We investigate a treelevel accurate action, D234c, on coarse lattices. For the improvement terms we use tadpoleimproved coefficients, with the tadpole contribution measured by the mean link in Landau gauge.
We measure the hadron spectrum for quark masses near that of the strange quark. We find that D234c shows much better rotational invariance than the SheikholeslamiWohlert action, and that meanlink tadpole improvement leads to smaller finitelatticespacing errors than plaquette tadpole improvement. We obtain accurate ratios of lattice spacings using a convenient “Galilean quarkonium” method.
We explore the effects of possible changes to the improvement coefficients, and find that the two leading coefficients can be independently tuned: hadron masses are most sensitive to the clover coefficient , while hadron dispersion relations are most sensitive to the third derivative coefficient . Preliminary nonperturbative tuning of these coefficients yields values that are consistent with the expected size of perturbative corrections.
1 Introduction
Lattice QCD remains the only complete implementation of nonperturbative QCD and so is essential for lowenergy QCD phenomenology. However, simulations of lattice QCD rely upon brute force Monte Carlo evaluations of the QCD path integral, and are very costly. In recent years it has been demonstrated that this cost is dramatically reduced by using coarse lattices, with lattice spacings as large as fm, together with more accurate discretizations of QCD. While highly corrected discretizations of gluon and heavyquark actions are now commonplace, less progress has been made with the much harder problem of constructing highly improved lightquark actions. The best lightquark actions in widespread use have finite errors proportional to , which are large compared with the errors for improved gluon actions. The problem is compounded by the fact that the effective lattice spacing for light quarks is rather than , because the lightquark action, unlike the others, involves firstorder derivatives. There are a number of problems, like relativistic heavyquark physics and highmomentum form factors, where improvement is crucial (in particular when used in conjunction with anisotropic lattices) for accurate results without more or less uncontrolled extrapolations over large mass and/or momentum regions. In this paper we take a step towards remedying this situation by presenting new results obtained using a highly corrected lattice action for lightquarks.
The finite errors can be removed, orderbyorder in , from a lattice lagrangian by adding correction terms:
(1.1) 
In principle, the coefficients of these correction terms can be computed using (weakcoupling) perturbation theory, but in lattice QCD there has been a longterm reluctance to rely on perturbation theory for any of the ingredients in QCD simulations. For most of the past twenty years this has meant that no correction terms were included in the action, which then has only one parameter, the bare quark mass; the mass is tuned nonperturbatively to give correct hadron masses. Recently a practical technique has been developed for nonperturbatively computing the coefficient of the correction[1]. The accurate quark action, originally discussed in [2], has led to substantial improvements over past work, but it is still of limited value for lattice spacings larger than 0.1–0.2 fm.
With only a few exceptions, it is very difficult to compute the coefficients for and higher corrections nonperturbatively. Thus a perturbative determination is the only practical alternative that permits further improvement. Given the advantages of very coarse lattices, we feel it is too restrictive to abandon perturbation theory completely. This is particularly the case since we now know that perturbation theory is generally quite reliable, provided one uses tadpoleimproved lattice operators [3, 4]. In particular, perturbation theory correctly predicted the relatively large renormalization of the correction to the quark action several years before it was confirmed in nonperturbative studies.
The coupling constant, , is larger on coarser lattices, and therefore perturbation theory is less convergent. This makes perturbation theory less practical for calculating such things as the overall renormalization factors relating lattice currents to continuum currents. The correction terms in the quark action, however, are suppressed by explicit powers of the lattice spacing. Consequently they require less high precision, and even loworder perturbation theory may suffice for results accurate at the few percent level.
In this paper we derive a tadpoleimproved accurate quark action, “D234c”. We compare its predictions with the those of the standard SheikholeslamiWohlert (SW) accurate action, and also with the original Wilson (W) action. To study finitelatticespacing errors it is not necessary to take the chiral limit, so we restrict our study to quark masses near the strange quark’s mass. Since finite errors tyically grow with quark mass, our results should improve for and quarks.
The important points in our analysis are:

We use the mean link in Landau gauge rather than the traditional plaquette prescription for calculating our tadpole improvement factor . Our reasons are: (1) it gives a more rotationally invariant static potential [5]; (2) it has been shown in NRQCD that it leads to smaller scaling errors in the charmonium hyperfine splitting [6]; (3) for Wilson glue, it gives a clover coefficient that agrees more closely with the nonperturbatively determined value [4]. These studies suggest that the meanlink tadpole prescription has smaller quantum corrections than the plaquette prescription. Of course, once higher order perturbative corrections are included the two prescriptions will come into agreement [4].

After tadpole improvement, the improvement coefficients are expected to have quantum corrections of order , which is on our coarsest lattice. In MonteCarlo simulations, we systematically study the effects of corrections of this size, and find that the clover coefficient is the only one whose quantum corrections will affect hadron masses significantly, and the third derivative coefficient is the only one that affects hadron dispersion relations significantly.

We perform various nonperturbative tests of the coefficients of the improvement terms. We measure the hadron dispersion relation (“speed of light”) to check the term; vector meson () scaling as a check on the relative weight of the clover and Wilson terms; dependence as an additional check on the relative weight of the Wilson and clover terms and the effects of ghost branches in the quark dispersion relation.

We perform a rough nonperturbative tuning of the two leading coefficients of the action, and discuss the comparison with perturbative expectations.

We set our overall scale from the charmonium splitting. However for comparisons of scaling it is ratios of lattice spacings that are important, and we introduce a simple method for determining these more accurately, and with less vulnerability to systematic errors. It consists of measuring in a fictitious heavy quark “Galilean quarkonium” state, i.e. using an NRQCD heavy quark action in which relativistic corrections are not included. Varying the quark mass changes the size of the state, and thereby tests for the presence of finite errors.
We have previously studied a plaquettetadpoleimproved accurate action on isotropic lattices [7], and plaquettetadpoleimproved accurate actions on anisotropic lattices [8, 9], and found good dispersion relations and scaling of mass ratios. In this paper we find that meanlink tadpole improved D234c has the same benefits, plus much smaller finite errors in hadron masses. This is as expected, because meanlink tadpole improvement gives a larger clover coefficient.
2 D234c Quark action
Following [9], we construct a quark action that is continuumlike (at tree level) through . We start with the continuum quark action:
(2.2) 
If we discretize this directly, our quark dispersion relation will contain unwanted doublers at the edges of the Brillouin zone. To avoid this, we perform a field redefinition, parameterized by , before discretizing:
(2.3) 
where
(2.4) 
Now , where the transformed continuum quark operator is
(2.5) 
We use
is the most local centered lattice discretization of the gaugecovariant ’th derivative [9, 10]; , and . The field strength consists of the standard clover term , and a relative correction [9] with coefficient :
At tadpoleimproved tree level, all links are divided by the Landau gauge mean link , and . We will explore the effects of deviations from these values.
The terms proportional to remove the doublers from the quark dispersion relations, so that for generic values of this is a doublerfree treelevel accurate quark action. The derivation can be straightforwardly generalized to anisotropic lattices [9, 10].
For there are three fairly high ghost branches in the free quark dispersion relation (Figure 1). To investigate the effect of redundant terms we will also study , for which one of the ghost branches moves down so that .
Note that the two leading terms both violate symmetries that will be restored in the continuum limit, and hence can be nonperturbatively tuned. The clover term violates chiral symmetry, and so can be tuned by imposing PCAC [1]. The term is the only rotational symmetryviolating term, and so its coefficient can be nonperturbatively tuned by imposing rotational invariance.
3 Gluon action and lattice spacing determination
We use a treelevel tadpoleimproved plaquette and rectangle glue action [11, 12, 13],
(3.6) 
where a Wilson loop goes one link in the direction, one link in the negative direction, etc.
This definition of is different from that of Ref. [13], where a factor of was absorbed into . We prefer the notation here because as in the original Wilson action. Furthermore the coupling
(3.7) 
is now tadpoleimproved and therefore roughly equal to continuum couplings like .
The tadpole improvement factor is the mean of the link operator in Landau gauge. At both our lattice spacings we found that an lattice was large enough for finite volume effects in to be of order . To fix to Landau gauge we maximize [4]
(3.8) 
We generated gluon configurations at two lattice spacings, and (see Table 1). The lattice spacing was determined in two ways.
Firstly, we performed NRQCD simulations of charmonium, using the experimental value of for the spinaveraged splitting; the results are given in Table 1 column 6 (details in Table 7). Note that the errors quoted are statistical, and do not reflect systematic uncertainties such as quenching, finite errors, or higherorder relativistic effects neglected in our NRQCD simulation. This lattice spacing determination is therefore not suitable for precise comparisons with data from other groups.
Secondly, a more accurate determination of ratios of lattice spacings is possible, since there is no need to simulate a known physical state. At each lattice spacing we can measure the mass of a fictitious state, whose properties are chosen for convenience. We chose “Galilean quarkonium,” a bound state of a quark and antiquark in a nonrelativistic world. We simulated this state using NRQCD with no relativistic corrections. By making the Galilean quarkonium lighter (and hence bigger) than charmonium we reduce the finite errors. In fact, we studied a range of quark masses down to about half the charm quark mass and found that lattice spacing ratios were all consistent with each other, within errors. For details see Appendix A and Table 6. Since our main goal is to compare our meanlinktadpoleimproved results with SCRI’s plaquettetadpoleimproved results, we used Galilean quarkonium to calculate the ratios of our lattice spacings to the SCRI lattice spacing. The results are in Table 6.
For convenience we want to give our results an absolute energy scale, so we take the SCRI lattice spacing to be , which corresponds to for their string tension. This gives the final column of Table 1, which is consistent with our charmonium measurements. Note that the error bars reflect the uncertainty in the ratio to SCRI’s lattice spacings, which is the relevant quantity for scaling comparisons. It does not reflect the overall uncertainty in the scale, which was introduced purely for convenience.
approx  charmonium  rel. to SCRI  

1.157  0.413  0.738  0.8196  0.40 fm  495(4) MeV  497(3) MeV 
1.719  0.278  0.797  0.8576  0.25 fm  790(10) MeV  785(6) MeV 
7.4 (SCRI)  0.24 fm  840(20) MeV  812(def) MeV 
For hadron spectrum measurements we used lattices of the same physical size () at both lattice spacings. We also performed a set of measurements investigating the volume dependence of hadron masses (see appendix, Table 8). We see that the lattice agrees with the and lattices within statistical errors.
4 Results
We subjected the D234c action to a series of tests to determine its viability at large lattice spacings. We examined the scaling of the vector meson mass and of baryon masses, and we measured hadronic dispersion relations. We also measured the sensitivity of these physical quantities to changes in the coefficients in the action.
4.1 Hadron masses
In Figure 2 we show how the vector meson mass varies with lattice spacing when the ratio of the pseudoscalar to vector meson masses is . (Full data is in Appendix B, along with data for , corresponding to a slightly larger quark mass; see tables 10, 9). We present data obtained using the treelevel D234c, with Landaulink tadpole improvement, at lattice spacings of 0.25 fm and 0.4 fm. These values are compared with results from SCRI obtained using the Wilson and SW actions [14], which can be extrapolated to give results, as indicated.
We also examined baryon masses; ratios of these to the vector mass are shown in Fig. 3. Our measured values at are within of the quadratic extrapolation of the SCRI values to the continuum.