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The running coupling from lattice gauge theory

arXiv:hep-lat/9310019v1 19 Oct 1993SWAT/13hep-lat/9310019October 1993The running coupling from lattice gauge theoryTalk at the Workshop on Field-Theoretical Aspects of Particle Physics,Kyffh¨auser, Germany, September 13th to 17th 1993P.W. StephensonDepartment of Applied Mathematics and Theoretical Physics, University of LiverpoolandDepartment of Physics, University College of SwanseaSingleton Park, Swansea, SA2 8PP, UK.∗AbstractI discuss some calculations of the running coupling in SU(N) gauge theories from lattice simula-tions, centering on the work of the UKQCD collaboration.

This talk is introductory in nature;full details have been published elsewhere.1. IntroductionAt the moment, lattice simulations are the most popular way of extracting truly non-perturbative resultsfrom quantum field theories.

Their commonest uses are, quite naturally, calculations with some directexperimental relevance, such as the spectroscopy of hadrons and the calculation of matrix elements.However, we are quite at liberty to examine more basic aspects of the discrete theory, in order toassure ourselves that the results are as we expect. (It would be more interesting if they were not, butin QCD this is increasingly unlikely.) In this talk I shall describe a (successful) attempt to look at thebehaviour of the running coupling of the field theory.

This allows us to make direct contact with thestandard formalism of perturbation theory and the renormalisation group.Phenomenologically, the most interesting field theory is quantum chromodynamics (QCD), with itsstrong interaction and significant scale-dependence over the region where the physics is most interesting.The proper lattice formalism is that of SU(3) gauge theory with dynamical quarks, incorporatingFeynman diagrams with internal quark loops. This is quite simply too difficult for us with our present-day technology and computer resources.

Even generating an equilibrated lattice (one on which samplesof the fields are guaranteed to be representative of the true vacuum with the right coupling) is verycostly and can take many months even on the largest machines.However, simulations in the quenched approximation, the theory without dynamical quarks butretaining fully non-perturbative gluonic contributions, are proving more tractable and we are seeingresults which in many cases agree with experiment at the level of 10% or so. One should of course becareful; at some stage the effect of quenching is likely to dominate our errors and in some calculationsto change the nature of the physics completely.

We are not yet at the stage where we can determine thelimits of the quenched approximation and in general the effect is not predicted. Preliminary indicationsare that where light quarks are unimportant the dominant effect is simply a uniform rescaling of theresults.This need not worry us if we are engaged in an exercise in field theory.

In fact, in that case we canmake one further simplification and use the gauge group SU(2) instead of SU(3), resulting in roughly∗Present address. Email: P.Stephenson@swan.ac.uk or PWS@UKACRL (BITNET).1

an order of magnitude reduction in the computing effort required for a similar standard of results. Weshall also see that the results for the two groups are qualitatively very similar.This talk deals for the most part in the pure gauge theory, which does not involve fermions at anystage.

This is not a further approximation beyond quenching: it simply means we are looking at thegluonic sector. In fact, the pure gauge theory is a true field theory (to the best of our knowledge) whilethe quenched fermionic theory is not since it involves an unnatural treatment of quarks on a backgroundgauge field with which they do not interact properly.The technology for producing pure-gauge results from lattice simulations is now quite well advanced.Using improved operators we can extract quite accurate numbers for, among other things, the interquarkpotential which will be our probe here.The method outlined in the major part of the talk first appeared in [1].

Full details of the calculationsby the UK QCD Grand Challenge (UKQCD) using this method are given in ref. [2] (for SU(2)) andref.

[3].2. The running coupling and asymptotic freedomIn QCD, or any quantum field theory, the physically-meaningful coupling — the one which relatesdirectly to the strength of the interaction — is the renormalised value.

This is commonly referred to asthe running coupling, as the process of renormalisation makes the coupling apparently change its valuein low-order perturbation theory. One way of looking at this new quantity is that it parametrises ourignorance about what is happening at very high energies when QCD is no longer valid, replacing it witha cut-offand an effective theory.In four dimensions, where the coupling is dimensionless, we see the rather surprising phenomenonknown as dimensional transmutation whereby the mechanics of renormalisation introduces a fixed scale,namely the lambda parameter, ΛQCD.

The “interesting” physics of QCD, by which I mean the battle-ground of different causes and effects, happens in processes involving momenta q ∼ΛQCD. The secondimportant consequence of dimensional transmutation is that the renormalised coupling is itself scale-dependent, gphys = gphys(q).

As we shall see later on, the connection between gphys (I shall drop the“phys” suffix) and the bare coupling g0 which appears in the original Lagrangian can be a bit obscure.Actually, this is true not just on the lattice: it is, after all, why we need renormalisation in the firstplace.The SU(N) theories have the feature that as the momentum scale is increased, the physical couplingdecreases (asymptotic freedom). This means that in the limit of large momentum the theory becomesperturbative.

That is how we shall connect the non-perturbative lattice results with analytic calculationsin perturbation theory.In SU(N) to two loops the running coupling is given byα(q) ≡4πg2 =14π (b0 log(q/Λ)2 + (b1/b0) log log(q/Λ)2)(1)where b0 and b1 depend only on N. This is for the pure gauge theory with no quarks; adding a fewspecies of quarks does not change the expression qualitatively, though asymptotic freedom is weakenedand eventually (with 17 quarks in SU(3)) the sign of b0 and the character of the theory change.Instead of a momentum scale, one can express the results in terms of an appropriate length scaleR ∼q−1. This is more appropriate to a static configuration like the one we use for extracting potentials.We need to change our renormalisation scheme to do this; the scheme appropriate to potentials usesthe separation R between the quark and antiquark, as described in the next paragraph, as the scaleparameter.

It turns out that this is close to MS in the sense that the Λ-parameters are related by asmall factor; this is not true of the usual scheme for lattice regularisation.2

In the perturbative limit, the force between a static (infinitely massive) quark and antiquark ofopposite colour (so the pair is colourless) a distance R apart is Coulombic:−FCoulomb = dVdR = Cfα(R)R2(2)(Cf is a combinatoric factor containing N; it derives from the quadratic Casimir operator). Hence if weform a dimensionless quantityαeff≡−FcalcR2Cf(3)a lattice calculation of Fcalc at small R is enough to tell us the running coupling.

We use the force insteadof the potential as it removes an uninteresting constant. On the lattice, the force is extracted from thepotential by a finite difference; this actually introduces negligible extra errors.

The significant thingabout the lattice calculation is that, although we are trying to show agreement with the perturbativeanalysis, the calculation is at every step fully non-perturbative.3. Lattices and scalesOur lattice formulation is the standard one of Wilson.

The gauge fields live on the links of a hypercubiclattice in Euclidean space-time, that is the lines joining nearest-neighbour points of the simple cubicstructure like a stack of wire-framed cubes. Each link has an associated matrix which is an elementof the gauge group.

Line integrals between points are replaced by simply matrix multiplication of theappropriate links; taking the trace of such a product which corresponds to a closed loop produces agauge-invariant object, the Wilson loop. These loops are essentially the only gauge-invariant quantityin the pure theory; the quarks, if present, would live at the sites of the lattice and act as sources andsinks of colour.The action for the theory is a sum over all plaquettes: the plaquette is the smallest possibleWilson loop consisting of four links about an elementary square of the lattice.

This preserves exactgauge invariance and in the continuum limit it goes over smoothly to the usual continuum action.The scale is fixed by the lattice spacing a, the length of one link. This is determined by the barecoupling g0 which we choose for our interaction (in fact, we usually work with β = 2N/g20, which appearsas the multiplier of the sum of plaquettes in the action).

The relationship between a at different valuesof the coupling is determined by the renormalisation group beta-function (two completely unrelated usesof the symbol beta, unfortunately); because of asymptotic freedom, a small g0 corresponds to a smalllattice spacing. Decreasing g0 therefore makes the lattice less and less important and takes us nearerthe real world where aΛ is small, or in other words the scale Λ is very much less than the momentumspace cut-off.One can think of a rectangular Wilson loop (as in figure 1) which extends R in a spatial directionand T in the time direction in the following way.

A source and a sink of colour in the fundamentalrepresentation of the gauge group, both infinitely massive, are created instantaneously a distance Rapart. They propagate in this fashion for time T and then are instantaneously annihilated.

We referto the colour sources as a (static) quark-antiquark pair even though the calculation does not involvefermions at any stage. We need some experimental input to determine a (or — which is equivalent intheory if not always in practice — ΛQCD) in physical units.3

q-q-SpaceqqaLTR(Euclidean) TimeFigure 1: The Wilson loop on the lattice.4. Measuring the quark-antiquark potentialTo extract the potential between the two, we simply need the expectation value of the loops.

Thisis related to the Euclidean propagator C(T ) ∼Pi ci exp(−ViT ) for the various energy levels Vi ofthe quark-antiquark system. At large T the main contribution is from the ground state and the ratioC(T )/C(T −1) tells us the potential V ≡V0.

Unfortunately the statistical errors increase with T , alsothe decay of the correlators is faster than typically found in (for example) calculations of light hadronmasses, so choosing a suitable value of T is something of an art form and introduces significant systematicerrors.We should note two other major sources of systematic error in our calculations, again deriving fromlimitations on computing resources. First of all the lattice size L is finite as we can only fit L/a siteson a side of our lattice.

If this is too small the fields will feel the effects of the boundaries of the lattice,being squashed into the box. In our case we have chosen sizes (from previous experience) such that thisis not expected to be a problem.Secondly, a is finite, generating cut-offeffects.

This is significant for us, since we are attemptingto probe the region at small R where perturbation theory is expected to become valid: our results arefor quark separations of only a few lattice spacings. We have used a one-parameter fit to smooth outthe bumpiness caused by finite a; the smoothness of the result, together with the agreement betweendifferent lattice spacings as described below and shown in figure 2, assures us that this has worked.Our Monte Carlo simulation generates different samples of the QCD vacuum for us to measure bylocal updating: changing one link at at a time until the set of all links is sufficiently different fromthe previous sample.

The process involves subjecting each link to two different procedures. The firstis a “heatbath”, closely analogous to the same concept in statistical mechanics, in which the link ismade aware of the coupling (corresponding to the temperature) of the links around it.

This includesthe element of randomness that drives our stochastic process, like the random kinetic motion in a real4

(R)R ⁄ a0.00.51.01.52.02.53.03.54.04.5051015202530αβ = 2.85Figure 2: The scaling properties of the potential in SU(2). Diamonds are β = 2.85data, triangles β = 2.7 and squares β = 2.4.

The R axis (only) is adjusted for the lattertwo.heatbath.The second procedure is “over-relaxation”. This uses the fact that the SU(2) group manifold is asphere and that we can flip the gauge element for any link about the (scaled) gauge element representingthe combined effect of the other plaquettes in which the chosen link appears without changing the action.Hence we can do this as often as we like without affecting the statistical-mechanical properties of thesimulation.In both procedures, each link of the lattice is updated in turn throughout the whole lattice (onesweep).

The SU(3) simulation is similar: for each link we perform a heatbath in each of the threepossible SU(2) subgroups in turn, and similarly for over-relaxation. Typically we perform four over-relaxation sweeps for every heatbath sweep; some other groups perform more, but our heatbath code iswell optimised so that the extra computer time over the over-relaxation code (which is simpler) is fairlylow.

This appears to be the most efficient way of generating distinct configurations at present.The sets of data we have used are in each case separated by several hundred sweeps (largely forlogistical reasons) and statistical correlations between different sets are found to be completely negligible.Having extracted values for the potential we should like to decide whether results on lattices atdifferent inverse couplings β1, β2 give equivalent results — in other words, whether the ratios of measuredquantities (say, masses) mX(β1)/mX(β2) are the same for all possible measurements X. If this happens,the results are said to scale and we know that any dependence on a in the expansion of the ratio hasdisappeared.

(However, we cannot necessarily make the stronger statement that the individual quantitiesmX have an expansion which behaves according to the RG beta function in low order. This requirement— asymptotic scaling — is mentioned later.

)One way of showing this from the potential data is by forming “αeff” as in eq. (3).

We do not requireat this point that this αeffshould be the true running coupling; we are simply using it as a convenient5

dimensionless physical quantity. If scaling holds we should be able to plot this against the separation Rand achieve a single curve simply by rescaling the R axis as appropriate.

Figure 2 shows this for threedifferent couplings in SU(2): β = 4/g20 = 2.4 [4], 2.7 [1]and 2.85 [2]. The scaling factor in R shows thatthe lattice spacing is some four times smaller at 2.85 than at 2.4.

This is a strong indication that wehave control over all finite lattice spacing effects — any scaling violation in the potential measurementshould show up clearly in this plot.5. The running couplingWe now proceed to the running coupling itself (following the method of ref.

[1]). The potential shown infigure 2 was over a wide range of scales which includes a linearly-rising potential V ∼KR that dominatesat large R (we confirm this by suitable fits to the data; the value of K is well-determined).

We concentrateon our smallest lattice spacing, with β = 2.85 and 48 lattice sites in each spatial direction, for smallseparations. As the lattice spacing is unphysical, we set the scale instead by using our measured valueof K on the same lattice.

This result is shown in figure 3 (the points with error bars). We show thecorresponding analysis for SU(3) in figure 4; in this case β = 6.5 and the lattice has 36 sites in eachspatial direction.We should like to compare this with analytic results.

We do this by choosing a value of aΛ that fitsour results. The lines show the running coupling from perturbation theory to two loops at the largestand smallest values of aΛ which seem consistent (this was done by eye).

When this is done our latticeeffective coupling seems to agree well with the perturbative expressions (note that to the right of thediagram non-perturbative effects are beginning to enter). In other words, we are seeing real perturbativefield theory from non-perturbative calculations on the lattice.We can use our chosen value of aΛ to extract Λ provided we can express a in physical units.

Todo this we turn back to K. In SU(3), we can equate this with the “string tension” in the Regge pictureof hadrons as a quark-antiquark pair connected by a tube of relativistic glue. (There is no more formaljustification for this; however, there is no good reason to suppose this is a bad way to set the scale either.

)In this picture√K = 440 MeV. From this, we extract the Λ parameter in the renormalisation schemeappropriate to the potential picture.

This is a small multiplicative factor away from the more familiarΛMS; we deduce that ΛMS = 256 ± 20 MeV. However, remember this is in SU(3) without dynamicalquarks.

We have not attempted to correct for this; we do not believe that in our case the systematicerrors are sufficiently under control.We can also use K to give more meaning to the horizontal scale. With the same value 440 MeV, a5 GeV momentum scale corresponds to R√K ∼0.9 on the lower scale of figure 4, so α ∼0.16.

This isvery rough because apart from all the other errors I have not bothered to do a proper conversion fromthe R scheme to MS, though the difference is small; the physics is in the running itself, not necessarilythe actual values we extract.6. Other calculationsThe same calculations have been performed in SU(3) by Bali and Schilling [6]; their results are verysimilar to ours, although they have more data.

Among their lattices are some with smaller physicalextent than ours; this does not change the running coupling behaviour, so it seems that the smalldistance physics is not strongly affected by finite size effects. This was not a priori obvious, though it isnot too surprising a result if one believes in the separation of physics at different scales.Other calculations (ref.

[7]; see also reviews in ref. [8]) for α have been performed using the char-monium potential (any such colourless, heavy quark-antiquark system will do as the results are nearlymass-independent).

Here too the potential is expected to be in the perturbative region. In this calcula-6

(R)R ⁄ aKR0.050.150.250.350.450.550.00.100.200.300.400.5001234560.050.100.150.200.250.300.35α√Figure 3: The running coupling in pure SU(2) lattice gauge theory. All results are atβ = 4/g20 = 2.85.

Error bars are both statistical and systematic; the latter are dotted.The upper scale on the horizontal axis shows the separation in lattice spacings; thelower scale relates it to the string tension. The upper and lower lines are two-loopperturbative predictions with aΛ = 0.044 and 0.038 respectively.

(R)R ⁄ aKR0.000.050.100.150.200.250.300.350.400.450.00.10.20.30.40.5012345α√β = 6.5Figure 4: The running coupling in pure SU(3) lattice gauge theory. The plot is similarto figure 3.

Here diamonds are data at β = 6.5 [3]and triangles at 6.2 [5]; the upper scaleon the horizontal axis is only appropriate to the former. The upper and lower lines inthis case are for aΛ = 0.070 and 0.060 respectively.7

tion the scale does not appear in an explicit non-perturbative fashion (as in our calculations) and onehas to make sure the scale is chosen appropriately, as emphasised by Stan Brodsky at this workshop,and that the problems with perturbation theory mentioned in the next section are correctly handled.These authors have chosen to correct for the effect of quenching (amongst other systematic effects), thatis to predict the effect of four light quarks using the renormalisation group; this increases the value ofα at 5 GeV from 0.140 ± 0.004 to 0.170 ± 0.010. Of course it can then be run in the same way to anyinteresting scale.7.

Lattices and perturbation theoryWhat might appear more surprising is that we have such good agreement with perturbation theory atall. The na¨ıve way of comparing with perturbative results is to insert our coupling g0 into perturbativeexpressions (such as the beta function) and see what comes out; in the asymptotic scaling limit we wouldfind agreement.

In practice this is not seen. For example, the beta function produces a ratio of latticespacings between a(β = 2.7)/a(β = 2.85) which is very different from our value (by almost 20%).The problem arises from the use of the bare coupling.

All our previous analysis of the runningcoupling was done without any perturbative input; β was used only to label our different lattices andso g0 never appeared in the calculations. There is clearly a problem with perturbation theory on thelattice which does not appear if one restricts oneself to physical quantities as we have done.This problem has been elucidated during the last couple of years by Lepage and Mackenzie [9].

Thelattice regularisation is somewhat unnatural for a field theory; large constants from tadpole diagramsare introduced into perturbative series which become only painfully convergent. Provided a physicalcoupling is used there is no problem.

Lepage and Mackenzie give methods for improving perturbativecalculations along these lines, using some non-perturbative input such as the average plaquette as arenormalisation factor.A further result of this is that attempts to extrapolate to zero lattice spacing (i.e. zero bare coupling)by some groups [6,10] has required more sophistication.One method of proceeding is to define animproved coupling with real physics in it.

The “βE” effective coupling scheme, descended originallyfrom a suggestion of Parisi [11,10], is one way of doing this; in fact, it is very much in the spirit of theLepage-Mackenzie programme. Any physically quantity (in this case the action, which is easily measuredon the lattice — in fact it very nearly emerges as a by-product of the way links are updated) may beexpressed as a power series in the coupling.⟨Splaq⟩= c1β + c2β2 + c3β3 + · · ·(4)This series may have a poor convergence.

However, one can truncate at some low order and invert it,expressing the coupling as some function of the action, and use this truncated expression to define aneffective coupling:β(1)E≡c1S(5)(this is the first order βE scheme; one can truncate at higher order). By missing out the higher terms, onehopes the new coupling is more appropriate to use in low order, having resummed any non-perturbativecontributions.

This does seem to help the link to asymptotic behaviour.8. The approach of L¨uscher et al.L¨uscher, Sommer, Weisz and Wolffimplemented a recursive strategy which involves measuring a suitableobservable on lattices of different sizes, allowing them to calculate the running coupling with no a prioriassumptions.

They have results for both SU(2) [12], and SU(3) [13].8

Increasing physical sizeIncreasingcouplingperturbativematchFigure 5: The strategy of L¨uscher, Sommer, Weisz and Wolff.The idea (see figure 5) is that one performs simulations on two lattices at a fixed coupling, so thatone can change the size of the lattice (used to set the scale) by an exact factor simply by adding morepoints in each direction. By inspired guesswork one can then find a lattice at a different (higher) couplingwhich is roughly the same physical size; any discrepancy can be handled with only small errors by therenormalisation group.

Hence the size can be increased stepwise. The smallest lattice used is well inthe perturbative region: a small box has small length scales which means it is perturbative by virtue ofasymptotic freedom.

Note that the choice of the measured observable is important: the response to achromoelectric field forced onto the lattice by the boundary conditions is used.This technique avoids our problem of having all the scales on the one lattice, so that having afinite cut-offis less of a problem. The obvious point against is that there are more technical details tounderstand and bring under control (for example, a lattice renormalisation of the chosen observable usedto extract the running coupling).The measured running couplings agree much better with na¨ıve perturbation theory than the UKQCDresults in both SU(2) and SU(3).

The reasons are not understood; it may simply be (as L¨uscher et al.observe [13]) that their chosen observable has particularly good asymptotic scaling properties.9. SummaryAs a summary of the results in SU(2) (as these are computationally easier to extract; those in SU(3)look very similar), here is a table of various estimates for the ratio of lattice spacings between two inversecouplings β = 2.7 and 2.85.

This deliberately uses the bare coupling, except where explicitly improved,to show the discrepancies; as explained above this is not generally the smartest thing to do. They rangefrom the UKQCD results, through our results improved by βE, the results of ref.

[12] (which can beread offfrom a graph in that paper using the authors’ own fit), to the na¨ıve perturbative result at twoloops. Apart from the UKQCD result (which includes an estimate of systematic errors), the errors arenegligible.9

Quantitya(2.7)/a(2.85)UKQCD: Full fit to the potential, including systematic errors1.60(6)Perturbation theory with first-order βE improvement1.53Ref. [12]1.48Ordinary perturbation theory at two loops1.4610.

ConclusionsAttempts to see the running coupling in lattice gauge theory and to relate it to perturbation theory havebeen successful; the necessity of dealing with the problems of perturbation theory on the lattice is nowclear. By setting a physical scale ΛQCD can be extracted.

As time goes by we hope to have more controlover the systematic errors involved. It is already claimed by analysts of the heavy quark data [8] thatthe lattice is a better place to extract the running coupling than experiment; lattice theorists will needto substantiate this by continuing to refine their methods.References[1] C. Michael, Phys.

Lett. B283 (1992) 103[2] UKQCD Collaboration, Nucl.

Phys. B394 (1993) 509[3] UKQCD Collaboration, Phys.

Lett. B394 (1993) 509[4] S.J.

Perantonis, A. Huntley and C. Michael. Nucl.

Phys. B326 (1989) 544[5] UKQCD Collaboration, Phys.

Lett. B284 (1992) 377[6] G.S.

Bali and K. Schilling, Phys. Rev.

D47 (1993) 661[7] A. El-Khadra, G. Hockney, A. Kronfeld and P. Mackenzie, Phys. Rev.

Lett. 69 (1992) 729[8] G.P.

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Mackenzie, Phys. Rev.

D48 (1993) 2250[10] J. Fingberg, U. Heller and F. Karsch, Nucl. Phys.

B392 (1993) 493[11] G. Parisi, Proceedings of the XXth International Conference on High Energy Physics, Madison,1980, eds. L. Durand and L.G.

Pondrom, American Institute of Physics, New York (1981) 1531[12] M. L¨uscher, R. Sommer, U. Wolff, P. Weisz Nucl. Phys.

B389 (1993) 247[13] M. L¨uscher, R. Sommer, P. Weisz, U. Wolff, “A precise determination of the running coupling inthe SU(3) Yang-Mills theory”, preprint hep-lat/9309005 (DESY 93-114)10

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