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PROGRESS IN QCD USING LATTICE

arXiv:hep-ph/9303305v1 25 Mar 1993FERMILAB-PUB-93/058-TPROGRESS IN QCD USING LATTICEGAUGE THEORYANDREAS S. KRONFELD and PAUL B. MACKENZIETheoretical Physics Group, Fermi National Accelerator Laboratory,P.O. Box 500, Batavia, IL 60510, USAMarch 1993KEY WORDS: hadron masses, quark mixing (CKM) matrix, weak matrix ele-ments, strong coupling constantto appear in Annual Review of Nuclear and Particle Science

Contents1Introduction22Lattice Methodology62.1Methods . .

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.72.3The quenched approximation . .

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. .93Establishing and Testing Lattice Methods103.1The ψ and Υ Systems .

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.103.2The Light Hadron Spectrum . .

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.164QCD Phenomenology184.1The Coupling Constant. .

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.184.2Quark Masses . .

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.215Weak-Interaction Phenomenology235.1Decay Constants . .

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.245.2Semi-Leptonic Decays. .

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. .285.3Neutral Meson Mixing .

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.336Summary and Prospects341IntroductionQuantum chromodynamics (QCD) is the only serious candidate for the theoryof strong interactions. It is supported by overwhelming qualitative evidenceand a growing body of quantitative evidence.

Lattice gauge theory is the onlyfundamental formulation of QCD allowing the calculation of all its consequencesin both the high and low energy regimes.Low energy QCD is worth studying not only for its own sake, but also forits role in understanding what lies beyond the standard model. At present, theonly experimental clues for this puzzle are the fundamental parameters of thestandard model.

Of these, the values of the strong coupling constant, all of thequark masses except the top quark mass, and most of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements either are now or soon will be dominated by2

theoretical uncertainties that can be attacked with lattice QCD. Table 1 con-tains a list of the most fundamental quantities in the standard model.

Whereappropriate it also indicates how lattice QCD will play an important role, andthe section(s) of this article containing relevant material. The central theme ofthis review is standard model phenomenology, with emphasis on lattice calcu-lations needed to determine the parameters and to understand the reliability ofthe determinations.When lattice gauge theory was first introduced by Wilson in 1974 [2], severalcalculational approaches were suggested, including strong coupling expansionsand various renormalization group methods.

Monte Carlo methods were firstapplied to pure gauge theory in 1979 [3, 4]. Methods for treating quarks inMonte Carlo calculations were introduced in 1981 [5].

Although these initialcalculations of the hadron spectrum had approximately the reliability of thenonrelativistic quark model, it was clear, at least in principle, how to developthem into genuine first principles QCD calculations. They initiated the waveof effort leading to the calculations described in this article.A very brief overview of lattice methods is given in Section 2.

That sec-tion also details the sources of error and uncertainty which must be understoodand eliminated as lattice methods evolve into true first principles calculations.Most of the calculations we discuss employ an approximation introduced inReferences [5], called the “quenched” or “valence” approximation. The formername is more common in the literature, but the latter one is, perhaps, moredescriptive: the quenched approximation treats valence quarks exactly and ig-nores the effects of sea quarks.In Section 3.1 we discuss lattice calculations of the the ψ and Υ systems.Solid error analysis is easiest to produce in these simple systems because of thepossibility of using nonrelativistic methods.

The 1P–1S splitting, ∆m1P–1S, inthese systems is insensitive to the most serious sources of error in lattice calcula-tions. This makes it the ideal quantity for setting the scale, i.e.

converting fromlattice to physical units. The calculation of the light hadron spectrum is one ofthe original goals of lattice gauge theory, and a completely reliable calculationis still a major piece of unfinished business.

For many years the progress wasincremental. As discussed in Section 3.2, however, recent developments mayrepresent a new standard in the thoroughness in the treatment of errors.Section 4 discusses applications of the spectrum calculations towards de-termining the fundamental parameters of QCD, the quark masses and strongcoupling constant.The lattice determination of the latter from the ψ andΥ spectra is already competitive with perturbative determinations from high-energy scattering experiments.

As with the “traditional” results for αS, thereis still a phenomenological component in the lattice determinations. The ma-3

jor uncertainty in the lattice QCD results come from modeling the effects ofsea quarks. However, in lattice QCD the path is clear towards eliminating themodeling completely.

Hence, in the long run, the most precise determinationof αS will likely come from lattice QCD.Although the spectrum calculations are indisputably essential to the veri-fication of QCD, many lattice calculations in weak-interaction phenomenologyare of even greater importance to the standard model. This is the subject ofSection 5.

The CKM matrix is responsible for (at least) four parameters, andone would like to overdetermine it to test whether there are further generations(with massive neutrinos). The good news is that some of the associated hadronicphysics, such as the kaon “B” parameter, can be calculated with comparableor greater reliability than the light hadron spectrum.Since an excellent introduction appeared in this series eight years ago [6],this article does not review the foundations of lattice field theory.Anotherpedagogical introduction is in Reference [7].

For an encyclopedic overview ofthe activity in lattice field theory, the reader can consult any recent proceedingsof the annual international symposium on lattice field theory [8]–[13].This review also omits several important applications of lattice field theory.The study of the deconfinement temperature in SU(3) gauge theory withoutquarks was influential. It was the first careful application of large scale MonteCarlo methods to a quantity whose value was not well known in advance [14, 15].More recent work with three light quarks suggests that the structure of thephase transition in QCD may depend sensitively on the mass of the strangequark [16].

For a review of these and other topics in QCD thermodynamics, seeReference [17]. Analytical and numerical methods of lattice field theory havebeen used to obtain upper bounds on the masses of the Higgs boson and ofheavy quarks in the standard model [18].

Most proposals for strongly coupledmodels of electroweak symmetry breaking require a lattice regularization forchiral fermions. This problem is still unsolved; the status of of current ideas forsolutions is reviewed in Reference [19].With one exception, all of the QCD entries in Table 1 are based on mesonproperties.

The bound on the strong CP-violating parameter θQCD, however,comes from the neutron electric dipole moment [20, 21]. Lattice QCD calcu-lations of such baryon properties are more difficult than comparable ones formesons, so there has been less systematic work.

Another QCD topic of vitalinterest is the study of glueballs and other bound states that would not ap-pear in the quark model. Despite the algorithmic improvements of recent years[22, 23], glueball mass calculations still suffer from a small signal-to-noise ratio.Therefore, it seems appropriate to postpone a review of baryon and glueballphenomenology.4

Table 1: Parameters of the standard model and lattice calculations which willhelp determine them. Ranges for CKM matrix elements assume unitarity butnot three generations.

Numerical values taken from Reference [1], except sin δand θQCD.parametervalue or rangerelated lattice calculationssection(s)αem1/137.036105GF1.166 GeV−2αMS(MZ)0.110–0.118∆m1P–1S; scaling3.1; 4.1mZ91.17 GeVmH> 48 GeVme0.51100 MeVmµ105.66 MeVmτ1.78 GeVmu2–8 MeVm2π, m2K3.2, 4.2md5–15 MeVm2π, m2K3.2, 4.2ms100–300 MeVm2K3.2, 4.2mc1.3–1.7 GeVmJ/ψ3.1, 4.2mb4.7–5.3 GeVmΥ3.1, 4.2mt> 91 GeV|Vud|0.974|Vus|0.220|Vub|0.002–0.007B →ρlν5.2|Vcd|0.179–0.228D →πlν5.2|Vcs|0.864–0.975D →Klν5.2|Vcb|0.032–0.054B →Dlν5.2|Vtd|0.0–0.14fB, BB; BK5.1, 5.3|Vts|0.0–0.45fBs, BBs5.1, 5.3|Vtb|0.0–0.9995sin δ̸= 0BK, BB, BBs5.3θQCD< 10−9dn5

2Lattice MethodologyBecause of our emphasis on standard-model phenomenology, we omit discussionof most technical details. For a more thorough introduction, see References [6,7].

We provide here only a schematic overview of lattice methods, plus a briefdiscussion for nonexperts of the most important sources of uncertainty in latticecalculations.2.1MethodsThe path integral formulation of quantum field theory is used to define latticeQCD:Z =ZDU Dψ D ¯ψ e−SG−SQ. (1)The integration is over each U variable (SU(3) matrices representing the gluonfields, defined on each link of the lattice) and each ψ and ¯ψ field (anticommutingvariables representing quark fields, defined on each site of the lattice).Thestandard action used in almost all lattice calculations isSG = β6Xx,µ,νPµν(x)(2)for the gluons.

The “plaquette” Pµν(x) is the trace of the product of the Umatrices around the elementary square at x in the µ-ν plane. There are twocommonly used formulations of lattice fermions.

Except for quark masses andBK, the calculations we discuss use the Wilson formulationSQ=−κXx,µ¯ψx[(1 −γµ)Ux,µψx+ˆµ + (1 + γµ)U †x−ˆµ,µψx−ˆµ](3)+Xx¯ψxψx.Wilson fermions allow the proper number of flavors at the expense of a dif-ficult handling of chiral symmetry. When chiral symmetry is crucial anotherformulation is available, “staggered fermions,” which maintain an exact chiralsymmetry, but then the number of flavors is a multiple of four.The parameters in the lattice action are β and the “hopping parameter”κ.

The bare lattice coupling constant is given by β = 6/g20. The bare quarkmass is related to κ by κ = 1/(8 + 2m0a).

Using these identifications it is easyto show that in the zero lattice spacing limit, this action reduces to the usualQCD action, SQ =R d4x ¯ψ(x)(∂µγµ + m0)ψ(x), and SG = (1/4g20)R d4x(F aµν)2,plus errors which vanish with the lattice spacing. The lattice spacing a is made6

to vanish by taking β →∞, keeping physical quantities fixed. Note, however,that much of the literature uses “lattice units,” where a = 1.The path integral formalism shows how the correlation functions of hadronoperators Φt(U, ψ, ¯ψ) (at time t) behave:⟨ΦtΦ†0⟩=1ZZDU Dψ D ¯ψ ΦtΦ†0e−SG−SF(4)=Xβ|⟨0|ˆΦ|β⟩|2e−tEβ.

(5)The hadronic correlation functions decay as sums of exponentials if the theoryis formulated in Euclidean space. The rates of decay Eβ of these exponentialsare the energies of the states β.

The coefficients of the exponentials are relatedto hadronic matrix elements. The Euclidean-space formulation has many nu-merical advantages.

For example, for t large enough, it is possible to isolatethe state with the lowest energy. However, the contributions to the correlationfunction of all the states β, other than the lowest state, produce errors whichmust estimated and eliminated.In integrals such as Equation 4 the integration over the Fermi fields canperformed explicitly, leaving a gauge-field integral.

This step expresses the cor-relation function in terms of quark propagators in a background gauge field. Theintegration over the gauge fields is evaluated by Monte Carlo with importancesampling, yielding ensembles of lattice gauge fields.

The gauge-field integral isapproximated as a finite sum, introducing a statistical error. New gauge fieldconfigurations are generated from previous ones and are correlated with them.This effect must be carefully accounted for in the statistical analysis.Quark propagators are solutions of the discrete Dirac equation, which isa sparse matrix equation.Sparse matrix methods are used to produce thepropagators for each lattice.

These algorithms must be employed much moreoften in full QCD than in the quenched approximation, to account for theback-reaction of the sea quarks on the gluons.2.2Error AnalysisThese sources of uncertainty in this section must be individually understoodif numerical lattice QCD is to become a widely accepted calculational tool. Afirst pass at a thorough enumeration has been attempted for only a few of thesimplest quantities, but there is a good hope that full error analysis (in thequenched approximation) will be extended to many more quantities over thenext two or three years, including some extremely interesting ones.

Therefore,we shall now discuss how the various sources of error can be reduced:7

Statistical errors.Any Monte Carlo procedure has statistical uncertainties.In lattice QCD these may be the errors which are currently under best control.A subtlety is to cope with the correlations among subsequent configurations.These correlations can extend over stretches in the Monte Carlo chain, especiallyfor the algorithms used in full QCD. Another subtlety is that the statisticaluncertainty of quantities calculated within a single ensemble of gauge fields arecorrelated.

Hence, ratios of similar quantities usually have smaller statisticalerrors than the quantities themselves.Finite lattice spacing errors.If the lattice action in is expanded in powersof the lattice spacing, one obtains the standard continuum action of QCD,plus an infinite series of unwanted, higher dimension operators whose effectson masses (or other quantities derived from the spectrum) vanish as powers ofthe lattice spacing. Their effects can be systematically eliminated by addinghigher dimension correction operators to the lattice action [24].

An order ofmagnitude estimate of their effects is aΛQCD to the appropriate power. Forβ = 6.0, a−1 ≈2 GeV, this is around 10–15% for the simple O(a) correction forWilson fermions, and 1–2% for the more complicated O(a2) errors of the quarkand gluon actions.The most serious of these errors, the O(a) error for Wilson fermions, can becorrected by the addition of a single term to the fermion action [25]δSQ = ig c2κXx,µ,ν¯ψxσµνFµνψx,(6)where σµν = i2[γµ, γν], the γµ are Euclidean gamma matrices, and Fµν reducesto the QCD field strength tensor as a →0.

Direct calculational evidence ofthe importance of this correction has been given in Reference [26] and in thecharmonium calculations described in Section 3.1. A more careful examinationreveals that the coefficient c depends on the bare coupling.

In perturbationtheory c = 1 + c1g20 + · · ·. The systematic program of adding corrections like∆SQ and calculating their coefficients is called “improvement” [24].Finite volume errors.Numerical calculations of lattice QCD are done ina finite volume, because then there is a finite number of degrees of freedom,which can be stored in the finite memory of a computer.

Finite volume errorsare nonperturbative properties of QCD, and thus more complicated to analyze.However, for periodic boundary conditions, they are expected to fall exponen-tially with lattice size. It is therefore a reasonable goal to increase the latticeuntil they are really negligible.

The asymptotic errors are known for the proton8

mass [27], and quite small for lattices of reasonable size. The functional formin the intermediate region is unknown and must be carefully determined bynumerical calculation.

For an excellent technical review of the state of the arton this and many other issues in hadron spectroscopy, see [28].The effects of higher mass states.In extracting the properties of theground state from correlation functions such as Equation 5, the contaminationfrom more massive states with the same quantum numbers must be estimatedand reduced. This is most often done by separating the creation and destructionoperators far enough that only a single exponential of the sum in Equation 5is visible within the statistical errors.

This approach has the drawbacks thatit is limited by increasing statistical errors as the operators are separated, andthat systematic uncertainty estimates are difficult. Another approach is to varythe operator or matrix of operators Φ to maximize the overlap with the desiredstate and minimize the overlap with the rest.The extrapolation to physical quark mass.Current lattice algorithms forsparse matrix inversion (and thus for the inclusion of the effects of sea quarks)become much more computationally demanding, and sometimes fail entirely, asthe quark mass is reduced toward its physical value.

Current calculations rarelygo below mπ/mρ ∼0.4, compared to the physical value of 0.18. Leading-orderchiral behavior is usually assumed in extrapolating to the physical quark mass(m2π and the masses of the other hadrons proportional to mq).The size ofdeviations from linearity for mesons of nearly the mass of the kaon are contro-versial among workers in chiral perturbation theory.

In lattice QCD they mustbe determined by numerical calculations.2.3The quenched approximationWhile gradual and systematic programs exist for the elimination of the abovesources of error, no better way is known to improve on the quenched approxi-mation than to include all effects of sea quarks at once. Formulas for the effectsof small numbers of internal quark loops may be derived in terms of correla-tions of hadronic operators with the fermionic effective action, but they appearto be even harder to handle than the exact formula.

Algorithms for inclusionof quark loops are much more computationally demanding than those whichomit them, so the analysis of the other sources of uncertainty is much cruderfor calculations which include them. This review will therefore concentrate oncalculations which omit them.

In a few cases, but not in general, it is possible9

to make phenomenological estimates of the accuracy of the quenched approx-imation. We will return in Section 6 to the general case of the effects of seaquarks.3Establishing and Testing Lattice Methods3.1The ψ and Υ SystemsThe discovery of charmonium, the bound states of c and c quarks, with theirclear positronium-like spectra, provided an important psychological boost tothe belief in the reality of quarks.

The success nonrelativistic potential mod-els [29] in accounting for these spectra provided a boost to the acceptance ofQCD as the theory of strong interactions, since the models became equivalentto leading order QCD in a well defined limit: the large quark mass limit. The ψand Υ systems are proving crucial in establishing the accuracy of lattice calcu-lations because nonrelativistic reasoning opens ways of checking and recheckingmethods of error analysis that are unavailable for the lighter hadrons.As Lepage [30], has emphasized, now that lattice methods are coming intofruition, it is these simple systems which will provide the best early tests oflattice methods.

There are some technical reasons for this. Since the quarksare heavy, the extrapolation to the physical light quark mass required in lighthadron calculations is unnecessary.

The propagators of heavy quarks are muchquicker to calculate on the lattice than those of light quarks. Further, since theheavy mesons are smaller than the light hadrons, smaller physical volumes suf-fice.

However, the most important fact making the properties of these mesonsthe easiest to calculate on the lattice is the one that made possible the goodphenomenological treatment of them twenty years ago: they are nonrelativis-tic systems. This means means that potential models and the nonrelativisticarguments justifying them can be used both to guide the physics expectationsof the lattice calculations, and to supplement the analysis of corrections anduncertainties in the lattice calculations.Potential models play an important role in defining physics expectationsfor lattice charmonium calculations.For example, the part of the hyperfineinteraction which is due to perturbative gluon exchange isHHF = 32π9αsm2qS1 · S2δ3(r).

(7)Evaluating this term perturbatively with nonrelativistic wave functions gives∆MHF = 32π9αsm2q|Ψ(0)|2(8)10

for the splitting between the ψ and the ηc. The spin-spin interaction in Equa-tion 7 arises from the exchange of transverse gluons between the heavy quarks.For massive quarks, the dominant effect of the O(a) correction for Wilsonfermions, Equation 6, is just such a gluon-spin interaction, so the hyperfinesplitting will be sensitive to this correction.

According to Equation 8, ∆MHFis also sensitive to the value of the quark mass, which is not determined on thelattice to perfect accuracy. We can therefore expect the hyperfine splitting tobe a sensitive test of lattice methods.On the other hand, the spin averaged splitting between the lowest angularmomentum (l = 0 and l = 1) levels of the ψ and Υ systems is a crucial onefor lattice QCD because nonrelativistic arguments tell us to expect it to beinsensitive to these important sources of error.Since it is a spin averagedquantity, it should be insensitive to uncertainties in the coefficient of the O(a)correction term.

Since it is virtually the same for the ψ and the Υ, it shouldbe insensitive to any imperfections in our knowledge of the quark mass. It istherefore a good quantity to use to extract information about QCD from latticemethods.

It may be the most accurate determination of the lattice spacingin physical units (a key component of the extraction of the strong couplingconstant using lattice methods).Because the systems are nonrelativistic, their Coulomb-gauge wave func-tions calculated on the lattice will give a good picture of the properties of thestates. Figure 1 shows the wave function of the J/ψ meson calculated on a 244lattice at β = 6.1 [31].

It has approximately the exponential shape of a Coulombwave function, but at large distances it falls offfaster due to confinement, andat short distances it rises more slowly due to asymptotic freedom. Halfwayacross the lattice, at r/a = 12, the effects of periodic boundary conditions areclearly seen.

Such wave functions have practical roles to play in lattice calcu-lations. They can be used to estimate finite lattice spacing and finite volumeerrors perturbatively.

They can be used to make improved operators to createand destroy the meson states.One of their most important roles, however,is the clear and simple demonstration that the lattice calculations are indeedproducing charmonium states.To the extent that the ψ and Υ systems are nonrelativistic, one can usepotential model arguments to estimate and correct for the effects of sea quarks.These effects are expected to be rather small, since, for example, the widths ofexcited ψ and Υ states into D or B mesons, 50–100 MeV, are only 10–20% oftypical energy splittings between states.If middle distance physics like the 1P–1S splitting is used to tune the bareparameters of the theory, the effective action at those distances will be aboutright. In a theory with too much asymptotic freedom, the effective coupling11

05101520r/a10-510-410-310-210-1ΨJ/ψ(r)Figure 1: The wave function of the J/ψ meson calculated on the lattice [31].at short distances will be a bit too small. Likewise, short distance quantitieswhich depend on it, like the wave function at the origin, will be too small.These effects may be estimated using the Richardson potential [32], whichincorporates the effects of asymptotic freedom at short distances.Figure 2shows two potentials resulting from fits to the charmonium spectrum: the lowerpotential having the correct one loop β function b0 = 11 −2nf/3 with nf = 3,and the upper one with the stronger β function for nf = 0 of quenched QCD.As expected, the two potentials agree almost perfectly in the middle distanceregion, but the nf = 0 potential is too soft at short distances.The spin-averaged 1P–1S splitting has been calculated by several groups[31, 33, 34].

The lattice spacing in physical units is obtained by the latticeresult obtained in lattice units with the physical answer. In the ψ system, forexample, ∆m1P–1S = mhc −(3mJ/ψ + mηc)/4 = 458.6 ± 0.4 MeV.

In the Υsystem, since the 1P1 is undiscovered, the splitting between the spin-averagedχb states and the spin-averaged 1S states may be used, ∆m1P–1S = 452 MeV.The values of the lattice spacing obtained from this splitting are shown inTable 2. They will be crucial components in the determination of the strongcoupling constant in Section 4.1.

They do not differ dramatically from those12

012345r (GeV)-1-1-0.500.5V(r) (GeV)Figure 2:Results from fits to the charmonium spectrum with asymptoticallyfree phenomenological potentials having the correct β function (bottom) andthe β function of quenched QCD (top).obtained from other quantities, such as the ρ mass [35] or the string tension[36]. It is the possibility of making better uncertainty estimates that makes thisan important way of determining the lattice spacing.In Reference [31], finite lattice spacing errors were treated by the explicitinclusion of the term in Equation 6, in the numerical calculations.

In Refer-ence [33], corrections operators were evaluated perturbatively using the Rich-ardson potential model wave functions, as the hyperfine splitting was evaluatedin Equation 8. The lattice action of nonrelativistic QCD (NRQCD) [30] wasused in this work, so O(v2) correction were also included.

These estimates couldalso be made without recourse to potential models, but still using using non-relativistic reasoning, by using wave functions calculated directly with latticemethods (see Figure 1).Both groups checked these corrections by verifying that the same answerwas obtained for several different lattice spacings. It would be desirable to havea calculation in which both methods were used in the same analysis in order totest carefully the method of directly including the correction operators in the13

Table 2: Inverse lattice spacings obtained from the 1P–1S splitting in the ψand Υ systems.βa−1 (GeV)SystemRef.5.71.15(8)ψ[31]5.91.78(9)ψ[31]6.12.43(15)ψ[31]5.71.14(4)ψ[33]5.71.26(14)Υ[33]6.02.11(7)Υ[33]simulation, since that is the only avenue for evaluating them available for thelight hadrons.Likewise, for the light hadrons the functional form of the finite volume errorsin the crucial intermediate distance region is not known, and must be calculatednumerically. Such a calculation in a nonrelativistic system supplemented by anonrelativistic wave function calculation of the finite volume errors would beuseful in illuminating the methods of error analysis for the light hadrons.The nonrelativistic picture tells us that the hyperfine splitting and leptonicdecay amplitude are short distance quantities, proportional to the square of thewave function at the origin.

The hyperfine splitting is also proportional to theshort distance coupling constant, and thus additionally suppressed. The size ofthese suppressions for the hyperfine splitting has been estimated as −30–40%using the Richardson potential [37].The hyperfine splitting can be used to check the effects of the O(a) correctionterm for Wilson fermions, Equation 6, which yields dominantly a spin-spincoupling for quarkonia.Compared with its physical value of 117 MeV andthe estimate of the quenched corrected value of 70 MeV, unimproved Wilsonfermions (c = 0 in Equation 6) produce splittings of as little as 10–20 MeV,depending on the lattice spacing.

Calculations with the tree level coefficientc = 1.0 yield around 50 MeV [38], and with a perturbatively corrected coefficientc = 1.4 yield around 90 MeV [37].The precision is insufficient to allow aphenomenological determination of the coefficient to supplant the perturbativeone, but does show clearly that the improved action yields reasonable resultswhile the unimproved action does not.A topic related to quarkonium is the lattice QCD calculation of the staticpotential. In Figure 3 we show results in the quenched approximation from a14

Figure 3:The heavy-quark potential (in lattice units) calculated in thequenched approximation [36, 39].324 lattice at β = 6.4 [36, 39]. It is fit very well by a Coulomb-plus-linear form.At short distances it agrees well with the predictions of lattice perturbationtheory [40].

The potential at large distances is well fit by a straight line. Thestring tension obtained obeys asymptotic scaling to about 20% if a renormalizedcoupling constant is used.

That is, the ratio √σ/Λ(0)MS varies by less than 20%when β > 5.7 (a < 0.2 fm). Earlier apparent evidence that scaling violationsas large as a factor of two and that much smaller lattice spacings were requiredhas been understood as an artifact of the use of the bare lattice coupling con-stant for the perturbative analysis [41].

This introduced poor behavior into theperturbation theory somewhat analogous to that resulting from attempting todo perturbative QCD phenomenology with the MS coupling constant αMS(q)rather than the MS coupling constant αMS(q). The remaining small scaling vi-olations arise from both logarithmic (in a) perturbative corrections and powerlaw finite a effects, so uncertainties associated with them cannot be removed15

K* φ N Ξ+Σ-N ∆ Σ* Ξ* Ω 11.522.5MX / MρFigure 4: The spectrum of the light hadrons. Error bars are from lattice calcu-lations in the quenched approximation [35], and + denotes experiment.cleanly by extrapolation.

A summary of various recent analyses [28] containsresults all falling in the range√σ/Λ(0)MS = 1.85 ± 10%. (9)3.2The Light Hadron SpectrumThis year Weingarten and collaborators took an important step forward inthe calculation of the light hadron spectrum in the quenched approximation[35].

This work, in a single, systematic calculation, attempted to analyze andextrapolate away the three major source of systematic error in the quenchedapproximation: extrapolation to zero lattice spacing, to infinite volume, and tophysical quark mass.The results are shown in Figure 4. The lattice spacing has been eliminatedusing mρ, and the bare quark masses using m2π and m2K.

The errors shownare statistical.The authors argue that the uncertainties involved in the ex-trapolations to infinite volume and physical quark mass are smaller than the16

statistical errors. They have not attempted to estimate the uncertainty in theextrapolation to zero lattice spacing.The extrapolation of m2π and the masses of the other hadrons to physicalquark mass was made linearly in mq in accordance with theoretical prejudice.The expected functional forms were seen in the data.

A version of the Gell-Mann–Okubo formula was used to argue that the error arising from this ex-trapolation was around 1%. This estimate could be supplemented by directnumerical investigation of the functional form.The extrapolation to infinite volume was made by performing the calcula-tions on the coarsest lattice at several volumes, and using the results to extrap-olate the calculations on the finer lattices to infinite volume.

The extrapolationused only two points. Much work is now by several groups [28] to determine thefunctional form of the volume dependence which should make the extrapolationreasonably solid.The results were extrapolated linearly in a to zero lattice spacing in accor-dance with theoretical prejudice, but the data were not precise enough to testthe accuracy of that prejudice.

This extrapolation can be improved by addingthe single additional term to the quark action which suffices to remove the O(a)error from Wilson fermions [25] and checking that the observed dependence ofthe results on the lattice spacing disappears.Possible problems with contamination from higher mass states in each had-ronic channel showing up in the work of other groups have been emphasizedby Ukawa [28].An important contribution toward reducing these problemswas made by the APE collaboration in 1988 [42] who pointed out that quarksspread out over roughly the size of a light hadron have a much larger overlapwith the light hadrons and a much smaller overlap with excited states than dothe local quark operators which had been in use up to that time. Much moresophisticated work along these lines is possible following the lead of glueballcalculations whose worse signal to noise problems have forced a more seriousexamination of this problem [43].The analysis of the uncertainties in a light hadron calculation is more de-manding than in a charmonium calculation, since nonrelativistic argumentsdo not help.

Nevertheless, there are further technical tools available, such asEquation 6, than have been applied so far in this calculation, which should makepossible the confirmation or improvement of these uncertainty estimates for thequenched light hadron spectrum with the present generation of computers.17

4QCD PhenomenologyQCD is supposed to describe high-energy perturbative phenomena, such as deepinelastic scattering, as well as low-energy nonperturbative phenomena, such ashadron masses. In QCD with nf quark flavors, there are nf + 1 parameters,the quark masses and the strong coupling constant.

The latter is equivalentto a standard mass to set the scale in MeV. Using the nonperturbative latticeformulation of QCD it is possible to compute them using nf+1 hadron masses asthe physics input.

If these masses are chosen unwisely, a cumbersome jugglingact must be performed, adjusting the bare parameters, computing the hadronmasses and working back to the renormalized parameters.As explained inSection 3.1 the 1P–1S splitting of quarkonium is insensitive to light and heavyquark masses. Consequently, it is ideal for converting from lattice units to MeV.Once this has been done, it is relatively straightforward to use meson massesto determine quark masses.Given αS one can then test whether the same QCD describes the strong in-teractions at all energies.

One simply inserts the nonperturbatively computedcoupling into the perturbative series for high-energy scattering and compareswith data. A favorable outcome will increase our quantitative confidence inQCD enormously.

At present lattice QCD can offer αS with systematic uncer-tainties comparable to deep inelastic scattering, although the analysis is lessmature. These results, and the theoretical ideas needed to reduce the uncer-tainties to a negligible level, are in Section 4.1.Because of confinement, quark masses cannot be measured directly.

How-ever, every serious theoretical construct that goes beyond the standard modelprovides MS quark masses as an output. Hence, good estimates of quark massesfrom lattice QCD should prove useful to builders of new physics models.4.1The Coupling ConstantThe numerical value of the strong coupling depends on the “scheme” chosento define it.

A scheme can be defined by a renormalization convention, suchas the MS scheme in dimensional regularization or the bare scheme in latticeperturbation theory. More generally, it can be defined by any physical quantitythat is equal to the bare coupling at the leading order of perturbation theory.For example, in QED the low-energy limit of Thomson scattering is used todefine the electromagnetic coupling.For QCD L¨uscher, et al., have delineated four criteria for a practical scheme[44].

The physical quantity should have a rigorous nonperturbative definition;otherwise it cannot be calculated nonperturbatively.In Monte Carlo simu-18

lations it ought to have a good signal-to-noise ratio, so that small statisticaluncertainties can be achieved in a reasonable amount of computer time. Fur-thermore, uncertainties arising from extrapolations in lattice spacing and quarkmass should accumulate slowly.

Finally, a perturbative calculation of the phys-ical quantity must be tractable, so that the nonperturbative coupling can beused in perturbative QCD.Once one has chosen a scheme s, one must relate the dimensionless cou-pling to the standard mass. This is done using the renormalization group.

Itis important to realize that renormalization-group calculations can be carriedout nonperturbatively.Because of asymptotic freedom, the nonperturbativeand perturbative q dependence must agree for large enough q. The region ofagreement provides a numerical value for g2s(q) that can be used in perturba-tive series for high-energy scattering.

The standard mass is needed to convertq from lattice units of the nonperturbative calculations to physical units. Forexample,q (MeV) =aqa∆m1P–1S458.6 (MeV);(10)the numerator and denominator of the fraction come from the lattice calcula-tion, and ∆m1P–1S = 458.6 MeV from experiment.In Reference [31], a perturbative relation (improved by mean field theory)was used to estimate the continuum coupling constant in terms of the barelattice coupling constant.In Reference [41] it was found that perturbativecalculations of short distance physical quantities in terms of a coupling constantestimated in this way were systematically lower than Monte Carlo calculationsby a few per cent.

This suggests that a slightly improved determination wouldbe given by extracting the coupling directly from Monte Carlo calculations.It remains to be checked that there are no substantial deviations between thecouplings determined in this way, from physics on scales from one to half adozen lattice spacings, and true continuum couplings [44].The work of Reference [31] gaveα(0)MS(5 GeV) = 0.140 ± 0.004,(11)where the superscript emphasizes the number of active quark flavors.Thiserror bar comes from the statistical uncertainty in the 1P–1S splitting used todetermine a, augmented somewhat by lattice-spacing effects. Complementaryanalyses based on the short-distance static potential [36, 45] yield values ofα(0)MS(5 GeV) consistent with these.The nf = 0 result can be converted to nf = 4 by appealing to the potentialmodels that describe quarkonia so well.

Choosing the 1P–1S splitting to set the19

scale is equivalent to adjusting the coupling of the quenched theory to repro-duce physics at intermediate energies. Since the coupling runs faster for fewerquarks, this adjustment makes the coupling at 5 GeV too small.

Correcting thequenched result for this effect yields [31]α(4)MS(5 GeV) = 0.174 ± 0.012. (12)For comparison with the compilation in Reference [1]α(5)MS(MZ) = 0.105 ± 0.004.

(13)The error bar in Equation 12 is three times larger than in Equation 11, becausethe matching energy is not known exactly, and because for charmonium it israther low. The bulk of the correction is due to the effects of light quarks onthe potential at short distances, which can be calculated in perturbation theory.However, a part of the correction arises from the effects of light quarks on thepotential at middle distances, which must be estimated phenomenologically.It is therefore significant that a similar analysis has also been carried outusing NRQCD in both the ψ and Υ systems [33].

Typical energy scales in theΥ are about twice those in the ψ. (For example, typical gluon momenta are400 MeV and 800 MeV in the J/ψ and Υ states, respectively.) The effects oflight quarks in the murky intermediate distance region may be expected to bequite different at the Υ than at the ψ.

Although some details of the systematicerror analysis is different, the ψ-system calculation agrees with Equations 11and 12. There are subtle differences in the determination of αMS from the Υsystem, which arise because the typical energy scales are higher.

As shown inTable 2 the lattice spacing determined by the Υ 1P–1S splitting is about 10%smaller. Propagating this change implies that α(0)MS(5 GeV) is somewhat larger.However, the correction for the quenched approximation is smaller, because thematching is done at somewhat higher energies.

If the argument used to computethe correction is valid, the two effects should cancel in α(4)MS. Reference [33] findsα(4)MS(5 GeV) = 0.170 ± 0.010(14)from the Υ system, which agrees remarkably well with the results from the ψsystem.The only way to eliminate the error from the quenched approximation inEquation 12 is to perform calculations in full QCD. The second-most importantuncertainty comes from the dependence on the lattice spacing.

Because the scaleq is tied to the cutoffin these calculations, it is impossible to separate the scalingdependence from any other a dependence. In other words, the criterion that20

uncertainties do not accumulate during extrapolation is not strictly respected.To clear things up, one must associate q with a physical scale. An elegant wayto do so is to take q = 1/L [46], where L is the linear size of the finite volume.References [44, 46] also suggest a class of schemes for which the extrapolationto the continuum limit is controlled.

So far these ideas have been applied to thepure SU(2) gauge theory [44, 46]. For the coupling chosen, the scaling behaviormatches two-loop perturbation theory at surprisingly low energies, perhaps evenas low as q = 1 GeV.In several years full QCD calculations with the scaling analysis of Refer-ences [44, 46] will have computed the strong coupling constant with a precisionof a few per cent.

The uncertainty will be due to finite statistics, compoundedsomewhat by extrapolations to zero lattice spacing and physical quark masses.There will be no uncertainty from truncating perturbation theory and no un-certainty from nonperturbative effects. The specific value for αs(q) will be com-plemented by an energy scale q, above which perturbative evolution is valid.Purely perturbative calculations can then be used to relate the nonperturba-tive scheme s to the MS scheme.

Rather than use this relation to determineαMS, one ought to eliminate it from high-energy perturbative series in favor ofαs. This is analogous to the strategy used in perturbative QED, where the MScoupling is used only as an intermediate step.4.2Quark MassesThe masses of the charm and bottom quarks are currently estimated frompotential model calculations.

Lattice calculations should eventually be able topin these down to a precision of perhaps 5%, limited by perturbation theory.The existing numerical data on the masses of the J/ψ and the Υ is alreadyquite adequate for this purpose. The remaining work required is short-distancelattice perturbation theory with massive quarks, which is rather complicated.The top quark is expected to decay weakly before it can form a QCD boundstate.

Lattice calculations are unlikely to be useful in determining its mass afterit is found.For the light quarks it is convenient to discuss the combinations ˆm = 12(md+mu), ∆m2du = m2d −m2u, and ms. Ratios of the light-quark masses are currentlybest estimated using chiral perturbation theory, a systematic description ofthe low energy, small quark mass limit of QCD [47].

To set the overall scalerequires a dynamical calculation. In lattice QCD, ˆm and ms can be extractedfrom the variation in the square of the pseudoscalar mass between m2π and m2K.The most difficult quark-mass combination is ∆m2du, which causes the isospin-violating part of the splittings in hadron multiplets.

Since chiral perturbation21

00.020.040.060.080.1a mq00.20.40.6a2 mπ2Figure 5: Pseudoscalar meson mass squared as a function of the quark mass,calculated in the quenched approximation [48]. The line is a linear fit to thedata.theory provides a formula for ∆m2du/m2s with only second-order corrections, itis likely that the best determination of ∆m2du will come from combining theformula with a lattice QCD result for ms.Uncertainties in the chiral estimates of these ratios arise from varying treat-ments of higher-order terms.

Existing lattice calculations either use very mas-sive sea quarks or ignore sea quarks entirely, so they also treat higher-orderhadronic effects somewhat incorrectly. We probably must wait for better calcu-lations including sea quarks correctly before lattice calculations can contributeto the determination of the ratios.

As illustrated in Figure 5, present calcu-lations show a linear relation between the quark mass and the square of themeson mass, as expected from lowest order chiral perturbation theory alone,up to surprisingly large values of the quark mass.The overall mass scale of the light quarks is currently determined by lessreliable phenomenological assumptions. A full error analysis of lattice determi-nations of this quantity has not been completed, but it is poorly enough knownfrom conventional phenomenology that it is worth discussing the state of the22

lattice results. Because staggered fermions have an exact chiral symmetry, theyare likely to be superior for these calculations.

As summarized by Ukawa [28],quenched results for ˆmMS(1 GeV) are in the range 2.4–3.0 MeV. This is outsidethe range of 3.5–11.5 MeV indicated in Table 1.

There is no evidence in thedata for large finite volume, finite lattice spacing, or statistical errors. Whenrelatively heavy sea quarks are added to the calculation no qualitative change isobserved: these results cluster around 2 MeV.

These results should not be takentoo seriously until a more complete error analysis exists, but the possibility thatthe conventional estimates are too high is intriguing.5Weak-Interaction PhenomenologyWe now turn to the role lattice QCD can play in determining the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the standard model the CKM matrixaccounts for four of the 19 parameters.

Furthermore, to test the standard modelone would like to extract all elements of the CKM matrix and verify that it isunitary. Because the observable consequences of the CKM matrix involve weaktransitions of hadrons, nonperturbative QCD enters immediately.

We shall fo-cus on processes that are especially amenable to lattice technology that also playa crucial role in determining the CKM matrix. For a review of weak-interactionphenomenology with emphasis on the CKM matrix see Reference [49]; for moretechnical reviews of the lattice technology see References [50, 51].

This sec-tion discusses leptonic decays in subsection 5.1, semi-leptonic decays in subsec-tion 5.2, and neutral meson mixing in subsection 5.3. A brief explanation ofthe difficulties with non-leptonic decays is in subsection 5.4.A lattice large enough to encompass the scales ΛQCD and mW (or mt) wouldhave several thousand sites on each side.

That is obviously not feasible. For-tunately, it is also not necessary.

Leptonic and semi-leptonic decay amplitudesfactor into a product of leptonic and hadronic matrix elements of electroweakcurrents.Lattice QCD is needed to calculate the hadronic factors ⟨0|Jµ|h⟩for leptonic decays and ⟨h′|Jµ|h⟩for semi-leptonic decays. For neutral mesonmixing and non-leptonic decays, the standard theoretical apparatus uses theoperator product expansion to disentangle contributions above and below ascale µ < mW (and mt).

This analysis leads to the effective weak Hamiltonian,which can be written schematically asHeff=XnCn(µ)O(n)(µ),(15)where, to leading order in m−2W , the O(n) are four-quark operators. Lattice QCD23

is needed to calculate hadronic matrix elements ⟨h′|O(n)|h⟩(µ), where h and h′are various hadronic states.Several conditions must be met before hadronic matrix elements of the four-quark operators can be applied to phenomenology. The µ dependence of thecoefficient functions and the four-quark operators must cancel.

Since the coef-ficient functions are determined perturbatively, the lattice calculations must beperformed with lattice spacings for which perturbation theory is applicable. Inthis way the lattice regulated matrix element ⟨h′|O(n)lat |h⟩(π/a) can be relatedto the renormalized matrix element ⟨h′|O(n)R |h⟩(µ) in the scheme R and at thescale µ for which the coefficient functions are available.

Similar, but simpler, re-lations apply to currents Jµ as well, see below. With our present understandingof lattice perturbation theory [41], this conversion should not introduce largeuncertainties.Independent of such scheme and scale dependence, the four-quark operatorsmust be defined nonperturbatively.

Interactions cause mixing with operatorswith the same (lattice) quantum numbers. When these other operators havethe same or higher dimension, it is presumably adequate to use the definitionsof perturbation theory.

It is at least consistent, because a similar classifica-tion already arises in the operator product expansion, Equation 15 which isestablished perturbatively. A more pernicious problem is mixing with lowerdimension operators.

Since the coefficients of these operators contain inversepowers of a there is no reliable method to remove them perturbatively. Theonly feasible way to remove them nonperturbatively is to insist on the correctscaling behavior and the restoration of continuum-limit symmetries.5.1Decay ConstantsLattice calculations of the decay constants are necessary both as tests and aspredictions of lattice QCD.

We shall follow the normalization convention thatfor pseudoscalar mesons⟨0|¯uγµγ5d|π−⟩= ipµfπ(16)and for vector mesons⟨0|¯uγµd|ρ−; λ⟩= iε(λ)µ mρfρ. (17)From leptonic decays one finds fπ = 131 MeV, fK = 160 MeV, and fρ =216 MeV.

On the other hand, the decay constants of heavy-light mesons (Dand B) are not known experimentally, and the measurements would be dif-ficult.Hence, even semi-quantitative lattice calculations of fD and fB are24

interesting, and, when the uncertainties are fully understood, quantitative lat-tice calculations will play an essential role in understanding D- and B-mesonphenomenology [52].Section 2 pointed out that the lattice artifacts of masses approach the contin-uum limit as a power of a. Hadronic matrix elements, such as decay constants,typically approach the continuum limit more slowly. The lattice operator andthe continuum operator are related as follows¯uγµγ5d|lat = ¯uγµγ5d|cont + c′a¯uDµγ5d + · · · ,(18)where a is the lattice spacing.

In one-loop perturbation theory one easily seesthat Dµ in the second term can absorb a gluon with momentum ∼1/a. Thiscontribution, together with analogous ones from the unwritten terms, yieldcg20a(1/a)/(16π2).

Generalizing to all orders¯uγµγ5d|lat = ¯uγµγ5d|cont"1 +∞Xν=1cν g2016π2!ν#+ O(a),(19)where equality holds for matrix elements of low-momentum states, and O(a)denotes terms that vanish as a power.For small a, g20 ∝(log a)−1.Hence¯uγµγ5d|lat approaches ¯uγµγ5d|cont rather slowly. Although we have used theaxial current as an example, composite operators generally obey an equationanalogous to Equation 19.There are several strategies for handling the lattice-spacing errors indicatedin Equation 19.

One can ignore the perturbative bracket and hope that theO(a) terms are the largest lattice artifact at accessible values of a. This wouldonly be sensible if the coefficients c1 were small, but explicit calculations inseveral papers [53, 54, 55] show that they are not.

One could acquire numer-ical data over a wide range of g20 to perform a correct extrapolation, but thatis impractical. Fortunately, it is possible to improve the situation.

First, ifone recasts perturbative series such as the one in Equation 19 in terms of arenormalized coupling constant, one expects the higher-order corrections to besmall [41]. Second, most of c1 comes from a certain class of diagrams (Feynmangauge tadpole diagrams) [56].

These contributions can be isolated and treatednonperturbatively [41]. Third, systematic improvement to the action [25] andthe operators [57] can reduce the O(a) terms.

With these three improvementsit should be possible to reduce lattice-spacing errors so that they are smallerthan the statistical uncertainties.The most systematic investigation of light-meson decay constants [58] usesthe same gauge configurations and quark propagators used to compute light-hadron masses in Reference [35].The extrapolation in quark mass and the25

Table 3: Summary of results for decay constants.The error bars for lightmesons [58] do not include errors estimates for the quenched approximationand plausibly small residual lattice-spacing errors. The error bars for heavy-light mesons [61] do not include, quenched, finite-volume, or non-zero latticespacing errors.

See the text for a discussion of these errors.fπ/mρfK/mρfρ/mρfD/fπfB/fπexpt.0.1710.2090.281lQCD0.129+0.040−0.0510.164+0.030−0.0340.245+0.055−0.0491.58 ± 0.151.43 ± 0.15Ref. [58][58][58][61][61]finite-volume corrections were handled in the same manner as for the hadronmasses (cf.

Section 3.2).In this case the finite-volume corrections increasethe error bars.The lattice-spacing extrapolation was done as follows.Thelogarithmic a dependence and some of the O(a) dependence was accounted foras specified in References [41, 59, 60], and the remaining a dependence wasassumed to be linear in a. The results of this analysis are tabulated in Table 3.One of the most eagerly pursued topics in lattice QCD is the calculationof heavy-light meson properties.When one of the quarks in the meson be-comes heavy, the dynamics simplifies considerably [62, 63].

In particular, formq ≫ΛQCD the typical momentum in a heavy-light meson remains small,p ∼ΛQCD. The energy scale mq decouples from the heavy quark dynamics,making it possible to derive effective theories [64]–[70].

For infinite mass thereare new symmetries among different spins and flavors of heavy quarks. Thesesymmetries have many interesting implications.

For example, mP = mV andfP = fV , where “P”and “V ” denote generic pseudoscalar and vector heavy-lightmesons, and the various form factors discussed in Section 5.2 can be expressedin terms of one universal function [66].For this section the most important result of heavy-quark symmetry is ascaling law for the pseudoscalar decay constant fP ∝M−1/2P[71]. The leadingsymmetry-breaking effect is at order M−1P , i.e.ΦP = fPpMP = Φ∞−Φ′∞M−1P .

(20)Because of the theoretical utility of heavy-quark symmetry, lattice QCD resultsfor Φ∞and Φ′∞are interesting, as well as the physical results fD and fB.The large mass is also an important technical issue for lattice QCD calcu-lations of heavy-light meson properties. At currently accessible values of the26

lattice spacing, charm and bottom lie in region mqa ≈1, and for the infinitemass limit one must reconcile mq →∞and a →0 in a compatible way. This isdone by formulating a lattice action of the effective theories, either static [52] ornonrelativistic [67, 70].

In analogy with eqs. (18) and (19), the currents of theeffective lattice theory must be matched to the relativistic continuum theory[72, 73, 74, 75].

Another approach is to use Wilson fermions and extrapolatetowards infinite mass. At first sight this seems risky.

However, it is possibleto show how the energy scale mq decouples in the lattice theory [76]. Such ananalysis shows how to interpret the Wilson theory as an effective theory, andhow it shares many features with the static and nonrelativistic theories [59, 60].Now let us discuss results from lattice QCD for Φ∞, Φ′∞, fD, and fB.

Mostof the work has focussed on one of two lines of attack. One is a systematicanalysis of the infinite-mass (or static) limit [77, 78, 79, 80], concentrating onΦ∞.

The important technical issues are optimizing the signal-to-noise ratio,and studying the lattice-spacing and finite-volume dependence of Φ∞.Theother line of attack is to concentrate on the mass dependence. Until now thishas meant combining numerical data from quark masses near the charm masswith the static-limit results, and interpolating [61, 82, 83, 84, 85].Results for fD and fB from Reference [61] are in Table 3.

(We cite Ref-erence [61] because it comes close to incorporating the mass effects derivedin References [59, 76].) Heavy-strange meson decay constants are fD/fDs =fB/fBs = 0.90 ± 0.05.

Taking meson masses and fπ from experiment, the scal-ing combinations are ΦD = 0.28 ± 0.03 GeV3/2 and ΦB = 0.43 ± 0.04 GeV3/2,which can be compared with the static limit Φ∞= 0.53 ± 0.10 GeV3/2 [61]. (This value is consistent with References [78, 85] and with Reference [77] whenscale-setting ambiguities are resolved.

)The systematic studies of the staticlimit [78] suggest that the extrapolation to infinite volume will change theseresults negligibly, and that the extrapolation to a = 0 may reduce the resultsby 10%. A more serious source of uncertainty comes from setting the scale.The results presented here use fπ to set the scale.

This may not be the bestchoice as a rule, but one might argue that the quenched approximation’s errorscancel to some extent in fP/fπ. For example, the ratio fK/fπ in Table 3 [58]agrees much better with experiment than the decay constants themselves.Although the numerical results may not yet be definitive, there are twoimportant conclusions to draw from these lattice results: First, Φ∞, fD, and fBare larger than many model calculations had suggested [86].

By combining thefirst column of Table 3 with the heavy-light results, one sees that discrepancyis even more dramatic using mρ as the standard of mass. Second, the 1/MPcorrections are large and phenomenologically important.This is not reallyunexpected from the heavy-quark symmetry arguments, since the correction is27

MM′lνs, c, bqsqdFigure 6:Spectator diagram for meson semi-leptonic decays. For the weakinteractions, the diagram may be interpreted as a Feynman diagram.

Howeverthe strong interactions binding quarks into mesons must be treated nonpertur-batively, as indicated by the grey shading.first order, and it need not be indicative of the size of second-order corrections.5.2Semi-Leptonic DecaysA generic semi-leptonic decay can be denoted A →Xlν, where A is a flavoredhadron. We shall focus on mesons, because they are easier than baryons tostudy, both experimentally and theoretically.

The process1 is depicted in Fig-ure 6. The flavored quark (strange, charm, or bottom) undergoes a weak decayby emitting a virtual W boson that subsequently turns into the lepton pair.The other quark (¯qs in Figure 6) does not take part in the weak decay, so itis called the spectator.

However, the QCD interaction between the spectatorquark and the decaying quark (s, c, b →qd in Figure 6) is the most difficultfeature of the decay to calculate. It is the part that requires lattice QCD.The amplitude for A →Xlν is proportional to the hadronic matrix element⟨X|Jµ|A⟩, where Jµ is the V −A charged current.

If the quark of flavor a turns1When the final state meson X is an isoscalar and A is charged, there is another diagramin which A annihilates into W and X emerges out of the glue. For simplicity we shall ignorethese decays.28

into flavor xJµ = ¯xγµ(1 −γ5)a. (21)It is convenient to express the amplitude in terms of form factors.

When X isa pseudoscalar meson one writes⟨X|Jµ|A⟩= f+(q2)(p + p′)µ + f−(q2)(p −p′)µ,(22)where p (p′) is the initial (final) state meson’s momentum and q = p−p′ = pl +pν. Similarly, when X is a vector meson there are four independent form factors.Decays correspond to the kinematic region m2l < q2 ≤q2max = (m −m′)2; inthe rest frame of the initial meson, the neutrino is soft at q2 = m2l , whereas thefinal state meson is at rest at the “endpoint” q2 = q2max.The interplay between experiment, lattice QCD, and the CKM matrix be-comes clear upon examining the differential decay rate.

For example, when Xis a pseudoscalar mesondΓdq2 = G2F λ3/2192π3m3AVaxf+(q2)2 ,(23)where Vax is the element of the CKM matrix associated with the quark-Wvertex in Figure 6, and λ = (m2A −m2X −q2)2 −4m2Xq2. The contribution off−to the rate is proportional to the lepton mass, so in most cases it can beneglected.

The exceptions are K →πµν and τ lepton final states. When X isa vector meson, the decay rate obeys a similar formula, with the contributionof one of the four form factors suppressed by one power of the lepton mass.Everything in Equation 23 is well-known or measurable except Vax and f+, soa measurement of dΓ/dq2 constitutes a measurement of |Vaxf+(q2)|.

Specificdecays and their CKM matrix elements are shown in Table 4.The form factor is calculable. The theoretical tools available are lattice QCDand symmetry arguments.

For example, chiral symmetry requires f K→π+(0) = 1,with second-order corrections estimated to be ∼< 1%. A combination of exper-imental measurements of the q2 dependence with this normalization conditiongives the best determination of Vus [87].

It is not likely that lattice QCD willcompete with this approach in the foreseeable future, especially since the smallquark masses in K →π pose additional technical difficulties for the latticecalculations. Nevertheless, a comparison of the q2 dependence of lattice andexperimental form factors, such as in Reference [88], could be used to get afeel for the reliability of the quenched approximation.

Similarly, heavy-quarksymmetry [66] requires f B→D+(q2max) = 1, again with second-order corrections[89]. Especially for the D-meson, the applicability of heavy-quark symmetry isnot guaranteed, but lattice QCD can be used to test it.29

Table 4: Semi-leptonic decays and the CKM matrix elements they determine.For brevity only pseudoscalar final states are listed; vector final states are ρ,K∗and D∗, as appropriate.A →XVaxcommentK →πVuscalibrate quenched approximationD →πVcduncertainty dominated first by BR(D →πeν), then by f+D →KVcsuncertainty dominated by f+B →DVcbtest corrections to heavy quark limitB →πVubvector final states useful; cf. textLattice calculations of semi-leptonic form factors, essentially using the strat-egy of Reference [90], have been carried out for D →π, K [88, 90, 91, 92, 93]and D →ρ, K∗[94, 95, 96].

Two groups are involved, which we shall abbreviateBKS [88, 90, 94, 96] and ELC [91, 92, 93, 95]. Both groups report statisticalerrors of roughly 15%.

BKS also estimate systematic errors, which introducean additional 30–40% uncertainty; presumably the systematic uncertainties ofthe ELC calculations are similarly large. Except for the quenched approxima-tion, however, the systematic uncertainties would be smaller if the statisticalerrors were smaller.

For example, the largest contributor to the systematic un-certainty is the lattice-spacing dependence of the form factors [88, 94]. Withbetter statistics over a wider range of lattice spacing, this component can bereduced by extrapolating.Table 5 summarizes lattice results for several form factors in semi-leptonicdecays of the D. Lattice results are most reliable at and near q2max, i.e.

whenthe spatial momentum of the hadrons is small. However, especially by givingthe initial-state meson non-zero momentum [91], it is possible to reach evenq2 < 0.

Experiments customarily quote results for the form factor at q2 = 0, soBKS and ELC do so too. The extrapolation to q2 = 0 is done by fitting to thepole-dominance formf+(q2) =f+(0)1 −q2/m2(24)where m is a suitable resonance mass.Both BKS and ELC find that theirnumerical calculations agree qualitatively with this form.

However, verificationof pole dominance is not essential to lattice QCD or to experiments. BKS stressthe utility of a direct comparison for vector-meson final states near the endpoint[96].Lattice calculations are most straightforward at q2max, but then vector30

Table 5: Some results for form factors f+(q2) in D →K semi-leptonic decaysand A1(q2) and A2(q2) in D →K∗semi-leptonic decays. Experimental resultsare from E691 and E653; their statistical and systematic errors have been addedin quadrature.

For BKS and ELC systematic errors are not listed. Based onthe estimates of BKS, it is reasonable to assign 30–40% systematic errors toform factors themselves and 20–30% to the ratio.f+(0)f+(q2max)A1(0)A2(0)A2/A1(0)A1(q2max)expt0.69(04)—0.46(07)—0.82(25)0.54(08)BKS0.90(08)1.64(36)0.83(14)0.59(14)0.70(16)1.23(16)ELC0.63(08)—0.53(03)0.19(21)——modes are preferable experimentally, because the phase-space suppression ofthe dominant form factor in the differential decay rate at the endpoint is onlyλ1/2 = 2mApX.Although the uncertainty estimates on the results presented in Table 5 arestill at a qualitative stage, it is important to realize that semi-leptonic decaysare not much more difficult to compute than the hadron masses and decayconstants.Since References [35, 58] have demonstrated the feasibility of asystematic, rather than incremental, approach, one can hope for a comparableanalysis of semi-leptonic decays in the near future.5.3Neutral Meson MixingSome of the operators on the right-hand side of Equation 15 induce neutralmeson mixing, e.g.

K0 ↔¯K0. For the kaon the four-quark operator isO∆S=2 = ¯saγµ(1 −γ5)da¯sbγµ(1 −γ5)db,(25)where a and b denote color indices.

The mixing amplitude is proportional tothe matrix element ⟨¯K0|O∆S=2|K0⟩. Similarly, the ∆B = 2 operator obtainedby the substitution ¯s 7→¯b induces B0- ¯B0 mixing.

On the other hand, D0- ¯D0mixing is expected to be too small to be interesting.Let us focus on the kaon. Phenomenologists use the so-called “vacuum sat-uration approximation” as a standard of comparison for ⟨¯K0|O∆S=2|K0⟩.

Thisapproximation treats the four-quark operator a product of two-quark operators,inserts a complete set of states, and then keeps only the vacuum contribution31

[97]. The result is⟨¯K0|O∆S=2|K0⟩VSA = 83m2Kf 2K.

(26)The factor 8/3 arises because there are two Fierz arrangements and becauseboth ¯s operators can act on the initial state.It is customary to define the“kaon B parameter”BK = ⟨¯K0|O∆S=2|K0⟩83m2Kf 2K. (27)In numerical lattice QCD the ratio BK is a convenient quantity, because thestatistical and systematic uncertainties of the ratio are under better controlthan those of numerator or denominator separately.A typical result using BK is the one for the parameter ǫ, which appears inthe analysis of CP violation in the K0- ¯K0 system.

Combining the measurementof |ǫ| with other experimentally known numbers, the standard model predicts(cf. Reference [49] and references therein)5.6 × 10−8 =−ˆBK |Vcb| Im Vtdh(η3f3(yt) −η1) yc|Vcd| + η2ytf2(yt)|Vcb| Re Vtdi,(28)where yq = m2q/m2W , V is the CKM matrix, the fi are kinematic functions,the ηi are perturbative QCD corrections, and ˆBK is a renormalization groupinvariant quantity related to BK.

Taking the one-loop anomalous dimension ofO∆S=2 into accountˆBK = (αS(µ))−2/9 BK(µ). (29)The combination of CKM matrix elements in Equation 28 depends on the CP-violating phase and (using unitarity constraints) on |Vub/Vcb|.As mentioned above, although there is no physical reason to prefer BK to thematrix element ⟨¯K0|O∆S=2|K0⟩, it makes better sense to quote BK from latticeQCD.

Because of correlations in the Monte Carlo, the statistical fluctuations ofthe numerator and denominator cancel to a large extent. Moreover, an analysisbased on chiral perturbation theory suggests that some effects of the quenchedapproximation also cancel in the ratio [98].

Finally, BK should be finite in thechiral limit (m2K →0), providing a consistency check on the numerical results.The most important reason why the lattice calculations of ⟨¯K0|O∆S=2|K0⟩are feasible is that there can be no mixing with lower dimension operators, be-cause O∆S=2 is the lowest dimension operator with ∆S = 2. Consequently, thenumerical calculations presented below are much more reliable than calcula-tions of analogous matrix elements of penguins and other denizens in the zoo of32

four-quark operators. Despite these advantages, there are still some difficulties.For Wilson fermions there are problems with chiral symmetry, making neces-sary a subtraction [99, 100] that ultimately decreases the signal-to-noise ratio.For staggered fermions chiral symmetry makes this subtraction unnecessary,but one must treat the extra flavors with care [101].The numerical results with the smallest uncertainties have been done withstaggered fermions [98].

At present the largest uncertainty comes from extrap-olating in a; it is uncertain whether the extrapolation should be taken in a ora2. The most recent quenched results [102] are ˆBK = 0.66 ± 0.06 after a linearextrapolation and ˆBK = 0.79 ± 0.03 after a quadratic extrapolation.

By com-parison, Wilson quarks yield 0.88 ± 0.13 [99, 103]. A calculation in full QCD iscompatible with the results from the quenched approximation, supporting thearguments that effects of the quenched approximation cancel in BK [104].These results for BK might foster the impression that the vacuum saturationapproximation gives a fair description, but that is misleading.

Separating thefour-quark operator into V V and AA terms, it turns out that the two have largecontributions that cancel in quenched lattice QCD. Conversely, the vacuumsaturation approximation would assert that the AA term contributes everythingand the V V term nothing.Mixing is also of great interest in the neutral B-meson system, because, likeǫ in the neutral kaon system, it gives insight into the third row of the CKMmatrix.

In the standard modelxd = (known factors) |V ∗tdVtb|2f 2BBB,(30)where xd = ∆MB0/ΓB0 = 0.66 ± 0.11 is a measure of the mixing. A similarformula applies to the Bs meson.

In addition to the decay constant, discussedabove, the B-meson B parameter is needed.Pilot lattice studies [84] yieldvalues of BB and BBs close to the vacuum saturation value of unity. The levelof technical detail in these calculations is not yet high enough to understand alluncertainties, but a better understanding will certainly emerge in the comingyears.5.4Non-Leptonic DecaysNon-leptonic decays, such as K →ππ processes, are also mediated by four-quark operators from Equation 15.

Many of the interesting operators sufferfrom the problem of mixing with lower dimension operators, which did notafflict the calculation of BK. A more serious obstacle to the treatment of non-leptonic decays is the presence of two (or more) hadrons in the final state.The technical aspect is the difficulty of separating the particles in the finite33

volume. The conceptual aspect is the determination of final-state phase shiftsfrom purely real quantities computed in Euclidean field theories [105, 106].

Itis rigorously known [107] how to determine the properties of the ρ resonance,which decays through an interaction in the QCD Hamiltonian. The stumblingblock is evidently the application of the ideas in Reference [107] when the par-ticle decays through an interaction being treated as a perturbation, as for weakdecays.

Note that these difficulties do not stem from the lattice cutoff, butfrom other features, finite volume and imaginary time, introduced to make thecomputational method tractable. Nevertheless, until these issues are resolved,lattice results for non-leptonic decays probably will not warrant attention fromnon-experts.6Summary and ProspectsThe coming generation of calculations will be done on computers with speedsof tens of gigaflops.

In a few years, computers with hundreds of gigaflops orperhaps a teraflop will probably be available [108]. These machines will makepossible crucial improvements in lattice calculations, but increases in comput-ing power alone with no methodological improvements will probably be in-sufficient to make possible first principles calculations with light sea quarks.Algorithms for the direct inclusion of sea quarks in QCD simulations madedramatic progress during the 1980’s.

The current best algorithms are orders ofmagnitude more efficient than those proposed for the first Monte Carlo spec-trum calculations around 1980. However, for large lattices and medium quarkmasses they still exhibit extremely long correlation times which are not under-stood theoretically, and whose scaling behavior in such quantities as the latticevolume and quark mass are not understood.

The current consensus is that oneorder of magnitude in computing power is likely to be too little to do definitivecalculations including light sea quarks, without further theoretical insight.What direction will lattice phenomenology take if there are no new algo-rithmic ideas? For heavy Q ¯Q mesons, nonrelativistic arguments should makepossible rock solid understanding of all errors aside from quenching errors, anddecent understanding of those.

For hadrons containing light quarks, it now ap-pears that good control of all errors aside from quenching errors is likely to beachievable in the coming generation of calculations. The uncertainties shownin Figure 4 [35], will be checked in the coming few years (and perhaps reducedto the point that the degree of disagreement of the quenched approximationwith the real world stands out more clearly).

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