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Fermi National Accelerator Laboratory

arXiv:hep-lat/9212018v1 14 Dec 19921Heavy Quark Physics ∗Paul B. MackenzieTheoretical Physics GroupFermi National Accelerator LaboratoryP. O.

Box 500Batavia, IL 60510 USA1. INTRODUCTIONPresent and future lattice calculations involv-ing b and c quarks include some of the most im-portant applications of lattice gauge theory tostandard model physics.

These include the heavymeson decay constants, the BB mixing ampli-tude, and various semileptonic decay amplitudes,which are all crucial in extracting CKM anglesfrom experimental data. They also include theextraction of αs from the charmonium and bot-tomonium spectra.Bound states of heavy quarks and antiquarks(quarkonia) have another crucial role to play inthe development of lattice gauge theory:theyprovide systems in which the estimation of theerrors inherent in current lattice calculations canbe done in a more reliable and robust way thanis possible for the light hadrons.

The reason isthat the quarks in these systems are relativelynonrelativistic.Coulomb gauge wave functionscalculated on the lattice may be used to aid inthe estimation of finite volume and finite latticespacing errors, and of the effects of quenching.We have a much better idea of what to expectin lattice calculations of these systems since po-tential models may be used to obtain the leadingbehavior in v2/c2. [1]Chris Sachrajda and I will split the subject ofheavy quark physics in these proceedings.

His re-view [3] will concentrate on the part of the subjectwhich involves the weak interactions. Mine willconcentrate on the part which does not.∗Review presented at Lattice 92, Amsterdam, Sept. 15–19, 1992.2.

LATTICEFORMULATIONSOFHEAVY QUARKSWhen the lattice spacing a is smaller than theCompton wave length of the quark 1/m, the stan-dard relativistic action of Wilson may be used.Cutoffeffects may be removed by taking the cut-off1/a to infinity. The bare lattice action mayalso be viewed as an effective field theory of QCDat the cutoffscale.

Cutoffeffects in an effectivefield theory are removed by adding higher dimen-sion interactions to the bare Lagrangian whilekeeping the cutofffixed.To remove the effectsof the cutoffto a finite order in a, a finite numberof interactions may be added to the bare latticeLagrangian.When aΛQCD ≪1, perturbationtheory may be used to calculate the required co-efficients of the new operators. [4] The ability toremove cutoffeffects perturbatively will probablybe spoiled eventually, perhaps at a small power ofa due to effects presently not understood, almostcertainly at a relatively large power of a due toinstantons.The dynamical scales in bound states are smallcompared to the fermion mass in QED and inQCD for the c and b quarks.

It is often advan-tageous in these systems to formulate the fieldtheory nonrelativistically as an expansion in 1/m[5–8], keeping the cutoffat or below m. The non-leading terms in nonrelativistic expansions havedimension higher than four. Loop corrections inthese effective field theories diverge if the cut-offis removed.

Cutoffeffects must be removedby adding higher dimension interactions to theLagrangian or by raising the cutoffto the newphysics scale (m), switching to the relativistic,

2renormalizable version of the theory, and thentaking the cutoffto infinity.When the kinetic energy of the heavy quark issmall compared to the typical interaction energies(as it is in bound states containing a single heavyquark), the kinetic energy may be treated as aperturbation. In this static approximation [7,6],the lowest order fermion action is justLstatic = φ∗iDtφ,(1)and the unperturbed quark propagator is just thetimelike Wilson line.In the general case, includ-ing quarkonia, the lowest order potential and ki-netic terms of the Lagrangian of NonrelativisticQCD (NRQCD) [8,5] (the terms on the first lineof Equation 2) must be included in the unper-turbed Lagrangian.LNRQCD = φ∗{iDt+D22m+g2mσ · B}φ+· · ·(2)Higher order terms in1m may be added as per-turbations.In processes such as the semileptonic decay ofheavy-light mesons, in which one heavy quark de-cays into another lighter (but still heavy) quarkwith a high velocity relative to the first, it is pos-sible, and useful, to formulate the static approxi-mation and nonrelativistic QCD as expansions inthe small internal quark momentum around somelarge, external meson momentum.

[9] Lattice im-plementations of this idea have been proposed [10]and are reviewed by Sachrajda.[3]2.1. The improvement program for Non-relativistic QCDIn a recent paper [11], Lepage et al.

have sys-tematically examined the improvement programfor NRQCD with the goal of reducing systematicsources of error from all sources to under 10%.This program involved the following elements:1) Since NRQCD has been formulated as a non-renormalizable effective field theory, cut-offef-fects are removed not by taking the cut-offto in-finity, but by adding additional operators to thebare Lagrangian (the dots in Eq. 2).

The infinitenumber of possible operators must be ordered ac-cording to expected size of their effects on thephysics.For heavy-light systems, the operatorordering is simply an expansion in 1/m, that is,in the dimensions of the operators. For heavy-heavy systems like quarkonia, the expansion iscomplicated by the presence of large quark ve-locities which do not fall to zero with the quarkmass.

Operators with the same dimension (suchas D22m andg2mσ · B ) are suppressed in their ef-fects on the physics by different powers of v (byv2 and v4, respectively, in this case).2) Once the operators required for a given accu-racy have been established, their coefficients mustbe determined by requiring that the NRQCD La-grangian reproduce the Green’s functions of ordi-nary QCD to this accuracy.3) Discrete forms of the required operatorsmust then be defined.As with light quark ac-tions, finite a errors must be estimated. If nec-essary, correction operators [4] must be added tothe action.4) The coefficients of the operators are modifiedby quantum effects.

Many corrections have beencalculated in mean field theory. [11] The correc-tions for the quark energy shift, mass renormal-ization, and wave function renormalization havebeen calculated in full one-loop perturbation the-ory.

[12] Deviations between the mean field andone loop results are rather small, from 0–10%.The result is a systematic correction programin v, a, and αs. The correction operators in vand a may be included directly in the simulationaction, or evaluated as perturbations using latticeor potential model wave functions.2.2.

A New Action for Four ComponentFermionsBecause coefficients of higher terms in theNRQCD Lagrangian such as D22m are explicit func-tions of 1/m, the quantum corrections describedin 4) above are also explicit functions of 1/m.These begin to diverge as ma is reduced belowa value of order one, making the nonrelativisticexpansion impractical. The Wilson action like-wise has been thought to have finite lattice spac-ing errors of order ma which blow up as ma israised above one.

Since the masses of the b and

3c quarks are such that ma is often O(1) at cur-rent lattice spacings, calculations of such cruciallyinteresting quantities as the heavy meson decayconstants fB and fD have often involved awk-ward interpolations between results in the staticapproximation and results using Wilson fermionsthrough a region where neither approximation iswell behaved. [3] While such an approach is proba-bly workable, it would clearly be desirable to havea method for lattice fermions which did not beginto break down right in the region of interest.To approach such a method, we consider a lat-tice version of LNRQCD with a few minor modifica-tions.

Like Wilson fermions (ψ), the fermions ofNRQCD contain four components per site: a two-component quark field (φ) and a two-componentantiquark field (χ).The bare mass is conven-tionally omitted in NRQCD calculations, but weare free to leave it in the theory. The usual Diraccoupling between quarks and antiquarks is absent(having been transformed into higher derivativeinteractions by the Foldy-Wouthhuysen transfor-mation), but we may add back a sufficiently sup-pressed amount of this interaction without spoil-ing the theory.

We thus consider the followingLagrangian:L=φ∗(c1 ∆−t + m0 −c22Xi∆+i ∆−i )φn+c3 φ∗Xiσi∆iχn+χ∗(−c1 ∆+t + m0 −c22Xi∆+i ∆−i )χn+c3 χ∗Xiσi∆iφn. (3)When c1 = 1 (times a correction factor whenma ≫1), c2 =1m, and c3 is negligible, it is agood Lagrangin for NRQCD.

The point of writ-ing the NRQCD Lagrangian in this particularform is that the action becomes precisely the stan-dard Wilson action with the choice of parametersc1 = c2 = c3 = 1. It is thus possible to adjust theparameters in such a way that as m0 is reduced,instead of blowing up, the theory turns smoothlyinto the Wilson theory.It is illuminating to expand the equation forWilson propagators nonrelativistically when themass is large.After normalizing the fields by1√1−6κ [13] (not1√2κ as is conventional) one mayobtainδn0 = [−E + M + (1 −U †n,0)(4)−12 1m0+1(1 + m0) (2 + m0) Xi(∆i)2]φn,where E is the energy eigenvalue obtained fromthe transfer matrix and M = Ep2=0 = ln(1+m0).This is a lattice Schr¨odinger equation not un-like the one obtained from NRQCD, but it hassome unusual features.

Most important, the two“masses” in the equation, M = ln(1 + m0) and1M =1m0 +1(1+m0) (2+m0), are completely differ-ent. If M is used to fix the fermion mass whenam ≫1, the dynamically more important masscondition ∂E/∂p2 =12m will be completely incor-rect.Kronfeld showed in his talk at this conference[14] that the two masses can be put back intoagreement with the use of the actionS=Xn[−¯ψnψn+κt ¯ψn(1 −γ0)Un,0ψn+ˆ0 + h.c.+κsXi¯ψn(1 −γi)Un,iψn+ˆi + h.c.](5)Thus, it seems that an action closely related tothe Wilson action is a member of the class ofactions suitable for NRQCD.

One can go evenfurther.In NRQCD and in the static approxi-mation, M plays no dynamical role. It can beignored, and is conventionally thrown away.

Thissuggests that the standard Wilson action itselfcan be used when am > 1 as long as M is ig-nored and ∂E/∂p2 =12m is used to fix the quarkmass, as is done in NRQCD.This proposal is obviously correct in free field,where we can calculate the behavior of quarkpropagators exactly to see that the proposed in-terpretation makes sense. It is certainly correctin mean field theory, too.Mean field improve-ment of these fermions, as of Wilson fermions, issimply the absorption of a “mean link” u0 (seeAppendix B) into an effective ˜κ ≡u0κ and thenproceeding as with free field theory.

(A plausi-ble estimate of the mean link in this context is

4probably u0 ∼1/8κc.) It remains to be shownwhether the theory is somehow spoiled by renor-malization.Perturbatively, Green functions must be ex-panded in p2 and αs.

Each term in the expansionis an explicit function of the quark mass, since thetheory must be solved exactly in ma. (The is alsothe case for the loop corrections of NRQCD.

[12]If these functions become singular or badly be-haved in some way, the theory could conceivablybreak down. The one loop perturbative correc-tions contain all of the ugliest features of Wilsonand NRQCD perturbation theory simultaneously,and have only been begun.

There is, however, onenumerical calculation by El-Khadra [15] indicat-ing that nothing too surprising occurs. The one-loop correction to the local current normalizationfor Wilson fermions with the naive normalizationis [16]⟨ψ|V loc4|ψ⟩=12κ(1 −0.17g2).

(6)The correct normalization with mean field im-provement is⟨ψ|V loc4|ψ⟩=1(1 −6κ8κc )(1 −0.06g2). (7)The remaining perturbative correction, 0.06g2,becomes an explicit (so far uncalculated) functionof m (or κ) in the new formalism which must notbecome singular if the theory is to make sense.Fig.

1 shows Eqs. 6 (upper curve) and 7 (lowercurve) along with a numerical calculation of thequantity at two values of κ (163x32 lattice, β =5.9).

It can be seen that for this quantity, notonly is the unknown function of m not singular,it is approximately equal to 1.Putting the new action on a secure footing willultimately require: 1) determination of the bareparameters of the action with mean field the-ory and full perturbation theory, 2) nonpertur-bative tests of the perturbative results, and 3)phenomenological tests of the resulting action incalculations of well understood physical quanti-ties. Not much of this program has yet been ac-complished.

However, as argued above, at largevalues of ma, the new action (and even the Wil-son action suitably reinterpreted) can be viewedFigure 1. Normalization of the local vector cur-rent as a function of κ.simply as unusual members of the general class oflattice actions proposed by Lepage and collabora-tors for NRQCD.

Quite a bit is now known aboutthe action for NRQCD. The discussion in points1) and 2) of Sec.

2.1 on operator classification isvalid for any method for treating heavy quarks,including this one. The fact that the mean fieldcorrections discussed in 4) reproduce the mass-dependent one loop corrections very well is en-couraging.Care will clearly be required in formulating nor-malization conditions which capture the most im-portant physics in both the relativistic and non-relativistic regions.

(Identifying ∂E/∂p2 ratherthan M as the fundamental mass condition is ex-ample number one of these.)3. PHENOMENOLOGYOFTHEJ/ψAND Υ SYSTEMSLike all phenomenological lattice calculations,calculations of the properties of heavy quark sys-tems serve a variety of purposes.Quantitieswhich are well understood experimentally, butwhich are very sensitive to lattice approxima-

5tions are good tests of lattice methods (Sec. 3.3).Quantities for which the lattice approximationsare well understood may be used to extract in-formation about the standard model (Sec.

3.1).A further purpose for lattice calculations is thedelineation of the limits and the reasons for thesuccesses of earlier models of hadrons (Sec. 4.3).I will discuss calculations in the ψ and Υ sys-tems by Davies, Lepage, and Thacker [17] usingNRQCD, and calculations in the ψ system bythe Fermilab group using Wilson fermions rein-terpreted as described in Sec.2.2 [18–22], andby UKQCD using Wilson fermions [23].

(See also[24].) Both groups studying the J/ψ system usedthe O(a) correction termδL = −ig c2ψΣµνFµνψ(8)of Sheikholeslami and Wohlert [25].UKQCDused the tree level coefficient c = 1.

The Fermi-lab group used a mean field improved coefficientc = 1.4 (see Appendix B).3.1. 1S–1P SplittingAn excellent determination of the lattice spac-ing in physical units is provided by the spin av-eraged splitting between the lowest angular mo-mentum (l = 0 and l = 1) levels of the ψ andΥ systems.

(In the charm system, for example,Mhc −(3Mψ + Mηc)/4 = 458.6 ± 0.4 MeV.) Thevalues of the lattice spacing obtained from thissplitting do not differ dramatically from those ob-tained from other quantities, such as the ρ mass[26] or the string tension [27].It is the possi-bility of making improved uncertainty estimatesthat makes this an important way of determin-ing the lattice spacing.

In quarkonia, error es-timates may often be made in several ways: bybrute force (e. g., by repeating the calculationseveral lattice spacings), by phenomenological ar-guments, and by direct calculation of correctionterms. Since determination of the lattice spacingis one of the key components of the determinationof the strong coupling constant from low energyphysics, it is important that these uncertainty es-timates be made rock solid.Preliminary results for this mass splitting werereported last year by the Fermilab group [18,19]and by Davies, Lepage, and Thacker [17].

Thisyear, El-Khadra [21] reported further work doneto check the corrections and error estimates givenin Ref. [18].

In [18], uncertainties due to an incor-rectly known quark mass and to O(a) errors aris-ing from an imperfectly determined coefficient cin the O(a) correction to the Lagrangian (Eq. 8)were taken to be less than 1% and omitted fromthe table of errors on the basis of the phenomeno-logical arguments.

(The splitting is expected tobe insensitive to small errors in the definition ofthe quark mass since it is almost identical in theψ and Υ systems. Likewise, in quark models thecontribution of the σ ·B interaction, which domi-nates δL nonrelativistically, to the spin averagedsplitting is zero.) These arguments were checkedthis year by repeat calculation at several valuesof the parameters and found to be correct withinstatistical errors.The O(a2) errors were argued to be small be-cause repeat calculations at β = 5.7, 5.7, and6.1 yielded almost the same result for α(5 GeV)(see next section).

An attempt was made to cor-rect for the small variation observed by extrap-olating to zero lattice spacing in a2.This ex-trapolation is not completely satisfactory, sincethe small a functional form is a messy combina-tion of O(a2) errors and perturbative logarithms.The way to improve this result, which has notyet be done, is to follow the example the NRQCDgroup [17] and evaluate directly the contributionsof the known correction operators to the splitting,thereby eventually obtaining zero measurable de-pendence on the lattice spacing. This group eval-uated the correction of the operators perturba-tively using the wave functions of the Richardson[28] potential model.

For the Υ at β = 6.0, for ex-ample, they obtained the rather small correctionsshown in Table 1. Wave functions directly calcu-lated by lattice gauge theory could also be used,eliminating the need for potential models.

Theyare easy to calculate to high statistical accuracy.Fig. 2 shows the Coulomb gauge wave functionof the J/ψ meson calculated on a 244 lattice atβ = 6.1.

Statisitical errors are negligible at smallseparations.Similarly, the estimate of the finite volume cor-rection needs to be bolstered by calculating themeson Coulomb gauge wave functions on the lat-

6Figure 2. The wave function of the J/ψ meson.Term∆M(1P −1S)%O(v2)-11 MeV-2%O(at)13 MeV3%O(a2x)-12 MeV-3%δSgluon-24 MeV-5%Total-33 MeV-7%Table 1NRQCD corrections to the 1P–1S splitting in theΥ system.

Finite lattice spacing corrections arefor β = 6.0.tice, and then calculating the overlap integral ofthe wave function with its periodically reflectedimage.3.2. Determination of αs from the 1S–1PSplittingThe most recent determinations of αs fromthe charmonium and bottomonium spectra us-ing NRQCD [17] and modified Wilson fermions[22] have error bars bracketing the region αs =0.103 −0.114 GeV.

They are somewhat below,but consistent with the world average given inthe review of QCD in the 1992 particle databook. They are inconsistent with the most recentLEP determinations, which are around 0.120 andabove.

[29]The determination of αs from the 1S–1P split-ting currently consists of three separate elements:the determination of the lattice spacing, the de-termination of a physical coupling constant at ascale measured in lattice spacing units, and, forthe time being, a correction for the absence oflight quarks. As discussed in the previous sec-tion, the uncertainties in the determination of αsarising from the determination of the lattice spac-ing seem to be in good shape right now, and thepath is clear to making them very solid.3.2.1.

Determination of the coupling con-stant.To determine the running coupling constant,one would like to combine the determination ofthe lattice spacing discussed above with a non-perturbative calculation of a physically definedcoupling constant, for example defined from thestatic quark potential at a given, fixed momen-tum transfer like 5 or 10 GeV.Since the largest cutoffmomenta for the exist-ing 1P–1S splitting calculations was π/a ≈7.5GeV, it was not possible in the existing calcula-tions to determine the continuum limit of a phys-ical coupling defined at short distances.In Ref. [18] a mean field improved perturbativerelation, (Eq.

27 in Appendix B) was used to ob-tain a renormalized coupling from the bare latticecoupling. This relation was tested over the pastyear on short distance Wilson loops.

[30] It didwell, but not perfectly: the loops calculated byMonte Carlo were systematically a few per centhigh. This suggests that Eq.

27 fixes most butnot all of the pathological relation between thebare lattice coupling constant and physical cou-plings, and that it is better to obtain physical cou-plings from short distance quantities calculatednonperturbatively. Using short distance Wilsonloops for this purpose (for example, via Eq.

28)raises the values of the renormalized couplings bya few per cent over those reported in Ref. [18].It remains to be determined how much cou-plings defined from continuum quantities differ

7from those defined from short distance quantitieslike the log of the plaquette. There is some rea-son to expect that this difference is small.

Thesecond order corrections to the short distance lat-tice static potential are within a few per cent ofthe continuum corrections. [31] Likewise, Creutzratios of Wilson loops up to six by six are quitewell behaved when expanded with a coupling de-fined from Eq.

28. [30]The Monte Carlo calculation of the static quarkpotential at a separation of one lattice spacingagrees to very high accuracy with perturbationusing the coupling of Eq.28.

(See Sec.4.1. )Therefore, a coupling constant obtained from thevery short distance static potential will give re-sults almost identical to those using Eq.

28. Aphenomenological method for estimating the con-tinuum coupling constant defined by the potentialusing short distance data has been proposed byMichael.

[32] It has been used by UKQCD [33] andby Bali and Schilling. [27] It yields results whichare quite close to those obtained with Eqs.

27and 28, and therefore to those which would beobtained directly from short distance lattice po-tential itself.A program to calculate explicitly a continuumcoupling constant using finite size scaling hasbeen proposed by L¨uscher et al. [34] To select theparticular coupling to focus on, they propose thecriteria that it: 1) be defined nonperturbatively,2) be calculable in perturbation theory, 3) be cal-culable in Monte Carlo simulations, and 4) havesmall, controllable lattice artifacts.These leadthem to propose the response of the QCD vacuumto a constant background color-electric field to de-fine the coupling constant.

The more phenomeno-logical choice of the static potential has poorersignal-to-noise properties in Monte Carlo calcula-tions approaching the continuum limit, and (theytell us) more difficult higher order perturbationtheory. An SU(2) calculation has been completed,which yields results similar to those obtained withEq.

27. An SU(3) calculation is in progress.3.2.2.

Correctionfortheeffectsofseaquarks.This is the greatest source of uncertainty in theresults quoted above. This correction is the mostphenomenological, and has the greatest likelihoodof having a problem.

Over the next few years, itwill be removed by direct inclusion of the effectsof sea quarks.The attempt to estimate the effect that the ab-sence of sea quarks has on this result is based onthree assumptions. They are, in order of decreas-ing rigor:• When certain physics quantities are used totune bare parameters in the quenched ap-proximation, the most important terms inthe effective Lagrangian at the dominantenergy scale for those quantities are givencorrectly.

The effective action at other en-ergy scales including the scale of the lat-tice cutoffwill be somewhat incorrect. Inparticular, if the effective coupling constantat the physics scale approximates that ofthe real world, the effective coupling at theshort distance cutoffwill be a bit small.• The most important term in the effectiveaction for quarkonia is the static potential.The phenomenological success of potentialmodels indicates that this assumption maybe valid to around 25%.

It is this assump-tion, which is certainly not valid for thelight hadrons, that leads us to dare to tryto make this correction for the charmoniumsystem when we would not try it for thelight hadrons.• The effects of light quarks on the static po-tential may be estimated by fitting char-monium data with a QCD based poten-tial model such as the Richardson potentialonce with the correct, nf = 3, β function inthe potential and again with the quenched,nf = 0, β function.The final assumption is certainly a good oneat short distances, which are responsible for mostof the difference in the evolution of the couplingfrom the middle distance charmonium physicsscale down to the lattice cutoffscale.It isalso reasonable at the less relevant large distancescale, since the lattice quenched string tensionand the string tension of Regge phenomenology

8are comparable. If, however, light quarks havea much greater effect on the potential in middledistances than they seem to at large and at smalldistpp ances, the assumption would fail.The naive expectation for the size of the cor-rection isβnf=00−βnf =30βnf =30∼20%.In Ref.

[18], a perturbative calculation was usedto bound the plausible size of the correction. Thisyear the estimated correction was checked [22] byfitting the charmonium spectrum with a poten-tial twice: once using a potential with the correctβ function and once using a potential with thequenched β function.

(See Sec. 3.3) The resultwas compatible with the one in Ref.

[18]. This,however, is not so much an independent check ofthe previous estimate as another quantification ofthe assumption that sea quarks have no more dra-matic effects on the potential at middle distancesthan they seem to at large and small distances.3.2.3.

Future prospects for determining αsErrors in the determination of the lattice spac-ing are already in good shape.The accuracy in the determination of the cou-pling constant needs further examination, but thecalculations of Ref. [30] suggests that the accu-racy to be expected of lattice perturbation theoryis greater than that expected of QCD perturba-tion theory in hadronic phenomenology.

In Ref. [30], discrepancies of about α2 were typically ob-served in comparisons of first order perturbationtheory with Monte Carlo calculations, and dis-crepancies of about α3 in comparisons of secondorder perturbation theory.

This amounts to only3-4% for calculations at the lattice cutoffat mod-erate β’s. In contrast, in QCD phenomenology,an accuracy of 10% is often taken to be opti-mistic.One difference may be that the latticecalculations of most interest are often quadrati-cally divergent integrals dominated by momentaof the order of the relatively well-defined latticecutoff.

They thus differ from calculations of un-ruly hadrons in collision, which insist on interact-ing on a wide range of momentum scales, piling uplarge logarithms from a nasty variety of sources.The aspect of the current determination of αswhich makes it no better than any other existingdetermination is the use of potential model argu-ments to estimate the effects of the absence of seaquarks. This is quite analogous to, for example,the phenomenological treatment of higher twistand fragmentation effects in determinations of αsin deep inelastic scattering.

All of the existingdeterminations have some phenomenological as-sumptions built into them. The difference is thatthe potential model estimate of quenched correc-tions will certainly be eliminated by brute com-puter force (if not by the use of more intelligentmethods) over the next few years, resulting in adetermination far more accurate than any of theexisting ones.3.3.

HyperfineSplittingandLeptonicWidthThese two quantities are very straightforwardto calculate on the lattice, and are good phe-nomenological tests of how well we understandthe parameters of the quark action. The hyper-fine splitting ∆m(J/ψ −ηc) and leptonic decayamplitude Vψ ≡m2ψ/fψ have been calculated inthe J/ψ system by the Fermilab group and byUKQCD.

They have been calculated in the Υ sys-tem by Davies, Lepage, and Thacker. [17]The potential model formula for the hyperfinesplitting is [1]∆m(J/ψ −ηc) = 32παs(mc)9m2c|Ψ(0)|2 .

(9)It arises from a coupling of the spins of the quarksto transverse gluons. It therefore should be ex-tremely sensitive to the value of the correctioncoefficient c. (It is not clear a priori whether toexpect strong sensitivity to the quark mass, since|Ψ(0)|2 should rise with the quark mass.

)The leptonic width to leading order isΓee = 16πα2e2cm2c|Ψ(0)|2 . (10)Nonrelativistically, the leptonic decay amplitudeis therefore simply the wave function at the origin,Ψ(0), properly normalized.

This quantity shouldbe quite sensitive to the mass of the quark.Before comparing existing lattice results withexperiment, we need to estimate the accuracy

9to expect in the quenched approximation. Bothquantities are proportional to |Ψ(0)|2, the prob-ability for the quarks to be at the same point(within one Compton wave length, say) and soare obviously short distance quantities.

With lat-tice parameters tuned to obtain the correct 1P–1Ssplitting, the coupling constant and potential atshort distances will be too weak. An analysis likethe one referred to in Sec.

3.2.2 yieldsα(0)s (mc)α(3)s (mc)= 0.81 ± 0.06 . (11)The incorrect weakness of the quenched po-tential at short distance may also yield a weak-ened wave function at the origin.

El-Khadra haschecked for this effect [21,22] using the Richard-son potentialV (q2) = CF4πβ(nf )01q2 ln (1 + q2/Λ2) ,(12)where β(nf )0= 11 −2nf/3 . Fitting the charmo-nium spectrum with once with nf = 3 in the βfunction parameter, and again with nf = 0, shefound that, indeed, The ratio of the wave func-tions at the origin wasΨ(0)(0)Ψ(3)(0) = 0.86 .

(13)This reduces our expectation for the hyperfinesplitting from the experimental result ∆m(J/ψ −ηc)exp = 117.3 MeV to around∆m(J/ψ −ηc)quenched ≃70 MeV . (14)These considerations reduce our expectations forleptonic matrix element by a smaller amount,from the experimental result V expψ=0.509GeV3/2 toV quenchedψ= 0.438 GeV3/2 .

(15)The results of UKQCD and Fermilab for thehyperfine splitting are shown in Fig. 3.

UKQCDset the value of the quark mass to obtain the ex-pected energy eigenvalue in the transfer matrix.In light of the arguments on the interpretation ofWilson fermions at large quark masses in Sec. 2.2,Figure 3.

∆m(J/ψ −ηc) calculated by UKQCD(diamond) and Fermilab (squares) compared withthe physical result (upper star) and an estimateof the quenched corrected result (lower star).the Fermilab group took this as unreliable and at-tempted to fix the quark mass by demanding thatthe leptonic width be correct. The UKQCD resultis slightly below the result expected on the ba-sis of the quenched correction.

This is consistentwith the mean field expectation that quantumcorrections boost the required value for the co-efficient of the correction term. Their results are,however, much closer to the physical answer thanearlier Wilson fermion calculations with no O(a)correction.

[35] The Fermilab results are slightlyabove the quenched expectation, but perhaps notvery significantly in light of the uncertainties inthe quenched correction and the statistical errors.4. THE STATIC QUARK POTENTIALHigh accuracy results for the static quarkpotential were reported this year by Bali andSchilling [36,27] and by UKQCD [33].Fig.

10Figure 4. The heavy quark potential, calculatedon the lattice in the quenched approximation.4 shows the potential calculated by Bali andSchilling in the quenched approximation on a 324lattice at β = 6.4.

The solid line is the fit to aCoulomb plus linear potentialV (R) = V0 −0.277(28)/R + 0.0151(5)R,(16)which fits quite well for R > 2√2. Comparisonof results on 164 with results on 324 lattice indi-cated that finite volume results are small.

A plotof results from β = 6.0, 6.2, and 6.4 with physi-cal units set by the string tension indicates goodscaling behavior.4.1. Short distance behavior.At such a large β we should expect the shortdistance part of the potential to agree very wellwith perturbation theory, and this is the case.I checked the value of V (1) given in Ref.

[27]against perturbative results for 324 lattices sup-plied by Urs Heller [37]. Using the “measured”coupling constant defined by Eq.

28, perturba-tion theory agreed with the Monte Carlo data towithin about 1%, perhaps fortuitously accurate,but still impressive. (This incidentally illustratesthat the potential is a natural candidate on thelattice as well as in the continuum to define im-proved coupling constants.

The coupling of Eq.28 was suggested mostly because the plaquette iseasy to measure and universally available. )Since perturbation theory agrees so well withthe Monte Carlo calculation of the potential, andsince perturbation theory implies a coupling con-stant rising with increasing R, it would be inter-esting to attempt to fit the data with an asymp-totically free Coulomb plus linear potential.

Thesize of the fit Coulomb term (0.277(28)/R) isquite close to the subleading long distance behav-ior of the potential (π/(12R) = 0.262/R). How-ever these two similarly-sized effects have nothingto do with each other, and we are not guaran-teed that, for example, the perturbative Coulombterm does not rise above 0.28 before the poten-tial settles back down to its asymptotic form.

Afit with an asymptotically free Coulomb plus lin-ear potential, for example a modified Richardsonpotential [38], might help to start exploring theextent to which the data support or rule out suchspeculation.It is easy to convince yourself with a ruler thatvalues of the string tension obtained by the fit arecompletely plausible. However, if the changeoverfrom the perturbative Coulomb potential to thenonperturbative long distance 1/R term is morecomplicated than we hope, a larger than expectedmiddle distance Coulomb term could be contam-inating the obtained string tensions more than isobvious from the current analysis.4.2.

Long distance behavior.The good scaling of the potential when thephysical scale is set by the string tension has al-ready been mentioned. Good asymptotic scalingof the string tension in terms of a physical cou-pling such as ΛMS is also observed in the newdata.

(Good means to perhaps 20%.) Folkloreto the contrary was based on the the search forscaling in terms of the bare lattice coupling con-stant.

The bare coupling has a highly patholog-ical, but reasonably well-understood relationshipto well-behaved physical coupling constants. Itwas pointed out long ago [39,40] that decent scal-ing is observed in terms of an effective couplingconstant defined from the plaquette.

It was em-phasized in Ref. [30] that such coupling con-stants are simply very close relations of the fa-miliar physical coupling constants such as αV andαMS of perturbative QCD.

11Figure 5. √σ/ΛMS as a function of the latticespacing.

The upper curve was obtained from thebare coupling constant. The lower curve was ob-tained from an effective coupling constant.Fig.

5 (from Ref. [40]) shows new and old datafor √σ/ΛMS plotted as a function of aΛMS.

Theupper curve was obtained via the bare couplingconstant. The much better behaved lower curvewas obtained via the effective couplingαeff = 3(1 −13T r(U))4π.(17)(Ref.

[30] advocates Eq.28, the logarithm ofT r(U), for this purpose on the grounds that log-arithms of Wilson loops have better perturba-tive behavior than the loops themselves.) Onlyabout 20% deviation from asymptotic scaling isobserved over the range of the data.

Part of thatdeviation is certainly perturbative, since the useof another reasonable perturbative scheme, Eq.28, changes the amount of deviation to 10%. [30]The fact that the ratio of the deconfinement tem-perature to the square root of the string ten-sion scales better than √σ/ΛMS is another in-dication that the deviation is more likely to beconnected to the determination of the couplingFigure 6.

The lattice quenched heavy quark po-tential (top curve) and the potentials of the Cor-nell model and the Richardson model.constant than to the string tension. Because theshort distance behavior is a mixture of pertur-bative logarithms and O(a2) errors, the extrapo-lation in a is not completely satisfactory and itis important to sort the origin(s) of the discrep-ancy: perturbation theory, O(a2) errors, or mea-surement errors.

However, the downward trendseems clear and the estimate√σ/ΛMS = 1.75 ± 15%(18)seems reasonable.4.3. Comparison with potential models.One useful task of first-principles calculationsis to support or destroy earlier phenomenologies.For example, it would be nice to be able to un-derstand if there is a reason that nonrelativisticquark models for the light hadrons work unrea-sonably well.

It is more straightforward to put thesuccess of potential models of heavy quark sys-tems on a rigorous footing using lattice methods.These systems are nonrelativistic and it is notsurprising that a nonrelativistic treatment yieldsrather accurate results.In Fig.6 the potential obtained by Baliand Schilling is compared with the potentials ofEichten et al. [41] and Richarson in the region

120.1 fm < R < 1.0 fm. The string tensions of thelattice and the phenomenological potentials aresimilar, but the Coulomb term required by phe-nomenology is about 1.8 times as large as thatyielded by the quenched lattice, seemingly a largediscrepancy.

The phenomenological potential isvery well known in this region between 0.1 and 1.0fm. Fits to the spectra of the charmonium andbottomonium systems with a wide variety of plau-sible and implausible functional forms yield po-tentials which differ by only a few per cent in thisregion.

On the other hand, the quenched latticepotentials are also rather convincing, especiallyat short distance, so what accounts for the dif-ference? First, we expect the quenched Coulombcoupling to be a bit smaller than the true QCDcoupling constant at short distances because ofthe incorrect β function of the quenched approx-imation.

(See Sec. 3.2.2.) This effect is in theright direction and is expected to be of orderβnf=00−βnf =30βnf =30∼20%.Second, the phenomenological potentials clearlyparameterize some of the effects of higher orderrelativistic corrections.These are roughly ex-pected to be of order v2/c2 ∼25% for charmo-nium.Some of these clearly have the effect ofstrengthening the attraction of the quarks, buta complete analysis of the spin-independent rel-ativistic corrections in potential models does notexist.

[42] A combination of these two effects couldthus easily explain as much as 1.5 out of the dis-crepancy of 1.8. A preliminary conclusion: thereis an interesting puzzle in this discrepancy, butno cause for alarm.5.

SUMMARYStatic Potential.• At short distances, the potential agrees withperturbation theory to a few per cent.• The string tension exhibits two loop asymp-toticscalingtoanaccuracyof20%.√σ/ΛMS is in the range 1.55–1.95.ψ and Υ systems.• The hyperfine splitting and leptonic widthsprovide good phenomenological tests of lat-tice methods.• The spin averaged 1P–1S splitting providesa very good determination of the latticespacing in physical units. Combined witha lattice determination of the renormalizedcoupling, it gives a determination of thestrong coupling constant which at presentis of comparable accuracy to that of con-ventional determinations.

When the effectsof sea quarks are properly included, its ac-curacy will be much better than any currentdetermination.Technical developments.• Lattice perturbation theory works very wellwhen renormalized coupling constants areused.• Minor changes to the actions of Wilson andof NRQCD may make it possible do cal-culations with a unified formalism at anyvalue of the quark mass, as long as thethree momentum is small. This will imply areinterpretation of calculations with Wilsonfermions at large quark mass.ACKNOWLEDGEMENTSI would like to thank Peter Lepage, EstiaEichten, Aida El-Khadra, Andreas Kronfeld, andChris Sachrajda for helpful discussions.Fermilab is operated by Universities ResearchAssociation, Inc.under contract with the U.S.Department of Energy.A.

NOTATIONWe use the forward, backward, and symmetricfinite difference operators∆+µ ψn≡Un,µψn+ˆµ −ψn,(19)∆−µ ψn≡ψn −U †n−ˆµ,µψn−ˆµ,(20)∆µψn≡∆+µ + ∆−µ2ψn. (21)

13The analogous continuum covariant derivative isdenotedDµ(22)The standard relation between the bare massm0 and the hopping parameter κ ism0 ≡12κ −4. (23)When considering nonrelativistic fermions, wedecompose the four-component Dirac field as twotwo-component fieldsψ = φχ,(24)and take as our representation of the Euclideangamma matricesγ0 = 100−1,γi = 0σiσi0.(25)B.

RESULTSFROMLATTICEPER-TURBATION THEORYThis section summarizes results from Ref. [30]which have been used in the text.B.1.

A sequence of improved coupling con-stantsIn Ref. [43] (1990) it was argued that latticeperturbation series are much more convergent andagree better with Monte Carlo data if they areexpressed in terms of a physical running couplingevaluated at a carefully chosen scale.

A good oneis αV , the one defined by the static quark poten-tial:1αV (q) =1αlat+ β0 ln( πaq ) −4.702. (26)The arguments were analogous to those leadingfrom the MS to the MS scheme in dimensionallyregularized QCD.In Ref.

[18] (1991) it was noted that the bulk ofthe correction coefficient in the previous equationis accounted for by a simple mean field argument.The coupling constant is enhanced at one loopby a coupling to the expectation value of the pla-quette induced by the higher order terms in theWilson action. Higher order analogues of this oneloop effect certainly exist.

This suggests that aneffective coupling constant which incorporates aMonte Carlo calculation of the plaquette expec-tation value, such as1αV (q) = ⟨T rUp⟩αlat+ β0 ln( πaq ) −0.513,(27)may yield improved accuracy.Over the past year (1992) we have tested thisassumption by calculating a variety of short dis-tance quantities using the mean field improvedcoupling constant and by Monte Carlo. [30] Wefound that, while using Eq.

27 significantly im-proved agreement between perturbation theoryand Monte Carlo, the Monte Carlo results tendedto be systematically slightly higher then the per-turbative results. This suggests that a couplingdefined directly from any of the Monte Carlo cal-culated quantities would yield improved predic-tions for the others.

A particularly simple one isthe coupling defined from the log of the plaquette:1αV (3.41/a)=1αeff−1.19≡−4π3 ln( 13T r(U)) −1.19(28)(The scale of the running coupling arises from anestimate of the typical momenta of gluons in thecalculation of the logarithm of the plaquette.[30])B.2. Mean field improvement of operatorsThe mean field argument leading to Eq.

27 maybe summarized as follows.The naive classicalrelation between the lattice and continuum gaugefieldsUµ(x) ≡eiagAµ(x) →1 + iagAµ(x). (29)is spoiled by tadpoles arising from the exponentialform of the lattice representation of the gaugefields.

Quantum fluctuations do not lead to anaverage link field close to 1.00000 as implied byEq. 29, but to something more likeUµ(x) →u0 (1 + iagAµ(x)),(30)where u0, a number less than one, represents themean value of the link.

In a smooth gauge, theMonte Carlo link expectation value can be used

14as an estimate of u0. A simple, gauge-invariantdefinition isu0 ≡⟨13TrUplaq⟩1/4.

(31)Other definitions based on κc or the static quarkself-energy may be used to fine tune mean fieldpredictions in particular situations.If naive definitions such as Eq.29 are usedto relate lattice and continuum operators, largecorrections will appear in quantum corrections.Much better behavior of loop corrections is ob-tained by taking Uµ(x)/u0 as the lattice approxi-mation to the continuum field. This implies thatthe lattice action˜Sgluon =X1˜g2u40Tr(Uplaq + h.c.).

(32)will approximate closely the desired continuumbehavior. This is the usual lattice action if weidentify˜g2=g2lat/u40 = g2lat/⟨13Tr(Uplaq)⟩.

(33)The perturbative result, Eq. 27, explicitly verifiesthat ˜g2 is a closer approximation to a standardcontinuum expansion parameter than g2lat is.

[44]The same considerations lead to the result that˜κ ≡κu0(34)produces a more continuum-like bare mass ( ˜m =12˜κ −4) and smaller quantum corrections in oper-ator renormalizations than does κ.Mean field arguments may be used both to es-timate perturbative predictions when the pertur-bative predictions are unknown, and also to im-prove known predictions.Just as we did withthe coupling constant in Eq. 27, we can improveperturbative predictions for operators involvingquark fields by substituting the Monte Carlo cal-culation of u0 in Eq.34 and including in theperturbative prediction only that part remainingafter the absorption of u0 into κ.

A possible finetuning in this case is to obtain u0 from κc ratherthan from the plaquette. This was the procedureused in obtaining Eq.

7.The cloverleaf approximations to Fµν used inthe Wohlert-Sheikholeslami O(a) correction tothe Wilson action [25] and in the magnetic spincoupling of NRQCD [11] contain four links each.The quark wave function normalization containsone link.We therefore expect the naive coeffi-cient of this operator to undergo quantum cor-rections of roughly a factor of u−30 , or about1 + 0.25g2 if we use the plaquette to estimateu0.This is in agreement with an unpublishedthesis calculation of Wohlert, 1 + 0.27g2. [45] Us-ing the plaquette calculated by Monte Carlo toestimate the correction term yields a factor of⟨13Tr(Uplaq)⟩−3/4 ∼1.4 −1.5 for β around 6.0.REFERENCES1.For a review on heavy quarkonia see for exam-ple: W. Kwong, J. L. Rosner, C. Quigg, Ann.Rev.

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