ) = − b x
11 + b b x + O ( x ) (12) (cid:39) x → − b x (cid:20) − b b x + O ( x ) (cid:21) , (13)and therefore the integral can be rewritten as (cid:90) ¯ g ¯ g dxβ ( x ) = (cid:20) b ¯ g − b ¯ g + b b log ¯ g − b b log ¯ g (cid:21) ++ (cid:90) ¯ g ¯ g d x (cid:20) β ( x ) + 1 b x − b b x (cid:21) . (14) In this section we will use massless renormalization schemes, where the β function is independent ofthe values of quark masses. See section 5 for a discussion of massive renormalization schemes. In general perturbative expansions in quantum field theories are asymptotic. Through this work afunction f ( x ) having an asymptotic expansion will be denoted by f ( x ) x → ∼ . . . O ( x ) when x →
0, and hence the integral is finite when the integration limit tendsto zero. After some algebraic manipulations Eq. (11) yields µ (cid:2) b ¯ g (cid:3) − b b e − b g exp (cid:26) − (cid:90) ¯ g d x (cid:20) β ( x ) + 1 b x − b b x (cid:21)(cid:27) = µ (cid:2) b ¯ g (cid:3) − b b e − b g exp (cid:26) − (cid:90) ¯ g d x (cid:20) β ( x ) + 1 b x − b b x (cid:21)(cid:27) . (15)The equality holds for any value of µ and µ , showing that the combination in Eq. (15)has units of mass, and is independent of µ . It is called the Λ-parameter and can beunderstood as the intrinsic scale of QCD that we were looking for. Note that the inte-gration of the renormalization group equation Eq. (8) is exact, and the Λ-parameter canbe defined as:Λ = µ (cid:2) b ¯ g ( µ ) (cid:3) − b b e − b g µ ) exp (cid:40) − (cid:90) ¯ g ( µ )0 d x (cid:20) β ( x ) + 1 b x − b b x (cid:21)(cid:41) . (16)This expression is valid beyond perturbation theory. Hadron masses, meson decay con-stants, or any other dimensionful quantity in QCD, can be measured in units of Λ, andare given by dimensionless functions of the renormalized coupling, and the renormalizedquark masses (also expressed in units of Λ). It is in this respect that we like to think ofthe Λ parameter as an intrinsic scale of QCD.The renormalized theory is defined by specifying the value of the renormalized cou-pling at a given scale, or equivalently by specifying the value of the Λ parameter. Notethat Eq. (16) is an implicit equation for ¯ g ( µ ), and therefore the running coupling is afunction of Λ /µ ; at high energies compared with Λ (i.e. µ/ Λ (cid:29)
1) the running of thecoupling is given by¯ g ( µ ) µ/ Λ (cid:29) ∼ b log( µ/ Λ) + b log log( µ/ Λ) + . . . . (17)At scales much larger than
Λ, ¯ g ( µ ) is small, QCD is weakly coupled and quarks behaveas almost free particles. There is quite some freedom when renormalizing QCD. In the framework of perturbativecomputations there are many valid ways to subtract the divergent parts of Feynmandiagrams that differ by finite terms. Non-perturbatively there are also multiple conditionsto use as a definition for renormalized coupling and quark masses. This freedom is called choice of scheme .The value of the strong coupling constant at high energies is a necessary input forthe study of all QCD cross sections at the Large Hadron Collider (LHC) and many otherhigh-energy experiments. For this reason it is convenient to quote its value in a schemethat can be easily used for phenomenological input. The so-called modified minimalsubtraction (MS) scheme [8] is by far the most widely-used choice. This scheme is definedin the context of perturbative computations; however the Λ-parameter extracted in thisconvenient scheme still has a non-perturbative meaning, as discussed below.9 f ¯ β ( α MS )3 1.0 + 0 . α MS + 0 . α + 0 . α + 0 . α . α MS + 0 . α + 0 . α + 0 . α . α MS + 0 . α + 0 . α + 0 . α . α MS − . α + 0 . α + 0 . α Table 1: β -function in the MS scheme. Here we write a series in α MS = ¯ g / (4 π ) and divide outthe leading order behaviour according to ¯ β ( α MS ) = − β (¯ g MS ) / ( b ¯ g ). Surprisingly, the perturbativecoefficients remain small up to five-loops. Most of the time we are going to deal with mass-independent renormalization schemes.Any modification of the theory that is performed in order to regularize and renormalizeQCD can always be made at energies much larger than the quark masses. From aperturbative point of view we can say that the UV divergences of Feynman diagrams areindependent of any quark mass. A nice property of mass-independent schemes is thatthe RG functions (like the β -function Eq. (8)) are independent of the quark masses. Ofcourse there is nothing fundamentally wrong with renormalization schemes that are notmass-independent, and we will study in detail the relation between mass-independentand mass-dependent renormalization schemes in chapter 5, but first let us state somebasic relations between massless renormalization schemes.By convention we normalize couplings in different schemes so that they agree toleading order. This implies that renormalized couplings in two schemes s and s (cid:48) arerelated perturbatively by ¯ g s (cid:48) ( µ ) ¯ g s → ∼ ¯ g s ( µ ) + c ss (cid:48) ¯ g s ( µ ) + . . . (18)with c ss (cid:48) a finite number. The β -function for the couplings ¯ g s ( µ ) and ¯ g s (cid:48) ( µ ) are different(i.e. β -functions are scheme dependent), but it is easy to check that the two leadingterms in its asymptotic expansion (10) are scheme independent: b and b are universal.Higher order coefficients b n with n > β -function is known up to five loops [11–15] (see table 1).On the other hand the Λ-parameter, defined in Eq. (16), is also scheme dependent.It is easy to see by using the one-loop relation between couplings, Eq. (18), that (cid:20) ¯ g s (cid:48) ( µ )¯ g s ( µ ) (cid:21) − b b ¯ g s → ∼ − b b c ss (cid:48) ¯ g s ( µ ) + . . . , (19a)1¯ g s (cid:48) ( µ ) − g s ( µ ) ¯ g s → ∼ c ss (cid:48) + . . . . (19b) In some cases massive renormalization schemes might be more convenient, like for example heavyquarks regulated on the lattice. In practice the lattice spacings that are currently accessible to simulationsprovide a UV cutoff that is not much larger than the mass of the heavy quarks c and b . In this contextmass-dependent renormalization schemes might have some advantageous properties. See e.g. [9, 10]. O (¯ g ), one can obtain an exact relation between Λ-parameters by taking the limit ¯ g s → s (cid:48) Λ s = exp (cid:18) − c ss (cid:48) b (cid:19) . (20)In other words, the relation of Λ-parameters in different schemes is exactly known viaa one-loop computation, as reported in Eq. (18). This observation, together with Eq. (16)allows a precise non-perturbative definition of the Λ-parameter even for schemes that areintrinsically defined in a perturbative context: even if MS is a “perturbative scheme”,Λ MS is a meaningful quantity beyond perturbation theory. Quarks are not massless particles. Every physical process in QCD depends not only onthe intrinsic scale of the strong interactions, but on the quark masses. In particular weexpect that if the process takes place at a some energy scale much lower than the massof some quark, this quark should “decouple” from all physical processes.How this decoupling takes place in mass-independent renormalization schemes is infact not trivial. The RG functions (and therefore the renormalized parameters of thetheory) do not depend on any quark mass: at any scale, the top and the up quarks givethe very same contribution to the β -function and therefore to the running coupling. Still,physical observables written in terms of these renormalized parameters should “know”when the energy scale of the physical process is large or small compared with some quarkmasses.We will return to these problems in detail in section 5, here it is sufficient to mentionthat decoupling in mass-independent renormalization schemes can be understood as amatching between different theories. At energy scales below the top quark mass, it is moreconvenient to use an effective 5-flavor QCD theory, without the top quark. The effectsof the top quark at low energies can be conveniently reabsorbed in a redefinition of thecoupling and quark masses of the 5-flavor theory (these can be computed perturbatively),with further corrections being power suppressed ∼ O (cid:0) (Λ /m t ) (cid:1) , O (cid:0) ( Q/m t ) (cid:1) .In a similar way, at energies much below the bottom (respectively charm) quark massthreshold, 4- (respectively 3-) flavor QCD is an excellent description of nature. Eachtheory has its own set of fundamental parameters, so e.g. the 4-flavor theory is completelydefined by the values of the 4-flavor coupling constant and of the quark masses. Theseeffective theories can be used to describe any physical process at energy scales muchbelow the corresponding thresholds m b ∼ m c ∼ . N f − N f active flavorsby the relation ¯ g ,N f − ( m (cid:63) ) = ¯ g ,N f ( m (cid:63) ) × ξ (¯ g ,N f ( m (cid:63) )) . (21)This expression neglects power corrections in the matching between theories, and ξ ( x )is a just a polynomial. In Eq. (21) m (cid:63) = m MS ( m (cid:63) ) is the MS mass at its own scale. Inthis case the one-loop term vanish and we have ξ (¯ g ) = 1 + c ¯ g + c ¯ g + c ¯ g + O (¯ g ) . (22)11ith the first 4 terms known [17–21].Eq. (21) allows to relate the values of the Λ parameters with a different number offlavors. For example, for processes at energy scales above the top mass threshold wewould need the value of Λ (6)MS . This can be obtained from Λ (5)MS by first determining thevalue of the five flavor coupling at the top mass threshold ( m (cid:63) t ≈
163 GeV)¯ g t , = ¯ g MS , ( m (cid:63) t ) , (23)from the implicit equationΛ (5)MS m (cid:63) t = (cid:2) b ¯ g , (cid:3) − b b e − b g , exp (cid:26) − (cid:90) ¯ g t , d x (cid:20) β ( N f =5) ( x ) + 1 b x − b b x (cid:21)(cid:27) . (24)Now this value of the coupling is transformed into the 6 flavor coupling by using thedecoupling relations Eq. (21), to obtain¯ g , ( m (cid:63) t ) = ¯ g , ξ (¯ g t , ) . (25)Finally the value of ¯ g t , = ¯ g MS , ( m (cid:63) t ) can be used to determine the value of the 6-flavorΛ parameterΛ (6)MS = m (cid:63) t (cid:2) b ¯ g , (cid:3) − b b e − b g , exp (cid:40) − (cid:90) ¯ g , d x (cid:20) β ( N f =6) ( x ) + 1 b x − b b x (cid:21)(cid:41) . (26)All this procedure can be summarized by defining ϕ ( N f ) ( t ) = (cid:2) b t (cid:3) − b b e − b t exp (cid:26) − (cid:90) t d x (cid:20) β ( N f ) ( x ) + 1 b x − b b x (cid:21)(cid:27) . (27)and using Λ ( N f +1) Λ ( N f ) = ϕ ( N f +1) (cid:16) ¯ g (cid:63) ξ (¯ g (cid:63) ) (cid:17) ϕ ( N f ) (¯ g (cid:63) ) . (28)Of course such a conversion suffers from several uncertainties. The expressions forthe β -functions ( i.e. β ( N f =5 , ) and the decoupling relations Eq. (21) are only known toa certain order in perturbation theory. This implies that the conversion of Λ-parametersEq. (28) carries a perturbative uncertainty. On top of that there are power correctionsthat have been neglected in the matching between the effective N f and fundamental N f +1theories (see section 5). However, we have now strong numerical evidence [22, 23] showingthat both the perturbative and power corrections are very small in the ratio Eq. (28).Even for the case of the decoupling of the charm quark (at a rather low energy scale m (cid:63) c ≈ . We will not discuss any subtleties in the determination of the top quark mass here. .2.3. Challenges in the determination of α MS ( M Z )From the previous discussion it seems logical to quote the intrinsic scale of QCD bygiving the value of Λ (5)MS , which is a well defined quantity, even beyond perturbationtheory. Together with Eq. (16) and the coefficients of the β -function in the MS schemereported in table (1), it can be used for high energy phenomenology. Moreover if one isinterested in a process at an energy scale above the top quark threshold (or below thebottom/charm thresholds), the procedure described in the previous section can be usedto determine the 3,4 or 6 flavor Λ-parameters.For historical reasons it is now standard to quote the intrinsic scale of the stronginteractions in an indirect way by referring to the value of the strong coupling in theMS scheme at a reference scale µ = M Z ( i.e. the mass of the Z vector boson M Z ≈ .
19 GeV). The current world average for α MS ( µ ) = ¯ g ( µ ) / π (29)quoted in the PDG [24] is: α MS ( M Z ) = 0 . , (30)with an uncertainty of around ∼
1% obtained by combining the uncertainty in thedetermination from several processes. Note that this coupling refers to the 5 flavor theorysince m b < M Z < m t . By using the five-loop asymptotic expansion of the β -function,the world average Eq. (30) is equivalent to Λ (5)MS = 0 . . (31)But, what are the challenges in a precise determination of the strong coupling? Tounderstand this subtle point, we have to look carefully at the fundamental equation usedto determine Λ in units of some reference scale µ ref :Λ = µ ref (cid:2) b ¯ g ( µ ref ) (cid:3) − b b e − b g µ ref) ×× exp (cid:40) − (cid:90) ¯ g ( µ ref )0 d x (cid:20) β ( x ) + 1 b x − b b x (cid:21)(cid:41) . (32)As already discussed above, this is the solution of a first-order differential equation, andΛ can be understood as an integration constant. In other words, knowing the value ofthe coupling at some reference scale is sufficient to determine Λ and hence to fix thecoupling value at all energies according to the RG running. In principle only one numberis needed to fully determine the value of the strong coupling. It could be for instancethe value of ¯ g ( µ ref ). The systematic errors in the determination of the strong couplingare better understood by discussing the determination of Λ. As we can see, (32) involvesthe integral of the β -function from 0 to ¯ g ( µ ref ). Sincelim µ →∞ ¯ g ( µ ) = 0 , (33) Note that due to the logarithmic running of the strong coupling a ∼
6% uncertainty in Λ (5)MS translatesinto an ∼
1% uncertainty in α MS ( M Z ). µ ref and infinity. In practice this limit is never reached, neither inexperimental processes nor in lattice simulations. At most we can determine the non-perturbative running in a limited range of scales, say from µ ref to µ PT . At energies higherthan µ PT one uses the perturbative approximation of the non-perturbative β -function.If we denote β ( (cid:96) ) the perturbative β function to (cid:96) -loops, we have that (cid:90) ¯ g ( µ ref )0 d x (cid:20) β ( x ) − β ( (cid:96) ) ( x ) (cid:21) = O (cid:16)(cid:0) ¯ g ( µ ref ) (cid:1) (cid:96) − (cid:17) . (34)And therefore this uncertainty propagates to the determination of Λ: (cid:90) ¯ g ( µ ref )0 d x (cid:20) β ( x ) + 1 b x − b b x (cid:21) ¯ g ( µ PT ) → ∼ (cid:90) ¯ g ( µ ref )¯ g ( µ PT ) d x (cid:20) β ( x ) + 1 b x − b b x (cid:21) + (cid:90) ¯ g ( µ PT )0 d x (cid:20) β ( (cid:96) ) ( x ) + 1 b x − b b x (cid:21) + O (¯ g (cid:96) − ( µ PT )) . (35)As shown in table 1 the β -function is known up to five loops in the MS scheme. Note,nevertheless, that in practice one never reaches this level of accuracy. In order to applyEq. (35) in the MS scheme, the value of the coupling ¯ g MS ( µ PT ) is needed. The latteris determined by matching an experimental quantity with its asymptotic perturbativeexpansion, typically known up to 3-4 loops. In this case the accuracy in the extractionof Λ MS will be limited by the limited knowledge in the perturbative expression of thephysical observable, and not by the perturbative knowledge in β MS (¯ g ).This phenomenon is present in one form or another in any extraction of the strongcoupling, not only the ones from lattice QCD, but also in phenomenological extractions.Even if Λ (5)MS is defined non-perturbatively, perturbation theory is needed for its deter-mination. Of course this does not mean that a truly non-perturbative determination ofthe Λ-parameter is impossible. The situation is conceptually very similar to many othersystematic effects present in any lattice determination; for example lattice calculationsare always performed at non-zero lattice spacing (and on a finite volume) and this doesnot prevent us to obtain values in the continuum (and in infinite volume). We needto simulate several lattice spacing (and several volumes) and perform an extrapolation .The situation here is very similar: the determination of Λ has to be understood as anextrapolation in ¯ g n ( µ PT ), with perturbation theory as a guide.The scale µ PT is usually called the scale of matching with perturbation theory . Onecrucial point to note is that the size of the missing terms is O (¯ g (cid:96) − ( µ PT )), where (cid:96) isthe number of loops included in the computation of the beta function. In order to havea significant change in the contribution of the missing terms, the matching scale µ PT hasto be changed substantially due to the slow logarithmic running of the strong coupling(cf. Eq. (17)). These issues play a central role in the determination of the systematicerror presented in section 3.2. 14 . Physical definitions of the strong coupling α MS ( M Z ) How is the value of the strong coupling constant extracted from experimental data?The generic procedure can be sketched as follows. Broadly speaking, the experimentalresults for a physical process P ( Q ) at high energies Q are compared with the perturbativeprediction (typically available up to some order n ), P ( Q ) = n (cid:88) k =0 c k ( s ) α k MS ( µ ) + O ( α n +1MS ( µ )) + O (cid:18) Λ p Q p (cid:19) , ( s = µ/Q ) . (36)Several subtle points are involved in this comparison. First we should notice that thecoefficients c k ( s ) grow logarithmically with s , and therefore the renormalization scale µ has to be chosen close to the physical scale of the process Q , in order to avoid largelogarithms and a poorly converging perturbative series. We should also note that once α MS ( µ ) is known at some energy scale µ ∼ Q , one can use the 5-loop β -function in theMS scheme to “run” this result either to a common reference scale (i.e. M Z ), or up toinfinite energy and quote the value of the Λ MS parameter. The considerations raised insection 2.2.3 also apply to the determinations that follow this approach. In this case therenormalization scale µ = sQ plays the role of µ PT : the energy scale at which we matchwith perturbation theory. One would like to extract α MS ( M Z ) (or Λ MS ) by using data atseveral values of µ = sQ , and take as the final result a suitable extrapolation sQ → ∞ .Since the value of s cannot be taken to be arbitrarily large, such a procedure requiresdata at several values of the physical scale Q in order to have a real constraining poweron the value of the coupling.In Eq. (36) we show the two types of corrections present in the perturbative expansionof a physical quantity. First the missing higher orders , due to the fact that we onlyknow a finite (typically n = 2 ,
3) number of terms in the perturbative expansion ofthe observable. Second, non-perturbative corrections (usually called power corrections ).These are of the form e − A/α MS ( Q ) ∼ O (cid:16) Λ p Q p (cid:17) with p = 2 A b and decrease faster thanany power of α MS . In order to keep the truncation and non-perturbative correctionssmall, the chosen process should be ideally inclusive and defined at high enough energies.High energy scales ensure that α MS ( Q ) is small. Inclusive measurements do not require aquantitative description of the strong interactions of hadronic states and therefore are lessaffected by systematic errors coming from models of hadronization and parton showers.Obtaining a precise value for the strong coupling with high energy experimentalinput has its own challenges. At high energies the strong coupling is small, which is justwhat is needed to have the truncation and power corrections under control, but at thesame time the effect that one is trying to measure is small. This usually translates inlarger uncertainties in α MS ( M Z ) from determinations based on data at high energies (seefigure 1). Extracting the value of the strong coupling at lower energies usually leads tosmaller uncertainties, although the estimation of the truncation uncertainties and thenon-perturbative effects become more challenging: clearly the extrapolation sQ → ∞ ismore difficult without data at large Q .Another point to take into account is that in contrast with the perturbative compu-tations, quarks are not the observed final states of any physical process. Hadronization15 . . . . . . . . .
91 10 100 M b M c M τ µ [GeV] δα MS ( M Z ) /δα MS ( µ ) δα MS ( M Z ) α MS ( M Z ) / δα MS ( µ ) α MS ( µ ) Figure 1: Error of the coupling at a scale µ compared with the error propagated to the reference scale M Z . Note that when the strong coupling is determined at low energy scales the result at the referencescale M Z becomes more precise. For example, the error in α MS ( M τ ) is reduced by almost one order ofmagnitude when the result is evolved to the scale M Z (solid purple line). This effect is not only dueto the reduction in the coupling itself: the relative error in the coupling is reduced approximately by afactor three (dashed green line). and other non-perturbative effects have to be taken into account when comparing exper-imental data with perturbative predictions, usually by using Monte Carlo generators. Extraction from data
A well-know example of the extraction of the strong couplingfrom experimental data is the extraction of α MS ( M Z ) using data for τ decaying intohadrons. We briefly summarise the procedure here, in order to highlight the main stepsand the sources of uncertainties, we refer the reader to an extensive review like e.g.Ref. [25] for a detailed discussion. The physical processes considered in this case arethe decays of τ leptons. More specifically, the ratio of the hadronic and leptonic decaywidths can be written as R τ,V + A = Γ( τ → ν τ + hadrons)Γ( τ → ν τ e − ¯ ν e ) = 3 | V ud | S EW (1 + δ P + δ NP ) . (37)In this case the typical energy scale of the process is set by the τ mass Q = M τ =1 . S EW ( Q ) is the electroweak contribution to R τ,V + A , and δ P , δ NP are the QCD perturbative and non-perturbative corrections to the process re-spectively. The non-perturbative (i.e. power) corrections are estimated to be very small δ NP ∼ − [25]. On the other hand the perturbative prediction δ P = (cid:88) k =1 r n ( s ) (cid:18) α MS ( µ ) π (cid:19) k + O ( α ) , ( s = µ/M τ ) . (38)is known up to four loops [26–28]. The impressive perturbative knowledge in the ratio R τ,V + A makes this quantity a good candidate to determine α MS . In fact such determina-tions are one of the most precise phenomenological determinations. On the other handthe scale at which α MS is determined is relatively low, and it cannot be changed, since M τ is what it is. 16 s ≈ . α s ≈ . α s ≈ . α s ≈ . .
001 0 .
01 0 . µ/ GeVΛ − ∼ L Λ − ∼ a . . . . µ − / fm Figure 2: Scales in a typical state of the art lattice computation. The volume of the simulation is a fewfm, while the cutoff 1 /a is a few GeV. Lattice QCD can resolve observables in this window of scales,where the strong coupling is α MS ∼ . The procedure with other observables is basically the same, although some details,like the number of known terms in the perturbative expansion or the size of the non-perturbative effects (i.e. δ NP in Eq. (37)), change from one observable to another.Combining multiple collider observables in a global fit provides a better lever-arm toconstrain α s together with the parton distribution functions (PDFs). Global fits thatinclude the wider ranges of data provide determinations of the strong coupling constantwith good statistical accuracy, see e.g. Refs. [29–31]. The challenges here stem fromcontrolling the systematic errors (both theoretical and experimental) in fits that involvevery large and diverse datasets and the relatively low energies involved (see for examplethe recent review[32]). Moreover, as recently discussed in Ref. [33], determinations ofthe strong coupling from hadronic processes should entail a simultaneous determinationof the parton distribution functions.
Extraction from lattice simulations
Lattice QCD offers an interesting alternativeto phenomenological determinations. Being a non-perturbative formulation of QCD, onecan combine input from well-measured QCD quantities – like for example the protonmass, or a meson decay constant – with the perturbative expansion of a short distanceobservable that does not need to be directly observable (like the quark anti-quark force).The advantage of this approach is that the experimental input comes from the hadronspectrum with a negligible uncertainty. Hadronization corrections are not needed, sincewe are working directly in a non-perturbative framework.Despite being very different approaches, both the phenomenological and the latticeQCD methods have to overcome similar challenges. Lattice extractions of the strongcoupling must be done at sufficiently high energies so that truncation and power correc-tions are well under control when matching to perturbative expansions. Due to the finitenature of computer resources, every lattice QCD simulation has two intrinsic scales: thetotal physical volume simulated L (IR cutoff), usually of a few fermi in order to keepfinite-volume corrections well under control, and the lattice spacing a (the UV cutoff ∼ .
04 fm in the most challenging present day simulations, which corresponds roughlyto a cutoff of 5 Gev in energies). Any lattice QCD simulation can only resolve a processif it is defined at a scale between these IR and UV cutoffs (see figure 2). The numberof lattice points in each direction is given by the ratio
L/a , viz. the separation of theUV and IR cutoffs determines the memory footprint and computing power, and hencethe computational cost, of the corresponding simulations, putting in practice a limit onthe energy scales that can be studied in any lattice simulation. While in principle lattice17 . . . . . . . .
088 0 0 .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 . L Λ M S α Final Result ν = − . ν = 0 ν = 0 . Figure 3: Determination of the Λ-parameter in units of a scale L ∼ / (4GeV). Different valuesof ν represent different choices of observable, and each point corresponds to a determination of thedimensionless product Λ MS L using a perturbative expansion (like Eq. (36)). The horizontal axes labelthe scale of matching with perturbation theory ( µ in Eq. (36)) parametrized in terms of the leadingcorrections O ( α ( µ )). The first error bar shows the statistical uncertainty. The second error bar showsthe total systematic plus statistic added in quadratures. The systematic uncertainty is determined byvarying the renormalization scale a factor of two above and below some specific value. For ν = − . α ∼ . − .
2, while in other cases ( ν = 0 .
3) the systematic uncertaintyoverestimates the true difference. See text for more details. (source [34, 35]). techniques can be used to compute non-perturbatively the running of the coupling untilthe perturbative regime is reached, in practice the range of scales that can be studied ina single lattice simulation is limited by computer resources. Reaching scales higher thana few GeV requires a dedicated approach. α MS The truncation performed in Eq. (36) neglects higher order terms (i.e. perturbativecorrections) and non-perturbative power corrections. When performing an extrapolationto Q → ∞ , these effects only affect how fast we approach the extrapolated value. Anexample of this behaviour can be seen in figure 3, where different observables (labeled by ν ) are used to estimate Λ MS by matching with perturbation theory at different physicalscales Q , which are translated in different values of α MS in the plot. Different observablespredict compatible results for Λ MS when the extrapolation Q → ∞ , corresponding to α → ν = − .
5) show a slow approach to the extrapolated value,with significant discrepancies even at energy scales Q ∼ −
10 GeV.In practice performing the extrapolation Q → ∞ is very difficult. Data over a largerange of energy scales is required in order to perform such an analysis. For example18he data in figure 3 involves precise lattice determinations of the target observables forenergy scales Q ∈ −
140 GeV. This is only possible with a dedicated approach (seesection 6.7).How do we estimate the systematics effects in the extraction of α MS when the datadoes not allow an extrapolation Q → ∞ ? Of course this is a complex subject in itself,that is of much relevance not only for the extraction of α MS , but also in the interpretationof many experimental data from hadron colliders. A possible estimate of the uncertaintydue to the missing terms is given by the last known term in the series c n ( s ) α n MS ( µ ). Gen-erally this results in significant theoretical uncertainties, and, perhaps more interestingly,correlations between experimental data.A common approach to estimate these uncertaintites exploits the fact that the trun-cated perturbative expansion to n th order P ( n ) ( Q, s ) = n (cid:88) k c k ( s ) α k MS ( µ ) , ( s = µ/Q ) . (39)still depends on µ , while the true value of the observable P ( Q ) does not. The truncationof the perturbative expansion introduces a spurious dependence on the renormalizationscale, which is in general an unphysical, arbitrary quantity. Higher-order effects areestimated by looking at the variation of P ( n ) ( Q, s ) when the renormalization scale µ ischanged by a factor two around some preferred value (for example µ = Q ). In principlethe relation between a variation in the renormalization scale µ and the size of the missinghigher-order terms given by δ n = | P ( Q ) − P ( n ) ( Q, s ) | is unclear, beyond the fact thatthe scale dependence in Eq. (39) is due to the truncation of the perturbative expansion.Under some assumptions on the size of the coefficients of the perturbative expansion( c n +2 ( s ) α ( µ ) (cid:28) c n +1 ( s )), it is possible to show that the scale variation yields a sensibleestimate of δ n (see for example Ref. [36]). Formally, µ d P ( n ) ( Q, s )d µ ∝ α n +1MS ( µ ) , (40)which implies that, at least parametrically, changes in µ capture the correct size of themissing terms. As an example, a recent comprehensive study of the theoretical uncertain-ties for numerous observables, based on scale variations, can be found in Refs. [37, 38].What about power corrections? they are not captured by this kind of analysis. Es-timating them requires to have access to different physical scales Q . Ideally one wouldlike to work at sufficiently high energies so that they are negligible compared with theaccuracy of the data. In practice this is not always the case. Note that the perturbativerunning is logarithmic, and distinguishing this perturbative running from a power-likebehaviour requires data that span large energy ranges.The assumptions that underlie the scale variation procedure constrain both the non-perturbative effects and the character of the perturbative series. In particular, the as-sumption that the first unknown term of the perturbative series is smaller than the lastknown one is implicit in any estimate that uses Eq. (40). Also the value of α MS ( µ ) is as-sumed to be small enough so that these uncertainties are meaningful. These assumptionsmight seem reasonable and mild, and often yield sensible estimates, but there are exam-ples in the literature where they have been shown not to be accurate. Let us mentionhere three relevant cases. 19 The convergence of the perturbative series in practice is not as good as we wouldlike. Due to the asymptotic nature of the PT series, one expects that at some pointthe coefficients in the perturbative series will grow factorially, see Ref. [39] for areview. • Extractions of the strong coupling from τ decays can be done by applying twoframeworks in perturbation theory, called fixed order perturbation theory (FOPT)and contour improved perturbation theory (CIPT). Using α MS ( m τ ) = 0 .
34 as thetypical value of the strong coupling at the scale set by the mass of the τ , thecontributions to both perturbative series look as follows δ FOPT = 0 . . . . . . , (42) δ CIPT = 0 . . . . . . . (43)The terms in both series decrease, and each of the expansions by themselves seemreasonable. Taking the last term as a measure of the uncertainty in the truncation,these values should be accurate with a precision ∼ .
01. But both approaches resultin values that differ by more than twice this amount. What is more worrisome, forthe highest orders the difference between both estimates grows as more terms areincluded in the expansion. • The scale variation approach to estimate truncation uncertainties (changing thevalue of the renormalization scale µ by a factor two around a preferred value)has been compared with non-perturbative data in a careful study for values of α MS ∼ . − . ν = − . factorial growth of the size of the terms in the series is expected, whichis deeply related to the non-perturbative effects of the theory. Note however that thestructure of the corrections have been investigated at length (see e.g. Ref. [39]). It isclear that there are at least two ingredients in the quality of any extraction of the strongcoupling.1. The value of α MS at which perturbation theory is used. Non-perturbative (power)corrections decrease very quickly with α MS . In order to make them negligible oneneeds to have access to high energy scales, hence, small values of α MS . The perturbative series has the form δ = (cid:88) n ( K n + g n ) (cid:16) απ (cid:17) n (41)where the K n coefficients are the same in FOPT and CIPT, while the g n coefficients are different inboth formulations. The coefficient K is unknown. Here we use an estimate K = 275. Note that thisdoes not affect the difference in δ between both formulations.
20. The extraction has to be performed over a range of values of α MS . This allows theunknown terms in the series to vary substantially, so that one can check that indeedthey are negligible. A reasonable requirement would be that the first unknown term α n +1MS varies significantly, say by a factor four.These are two criteria that are usually relevant in any extrapolation. As shown in theexamples above, they will impact the quality of any extraction of the strong coupling.Of course some determinations cannot really change the value of the momentum scale atwhich perturbation theory is used. A good example is the above mentioned extractionfrom τ decays, since the mass of the τ is what it is and sets the overall energy scale tothe process. For the case of lattice simulations, changing the values of α MS at whichperturbation theory is used is challenging, yet feasible. We shall keep these two criteriain mind when describing any lattice computation/method, and not only the quotedtheoretical uncertainties.We would like to end this section recommending the reader the recent contributionto the Lattice Field Theory Symposium by M. Dalla Brida [41], where these issues arealso discussed in detail.
4. Lattice field theory
This section summarizes, briefly, the ideas that underlie lattice QCD. While it does notprovide an extensive discussion of lattice QCD, it is intended to present the framework oflattice simulations for the non-expert reader in a self-consistent form, setting the notationfor the following sections, and providing references for further reading. Hoperfully itwill yield the foundations to better understand the sources of systematic errors that arediscussed in what follows. It can safely be skipped unless the reader is actually interestedin the details of the lattice simulations.The formulation on a discretized lattice provides a non-perturbative definition of aQuantum Field Theory (QFT). The starting point is the path integral of the theory inEuclidean space Z = (cid:90) D ¯ ψ D ψ D A µ e − S QCD [ ¯ ψ,ψ,A µ ] . (44)where S QCD [ ¯ ψ, ψ, A µ ] = (cid:90) d x − g Tr { F µν F µν } + N f (cid:88) f =1 ¯ ψ f ( D µ γ u + m f ) ψ (45)Lattice field theory gives a precise definition to the path integral in Eq. (44) bydiscretizing the spacetime in a hyper-cubic lattice with spacing a . In this approachmatter fields are defined at the lattice sites x µ = an µ for n µ = 1 , . . . , L µ /a , where L µ is the physical size in direction µ . After this discretization the path integral Eq. (44) issimply an integral over very many degrees of freedom (the value of each of the fields at21ach spacetime point). For example the measure for the fermion fields becomes D ¯ ψ ( x ) −→ L µ /a (cid:89) n µ =1 d ¯ ψ ( an µ ) , (46) D ψ ( x ) −→ L µ /a (cid:89) n µ =1 dψ ( an µ ) . (47)As already discussed the lattice spacing a , provides the UV cutoff of the theory.The most appealing characteristic of the lattice formulation of QCD is that it allowsquantitative computations to be performed using numerical simulations. Field correlatorsare computed as integrals in spaces of very large (but finite) dimensions using MonteCarlo techniques.A crucial step in lattice field theory consists in defining a “discretized” version ofthe continuum action S QCD . When constructing lattice actions, one has to pay specialattention to the symmetries of the theory. Ideally the lattice action should preserve exactly as many of the symmetries of the continuum action as possible. The case ofgauge symmetry plays a special role, since it is crucial to guarantee the renormalizabilityof the theory. Another common requirement for the lattice action is to reduce to thecontinuum action in the naive classical limit a → . A naive discretization of the pure gauge action, obtained by substituting the derivativesin the continuum action by finite differences, results in discretization effects that breakgauge invariance. The way to construct lattice actions for gauge theories is rooted inthe geometric interpretation of gauge invariance, and was first proposed by Wilson [43].Since the gauge field acts as an affine connection in the continuum theory, its latticecounterpart is the parallel transporter along the links of discretized spacetime. Hencethe key idea is to work with link variables U ( x, µ ) = e aA µ ( x ) , (48)where the pair ( x, µ ) uniquely identifies the link that originates from point x in thepositive µ direction. These link variables can be seen as a discretization of a continuumWilson line, the parallel transporter mentioned above, linking the points x and x + a ˆ µ U ( x, µ ) = P exp (cid:26) a (cid:90) d t A µ ( γ ( t )) (cid:27) + O ( a ) . (49)Here γ ( t ) is a path that links the point x with x + a ˆ µ , e.g. γ ( t ) = x + at ˆ µ , t ∈ [0 , . (50)A product of link variables along a closed loop is called a Wilson loop . Recently several works have pointed out that universality might still give the correct continuumresults, even in cases where the lattice action does not reproduce the continuum action in the naive limit a →
0. The interested reader should consult the seminal work [42]. The four-vector ˆ µ has all components equal to zero, except the coordinate µ , that has a value 1. a) S (b) S Figure 4: Traces of the plaquette ( S ) and the 2 × S ) can be used to construct differentlattice actions. Link variables transform under a gauge transformation ω ( x ) ∈ SU (3) as U ( x, µ ) → ω ( x ) U ( x, µ ) ω ( x + a ˆ µ ) † , (51)which implies that the trace of a Wilson loop is gauge invariant. Lattice actions are con-structed by combining these traces, with different choices yielding the same low-energyphysics (compared to the cutoff scale), with different lattice artefacts. The simplestchoice, first proposed by Wilson, consists in using the smallest possible loop (the plaque-tte) S W = 1 g (cid:88) W∈S tr { − U ( W ) } , (52)where the sum runs over all oriented Wilson loops of type S (see figure 4). It is easy tocheck that for a given plaquette W x in the µ, ν plane with left lower corner at point x ,we have Re tr { − U ( W x ) } = 12 a tr { F µν ( x ) F µν ( x ) } + O ( a ) (53)And therefore the Wilson action reduces to the continuum YM action in the naive limit a → Improved actions
This is not the only option. By including in the definition of thelattice action larger loops the rate of convergence to the continuum can be improved, i.e. the size of lattice artefacts can be reduced. This is a general programme that goes underthe name of improvement , and extends to the fermionic part of the action. Improvedactions play an important role in reducing lattice artefacts and therefore providing moreprecise extrapolations to the continuum limit. Although the literature covers a widerange of lattice actions, most lattice simulations are performed using some particularchoice of the one-parameter family that can be constructed from the plaquette and the2 × S latt = 1 g (cid:88) i =0 , c i (cid:88) W∈S i tr { − U ( W ) } . (54)23he constants c , c have to obey the constraint c + 8 c = 1 , (55)in order to recover the classical continuum limit, but otherwise can be chosen at will.Clearly the simple Wilson action, S W in Eq. (52), is recovered by choosing c = 1 , c = 0.Other popular choices include the Symanzik tree-level improved action ( c = 5 / , c = − / c = 3 . , c = − . α .For the path integral to be fully specified, it is also necessary to define the integrationmeasure of the gauge link variables U µ ( x ). In order to ensure the gauge invariance of thepath integral, the measure d U of each link variable needs to be invariant under both leftand right multiplication by elements of the group:d U = d( gU ) = d( U g ) , (56)where g is a generic element of the gauge group. Imposing the normalization condition (cid:90) dU = 1 , (57)the integration measure is uniquely defined to be the Haar measure on the group. Thereader interested in more details can consult any standard reference on compact topolog-ical groups. When integrating over all the lattice link variables, we will use the shorthandnotation d U = (cid:89) x,µ d U ( x, µ ) . (58) Fermions in the functional integral language are represented by Grassmann (anti-commuting)variables { ψ i , ψ j } = { ψ i , ¯ ψ j } = { ¯ ψ i , ¯ ψ j } = 0 . (59)Obviously ψ i = ¯ ψ i = 0, so that any function of Grassmann variables is defined by itsTaylor expansion up to second order. Integrals over Grassmann variables are defined by (cid:90) d ψ = (cid:90) d ¯ ψ = 0 , (cid:90) d ψψ = (cid:90) d ¯ ψ ¯ ψ = 1 , (60)Computing integrals of a function of Grassmann variables f ( ψ, ¯ ψ ) is therefore a problemin combinatorics. When computing the integral over several Grassmann variables we will24se the shorthand notation d ψ = (cid:89) i d ψ i , d ¯ ψ = (cid:89) i d ¯ ψ i . (61)A key role is played by the integrals (cid:90) d ψ d ¯ ψ e − ¯ ψ i M ij ψ j = ( − n ( n − det M , (62) (cid:90) d ψ d ¯ ψ ψ k ψ k · · · ¯ ψ l ¯ ψ l · · · e − ¯ ψ i M ij ψ j = (cid:88) P ( − σ P ( M − ) k a l b · · · ( M − ) k a l b with the sum is over all permutation of the k, l indices.As discussed above, matter fields on the lattice are associated to sites, denoted bythe suffices in the equations above. The fermionic action is quadratic in the fermionfields, and different discretizations can be cast into different choices for the matrix M .A discretized version of the derivative ∂ µ f ( x ) = f ( x + a ˆ µ ) − f ( x ) a , (63) ∂ ∗ µ f ( x ) = f ( x ) − f ( x − a ˆ µ ) a , (64)enters in the non-diagonal elements of this matrix M . Moreover, for fermions minimallycoupled to the gauge field, the discrtized derivative needs to be replaced by its covariantversion. Hence in QCD the matrix M depends on the gauge field configuration U µ ( x )and the mass of the fermions m f . We will use the notation D f = D [ U, m f ] , (65)to denote the lattice Dirac operator for one fermion species, the latter being identifiedby the index f.Although the simulation of Grassmann variables on a computer is possible, it iscomputationally very inefficient. Instead, the previous relation is used to directly definethe path integral of lattice QCD as (cf. Eq. (62)) Z = (cid:90) d U (cid:34) N f (cid:89) f=1 det( D f ) (cid:35) e − S G [ U ] . (66)Note that by integrating out the fermions fields exactly, one is effectively simulating anon-local theory. The Euclidean action for a single free fermion in the continuum reads S F [ ψ, ¯ ψ ] = (cid:90) d x ¯ ψ ( γ µ ∂ µ + m ) ψ . (67)A naive attempt to discretize this action leads to the so-called doubling problem: insteadof describing a single fermion, the lattice action describes 2 fermion flavors. In fact this25henomenon is intimately related with chiral symmetry. In the absence of a mass termthe fermion action is invariant under chiral transformations ψ (cid:55)→ e ıθγ ψ , ¯ ψ (cid:55)→ ¯ ψe ıθγ . (68)This is just a consequence of the kernel of the fermion bilinear being proportional to γ µ . The Nielsen-Ninomiya theorem [44–46] shows that any local lattice hermitian actionthat preserves translational invariance and chiral symmetry describes an equal numberof positive- and negative-chirality fermions. In the case of the naive fermion action,the Nielsen-Ninomiya theorem is satisfied because of the 16 fermions, 8 have positivechirality and the remaining 8 have negative chirality. It is possible to reduce the numberof doublers from 15 to just 1, but in order to describe a single fermion one has to break oneof the hypotheses of the Nielsen-Ninomiya theorem. Giving up locality leads to seriousdifficulties for the renormalization of the theory. Therefore most efforts have focused onfour particular approaches. Wilson fermions
One of the most popular choices follows Wilson’s original proposalto break chiral symmetry at finite lattice spacing by an irrelevant operator [47].This is implemented by adding a dimension 5 term to the Lagrangian that is justa suitable discretization of L = r ψ∂ ψ . (69)The addition of this irrelevant operator has nevertheless an important impact inthe spectrum of the theory by removing all the doublers.There are two unpleasant effects of this extra term in the action. Firstly the fermionmass is no longer protected by chiral symmetry in the regularised theory, andtherefore acquires an additive renormalization. As a consequence the masslesstheory can only be obtained by fine tuning the bare mass in the action. Secondlythe scaling violations are linear in the lattice spacing O ( a ). The massless theorycan be non-perturbatively improved by adding the so called Sheikholeslami-Wohlertterm [48] L (cid:48) = c SW ¯ ψF µν [ γ µ , γ ν ] ψ . (70)If the coefficient c SW is chosen appropriately (see [49]), all remaining linear cutoffeffects are proportional to the quark masses O ( am ). This last term is usuallycalled clover term in the lattice jargon, and the discretization is referred as Wilson-clover fermion action.
Twisted mass fermions
A close relative of Wilson clover fermions are twisted massfermions. In this case one uses the same 5-dimensional operator to break chiralsymmetry, except that in this case the mass term is of the form m + ıµγ τ , (71)where τ is the third Pauli matrix acting in flavor space. Twisted mass latticeQCD always describes multiples of two fermion flavors with a mass given by a These terms can be further eliminated. See [50] for a comprehensive review of the improvementprogramme. m and τ . Note that in the continuum one can always set µ = 0(with the help of a non-anomalous chiral transformation), recovering the usual massterm.On the other hand, at non-zero values of the lattice spacing, the twisted mass termcannot be reduced to the standard Wilson form, because of the explicit breakingof chiral symmetry of the Wilson term Eq. (69). The main advantage of thisformulation is that, for a specific choice of the mass parameter m (called maximallytwisted ) all physical observables are automatically O ( a )-improved [51]. In twistedmass lattice QCD there is no need of tuning the c SW term of Eq. (70). On theother hand, parity and flavor symmetries are broken at finite lattice spacing, and,as already mentioned, one can only simulate an even number of quarks. The readermight be interested in the review [52]. Staggered fermions
One can live with some of the doublers and in fact use them inone’s favor [53, 54]. This is the approach taken by the staggered formulationsof QCD: some of the doublers are used to represent the 4 spin components ofthe fermion. The staggered fermion formulation therefore reduces the amount ofdoublers to 4. The main advantage of this approach is that it preserves an exact U (1) symmetry that is enough to guarantee, among other things, that scalingviolations are proportional to a . Moreover they are computationally cheap.The main drawback of this particular fermion formulation is that the 4 remainingdoublers are degenerate in mass, while the 4 lightest quarks in nature have verydifferent masses. In order to describe single flavors, the staggered fermion formu-lation uses a rooting prescription: the fermion determinant that describes the 4doublers is replaced with its fourth root with the hope that this will describe asingle flavor [55]. This rooting prescription has the unpleasant effect of breakinglocality. It has been argued that locality is recovered in the continuum. If this isactually the case or not has been the subject of many heated discussions in thepast, although the issue has never been completely resolved [56–59]. Domain Wall fermions
One way to circumvent the Nielsen-Ninomiya theorem is torequire that the fermion action, at finite lattice spacing, is invariant under a set ofmodified chiral transformations, ψ (cid:55)→ e iαγ [1 − ζaD ] ψ , and ¯ ψ (cid:55)→ ¯ ψe iαγ [1 − − ζ ) aD ] , (72)where ζ is a free parameter [60]. A lattice Dirac operator that is invariant underthese transformation satisfies { γ , D } = 2 aDγ D . (73)This relation is known as Ginsparg-Wilson relation, and was derived from renormal-ization group arguments in Ref. [61] long before the symmetry above was suggested.One solution of Eq. (73) is the overlap operator obtained in Ref. [62]: aD N = 12 [1 + γ sgn ( H )] , (74)27here H = γ ( D W − M ), where D W is a lattice Dirac operator that has the correctnaive continuum limit, and M ∼ /a is a mass parameter.The operator D N is a representation in terms of four-dimensional fields of five-dimensional domain-wall fermions (DWF) [63]. In DWF formulations, a fermioniczero-mode with definite chirality, localised on the boundary of the (semi-infinite)extra dimension, plays the role of a four-dimensional chiral fermion, while the otherstates decouple from the low-energy dynamics. As it turns out, the five-dimensionalformulations of DWFs presented in Refs. [64, 65] has become the method of choicefor simulating chiral fermions on the lattice. Without entering into a detailedexplanation of the DWF construction on the lattice, we write down explicitly theaction for DWF, and discuss some of its features, while referring the interestedreader to the literature for detailed discussions [66, 67]. Denoting the coordinatein the fitfth direction by s ∈ [0 , ∞ ), the DWF action in Ref. [64] reads S = (cid:88) x,s,µ ¯ ψ ( x, s ) γ µ ∂ µ ψ ( x, s ) + M ¯ ψ ( x, s ) ψ ( x, s ) + r (cid:88) x,s,µ ¯ ψ ( x, s ) ∂ µ ψ ( x, s ) ++ (cid:88) x,s> ¯ ψ ( x, s ) γ ∂ ψ ( x, s ) + r (cid:88) x,s> ¯ ψ ( x, s ) ∂ ψ ( x, s ) ++ 12 (cid:88) x ¯ ψ ( x, γ ψ ( x,
1) + r (cid:88) x ¯ ψ ( x,
0) [ ψ ( x, − ψ ( x, . (75)The index µ runs over the first four Euclidean directions, and the derivatives ∂ µ become covariant derivatives when the fermions are coupled to a four-dimensionalgauge field. In actual simulations the fifth dimension has a finite size, while thelattice chiral symmetry is only recovered when the size of the fifth dimension goesto infinity. In practice a balance must be found between the breaking of chiralitydue to the finite fifth dimension and the growing cost of the simulations as the sizeof this dimension is increased. Quantum Field Theories provide the machinery to evaluate field correlators, i.e. theexpectation value of functions of the elementary fields that appear in the definition of thepath integral. Specific physical properties can then be extracted from these correlatorsby means of dedicated analyses. It is interesting to discuss a few explicit examples inorder to introduce some of the quantities that are used later in this review.Let us begin with a two-point correlator C ΓΓ (cid:48) ( t ) = (cid:88) x , y (cid:104) O Γ ( t, x ) O † Γ (cid:48) ( t, y ) (cid:105) , (76)where O Γ is a quark bilinear O Γ ( t, x ) = ¯ q ( t, x )Γ q (cid:48) ( t, x ) . (77)The matrix Γ determines the spin structure of the bilinear, which we assume to be acolor singlet. In Eq. (77) we have suppressed the flavor indices, for simplicity, we assumethe bilinear to be a non-singlet with respect to flavor transformations.28sing Wick’s theorem, the correlator can be rewritten in terms of quark propagators: C ΓΓ (cid:48) ( t ) = (cid:88) x , y tr [Γ (cid:48) S (0 , y ; t, x )Γ S (cid:48) ( t, x ; 0 , y )] , (78)where S ( x ; y ) and S (cid:48) ( x, y ) are the propagators of the quarks q and q (cid:48) respectively, com-puted as the inverse of the Dirac operator in the gauge background. The expectationvalue is computed by averaging the trace above over an ensemble of gauge configurationsgenerated by Monte Carlo methods.In order to extract physical quantities from a two-point function, we insert a completeset of hadronic states, 1 = (cid:88) n (cid:90) d p (2 π ) E n | n, p (cid:105)(cid:104) n, p | . (79)A few lines of algebra show that the sum over the spatial coordinates implements aprojection over zero-momentum states, and therefore C ΓΓ (cid:48) ( t ) = (cid:88) n E n (cid:104) | O Γ (0) | n, p = 0 (cid:105)(cid:104) n, p = 0 | O Γ (cid:48) (0) † | (cid:105) e − E n t . (80)The two-point function can be trivially extended to project on eigenstates of the spatialmomentum.Eq. (80) shows the main features that allow the extraction of physical observablesfrom field correlators. In Euclidean space the time dependence of correlators is a sumof exponentials, whose decay rates are determined by the energies of the states thathave a non-vanishing matrix element (cid:104) | O Γ (0) | n, p = 0 (cid:105) . For this reason these operatorsare often called interpolating operators . For large time separations, the correlators aredominated by the lowest energy state, and the time dependence becomes a simple expo-nential. The prefactors multiplying the exponential yield various combinations of matrixelements of interest.The so-called effective mass is defined as the time-derivative of the two-point function: M eff ( t ) = − ddt log C ΓΓ (cid:48) ( t ) = E + c e − ( E − E ) t + . . . . (81)For large times t , M eff ( t ) tends to a constant, which is the mass of lowest-energy state.However for the range of separations that can be achieved in practice, it is always impor-tant to check that the contamination from excited states is sufficiently small, or otherwiseunder control, since this is one of sources of systematic error that affect the computationof physically interesting observables.Another interesting example is the computation of the PCAC mass. Specialising theabove interpolating operators to the case of two degenerate light quarks ( u and d ), andfollowing the notation in Ref. [68], we define P ud ( x ) = ¯ u ( x ) γ d ( x ) , and A udµ ( x ) = ¯ u ( x ) γ µ γ d ( x ) , (82)and the two-point correlators C P P ( t ) = (cid:88) x (cid:104) P ud ( t, x ) P du (0) (cid:105) , (83) C AP ( t ) = (cid:88) x (cid:104) A ud ( t, x ) P du (0) (cid:105) . (84)29ollowing the arguments above, C P P ( t ) at large distances is dominated by a a singlepseudoscalar state. Denoting by M PS the mass of the pseudoscalar state, and by G PS itsvacuum-to-meson matrix element, we obtain C P P ( t ) = − G M PS e − M PS t + . . . . (85)Interestingly the ratio m PCAC = (cid:18)
12 ( ∂ + ∂ ∗ ) C AP ( t ) + c A a∂ ∗ ∂ C P P ( t ) (cid:19) /C P P ( t ) , (86)tends to the PCAC mass defined in the continuum theory through the axial Ward iden-tity. The interest in the PCAC mass is two-fold. For fermionic formulations that breakexplicitly chiral symmetry at finite lattice spacing, like e.g. the Wilson fermions describedabove, the bare parameters in the action need to be tuned to approach the chiral limit.In particular an additive renormalization of the bare fermion mass is required. The chiraltheory is defined by requiring the PCAC mass to vanish. After renormalization, the rateof convergence of m PCAC will be proportional to the lattice spacing a if the theory is notimproved, while it becomes proportional to a when c SW in Eq. (70) and c A in Eq. (86)are properly tuned.Note that the decay constant of the PS state, defined as (cid:104) | A µ (0) | PS (cid:105) = iZ A F PS p µ , (87)can be computed directly from fitting C AP ( t ) and C P P ( t ), which yield the matrix ele-ments (cid:104) | A (0) | PS (cid:105) and (cid:104) | P (0) | PS (cid:105) . Alternatively, the decay constant can be evaluatedfrom the quantities defined above using F PS = Z m m PCAC M G PS . (88)It is interesting to remark that the two definitions differ by lattice artefacts. While ingeneral this is not necessarily a cause for concern, it shows that definitions that areequivalent in the continuum limit do differ at finite lattice spacing. This is something tokeep in mind in choosing observables when aiming for high precision measurements. Any lattice QCD computation has several sources of systematic uncertainties that haveto be kept under control in order to be able to quote accurate results. Since lattice QCDis a first principle definition of QCD, these sources of systematic uncertainties reflect thecurrent limitations in computer power, or in our knowledge of efficient algorithms. Sinceboth computer power and our knowledge of efficient algorithms are constantly improving,lattice QCD is able to solve now problems that were basically impossible just a few yearsago. The renormalization constants Z A and Z m need also to be computed. We point the reader to thereviews [50, 69] for more details on the topic. L/a ) × ( T /a ) simulated (the lattice volume). The com-puter cost increases at least linearly with this lattice volume, and sets a basic compromisebetween simulating a large physical volume, and a small lattice spacing. Computing re-sources also limit the range of quark masses that can be simulated, even though it hasbecome common to have simulations with physical quark masses. In summary we havethe following main sources of systematic error.
Finite volume corrections:
QCD quantities determined on a large but finite boxsuffer from finite volume effects. These are exponentially suppressed with thesmallest mass present in the spectrum of the theory (i.e. the pion M π ) [70] ∝ e − M π L . (89)Usually M π L > M π L > e.g. meson decay constants) chiral perturbation the-ory yields an estimate of the size of the finite volume corrections, allowing tosubtract them from the data in some cases.Since one typically uses a short distance observable to determine the strong cou-pling, these determinations are normally affected very little by finite volume effects.Nevertheless every determination of the strong coupling needs a determination ofthe scale (see section 4.5.2), where finite volume corrections can be substantial.
Continuum extrapolation:
All determinations on the lattice require a continuumextrapolation to reproduce QCD results. Lattice artefacts are small only if we canachieve a significant separation between the scales at which observables are definedand the lattice cutoff. We will examine in detail the process of taking the continuumlimit in section 4.5. Here we just mention that this point is particularly delicate forthe determinations of the strong coupling. As we try to use observables computedat high energies in order to define the strong coupling, we necessarily need to facethe issue of larger cutoff effects. Typical current large volume simulations use a ≈ . − . Chiral extrapolation:
Many lattice QCD computations used to be performed at non-physically heavy values of the quark masses. There are two reasons for this. First,lattice QCD simulations become more expensive at lighter quark masses. The gapin the spectrum of the Dirac operator depends on the mass of the lightest quark inthe simulation. This has the effect of making simulations close to physical values ofthe quark masses computationally very expensive. Second, close to physical valuesfor the quark masses finite volume effects are larger, and therefore there is an extracost due to the need to simulate in larger physical volumes.For example, simulating physical values for the quark masses ( M π ≈
135 MeV)with sub-percent finite volume effects M π L = 4 and a very fine lattice spacing a ≈ .
05 fm requires a lattice with
L/a ≈
120 points in each direction. At the timeof writing this report this is right at the edge of current capabilities for most choices31f lattice action. Algorithmic developments have made it possible to simulatedirectly at the physical point, but these physical regimes of the parameters areusually simulated on coarser lattices, making the chiral extrapolation a crucialingredient of any lattice QCD computation.
Any lattice action has N f + 1 free parameters: the bare quark masses in lattice units am , and the bare coupling g (usually the lattice community uses β = 6 /g as inputparameter for the simulations). While the role of the bare quark masses is clear (theydirectly affect the values of the quark masses), the role of the bare coupling is less obvious.In fact the bare coupling is tuned in order to approach the continuum limit. Naively thecontinuum limit amounts to take a →
0. But in the lattice action there is nowhere anyreference to the lattice spacing a (or any other parameters with dimensions), raising thequestion about how to actually take the continuum limit.Since all lattice input parameters are dimensionless, lattice QCD by itself only givespredictions of dimensionless quantities. For example, the study of the proton correlatoryields the proton mass in lattice units aM p . The key idea in order to make contact withphysical, dimensionful, quantities is to choose one quantity as a reference scale. Everyother dimensionful quantity is computed in units of this reference scale. For example,one can take as reference scale the proton mass M p . Any other quantity, say for instancethe Ω baryon mass is measured in units of this reference mass – in practice in a latticesimulation we determine the dimensionless ratio ( aM Ω ) / ( aM p ) = M Ω /M p .If we focus on the case of N f = 2 + 1 simulations (two degenerate light quark massesplus the strange quark), a prediction for the value of M Ω would conceptually proceed asfollows:1. Choose a value of the bare coupling g . Measure the values of the masses of the π and K mesons and the reference scale in lattice units (i.e. aM π , aM K , aM p ). Tunethe bare quark masses such that the ratios of the pseudo goldstone bosons massesto the reference scale ( aM π ) / ( aM p ) and ( aM K ) / ( aM p ) are equal to the physicalvalues M exp π /M exp p and M exp k /M exp p , see e.g. the values reported by the PDG [71].This procedure fixes the values of the bare quark masses am for each choice of g .The lattice spacing is then a = ( aM p ) /M exp p , where the numerator is the output ofthe numerical simulation, and the denominator is the reference scale.2. Repeat the process for several values of g . The final prediction has to be takenas the limit where the reference scale in lattice units is much smaller than one, i.e. aM p (cid:28) M Ω M p = lim aM p → aM Ω aM p . (90)By using the experimental value of the proton mass M p ≈
940 MeV, this lastprediction of a dimensionless ratio can be translated in a prediction for M Ω .The first step sets up a line of constant physics (LCP). It requires one experimentalinput per quark mass. Pseudo-goldstone bosons are the natural candidates, since theirmass depends strongly on the values of quark masses. The value of the reference scale( M p in the example) is an extra input, used to convert lattice dimensionless predictions in32imensionful quantities. Note that although usually one takes the values of the quantitiesto fix from experiment, one can follow the same procedure to set up a line of constantphysics at arbitrary non-physical values of the quark masses (for example to investigate aworld with mass degenerate u, d, s quarks): lattice QCD also allows to make unambiguous predictions of dimensionless quantities for unphysical values of the quark masses. Thesecond step takes the continuum limit . Different LCP (for example by choosing a differentreference scale) would result in different approaches to the same universal continuumvalues .These two steps together define a non-perturbative renormalization scheme for thetheory. The bare parameters are tuned in order to reproduce some physical world (valuesof the quark masses and reference scale) and quantities are computed in the limit wherethe UV cutoff (1 /a ) is much larger than the energy scales of interest ( M p , M Ω , . . . ).Asymptotic freedom can be used as a guidance in order to take this continuum limit. Atshort distances the theory is weakly coupled, which suggests that a series of decreasingvalues of the bare coupling g will successively approach the continuum limit.Even though the procedure sketched above is perfectly correct from a conceptualpoint of view, it misses some fine details that are crucial in order to obtain the precisionachieved nowadays. Let us briefly mention them here. • As mentioned above lattice QCD simulations become very expensive at the bareparameters that correspond to the physical values of the quark masses. There aretwo reasons for this. First is the increasing numerical cost of simulating quarks atsmall values of the bare quark mass. Second, lighter values of the quark massesresult in lighter values of the π meson masses. Since M π L is the quantity thatdictates the size of finite volume effects, simulations at lighter quark masses requiresto simulate larger physical volumes. In many practical situations the physical pointis only reached by extrapolating from simulations at heavier quark masses, althoughthis situation is changing fast with the increasing computer power and improvedsimulation algorithms. • The experimental values of the physical hadronic quantities are affected by theelectromagnetic interactions. This has implications for the determination of theLCP, since the use of the experimental values as input for a lattice QCD computa-tion has to be done with care, usually correcting them for isospin breaking effects.One has to make sure that the size of the electromagnetic effects has a negligibleeffect in the determination of physical quantities. When this is not the case, a first-principles prediction requires to simulate both the strong and the electromagneticinteractions in order to make contact with the physical world. We anticipate herethat for the case of the the determination of α s , isospin breaking effects are notparticularly relevant at the current level of precision (cf. section 7). Since short-distance observables are more susceptible to show large cutoff effects, thecontinuum extrapolation plays a key role in the extraction of the strong coupling. Here wewant to show in some detail how to asses the quality of these continuum extrapolations.Most lattice QCD simulations choose an improved discretization where the leadingscaling violations that we discussed above are O ( a ). This is achieved either by choosing33 fermion formulation where any O ( a ) violations are forbidden by symmetry arguments(like domain-wall fermions, twisted mass at maximum twist, or staggered fermions), orby tuning the parameters of the action (the so-called Wilson-clover fermions). In thissituation any dimensionless quantity D has an asymptotic expansion of the form D a → ∼ D cont + . . . , (91)where the dots represent the scaling violations, with an asymptotic expansion with lead-ing term O ( a ). But the functional form of these scaling violations is in general verycomplicated. The corrections include logarithmic terms of the form (i.e. are of the form a n log k a ). How does these generic statements affect in practice the extrapolation of lat-tice data? It is clear that corrections need to be small in order for the precise functionalform to have a negligible effect on the extrapolation. . . . . . . . . . . . .
001 0 .
002 0 .
003 0 .
004 0 .
005 0 .
006 0 . D a Fit D cont + O ( a ) Fit [1 /D cont + O ( a )] − Set 1Set 2Extrapolations
Figure 5: Continuum extrapolation of a dimensionless quantity. Without entering in the details of thedefinition of the specific quantity in the plot, it is worth noting that these data correspond to actuallattice simulations, see [72] for more details.
In order to be more precise it is best to look at one example. Figure 5 shows thecontinuum extrapolation of a dimensionless quantity using two different discretizations.These are labeled 1 and 2 respectively in the legend. The extrapolations of both datasetshave very different scaling violations, and this has an impact in the accuracy of thecontinuum value. These differences are:
Range of lattice spacings:
Dataset 1 includes simulations that span a factor a max /a min ≈ . a max /a min ≈ . . ize of the extrapolation: Dataset 1 has very small scaling violations: the finestlattice spacing is compatible within errors with the continuum value. On the otherhand dataset 2 shows large scaling violations: the data at the finest lattice spacingis about 5 standard deviations away from the extrapolated value.These two points are crucial to understand the assumptions that are behind the extrap-olation of these two datasets. What we need is a way to measure the impact of the terms beyond the leading O ( a ) scaling violations. In order to do so, it is useful to considernot only the extrapolation of D , but also the extrapolation of different functions f ( D ).In principle this function seems completely redundant, since f ( D ) a → ∼ f cont + . . . , ( f cont = f ( D cont )) , (92)one can, in principle, recover the same D cont from any choice of function by using D cont = f − ( f cont ) . (93)However, when the range of lattice spacings available is not very large and/or the ex-trapolation is quite sizable, the choice of function can change substantially the result ofthe extrapolation. This is illustrated in figure 5, with two simple choices f ( x ) = x (i.e.extrapolating the data itself) and f ( x ) = 1 /x (i.e. extrapolating the inverse of the data).In set 1, these different choices result in practically identical values for the extrapolatedvalue. On the other hand in set 2 the different choices of f result in extrapolations thatdiffer by several standard deviations. Obviously the difference between these two choicesfor the extrapolation are higher order terms, since1 A + Ba = 1 A − BA a + B A a + . . . . (94)These results suggest that the dependence of the extrapolation on the choice of function f is equivalent to a dependence on the non-leading scaling violations . It is important tonote that discriminating among different functions cannot always be done by looking atthe quality of the fit. In the quoted example of figure 5 the fits for the two choice of f are equally good.Another important point worth mentioning is that this kind of analysis usually onlyexplores the non-leading power corrections ( i.e. a in the example). Logarithmic cor-rections are much more difficult to estimate, although one should be very careful withneglecting them (the reader interested in this topic will enjoy the puzzle described inRef. [73], and the recent discussion in the context of lattice QCD [74]). These top-ics are also very nicely discussed in a recent contribution to the Lattice Field TheorySymposium [41], a reference that we recommend to the reader will enjoy.Summarising these results, it is crucial to remember that any extrapolation is basedon some assumptions. In the case of the continuum extrapolations, these assumptionsdepend crucially on three characteristics of the dataset: the number of lattice spacingssimulated, the ratio of the coarser and the finest lattice spacings, and the size of theextrapolation. This process of taking the continuum limit is usually seen by the lattice community froman equivalent perspective, which comes under the name of scale setting . As discussed35bove some reference scale M ref is used to determine the value of the lattice spacing a . This is simply done by setting up a line of constant physics and declaring that a ≡ ( aM ref ) M expref , (95)where the parentheses emphasise that ( aM ref ) is the quantity that is actually computedin a lattice simulation, and M expref is the experimental value of the reference scale. Nowany other quantity Q with units of mass can be determined by the expression Q = lim a → ( aQ ) a . (96)Obviously this is just a rephrasing of Eq. (90). Note that determinations of a are in-trinsically entangled with the LCP and the experimental quantity that is used to setthe scale (the particular choice of M ref ). Different LCP and/or physical quantities willresult in different values for the lattice spacing. This is however not a problem since any such determinations will give the same predictions for any physical quantity afterthe continuum limit is taken . This procedure explains the usual jargon of the field: oneexperimental input is used to determine the lattice spacing.Scale setting is a key ingredient in any lattice determination of the strong coupling.The Λ parameter is a quantity with units of mass, and therefore the error in the scaletranslates into an equal relative error in the determination of Λ. What is more important any unaccounted systematic in the determination of the scale propagates into an effectof the same relative size to Λ. What quantities are used as reference scales? Ideallyone would like some quantity that has a clean and precise experimental determination,and that can be determined with high precision and accuracy on the lattice. The readermight be surprised to read that no such clean quantity exist. For example the abovementioned proton mass, that would look as a natural quantity, has a large signal to noiseproblem (see appendix A), especially close to the physical point. In practice differentquantities are used, with each choice having pros and cons. Let us briefly review thecharacteristics of some of them.
Meson decay constants ( F π , F K ) These quantities are clean from a lattice point of view, which explains why they area popular choice in the lattice community. They are determined from meson 2-pointfunctions, as discussed above. For example F π is obtained thanks to the relation (cid:10) | ¯ uγ γ d (0)[¯ uγ γ d ( x )] † | (cid:11) ∼ aM π ( aF π ) e − aM π x + . . . , (97)which is free from the infamous signal-to-noise problem, leading to these decay con-stants in lattice units aF π , aF K being determined with a precision of a few permille.One problem is that the chiral corrections on the decay constants (especially F π ) is notsmall. Recent determinations are performed at values of the quark masses very closeto its physical values, so in principle this has become a lesser issue in state of the artcalculations. For the sake of the presentation we will assume that the reference quantity has units of mass.
36n the other hand decay constants are not that clean from the experimental pointof view. First, there is a theoretical issue in the definition of these quantities. They areunambiguous in QCD, but the electromagnetic interactions render these quantities illdefined – the quantity in Eq. (97) is not even invariant under a U (1) gauge transformation.The actual experimental observable is the photon-inclusive decay rate Γ P l , defined asΓ P l = Γ( P → lν ) + Γ( P → lνγ ) , (98)with P = K ± , π ± . Most lattice QCD calculations compute the isospin symmetricquantity F P which parametrizes the decay of the pseudoscalar in a world where electro-magnetic interactions are switched off, and isospin is unbroken:Γ( P → lν ) (cid:12)(cid:12)(cid:12) α EM =0 ,m u = m d = G F | V P | F P π M P m l (cid:20) − m l M P (cid:21) . (99)In order to relate these two quantities we need to estimate the EM effects. The masterformula, as discussed in Ref. [75], isΓ P l = Γ( P → lν ) × [1 + δ EW ] × (cid:0) δ P EM (cid:1) , (100)where the first bracket term is the universal electroweak correction δ EW = 0 . ≈ α EM π log (cid:18) m Z m ρ (cid:19) , (101)a short distance contributions that affects all semileptonic charged current amplitudeswhen expressed in terms of the Fermi constant. Finally the δ P EM piece can be written as δ P EM = − α EM π (cid:26) − F (cid:18) m l m P (cid:19) + 32 log (cid:18) m ρ m P (cid:19) + c P + m l m ρ (cid:20) c P log (cid:18) m ρ m l (cid:19) + c P + c P (cid:18) m l m P (cid:19)(cid:21) − ˜ c P m P m ρ log (cid:32) m ρ m l (cid:33)(cid:41) . (102)In this expression, the first term is the universal long distance electromagnetic correction,computed to one loop assuming that the π, K are point-like particles, while the termsproportional to c P , , , , ˜ c P parametrize the structure dependent part, c P being the leadingone. A conservative estimate of the electromagnetic uncertainties consists in taking thefull difference between the point particle approximation – where all c Pi , ˜ c Pi are zero –and the best phenomenological determination available in the literature, based on χ PT.This gives an uncertainty ∼ .
27% for the case of F π and ∼ .
21% for F K . Thesefigures set a conservative limit on the precision that can be quoted for these decayconstants, and therefore a limit on the precision of the scale determination for any latticecomputation that relies on them to set the scale. Going beyond this precision requiresthe inclusion of electromagnetic effects on the lattice and a direct computation of thedecay rate, something that nowadays is challenging from the theoretical point of view,but an impressive progress has been achieved recently [76, 77]. ... albeit not all! F K one has also to take into account the strong isospinbreaking effects, which vanish at leading order for the case of F π . This is only a practicalproblem, since the leading corrections ∝ ( m u − m d ) can in principle be determined.Finally, the relation between the experimentally measured decay rates and the decayconstants involves some CKM matrix elements. In the case of F π the relevant term is V ud , that is very well determined experimentally from super-allowed β -decays, but thecase of F K needs a determination of V us , which usually involves lattice input and assumesCKM unitarity relations.All in all, the pion decay rate ( F π in pure QCD) remains an attractive quantity forscale setting, especially nowadays that we can simulate quark masses close to their phys-ical values. It can be determined very accurately on the lattice. The model dependentelectromagnetic corrections are below the 0.3%, the leading strong isospin breaking effectsvanish, and the necessary CKM matrix element is cleanly determined experimentally. The Ω mass The mass of the Ω − baryon is also a common choice for scale setting. This particleis stable under the strong interactions and its mass is known very precisely. Being madeof three strange valence quarks, the dependence on the value of the light quark masses isonly induced via loop effects, which translates in a mild chiral extrapolation if simulationsare performed at constant value of the strange quark mass.Contrary to the case of the decay constant, what is measured by the experimentis directly related with what is measured on the lattice. Strong isospin breaking andelectromagnetic corrections on the Ω mass are also small, with some recent lattice studiespointing to corrections below the 0.3%.Unfortunately the determination of the Ω − mass on the lattice is challenging. Likeall baryons, the Ω − is affected by the signal-to-noise problem (see appendix A).The current precision in the scale determined from M Ω ranges from 2% to 0 . ≤ .
5% isospin breaking corrections, which makes it an attractive quantity for scalesetting. Nevertheless, very precise determinations of the scale require to control theexcited states contamination in a correlator affected from a strong signal to noise problem.How to achieve this in practice without assumptions on these excited states is currentlyanother hot research topic.
An ideal quantity to be used as a reference scale must have some particular characteristics.First it must have a weak dependence on the quark masses. Having a simple chiraldependence is crucial for those lattice QCD simulations that are performed at unphysicalvalues of the quark masses, and reach the physical point only by extrapolation. Second,38he quantity must be clean from the computational point of view. A quantity that iscomplicated to compute on the lattice, as a result of several extrapolations or someinvolved fits of the lattice raw data are better left as predictions, and not as referencescales. Third, the quantity must have a clear experimental determination, ideally with aweak dependence on strong isospin breaking effects.In the previous sections we have seen the typical quantities used for scale setting(decay constants and M Ω ), and the pros and cons of each choice. There exists interestingalternatives, with a very weak chiral dependence and that are straightforward to computefrom the lattice point of view. The drawback is that they are not quantities that canbe accessed by experiments (hence the name “theory scales”). Nevertheless they arevery useful as intermediate reference scales (i.e. to “determine the lattice spacing” as inEq. (95)). Scales derived from the static potential
One example is the theory scale r , which is derived from the force between staticquarks, as discussed in section 6.2. This force F ( r ) has dimension of mass squared, andtherefore the quantity r F ( r ) , (103)is a dimensionless function of the distance between the static quarks. This suggests todefine a reference scale r by the condition [78] r F ( r ) (cid:12)(cid:12)(cid:12) r = r = 1 . . (104)The particular value 1 .
65 is chosen so that r ∼ . r ∼ . r F ( r ) = 1 is also commonly used in order to improvethe precision, at the cost of larger cutoff effects [79].The extraction of the scales r , r from lattice data is not completely free of challenges.The quantity from which r , r is extracted has a divergent signal-to-noise ratio whenone approaches the continuum, these challenges have been overcome by a combinationof several techniques (see [80] for an overview). At the current values of the simulationparameters a precision <
1% can be achieved in these quantities.The advantages of scales derived from the static potential are clear. Being gluonicquantities, their dependence on the value of the quark masses is very small. The chiraldependence of these quantities is very mild. Even if its extraction is not completelytrivial , the lattice community has vast experience determining the static potential. Scales derived from the gradient flow
Recently even better theory scales have been proposed. They are derived from thegradient flow [81, 82], a diffusion like process for the gauge field in a fictitious timecoordinate t called flow time. The gauge field evolves in flow time according to theequation ∂ t B µ ( t, x ) = D ν G νµ ( t, x ) , B µ (0 , x ) = A µ ( x ) , (105) In fact the literature has seen discrepancies in the values of these scales, although it is not clearthat the fault was the evaluation of the static force, and not the conversion of r , r to physical units.See [80]. D µ = ∂ µ + [ B µ , · ] is the covariant derivative with respect to the field B µ , and G µν = ∂ µ B ν − ∂ ν B µ + [ B µ , B ν ] , (106)is the corresponding field strength tensor. Note that here x denotes the four-dimensionalspacetime coordinates, while the flow time t has units of length squared. The field B µ ( t, x ) can be seen as a smoothed version of the original gauge field A µ ( x ) over a lengthscale ∼ √ t . Gauge invariant quantities constructed from the flow field B µ ( t, x ) do notneed renormalization at t >
0, beyond the usual renormalization of the bare parametersof the Lagrangian [83]. For example, the action density (cid:104) E ( t, x ) (cid:105) = − (cid:104) tr { G µν G µν ( t, x ) }(cid:105) (107)is a renormalized observable - i.e. it has a finite continuum limit. By dimensional analysis,the quantity t (cid:104) E ( t, x ) (cid:105) is dimensionless, but its value depends on the scale √ t . Similarto what is done for r , a convenient scale can be defined by the condition t (cid:104) E ( t, x ) (cid:105) (cid:12)(cid:12)(cid:12) t = t = 0 . , (108)which results in a hadronic scale ( √ t ∼ . t is the w scale [84], initially introduced with the aim of reducing the size of the cutoff effects in t and defined by the condition t dd t t (cid:104) E ( t, x ) (cid:105) (cid:12)(cid:12)(cid:12) t = w = 0 . . (109)Flow scales have many advantages. First, as for the case of r , r they are simplegluonic observables, so that their chiral dependence is very mild. But in contrast withthe scales r , r they are given directly by an expectation value. Their computationin lattice simulations only involves integrating the flow equation (105), something thatcan be done in practice with arbitrary precision. There is no need to look to the largeEuclidean time behavior of a correlator, to perform any fit or to deal with any signal tonoise issue. Moreover flow observables have a very small variance, making the statisticalerrors in the computation of such quantities very small. Recent scale determination of t in the pure gauge case have reached a precision ∼ .
2% in very fine lattice spacings [85].Of course the drawback of any of these theory reference scales is that ultimately theyneed to be computed in terms of a real experimental observable if one aims at makinga full prediction. But they are invaluable as intermediate reference scales, especially ifone takes into account that they can be quoted at quite unphysical values of the quarkmasses.
As was discussed at the beginning of this section, numerical lattice QCD is based onthe fact that the path integral, after discretization, is an integral in a large, but finite,dimensional space Z latt = (cid:90) e − S [ U ] d U . (110)40n typical state of the art current simulations, expectation values are integrals in d ≈ dimensions. They are computed with a sub-percent precision using a few ( N ∼ O (1000))gauge fields ( U (1) , . . . , U ( N ) ) that are drawn with probability distributiond P ( U ( k ) ) ∼ e − S [ U ] Z latt d U , (111)where d U is the Haar measure on SU (3).Drawing representative ensembles in lattice QCD uses the techniques of Markov chainMonte Carlo. For the case of the pure gauge theory very efficient local link update al-gorithms exists, but almost every lattice QCD simulation is performed with some vari-ant of the Hybrid Monte Carlo (HMC) algorithm [86]. Once a representative ensemble { U ( k ) } Nk =1 is available, estimates of any observable are determined by averaging over theensemble (cid:104) O (cid:105) = 1 N N (cid:88) k =1 O ( U ( k ) ) + O (cid:16) / √ N (cid:17) . (112)A crucial step in any lattice QCD work is the estimate of the statistical uncertainty. i.e. how much does the estimate ¯ O = 1 N N (cid:88) k =1 O ( U ( k ) ) (113)deviates from the exact value of the expectation value (cid:104) O (cid:105) . An estimate of this uncer-tainty δ ¯ O is given by the variance of the mean( δ ¯ O ) = Var (cid:34) N N (cid:88) k =1 O ( U ( k ) ) (cid:35) . (114)There are two key points in estimating this uncertainty1. The properties of Markov chains ensure that the variance of each of the terms in thesum Eq. (114) is the same for all samples U ( k ) and in fact given by an expectationvalue with the same probability distributionVar (cid:104) O ( U ( k ) ) (cid:105) = (cid:104) ( O − (cid:104) O (cid:105) ) (cid:105) = σ . (115)2. Any Markov chain Monte Carlo algorithm works by producing the next sample( U ( k +1) ) from the current one ( U ( k ) ). Subsequent measurements of an observable O ( U ( k ) ) are correlated . Note that this correlations have the unpleasant effect ofincreasing the uncertainties: the error estimate of an observable (Eq. (114)) is givenby ( δ ¯ O ) = σ N N (cid:88) i>j Γ( i − j ) σ . (116)where Γ( i − j ) = Cov (cid:16) O ( U ( i ) ) , O ( U ( j ) ) (cid:17) . (117)41he first term in the bracket of Eq. (116) accounts for the error estimate if the datawere uncorrelated. The second term is due to the correlations. Usually the previousformula for the error of an observable is written using the integrated autocorrelationtime τ int = 12 + 1 N (cid:88) i>j ρ ( i − j ) , (cid:18) ρ ( t ) = Γ( t ) σ (cid:19) , (118)as ( δ ¯ O ) = σN (2 τ int ) . (119)It is clear that τ int = 1 / τ int for a particular observable O , the larger the uncertainty for someensemble of fixed length N . The problem is that τ int has to be estimated from the dataitself. The sum in Eq. (118) has no upper limit ( i − j ) → ∞ , while in practice it hasto be truncated at some finite value. This means that the estimated value of τ int willnaively be systematically lower than the correct value (see Fig. 6).This is not a mere academic observation. The properties of Markov chain MonteCarlo ensures that the autocorrelation function Eq. (117) is a sum of exponentialsΓ( t ) = (cid:88) k A k e − t/τ k −−−→ t →∞ A (cid:48) e − t/τ exp . (120)At large MC times t → ∞ , the dominant contribution to the autocorrelation functionis given by the slowest mode of the Markov operator. This is usually called exponentialautocorrelation time ( τ exp ). Clearly the number of measurements must be large comparedwith τ exp in order to have sensible error estimates and ensure the ergodicity of thesimulation. What do we know about τ exp that is relevant for the lattice determinationsof the strong coupling?1. At fixed physical volume, one expects τ exp to increase proportional to 1 /a . Atfine lattice spacing, one need large statistics in order to estimates the uncertaintiescorrectly .2. The values of τ exp are not very sensitive to the fermion masses. In fact evenpure gauge simulations show similar values of τ exp as simulations with dynamicalfermions when similar algorithms are used.3. A reasonable estimate of the order of magnitude for τ exp can be made by tak-ing τ exp ∼
70 MDU at a ≈ simulations without topology freezing (see [89, 90]). Values at finer lattice spacing can be estimated using an approxi-mate a scaling. This simple estimate of the order of magnitude is already tellingus that estimating statistical uncertainties for a ∼ .
03 fm, where τ exp ∼
350 re-quires substantial statistics ( i.e. one would not feel comfortable with less than4000MDU’s). In practice the situation might be even more delicate due to a phenomena called topology freezing .see appendix A and the original works [87, 88]. MDU stands for Molecular Dynamics Unit, and measure simulation “Monte Carlo” time in HMCsimulations. The typical spacing between measurements in realistic simulations is between 1 and 2MDU’s. A simulation of 1000 MDUs would allow to have between 500-1000 measurements. . . . . .
81 0 20 40 60 80 100 ρ ( t ) t ρ ( t ) Ensemble 1 ρ ( t ) Ensemble 2
Figure 6: Two replica of a Monte Carlo simulation with τ exp = 100. Ensemble 1 has length 20000, whileensemble 2 has length 500. A long Monte Carlo run allows to determine the normalized autocorrelationfunction ρ ( t ) ≡ Γ( t ) /σ more precisely (see Eq. (117)). This results in a more solid estimate of thestatistical uncertainties. Let us end this section commenting that the most common analysis techniques inlattice QCD involves binning the data: the Monte Carlo measurements are averaged ingroups of size N bin . The data bins are treated as independent measurements and a naiveerror estimate is performed (usually by resampling). This alternative analysis methoddoes not improve the determinations of the statistical uncertainties over the methodsthat directly determine the autocorrelation function. It is clear that bins of data are lesscorrelated than the data itself, but it has been shown that the decrease in the correlationsis slow ∼ /N bin [91]. Moreover, in the fairly common case that N bin cannot be takenmuch larger than the exponential autocorrelation time τ exp , there are no known methodsto explicitly include the slow modes of the Markov operator in the binning analysis. Onthe other hand a direct analysis of the autocorrelation function allows to include theseeffects in the error estimates [92, 88] (see also the summary in [93]).In summary, it is important to point out that in contrast with other numerical fieldswhere the number of MC samples is very large, lattice QCD simulations are performedin the uncomfortable situation that the number of samples is not much larger than therelaxation time of the Markov operator. In this situation the estimates of statisticaluncertainties can be challenging. This observation is especially relevant for the deter-minations of the strong coupling, since τ exp scales like 1 /a , and fine lattice spacing areneeded in order to study observables at short distances.43 . Decoupling of heavy quark and matching accrossthresholds Following a Wilsonian approach to field theory, it is natural to imagine that the low-energy dynamics is not sensitive to the details of the theory at high energies – thehigh-energy degrees of freedom are integrated out . More precisely, when consideringobservables defined at some low-energy scale µ , the effects of heavy states of mass m areexpected to be encoded in a redefinition of the couplings, or suppressed by powers of µ/m in the limit where µ/m (cid:28) on-shell renormalization scheme, and denote therenormalized mass and coupling constant ¯ m os and ¯ g os respectively. The gluon propagator,the three-gluon vertex and the fermion propagator are given respectively by D abµν ( k ) = δ ab k (cid:18) g µν − k µ k ν k (cid:19) d (cid:18) k µ , ¯ m µ , ¯ g os ( µ ) (cid:19) , (121) i Γ abcµνσ ( p, q, r ) = f abc (cid:104) ( p − q ) µ g νσ + ( q − r ) ν g σµ + ( r − p ) σ g µν (cid:105) ×× G (cid:18) k µ , ¯ m µ , ¯ g os ( µ ) (cid:19) , (122) S ( p ) = 1 p − m (cid:20) a (cid:18) p µ , ¯ m µ , ¯ g os ( µ ) (cid:19) /p + b (cid:18) p µ , ¯ m µ , ¯ g os ( µ ) (cid:19) m (cid:21) , (123)where in Eq. 122, the 1PI vertex is evaluated at a symmetric point p = q = r = k .The mass counterterm is adjusted by imposing that the fermion propagator has a polefor p = ¯ m , and the remaining counterterms are defined by providing renormalizationconditions at the scale µ , e.g. d (cid:18) − , ¯ m µ , ¯ g os ( µ ) (cid:19) = 1 , (124) G (cid:18) − , ¯ m µ , ¯ g os ( µ ) (cid:19) = 1 , (125) a (cid:18) − , ¯ m µ , ¯ g os ( µ ) (cid:19) = 1 . (126)44he running coupling constant at a generic scale k is defined as¯ g os (cid:18) k µ , m µ , ¯ g os ( µ ) (cid:19) = ¯ g os ( µ ) G (cid:18) k µ , m µ , ¯ g os ( µ ) (cid:19) d (cid:18) k µ , m µ , ¯ g os ( µ ) (cid:19) / , (127)and its dependence on the scale is described by the beta function: k ddk ¯ g os = β (cid:18) ¯ m − k , ¯ g os (cid:19) . (128)The specific value of the coupling at the scale µ that was used in the renormalizationprocess serves as the initial condition for integrating the beta function. Note that, aspointed out in the original derivation in Ref. [94], the beta function depends on the renor-malized mass ¯ m os through the ratio ¯ m /k – this is a direct consequence of working inan on-shell scheme, where the mass of the particle enters explicitly in the renormalizationconditions and therefore in the running of the coupling constant.In this specific scheme, Appelquist & Carazzone show that diagrams containing heavypropagators are suppressed by powers of k/ ¯ m os or µ/ ¯ m os , where k is the typical scaleof the external momenta of the diagram. The second, important, result is that, in thedecoupling limit, the beta function of the theory in Eq. 128 reduces to the beta functionof the theory where the heavy particle has been integrated out, i.e. in this particularcase the beta function of pure Yang-Mills theory; the effects of the heavy states simplyshow up as power corrections that interpolate between the two theories. It was noted in Refs. [95, 96] that the decoupling theorem does not apply in minimalsubtraction (MS), since all loops contribute to the beta function independently of themass of the state. The same problem exists for generic mass-independent schemes [97].The solution to this problem is found by matching at low energies the theory with heavyparticles to an effective theory containing only the light degrees of freedom, i.e. bytuning the couplings of the effective theory so that it reproduces the field correlators ofthe full theory at low energies up to corrections that are suppressed by the ratio of thescales that characterise the light and heavy degrees of freedom respectively. Matchinggauge theories across thresholds is first discussed in Ref. [18], then analysed in detail inRefs. [98, 19]. It has become the method of choice to define the coupling constant atenergies that span a wide range and hence cross several mass thresholds.Once again we find it useful to give a pedagogical review of the main steps in theprocedure. We consider a theory with n f light fermions and only 1 heavy fermion ,whose dynamics is specified by a lagrangian L and a set of bare couplings and fields: g, m, m h , ξ, ψ, A µ , c . (129)Using a familiar notation for QCD, g is the gauge coupling, m the mass of the lightfermions, m h the mass of the heavy fermion, and ξ is the gauge parameter. The fields ψ, A µ , c describe the fermions, the gauge field and the ghosts respectively. The argument can be readily extended to the case where more than one particle is integrated out. L (cid:48) which only involves the light degrees of freedom, and a set of rescaled couplings andfields. Following Ref. [99], the bare couplings and fields of the effective theory are denotedby primed letters, and are connected to the bare couplings and fields of the full theorythrough the so-called decoupling constants ζ i : g (cid:48) = ζ g g , m (cid:48) = ζ m m , ξ (cid:48) − ζ ( ξ − , (130) ψ (cid:48) = (cid:112) ζ ψ , A (cid:48) µ = (cid:112) ζ A µ c (cid:48) = (cid:113) ˜ ζ c . (131)It is easy to argue on symmetry grounds that L (cid:48) must have the same form as L , butcontain only the light degrees of freedom: L (cid:48) ( g, m, ξ, ψ, A µ , c ; ζ i ) = L (cid:0) g (cid:48) , m (cid:48) , ξ (cid:48) , A (cid:48) µ , c (cid:48) (cid:1) . (132)Higher-dimensional operators can appear in L (cid:48) , but are suppressed by inverse powersof the heavy mass. The decoupling constants ζ i are determined by computing fieldcorrelators in both theories, and matching them up to power contributions.The matching procedure yields the relation between the renormalised coupling ¯ α inthe two theories: ¯ α (cid:48) ( µ ) = (cid:18) Z g Z (cid:48) g ζ g (cid:19) ¯ α ( µ ) = ζ Rg ¯ α ( µ ) , (133)where Z g and Z (cid:48) g are respectively the renormalization constants for the coupling in thefull and in the effective theory. The decoupling constant ¯ ζ g has a perturbative expansion¯ ζ Rg = 1 + ∞ (cid:88) (cid:96) =1 ¯ α ( µ ) (cid:96) C (cid:96) ( x ) , (134)where the coefficients C (cid:96) are functions of the logarithm of the ratio of scales x =log (cid:0) µ / ¯ m h ( µ ) (cid:1) , and ¯ m h ( µ ) is the mass of the heavy fermion in MS.It is interesting to recall here that the functional dependence of the coefficients C (cid:96) on x is dictated by the renormalization group equations, as discussed eg. in Ref. [19].Keeping in mind that µ ddµ x = 1 + γ = 1 + ∞ (cid:88) k =1 γ k ¯ α ( µ ) k , (135)where γ is the mass anomalous dimension, we can take the derivative of Eq. 133 withrespect to the logarithm of µ , and solve the resulting equation order by order in ¯ α ( µ ).This procedure yields a set of differential equations that the functions C (cid:96) must satisfy, viz. ddx C ( x ) = β (cid:48) − β , (136) ddx C ( x ) = 2 ( β (cid:48) − β ) C + β (cid:48) − β − γ ddx C , (137) . . . (138)46 h [GeV] c × c × c × δα MS ( M Z ) [%]3 → m c ≈ . → m b ≈ . Table 2: Coefficients for the decoupling of the charm quark (3 →
4) and the bottom quark (4 →
5) forthe case µ = m h (see Eq. (140)) [17, 21, 101]. The last column quotes the effect of the last known termin the series in the value of the strong coupling at the scale M Z . The structure of these equations implies that C (cid:96) is a polynomial of degree (cid:96) . The coeffi-cients of these polynomials are functions of the coefficients β k , β (cid:48) k and γ k . On top of that,for each differential equation, an integration constant C (cid:96), appears, which is determinedby matching a vertex function at (cid:96) loops. The simplest example is the integration ofEq. 136, which yields C ( x ) = ( β (cid:48) − β ) x + C , . (139)As discussed above, C is a linear function of x , the slope of the function is given bythe difference of the first coefficients of the beta functions in the two theories, β and β (cid:48) , while the integration constant C , needs to be computed from a one-loop matching.Recent matching calculations up to four loops are available [21, 17].The final result for the matching of couplings across mass thresholds is reported inthe PDG [100]; using the PDG notation and for the particular case µ = m (cid:63) h , we have α ( n f +1) s ( µ ) = α ( n f ) s ( µ ) (cid:32) ∞ (cid:88) (cid:96) =2 c (cid:96) (cid:104) α ( n f ) s ( µ ) (cid:105) (cid:96) (cid:33) , (140)here α ( n f ) s is the coupling in the MS scheme with n f massless fermions, and m (cid:63) h is themass of the heavy fermion, also defined in the MS scheme, at the energy scale given by themass itself. Note that for this particular choice of scales the O ( α ) term in the relationbetween couplings vanish. The coefficients c n for n ≤ → → c (cid:104) α ( n f ) s ( µ ) (cid:105) has in the determination of the strong coupling. As the reader cansee, the truncation of the perturbative series in the decoupling relations has a completelynegligible effect on the extraction of α s ( (cid:46) .
2% for the case of the charm quark). Notethat this analysis does not exclude potentially large non-perturbative corrections in thematching between theories (see next section).
As noted before in Sect. 2.1, the running of the coupling constant with the energy scale isdetermined by the knowledge of the beta function and the Λ-parameter of a given theory.Hence the matching of the coupling between theories described in the sections above canbe reformulated in terms of the matching of the Λ-parameters, which leads naturally toa framework where decoupling can be discussed beyond perturbation theory. We presenthere a brief summary of the ideas that were originally developed in Ref. [102].47ollowing the notation introduced above, we denote quantities in the theory withheavy particles with unprimed variables, while primed variables always refer to the ef-fective theory that includes the light degrees of freedom only. The relation between theΛ-parameters is fixed by requiring that low-energy quantities are matched up to powercorrections; without loss of generality we can writeΛ (cid:48) = f (Λ , M ) , (141)where M is the RGI mass of the heavy particle in the full theory. Furthermore, we canargue on dimensional grounds thatΛ (cid:48) / Λ = P ( M/ Λ) . (142)The dependence of P on the mass M is encoded in η ( M ) = 1 P M ∂∂M P (cid:12)(cid:12)(cid:12)(cid:12) Λ , (143)which can be reliably evaluated in perturbation theory only for large values of Mη ( M ) M →∞ ∼ η + η ¯ g ( M ) + . . . . (144)Introducing the variable τ = log ( M/ Λ), and integrating Eq. 144, yields P ( M/ Λ) = 1 k exp ( η τ ) τ η / (2 b ) × (cid:18) O (cid:18) log ττ (cid:19)(cid:19) , (145)where k is a constant that can be computed given the conventions for Λ and M .A nonperturbative matching condition requires that some hadronic scale remains thesame in the two theories, ie. m (cid:48) had (Λ (cid:48) ) = m had (Λ , M ) + O (cid:18) Λ M (cid:19) . (146)In the equation above Λ and M are dimensionful quantities that define the coupling andthe masses in the full theory, and are given. The matching condition then determinesΛ (cid:48) . As a consequence we find: m had (Λ , M )Λ = m (cid:48) had (Λ (cid:48) )Λ (cid:48) Λ (cid:48) Λ + O (cid:18) Λ M (cid:19) , (147)and hence m had (Λ , M ) m had (Λ ,
0) = Q had P ( M/ Λ) + O (cid:18) Λ M (cid:19) , (148)where Q had = m (cid:48) had (Λ (cid:48) )Λ (cid:48) Λ m had (Λ , . (149)48he left-hand side of Eq. 148 can be measured by performing MC simulations of the fulltheory with different values of the mass M , while Q had requires an extra simulation inthe effective theory.References [102, 22] study the effect of heavy quarks along the lines described above.Making a long story short, they estimate the non-perturbative effects in the matchingbetween theories with the conclusions that for the most important case of the charmquark, they affect the extraction of α s below the 0 .
4% level. The interested reader isinvited to consult the original works.
6. Convenient observables for a coupling definition
In principle there are very many possibilities to define the strong coupling on the lattice.The only thing that is needed is a dimensionless finite observable that depends on a singlescale . Our discussion on scale setting (in particular the discussion about theory scalesin section 4.5.3) has already introduced some observables with these characteristics. Inthis section we will introduce the observables that are more commonly used to computethe strong coupling from the lattice, with special emphasis on the systematic effectsassociated with the truncation of the perturbative series.We recall that the general method to extract the value of the strong coupling con-sists in comparing a perturbative theoretical prediction with lattice measurements (i.e.eq. (36)). It is convenient to introduce non-perturbative definitions of the coupling con-stant, where the value of the observable is used for the definition of the strong coupling.We take a dimensionless observable with perturbative expansion O ( µ ) α MS → ∼ (cid:88) k =1 c k ( µ/µ (cid:48) ) α MS ( µ (cid:48) ) , (150)and define α O ( µ ) = O ( µ ) c (1) . (151)This equation defines a particular renormalization scheme, which we refer to as O -scheme.When computed on the lattice, O ( µ ) can be determined at all energy scales, thus pro-viding a non-perturbative renormalization scheme.Note that the RG equation for the coupling in the O -scheme defines a β -function,that can also be computed beyond perturbation theory µ d¯ g O ( µ )d µ = β O (¯ g O ) g O → ∼ − g O ( b + b g O ) + . . . . (152)This β -function has all the properties that one would expect, starting from a perturbativeexpansion with the usual first two universal terms. Examining the convergence of theperturbative series of this beta function provides a piece of information on the size ofthe truncation effects in the determination of the strong coupling. If these are small, thebeta function β O ( x ) has to be well approximated by its perturbative expansion. Notealso that our fundamental relation Eq. 16 allows the determination of the Λ-parameterin this scheme (Λ O ). Figure 7 shows the β -function for the typical choices of observablesthat we are going to explore in detail later.49 . − . − . − . − . − . − . − .
055 0 0 .
05 0 . .
15 0 . .
25 0 . . β ( g ) / g α ≡ g / (4 π ) Universal3-loop MS W Figure 7: β -function in three flavor QCD in different schemes (see Eq. 152). The 2-loop universal behavioris compared with the 3-loop β -function in different schemes: α HQ , (Eq. (171)), α qq (Eq. (163)), α SF (Eq. (196)) and α W (Eq. (175)). With the exception of the scheme based on Wilson loops, wherethe 3-loop coefficient is significantly larger than in the MS scheme, all schemes seem well behaved froma perturbative point of view. Note that for the case of the static potential, one more order is knownperturbatively. We have explained in detail (see sections 2.2.3, 3.2) the particular systematic effectsthat affect the lattice determinations of the strong coupling. Let us now focus on ex-plaining how the different methods to extract the strong coupling face these challenges.In particular we will pay special attention to the following issues.
Perturbative truncation effects:
We already insisted on the fact that the determi-nation of the strong coupling has to be understood as an extrapolation (cfr. sec-tion 2.2.3 and 3). An important criterion is the range of scales µ PT at which onematches with perturbation theory. More precisely, we are interested in the valuesof the coupling α ( l − ( µ PT ) at such scales, where l is the number of analyticallyknown loops in the asymptotic expansion of the β function, since they parametrizethe leading perturbative truncation effects. Non perturbative effects:
Power corrections are understood to arise from the Oper-ator Product Expansion (OPE). The short distance observable used to extract the50bservable l α MS ( µ PT ) µ PT [GeV] Power correctionsQCD vertices 3 0 . − .
30 2 − ∼ /µ , /µ Static Potential 3 0 . − . (cid:63) . − (cid:63) - (cid:63) HQ correlators 2 0 . − . † ¯ m † c − m † c -Wilson loops 2 0 . − .
40 1 /a = 1 . − . ∼ /µ ‡ Vacuum polarization 4 0 . − .
31 1 . − ∼ /p k ( k = 1 , ..., . − .
23 4 −
140 -
Table 3: We summarize some key numbers in different techniqes used to determine the strong coupling.Column labeled l gives the number of loops known in the perturbative expansion of the β -function. Thesecond and third columns shows the range of couplings that have been explored in the literature, and thecorresponding energy scale (note however that not all individual works explore the same energy scales ).Finally we show if power corrections are needed to describe the data, and how many terms are used. + Most of the determinations find it neccesary to add these power corrections in order to describe thelattice data. Nevertheless, see [103, 104]. (cid:63)
Most of the works expore this range of scales without any need for power corrections. In [105] powercorrections are added to the analysis, allowing to use data down to 0.6 GeV. † Most determinations are performed at the charm mass, but some works [106, 107] explore larger masses.See section 6.3 for more details. ‡ In this approach cutoff effects manifest as power corrections. See 6.4 for more details. strong coupling O ( µ ), typically has an OPE that schematically can be written as O ( µ ) = (cid:88) k =0 d k ( µ ) O ( k ) µ k , (153)where O ( k ) are operators of dimension k . Note that the perturbative expansion ofthe observable Eq. (151) neglects all power corrections (terms with k > , but estimated from the samedata for O ( µ ) by fitting them. Distinguishing the perturbative running from thenon-perturbative corrections given the limited range of scales that are available inmany extractions is always challenging.Ideally one would like to work with observables and energy scales where the powercorrections are negligible. In practice this is not always the case. The definition of the strong coupling using QCD vertices is the one that is more similar tothe type of computation that is usually done in the context of perturbation theory. Thereare several issues in the extraction of these QCD vertices from lattice QCD simulations.In practice the coupling is extracted from the gluon/ghost two-point functions, but theseare not gauge invariant, and therefore this scheme can only be implemented by fixing the The higher dimensional operators O ( k ) in Eq. (153) (i.e. “condensates”) are composite operatorsthat typically mix other lower dimensional operators. Computing them (i.e. their non-perturbativerenormalization) is probably more complicated than the determination of the strong coupling. beyond perturbationtheory ( i.e. Gribov ambiguities [108]), have been discussed at length in the literature (seefor example [109]), and we can add very little to the discussion, except pointing out thatthe issue of Gribov ambiguities is also present in other lattice QCD calculations. Oneof the most widely used methods of non-perturbative renormalization (i.e. “RI/MOM”schemes, see [110]) also requires to fix the configurations (typically the Landau gauge isused). It is believed that at the relatively high energy scales where α s is extracted, thisis not a serious issue.In principle there are several options to extract the strong coupling, since it canbe defined from different three- and four-point functions. All the methods define thecoupling by requiring that some vertex is equal to its tree-level value. The momentaentering in the vertex are part of the definition of the scheme. The most popular choicesset one of the momenta to zero, and are usually labeled (cid:94) MOM schemes.Nowadays the most common coupling definition uses the un-renormalized ghost-ghost-gluon vertex, which can be constructed from the gluon and ghost two-point func-tions. The non-perturbative coupling definition reads α T ( µ ) = lim a → F lat ( p, a ) D lat ( p, a ) g π (cid:12)(cid:12)(cid:12) µ = p . (154)This is usually referred to as the Taylor scheme, as indicated by the suffix. Here D latt ( p, a )and F latt ( p, a ) are the “dressing functions” of the lattice gluon and ghost two-pointfunctions and g is the bare coupling used for the simulation. The main advantage ofthis scheme is that one does not need to determine any three- or four-point function, sincethe coupling is directly defined from the computation of the propagators. We are goingto focus the discussion on this particular choice, although most of what we are going tostate is also valid for other, similarly defined schemes. The perturbative expansion for α T ( µ ), α T ( µ ) µ →∞ ∼ α MS ( µ ) + t α ( µ ) + t α ( µ ) + t α ( µ ) + O ( α ) , (157)is known up to three-loops [111].There are two important issues that affect generally these extractions and that areworth mentioning. Non perturbative corrections
Several studies have concluded that including non per-turbative corrections in the analysis is mandatory to find consistent results amongobservables and range of scales used to determine the Λ-parameter (see refer-ence [112] for a detailed study in pure gauge and energy scales in the range 3 − In the continuum the relation of the propagators F ab ( p ) , D abµν ( p ) with the dressing functions F, D is F ab ( p ) = − δ ab F ( p ) p , (155) D abµν ( p ) = − δ ab (cid:18) δ µν − p µ p ν p (cid:19) D ( p ) . (156)On the lattice the relation is similar, but the momentum p is substituted by a function of p and thelattice spacing a that only reduces to p in the continuum limit and depends on the particular choice ofdiscretization. ∼ g (cid:104) A (cid:105) p . (158)These corrections are included in the analysis either by using an estimate for (cid:104) A (cid:105) (like for example [113]) or by fitting their data including such a term. Higher ordernon-perturbative corrections (i.e. ∼ p − x for different values of x ) are also typicallyneeded to match the lattice data with the perturbative running [114–116, 113] (seeFig. 8). Cutoff effects
The range of scales where α T ( µ ) can be described by its perturbativeexpansion (and the non-perturbative contributions) is typically about µ ∼ − µ . Therefore the common approach is to include several terms to either subtractor fit the cutoff effects (see [114–117]).Figure 8 shows these two points exemplified in the results of one particular work [116].Panel (a) shows that the raw measurements for the Taylor coupling at two different valuesof the lattice spacing differ by approximately 50% after subtracting the H (4) breakingcutoff effects (see [116] for details). The remaining scaling violations have an asymptoticexpansion that includes terms ∼ O ( a p ) and are noticeable compared with the statisticalprecision of the data. Panel (b) shows the extraction of the Λ-parameter after matchingwith its perturbative expansion at the scale µ [113]. The plot shows the values of Λ MS as a function of α (the leading correction to the extraction). The data with only 1 /µ corrections subtracted shows a correction compatible with a large α perturbative term.But if another non-perturbative term 1 /µ is included as a fit parameter, the data seemto be well described by the perturbative expression in Eq. (157).In summary, the extraction of α s from QCD vertices is hampered by a slow rate ofconvergence to the perturbative behavior. Several non-perturbative corrections need tobe fitted at the same time, since they are noticeable. Even after fitting for the non-perturbative corrections the range of energies that can be used to determine the strongcoupling is limited. Large energies scales are needed, where most data come typicallyfrom a single lattice spacing.These extractions also show that distinguishing the perturbative and non-perturbativecorrections is, in practice, very difficult. Figure 8 (b) shows the difficulty in distinguishinga correction of order α ( µ ) from a 1 /µ non perturbative correction when we have onlyaccess to a limited range of scales. Indeed we see that a variation of the order of 10 MeVin Λ can be reabsorbed by higher-order power corrections.We believe that a dedicated study in pure gauge theory with the aim of reachingenergy scales where the non-perturbative data is described by the perturbative prediction without fitting any non-perturbative terms would be very interesting. A pure gaugesimulation would also allow a detailed investigation of the continuum extrapolationsusing several fine lattice spacing. 53 a( β ) q α T ( a q ) β =2.10, a µ l =0.002 β =1.90, a µ l =0.005 β =1.90, a µ l =0.004 β =1.90, a µ l =0.003 (a) . . . . . .
325 0 0 .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 .
03 0 .
035 0 . Λ M S α ( µ ) With /µ power correctionsWith /µ and /µ power corrections (b) Figure 8: Extraction of the strong coupling from QCD vertices. (a) Determination of the strong couplingin the Taylor scheme. The figure shows the raw values of the coupling for two different lattice spacing( a ≈ .
089 fm for β = 1 .
90 and a ≈ .
06 fm for β = 2 .
10) and different values of the quark masses. Asthe reader can see, the chiral dependence is mild, but the cutoff effects are substantial. (source [116]).(b) Extraction of the Λ-parameter in the Taylor scheme. Including only 1 /µ power corrections is notenough to to reach the perturbative running. In this particular work [113] , the extraction of Λ seemsconsistent in the region 0 . < α < .
025 by fitting the data with an additional 1 /µ power correctionterm. The force between static color charges has been traditionally one of the first observablesto be studied in lattice QCD [118]. The potential at distance r can be extracted fromWilson loops, that behave asymptotically as W r × T ∼ λ e − V ( r ) T + (cid:88) k λ k e − V n ( r ) T . (159)The potential V ( r ), given by the ground state (i.e. the leading decaying exponential), isformally computed as V ( r ) = lim T →∞ T log (cid:104)W r × T (cid:105) . (160)In practice it is extracted at large, but finite, values of T , and therefore several techniquesare needed in order to enhance the overlap with the ground state and to distinguish theleading exponential from the excited state contamination. When computed on the lattice,the static potential is power divergent ∼ /a . This is related to the ambiguity in theoverall magnitude of V ( r ): only energy differences are physical. The cleanest way to dealwith this linear divergence is to define the coupling via the static force F ( r ) = d V ( r )d r . (161)The derivative with respect to r removes the linear divergence but requires to performa numerical derivative of the potential. This is implemented by some finite differenceexpression. It is convenient to define the force by F ( r I ) = V ( r ) − V ( r − a ) a , (162)54 à à à à à à à à à à à à à à à à à à à - - - - - - - - - - r (cid:144) r r E + c on s t . Β= -- Nref = -- + lead. us. (a) à à à à à à à à à à à à à à à à à à à à - - - - - - - - r (cid:144) r d a t a - t h e o r y Β= -- Nref = -- + lead. us. (b) Figure 9: (a) Comparison of the perturbative prediction and the lattice data at the finest lattice spacings( a ≈ .
04 fm) for the static energy E ( r ). (b) Residuals of the fit. with r I chosen so that the force has no cutoff effects to leading order in perturbationtheory. This has been shown to reduce the cutoff effects in the force [119].A renormalized coupling constant can be defined non-perturbatively using the staticforce. The non-perturbative definition of the coupling and its perturbative expansion inpowers of the coupling in the MS-scheme reads α qq ( µ ) = 43 r F ( r ) (cid:12)(cid:12)(cid:12) µ =1 /r r → ∼ α MS ( ν ) + c (1) qq ( s ) α ( ν ) + c (2) qq ( s ) α ( ν )+ c (3) qq ( s ) α ( ν ) + . . . ( s = µ/ν ) . (163)The relation of this coupling to the MS scheme is known up to three loops, but theobservable suffers from IR divergences, that manifest themselves in the naive perturbativeexpansion of eq. (163) being divergent. These so-called soft and ultra-soft divergencescan be re-summed and produce logarithmic corrections to the perturbative series. Theleading one is ∝ α ( ν ) log α MS ( ν us ). This re-summation process introduces an arbitraryenergy scale (so called ultra-soft scale ), its natural value being ν us = α ( ν ) /r . In principlethis means that not only the scale ν ≈ /r has to be large, but also ν us = α ( ν ) /r hasto be large. The additional factor α is not negligible taking into account that theseextractions take place at a few GeV.All in all, the perturbative expansion, including terms α , α log α, α log α, α log α ,is known [120–127] (see [128] for a summary on the perturbative expressions).Regarding the perturbative behavior, figure 7 shows that the β -function in this schemeis well behaved, with a 2-loop coefficient of a similar size as in the MS scheme.Several works extract the value of the strong coupling not from the force, but directlyfrom the potential. In particular they examine the dimensionless quantity rV ( r ). Basi-cally the same considerations apply for these works: the perturbative expansion in theMS scheme is known up to order O ( α ), and the soft and ultra-soft gluons give rise tologarithmic corrections in the perturbative expansion, starting at order α . The maindifference is in the role of the linear divergence in the potential V ( r ) ∼ /a .Of course there are many different ways to use the perturbative expansion Eq. (163)(or a similar expression for the potential rV ( r )) to extract α s . Different works, althoughsimilar in spirit, use different approaches to fit the lattice data and deal with the additiverenormalization (in case they use V ( r ) for the extraction). Here we will focus on one55articular work [129] to show the details of such analysis. Ref. [129] uses the expressionof the static force and determines the integral up to a reference distance r ref , E ( r ) = (cid:90) rr ref d x F ( x ) . (164)The perturbative expansion for the force gives a similar perturbative expansion for E ( r ).Note that this quantity is similar to the static potential V ( r ), with the difference thatit is free of the linear divergence ∼ /a (the relation E ( r ref ) = 0 is exact for all latticespacings), and that depends now on two scales ( r and r ref ). The perturbative expressionfor E ( r ) is fitted to the lattice data. Fig. 9 shows the result of such an analysis. Non-perturbative corrections do not seem to be needed to describe the data at least fordistances in the range r (cid:46) . r (cid:46) . LO perturbative knowledge,together with the fact that data in this scheme seem to follow the perturbative predictionsat scales as low as 1 . The idea of using correlators of heavy quarks to extract the value of the strong couplinghas its origins in a phenomenological determination of α s : moments of quarkoniumcorrelators in the vector channel can be compared with experimental data for e + e − → hadrons.As was first noted by the HPQCD collaboration [132], the strong coupling and thecharm quark mass can be extracted on the lattice from correlators of the pseudoscalardensity involving two heavy quarks, G ( x ) = a ( am ) (cid:88) x (cid:104) ψγ ψ ( x , x ) ψγ ψ ( , (cid:105) , (165)where am is the bare quark mass. This correlator has a short distance divergence ∼ /x , but if one uses a fermion formulation that preserves some chiral symmetry (like56taggered or domain wall fermions), the PCAC relation ensures that moments of G ( x ),defined as G n = (cid:88) x (cid:16) x a (cid:17) n G ( x ) , (166)are dimensionless quantities with a well defined continuum limit for n ≥
4. The maincontribution to these moments comes from Euclidean times x ∼ /m .The extraction of the strong coupling is performed using the reduced even moments r = G G (0)4 , (167) r n = ¯ m ( µ ) am η am (cid:20) G n G (0) n (cid:21) , ( n ≥
6) (168)where G (0) n denotes the leading order prediction for G n in bare lattice perturbation theory.This normalization is introduced in order to reduce cutoff effects (i.e. to leading order, r n is free of lattice artifacts). Here m η denotes the mass of the η c meson, and r n admitsa perturbative expansion that allows an extraction of the strong coupling constant: r n α MS ( µ ) → ∼ r n, α MS ( µ ) + r n, ( s ) α ( µ ) + r n, ( s ) α ( µ ) + . . . (169)where s = µ ¯ m ( µ ) . (170)A non-perturbative definition of the strong coupling at the scale ¯ m ( ¯ m ) is given by α HQ , n ( µ ) (cid:12)(cid:12)(cid:12) µ = ¯ m ( ¯ m ) = r n − r n, α MS ( µ ) → ∼ α MS ( ν ) + r n, ( s ) r n, α ( ν ) + r n, ( s ) r n, α ( ν ) + . . . , (171)with s = ν/ ¯ m ( ¯ m ) = ν/µ . The first three coefficients ( r n, , r n, ( s ) , r n, ( s )) are analyti-cally known (see [133–136]. Reference [107] has values tabulated for N f = 3).There are two crucial points in these determinations: the estimate of the truncationuncertainties, and controlling the continuum extrapolations. Note that these two pointshave competing interests: the continuum extrapolation is more easily kept under controlfor a quantity measured at larger distances, and therefore the “high” moments r , , have milder continuum extrapolations. On the other hand the truncation uncertaintiesare smaller for a short distance quantity: the first moment r has a better behavior. Thisis just another manifestation of the “window problem” (see figure 2).The estimates for these uncertainties vary significantly across different studies, andwe will comment in detail on the issue of the truncation uncertainties in section 7. Herewe will focus on the more technical issue of the continuum extrapolation. Figure 10shows the continuum extrapolation of r at scale m c of the works [137, 106]. As thereader can see, the scaling violations are significant and have a complicated functionalform. Different works in the literature deal with these complicated cutoff effects in verydifferent ways With Wilson fermions an extra finite renormalization for the axial current would be needed. . . . . . . . .
05 0 . .
15 0 . .
25 0 . . r ( am h ) (a) .
02 0 .
04 0 .
06 0 .
08 0 . .
12 0 .
14 0 . Λ M S [ M e V ] α ( µ ) (b) Figure 10: (a) Continuum extrapolation of r at the lower energy scale µ ∼ ¯ m c . Scaling violations aresignificant, even at the smaller quark masses used in the study. The extrapolations is performed usingboth a 5 th degree polynomial, or a second degree one with a restricted fitting window (source [106]). (b)Dependence of the Λ MS parameter extacted using heavy quark correlators on the value of the couplingat the matching scale with perturbation theory ( α MS ( µ )) (source [106]). JLQCD collaboration [138]:
In this case they prefer to only perform extrapolationslinearly in a . Their data for r does not allow such an extrapolation, and thereforeit is excluded from their analysis. HPQCD collaboration [107, 139]:
In these works all moments are used, and massesabove the charm quark mass are used, including data with am ∼ .
9. Cutoff effectsare large and the data is contaminated by effects ∼ ( am c ) p . Therefore their fitAnsatz includes terms a p with p up to 10. These fits typically have more termsthan data, and require to include an estimate of the size of these coefficients asBayesian priors. Ref. [106]:
Here energy scales larger than the physical charm quark mass are explored,but the continuum extrapolations are difficult and the data usually has associatedlarge uncertainties (see figure 10).The main drawback of this approach is the large cutoff effects that affect the quantityused to extract the strong coupling. This makes it very challenging to explore energyscales larger than the physical charm quark mass m c ∼ . m c − m c ,but the continuum extrapolation is very challenging already at µ (cid:38) m c . Even at thescale of the charm quark mass, scaling violations are significant and have a complicatedfunctional form (see Figure 10). Together with the fact that the perturbative relationis known only to NNLO the situation is far from ideal: truncation uncertainties atthe energy scales reached by current simulations are not small (see detailed discussionin section 7.2.3). This might change in the future, as smaller lattice spacings can besimulated, allowing a reduction of the discretization effects. A detailed study in puregauge and reaching energy scales significantly larger than the charm quark mass, wouldprobably give very important information on the systematics of this method. Note that the leading cutoff effects in this approach are O ( a α s ), while in other extractions (like forinstance the one based on the static potential) they are suppressed by an extra factor of α s . .4. Observables defined at the scale of the cutoff Lattice QCD offers the interesting possibility of extracting the strong coupling fromexpectation values computed at non-zero lattice spacing. This approach is fundamentallydifferent from the strategies outlined above, where a quantity is computed in QCD (i.e.extrapolated to the continuum), and then compared with a perturbative prediction. Theobservables that we are going to discuss in this subsection are defined at a scale given bythe lattice spacing 1 /a . Lattice bare perturbation theory is able to relate these purelylattice observables with a power series in the renormalized coupling. The usual problemthat a naive approach has to face is that bare lattice perturbation theory is just terrible.Absurdly small values of the lattice coupling α latt = g / π have to be used in orderto reach the domain of apparent convergence. It has been argued that this apparentfailure of lattice perturbation theory is just a due to the choice of the bare coupling g asexpansion parameter [140]. If lattice quantities are expressed as a perturbative series ina renormalized coupling, like α MS ( µ ), their perturbative behavior improves substantially(see [140]).The HPQCD collaboration has pursued the systematic study of several Wilson loopsof size n × m (denoted (cid:104) W nm (cid:105) ) and used them to extract the value of the strong coupling.In these analyses lattice quantities are expressed as a perturbative series in terms of therenormalized coupling α V ( µ ); for an SU(3) gauge theory coupled to N f fermions in thefundamental representation, the latter coupling is defined by [123] α V ( µ ) = α MS (¯ µ ) + 2 . − . N f π α (¯ µ ) + 53 . − . N f + 0 . N π α (¯ µ ) , (172)with ¯ µ = exp( γ − / µ ≈ . × µ . Wilson loops have a perturbative expansion − log (cid:104) W nm (cid:105) a → ∼ w α V ( µ ) + w α ( µ ) + w α ( µ ) + . . . . (173)Alternatively one can use Creutz ratios or tadpole improved Wilson loops. This latterchoice − log (cid:32) (cid:104) W nm (cid:105) (cid:112) (cid:104) W (cid:105) n + m (cid:33) a → ∼ w b1 α V ( µ ) + w b2 α ( µ ) + w b3 α ( µ ) + . . . . (174)is supposed to lead to smaller truncation uncertainties. In all cases the perturbativecoefficients are known for several choices of n, m [141]. The scale is given by µ = d/a ,where d ≈ π with the exact value depending on the choice of Wilson loop. Note thatnon-perturbative couplings can be defined by expressions α W nm (1 /a ) = − log (cid:104) W nm (cid:105) w a → ∼ α V ( µ ) + w w α ( µ ) + w w α ( µ ) + . . . . (175)Several quantities are fitted to the previous perturbative expressions, with the valueof α V ( µ ref ) as fit parameter. The values of α V ( µ ) at other scales are obtained from α V ( µ ref ) and the RG equation µ d α V ( µ )d µ = − α
2V 3 (cid:88) k =0 β k α k V . (176)59he final value of α V ( µ ref ) can be converted to the more convenient MS scheme usingEq. (172).An example of such extractions are the HPQCD works (see [132]). A total of 22quantities (all like Eq. (173)) are fitted to their perturbative expression. Unfortunatelythe truncated perturbative expression Eq. (173) does not describe the data well, andseveral extra terms (up to α ) are necessary in order to obtain a sensible fit. Thecoefficients of these terms are constrained with Gaussian priors, which eventually leadto stable fits, with a consistent determination of the strong coupling using any of the 22quantities (although some of them require to also fit several power corrections).The main advantage of methods based on observables defined at the cutoff scale, isthat high energies can be reached without having to worry about the continuum extrap-olation. The statistical accuracy is excellent, since the observables entering the deter-mination Eq. (173) have a very small variance. On the other hand the uncertainty inthese determinations is dominated by the truncation of the perturbative series. The factthat several higher order terms have to be fitted (and constrained with Gaussian priors)in order to describe the data is not ideal. It is clear that expressing lattice quantitiesas a power series in renormalized couplings , as suggested in [140] greatly improves thepredictive power of perturbation theory (bare perturbation theory is just useless). Stillthese lattice observables are far from ideal from a perturbative point of view: even ifenergy scales 1 /a ≈ α P (1 /a ).This can be easily understood by noting that a ∼ exp (cid:26) − π b α W (1 /a ) (cid:27) , (177)Of course in any extraction of the strong coupling based on these methods, these effectsare not parametrically the leading ones, since the truncation of the perturbative seriesEq. (175) misses terms of order O ( α nW ) ∼ log n a . However in practice it is not clearwhich effects dominate (the O ( a ) cutoff effects or the log n a from the truncation of theperturbative series), and this might even depend on the particular observable used to setthe scale. The hadronic vacuum polarization function (HVP) is defined from two-point functionsof the vector and axial-vector currents V aµ ( x ) = ¯ ψ a γ µ ψ a ( x ) , (178) A aµ ( x ) = ¯ ψ a γ γ µ ψ a ( x ) , (179)after a decomposition in Fourier space (with J µ = V µ , A µ ) (cid:90) d x e ıpx (cid:104) J aµ ( x ) J aν (0) (cid:105) = ( δ µν p − p µ p ν )Π (1) J ( p ) − p µ p ν Π (0) J ( p ) . (180)60he quantity Π( p ) = Π (0) V ( p ) + Π (1) V ( p ) + Π (0) A ( p ) + Π (1) A ( p ) . (181)is dimensionless and has a perturbative expansionΠ( p ) p →∞ ∼ c + (cid:88) k =1 c k ( s ) α k MS ( µ ) + O ( α ) . ( s = p/µ ) . (182)known up to 5-loops. The constant term c ( s ) is divergent, so that the strong couplingis usually extracted from the difference Π( p ) − Π( p ), or the Adler function D ( p ) = p dΠ( p )d p . (183)The recent work [142] determines the finite difference∆( p , p ) = Π( p ) − Π( p )log( p/p ref ) . (184)at high energies in order to make contact with the perturbative running. They use severalvalues of p ∼ − p and values of p ref toextract α MS ( M Z ).The main issue with extractions based on the HVP is that power corrections aresignificant even for large momenta [143]. In fact, as discussed in detail in Ref. [142],these corrections show a very poor convergence. One would expect that higher powercorrections become negligible as the momentum increases, but this is not the case dueto accidental cancellations between condensates of different dimensions (up to 1 /µ ).Ref. [142] pushes the determination to high energies, so that the data can be describedwithout any power corrections, but then cutoff effects become larger and the windowof scales to obtain the strong coupling decreases. Despite the impressive perturbativeknowledge in this scheme (5 loops), the authors think that more work is needed in orderto convincingly show that contact with the perturbative running has been made, and thatthe continuum extrapolations are under control. Being a relatively new technique, thereis not a single work for the pure gauge theory. Once again, we would like to stress thata detailed study in this simpler case, where very fine lattice spacings can be simulated,would shed some light on many of these issues. Recently a novel approach to extract the strong coupling has been proposed. It uses thespectral density of the continuum Dirac operator ρ ( λ ) = 1 V (cid:42)(cid:88) k [ δ ( λ − ıλ k ) + δ ( λ + ıλ k )] (cid:43) , (185)and its perturbative expansion ρ ( λ ) = 3 λ π (cid:0) − ρ ( s ) α MS ( µ ) − ρ ( s ) α ( µ ) − ρ ( s ) α ( µ ) + O ( α ) (cid:1) . ( s = µ/λ ) . (186)61 non-perturbative coupling definition can be defined by using α D ( λ ) = ρ ( s ) − (cid:20) π λ ρ ( λ ) − (cid:21) . (187)with a β -function known up to 3-loops.Alternatively one can use the derivative F ( λ ) = ∂ρ ( λ ) ∂ log λ = 3 − F ( s ) α MS ( µ ) − F ( s ) α ( µ ) − F ( s ) α ( µ ) − F ( s ) α ( µ )+ O ( α )(188)to also define the strong coupling whith a perturbative expansion known up to 4-loops.Only one work in the literature uses this method to extract the strong coupling [144].Naively the truncation error at the energy scales used to extract α s ( i.e. λ ≈ . − . ∼
20% in F ( λ )). The renormalization scale is pushedto higher values by using µ/λ = 5. For these values of µ truncation effects are expectedto be reduced to a percent level, while the perturbative expansion for the strong couplingstill shows good convergence properties.It is important to note that the dominant uncertainty from the truncation of theperturbative series is determined by an estimate of the leading missing coefficient in theperturbative series Eq. (186). Due to the low energy scale of the determination, the usualprocedure of varying the renormalization scale by a factor 2 below and above λ wouldresult in a substantially larger truncation error. On the other hand the work [144] doesnot seem to need any power corrections to describe the data at energy scales 0 . − . aλ < . O ( a )scaling violations. This restricts the energy scales that can be reached with their data-set (with lattice spacing a − = 2 . , . . λ < . Here we will describe a theoretical idea to overcome the fundamental limitation of anyof the previous lattice determination of the strong coupling , namely the compromisebetween reaching large energy scales to control perturbative truncation effects, and andhaving a clear separation between these energy scales and the lattice cutoff so thatcutoff effects in observables measured at small distances are kept under control. Thisfundamental compromise present in any single lattice computation – the scales µ thatcan be probed have to obey 1 /a (cid:29) µ (cid:29) m π – originates from the fact that we want to The ideas of finite size scaling are able to deal with many other multiscale problems, like the de-scription of heavy quarks in lattice simulations, see Ref. [145] for example, or the renormalization ofcomposite operators. had ∼
100 MeV µ PT ∼
100 GeV µ had M Ω µ PT µ had Λ µ PT Λ = M expΩ × µ had M Ω × µ PT µ had × Λ µ PT Large volume ↔ finite volume matching Determination of step scaling function σ s ( u ) Perturbation theory Figure 11: A product of dimensionless ratios together with a single dimensionful experimental input isused to determine Λ. At low energies an experimental quantity ( M Ω in the example) is computed interms of µ had , a hadronic scale defined in a massless finite volume scheme (see section 6.7.3). The finitevolume scheme is used to connect non perturbatively the hadronic ( µ had ) and perturbative ( µ PT ) regimesof QCD by determining the step scaling function (see section 6.7.2, Eq. (205) and Eq. (207)). Finallyusing perturbation theory one can determine the Λ parameter in units of the high energy perturbativescale ( µ PT ). accommodate in a single lattice computation the scales used to match with perturbationtheory µ PT and the scales used to describe hadronic physics. Finite size scaling solvesthis problem by adopting a different strategy. Only a finite range of scales is resolvedin any single lattice computation, and a recursive procedure allows to relate the energyscales explored with different simulations.This idea is implemented by using a finite volume renormalization scheme [146]. Theobservable used to define the coupling O ( µ ) is defined at a scale linked with the finitevolume of the simulation µ ∝ /L and the coupling will run with the size of the system.Very small physical volumes – the so-called femtouniverse – are simulated in order toreach high energy scales. The only constraint for these simulations is that the energyscale explored has to be far away from the cutoff (1 /a (cid:29) µ ∼ /L ). This is easilyachieved by using lattices of moderate size L/a ∼ − • We need a way to relate our finite volume simulations with experimental data. • We need a way to match simulations in different volumes. • We need to match our results in very small volumes (high energies) with a pertur-bative computation.Figure 11 shows a schematic representation of the procedure. Note that we are ef-fectively separating the problem of solving non-perturbatively the RG equations anddetermining an overall hadronic scale. But before explaining in detail these three ingre-dients, let us discuss some technical details of QCD in small physical volumes.
Typical lattice simulations aim to explore QCD in the infinite volume limit. One isusually interested in the determination of hadronic quantities like the spectrum or somedecay form factors, which can be reliably computed only using large volume simulations,with say Lm π >
4. In these simulations the choice of boundary conditions is not very63elevant, since we are supposed to describe the infinite volume physics within a percentaccuracy.On the other hand, when studying QCD on a finite volume we are exploring a com-pletely unphysical regime of the theory with then aim of solving the RG equations. Thesize of the system is just part of the renormalization prescription, and we are free tochoose it at will. But in this particular situation the boundary conditions play a funda-mental role. Different choices of boundary conditions have to be understood as differentrenormalization schemes, and not as small corrections. The choice of boundary conditionsis part of the definition of the observable itself.One would naturally think that periodic boundary conditions are a natural choice,but it was soon found [147] that QCD observables computed in a finite periodic boxhave complicated perturbative expansions. Perturbative expansions derive from a saddlepoint expansion of the path integral Z = (cid:90) D A µ e − S [ A µ ] (189)around the minimum. In infinite volume this minimum ( A µ = 0) is unique, up togauge transformations. On the other hand fields with periodic boundary conditions ona finite volume do not have a unique minimum of the action. All zero momentum fields A µ = constant have zero action. Note that these configurations, in a finite volume, arenot all related to each other by gauge transformations. Gauge transformationsΩ( x ) = exp ( ω a ( x ) T a ) (190)have to be single valued functions, imposing the condition ω a ( x + 2 π ) = ω a ( x ). It is easyto see that this implies that gauge transformation can only shift the zero mode of thegauge fields by a multiple of 2 π , A aµ → A aµ + 2 πn . (191)Naively these zero momentum field configurations produce flat directions in the pathintegral Eq. (189), making the integral divergent. Expectation values are finite, butin general the perturbative expansion can no longer be written as a power series in α .Fractional powers (like α / ) or even logarithmic contributions ( α log α ) appear in theperturbative expansion . These non-analytic terms in the perturbative expansion of ob-servables defined in a periodic box result in coupling definitions (defined via Eq. (151))that generically do not even share the universal coefficients of the β -function. The Λ-parameter can not be defined in schemes defined from these observables (readers inter-ested in this topic can consult the review [148]).Of course one could still match these observables with the perturbative expansion,including the non-analytic terms , but fortunately there are better solutions. Since thesenon-analytic perturbative expansions are just a consequence of our choice of boundaryconditions, we can choose the latter wisely in order to avoid these complications, andthere exists several options to accomplish this goal. The concrete form of these terms depends on the number of periodic directions and the rank of thegroup. The matching with the asymptotic perturbative behavior for these kind of observables might bedelicate, and require access to substantially larger energy scales. wisted boundary conditions Demanding physical quantities to be periodic does notnecessarily require a periodic gauge field A µ ( x ). It is enough for A µ ( x ) to be periodic modulo a gauge transformation [149] A µ ( x + L µ ˆ µ ) = Ω µ ( x ) A µ Ω † µ ( x ) + ı Ω µ ( x ) ∂ µ Ω † µ ( x ) . (192)The matrices Ω µ ( x ), known as transition matrices, can be chosen in order to guar-antee that the action has a unique minimum up to gauge transformations [147].In some sense, twisted boundary conditions are the most natural choice, sincetranslational invariance is fully preserved. Fermions can be added without majorconceptual problems. However, for fermions in the fundamental representation ofthe gauge group, boundary conditions impose that the number of flavors to beincluded in the model have to be a multiple of the rank of the group ( i.e. N f ∝ SU (3)). For QCD applications this might be a problem, since onlythe three and six flavor theories can be formulated with this choice of boundaryconditions.The first applications of twisted boundary conditions in finite size scaling useda ratio of Polyakov loops as definition of the renormalized coupling [150]. Thiscoupling definition suffers from large statistical errors, due to the large varianceof this particular observable. Most recent works use the gradient flow, which weintroduced in section 4.5.3, to define the renormalized coupling [151]. Schr¨odinger Functional boundary conditions
The most common choice of bound-ary conditions to study QCD on a finite volume are called Schr¨odinger Functionalboundary conditions [152, 153]. Dirichlet boundary conditions are imposed on thespatial components of the gauge field at Euclidean times x = 0 , TA i ( x ) (cid:12)(cid:12)(cid:12) x =0 = C i ( x ); A i ( x ) (cid:12)(cid:12)(cid:12) x =0 = C (cid:48) i ( x ) . (193)The time component of the gauge field inherits its boundary conditions from thegauge fixing condition. If the boundary fields C i ( x ) , C (cid:48) i ( x ) are chosen appropriately,the minimum of the action is unique up to gauge transformations. There are twocommon choices in the literature. First one can choose C i ( x ) , C (cid:48) i ( x ) to be constantdiagonal matrices. For the case of SU (3), which has two diagonal generators, thegeneric form of the the background field is C k = iL diag (cid:26) η − π , η (cid:16) ν − (cid:17) , − η (cid:16) ν + 12 (cid:17) + π (cid:27) , (194) C (cid:48) k = iL diag (cid:26) − η − π, η (cid:16) ν + 12 (cid:17) + π , − η (cid:16) ν − (cid:17) + 2 π (cid:27) , (195)where the parameters η, ν can be chosen at will. An important advantage of thissetup is that derivatives of the effective action with respect to these boundaryparameters can be used to define a renormalized coupling at a scale given by thevolume of the system: 12 π ¯ g ,ν ( µ ) = (cid:28) ∂S∂η (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) η =0 , ( µ = 1 /L ) . (196)65ifferent choices of ν represent different renormalization schemes (different couplingdefinitions). Conveniently, the values of ¯ g SF ,ν for all values of ν can be calculatedfrom expectation values evaluated at ν = η = 0. These couplings have been thepreferred choice for finite size scaling studies in QCD (see [154–157]), but theyare gradually being replaced by the new coupling definitions based on the gradientflow [158]. These are more conveniently defined with zero background field (i.e. C i ( x ) = C (cid:48) i ( x ) = 0 in eq. (194)).Schr¨odinger functional boundary conditions can be easily simulated on the lattice,but breaking translational invariance has the unpleseant effect of producing linear O ( a ) cutoff effects (even in the pure gauge theory [152]). These can in principle beremoved by tuning boundary counterterms. They are only known in perturbationtheory and the effect of the higher order corrections have to be studied in detail inany step scaling study. Open-SF boundary conditions
For a long time, topology freezing was thought notbe an issue in small volume simulations since non-trivial topological sectors arehighly suppressed on small volumes. But nowadays it is clear [159] that thesesimulations are also suffering from this effect when the physical volume is ∼ . A way to overcome thisissue is just to use definitions in the sector of zero topological charge, as suggestedin Ref. [159].Using open boundary conditions in Euclidean time provides a solution to the prob-lem: the topological charge is no longer quantized and transitions between differ-ent topological sectors are allowed [160]. As discussed previously, the breaking oftranslational invariance leads also in this case to linear O ( a ) cutoff effects near theEuclidean time boundaries.Whatever the choice of boundary conditions, simulations on small physical volumeshave one more crucial ingredient that makes them attractive: the possibility to directlysimulate massless quarks on the lattice. This is possible because in this regime the sizeof the system acts as IR cutoff ( ∼ /L ). The Dirac operator has a gap, even at m = 0.This possibility removes a source of systematic effect in the lattice determinations of thestrong coupling, since the matching with the perturbative regime is usually only knownto high accuracy in massless renormalization schemes. There is no need to perform anextrapolation to zero mass. Finite volume renormalization schemes work by providing coupling definitions that de-pend on a single scale given by the volume of the system ( µ ∼ /L ). It is clear that largeenergy scales can be easily reached simulating a small physical volume, requiring only amodest lattice size ( L/a ∼ − There exists an index theorem with twisted boundary conditions that guarantees that when simu-lating massless quarks only topologically trivial sectors contribute to the path integral. This opens thedoor for three flavor QCD to actually determine the running coupling without having to worry aboutany topology freezing problems. L = 2 L ¯ g ( µ ) σ ( u ) = ¯ g ( µ ) (cid:12)(cid:12)(cid:12) u =¯ g ( µ ) ,µ = µ / L → L Figure 12: In finite volume renormalization schemes the coupling ¯ g ( µ ) is defined at a scale givenby the physical size of the system ( µ = 1 /L ). If at some scale µ = 1 /L the coupling has value u = g ( µ ) (cid:12)(cid:12)(cid:12) µ =1 /L , and we measure the coupling in a volume two times larger (2 L ), we obtain thestep scaling function σ ( u ) (see Eq. (197)). This is a discrete version of the β -function that measureshow much the coupling changes when the scale is changed by a factor 2. The step scaling function σ s ( u ) is the key quantity that makes it possible solvingthe RG equations (see figure 12). It measures the change in the coupling when therenormalization scale changes by a factor s , and therefore can be understood as a discreteversion of the β -function, σ s ( u ) = ¯ g ( µ ) (cid:12)(cid:12)(cid:12) ¯ g ( µ/s )= u , ( µ = 1 /L ) . (197)The values s = 2 , / g , am ) and the lattice spacing a . In massless schemes we simulate with ¯ m = 0, andthe lattice spacing a is set solely by the value of the bare coupling g . This means thatif we keep the same bare simulation parameters and just multiply the number of latticepoints by a factor s , we will obtain a lattice estimate of the step scaling function L/a → sL/a = ⇒ u = ¯ g → Σ s ( u, a/L ) . (198)Of course the previous process still depends on the value of the cutoff a , but this resultcan be repeated with several pairs of lattices in order to obtain a continuum resultlim a → Σ s ( u, a/L ) = σ s ( u ) . (199)By repeating a series of continuum extrapolations at different values of the coupling u , the function σ s ( u ) can be determined. Once the step scaling function is known, theRG problem is solved non-perturbatively. Note that the relationlog s = − (cid:90) √ σ s ( u ) √ u d xβ ( x ) , (200)67 . . . . . .
002 0 .
004 0 .
006 0 .
008 0 .
01 0 .
012 0 .
014 0 . Σ ( u , a / L ) / u ( a/L ) Fit Σ Fit / Σ Continuum (fit Σ )Continuum (fit / Σ )Data Figure 13: The bare coupling g is fixed on several lattice sizes L/a = 8 , , ,
16 so that the renormalizedcoupling is equal to ¯ g = u i for nine values u i ≈ . , . , . , . , . , . , . , . , .
54. Thismeans that all simulations have the same physical volume L , up to scaling violations. By computingthe coupling on lattices twice as large, one determines a lattice approximation Σ s ( u, a/L ) of the stepscaling function σ ( u ). This plot shows the continuum extrapolation of the lattice step scaling functionΣ s ( u, a/L ) (see Eq. (199)). The continuum values can be used to parameterize σ ( u ) for u ∈ [2 . , . β -function. (Source [161]).
68s exact. The determination of the Λ-parameter Eq. (16) uses the previous relation tobreak the fundamental integral of the β -function (cid:90) ¯ g ( µ had )0 d xβ ( x ) , (201)into several pieces. The scale µ had is a low-energy scale, of the same order as the referencescale used to “set the scale”, as discussed in section 4.5.2. With the help of the step-scaling function one can produce a series of couplings ¯ g k such that¯ g ≡ ¯ g ( µ PT ); ¯ g k − = σ s (¯ g k ) . (202)Note that since the step scaling function changes the scale by a factor s , we have ¯ g k =¯ g ( s k µ had ). Recalling the basic relation Eq. (200) the integral of Eq. (201) can now bewritten as (cid:90) ¯ g ( µ had )0 d xβ ( x ) = (cid:90) ¯ g ( s n µ had )0 d xβ ( x ) + n log s . (203)With moderate values of n ( i.e. n ∼
10) one can reach high energy scales, since s n µ had ∼
100 GeV. The remaining integral in Eq. (203) can be very well approximated by usingperturbation theory (cf. section 3.2). (cid:90) ¯ g ( s n µ had )0 d xβ ( x ) n →∞ ∼ (cid:90) ¯ g ( s n µ had )0 d xβ PT ( x ) + . . . . (204)The scale s n µ had is the scale at which the result is matched with perturbation theory µ PT = s n µ had . (205)One of the big advantages is that finite size scaling allows to vary this scale substantially:it can be pushed to very large values with a moderate computational effort. The Λ-parameter can be determined by matching with perturbation theory at different energyscales. We have already seen an example of such analysis in figure 3. Direct determination of the β -function Recent works use the step scaling function to directly determine the β -function. Thisapproach has some technical advantages like making the determination of arbitrary ratiosof scales µ /µ possible, or allowing different scales factors s to be used to fix the same setof coefficients. Several parametrizations are possible. At high energies the most naturalone is just to write β ( x ) = − x N (cid:88) n =0 b n x n , (206)with the first few coefficients fixed by the perturbative prediction. In this case, once the β -function is known, one can use µ PT µ had = exp (cid:40) − (cid:90) g ( µ had ) g ( µ PT ) d xβ ( x ) (cid:41) (207)to connect the perturbative and hadronic scales similarly to Eq. (205).Another advantage is that it allows a direct comparison of the data with perturbationtheory by comparing the non-perturbatively determined β -function and its perturbativeexpansion (see [34, 161] for some examples).69 .7.3. Matching to an experimental quantity In the previous section we have seen how finite size scaling techniques are able to solvenon-perturbatively the RG equations: ratios of scales defined in a massless scheme, like µ PT /µ had can be determined precisely. But what we really need are the values of thesescales in physical units. Following the discussion of section 4.5.2, we need to determine µ had /M ref , where M ref is a reference scale (for example the Ω mass) determined in largevolumes and with physical values of the quark masses ( i.e. in a setup that can be matchedwith an experimental input).In detail, the procedure is as follows: first one fixes the value of the coupling in agiven massless finite volume renormalization scheme to some particular value¯ g ( µ had ) = fixed , (208)for several values of L/a . Since the coupling depends only on one scale µ had ∝ /L had ,the physical volume of all these simulations is the same, up to scaling violations (i.e. thedifferent values of L/a ∝ / ( aµ ) are really different values of a at the same L = L had ).Second, one determines in large volume simulations the value of some low-energy scale at the same values of the bare coupling g and for physical values of the quark masses.This low energy scale is typically whatever reference quantity is used to set the scale.For example let us assume that it is the mass of the Ω baryon M Ω . The large volumelattice simulations yield values of the mass in lattice units, we denote this dimensionlessquantity ˆ M Ω ( a ) where we have written explicitly its dependence on the lattice spacing a . Since the bare coupling has been kept the same, the values of a in our determinationof aµ had = a/L had are the same, up to scaling violations, as the values of a in ourdeterminations of ˆ M Ω ( a ). This allows to determine the ratio µ had M Ω = lim a → aµ had ˆ M Ω ( a ) . (209)Note that the fact that different values of the quark masses or physical volumes are usedfor the determination of aµ had and aM Ω is not an issue once the continuum extrapolationis performed [49] . Once this ratio is known in the continuum, the experimental valueof M Ω can be used to determine µ had in physical units µ had = M expΩ × µ had M Ω . (210)We suggest the reader to look again to Fig. 11 for a schematic summary of the fullprocedure: the Λ-parameter, and therefore the strong coupling constant, is determinedfrom just one experimental dimensionful quantity (like M expΩ ). Perturbation theory isonly needed at scales larger than µ PT = s n µ had . This scale can be made (almost)arbitrarily large with a modest (but dedicated) computational effort.70 α MS ( M Z ) δ Λ (3)MS EM uncertaintyin the scale ( f π , M Ω ) PDG non-lattice FLAG average
Charm decouplingCharm effectsin the scale
Figure 14: Current uncertainty in the determination of the strong coupling. A 5% uncertainty in thethree flavor Λ parameter translates in a sub-percent precision in the strong coupling at the electroweakscale. Electromagnetic uncertainties in the scale (section 4.5.2) and charm quark effects (section 5) arefar from being the limiting factor in the precision (note the gap in the axes). See text for more details.
7. Present and future of lattice determinations of α s In this review we have focused on the determination of the Λ-parameter, which in ouropinion should be considered the fundamental parameter from which the strong couplingconstant can be deduced. From our point of view it has several conceptual advantages,first and foremost the fact that it is defined non-perturbatively, and therefore allows aclean separation of the role of perturbation theory. Moreover since Λ has units of mass, itsconnection with scale setting is transparent. Many determinations of the strong couplingclaim a precision below 1%. This is impressive, but if translated to an uncertainty in theΛ-parameter the achievement looks quite modest compared with other state of the artlattice QCD computations: a 5% uncertainty in Λ (3)MS is enough to achieve a 1% precisionin α MS ( M Z ). In an era where many lattice computations of decay constants, the hadronspectrum, or the anomalous magnetic moment of the muon reach a sub-percent precision,the current ≈
5% uncertainty in the Λ-parameter is a clear testimony to the difficultiesinvolved in solving this multi-scale problem.As a consequence many of the fine details needed in current state of the art latticeQCD computations are not crucial for improving the current determination of Λ MS . Letus illustrate this point in detail by discussing the effect of electromagnetic correctionsand massive sea quarks. Electromagnetic effects
They enter in any determination of the strong coupling intwo different ways.First there are the electromagnetic effects in whatever quantity is used to set thescale. If the pion decay constant is used to set the scale, one can claim that theseeffects are of the order of 0 .
3% in f π (see discussion in section 4.5.2). The size of If one uses a fermionic action that violates chiral symmetry, like Wilson fermions, the scaling viola-tions might naively be O ( a ), unless the bare coupling g is shifted by an term ∝ am q . This is a technicalissue only of practical importance, that does not change the validity of the above statements. .
05% level: well below the precision of any foreseeable result in the nearfuture.The second point where electromagnetic corrections affect the lattice determina-tions of the strong coupling is in the actual running. The running of the strongcoupling is affected by the fact that quarks are electrically charged and couple tophotons. These effects are estimated in perturbation theory (see for example [163]for a calculaiotn up to three loops) with the result that the leading effect is an O ( α s α EM ) term that can be interpreted as a correction in the leading b termof the β function. This correction is below the percent level and translates in aper-mille effect in Λ, i.e. a completely negligible effect in α s ( M Z ) Charm sea quark effects
Some lattice collaborations nowadays include a dynamicalcharm quark in their simulations (these simulations are usually labeled as N f =2 + 1 + 1), while other collaborations still ignore the effects of a dynamical charmquark ( N f = 2 + 1) . A naive estimate of the contribution of charm quark loopscan be obtained from the large- N c counting rules, which suggest that quark loopeffects are 1 /N c suppressed. Moreover decoupling arguments show that these arefurther suppressed by a factor ( E/M c ) where E is the typical energy scale of anhadronic quantity, and M c is the RGI-invariant charm quark mass. For the sake ofthe argument we will take here E ≈ . r , √ t ≈ . N c × (cid:18) EM c (cid:19) −→
3% effect . (211)One can argue that there is an additional suppression by a factor α MS ( M c ). In thiscase the overall effect will be O (1%).Either way, we have strong evidence that charm quark loop effects are dynamicallysuppressed, resulting in overall effects much smaller than the uncertainty quotedabove. • The FLAG report does not find any significant difference between lattice N f =2 + 1 and N f = 2 + 1 + 1 computations in several quantities (decay constants,quark masses, . . . ) [1]. Some of these computations agree with a sub-percentprecision. • Charm quark effects in dimensionless ratios of gluonic quantities (ie r / √ t , w /r , . . . )have been estimated recently [102] by comparing N f = 2 lattice simulations Note that it is not clear what setup is better at the current values of the simulation parameters . Adynamical charm quark has the unpleasant effect of enhancing cutoff effects due to the fact that am c istypically not very small at the simulated lattice spacing. i.e. r , t , f π , . . . ) propagateslinearly into Λ, and these effects are well below the 0 .
1% uncertainty in α s ( M Z ).Charm quark effects also affect the decoupling relations used to translate Λ (3)MS → Λ (4)MS → Λ (5)MS , which are needed to quote the value of the strong coupling at theelectroweak scale. The perturbative relations have non-perturbative corrections O (Λ /M c ), and these effects have been recently estimated to be about 0 .
2% [22].This estimate is a result of comparing N f = 2 lattice simulations with two heavyquarks and the pure gauge theory ( N f = 0). The same reference adds a factor twoto this estimate, resulting in a conservative uncertainty in Λ of about 0 . τ -decays, DIS,QCD Jets, hadron collider data and electroweak precision fits. For the lattice results weuse the FLAG averages .Without duplicating the detailed work done in the FLAG review, in what follows weshall try to assess the limiting factors in the different extractions of the strong coupling.The first naive observation is that figure 15 does not include lattice results for some ofthe methods reviewed in section 6. Let us then begin by discussing these methods. FLAG applies several quality criteria to determine which works enter in the average.These quality criteria aim at covering all the possible sources of systematic effects in thecalculation. Data sets that are unlikely to allow for a reasonable control of systematiceffects are excluded from final averages. The methods that do not qualify to enter in thefinal average are discussed in this subsection.
QCD vertices
There are three lattice works in the latest FLAG review that extractthe value of the strong coupling from QCD vertices [114–116]. In all cases the dataset does not allow for a controlled continuum extrapolation, which prevents theseworks from contributing to the average. Another work [113] has appeared since theFLAG review was published, but the extraction of the strong coupling is still donefrom simulations at a single lattice spacing. It is important to point out that each data point by the PDG in figure 15 is the average of severalworks. Some of these individual works claim smaller errors than the average. This is very similar to thesituation in the lattice methods despite the fact that the procedure for averaging is very different. Inthis review we will just take the averages at face value, leaving the reader with the task of making theirown opinion on each averaging procedure. .
11 0 .
115 0 .
12 0 .
125 0 . LATTICEPDG
FSSPotentialHQ CorrelatorsWilson Loops α MS ( M Z ) τ -decayDISJetsHadron collidersEW precision fitsPACS-CSALPHAJLQCDHPQCD 14HPQCD 10Petrezky 19HPQCD 10MaltmanBazavov 14Bazavov 19Ayala 20 Figure 15: Summary of lattice and non-lattice result for the strong coupling using different methods. Fornon-lattice results we use the PDG averages [100]. For the lattice results we quote the works that enterin the FLAG average [1] and the recent updates that have been published after the FLAG review. Thegray boxes are the FLAG average for each method. The recent updates (Petrezcky 19 [106] and Bazavov19 [164]) are not included in the current FLAG averages because they were published after the FLAGupdate. See text for more details. (References: Maltman [165], HPQCD 10 [107], Petrezky 19 [106],HPQCD 14 [139], JLQCD [138], Ayala 20 [130], Bazavov 19 [164], Bazavov 14 [129], ALPHA [166],PACS-CS [156]).
The limiting factor in these determinations is the ability to reach energy scalesthat are sufficiently high to make contact with perturbation theory while havingthe continuum extrapolation under control.
HVP
Two works extract the strong coupling from the vacuum polarization [143, 142].Nevertheless these works do not qualify for the FLAG average because the contin-uum extrapolation is not under control.Here we face the same issues: the difficulty to reach the perturbative region whilehaving the continuum extrapolation under control.
Eigenvalues of the Dirac operator
There is only a single work that determines thestrong coupling from the eigenvalues of the Dirac operator [144]. The extraction isperformed at a very low energy scale, where α ≈ . − .
4. According to the FLAGcriteria, this does not allow to control the matching with perturbation theory.74ll in all, these methods fail to show convincing evidence for a safe contact with theasymptotic perturbative behavior and/or fail to show that the continuum limit can bereached at the energy scales needed to extract the value of α s . At the time of writing,we think that it would be better to use the value of the strong coupling as an inputto investigate the physics related with these phenomena, instead of using these physicseffects to perform a precision determination of the fundamental parameters of the SM.Of course this situation might change in the future. More powerful computers mightallow to reach higher energies. Eventually a safe contact with perturbation theory in thecontinuum can lead to precise values of α s , but this is not the case at the moment. Next we will comment critically on the methods that actually enter in the final FLAGaverage (see Fig. 15). These methods show compelling evidence of producing robustresults in the continuum limit (by using several lattice spacing to extrapolate their results)and reach energy scales where the perturbative matching is convincing.A crucial point in the following discussion is the estimate of the truncation error, i.e. the uncertainty that is introduced in the determination of the coupling by workingat a given order in perturbation theory. As we shall see below, this is the main sourceof uncertainty in most of the determinations of the strong coupling . Here we willlook at these uncertainties in detail using the scale variation method. In particularwe will examine how a given measurement of an observable in three flavor QCD at aphysical scale Q produces different determinations of α MS ( M Z ) when the ratio betweenthe renormalization and physical scales is varied. Without loss of generality, we willassume that the observable is determined in N f = 3 QCD and normalized as a coupling(as discussed at the beginning of section 6) α ( Q ) = α MS ( sQ ) + n (cid:88) k =2 c k ( s ) α k MS ( sQ ) . (212)The procedure that we employ to estimate the uncertainty in our determination of α MS ( M Z ) is detailed in appendix B. The reader interested in numerical values should con-sult the freely available package https://igit.ific.uv.es/alramos/scaleerrors.jl .In summary we proceed as follows:1. Fix the value α ( Q ) using a canonical value Λ (3)MS = 341 MeV (equivalent to α MS ( M Z ) ≈ . ) for some value of the scale µ = s ref Q . This is achieved by reverse engi-neering , i.e. by computing α ( Q ) = α MS ( µ ) + n (cid:88) k =2 c k ( s ref ) α k MS ( µ ) , ( µ = s ref Q ) . (213) The FLAG estimate of the truncation uncertainties for the static potential, heavy quark correlatorsand small Wilson loops determinations are larger that the uncertainties quoted by some of the worksthat enter in the average, in some cases by more than a factor two. This has in fact produced somecontroversies (see for example [167]). Note that since we assume that the observable is determined in three-flavor QCD, we need to crossthe charm and bottom thresholds. Q [GeV] FLAG error [%] δ (cid:63) (4) [%] δ (2) [%] δ (cid:63) (2) [%]1.5 1.4 2.6 2.7Potential 4 2.5 0.9 1.5 1.55.0 0.4 0.8 0.8HQ r m c r m c r m c r m c − log W − log W /u Table 4: Summary of truncation uncertainties on α MS ( M Z ) estimated by varying the scale. We comparethe error quoted by flag with a change of scale by factors two and four around s = 1 or s = s (cid:63) (thevalue of fastest apparent convergence). For each method we quote the number of known loops in therelation between the observable and the MS scheme according to h counting in section 6. Details on thedifferent types of extractions can be found in section 7.2.2 (potential), section 7.2.3 (HQ), section 7.2.4(log W , log W ) and section 7.2.1 (FSS). where α MS ( µ ) is the value of the three flavor coupling at scale s ref Q obtained fromΛ (3)MS = 341 MeV.2. Use Eq. (212) again, in order to determine the values of α MS ( sQ ), by solving α ( Q ) = α MS ( sQ ) + n (cid:88) k =2 c k ( s ) α k MS ( sQ ) . (214)at the values s = s ref / , s ref .3. Use the 4- and 5-loop β -function to run the values of α MS ( sQ ) obtained in step 2. tothe reference scale M Z (crossing the charm and bottom thresholds). A comparisonbetween the values of α MS ( M Z ) is a measure of the truncation uncertainty due tothe scale variation.The usual procedure in phenomenology is to vary the renormalization scale by a factor2 above and below some reference scale. Estimates of the truncation uncertainties thatuse renormalization scales below 1 GeV tend not to be reliable. Since many extractions ofthe strong coupling are performed at relatively low scales, the above mentioned proceduremight lead to unreasonable uncertainties. For this reason, the uncertainty resulting fromcomparing the change s ref → s ref provides complementary information on the size ofthe truncation uncertainties, specially if one explores the dependence with s ref in a range1 −
2. In order to get a more quantitative understanding of these effects, we will use thefollowing quantities. δ (4) ( s ref ) : Change the renormalization scale by a factor two above and below some refer-ence scale s ref Q . We quote a symmetric error by averaging the difference between76 . . . . . . . . . . . . . . α M S ( M Z ) s FLAG Q = 80 . GeV ( ν = 0 ) Q = 80 . GeV ( ν = − . ) (a) . . . . .
01 1 1 . . . . δ α M S ( M Z ) [ % ] s ref FLAG δ ref(2) δ ref(4) (b) Figure 16: Truncation effects in α s extractions from finite size scaling. Note that in this methods theextraction is performed at very high energies Q ≈
80 GeV. The scale dependence of the coupling at Q = 8 GeV in plot (a) is just for reference. Plot (b) shows that truncation uncertainties are typicallymuch smaller than the quoted uncertainties. the scales s ref Q and 2 s ref Q , and the difference between the scales s ref / Q and s ref Q . Note however that in some cases the error is markedly asymmetric. δ (2) ( s ref ) : Change the renormalization scale by a factor two above the reference scale s ref Q only.We will show explicitly in the computations below how the two measures δ (4) ( s ref )and δ (2) ( s ref ) depend on the choice of s ref . In principle any number O (1) is a reasonablechoice for s ref , but there can be significant differences in the results depending on itsactual value. For this reason we will explore two common choices. • Take s ref = 1. i.e. the renormalization and physical scales are the same. In thiscase the uncertainties will be labeled δ (2) , δ (4) . • Take s ref = s (cid:63) as the value of fastest apparent convergence . This value is determinedwith the condition c ( s (cid:63) ) = 0 (215) i.e. the NLO coefficient in the relation between α ( Q ) and α s vanishes (see Eq. (212)).In this case the uncertainties will be labeled δ (cid:63) (2) , δ (cid:63) (4) .A summary of the results is presented in table 4. One can readily see that thetruncation uncertainties obtained with this method are in the same ballpark as theFLAG uncertainties, except for the case of the Wilson loops, where our estimates aresubstantially larger. Once again we would like to end with a warning: estimates of thetruncation uncertainties based on the scale variation can (and have been shown to) failin some cases (see discussion in section 3.2 and specifically figure 3). The FLAG average is the result of Refs. [156, 166], wich are in good agreement with eachother. Perturbation theory is used at very high energies ( Q ≈
80 GeV), where perturba-77 . . . . . . . .
123 0 . . . α M S ( M Z ) s FLAG Q = 1 . GeV Q = 2 . GeV Q = 5 . GeV (a) . . . . .
01 1 1 . . . . δ α M S ( M Z ) [ % ] s FLAG δ ref(2) ; Q = 2 . GeV δ ref(4) ; Q = 2 . GeV δ ref(2) ; Q = 5 . GeV δ ref(4) ; Q = 5 . GeV (b)
Figure 17: Truncation effects in α s extractions from the static potential. Note that the high energy scale Q = 8 GeV is only reached in the last work [164]. The FLAG error is determined for tive estimates of the truncation uncertainties are reliable. They affect the extraction of α s only at the 0 . − .
2% level (see figures 16), well below the quoted uncertainties.The most recent work [166] explores the dependence of the extractions of Λ on thephysical energy scale over a large range of energy scales Q ∼ −
140 GeV (see Figure 3and the related discussion). This study compares their extraction of the coupling α s withthe extraction performed with several observables after extrapolating the renormalizationscale at which perturbation theory is used µ → ∞ (see discussion in section 3.2). All inall, results based on finite size scaling do not depend on estimates of the perturbativeuncertainties done in perturbation theory, although figure 16 show that they tend to berather small, as expected given that the electroweak scale is reached by non-perturbativesimulations. We should also point to the detailed study in [35].The continuum extrapolation is the main source of systematic uncertainty. Accordingto our discussions in sections 4.5 and 6.7 the method allows to to perform a controlledcontinuum extrapolation by using several values of the lattice spacing at each energyscale. In particular, the most delicate continuum extrapolations in [161] uses three valuesof the lattice spacing that vary by a factor two at each value of the scale.The statistical precision of the non-perturbative running is the main limiting factorof these determinations. Note that coupling definitions using the gradient flow – seesection 4.5.3 and 6.7 – have a very small variance. These couplings have been used inRef. [166], but not in Ref. [156] (these techniques were not known at the time), explainingthe large difference in the error between the two works: the result in [156] quotes a 2.5%error in α MS ( M Z ), while the result [166] quotes a 0.7% uncertainty. The FLAG average is basically the result of Ref. [129] with an added uncertainty becauseof the perturbative truncation errors. Since the publication of the FLAG report, two newdetermination of the strong coupling by the same group has been published [164, 130].In these last works the determination of the strong coupling is extracted at energy scales78 ≈ . − , improving significantly the previous determination, which used Q ∈ . − . ≈ . α are evaluated (the ultra-soft scale) is significantly smaller (see discussion in section 6.2).Figure 17 shows the scale dependence of the strong coupling and the truncationerrors δ ref(2) , δ ref(4) as a function of s ref . First it is important to point out that the errorestimate by FLAG, even if it comes from a completely different argument is in the sameballpark as our estimates. For the relevant energy scales used in present works [129, 164]( Q ∈ . − δα MS ( M Z ) ≈ . − . Ref. [129]:
The perturbative truncation error is estimated by changing the renormal-ization scale a factor √ α MS ( M Z ) = 0 . +0 . − . , (216)has an uncertainty between − .
7% and +1%, that is actually dominated by theperturbative truncation uncertainty. This uncertainty is smaller than the quoteduncertainty by FLAG (1 . Ref. [164]:
The estimate comes from a similar analysis, but this time the renormaliza-tion scale is varied a factor 2 above and below the physical scale. Moreover theyalso include in their estimate the effect of the different treatment of the logarith-mic corretions in the perturbative expansion (that we have ignored in our analysis).This is very similar to our approach, and the uncertainty is similar to our quoted δ (4) in table 4, with the exception that they do not symmetrize the error. Theirresult α MS ( M Z ) = 0 . +0 . − . , (217)has a very similar uncertainty than the previous work, see Eq. (216), despite thefact that they reach significantly higher energies. Obviously this is a result of themore conservative approach to the estimate of the truncation uncertainties. The same work also extract the strong coupling at shortes distances in a finite temperature setup,finding good agreement.
Ref. [130]:
This work uses the same data as reference [164], but they use the knownterms in the perturbative series to fix the normalisation of the renormalon ambigu-ity and subtract some non-perturbative ( i.e. power) corrections. Their final result α MS ( M Z ) = 0 . , (218)quote a similar uncertainty as [164], but their central value is significantly larger.In summary, the estimate by FLAG of these uncertainties (1.4%) is reasonable. Itis basically the same as the difference in central values between the two most recentworks [164] and [130], that use the same data but different strategies to match withperturbation theory. This uncertainty is also larger than the total quoted uncertainty inboth works [130, 164].Determinations of the strong coupling from the static potential have recently im-proved significantly. Also the estimate of perturbative uncertainties is more conservativecompared with previous works. These determinations are in very good shape, but thefollowing points should be better understood: • The most important point to be understood is if the perturbative region is reachedin current extractions. The significant difference in central values between extrac-tions using the same dataset but different power corrections ( i.e.
Refs. [130, 164])needs a better understanding.This difference between central values is about 1 . does not address the issue of the logarithmic corrections to the perturbativeseries, related with the IR divergences of the static potential. • Another manifestation of the same problem is the strong dependence on the valueof Λ ( N f =0) that has been observed for extractions based on the static potential forvalues of the coupling α s (cid:46) .
01 [168]. This effect has been studied in [164] andthey observe a much milder effect.Note however that in this last reference Λ is extracted as a fit over a range of energyscales. This procedure makes it much more difficult to see any dependence in theextracted value of Λ, and in fact means that all the different extractions of Λ (withthe exception of one point) use data with α > . • The effect of the bad sampling of the topological charge has still to be investigatedin detail. The ensembles at the finest lattice spacing have the topology completelyfrozen (see [169]). The impact of frozen topology can be studied by computing thescale r from ensembles that are stuck in distinct topological sectors. The numericalresults so far are all compatible within errors. Note however that the values of thetopological charge simulated (basically 2 and 0) are small. A dedicated quenchedstudy could shed some light on these issues.80 . . . . . . .
128 0 . . . α M S ( M Z ) s FLAG r r r r (a) . . . . .
01 1 1 . . . . δ α M S ( M Z ) [ % ] s ref FLAG δ ref(2) ; Q = m c δ ref(4) ; Q = m c δ ref(2) ; Q = 2 m c δ ref(4) ; Q = 2 m c (b) Figure 18: Truncation effects in α s extractions from heavy quark correlators. • A related problem is that the physical volumes are relatively small (with m π L ≈ . r , but oneshould keep in mind that the uncertainty in the scale has a small effect on the un-certainty of α s ( M Z ). It would be interesting to quantify at which level of precisionfinite volume effects start to be a concern for these determinations. The average in figure 15 is the result of the works [107, 139, 138]. Once again the mostdelicate sources of uncertainty are the perturbative error and controlling the contin-uum extrapolation. It is instructive to see how different works estimate that particularsystematic error.
JLQCD collaboration [138]:
The authors use a scale variation method, very similarto our procedure (see figure 18). A crucial difference is that they vary independentlyboth the renormalization scale in the quark mass and in the strong coupling inthe range 2 − r . This is the quantity with the bestperturbative behavior (see figure 18), but they find the continuum extrapolationvery challenging. Their result α MS ( M Z ) = 0 . , [2 . , (219)claims a ∼
2% error, mostly dominated by the perturbative truncation uncertain-ties.
Ref. [137]:
They estimate the truncation uncertainty by estimating the α ( µ ) term inthe perturbative expansion of r n . They use a range of values c (1) = ± c (1) (seeEq. (212)), which yields a perturbative truncation uncertainty that is much smaller81han the one obtained by the scale variation method. The truncation uncertaintyrepresents a negligible contribution to the uncertainty of their final result α MS ( M Z ) = 0 . , [0 . , (220)which is 3 times more precise than the JLQCD result. Ref. [106]:
This work can be considered an update of [137]. Again the perturbativeuncertainty is computed by estimating the size of the α ( µ ) term in the perturbativeexpansion of r n , but this time they allow a larger range of values for the unknowncoefficient c (1) = ± c (1) (see Eq. (212)). Their updated result α MS ( M Z ) = 0 . , [1 . , (221)has in fact a larger uncertainty. Ref [139]:
The HPQCD collaboration analyze data close to the physical charm quarkmass ( Q = m c ). Their analysis includes higher order terms in the perturbativeexpansion of their data amongst the fitted parameters. These unknown terms (upto powers α ) are constrained using Bayesian priors. The perturbative truncationerrors are the main source of uncertainty, but their final result α MS ( M Z ) = 0 . , [0 . , (222)claims an uncertainty 4 times smaller than our estimates of the truncation uncer-tainties at the scale Q = m c . This uncertainty is estimated by varying the numberof terms added to the fit. It is not clear to the authors why this estimate should be areliable estimate of the truncation uncertainties. In particular these error estimatesare substantially smaller than the usual estimates coming from scale variation.FLAG quotes a 1.2% truncation uncertainty for extractions performed at the scale m c . Our scale variation method tends to point to even larger values for the truncationuncertainty (see figures 18): about 2-3% for extractions at m c and between 1-2% forextractions at 2 m c .It is easy to see why the estimates based on the value of the unknown coefficient c (1)lead to small uncertainties. The last known coefficient in the series is given by c ( s ) = 0 . .
588 log( s ) + 2 .
052 log ( s ) . (223)This coefficient is anomalously small for s = 1. Even when multiplied by a factor 5, itleads to a small estimate for the coefficient | c | ≈ .
4. On the other hand c ( s ) can beestimated by other means. The scale dependence of c ( s ) is fully predicted by the RGequations c ( s ) = c (1) + 0 .
865 log( s ) + 2 .
425 log s + 2 .
939 log s . (224)( i.e. only the non-logarithmic dependence c (1) is unknown). This logarithmic depen-dence alone predicts a coefficient | c ( s ) | ≈ s = 2.This value for the c term is 7.5 times larger than the estimate used in the last workRef. [106] and almost 20 times larger than the estimate of [137]. There is an extra sup-pression α s ( sQ ) that makes the uncertainty smaller when s >
1, but this effect can only82 . . . . .
025 0 . . . . . δ α ( m c ) sc ( s ) α ( sm c ) c ( s ) α ( sm c )5 c (1) α ( m c ) Figure 19: Truncation effects in α s extractions using the last term in the perturbative series. Thehorizontal dashed line represent the estimate of Ref. [106] δα = 5 c (1) α ( m c ). This uncertainty issignificantly smaller than the last known term in the series (dashed curve) unless the renormalizationscale is chosen very close to the physical scale ( i.e. s ≈ only from the scale dependence of c (solid curve). account for a factor between 2.8 (for Q = m c ) and 4.5 (for Q = 2 m c ): clearly insuffi-cient to compensate the factor 7.5 or 20 in the estimate of c . Moreover, note that thissimple estimate of c ( s ) neglects completely c (1). (See Figure 19). This explains whyuncertainties based on scale variation are generally larger. Even assuming that one ordermore is known, and that c (1) = 0, the scale variation approach results in ≈ .
5% errorfor r at the charm scale.Estimates of the truncation uncertainties based on varying the number of fit param-eters constrained by priors and using Bayesian methods lacks a solid theoretical basis.We suggest that estimates based on scale variation should be preferred.These considerations, together with the complicated scaling violations (see discussionin section 6.3) make it very challenging to obtain α s with less than a 1.5% uncertaintyusing these methods. Quark masses significantly larger than the physical charm quarkmass would be required (and one would need to deal with complicated continuum ex-trapolations).There are two interesting directions for future research. First, the next order inthe perturbative relation of the heavy quark moments could be very useful in futureextractions. Second a dedicated pure gauge study, where significantly larger energy scalescould be explored, would shed some light on the difficulties associated with the truncationuncertainties and the approach to the continuum in this type of lattice determinations. There are two studies that contribute to this average [165, 107] . The main contributionto the uncertainty in these determinations is purely systematic, with the perturbativeuncertainties playing a leading role. Note that an advantage for these observables is thatthere is no need to perform a continuum extrapolation. It is very difficult to obtain anindependent estimate of the truncation uncertainties for these observables. The reasonis that the data does not follow the known perturbative prediction in the range of energy Reference [107] simply updates the analysis of [170] with a more precise determination of the scale. . . . . . . . . . . . . . α M S ( M Z ) s FLAG α W α W (a) . . . . .
01 1 1 . . . . δ α M S ( M Z ) [ % ] s ref FLAG δ ref(2) α W δ ref(4) α W δ ref(2) α W δ ref(4) α W (b) Figure 20: Truncation effects in α s extractions from quantities at the scale of the cutoff. scales reached in the numerical simulations [165, 107]. Several terms are added to theperturbative expansion, up to terms α (see Eq. (175)), and the higher-order coefficientsare fixed by fitting the lattice data to the expression α W nm (1 /a ) = α V ( µ ) + c α ( µ ) + c α ( µ ) + (cid:88) k =4 d i α i V ( µ ) . (225)Here α V is the coupling in the potential scheme (see Ref. [140]) and µ ≈ π/a with theexact value depending on the observable). A crucial point in these extractions is thediscretization used. We will focus in the same discretization as used in the works [170,107].At least one extra term, with an unknown coefficient, is necessary in order to obtain asatisfactory description of the data. Moreover, the data is not precise enough to determineall the extra coefficients, so Bayesian priors are used in order to constrain the size of thecoefficients d i ≈ ± .
5. The HPQCD analyses [107] estimate the perturbative truncationerror by varying the number of terms in expression Eq. (225). Firstly let us note thatthe perturbative expansion of α W nm is only expected to be asymptotic. In particular thecoefficients in the perturbative expansion should eventually grow. Constraining the sizeof all coefficients to the same size O (1) is not justified based on theoretical arguments.Following the exact same procedure that we used for the other observables, we canestimate the truncation error by performing a scale variation analysis. For this purpose,we describe the observable using the known perturbative coefficients in the perturbativeexpansion in terms of the renormalized MS coupling: α W nm (1 /a ) = α MS ( µ ) + c α ( µ ) + c α ( µ ) + . . . , (226)with µ ≈ . /a (the exact relation depending on the concrete quantity). For illustration,let us focus here on the shortest distance object (the plaquette), and the 1 × α W ( a ) = − c (11)1 log W = α V ( d/a ) + c (11)2 c (11)1 α V ( d/a ) + c (11)3 c (11)1 α V ( d/a ) + . . . (227) α W ( a ) = − c (12)1 log W u = α V ( d/a ) + c (12)2 c (12)1 α V ( d/a ) + c (12)3 c (12)1 α V ( d/a ) + . . . . (228)These quantities show a very strong dependence on the scale (see figures 20), pointingto truncation uncertainties of the order 3%.FLAG uses the HPQCD fit result, d ≈
2, to estimate the truncation uncertaintyas 2 α . This procedure results in a smaller value than the one obtained form the scalevariation approach δα MS ( M Z ) ≈ . , [1%] . (229)This last uncertainty is about 2.5 times larger than the estimate of HPQCD.A last piece of information comes from comparing the results in Refs. [165, 170].They use basically the same dataset (reference [165] uses a subset of the 22 quantitiesused in [170]), but analyse it using different perturbative expressions and deal with thenon-perturbative corrections in different ways. Their results for α MS ( M Z ) differ by a0.6%–1.2%.The overall conclusion is that truncation uncertainties based on varying the number ofterms constrained with Bayesian priors underestimate significantly the true uncertainties.The same phenomena have been observed in the case of determinations based on currentsof heavy quark correlators. Moreover it is important to recall that the lattice datafor these short distance quantities do not follow the perturbative prediction if only theknown coefficients are used, making it mandatory to fit several additional terms in theperturbative expansion constrained by Bayesian priors. We think that this method needsfurther study. In particular it is mandatory to find a solid estimate of the truncationuncertainties based on the same techniques that are common in the estimates of theperturbative QCD uncertainties. A detailed study in pure gauge could shed some light onthe issue. Without these insights the claimed uncertainties of some of these computations(0.6%) seem an underestimate of the true uncertainties. What can we conclude on the status of the lattice determinations of the strong coupling?First we should only consider extractions with a dataset that allows a proper control overthe different sources of systematic effects. This is the approach taken for example bythe FLAG review. For the particular case of the determination of the strong coupling,the most delicate points in the extractions of the strong coupling is the estimate of theuncertainties related with the truncation of the perturbative series and the large scalingviolations in short distance quantities. Note that controlling both sources of systematicuncertainties is challenging: cutoff effects are larger for short distance quantities, whileperturbative truncation errors are larger for low energy quantities, the window problem described in section 2.2.3 is again the limiting factor. This is why we have insisted onthese points along the review. 85any lattice methods do not allow a simultaneous control of these two sources of sys-tematic effects with current computer resources. Typically in these cases the extractionsof the strong coupling are performed at a single lattice spacing, and/or the extraction isperformed at energy scales where significant contribution from non-perturbative effectsare present. Our analytic understanding of these effects is, at best, very limited. Inpractical terms we have to deal with them by fitting these contributions. This is verydelicate, since distinguishing the perturbative running from the non-perturbative contri-butions when the data is available only in a restricted range of scales is far from easy. Wetherefore prefer to focus on methods where the data follows the perturbative predictionand non-perturbative corrections are smaller than the uncertainties.Still, with this ambitious aim, several methods that allow a reliable extraction thestrong coupling have been developed in lattice QCD. These methods differ in their con-trol over the systematic errors associated with the continuum extrapolation and thetruncation uncertainties. • Finite size scaling is the only method that offers a solution to the window probleminstead of trying to find a compromise . Arbitrarily large energy scales can bereached, and the continuum extrapolation can be performed by using several latticespacing at each constant renormalization scale.This strategy trades the systematic errors associated with the truncation of theperturbative series at relatively low scales with the statistical error accumulatedwhen computing the non-perturbative running. It remains challenging to obtainprecise results, but thanks to several recent developments, a sub-percent precisionhas been reached by these kind of determinations. The truncation uncertaintiesare negligible, since perturbation theory is typically applied at the electroweakscale. Several observables have been studied non perturbatively and in some casesagreement with perturbation theory is achieved over a large range of scales (from4 to 140 GeV) [35, 166].Our reservation with this approach is that until now only two groups have used itto determine the strong coupling, and with very similar setups. A new independentdetermination would be welcome. • Determinations based on the static potential have several advantages over mostother determinations. First, the relevant perturbative relations are known to N LO,one order more than what is typically known for other observables. Second, thenon-perturbative lattice data seems to follow the perturbative prediction for energyscales as low as 2 GeV. This energy scale can be reached at several values of thelattice spacing and the data can be extrapolated to the continuum.At the moment of writing, two new studies [164, 130] have been published. Oneof them can be considered an update of some of the works evaluated by FLAG.This new determination improves significantly in the energy scales that they areable to reach. What is more important, they treat the perturbative uncertaintiesmore conservatively than in previous works and claim a sub-percent precision inthe strong coupling. The other recent work (Ref. [130]) uses the same dataset buta different treatment of the lattice data and matching with perturbation theory.They obtain a result for α MS ( M Z ) about 1.3% larger, rising some doubts on theclaimed sub-percent accuracy of these works.86ccording to our analysis using the scale variation approach, the uncertainty quotedby FLAG is reasonable, although our method neglets the delicate issue of logarith-mic corrections to the perturbative series.As with all large volume determinations, there are reservations with this methodbeyond the estimates of perturbative uncertainties, related with the compromisesthat are made. Determinations reach high energy scales (up to 8 GeV), but theenergy scales where the continuum limit can be taken with several values of thelattice spacing are substantially lower. Some of the volumes simulated are rela-tively small ( m π L ≈ . even when a conservativeapproach is taken . These points should (and will) be investigated further, but be-yond any doubts determinations coming from the static potential have reached aconsiderable level of maturity and precision. • Some extractions based on currents of heavy quark correlators are among the mostprecise determinations of the strong coupling. Scaling violations have a complicatedfunctional form, but at least the continuum limit can be explored with several lat-tice spacing. On the other hand, and compared with extractions based on thestatic potential, our current perturbative knowledge in these extractions is oneorder less and cutoff effects seem larger. In summary, this method is more chal-lenging than the static potential from both sides of the window problem. Mostof the determinations are performed at the charm quark mass M c GeV. At theselow scales, truncation uncertainties estimated using the scale variation method areabout 2 −
3% in α MS ( M Z ). The FLAG 2019 review quotes a smaller uncertainty(1 . M c with full control over thecontinuum extrapolations. In any case, until the perturbative knowledge is knownand larger energy scales have been studied, it seems difficult to claim a smalleruncertainty than what the FLAG review assigns to these extractions. • Finally, there is still work to do in order to better understand the determinationsbased on lattice observables defined at the cutoff scale. Despite these methodsclaiming the smallest uncertainties, this claim is not backed up by a solid analysisof the truncation uncertainties. A scale variation approach points to significantlylarger uncertainties. In fact the truncation uncertainties in these extractions seemto be significantly larger when using several different methods. Our numericalinvestigation suggests that even the FLAG error is still underestimating this sourceof systematic uncertainty.One should also point out that in pure gauge there is a significant discrepancybetween some recent extractions based on observables defined at the cutoff scale87nd a recent extraction using finite size scaling (see [171]).Let us end this section with a general comment about figure 15. Basically everymethod to extract the strong coupling using lattice QCD seems to be able to reach abetter precision than any phenomenological extraction. This seems at least in some cases,to be fully justified based on the general principles on which we have insisted along thereview: the most precise phenomenological determination, the extraction of α s from τ decays, is performed at a fixed energy scale m τ ≈ . µ ≈ − α s Now that we understand the main limitations of the different lattice methods to extractthe strong coupling we are in a good position to discuss what is needed in order tosubstantially reduce the current uncertainty in the strong coupling.From a general point of view we must realize that, with the exception of finite sizescaling methods, the limitations of the lattice determinations of the strong couplingare a direct consequence of the window problem : the fundamental compromise betweenreaching large energy scales, where the perturbative behavior is better, and using lowrenormalization scales, where the continuum extrapolation is well under control. Oneshould also be aware that truncation uncertainties decrease with powers of the coupling,and therefore slowly ( i.e. logarithmically) with the scale (see figure 21). This makes thewindow problem hard to solve by brute force.What progress can we expect on this front? Since the window problem is basically acomputational limitation we expect to see improvements in the future by just waiting.Computer power has been steadily improving during the last 50 years, and most probablywill continue to do so, with exascale machines looming on the horizon. Pushing the UVcutoff of a simulation ( i.e. reducing the lattice spacing of the simulation) by a factor twokeeping the physical parameters fixed multiplies the computational cost by a factor = 128. We can therefore expect that on exascale machines the state of the art latticesimulations will be performed on lattice spacing reaching down to a ≈ .
02 fm, assumingthat topology freezing is still under control on such fine lattices. Beyond the trivial ob-servation that smaller lattice spacing would allow to check the extrapolations thoroughly,the expected increase in computing power would allow the following improvements.1. Determinations based on QCD vertices and the hadron vacuum polarization, thatare nowadays limited by the size of scaling violations, will be able to perform a Naively the simulation scales with the lattice volume ∝ ( L/a ) . At the same time simulations atsmaller lattice spacing show larger autocorrelations, that are expected to scale like a in absence oftopology freezing. On top of these six powers of a , the integration of the molecular dynamic equationsrequires to reduce the step size at smaller lattice spacing, which brings down another power of a to thescaling. . . . . . µ [GeV] δ α M S ( M Z ) [ % ] .
04 fm < a < . .
02 fm < a < .
04 fm
SFStatic Potential
HQ correlators ( r )HQ correlators ( r )HQ correlators ( r )HQ correlators ( r ) W × W × Figure 21: Scale truncation uncertainties for different lattice methods (we quote δ (2) , see section 7.2).The dark shaded region is the current accessible range for large volume lattice simulations nowadays.The light shaded region represents the accessible scales if a reduction by a factor two in the lattice spacingis possible in the future. Note that most methods require a continuum extrapolation with several latticespacings: in practice the accessible renormalization scales for all methods except finite size scaling (SF)is at least factor two/four smaller than the lattice spacing. continuum extrapolation at fixed renormalization scale with several lattice spacing.This will bring the scaling violations under control.As of today these methods require to include power corrections as fit parameters todescribe the lattice data. Having access to shorter distances would clarify whetherthe perturbative region can be accessed without having to include these powercorrections.2. Pushing the UV cutoff a factor two higher would allow to match with the pertur-bative regime at larger renormalization scales.For determinations based on the static potential, whose β -function is known up to4-loops, one expects truncation errors O ( α ) to be reduced. Note however that thelatest works have reached high energy scales Q ≈ β -function is only known to 3-loops (with truncation errors formally O ( α )), butthe scale typically used in the extractions is significantly lower (we use Q ≈ ¯ m c ).At these low scales the strong coupling runs faster. Using our estimates for theperturbative truncation effects, these methods could reach a precision ∼
1% witha solid and conservative estimate of the truncation uncertainties.89eterminations based on observables defined at the cutoff scale have a more uncer-tain future. Truncation uncertainties estimated using the scale variation approachpoint to a not very well behaved perturbative series for these observables. Themain benefit would come from the possibility of reaching the perturbative domain without having to add any additional terms to the known perturbative expansion( i.e. reaching comfortably the perturbative region).3. Determinations based on finite size scaling are not limited by the systematic errorassociated with the continuum extrapolation or the truncation of the perturbativeseries, but by statistics. An increase in computer power by a factor 2 would help inreducing the statistical uncertainty dramatically. Naively one expects that the non-perturbative data would become an order of magnitude more precise , decreasingthe uncertainty to the level of 0 . seems impressive, but this will be eaten by just simulating values of the lat-tice spacing two times smaller. Some compromise between reducing the statisticalerrors, and improving the continuum extrapolations would be needed. Moreoversome ingredients of the determination ( i.e. the scale setting), that nowadays repre-sent a very small contribution to the total uncertainty, would need to be computedon large volumes where the uncertainties are not merely statistical.Nevertheless, it is clear that such an increase in computer power would allow toreduce the uncertainty in Λ by a factor two to three. A determination of thestrong coupling with an uncertainty (cid:46) .
3% would in principle be possible. Ofcourse at such level of precision one would have to think about other problems, likeelectromagnetic contributions to the scale and the running.Beyond the improvement coming from the increase in computational power, thosedeterminations that are limited by the truncation of the perturbative series could benefitfrom a better perturbative knowledge. This is especially true for the case of the momentsof heavy quark correlators and the observables defined at the cutoff scale, since therelation to the MS scheme is only known to 2-loops, and increasing this to 3-loops seemsfeasible . This would naively suppress the perturbative truncation effects by an extrapower of α , so potentially a reduction of the error by a factor 3 − ifthe size of the perturbative coefficients does not increase . However this assumption is notcompletely innocent. One should never forget the asymptotic nature of the perturbativeseries: higher order is never as good as higher renormalization scales .All in all, experience shows that progress in Lattice QCD comes always from twofronts: computer power, and the new methods and better understanding of the prob-lems by the community. We can speculate the improvements that more computer powerwould bring, but the most interesting and potentially better improvements coming fromthe new methods are much more difficult to predict. On this front there is a recent pro-posal [172] that allows to determine the N f -flavor Λ-parameter from the pure gauge one. As with all Monte Carlo methods, the statistical uncertainties of a lattice computation decrease ∝ / √ N where N is the number of configurations. The computer requirements are proportional to N . Note however that for the case of observables defined at the cutoff scale, this requires a perturbativecomputation in
Lattice perturbation theory , that is substantially more complicated. α s using this method yet. We have discussed that the limitations usually found in the determination of the strongcoupling are not the same as those of other state of the art lattice QCD computations.As a consequence of the window problem, the truncation of the perturbative series andthe large scaling violations present in short distance observables represent the limitationof all lattice methods to determine the strong coupling, with the exception of finite sizescaling. Statistical accuracy is the main limitation of finite size scaling.The determination of the Λ parameter in the pure gauge theory faces exactly thesame limitations. On the other hand, computationally the pure gauge theory is muchmore tractable than QCD. Simulations algorithms for the pure gauge theory are muchmore efficient, and smaller physical volumes are acceptable, since finite volume effectsare suppressed by the larger mass of the glueballs.We do not need to wait 10 years to simulate the pure gauge theory at a latticespacing of a = 0 .
02 fm. Testing what precision in the strong coupling could potentiallybe achieved with 100 times more computer time can be done just by determining the Λparameter in pure gauge.In fact the situation in pure gauge is not that clear, with some recent determinationsclaiming an uncertainty ≈ .
5% in the Λ parameter, but disagreeing by about a 5% .A very detailed and precise study is been performed using the static potential [168, 131],but at the moment the preliminary results are inconclusive. There is not a single puregauge determination using heavy quark correlators.Given the similarities in the challenges, we hope that the lattice community takes thisdiscrepancy seriously and does not neglect the issue due to the lack of phenomenologicalinterest. Pure gauge studies do not enter in the world average, but in our opinion theywill be crucial to help us understand the problems that we have to face and how to solvethem.
8. Conclusions
Lattice QCD is, in principle, ideally suited to compute the non-perturbative quantitiesthat are necessary in order to extract the fundamental parameters of the standard model.Being a non-perturbative formulation of QCD, it allows a determination of the value ofthe strong coupling at energy scales that are measured in units of some well-known QCDspectral quantity ( i.e. the proton mass, or the meson decay constants). The determinations [173–175], based on observables defined at the cutoff scale point to a value r Λ ≈ . ≈ . α s . Theextraction of the strong coupling requires the application of perturbation theory, andideally one would like to use perturbation theory at very high energy scales, where goodconvergence is expected. This wish is in conflict with an intrinsic limitation of anynumerical simulation: with finite computing resources only a limited range of scales canbe resolved by a single lattice QCD simulation. Datasets that have been generated inorder to study the low-energy properties of the strong interactions ( i.e. the spectrum,decay form factors, the anomalous magnetic moment of the muon, . . . ) are limited toreach a few GeV in energy scales at most. This explains why most lattice QCD extractionsof α s are limited by the uncertainties associated with the application of perturbationtheory and the truncation of the perturbative series at relatively low-energy scales.In writing this review we have focused on explaining our vision of the field. We havetried to highlight the challenges that lattice determinations of the strong coupling have toface, and insisted on the specific difference between these and other lattice computations.In the last 15 years lattice QCD has made enormous progress. We have witnessed thesurge of dynamical simulations, the values of the lattice spacing has been pushed downto a ∼ .
05 fm, some simulations reach physical volumes of 7 fm and simulations at thephysical point, that once seemed out of reach, are nowadays common.This progress has made it possible to produce solid, first principle predictions in thelow-energy regime of the strong interactions, with a tremendous impact in flavor physics,searches of beyond the standard model effects and many other topics in high energyphysics. It is this very same progress that has pushed the lattice determinations of thestrong coupling to the top of the podium: as the overall quality of lattice simulationshas improved, the determination of the strong coupling has become dominated by latticeQCD results.But this situation is changing right now. The limitations in current lattice studies oflow-energy hadronic processes are related to the inclusion of charm effects, electromag-netic corrections, and how to deal with the large (power-like) finite volume effects thatappear in QED. Solving these issues will have little benefit for the lattice determinationsof the strong coupling.Progress in lattice determinations of α s will come from dedicated approaches . Mereupdates of lattice determinations in new ensembles that are generated with the aimof studying electromagnetic corrections, or any other relevant problem of low-energyQCD, will very soon be irrelevant. The lattice community should embrace this as anopportunity to make real progress. The current problems of topology freezing, newtechniques to simulate very fine lattices, and more important than anything, the studyof alternative techniques to solve multi-scale problems should be taken seriously .We hope that this review work will further stimulate the lattice community to thinkabout these issues.The high-luminosity upgrade of the LHC will require high-precision QCD predictionsin order to really benefit from the increase in statistics at the experiments. With thedominance of lattice determinations of the strong coupling in the world average, andthe relevance of α s for precision phenomelogy at the LHC, it is important for the phe-nomenology community to understand the problems involved in lattice determinations ofthe strong coupling, to have some appreciation of the systematic errors that limit thesestudies, and to foster the progress in the lattice community. Providing a self-containedintroduction to the topic for non-lattice specialists has been one of the focal points of92ur work. We have given an introduction to lattice QCD with an emphasis in the areasthat are more relevant in the extractions of the strong coupling: the continuum extrap-olation of lattice data, the process of scale setting, and the subtleties involved in theanalysis of lattice QCD data at fine lattice spacings. Moreover we have dissected all dif-ferent approaches to extract α s via lattice simulations, trying to be as critical as possible.With one important exception, lattice extractions of α s based on finite size scaling (seesection 6.7), all lattice determinations of the strong coupling represent a compromisebetween the potentially large cutoff effects present in every lattice determination of ashort distance quantity and the potentially large systematic effects associated with theuse and truncation of perturbation theory at energy scales of a few GeV. Different latticemethods take different compromises and suffer from these effects in very different ways.Our point of view is that only methods that show convincing evidence to have reachedthe perturbative running and are able to explore the continuum limit with several latticespacing should be used to determine the fundamental parameters of the standard model.We have studied the perturbative truncation effects of the most reliable approaches indetail, trying to provide a broad picture of each of the methods. Our conclusions are thatthere are at least three methods that, as of today, allow a reliable extraction of the strongcoupling: finite size scaling (section 6.7), the static potential (section 6.2) and momentsof heavy quark correlators (section 6.3), although with very different degrees of reliance.The precision of these extractions, as well as the potential to improve substantially thecurrent determinations, have been analyzed.On the other hand we have not tried to evaluate each publication. The interestedreader in this fine detailed perspective should consult the excellent work of the FLAGreview.Let us end this review with a comment on the status of the world average. The mostprecise lattice determinations show a good agreement. The world average value of thestrong coupling performed by the PDG [100] includes all phenomenological and latticecomputations. In the PDG average individual works are classified according the methodof extraction, and pre-averaged, before determining the final world average. The list ofmethods includes several phenomenological procedure (DIS, τ decays, e + − e − , . . . ). Therationale behind such a procedure is that different works within the same pre-averagehave to face the same systematic effects. As we have seen different lattice strategiesfor the extraction of the strong coupling are limited by very different reasons. In fact, once the continuum extrapolations are under control , lattice QCD and phenomenologicaldeterminations stand on the same footing, and have to face very similar challenges (seealso [32]). Many lattice QCD extractions using moments of heavy quark correlators aresomehow similar to extractions using τ decays: the extraction is performed at a relativelylow scale and the systematics associated with the use of perturbation theory at such lowscales dominate the error budget . Lattice QCD extractions of the strong couplingbased on finite size scaling are similar to phenomenological extractions based on Z bosondecays in the sense that they are performed directly at the electroweak scale and limitedby statistics.We would like to see a world average of α s that groups different extractions basedon similar systematics. Questions like at What scale is perturbation theory applied?, Note however that the perturbative expansion of the R τ,V + A is known up to four loops, while theperturbative expansion of the heavy quark correlators is known only up to 3-loops. α s , regardless of it being a lattice or a phenomenology extraction.We hope that this work is also useful to the brave that attempt to produce such a worldaverage. Acknowledgments
The authors want to thank G. Salam for interesting discussions and a critical reading ofan earlier version of this manuscript. We also thank him for hospitality in Oxford.AR has a large debt with the members of the ALPHA collaboration for a fruitfulcollaboration in several works, spanning several years, about many issues covered in thisreview. AR also wants to show his gratitude to Rainer Sommer and Stefan Sint for themany discussions on several topics covered in this review.We warmly thank Peter Petreczky for sharing his data and scripts to produce fig-ures. 10, the authors of [113], in special Jose Rodriguez Quintero and Savvas Zafeiropou-los, for sharing and explaining the data used to produce figure 8 (b). Rainer Sommerfor sharing figure A.23, and Christopher Kelly and the RBC/UKQCD collaboration forpermission to reproduce figure A.25.LDD is supported by an STFC Consolidated Grant, ST/P0000630/1, and a RoyalSociety Wolfson Research Merit Award, WM140078. AR is partially supported by theGeneralitat Valenciana (CIDEGENT/19/040) and was partially supported by the H2020program in the Europlex training network, grant agreement No. 813942.
A. Challenges in Lattice QCD
A.1. Topology freezing and large autocorrelation times
Most definitions of the topological charge are not quantized on the lattice. It is onlywhen the lattice spacing of a simulation is small enough ( a < .
05 fm), that the valuesof the topological charge measured on the lattice cluster close to integer values. Thedifferent topological sectors emerge as the simulation approaches the continuum limit.As was first realized in [87], at the same time the transition between different topologicalsectors also becomes less and less frequent at small lattice spacings (see fig. A.22) .This has implications for the error estimation of observables that couple stronglywith the topological charge. Lattice measurements of these observables are correlatedfor very long simulation times, since they feel the topological sector in which they havebeen measured. Getting a solid estimate of the statistical error of such observables isreally difficult since the topological charge is not well sampled.The most clean solutions to the problem improve the sampling of different topolog-ical sectors by changing the boundary conditions of the lattice simulation in Euclidean This is expected for the kind of algorithms used in all lattice QCD simulations, since they are basedon a continous change of the fields (i.e. the HMC). Local update algorithms (such as those used in thesimulations of fig. A.22) still suffer from topology freezing at small lattice spacings. . . . . . . . . . − − − − Q (a) 32 with a = 0 .
08 fm. . . . . . . − − − − Q (b) 48 with a = 0 .
03 fm. − − − − − Q MC time (c) 32 with a = 0 .
08 fm. − − − − − Q MC time (d) 48 with a = 0 .
03 fm.
Figure A.22: The topological charge is not quantized on the lattice (here we use the definition based onthe Gradient flow), but in practice values for Q in simulations with very fine lattice spacing cluster veryclose to integer values. At the same time the transition between different topological sectors becomesless and less frequent in simulation time. time [176, 160]. In these cases the topological charge is no longer quantized (even in thecontinuum), and can fluctuate. In other cases the problem can be bypassed. One canperform the simulations at fixed topological sector and deal with power law finite volumeeffects if necessary [177]. Recently simulations on very large physical volumes so that thefinite volume effects of a fixed topology become irrelevant have been performed in the95ure gauge theory[85].All in all it is pretty uncomfortable that no algorithm is able to sample correctly thetheory at fine lattice spacings. It remains an algorithmic challenge to find an efficientalgorithm for the simulation of QCD at fine lattice spacings. In the meantime, theaccuracy of the statistical errors quoted for simulations at fine lattice spacings withoutthe use of some variant of open boundary conditions has to be taken with a grain of salt. A.2. Signal to noise problem ππ πππ
Figure A.23: The variance of a meson at large Euclidean times is dominated by two mesons propagating.On the other hand the variance of a nucleon two-point functions at large Euclidean times is not dominatedby the propagation of two baryons, but by the propagation of three meson states.
Every observable ( O ) in lattice QCD is computed as an avegare over configurationsgenerated by Monte Carlo sampling (cid:104) O (cid:105) = 1 N N (cid:88) t =1 O [ U t ] + O (1 / √ N ) . (A.1)Here U t represents the configuration at Monte Carlo time t and O [ U t ] is the value ofthe observable measured on such a configuration. All observables carry an statisticaluncertainty O (1 / √ N ) due to this stochastic estimation. The error in the observable isproportional to the variance ( δO ) ∝ (cid:104) O (cid:105) − (cid:104) O (cid:105) . (A.2)The signal to noise problem refers to the particular behavior of the error in some corre-lators. If we examine the case of a meson correlator, like the pion, we have (cid:104) [¯ uγ d ( x ) ¯ dγ u (0)] (cid:105) x →∞ ∼ Ae − M π x + . . . (A.3) (cid:104) [¯ uγ d ( x ) ¯ dγ u (0)] (cid:105) x →∞ ∼ A (cid:48) e − M π x + . . . (A.4)The first equation is clear, since this is just the squared π propagator. The secondequation becomes clear when one realizes that the Wick contractions in the expectationvalue (cid:104) O (cid:105) can be understood as 2 π mesons propagating between Euclidean times 0 and96 .5 1 1.5 2 2.5 3 3.5 4 4.50.40.50.60.70.80.911.1 x / r N m e ff − o ff s protonOmegaV(r )V(r )F π eff Figure A.24: Several meson and baryon correlators, as well as the static potential at different distances(shifted vertically). Meson correlators, do not have a signal to noise problem, and its value can bedetermined with high precision at large Euclidean times. (Source: m p [178], m Ω [179], V ( ≈ r ) , V ( ≈ r ) [180], f π [181]). x (see fig. A.23). On the other hand for the case of a baryon, like the proton p , thecorrelators have the form ( a, b, c are color indices and C the charge conjugation matrix) (cid:104) u Ta Cγ u b d c (cid:15) abc ( x ) [ u Ta Cγ u b d c (cid:15) abc (0)] † (cid:105) x →∞ ∼ Ae − M p x + . . . (A.5) (cid:104) (cid:8) u Ta Cγ u b d c (cid:15) abc ( x ) [ u Ta Cγ u b d c (cid:15) abc (0)] † (cid:9) (cid:105) x →∞ ∼ A (cid:48) e − M π x + . . . (A.6)In this case the dominant contribution at large Euclidean times for the expectation value (cid:104) O (cid:105) comes from three π meson propagating between Euclidean times 0 and x (fig. A.23).The term with two protons propagating also contributes to this expectation value, butwith a term ∝ e − M p x that decays much faster. See also [183] for more details.In summary the ratio signal-to-noise shows a quite different behavior in both cases(see fig. A.23) C π ( x ) δC π ( x ) x →∞ ∼ , (A.7) C p ( x ) δC p ( x ) x →∞ ∼ e ( M p − M π ) x . (A.8)(A.9)This exponential increase in the signal to noise has in practice important implications:it is almost impossible to determine the value of a baryon correlator with high precisionat distances of 1 fm and larger. Precise determinations of baryon masses extract such97 t/a a m e ff hhh ( t ) LWLZ B Figure A.25: Precise values of baryon masses require to fit the correlators at small Euclidean times,where there are significant contributions from excited states. (Source [182]) quantities from the values of the correlator at relatively small Euclidean times, wherethere is significant contamination of excited estates. The use of different interpolatingoperators (with different excited state contamination) and the use of several numericaltechniques are common to deal with this situation (see figs. A.24 and A.25).Once more, the situation is far from ideal. Such basic quantities as baryon massesideally should be determined without dealing with complicated systematic effects in thedetermination of the effective mass plateau, but currently there no known techniquesthat allow the precise extraction of baryon masses at distances ∼ B. Scale variation estimation of truncation errors
The definition of a non perturbative coupling in lattice QCD uses an observable O ( µ )that depends on only one scale ( µ ) and can be determined non perturbatively. Thisobservable has a perturbative expansion when µ (cid:29) Λ, that after a proper normalizationcan be written as O ( µ ) µ →∞ ∼ α MS ( µ ) + (cid:88) k> c k α k MS ( µ ) . (B.1)98he RG evolution of α MS ( µ ) is given by the β function, and can be used to obtain anestimate the truncation effects of the perturbative series, which is valid under certainassumptions on the size of the unknown coefficients. The truncated series O ( n ) ( µ ) = α MS ( µ ) + n (cid:88) k =2 c k α k MS ( µ ) + O ( α n +1MS ) , (B.2)can be rewritten, with the same level of precision, as O ( n ) ( µ, s ) = α MS ( µ (cid:48) ) + n (cid:88) k =2 c (cid:48) k ( s ) α k MS ( µ (cid:48) ) + O ( α n +1MS ) , ( s = µ (cid:48) /µ ) . (B.3)Note that the dependence of O ( n ) ( µ, s ) on s is only due to the truncation of the pertur-bative series. The coefficients c (cid:48) k ( s ) = k − (cid:88) l =0 c (cid:48) k,l log l ( s ) , (B.4)can all be determined from c k that appear in Eq. (B.2) thanks to the recursion c (cid:48) k, = c k , (B.5) c (cid:48) k,l = 2 l k − (cid:88) j =0 j (4 π ) k − j b k − j − c (cid:48) j,l − , (B.6)where b n are the coefficients of the β function Eq. (9).Given a value for the observable O ref = O ( µ ref ) determined on the lattice at somevalue of the scale µ ref , we can compare the determination of α MS ( M Z ) coming fromusing Eq. (B.3) for different values of s , and use them as an estimate of the truncationuncertainties. Note that usually lattice works with N f = 2 + 1 or N f = 2 + 1 + 1 QCD,which means that in order to obtain α MS ( M Z ) one needs to match the couplings throughthe bottom and/or charm thresholds.For numerical results the reader should look at the RunDec package [184, 101]. Wealso provide a freely available software
ScaleErrors to determine the truncation un-certainties in α s as we detail below. B.1. The case of the Schr¨odinger Funcional coupling in full detail
In this section we describe in full detail our procedure for estimating the impact of scalevariations in the case of works based on finite size scaling. The implementation for theother observables is sketched briefly in the rest of the appendix.Our starting point is the value of the three flavor theory, that we assume to beΛ (3)MS = 341 MeV , (B.7) https://igit.ific.uv.es/alramos/scaleerrors.jl µ = 1 /L . For N f = 3 QCD we have α SF ( µ ) = α MS ( sµ ) − [1 . − . s )] α ( sµ ) (B.8)+ [0 . − . s ) + 2 . s ) ] α ( sµ ) . where the log terms can be computed from expression (B.3) and the perturbative compu-tation [185, 186]. We point that the routine scale errors from the package ScaleErrors.jl (see footnote 40), takes as input the perturbative coefficients c k (1) = { . , − . , . } , (B.9)and determines the truncation uncertainties exactly as described below.We proceed as follows:1. Chose a reference value for the scale s ref and solve the non linear equationΛ (3)MS s ref µ = (cid:2) b ¯ g ( s ref µ ) (cid:3) − b b e − b g s ref µ ) exp (cid:40) − (cid:90) ¯ g ( s ref µ )0 d x (cid:20) β MS ( x ) + 1 b x − b b x (cid:21)(cid:41) . (B.10)in order to obtain α MS ( s ref µ ) ≡ ¯ g ( s ref µ ) / (4 π ). For example, using µ = 80 GeV, s ref = 2 and the 5-loop β MS function together with our reference value of Λ (3)MS = 341MeV we get α MS ( s ref µ ) = 0 . . (B.11)2. This value is plugged in Eq. (B.8) in order to obtain α SF (80 GeV) = 0 . . (B.12)3. We can now solve the polynomial equation (B.8) for different values of s and thel.h.s fixed to the value in Eq. (B.12). i.e. Using s = 4 ( sµ = 320 GeV), we solve0 . α MS ( sµ ) − [1 . − . s )] α ( sµ ) (B.13)+ [0 . − . s ) + 2 . s ) ] α ( sµ ) , To obtain α MS (320 GeV) = 0 . . (B.14)4. The two values of the coupling Eqs. (B.11), (B.14) should be equivalent, except forthe truncation uncertainties. In order to compare them, we run the two values toa common scale ( M Z ), crossing the charm and bottom thresholds. We get α MS (80 GeV) = 0 . ⇒ α ( N f =5)MS ( M Z ) = 0 . , (B.15a) α MS (320 GeV) = 0 . ⇒ α ( N f =5)MS ( M Z ) = 0 . . (B.15b)Let us give a few details on how this is done. Using the 5-loop three flavor betafunction and the charm quark mass m (cid:63) c = m c ( m c ) = 1275 . (cid:18)
80 GeV m (cid:63) c (cid:19) = (cid:90) ¯ g (80 GeV)¯ g ( m c ) d xβ ( N f =3)MS ( x ) . (B.16)100n order to obtain the value of the three flavor coupling at the charm scale. Weobtain α MS (80 GeV) = 0 . ⇒ α MS ( m (cid:63) c ) = 0 . . (B.17)We now “cross the charm threshold” by using the decoupling relations [17–21] todetermine the four flavor coupling at the scale m (cid:63) c . α MS ( m (cid:63) c ) = 0 . ⇒ α ( N f =4)MS ( m (cid:63) c ) = 0 . . (B.18)A similar procedure is used to run the coupling to the bottom quark threshold m (cid:63) b = m b ( m b ) = 4198 . α ( N f =4)MS ( m (cid:63) c ) = 0 . ⇒ α ( N f =4)MS ( m (cid:63) b ) = 0 . ⇒ α ( N f =5)MS ( m (cid:63) b ) = 0 . . Finally, one uses Eq. (B.16) using the five flavor β MS function to run the couplingto the scale M Z .5. Finally, the difference between values of α ( N f =5)MS ( M Z ) in Eqs. (B.15) δα ( N f =5)MS ( M Z ) = 8 . × − [0 . . (B.19)can be used as an estimate of the truncation uncertainties. B.2. The static potential
We start from the perturbative expression of the potential V ( r ) V ( r ) = − r (cid:88) n =0 P n (cid:16) α s π (cid:17) n , (B.20)where α s = α MS (1 /r ) is the strong coupling. Note that we ignore the logarithmic cor-rections ∝ log α due to the IR divergent nature of V ( r ). See section 6.2 for more details.Using the RG equations, we haved α s d r = 2 α s r (cid:88) i =0 (4 π ) i b i α is , (B.21)which we can use to evaluate the derivative of the static potential with respect to thescale r : F ( r ) = d V ( r )d r = 43 r (cid:88) n =0 P n (4 π ) n α n +1 s − r (cid:88) n =0 n + 1) P n (4 π ) n α n +2 s (cid:88) j =0 (4 π ) j +1 b j α js . (B.22)Collecting the coefficients in the equation above, we get an expression for the perturbativeexpansion of the force, as a function of the number of flavors N f . We use the force as theobservable that determines the strong coupling constant, with the typical scale associated101o the observable being µ = 1 /r . In N f = 3 QCD the known terms in the perturbativeseries [120, 187, 125, 188] together with expression Eq. (B.3) allows to write α qq ( µ ) µ →∞ ∼ α MS ( sµ ) + [ − . . s )] α ( sµ )+ [0 . . s ) + 2 . s ) ] α ( sµ )+ [0 . . s ) + 2 . s ) + 2 . s ) ] α ( sµ )+ . . . . (B.23)The scale of fastest apparent convergence is reached at s (cid:63) = 1 . ∝ α ( Q ). If we assume a fixed value for the ultra-softscale ν us = 2 GeV, the last term would read instead: · · · + α ( sQ )[0 . . s ) + 2 . s ) + 2 . s ) ] . (B.24)This has an effect in the truncation uncertainties at the 20% level. B.3. HQ correlators
The perturbative expansions for the ratios of moments α HQ , n is given by[136, 189, 190]: α HQ , ( µ ) µ →∞ ∼ α MS ( sµ ) − [0 . − . s )] α ( sµ ) (B.25)+ [0 . . s ) + 2 . s ) ] α ( sµ ) + . . . .α HQ , ( µ ) µ →∞ ∼ α MS ( sµ ) + [0 . . s )] α ( sµ ) (B.26) − [0 . − . s ) − . s ) ] α ( sµ ) + . . . .α HQ , ( µ ) µ →∞ ∼ α MS ( sµ ) + [1 . . s )] α ( sµ ) (B.27)+ [0 . . s ) + 2 . s ) ] α ( sµ ) + . . . .α HQ , ( µ ) µ →∞ ∼ α MS ( sµ ) + [1 . . s )] α ( sµ ) (B.28)+ [0 . . s ) + 2 . s ) ] α ( sµ ) + . . . . B.4. Wilson loops
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