COLO-HEP-396 Structure of the QCD Vacuum As Seen By 8 9 Lattice Simulations 9 1 n T. DeGrand, Anna Hasenfratz, Tama´s Kov´acs a J Physics Department, University of Colorado, 7 Boulder, CO 80309 USA 2 1 Abstract v 7 Thistalkisareviewofourstudiesofinstantonsandtheirpropertiesas 3 0 seeninourlatticesimulations ofSU(2) gaugetheory. Wehavemeasured 1 thetopologicalsusceptibilityandthesizedistributionofinstantonsinthe 0 QCDvacuum. Wehavealsoinvestigatedthepropertiesofquarksmoving 8 ininstantonbackgroundfieldconfigurations,wherethesizesandlocations 9 of the instantons are taken from simulations of thefull gauge theory. By / themselves, these multi-instanton configurations do not confine quarks, t a butthey inducechiral symmetry breaking. l - p e Talk presented by T. DeGrand at the 1997 Yukawa International Seminar h “Nonperturbative QCD–Structure of the QCD Vacuum” : v i X r a 1 Introduction What features of the QCD vacuum are responsible for confinement or for the generation of the observed structure of hadron spectroscopy? This question might, in principle, be answered by lattice simulations of non-Abelian gauge theories. WehavebeenstudyingthepropertiesofinstantonsfromSU(2)lattice simulations. ThistalkisasurveyofourworkaspresentedinRefs.[1,2,3,4,5]. The study of the QCD vacuum using lattice Monte Carlo is complicated by two problems. The first one is that the dominant features of the QCD vacuum as seen in lattice simulations are short distance fluctuations (as they would be for any quantum field theory). They are basically uninteresting noise. The so- lution to this problem is to invent operators which are insensitive to the short distance behavior of the field variables. This brings the second problem: The separation of vacuum structure into short distance and long distance parts is ambiguous, and what one sees can depend strongly on the operators one uses. Alldirectsmoothingtransformations[6]distorttheoriginallatticeconfiguration. Thismakestheextractionof(continuum)shorttomediumdistancephysics,like observations of topological objects, very delicate. If any space-time symmetric smoothing transformation is repeated enough times, all the vacuum structure in any finite volume, including the simulation volume, will be washed away. Thus, it does not make sense to extrapolate one’s results to the limit of a very large number of smoothing steps. The only measurements which are physically meaningful are measurements which are extrapolated back to the original lat- tice, that is, back to zero smoothing steps. This requires careful monitoring of observables over the whole history of smoothing transformations. In QCD instantons may be responsible for breaking axial symmetry and resolving the U(1) problem. [7] The relevant observable is the topological sus- ceptibility χ , defined as the infinite volume limit of t 2 hQ i 4 χ =h d xQ(x)Q(0)i= (1) t Z V where Q is the topological charge and V the space time volume. In QCD χ t is a dimension-4 object with no weak coupling expansion, and a calculation of χ in physical units in the continuum requires nonperturbative techniques. In t the large-N limit the mass of the η′ is (probably!) related to the topological c susceptibility through the Witten-Veneziano formula[8] f2 2Nπ (m2η′ +m2η−2m2K)=χt. (2) f The left hand side of this equation is equal to (180 MeV)4 in the real world. 2 Based on phenomenological models, it has been argued that instantons are largelyresponsibleforchiralsymmetrybreakingandthelowenergyhadronand glueball spectrum. [9, 10] Instanton liquid models attempt to reproduce the topologicalcontentof the QCD vacuum andconclude that hadroniccorrelators in the instanton liquid show all the important properties of the corresponding full QCD correlators. These models appear to capture the essence of the QCD vacuum, but their derivationsinvolvea number ofuncontrolledapproximations and phenomenological parameters. Instantonphysicsonthelatticeisasfullofcontroversyascontinuuminstan- tonphysics. Therearepresentlythreedifferentwaysofmeasuringatopological charge. The “geometric” definition[11, 12] reconstructs a fiber bundle from the lattice gauge field and identifies the second Chern number of this bundle with the topological charge. It will always give an integer, but if the configuration is sufficiently rough, it can fail catastrophically. “Algebraic” definitions[13] in- troduce some lattice discretizationof Q, as a sum ofclosed paths of loops. The worst aspect of the algebraic definitions is that the topological charge can mix with quantum fluctuations. Finally, one can define Q through fermionic opera- tors. (For a recent example, see Ref. [14].) All these definitions have a cutoff: they cannotsee instanton(like)configurationswhen the instantonradius is too small (typically ρ/a ≃ 1−2). It is usually not possible to specify this cutoff precisely, and it can contaminate lattice measurements. 2 Finding Instantons We have explored three different methods for extracting information about topology from lattice simulations. Each has its own strengths and weaknesses. 2.1 Inverse Blocking “Inverse blocking” does not distort the original configuration. This technique has been introduced by Hasenfratz[15] in his lectures. Imagine beginning with a set of lattice variables {V} on a lattice whose spacing is a and lattice size is L. The lattice action is a fixed-point (FP) action [16] SFP(U). The inverse blocking transformationconstructs a set of fine lattice variables{U} occupying alatticeoflatticespacinga/2andlatticesize2L,bysolvingthesteepest-descent equation SFP(V)=min SFP(U)+κT(U,V) , (3) {U}(cid:0) (cid:1) 3 whereκT(U,V)istheblockingkernel. Inverseblockingidentifiesthesmoothest among the configurations that block back to the original configuration. Since for fixed point actions, topology is unchanged by inverse blocking, the strategy is to take a roughconfiguration(generatedby Monte Carlousing a FP action), inverseblockit,andmeasurethechargeonthefinelattice,wheretheinstantons are twice as big and charge measurement is more reliable. We did this[2] and found χ = (235(10) MeV)4 for SU(2). This created a low-level controversy, t since our number was larger than other measurements.[14, 18, 19] Inverse blocking is reliable, but it is very expensive. In four dimensions, one can only perform it once (to go from lattice spacing a to a/2). The fine configurations are still too rough to identify individual instantons. 2.2 Cycling Our smoothing mechanism for seeing instantons is called “cycling.” [3] One first performs an inverse blocking from a coarse lattice to a set of fine lattice variables by solving Eqn. 3. Now the original lattice occupies one of the 16 sublattices of the fine lattice. Next, we perform a blocking transformation to a set of coarse variables {W} based on one of the other sublattices. The delicate coherence among the fine variables is broken and the new coarse variables are strongly ordered on the shortest distance scale (as measured, for example, by the expectationvalue ofthe plaquette) while retaining alllong distance physics (because they are generated by a RG blocking transformation). This is the second part of the cycling transformation V (x) → U (x) → W (x). Cycling µ µ µ stepscanbeiterated,andafewcyclingstepscanreducetheplaquettetowithin 0.001 of its free-field value. Individual instantons can be seen after a few cycling steps. Their sizes drift with smoothing (see the next section for pictures) but the drift is small enough that we can extrapolate their properties back to zero cycling steps. We found χ1/4 =230(10)MeV forSU(2),aninstantondensity ofabouttwoper fm4, and t ameaninstantonradiusofabout0.2fm. Again,thesusceptibilityishigherthan others’, and now the mean instanton size is smaller than other measurements, althoughitagreeswiththepredictionsoftheinteractinginstantonliquidmodel. [9, 10] Cycling is still quite expensive because of the inverse blocking step. This makes it hard to push to small lattice spacing and to test scaling. 4 2.3 RG Mapping Cycling produces a sequence of coarse ({V}) and fine ({U}) lattices {V} → {U1}→{V1}→{U2}→{V2}→.... RGmappingisatechniqueforeliminating the inverse blocking step and generating a sequence of coarse lattices {V} → {W1} → {W2} → ... where {Wn} is an approximation to {Vn}. The idea is that, while formally the inverse blocking is non-local, for local FP actions the dependence ofthe fine links onthe originalcoarselinks diesawayexponentially with their separation[16] and the mapping {V}→{U1(V)}→{V1(V)} can be considered local. The {W } lattice can be constructed from the originalcoarse n {V} latticeasa sumofloopsof{V}’swhicharedesignedto reproducea cycled {V }. We discoveredthat this could be done by APE-smearing: [20] froma set n {V} construct a new set of links {X} by X (x)=(1−c)V (x) + c/6 (V (x)V (x+νˆ)V (x+νˆ)† µ µ ν µ µ νX6=µ + V (x−νˆ)†V (x−νˆ)V (x−νˆ+µˆ)), (4) ν µ µ with X (x) projected back onto SU(2) to generate W (x). We found that a µ µ series of steps with c=0.45 was a good choice to mock up cycling. RG-mapping is very cheap, but the price is that everything which is mea- suredmustbe monitoredcarefully,andextrapolated(if necessary)backto zero mapping steps. Theproblemisthatanyapproximationtothefixedpointchargeisexpected to distort the charge density profile. We have to correct this distortion. This canbedonebymonitoringhowthechargedensity–measuredbytheFPcharge– changesinthe courseofsmoothingandextrapolatingthis backtozerosmooth- ing steps. Ofcoursewe cannotdirectly measurethe FP chargesince this would involve severalinverse blocking steps. Howeverafter a few smoothing steps the configurations become smooth enough so that our improved charge operator is very close to the exact FP charge and data taken after further smoothing can be used for extrapolating back. Asanexample,weconsidera164configurationgeneratedwithWilsonaction β = 2.5. Fig. 1 shows the size of the 8 “stable” objects as a function of the APE-smearingsteps. Fromthe 4instantons (diamonds)3increaseinsize while one decreases,but allvary linearlywith smearing steps. The slope ofthe linear change for all of them is small. Three of the anti-instantons (bursts) behave similarly, though one has a slightly larger slope. The fourth anti-instanton (crosses)startsto growrapidly after 18smearing steps and will disappear after a few more steps. This object is likely to be a vacuum fluctuation, not an instanton. 5 Figure 1: Radius versus APE-smearing steps of instantons (diamonds) and anti-instantons (bursts and crosses) on a 164 β =2.5 configuration. In the early stages of cycling (1-2 steps) we see many lumps of charge. Most of them quickly grow or shrink away. The locations of what we call true topological objects are stable over many smearing steps and their size changes slowly. To identify them on the lattice one has to track them over several smearing steps and monitor their behavior. Instantons present in QCD simulations differ from hand crafted instantons in trivial background configurations in that the former usually grow while the latter objects always shrink under APE smearing. Neither cycling nor RG-mapping affect the string tension, but the short distance part of the potential is distorted. Oneofourgoalswastocompareresultsobtainedwiththecycling/RGmap- ping method with results published using other algorithms [17, 18, 19], so we used the Wilson action in conjunction with RG-mapping. We observe, as expected, a small systematic decrease in the susceptibility as we increase the number of smearing steps. At large β the change is small andstatisticallyinsignificant. Onlyatβ =2.4,wheretheconfigurationsarethe 6 roughest, do the 12 and 24 smearing steps results differ by about a standard deviation. The susceptibility increases by 10% from β = 2.4 to β = 2.5 but stabilizes afterthatatthevalueχ1/4 =220(6)MeV.Weinterpretthechangebetweenthe t extrapolatedβ =2.4resultandthelargerβresultsasduetotheabsenceofsmall instantons at β = 2.4 because of the larger lattice spacing. This interpretation will be supported by the instanton size distribution result discussed below. We have identified individual instantons after every 2 APE-smearing steps between12and24steps. Sincethese configurationsarestillrough,manyofthe objects identified as instantons are in fact vacuum fluctuations and disappear after more smoothing steps. Figure 2 shows the observed instanton size distri- bution on the 12 and 24 times smeared lattices at β = 2.5. For comparison we also plot the result of Ref. [17] which corresponds, in our normalization, to about 100 times smeared lattices. It is obvious from the figure that the total number of identified objects decreases as we increase the smoothing. The den- 4 4 sityofidentifiedobjectsis4.6perfm after12smearingsteps,3.0perfm after 24 smearing steps, and about 2 per fm4 in Ref. [17]. These density values are considerably largerthan the expected value of about 1 per fm4. The maximum of all 3 distributions is around ρ¯ ≈ 0.3 fm, but that does not mean that on the original lattice ρ¯ ≈ 0.3 fm since instantons usually grow under smearing. This growth can also be observed from the increasing tail of the distribution, especially after the many blocking steps of Ref. [17]. Our final result for the instanton size distribution (after extrapolation back to zero smoothing steps) is shown in Figure 3, where we overlay the data ob- tainedatβ =2.4(octagonson124,squareson204lattices),β =2.5(diamonds), and β = 2.6 (bursts). Since the smoothing method cannot identify instantons with ρ≤1.5a, we chose the bins such that the second bin for each distribution starts at ρ=1.6a. That means that we expect the second bin of each distribu- tion to be universal. The first bins on the other hand contain only some of the small instantons and their value is not expected to be universal. Thefourdistributionsformauniversalcurveindicatingscaling. Theβ =2.4 curvescoveronlytheρ>0.2fmregion,andsmallinstantonsareobviouslymiss- ing. The agreementbetween the 124 and 204 configurationsat β =2.4 indicate that a linear size of about 1.4 fm is sufficient to observe all the topological objects. The β = 2.5 and 2.6 distributions have most of the physically rele- vant instantons, supporting the scaling behavior observed for the topological susceptibility. The instanton liquid model predicts a very similar picture to ours. In 7 Figure 2: The size distribution on 12 (bursts) and 24 (crosses) times APE smeared lattices at β = 2.5. The square symbols are the results of Ref. 11 rescaled appropriately. 8 Figure 3: The size distribution of instantons. Octagons correspond to β = 2.4 4 4 12 , squares to β = 2.4 20 , diamonds to β = 2.5 and bursts to β = 2.6. The first bin of each distribution is contaminated by the cut-off. The solid curve is a twoparameterfitto the data points accordingtothe formulainRef. 23. The dashed curve is a similar fit from Ref. 23 which describes the instanton liquid model quite closely. 9 Ref. [21] Shuryak predicted an instanton size distribution that peaked around ρ=0.2fm. ThecurveinFig. 3isafitusingthetwoloopperturbativeinstanton distribution formula with a “regularized”log 8π2 SI = g2(ρ) =b0L+b1logL (5) 1 L= log[(ρΛ )−p+Cp] (6) inst p where b0 and b1 are related to the first two coefficients of the perturbative β function and p and C are arbitrary parameters. The solid curve is a fit to our data while the dashed curve is the fit given in Ref. [21] which describes the instanton liquid model quite closely. The difference between the two fits is significant at large ρ values only. We do not know if changing the parameters of the interacting instanton liquid model slightly would change the predictions of that model improving the agreement with the Monte Carlo data. To make a long story short, we believe that there are two reasons why our instanton numbers are different from those of Refs. [17, 18, 19]. First, we are sensitive to instantonsof smallerradius. As we cutlargerandlargerinstantons from our sample, we see χ fall. Second (and most important) we extrapolate t our measurements back to zero smoothing steps. Instantons generally grow under smoothing, so our extrapolated sizes are smaller. 3 What Do Instantons Do? To test what role instantons play in the QCD vacuum, we took a set of SU(2) configurations with lattice spacing a ≃ 0.14 fm and smoothed them enough to identify the instantons. These smoothed configurations have essentially the same string tension as the original configurations, even though about seventy per centof their vacuum actionis carriedby the instantons. We then identified the sizes and locations of the instantons in the configurations and built multi- instanton configurations from them. We built both “parallel” and randomly oriented instanton configurations. Notice that the instanton locations are not random, and the sizes are not taken from a model distribution–they come from the (lattice) QCD vacuum. In Fig. 4 the heavy-quark potentials obtained from the three ensembles are compared. Wecanconcludethatneithertheparallelnortherandomlyoriented instantons confine. It seems that instantons by themselves are not responsible for confinement. 10