QCD for Coarse Lattices

TL;DR

This paper reviews recent advances in simulating QCD on very coarse lattices, which could significantly impact nonperturbative low-energy QCD studies.

Contribution

It summarizes progress in coarse lattice QCD simulations, highlighting potential for more efficient nonperturbative calculations.

Findings

Enhanced accuracy in coarse lattice QCD simulations

Potential for revolutionary implications in low-energy QCD

Progress towards practical nonperturbative QCD computations

Abstract

This is the Lattice '95 review of recent progress towards accurate QCD simulations using very coarse lattices. This development could have revolutionary implications for low energy (nonperturbative) QCD.

Tables

Table 1. Table 1: The critical temperature for the QCD deconfining phase transition computed on lattices with a ​ T c = 1 / N t 𝑎 subscript 𝑇 𝑐 1 subscript 𝑁 𝑡 a\,T_{c}=1/N_{t} using the improved gluon action. The scale is set using the 1 ​ P − 1 ​ S 1 𝑃 1 𝑆 1P\!-\!1S  splitting in charmonium.

$N_t$	${\beta }_{\mathrm{pl}}^c$	$T_c$ (MeV)
2	6.96	273 (10)
3	7.45	285 (15)

Table 2. Table 2: Twice the ratio of the square root of the torelon mass to the critical temperature for the Wilson action and for the approximately perfect QCD action. The lattice spacing is 1 / N t ​ T c 1 subscript 𝑁 𝑡 subscript 𝑇 𝑐 1/N_{t}T_{c} and the torelons are for lattices with 2 ​ N t 2 subscript 𝑁 𝑡 2N_{t}  sites in the spatial direction.

$N_t$	Wilson	Perfect Action
2	3.38 (4)	3.09 (3)
3	3.15 (5)	3.05 (5)
4	2.98 (3)	3.08 (4)

Equations

$$\text{mbox}cost\propto {\left(\frac{L}{a}\right)}^4{\left(\frac{1}{a^2}\right)}^{\delta }{\left(\frac{1}{m_{\pi }^{2}a}\right)}^{ϵ},$$

$$U_{\mu }(x)=\mathrm{P}\exp i{\int }_x^{x+a\hat{\mu }}gA\cdot dx$$

$${\varphi }_i=\varphi (x=x_i).$$

$$\lambda \varphi (x{)}^3⟶\lambda {\varphi }_i^3.$$

$${\partial }^2\varphi (x_i)⟶\left({\Delta }^{(2)}-\frac{a^2}{12}{\left({\Delta }^{(2)}\right)}^2+\cdots \right){\varphi }_i$$

$${\Delta }^{(2)}\varphi \equiv \frac{{\varphi }_{i+1}-2{\varphi }_i+{\varphi }_{i-1}}{a^2}.$$

$${\partial }^2\varphi (x_i)\to -\sum _j{d}_{\mathrm{pf}}^{(2)}(x_i-x_j){\varphi }_j$$

$$d_{\mathrm{pf}}^{(2)}(x_i-x_j)\equiv \int \frac{dp}{2\pi }{p}^2{\mathrm{e}}^{ip(x_i-x_j)}\theta (|p|<\pi /a).$$

$${\varphi }_i$$

$$=$$

$$\Pi (p)\equiv \frac{2\sin (pa/2)}{pa}$$

$$⟨{\varphi }_i{\varphi }_j⟩$$

$$=$$

$${\int }_{-\pi /a}^{\pi /a}\frac{dp}{2\pi }{\mathrm{e}}^{ip(x_i-x_j)}\sum _n\frac{{\Pi }^2(p+2\pi n/a)}{(p+2\pi n/a{)}^2+m^2},$$

$$S^{(2)}={\mbox{$\frac{1}{2}$}}\sum_{i,j}\phi_{i}\,d^{(2)}_{\rm pa}(x_{i}\!-\!x_{j},m)\,\phi_{j},$$

$${\int }_{-\pi /a}^{\pi /a}\frac{dp}{2\pi }{\mathrm{e}}^{ip(x_i-x_j)}{\left(\sum _n\frac{{\Pi }^2(p+2\pi n/a)}{(p+2\pi n/a{)}^2+m^2}\right)}^{-1}.$$

$$\varphi {\partial }^2\varphi \to c_0\varphi {\Delta }^{(2)}\varphi -c_1\frac{a^2}{12}\varphi {\left({\Delta }^{(2)}\right)}^2\varphi +\cdots$$

$${\tilde{U}}_{\mu }(x)\equiv \frac{U_{\mu }(x)}{u_0},$$

$$H_0=-\frac{{\mathbf{D}}^2}{2M}+ig{A}_0,$$

$$\delta H=-c_1\frac{{\left({\mathbf{D}}^2\right)}^2}{8M^3}-c_2\frac{g\sigma \cdot \mathbf{B}}{2M}-\cdots .$$

$$E_{{\rm Y}}(\mathbf{p})=M_1+\frac{{\mathbf{p}}^2}{2M_2}+\cdots .$$

$$\begin{array}[]{r@{~}c@{~}l}1-{\textstyle{1\over 3}}{\rm Re}\,\hbox{Tr}U_{\rm pl}~{}&=\hfil~{}&\displaystyle r_{0}\sum_{\mu,\nu}\hbox{Tr}(F_{\mu\nu}F_{\mu\nu})\\[10.76385pt] ~{}&+\hfil~{}&a^{2}\Bigl{[}r_{1}R_{1}+r_{2}R_{2}+r_{3}R_{3}\Bigr{]}\\[10.76385pt] ~{}&+\hfil~{}&{\cal O}(a^{4})+\hbox{total derivatives},\end{array}$$

$$\begin{array}[]{r@{~}c@{~}l}R_{1}~{}&=\hfil~{}&\sum_{\mu,\nu}\hbox{Tr}(D_{\mu}F_{\mu\nu}D_{\mu}F_{\mu\nu}),\\[10.76385pt] R_{2}~{}&=\hfil~{}&\sum_{\mu,\nu,\sigma}\hbox{Tr}(D_{\mu}F_{\nu\sigma}D_{\mu}F_{\nu\sigma}),\\[10.76385pt] R_{3}~{}&=\hfil~{}&\sum_{\mu,\nu,\sigma}\hbox{Tr}(D_{\mu}F_{\mu\sigma}D_{\nu}F_{\nu\sigma}).\end{array}$$

$$A_{\mu }\to A_{\mu }+a^2{\alpha }_{s}f({\alpha }_s)\sum _{\nu }{D}_{\nu }{F}_{\nu \mu },$$

$$U_{\rm rt}=\begin{picture}(60.0,30.0)(0.0,15.0)\put(10.0,10.0){\vector(0,1){12.5}} \put(10.0,10.0){\line(0,1){20.0}} \put(10.0,30.0){\vector(1,0){12.5}} \put(10.0,30.0){\vector(1,0){32.5}} \put(10.0,30.0){\line(1,0){40.0}} \put(50.0,30.0){\vector(0,-1){12.5}} \put(50.0,30.0){\line(0,-1){20.0}} \put(50.0,10.0){\vector(-1,0){12.5}} \put(50.0,10.0){\vector(-1,0){32.5}} \put(50.0,10.0){\line(-1,0){40.0}} \end{picture},\quad U_{\rm pg}=\begin{picture}(60.0,30.0)(0.0,15.0)\put(10.0,10.0){\vector(0,1){12.5}} \put(10.0,10.0){\line(0,1){20.0}} \put(10.0,30.0){\vector(2,1){10.0}} \put(10.0,30.0){\line(2,1){15.0}} \put(25.2,37.6){\vector(1,0){12.5}} \put(25.2,37.6){\line(1,0){20.0}} \put(45.2,37.6){\vector(0,-1){12.5}} \put(45.2,37.6){\line(0,-1){20.0}} \put(45.2,17.6){\vector(-2,-1){10.0}} \put(45.2,17.6){\line(-2,-1){15.0}} \put(30.0,10.0){\vector(-1,0){12.5}} \put(30.0,10.0){\line(-1,0){20.0}} \end{picture}.$$

$$S[U]$$

$${\beta }_{\mathrm{rt}}$$

$${\beta }_{\mathrm{pg}}$$

$$u_0$$

$${\alpha }_s$$

$$r_{\mathrm{hadron}}\approx a\Rightarrow \text{cases}\mathcal{O}(a^2)\text{mbox}corrections\approx \text{mbox}15--20\%\text{cr}\text{mbox}\text{cr}\mathcal{O}({\alpha }_s{a}^2,a^4)\approx \text{mbox}2--4\%$$

$${\Delta }^{(1)}\psi \equiv \frac{\psi (x+a)-\psi (x-a)}{2a}$$

Full text

QCD for Coarse Lattices††thanks:

Invited talk at Lattice ’95, Melbourne, Australia (1995).

G. Peter Lepage

Floyd R. Newman Laboratory of Nuclear Studies

Cornell University, Ithaca, NY 14853

gpl@hepth.cornell.edu+

(September 10, 1995)

1 A Revolution for Lattice QCD?

In this lecture I discuss the prospects for simulating QCD accurately on very coarse lattices. This subject is very important because the cost of a Monte Carlo simulation of QCD is very sensitive to the lattice spacing. Simulation cost varies as

[TABLE]

where $a$ is the lattice spacing, $L$ is the length of one side of the lattice, $m_{\pi}$ is the pion’s mass in the simulation, and $\delta$ and $\epsilon$ are exponents between 0 and 1 that are related to algorithm performance. With the best algorithms the cost of a full QCD simulation QCD increases like $1/a^{6}$ as $a$ is decreased, making lattice spacing by far the most important determinant of the cost. The very large exponent means that we that we should do everything we can to keep $a$ as large as possible.

Until recently most of us felt that lattice spacings of order .05–.1 fm or less were necessary before accurate simulations became possible. In this lecture I show new Monte Carlo results from several groups indicating that accurate results can be obtained even with lattice spacings as large as .4 fm. Given that the cost is proportional to $1/a^{6}$, an $a\!=\!.4$ fm simulation costs $10^{3}$–$10^{6}$ times less than one with conventional lattice spacings. This is a very dramatic improvement — far larger, for example, than the difference between a high-end PC and a typical supercomputer. If the coarse lattices really work, lattice QCD is about to become vastly more productive.

2 Improved Actions

A major obstacle to using large lattice spacings is that traditional discretizations of QCD are not very accurate when $a$ equals, say, .4 fm. We need improved discretizations for actions and operators. There are two standard approaches to the design of improved actions and operators. We examine each of them, first for classical field theory and then taking account of quantum effects.

2.1 Perfect Fields (Classical)

The first approach to discretizing a classical field is based upon standard finite-difference approximations from numerical analysis; in the context of lattice QCD this approach is frequently associated with Symanzik’s work [1]. In this approach the lattice theory is designed so that the lattice field $\phi_{i}$ at node $x_{i}$ equals the corresponding continuum field $\phi(x)$ evaluated at that node:111The analogue of this equation for the QCD gauge field is

$U_{\mu}(x)={\rm P}\,\exp\,i\int_{x}^{x+a\hat{\mu}}\!\!\!\!gA\cdot dx$

where $A_{\mu}$ is the continuum vector potential and ${\rm P}$ denotes path ordering.

[TABLE]

Given this identity the lattice version of nonlinear terms in the field equations is exact: for example,

[TABLE]

Continuum derivatives, however, must be approximated. Using finite differences, for example,

[TABLE]

where

[TABLE]

The expansion of $\partial^{2}\phi(x_{i})$ is an infinite series in powers of $a^{2}$. Normally only a couple of terms are needed. For example, if we are modelling a bump in $\phi(x)$ that is 3 or 4 lattice spacings across, the $a^{2}$ term in this expansion is typically only of order 10–20% the leading term, while the $a^{4}$ term is only a few percent. The terms of order $a^{4}$ and higher would be unnecessary for most applications in lattice QCD.

When one wishes to reduce the finite-$a$ errors in a simulation, it is usually far more efficient to improve the discretization of the derivatives (by including more terms in the expansion) than to reduce the lattice spacing. For example, with just the first term in the approximation to $\partial^{2}\phi$, cutting the lattice spacing in half would reduce a 20% error to 5%; but the cost would increase by a factor of $2^{6}\!=\!64$ in a simulation where cost goes like $1/a^{6}$. On the other hand, including the $a^{2}$ correction to the derivative, while working at the larger lattice spacing, achieves the same reduction in error but at a cost increase of only a factor of 2.

Although not necessary in practice, it is possible to sum all the terms in the $a^{2}$ expansion of a continuum derivative using Fourier transforms: we replace

[TABLE]

where

[TABLE]

We do not do this because $d^{(2)}_{\rm pf}(x_{i}\!-\!x_{j})$ falls off only as $1/|x_{i}\!-\!x_{j}|^{2}$ for large $|x_{i}\!-\!x_{j}|$, resulting in highly nonlocal actions and operators that are very costly to simulate, particularly when gauge fields are involved. So with “perfect fields” it is generally far better to truncate the $a^{2}$ expansion than to sum to all orders. However this analysis suggests a different strategy for discretization that we now examine.

2.2 Perfect Actions (Classical)

The slow fall-off in the all-orders or perfect derivative $d^{(2)}_{\rm pf}$ of the last section is caused by the abruptness of the lattice cutoff (the $\theta$ funtion in Eq. (7)). Wilson noticed that $d^{(2)}$ can be made to vanish faster than any power of $1/|x_{i}-x_{j}|$ if a smoother cutoff is introduced [2]. In his approach the lattice field at node $x_{i}$ is not equal to the continuum field there, but rather equals the continuum field averaged over an interval centered at that node: for example, in one dimension222The smearing discussed here is conceptually simple, but better smearings, resulting in improved locality, are possible. In [3] the authors use stochastic smearing in which $\phi_{i}$ equals the blocked continuum field only on average. This introduces a new parameter that can be tuned to optimize the action.

[TABLE]

where, in the transform, smearing function

[TABLE]

provides a smooth cutoff at large $p$. The lattice action for the smeared fields is designed so that tree-level Green’s functions built from these fields are exactly equal to the corresponding Green’s functions in the continuum theory, with the smearing function applied at the ends: for example,

[TABLE]

By rewriting the propagator $\langle\phi_{i}\,\phi_{j}\rangle$ in the form

[TABLE]

we find that the quadratic terms of this lattice action are

[TABLE]

where $d^{(2)}_{\rm pa}(x_{i}\!-\!x_{j},m)$ is

[TABLE]

It is easily shown that, remarkably, $d^{(2)}_{\rm pa}(x_{i}\!-\!x_{j},m)$ vanishes faster than any power of $1/|x_{i}\!-\!x_{j}|$ as the separation increases. So the derivative terms in such an action are both exact to all orders in $a$ and quite local — that is, the classical action is “perfect.” Again, this is possible because the lattice field $\phi_{i}$ is obtained by smearing the continuum field, which introduces a smooth UV cutoff in momentum space rather than the abrupt cutoff of the last Section.

The complication with this approach is that it is difficult to reconstruct the continuum fields $\phi(x)$ from the smeared lattice fields $\phi_{i}$. Consequently nonlinear interactions, currents, and other operators are generally complicated functions of the $\phi_{i}$, and much more difficult to design than with the previous (“perfect field”) approach, where the lattice field is trivially related to the continuum field. This is an important issue in QCD where phenomenological studies involve a wide range of currents and operators. It also means that in practice the “perfect action” used for an interacting theory is not really perfect since the arbitrarily complex functions of $\phi_{i}$ that arise in real perfect actions must be simplified for simulations. (Although recent numerical techniques developed by Hasenfratz, Niedermayer and their collaborators[4] seem to produce accurate approximations to the classical perfect action even for complicated theories like QCD.)

In general there is no clear choice between the two discretization techniques. In the “perfect field” approach operators and actions are trivial to design, but this simplicity may be at the expense of locality or accuracy. In the “perfect action” approach the action has no $a^{n}$ errors, but this perfection requires a very considerable (and costly) design effort, even at the classical level. As we shall see both approaches have been very successful applied to QCD.

2.3 Quantum Effects

The techniques described in the previous two sections have long been understood, and yet they are far from commonplace in lattice QCD analyses. This is because of quantum effects. Quantum fluctuations with momenta $p\!>\!\pi/a$ are excluded or badly mutilated by the lattice in either of the approaches we have discussed. In principle we can compensate for this by renormalizing the couplings in our quantum opertors. Thus, for example, $\phi\,\partial^{2}\phi$ is discretized by the replacement

[TABLE]

where the coefficients $c_{0}$ and $c_{1}$ equal unity in a classical theory but are renormalized in a quantum theory: $c_{i}=1+\alpha_{s}(\pi/a)c_{i}^{(1)}+\cdots$. The renormalizations are usually context specific and so must be recomputed for each new application. However, if $a$ is small enough, $p\!>\!\pi/a$ physics is perturbative and the renormalizations can be calculated perturbatively.

These considerations indicate that improved actions are practical only for lattice spacings that satisfy the following criteria:

• •

$a\!<\!(\mbox{important length scales})$: This requirement is obvious and applies in classical as well as quantum problems. To study the static properties of hadrons, for example, one cannot use lattice spacings much larger than about .4 fm, since light hadrons have radii of order .8 fm. On the other hand lattice spacings much smaller than this should not be necessary if improved operators are used.

• •

perturbation theory works at distances of order $a$ or less: The dynamics can be perturbative at distances of order the lattice spacing only if the coupling $\alpha_{s}(\pi/a)$ is small; and if the coupling is small then the coefficients in an expansion like Eq. (16) are well approximated by their classical values. Furthermore the small quantum corrections can be systematically computed using (analytic) perturbation theory. However, if the lattice spacing is too large the many extra couplings in improved actions and operators must be determined nonperturbatively (using, for example, the Monte Carlo Renormalization Group), making improvement very costly.

The second of these criteria has long delayed the widespread introduction of improvement techniques into lattice QCD. Until recently it was generally believed that perturbation theory was useless except at very small distances ($<\!.1$ fm); and therefore improved actions at, say, $a\!=\!.4$ fm were also useless, or, at best, prohibitively costly to generate. This view changed completely a few years ago when Paul Mackenzie and I noticed that perturbation theory, done correctly, works very well even at distances as large as $1/2$ fm[5].

Perturbation theory is indeed almost useless for lattice calculations unless one deals with the “tadpole problem.” The nonlinear relation between the link field $U_{\mu}(x)$, from which lattice operators are built, and the continuum vector potential $A_{\mu}(x)$ leads to tadpole diagrams in lattice perturbation theory that do not arise in continuum calculations. These tadpole contributions result in large renormalizations — often as large as a factor of two or three — that spoil naive perturbation theory. However tadpole contributions are generically process independent and so it is possible to measure their contribution in one quantity and then correct for them in all other quantities. Mackenzie and I developed a two-part prescription for doing this [5]:

• •

renormalized perturbation theory: A renormalized coupling constant like $\alpha_{V}(q^{*})$ should be used in perturbative calculations, not the bare coupling $\alpha_{\rm lat}$. It is important that this coupling be measured in a simulation, for example, by extracting its value from that of the expectation value of the plaquette; and it is important that the scale $q^{*}$ be chosen appropriately for the quantity being studied (see [5] for details).333In practice, this procedure is significantly more accurate than simply boosting the bare coupling by dividing by the plaquette. By measuring the coupling we automatically include any large renormalizations of the coupling due to tadpoles.

• •

tadpole improvement: Each $U_{\mu}(x)$ in a lattice operator should be replaced by

[TABLE]

where $u_{0}$ is the scalar mean value of the link, defined to be the fourth root of the expectation value of the four-link plaquette (as measured in the simulation). The $u_{0}$ cancels tadpole contributions, making lattice operators far more continuum-like in their behavior.

This prescription for removing tadpole contributions is now widely used, and its success is well documented. I give here only two illustrations. In Figure 1 I compare (nonperturbative) Monte Carlo results for the critical quark mass $m_{c}(a)$ with results from first-order perturbation theory [5]. The critical quark mass is the bare mass in Wilson’s quark action for which the pion is massless. This mass diverges like $1/a$ in Wilson’s theory and so should be quite perturbative. However, naive perturbation theory, in terms of $\alpha_{\rm lat}$, fails by almost a factor of two even at lattice spacings as small as $1/20$ fm. Reexpressing perturbation theory in terms of the renormalized coupling $\alpha_{V}(q^{*})$ leads to excellent results, especially considering that this is only a first-order calculation. But even these last results are significantly improved by first tadpole-improving the quark action, and then computing $m_{c}$ using first-order renormalized perturbation theory.

In Figure 2 I compare Monte Carlo results for $-\ln W_{2,2}$ at several lattice spacings with predictions from third-order (renormalized) perturbation theory [6]; $W_{2,2}$ is the expectation value of a $2\times 2$ Wilson loop. I plot these results versus the value of the coupling constant $\alpha_{V}(q^{*}\!=\!2.65/a)$ used in the perturbation theory. This figure shows that perturbation theory is still useful even when $\alpha_{V}\!\approx\!1/2$. Similar results are found for all sorts of other quantities [5]. Note that the $2\times 2$ loops in these simulations range from $10^{-6}$ fm to almost $1.2$ fm in size.

These examples and many others show that perturbation theory can usually be trusted out to distances of order $1/2$ fm. I say usually here because there are undoubtably situations or quantities for which perturbation theory fails to converge or nonperturbative effects are unusually large. However experience indicates that these are the exception rather than the rule. Consequently our criteria for using improved actions are satisfied if we choose lattice spacings of order $.4$ fm or less; even classically (tree-level) improved actions, provided they are tadpole improved, should give accurate results for such lattices.

3 Monte Carlo Results

In the previous sections we have argued that improved actions ought to be very effective on coarse lattices. Here we review recent tests of this idea. The central questions are: Do the corrections in an improved operator really work? And, are the simulations really faster?

3.1 Heavy Quarks — NRQCD

One of the most thoroughly studied actions that relies upon perturbative improvement is the NRQCD action for heavy quarks [7]. In NRQCD the basic dynamics is given by nonrelativistic Schrödinger theory,

[TABLE]

while relativistic effects and finite-$a$ are included by using correction terms:

[TABLE]

The corrections were designed using the “perfect field” or Symanzik approach. The couplings are computed using perturbation theory, but only after the operators are tadpole improved; almost all work to date has used just the tree-level values for the couplings ($c_{i}\!=\!1$). Consequently there are only two parameters in the NRQCD action: the coupling $g$ and the bare quark mass $M$.

Many of the results obtained from NRQCD depend upon $\delta H$ and so provide nontrivial tests of the improvement program. For example, spin-splittings in the upsilon family of mesons vanish if $\delta H$ is omitted. In Figure 3 I show NRQCD results for the splittings between the $1P$ states of the upsilon [8]. The results are in excellent agreement with experiment, particularly when extrapolated to the correct number of light-quark flavors, $n_{f}\!=\!3$. Note that all of the parameters in NRQCD are tuned from the spin-averaged spectrum, and so there are no free parameters in this determination of the spin splittings. Note also that the spin terms in $\delta H$ all involve either chromoelectric or chromomagnetic field operators. These operators are built from products of four link operators and so tadpole improvement increases their magnitude by almost a factor of two at $\beta=6$. Without tadpole improvement the spin splittings are much too small.

The correction terms also affect the low-momentum dispersion relation of the upsilon,

[TABLE]

In a relativistically invariant theory $M_{1}\!=\!M_{2}$ is the upsilon mass. However in a purely nonrelativistic theory $M_{2}$ equals the sum of the quark masses, which differs from $M_{1}$ since $M_{1}$ also includes the quarks’ binding energy. Only when relativistic corrections are added is $M_{2}$ shifted to include the binding energy. Simulations tuned to the correct mass $M_{1}\!=\!9.5(1)$ GeV give $M_{2}\!=\!8.2(1)$ GeV when $\delta H$ is omitted. When $\delta H$ is included, the simulations give $M_{2}\!=\!9.5(1)$ GeV, in excellent agreement with $M_{1}$ [9].

3.2 Gluons — Perfect Field

The standard Wilson action for gluons has finite-$a$ errors of order $a^{2}$:

[TABLE]

where the $r_{i}$ are constants and

[TABLE]

Only operator $R_{1}$ arises at tree-level; it also breaks rotation invariance. The other two $a^{2}$ operators are generated by quantum corrections. Since operator $R_{3}$ can be removed by a field redefinition,

[TABLE]

it doesn’t affect spectral quantities and can be ignored.

My collaborators and I have been studying “perfect field” (Symanzik) techniques for improving the gluon action; most of the new results I present in this section are from that collaboration and are descibed in [10]. Following earlier work, we added new Wilson loops to the gluon action to remove $R_{1}$ and $R_{2}$. There are many possible choices for the new loops, but among the simplest are the rectangle and “parallelogram”:

[TABLE]

The improved action is [11, 12]

[TABLE]

with $\beta_{\rm pl}$ given as an input, and $\beta_{\rm rt}$ and $\beta_{\rm pg}$ computed in tadpole-improved perturbation theory to cancel out $R_{1}$ and $R_{2}$. At tree-level, the $\beta$’s are readily calculated by combining expansions like Eq. (21) for each of the three loops. They are tadpole-improved by dividing each Wilson loop having $L$ links by $(u_{0})^{L}$. One-loop corrections have also been computed [13], but must be adjusted to account for the tadpole improvement. We found:

[TABLE]

Here $u_{0}$ and $\alpha_{s}$ are calculated from the measured expectation value of the plaquette:

[TABLE]

As in NRQCD, there is no tuning of the couplings for the correction terms: tadpole-improved perturbation theory determines them in terms of the single bare coupling $\beta_{\rm pl}$.

The effects of improvement are immediately apparent if one computes the static quark potential using a lattice spacing of .4 fm. Our results for both the Wilson theory and the improved theory are shown in Figure 4. Off-axis points deviate by as much as 40% from a fit to the on-axis points for the Wilson theory; these errors are reduced to 4–5% in the improved theory. The potential from the improved theory is essentially unchanged if the $\alpha_{s}$ corrections are dropped from the action, leaving just the tree-level improved theory (with tadpole-improved operators). This suggests that ${\cal O}(\alpha_{s}a^{2})$ corrections, together with ${\cal O}(a^{4})$ corrections, are relatively unimportant. Our simulations also show that the errors are 2–3 times larger without tadpole improvement even if $\alpha_{s}$ corrections are retained.

Morningstar and Peardon, in their presentation to this conference, show the first glueball calculations with this improved action. They find a dramatic reduction in the lattice-spacing dependence of the glueball mass relative to the Wilson theory. Also Bliss, Hornbostel and I have completed preliminary measurements of the temperature $T_{c}$ of the deconfining phase transition in gluonic QCD [14]. Results obtained from lattices with $a$ equal to $1/2T_{c}$ and $1/3T_{c}$ agree to within the statistical errors of a few percent, showing no sign of finite-$a$ errors (see Table 1).

The spin-averaged spectrum of the $\psi$ family of mesons is also independent of the lattice spacing for spacings of .4 fm and less [10]. This is illustrated by the simulation results for the spectrum in Figure 5; the scale for each result is set by the $1P\!-\!1S$ splitting. The $a$-independence (scaling) of these results is also evident if they are compared with the slope of the static potential $V^{\prime}$ evaluated at .6 fm, as shown in Figure 6. The results shown for the Wilson theory, however, change by 40% as $a$ is decreased, showing large $a^{2}$ errors.

As final evidence of the quality of the improved actions, I show the $1S$ and $1P$ radial wavefunctions for charmonium, computed on lattices with two different lattice spacings, using these actions (Figure 7). The most striking observation from these pictures is that the charge radius of the $\psi$ is almost exactly equal to one lattice spacing on the coarser lattice, and yet the results from the coarser lattice are essentially identical to those from the finer lattice. These data show that in general one needs only a few lattice points per direction within a hadron to obtain accurate results (few percent) from an $a^{2}$-accurate action. Numerical experiments with the NRQCD action, where $a^{2}$ corrections are easily turned on and off, suggest that

[TABLE]

3.3 Gluons — Perfect Action

DeGrand, Hasenfratz, Hasenfratz and Niedermeyer have recently computed the classically perfect gluon action for QCD using analytic techniques for terms quadratic and cubic in $A_{\mu}$ and numerical techniques for the rest of the action [4]. This action is too complicated to use in simulations but they have designed a much simpler, eight-parameter action that approximates the perfect action to within a few percent for any field configuration. This approximately perfect action is built from the plaquette and parallelogram loops and powers of these. (The powers are important.)

To test their action these authors computed the phase-transition temperature $T_{c}$ for lattice spacings ranging from $1/6T_{c}$ to $1/2T_{c}$ using their action and using the Wilson action. They compare $T_{c}$ with the torelon mass $\sigma$ (that is, the string tension on lattices of a fixed spatial size). I show some of their results for $2\sqrt{\sigma}/T_{c}$ in Table 2. Results for the Wilson action show finite-$a$ effects of order 10%; results from their improved action are $a$-independent to within statistical errors of a couple percent. There is insufficient data from other sources to know whether their results are the correct $a\!=\!0$ results.

They also computed the static-quark potential for two different lattice spacings. The results, in Figure 8, show large-$r$ behavior that is independent of the lattice spacing. At smaller radii, results from the coarser lattice show some failure of rotational invariance. These finite-$a$ errors are roughly twice as large as those we find with the Symanzik-improved gluon action of the previous section. DeGrand and collaborators suggest that the difference is because they used the blocked gluon operator to compute the potential, which introduces additional finite-$a$ errors. I am not certain this is the explanation since the $a^{2}$ errors in their measurement are rotationally invariant, and the $a^{4}$ errors ought to be pretty small. Another possibility is that the errors in the potential are due to tadpole effects. The finite-$a$ errors in their potential are almost identical to what we obtain from the Symanzik-improved action if we omit tadpole improvement (with or without $\alpha_{s}$ corrections). This suggests that a tadpole-improved version of the perfect action should be tried.

3.4 Light Quarks

All standard relativistic quark actions are built using a first-order centered difference operator,

[TABLE]

(ignoring gauge fields). Consequently the effective lattice spacing in light-quark actions is $a_{\rm eff}\!=\!2a$, and not $a_{\rm eff}\!=\!a$ as in gluon and heavy-quark (NRQCD) actions. This makes the light-quark actions the least accurate component in lattice simulations and the most in need of repair. Based on our experience with NRQCD (Eq. (30)), we expect order $a^{2}$ corrections to a light-quark action to be of order 20% when $a\!=\!.4$ fm, which corresponds to an $a_{\rm eff}$ of order the light-hardron radius. To get errors down to a few percent with such a coarse lattice requires an action corrected for ${\cal O}(a^{2})$ effects and possibly also for ${\cal O}(a^{3})$.

The most accurate light-quark action in common use today is the tadpole-improved version of the Sheikholeslami-Wohlert (SW) action:

[TABLE]

where parameter $r\!=\!1$ is usually chosen. There is now considerable evidence showing that this action, which has ${\cal O}(a^{2})$ errors, is much more accurate than the Wilson action for quarks, which has ${\cal O}(a)$ errors. At this conference Brian Gough presented data from Fermilab on quark masses that show a significant reduction in finite-$a$ errors with the SW action. Also Hugh Shanahan presented new UKQCD spectrum results obtained using this action. I compare his $\rho$ mass with masses computed by Weingarten and collaborators [15] using the Wilson quark action in Figure 9. I use the QCD scale parameter $\Lambda_{V}$ determined from the measured value of $\alpha_{V}(3.4/a)$ as a reference scale.444Some people prefer to use the string tension or other infrared quantities to fix relative scales. The problem with that approach is that most such quantities have significant $a^{2}$ errors themselves, unless they are highly corrected as are the NRQCD charmonium and upsilon spectra. $\Lambda_{V}$, on the other hand, when defined in terms of the plaquette has no $a^{n}$ errors. The only errors in $\Lambda_{V}$ are from higher-order perturbation theory, and these are negligible. Results from the Wilson quark action show large finite-$a$ errors, while the SW results are $a$-independent to within (large) statistical errors. The two calculations agree within errors on the $a\!=\!0$ mass. Note that tadpole improvement almost doubles the coefficient of $\sigma\cdot F$ in the SW action and so is very important to the success of this action [16].

These results show that the SW quark action is probably sufficiently accurate for most of our needs provided $a$ is less than about $1/6$ fm ($\beta\!=\!5.7$ or higher for quenched simulations). However the $a^{2}$ errors in this action become noticable at larger lattice spacings. This is clear in the SCRI results presented at this conference by Robert Edwards for the $\rho$ mass compared with the string tension. His results show finite-$a$ errors of order 20–25% at $a\!=\!.4$ fm, which is consistent with my estimate inspired by NRQCD analyses. Alford, Klassen and I noticed that the leading correction to SW,

[TABLE]

violates relativistic invariance. We test for this by computing the square of the speed of light using the pion’s dispersion relation:

[TABLE]

We find that $c^{2}$ is 30–40% too small when we use the SW action with $a\!=\!.4$ fm [17].

The SW action must be replaced by a more accurate action before useful simulations involving light quarks are possible at $a\!=\!.4$ fm. Alford, Klassen and I are examining a variety of such actions. One of the first we studied is the “D234 action:”

[TABLE]

where $\Delta^{(3)}_{\mu}$ is a discretization of $D_{\mu}^{3}$, and $\Delta^{(4)}$ is a discretization of $\sum_{\mu}D_{\mu}^{4}$. This action made the error in $c^{2}$ three or four times smaller, and had several other successes as well [17]. I worry that quark doublers introduced by the higher-order differences may be a problem for this action, but there are several ways of dealing with this problem that we are exploring.

I have no doubt that actions substantially more accurate than the SW action will be available shortly. The actions discussed above are all of the “perfect field” or Symanzik variety. Work has also begun on “perfect actions” for the quarks, but is still in its infancy [18].

3.5 Computing Cost

The use of improved actions on coarse lattices should lead to substantial reductions in simulation costs. My own experience is that the reductions are spectacular. Almost all of the $a\!=\!.4$ fm results for gluons and heavy quarks that I discuss in this paper were generated on an IBM RS6000/250 workstation, which performs like a mid to high-range PC. These simulations typically involved lattices with only 6 points on a side, which fit easily in the 8–12 MB of memory common on PC’s. Quantities like the charmonium wavefunction can be computed from scratch in an hour or two on such a workstation; all previous work on charmonium was done with nothing smaller than a CRAY, and turnaround times were generally quite a bit longer than an hour.

Beyond the savings associated with smaller grids, a large part of the cost reduction in these simulations is due to the near absence of critical slowing down on coarse grids. Field configurations decorrelate after 10–20 sweeps even with slow algorithms like Metropolis. Note also that the asymptotic exponential behavior in propagators usually sets in after only one or two lattice spacings; this is partly because operators on a coarse lattice are automatically smeared (there are no high momentum states), and partly because the asymptotic region occurs at distances of order $1/2$–1 fm, which is only one or two lattice spacings on a coarse lattice. The added cost of simulating with a more complicated action is negligible (factor of two or three) compared with these savings.

4 Conclusions

The results I have reviewed here argue persuasively, I think, that classically improved actions can be made accurate to a few percent for lattice spacings as large as .4 fm. Furthermore classical (i.e., tree-level in perturbation theory) improvement is sufficient for this purpose, provided lattice operators are tadpole-improved. Without tadpole improvement, finite-$a$ errors are only half removed.

Some of the features of the gluon actions I have discussed were anticipated in actions obtained using numerical techniques like the Monte Carlo Renormalization Group (see work by Gupta, Iwasaki and others). But the high cost of these techniques greatly limits their general utility. Renormalized perturbation theory and tadpole improvement allow us to realize the same goals but with negligible costs.

Both gluon and heavy-quark actions have been used with great success at .4 fm, and work is proceeding on the development of new, more accurate actions for the light quarks. Even without new light-quark actions, there is now evidence that the Sheikholeslami-Wohlert quark action is sufficiently accurate for state-of-the-art work at $a\!=\!1/6$ fm ($\beta\!=\!5.7$ quenched QCD), which is already a major advance.

As we shift to coarse lattices, numerical QCD will be redefined by a $10^{3}$–$10^{6}$ speedup in simulations. This will have far reaching implications for both the sociology of our field and the physics it produces. The entry-level computer needed for serious QCD simulations is no longer a CRAY; a personal computer or desktop computer will do — lattice QCD is now accessible to almost anyone. The cycle time for ideas is reduced from months to hours — we can dare to be much more adventuresome than we have in the past. In the short term we should be capable of an abundance of 10% accurate results for spectra, form factors, decay constants, and all sorts of other phenomenologically relevant quantities. By using our new techniques in simulations on our largest computers, we should finally obtain reliable callibrations of the various approximations we make in lattice QCD (especially quenching). By combining rapid simulations on coarse grids with analytic methods, like strong coupling expansions, we may move closer to analytic solutions of low-energy QCD — that is, nonperturbative QCD without computers.