Probing QCD perturbation theory at high energies with continuum extrapolated lattice data
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Probing QCD perturbation theory at high energieswith continuum extrapolated lattice data
Stefan Sint , a for the ALPHA collaboration School of Mathematics & Hamilton Mathematics Institute,Trinity College Dublin, Dublin 2, Ireland
Abstract.
Precision tests of QCD perturbation theory are not readily available from ex-perimental data. The main reasons are systematic uncertainties due to the confinementof quarks and gluons, as well as kinematical constraints which limit the accessible en-ergy scales. We here show how continuum extrapolated lattice data may overcome suchproblems and provide excellent probes of renormalized perturbation theory. This workcorresponds to an essential step in the ALPHA collaboration’s project to determine the Λ -parameter in 3-flavour QCD. I explain the basic techniques used in the high energyregime, namely the use of mass-independent renormalization schemes for the QCD cou-pling constant in a finite Euclidean space time volume. When combined with finite sizetechniques this allows one to iteratively step up the energy scale by factors of 2, therebyquickly covering two orders of magnitude in scale. We may then compare perturbationtheory (with β -functions available up to 3-loop order) to our non-perturbative data for a1-parameter family of running couplings. We conclude that a target precision of 3 percentfor the Λ -parameter requires non-perturbative data up to scales where α s ≈ .
1, whereasthe apparent precision obtained from applying perturbation theory around α s ≈ . Lattice QCD is usually thought of as a tool to extract non-perturbative information about QCD in thehadronic regime. Therefore it is widely believed that lattice QCD is limited to the low energy regimewhere the cuto ff (i.e. the inverse lattice spacing, 1 / a ) sets the limit for accessible scales, typically at afew GeV. In this talk I would like to dispel this myth and draw the wider QCD community’s attentionto the fact that lattice QCD provides excellent ways to probe perturbation theory at high energies andindeed covering a range of energy scales orders of magnitude apart (cf. [1] and references therein).To understand the origin of the above mentioned misconception we first need to remind ourselvesthat hadronic physics is done in physically large volumes such that the linear extent of the volume, L , is large in units of the Compton-wave length of the pion, 1 / m π , which is the lightest particlearound. At least for single particle states, the infinite volume limit is reached exponentially fast [2]and, depending on the target precision and the particular observables under study, it may be su ffi cient a e-mail: [email protected] a r X i v : . [ h e p - l a t ] D ec PJ Web of Conferences to require m π L > .
04 - 0 . L / a = ff ects are then not dominated by pion physics and might still be under control for theperturbative observables under study. One may take this a step further by making the finite volume apart of the definition of the perturbative observable [4]. In this case there is no need to extrapolate toinfinite volume and one may freely move up and down the energy scale, thus reaching high energiesof O(100) GeV. The main drawback is that the finiteness of the volume becomes a defining propertyof the observable, so that perturbation theory must be done in finite volume, too. Depending on thechosen boundary conditions this may substantially enhance the technical di ffi culties of perturbationtheory. Observables in QCD are defined in terms of renormalized correlation functions of gauge invariantcomposite fields via the QCD path integral, (cid:104) O (cid:105) = Z − (cid:90) D [ A , ψ, ¯ ψ ] O [ A , ψ, ¯ ψ ] exp ( − S ) , (1)where A µ , ψ and ¯ ψ denote the gluon, quark and anti-quark fields and we have here assumed theEuclidean framework. Many such correlation functions have a well-defined perturbative expansionsin powers of a renormalized coupling, α s ( µ ) = ¯ g ( µ ) / π , (cid:104) O (cid:105) = c + c α s ( µ ) + c α s ( µ ) + . . . , (2)where the renormalization scale µ is a priori arbitrary. However, the perturbative series behaves best if µ is chosen close to the relevant physical scales involved in the correlation function, given typically byparticle masses, energies or momenta. Moreover, for the perturbative description to become accurate, µ must be in the “perturbative regime", µ (cid:29) Λ , where the Λ -parameter is around a few hundred MeVin the MS scheme. How can a statement about the accuracy of perturbation theory be made morequantitative? Besides non-perturbative data, obtained either from experiment or from the lattice, onewould like to have several orders of the perturbative series available. However, due to the asymptoticnature of the series it is probably even more important to vary the size of the expansion parameter, α s ( µ ), which is tantamount to varying the scale µ .For observables defined as in eqs. (1,2) one may normalize the perturbative expansion by definingan “e ff ective coupling", through α O ( µ ) = (cid:104) O (cid:105) − c c = α s ( µ ) + c (cid:48) α s ( µ ) + c (cid:48) α s ( µ ) + . . . , (3)so that its expansion starts with α s ( µ ). In principle, α O ( µ ) can be derived from an experimentallymeasured observable. However, some practical problems tend to limit the accuracy:1. While perturbation theory takes the path integral as starting point, its non-perturbative connec-tion to the experimental observable requires some assumption about the transition from quarkand gluon to hadronic degrees of freedom. The root of the problem is confinement of quarks inhadrons, an inherently non-perturbative phenomenon. ONF12
2. The scale µ is usually constrained by the kinematics of the experiment under consideration.For example in τ -decays, µ is essentially determined by the τ -lepton’s mass [5, 6]. It is thennot possible to vary the energy scale and the only control over the perturbative expansion isobtained by studying the apparent convergence of the asymptotic series.3. For observables defined at energies of a few GeV, the dependence of the e ff ective coupling onthe charm and bottom quark masses cannot be ignored so that one uses e ff ective theories with N f = , , ff " the charm and bottomthresholds. As a result the e ff ective coupling becomes a non-perturbatively defined running couplingin a quark mass independent renormalization scheme [7] for 3-flavour QCD, and its scale dependencecan be traced over a range of scales several orders of magnitude apart. An attractive class of finite volume couplings can be obtained by imposing Dirichlet boundary condi-tions on the fields at Euclidean times x = x = T . Due to its relation with the Schrödinger rep-resentation in Quantum Field Theory [8], the path integral in this case is referred to as the SchrödingerFunctional (SF) [9, 10]. The SF can be considered a functional of the boundary values of the gaugefield, whereas homogeneous Dirichlet conditions are imposed on (half the components of) the quarkand antiquark fields [10]. In a continuum language the spatial components of the gauge potential, A µ ,are set to abelian, spatially constant boundary fields, C k , C (cid:48) k , A k ( x ) | x = = C k , A k ( x ) | x = T = C (cid:48) k . (4)The boundary fields are chosen to depend on 2 real parameters [11], η, ν , corresponding to the 2abelian generators of SU(3), C k = iL diag (cid:32) η − π , η (cid:32) ν − (cid:33) , − η (cid:32) ν + (cid:33) + π (cid:33) , (5) C (cid:48) k = iL diag (cid:32) − π − η, η (cid:32) ν + (cid:33) + π , − η (cid:32) ν − (cid:33) + π (cid:33) , (6)independently of the direction k = , ,
3. For fixed parameters and up to gauge equivalence it has beenrigorously shown in [9] that the absolute minimum of the action corresponds to an abelian spatiallyconstant background field, B µ , B k ( x ) = C k + x T (cid:16) C (cid:48) k − C k (cid:17) , B = . (7)with classical action S [ B ] = π + η ) /g . The e ff ective action is then unambiguously definedthrough, e − Γ [ B ] = (cid:90) D [ A , ψ, ¯ ψ ]e − S [ A , ψ, ¯ ψ ] , (8)with its perturbative saddle point exansion given as usual by Γ [ B ] = g Γ [ B ] + Γ [ B ] + O( g ) , Γ [ B ] = g S [ B ] . (9) PJ Web of Conferences
A family of couplings in the SF scheme is thus obtained by defining [1, 12]1¯ g ν ( L ) = ∂ η Γ [ B ] ∂ η Γ [ B ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η = = (cid:68) ∂ η S (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) η = π = g ( L ) − ν × ¯ v ( L ) (10)Note that the η -derivative produces an expectation value of an observable which can be measured in anumerical simulation. While we set η =
0, the dependence on the parameter ν is completely explicit,so that a calculation at ν = g ν , in terms of ¯ g = ¯ g ν = and a second observable, ¯ v . The η -derivative can be interpreted as a variation of the backgroundfield, and the SF couplings are thus defined by the response of the system to a change of such a colourelectric field. Finally, the quark masses are set to zero and any remaining dimensionful quantities suchas the Euclidean time extent, T , or the strength of the background field are scaled proportionally to L . The couplings thus depend on a single scale which is naturally identified with the renormalizationscale, i.e. µ = / L . In perturbation theory the SF coupling [11, 13] has been matched to the MS coupling up to 2-looporder [14–17] and therefore its 3-loop β -function can be inferred from the MS -scheme [18, 19]. Inour conventions we have β (¯ g ) = − L ∂ ¯ g ( L ) ∂ L , β ( g ) = − b g − b g + . . . , (11)with universal coe ffi cients, b = (cid:16) − N f (cid:17) / (4 π ) , b = (cid:16) − N f (cid:17) / (4 π ) , (12)and the 3-loop coe ffi cient(s), b ,ν = ( − . − ν × . / (4 π ) . (13)For tests of perturbation theory the definition of our observables in a finite Euclidean space-time vol-ume represents a real advantage. In particular, the infrared cuto ff by the finite volume prevents anyrenormalon issues. The fact that the minimum action configuration is unique makes the saddle pointexpansion straightforward. However, as is always the case with asymptotic expansions, exponentiallysmall corrections to the series are neglected. Their size depends on the choice of observable and thevalue of the coupling. In our case such terms originate from secondary minima of the classical actionand one would expect such contributions to be suppressed by exp( − ∆ S ) with ∆ S the di ff erence be-tween the classical action taken at a secondary minimum and at the absolute minimum, respectively.We have investigated this issue [20] and found the nearest stable stationary point of the action corre-sponds to g ∆ S = π /
3. Hence, for the range of couplings used in our work such contributions arecompletely negligible. Λ -parameter In order to define a target accuracy for the comparison with perturbation theory it is useful to referto the Λ -parameter, which, in a mass-independent renormalization scheme, is defined as an exactsolution to the Callan-Symanzik equation, viz. (cid:32) µ ∂∂µ + β ( g ) ∂∂g (cid:33) Λ = . (14) ONF12
Non-perturbatively defined couplings imply the non-perturbative definition of the corresponding β -function and one obtains the exact solution ( µ = / L ), L Λ = ϕ (¯ g ( L )) , ϕ (¯ g ) = (cid:2) b ¯ g (cid:3) − b b e − b g exp (cid:26) − (cid:90) ¯ g d g (cid:20) β ( g ) + b g − b b g (cid:21)(cid:27) . (15)The coupling, its β -function, and thus ϕ and Λ depend on the renormalization scheme. However, thescheme dependence of the Λ -parameter is almost trivial: assuming 2 schemes X and Y the matchingof the respective couplings to one-loop order entails the exact relation, g ( µ ) = g ( µ ) + c XY g ( µ ) + ... ⇒ Λ X Λ Y = e c XY / b , (16)so that we may use Λ = Λ
SF, ν = as reference. Introducing a reference scale 1 / L through,¯ g ( L ) = . ⇒ g ν ( L ) = . − ν × ¯ v ( L ) (cid:124)(cid:123)(cid:122)(cid:125) eq. (30) , (17)we consider the reference quantity, L Λ = Λ / Λ ν (cid:124)(cid:123)(cid:122)(cid:125) exp( − ν × . × L / L (cid:124)(cid:123)(cid:122)(cid:125) non-perturbative × ϕ ν (¯ g ν ( L )) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) perturbative . (18)Hence, if we use non-perturbative results for the running coupling in the range 1 / L ≤ µ ≤ / L , andapply perturbation theory for µ > / L by replacing β ν ( g ) by its 3-loop approximation, β ν, ( g ), ϕ ν (¯ g ν ( L )) ∝ exp (cid:26) − (cid:90) ¯ g ν ( L )0 d g (cid:20) β ν, ( g ) + b g − b b g (cid:21)(cid:27) , β ν, ( g ) = − b g − b g − b ,ν g , (19)we should find that L Λ is independent of the choice of L and ν , up to perturbative errors at the scale µ = / L . Moreover, we have some idea of the target precision for the Λ -parameter: if we aim for,say 0.5 percent accuracy for α s ( m Z ) then the 3-flavour Λ -parameter should be determined to betterthan 3 percent accuracy [21]. Imposing this criterion on L Λ thus gives us a handle to assess theaccuracy of perturbation theory when applied at scales µ > / L . We just have to explain how to tracethe non-perturbative running of the SF coupling from 1 / L to 1 / L , which should be a range coveringa couple of orders of magnitude. L / L The starting point is the so-called step-scaling function [22], σ ( u ) = ¯ g (2 L ) (cid:12)(cid:12)(cid:12) u = ¯ g ( L ) , (20)which determines how the coupling at scale 1 / (2 L ) depends on the coupling at scale 1 / L . Hence oneconsiders scales which are separated by a factor 2 rather than infinitesimally as for the β -function.The relation to the latter is defined by (cid:90) √ σ ( u ) √ u d gβ ( g ) = − ln 2 . (21) PJ Web of Conferences
Figure 1.
Illustration of the computation of the stepscaling function, Σ ( u , a / L ), at a fixed value of thecoupling u = ¯ g ( L ) and for 2 lattice resolutions L / a = , The step-scaling function σ ( u ) is defined in the continuum limit. In order to obtain this function at afixed value of its argument, one measures the coupling on pairs of lattices with extent L / a and 2 L / a in each direction. Since all quark masses are set to zero, the only parameter to be tuned is the barecoupling which is equivalent to the lattice spacing a . Hence if we start with a lattice size L / a = u = ¯ g ( L ) in a numerical simulation and then measure ¯ g (2 L ) by doubling the lattice size to2 L / a (at the same value of the bare coupling g ), we obtain a first approximant, Σ ( u , / σ ( u ),where the second argument is the resolution a / L . We now want to keep L constant in physical unitsbut choose an L / a = g (cid:48) suchthat the previous u -value is matched (which implicitly fixes L ). Then, at the same g (cid:48) one doubles thelattice size and measures the SF coupling to obtain Σ ( u , / σ ( u ) = lim a / L → Σ ( u , a / L ) , (22)at the given value of u . Repeating the same procedure for a range of u -values one obtains the function σ ( u ) for this range with a certain error, due to both statistics and systematic e ff ects from the continuumextrapolation. Once σ ( u ) is available for a range of values u ∈ [ u min , u ] one may iteratively step upthe energy scale: u = ¯ g ( L ) , u n = σ ( u n + ) = ¯ g ( L n ) = ¯ g (2 − n L ) , n = , , ... (23)In particular, by construction the scale ratios are L / L n = n , where n is the number of steps. Σ ( u , a / L ) The lattice step-scaling function Σ ( u , a / L ) is expected to be a smooth function of u . Lattice e ff ects are,up to slowly varying logarithmic terms, polynomial in a / L . Moreover, terms linear in a / L are removedin the bulk by the use of the non-perturbatively O( a ) improved action [23], and highly suppressed at ONF12 .
005 0 .
01 0 .
015 0 .
02 0 . . . . . . . .
22 ( a/L ) Σ ( u , a / L ) / u Figure 2.
Continuum extrapolation of the step scalingfunction. Some data points have been slightly shiftedin order to keep constant u , which is not required forthe global fit. The leftmost points are the continuumvalues, whereas the stars are obtained fromperturbative scale evolution using the 3-loop β -function. the time boundaries by using perturbative estimates of the counterterm coe ffi cients. As a safeguardagainst any residual O( a ) e ff ects we treat a variation of the 2 counterterm coe ffi cients as a systematice ff ect and propagate it to the data. Rather than extrapolating the step-scaling function separately forindividual values of u it is more practical to use a global fit ansatz. A typical example is Σ ( u , a / L ) = u + s u + s u + c u + c u + ρ u a L + ρ u a L (24)with s , s fixed to perturbative values, s = b ln 2 , s = s + b ln 2 . (25)Our data for Σ ( u , a / L ) has L / a = , , u -values we have L / a =
12. As a safeguardwe omit the coarsest lattices with L / a = c , c , ρ , ρ . Fig. 2 shows the data together with the fit function. The χ / d.o.f. ≈ ff erent fit ansätze indicates that we have a good control over the continuumlimit. The continuum SSF is then represented by σ ( u ) = u + s u + s u + c u + c u , in the fitrange [1 . , .
02] and with numerical values for c , c , together with their errors and correlation. Thecontinuum step-scaling function can be compared with earlier results in [24]. where an attempt ismade to reach larger couplings so as to match hadronic scales. In the high energy regime we havesignificantly improved on the precision both in terms of statistical and systematic errors, for instancethrough a very precise tuning to zero quark mass [25]. The step scaling functions have been analysed for a number of ν -values of O(1). Using eq. (18) for L Λ we use non-perturbative running between L and L n = − n L , with n = , , . . . ,
5. In physicalunits 1 / L is about 4 GeV, so that we cover a range from 4 to 128 GeV. For a given n we then integratethe integral in the exponent using the β -function to 3-loop order. In fig. 3 the results obtained atvarious n and the 3 values ν = , . , − . α s , which is the order of the neglected terms.Up to such terms all the data points should agree within errors. The expected asymptotic behaviour isindeed observed. We see that for ν = . α s is essentially zero, whereas it is rather largeat ν = − .
5. From fig. 3 and a variety of further fits not shown here we conclude that all results agreearound α = . L Λ = . ⇔ L Λ N f = = . . (26) PJ Web of Conferences .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 . . . . . . α L Λ Figure 3.
The extraction of the Λ -parameter using perturbation theory atvarious values of α s , plotted vs. α s . Thedata points are, from top to bottom, for ν = − . , , . n = , , . . . , α s ≈ . While for ν = . α s , this is clearly not the case for ν = − .
5. To further assess the accuracy of perturbation theory it is instructive to directly look at thesecond observable ¯ v ( L ) = ω ( u ) | u = ¯ g ( L ) , (27)which allows us to study the couplings for all ν -values (10) We have extrapolated the non-perturbativedata to the continuum limit using similar global fits as for the step-scaling functions. However, herethese fits are more constrained as there is no doubling of the lattice size involved and lattice sizes thusrange from L / a = L / a =
24. Two resulting fit functions in the continuum limit of the form ω ( u ) | fit 1 = v + v u + d u + d u + d u , (28) ω ( u ) | fit 2 = v + d u + d u + d u + d u , (29)with 3 and 4 fit parameters d k , k = , . . . , respectively, are shown with their error bands in fig. 4. Bothfits agree perfectly well in the whole range of the available non-perturbative data. The continuumresult at ¯ g ( L ) = . v ( L ) = ω (2 . = . , (30)is obtained from these fits and defines the starting values for the step-scaling procedure for ν (cid:44) .
02 0 .
04 0 .
06 0 .
08 0 . .
12 0 .
14 0 .
16 0 . . . . . α ω d , . . . , d d , . . . , d two-loop PT Figure 4.
The observable, ω ( u ) = ¯ v ( L ),plotted vs. α s , together with the fitseqs. (28,29) and the two-loop result(cf. text). ω ( u ) = ¯ v | ¯ g (1 / L ) = u , m = = v + v u + O( u ) , (31)where the coe ffi cients v , v can be found in [26]). The non-perturbative data clearly breaks awayfrom two-loop perturbation theory at larger couplings. To quantify this deviation we choose the value α s = .
19 and measure an e ff ective 3-loop coe ffi cient as follows( ω (¯ g ) − v − v ¯ g ) /v = − . α s . (32) ONF12
Indeed this e ff ective coe ffi cient seems too large for perturbation theory to be trustworthy at this valueof the coupling. We come to the conclusion that α ≈ . We have pointed out that lattice observables can be defined at high energies if one gives up the require-ment that volumes should be large enough to fit hadronic states. By defining observables in a finitevolume it is possible to obtain non-perturbative precision data over a wide range of scales. Moreover,the heavy quark thresholds for charm and bottom can be “switched o ff " on the lattice, thereby remov-ing an important source of systematic errors. The main drawback is that perturbation theory mustmatch this situation and take the finite volume into account. This implies some technical di ffi cultieswhich depend on all the details of the chosen set-up. With the SF scheme chosen here there existsa 2-loop calculation matching the SF couplings to the MS-scheme and thus the 3-loop β -function isknown for a whole 1-parameter family of SF couplings. This provides excellent opportunities to testthe accuracy of perturbation theory. As it turns out, a precision of 3 percent for the Λ -parameter canbe quoted with confidence if perturbation theory is restricted to couplings around α s ≈ . α s ≈ . Λ -parameter, eq. (26), is an essential step in the ALPHA collaboration’s projectto determine the Λ -parameter in 3-flavour QCD in units of a hadronic scale such as the kaon andpion decay constants, F K ,π . These are determined in large volumes [27] on gauge configurationsproduced through the CLS e ff ort [28]. Preliminary results have been presented by M. Dalla Brida atthis conference and by R. Sommer in [29]. An estimate of α s ( m Z ) is given there, assuming that thestandard perturbative treatment of the charm and bottom quark thresholds is reliable. While we havefocussed here on the Λ -parameter and running couplings, an analogous study can be made for therunning quark masses and first results have been presented by P. Fritzsch at this conference [30]. Acknowledgments
This work was done as part of the ALPHA collaboration research programme. I would like to thankthe members of the ALPHA collaboration and particularly my co-authors of ref. [1] for the enjoyablecollaboration on this project and for comments on the manuscript. Computer resources by the com-puter centres at HLRN (bep00040) and NIC at DESY, Zeuthen, as well as financial support by SFIunder grant 11 / RFP / PHY3218 are gratefully acknowledged.
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