# Highly Improved Staggered Quarks on the Lattice,

with
Applications to Charm Physics.

###### Abstract

We use perturbative Symanzik improvement to create a new staggered-quark action (HISQ) that has greatly reduced one-loop taste-exchange errors, no tree-level order errors, and no tree-level order errors to leading order in the quark’s velocity . We demonstrate with simulations that the resulting action has taste-exchange interactions that are at least 3–4 times smaller than the widely used ASQTAD action. We show how to estimate errors due to taste exchange by comparing ASQTAD and HISQ simulations, and demonstrate with simulations that such errors are no more than 1% when HISQ is used for light quarks at lattice spacings of fm or less. The suppression of errors also makes HISQ the most accurate discretization currently available for simulating quarks. We demonstrate this in a new analysis of the mass splitting using the HISQ action on lattices where and 0.66, with full-QCD gluon configurations (from MILC). We obtain a result of 111(5) MeV which compares well with experiment. We discuss applications of this formalism to physics and present our first high-precision results for mesons.

###### pacs:

11.15.Ha,12.38.Aw,12.38.GcHPQCD, UKQCD

## I Introduction

The reintroduction of the staggered-quark discretization in recent years has transformed lattice quantum chromodynamics, making accurate calculations of a wide variety of important nonperturbative quantities possible for the first time in the history of the strong interaction ratio-paper ; fk-fpi ; alphas-paper ; fd ; mb-paper . Staggered quarks were introduced thirty years ago, but unusually large discretization errors, proportional to where is the lattice spacing, made them useless for accurate simulations. It was only in the late 1990s that we discovered how to remove the leading errors, and the result was one of the most accurate discretizations in use today. Staggered quarks are faster to simulate than other discretizations, and as a result have allowed us for the first time to incorporate (nearly) realistic light-quark vacuum polarization into our simulations. This makes high-precision simulations, with errors of order a few percent, possible for the first time (see ratio-paper for a more detailed discussion). In this paper we present a new discretization that is substantially more accurate than the improved discretization currently in use. With this new formalism accurate simulations will be possible at even larger lattice spacings, further reducing simulation costs.

The discretization errors in staggered quarks have two sources. One is the usual error associated with discretizing the derivatives in the quark action. The correction for this error is standard and was known in the 1980s naik-paper . The second source was missed for almost a decade. It results from an unusual property of the staggered-quark discretization: the lattice quark field creates four identical flavors or tastes of quark rather than one. (We refer to these unphysical flavors as “tastes” to avoid confusion with the usual quark flavors, which are not identical since quarks of different flavor have different masses.) The missing error was associated with taste-exchange interactions, where taste is transferred from one quark line to another in quark-quark scattering. The generic correction for this type of error involves adding four-quark operators to the discretized action.

Taste is unphysical. Instead of one , for example, one has sixteen. Taste is easily removed in simulations provided that different tastes are exactly equivalent, which they are, and provided there are no interactions that mix hadrons of different taste. Taste-exchange interactions, however, cause such mixing, and therefore it is crucial that we understand such interactions and suppress them. Furthermore, past experience suggests that the errors due to residual taste exchange are the largest remaining errors in current simulations.

The corrections to the lattice action that suppress taste exchange were missed initially because four-quark operators are not usually needed to correct discretization errors in lowest order perturbation theory (that is, at tree level). They were first discovered empirically from simulations in which the gluon fields in the quark action were replaced by smeared fields, which happen to suppress taste-changing quark-quark scattering amplitudes doug-papers . This discovery led to a proper understanding of the taste-exchange mechanism lepage1-paper ; sinclair-paper , and to the first correct analysis of the errors in staggered-quark formalisms lepage1-paper ; lepage2-paper . The resulting “ASQTAD” quark action has provided the basis for almost all simulations to date that include (nearly) realistic light-quark vacuum polarization.

While it is possible to remove all tree-level taste exchange by smearing the gluon fields appropriately, higher-order corrections can only be removed by adding four-quark operators. Recent numerical experiments suggest, however, that even these corrections can be significantly suppressed through additional smearing hyp-paper . These studies, while very important, were limited in two ways. First, they used the mass splittings between the sixteen tastes of pion as a probe for taste exchange; there are many taste-exchange interactions that do not affect this spectrum. Second, smearing the gluon fields introduces other types of error which undo the advantage of an improved discretization (that is, accuracy at large values of ).

In this paper we present the first rigorous analysis of one-loop taste exchange. Based upon this analysis we develop a simple smearing scheme that suppresses one-loop corrections by an order of magnitude on average. Our smearing scheme is simpler than that in hyp-paper and, unlike that scheme, it does not introduce new errors. The result is a new discretization for highly-improved staggered quarks which we refer to by the acronym HISQ.

An added advantage of a highly improved action is that it can be used to simulate quarks. Heavy quarks are difficult to simulate using standard discretizations because discretization errors are large unless , where is the lattice spacing and the quark mass. To simulate quarks, for example, one would require lattice spacings substantially smaller than fm — much too small to be practical today. Consequently high-quality simulations of quarks rely upon rigorously defined effective field theories, like nonrelativistic QCD nrqcd , that remove the rest mass from the quark’s energy. While this approach works very well for quarks, it is less successful for quarks because the quark’s mass is much smaller, and consequently quarks are much less nonrelativistic. The smaller quark mass, however, means that for lattice spacings of order fm, which are typical of simulations today. As we demonstrate in this paper, we can obtain few-percent accuracy for charm quarks when provided we use a highly improved relativistic discretization like HISQ.

In Section II we review the (perturbative) origins of taste-exchange interactions in staggered quarks, and their removal at lowest order (tree level) in perturbation theory, using the Symanzik improvement procedure. We also extend the traditional Symanzik analysis to cancel all errors to leading order in the quark’s velocity, (in units of the speed of light ). The additional correction needed is negligible for light quarks, but significantly enhances the precision of -quark simulations.

In Section III, we extend our analysis of taste exchange to one-loop order, and show how to suppress such contributions by approximately an order of magnitude, thereby effectively removing one-loop taste exchange from the theory. The resulting quark action (HISQ) is the first staggered-quark discretization that is both free of lattice artifacts through and effectively free of all taste exchange through one-loop order. We also identify the order corrections to the action that are important for high-precision physics.

We illustrate the utility of our new formalism, in Section IV, first by examining its effect on taste splittings in the pion spectrum. We also show how to directly measure the size of taste-exchange interactions in simulations, by comparing ASQTAD and HISQ simulations. There has been much discussion recently about the formal significance of taste-exchange interactions creutz-paper-etc . With our procedure it is possible to directly measure the effect of these interactions on physical quantities. We demonstrate the procedure by showing that residual taste-exchange interactions in HISQ contribute less than 1% at current lattice spacings to light-quark quantities such as meson masses and meson decay constants.

We report on a new simulation of quarks and charmonium using the HISQ formalism in Section V. This is the most severe test of the HISQ formalism since for the lattice spacings we use, but we show that the formalism delivers results that are accurate to a couple of percent, making it the most accurate formalism there is for simulating quarks. We demonstrate this in a new high-precision determination of the – splitting. This analysis confirms our expectation that the largest errors in the ASQTAD action are associated with taste exchange, and we show that these are greatly reduced by using the HISQ action instead.

The moderately heavy smearing used in the HISQ formalism complicates unquenched simulations. In Section VI, we discuss how these complications can be dealt with. Finally, in Section VII, we summarize our results and discuss their relevance to future work. Here we also present our first physics results from the HISQ action.

Much of our discussion is framed in terms of the “naive” discretization for quarks, which is conceptually equivalent to the staggered-quark discretization (see Appendix B). We usually find naive quarks to be more intuitive in analytic work than staggered quarks, while the latter formalism is definitely the more useful one for numerical work. We outline the formal connection between the naive-quark and staggered-quark formalisms in a series of detailed Appendices. We use these results in our one-loop analysis of taste-exchange interactions, which is described in detail in Appendix F.

We used several sets of gluon configurations in this paper. The parameters for these various sets are summarized in Table 1.

(fm) | Gluon Action | ||||
---|---|---|---|---|---|

UKQCD | Wilson | – | – | 16 | |

MILC | Lüscher-Weisz | 0.01 | 20 | ||

MILC | Lüscher-Weisz | 28 |

## Ii Symanzik Improvement for Naive/Staggered Quarks

### ii.1 Naive Quarks, Doubling, and Taste-Changing Interactions

We begin our review of staggered quarks by examining the formally equivalent “naive” discretization of the quark action (see Appendix B):

(1) |

where is a discrete version of the covariant derivative,

(2) |

is the gluon link-field, the lattice spacing, and is the bare quark mass. The gamma matrices are hermitian,

(3) |

where indices and run over 0…3. A complete set of sixteen spinor matrices can be labeled by four-component vectors consisting of 0s and 1s (i.e., ):

(4) |

A set of useful gamma-matrix properties is presented in Appendix A.

The naive-quark action has an exact “doubling” symmetry under the transformation:

(5) | |||||

Thus any low energy-momentum mode, , of the theory is equivalent to another mode, , that has momentum , the maximum allowed on the lattice. This new mode is one of the “doublers” of the naive quark action. The doubling transformation can be applied successively in two or more directions; the general transformation is

(6) |

where

(7) | |||||

and is a vector with , while is “conjugate” to (see Appendix A):

(8) |

Consequently there are 15 doublers in all (in four dimensions), which we label with the fifteen different nonzero ’s.

As a consequence of the doubling symmetry, the standard low-energy mode and the fifteen doubler modes must be interpreted as sixteen equivalent flavors or “tastes” of quark. (The sixteen tastes are reduced to four by staggering the quark field; see Appendix B.) This unusual implementation of quark tastes has surprising consequences. Most striking is that a low-energy quark that absorbs momentum close to , for one of the fifteen ’s, is not driven far off energy-shell. Rather it is turned into a low-energy quark of another taste. Thus the simplest process by which a quark changes taste is the emission of a single gluon with momentum . This gluon is highly virtual, and therefore it must immediately be reabsorbed by another quark, whose taste will also change (see Fig. 1). Taste changes necessarily involve highly virtual gluons, and so are both suppressed (by ) and perturbative for typical lattice spacings. One-gluon exchange, with gluon momentum , is the dominant flavor-changing interaction since it is lowest order in and involves only 4 external quark lines. (Processes with more quark lines are suppressed by additional powers of where is a typical external momentum.) This observation is crucial when trying to improve naive quarks by removing finite- errors, as we discuss below (see also lepage1-paper ; sinclair-paper ; lepage2-paper ).

Sixteen tastes of quark from a single quark field is fifteen too many. In the absence of taste exchange, factors of are easily inserted into simulations to remove the extra copies. In particular, quark vacuum polarization is corrected by replacing the quark determinant in the path integral by its root,

(9) |

or, equivalently, by multiplying the contribution from each quark loop by . Recent empirical studies of the spectrum of , and improved versions of it, show how eigenvalues cluster into increasingly degenerate multiplets of sixteen as the lattice spacing vanishes, which is necessary if this procedure is to be useful for quark vacuum polarization Dslash-spectrum . The treatment of valence quarks is illustrated in Appendix G.

It is easy to see how the root corrects for taste in particular situations. For example, consider the vacuum polarization of a single pion in a theory of only one massless quark flavor. Ignoring taste exchange, the dominant infrared contributions come from the quark diagrams shown in Fig. 2 (gluons, to all orders, are implicit in these diagrams). The first diagram, with its double quark loop, has contributions from identical massless pions, corresponding to configurations where the quark and antiquark each carries a momentum close to an integer multiple of (and is therefore close to being on shell). This diagram is multiplied by since there are two quark loops, thereby giving the contribution of a single massless pion. The second diagram involves quark annihilation into gluons, and has contributions from identical massless pions, corresponding to configurations where the quark and antiquark each carries a momentum close to the same (so they can annihilate into low-momentum gluons). This diagram is multiplied by since there is only one quark loop, thereby giving the contribution again of a single massless pion but now with an insertion from the gluon decay. Inserting additional annihilation kernels results in a geometric series of insertions in the propagator of a single massless pion that shifts the pion’s mass away from zero, as expected in a flavor theory.

Such patterns are disturbed by taste-exchange interactions like that in Fig. 1. For example, these interactions cause mixing between the different tastes of pion in Fig. 2. This mixing lifts the degeneracy in the pion masses so that different tastes of pion are only approximately equivalent. Consequently the “-root rule” gives results that correspond only approximately to a single pion. Taste-changing interactions are suppressed by , however, and detailed analyses, using chiral perturbation theory, demonstrate that the impact on physical quantities is therefore also suppressed by aubin . Consequently, while taste exchange can lead to such anomalies as unitarity violations, these are suppressed in the continuum limit. They are further, and more efficiently, suppressed through Symanzik improvement of the action, as we now discuss.

### ii.2 Tree-Level Symanzik Improvement

The discretization errors in the naive-quark action come from two sources. The more conventional of these corrects the finite-difference approximation to the derivatives in the action: one replaces naik-paper

(10) |

in the naive-quark action (Eq. (1)). The correction is often referred to as the “Naik term.”

Less conventional is a correction to remove leading order taste-exchange interactions lepage1-paper ; sinclair-paper ; lepage2-paper . As discussed in the previous section, these interactions result from the exchange of single gluons carrying momenta close to for one of the fifteen non-zero s (). Since these gluons are highly virtual, such interactions are effectively the same as four-quark contact interactions and could be canceled by adding four-quark operators to the quark action. These operators affect physical results in since they have dimension six. A simpler alternative to four-quark operators is to modify the gluon-quark vertex, , in the original action by introducing a form factor that vanishes for (taste-changing) gluons with momenta for each of the fifteen nonzero s. In fact the form factor for direction need not vanish when since the original interaction already vanishes in that case. Consequently we want a form factor where

(11) |

We can introduce such a form factor by replacing the link operator in the action with where smearing operator is defined by

(12) |

and approximates a covariant second derivative when acting on link fields:

(13) |

This works because (and vanishes) when acting on a link field that carries momentum . This kind of link smearing is referred to as “Fat7” smearing in doug-papers .

Smearing the links with removes the leading taste-exchange interactions, but introduces new errors. These can be removed by replacing with lepage2-paper

(14) |

where approximates a covariant first derivative:

(15) |

The new term has no effect on taste exchange but (obviously) cancels the part of . Correcting the derivative, as in Eq. (10), and replacing links by -accurate smeared links removes all tree-level errors in the naive-quark action. The result is the widely used “ASQTAD” action lepage2-paper ,

(16) |

where, in the first difference operator,

(17) |

In practice, operator is usually tadpole improved tadpole ; in fact, however, tadpole improvement is not needed when links are smeared and reunitarized tsukuba ; schladming .

### ii.3 Quarks

The tree-level discretization errors in the ASQTAD action are , which are negligible (%) for light quarks. When applied to quarks, however, these errors will be larger. The most important errors in this case are associated with the quark’s energy, rather than its 3-momentum, since quarks are typically nonrelativistic in s, s and other systems of current interest. Consequently and the largest errors are or 6% at current lattice spacings (). Such errors show up, for example, in the tree-level dispersion relation of the quark where:

(18) | |||||

This dominant (tree-level) error can be removed by retuning the coefficient of the (Naik) term in the action:

(19) |

where parameter has expansion

(20) |

With this choice, . Only the first term in ’s expansion is essential for removing errors; the remaining terms remove errors to higher orders in but leading order in , the quark’s typical velocity. In practice, terms beyond the first one or two are negligible though trivial to include.

Note that the bare mass in the quark action is related to the tree-level quark mass by

(21) |

for this choice of . This formula is useful for extracting the corrected bare quark mass, , from a tuned value of .

To examine the tree-level errors in this modified ASQTAD action, it is useful to make a nonrelativistic expansion since, as we indicated, heavy quarks are generally nonrelativistic in the systems of interest. This expansion is easy because the action is identical in form to the continuum Dirac equation, and consequently has the same nonrelativistic expansion but with

(22) |

At tree level,

(23) |

when we replace

(24) |

where is the quark mass. This redefinition removes the rest mass from all energies. Expanding in powers of , we then obtain the tree-level nonrelativistic expansion of the modified ASQTAD action:

(25) |

where

(26) | ||||

(27) |

The coefficients of all terms through order have no errors at tree level. In particular the term is correct at tree level once is tuned (nonperturbatively). The coefficients of the remaining terms shown are off by less than 6% and 1% for charm quarks on current lattices (), which should lead to errors of order % or less in charmonium analyses and less than 0.5% for physics. (Errors should actually be considerably smaller than this because of dimensionless factors like the in the term.) Other tree-level errors, which are order and , should also be less than 1% for fm.

## Iii Symanzik Improvement at One-Loop Order

One-loop errors, which are , should be no more than a percent or so for light quarks and current lattice spacings. Taste-changing errors, however, have generally been much larger than expected. These errors also bear directly on the validity of the -root trick for accounting for taste in the vacuum polarization. For this reason it is worth trying to further suppress taste-exchange errors beyond what has been accomplished at tree level. We show how to do this in this section.

The only other one-loop effects that are important at the level of a few percent are for heavy quarks, and, in particular, the charm quark. We show how to correct for these errors in the Section III.2.

### iii.1 Light Quarks

One-loop taste exchange comes only from the diagrams shown in Fig. 3; other one-loop diagrams vanish because of the tree-level improvements. Again the gluons in these diagrams transfer momenta of order , and so are highly virtual. Consequently, the correction terms in the corrected action that cancel these will involve current-current interactions that conserve overall taste (because of momentum conservation). There are 28 such terms in the massless limit for the staggered-quark action (see Appendix F):

(28) | |||

where sums over indices are implicit and . The currents in the first 14 operators are color-singlets with taste and spinor structure , while the last 14 operators are the same but with color octet currents. The precise definitions of these currents are given in Appendix E.

Unimproved Gluons | Improved Gluons | |||
---|---|---|---|---|

ASQTAD | HISQ | HYP | ASQTAD | |

Octet: | ||||

1.41 | 0.19 | 0.02 | 0.77 | |

0.46 | 0.00 | 0.08 | 0.25 | |

0.23 | 0.01 | 0.00 | 0.16 | |

0.38 | 0.02 | 0.01 | 0.27 | |

0.34 | 0.08 | 0.04 | 0.19 | |

0.35 | 0.03 | 0.02 | 0.19 | |

0.08 | 0.00 | 0.01 | 0.05 | |

0.20 | 0.00 | 0.01 | 0.13 | |

0.20 | 0.01 | 0.02 | 0.11 | |

0.31 | 0.01 | 0.01 | 0.17 | |

0.06 | 0.00 | 0.00 | 0.04 | |

0.07 | 0.00 | 0.01 | 0.04 | |

0.17 | 0.00 | 0.01 | 0.09 | |

0.06 | 0.00 | 0.00 | 0.04 | |

Singlet: | ||||

0.76 | 0.10 | 0.01 | 0.41 | |

0.12 | 0.00 | 0.00 | 0.09 | |

0.18 | 0.05 | 0.02 | 0.10 | |

0.04 | 0.00 | 0.00 | 0.03 | |

0.11 | 0.01 | 0.01 | 0.06 | |

0.03 | 0.00 | 0.00 | 0.02 | |

0.09 | 0.00 | 0.00 | 0.05 | |

Avg , | 0.23 | 0.02 | 0.02 | 0.13 |

0.023(1) | 0.011(1) | 0.009(1) |

We have computed the coupling constants and for the tree-level improved theory; the results are presented in Table 2 (see also Appendix F). Armed with these one could incorporate into simulations, removing all one-loop taste exchange. This procedure is complicated to implement, however. A simpler procedure is suggested by the results in hyp-paper . That analysis shows that repeated smearing of the links further reduces the mass splittings between pions of different taste.

This result is not surprising given the perturbative origins of taste exchange. The dominant taste-exchange interactions (in ASQTAD) come from the one-loop diagrams in Fig. 3. The largest contributions to these diagrams come from large loop momenta, . Smearing the links introduces form factors that suppress high momenta, leaving low momenta unchanged, and therefore should suppress this sort of one-loop correction. The links in the ASQTAD action are smeared, but the operator , which smears links in the direction, introduces additional links in orthogonal directions (to preserve gauge invariance) that are not smeared. Smearing the links multiple times guarantees that all links are smeared, and high momenta are suppressed.

There are three problems associated with multiple smearing. One is that it introduces new discretization errors, and these errors grow with additional smearing. The action presented in hyp-paper suffers from this problem. The errors can be avoided by using an -improved smearing operator, such as (Eq. (14)). Replacing smearing operator

(29) |

in the quark action, for example, smears all links but does not introduce new tree-level errors.

The second problem with multiple smearings is that they replace a single link in the naive action by a sum of a very large number of products of links. This explosion in the number of terms does not affect single-gluon vertices on quark lines, by design, but leads to a -growth in the size of two-gluon vertices where is the number of terms in the sum. The -growth enhances one-loop diagrams with two-gluon vertices, canceling much of the benefit obtained from smearing. This problem is remedied by reunitarizing the smeared link operator: that is we replace

(30) |

where operator unitarizes whatever it acts on. A smeared link that has been unitarized is bounded (by unity) and so cannot suffer from the problem. Although we verified that it is unnecessary to make the unitarized link into an matrix — simple unitarization is adequate — the results given in this paper use links that are projected back onto . Reunitarizing has no effect on the single-gluon vertex, and therefore does not introduce new errors.

The doubly smeared operator is simplified if we rearrange it as follows

(31) |

where the entire correction for errors is moved to the outermost smearing. Our new “HISQ” discretization of the quark action is therefore

(32) |

where

(33) |

and now, in the first difference operator,

(34) |

while in the second,

(35) |

The third problem with smearing is that previous work has only demonstrated its impact on the mass splittings between pions with different tastes. For small masses, there are only five degenerate multiplets of pion. Consequently showing that smearing reduces the splittings between the pion multiplets does not guarantee that all 28 taste-exchange terms in the quark action (Eq. (28)) are suppressed; and, therefore, it does not guarantee that taste-exchange effects are generally suppressed (other than in the pion mass splittings).

To examine this issue, we recomputed the one-loop coefficients for all of the leading taste-exchange correction terms in the action (Eq. (28)) using the HISQ action, as well as the HYP action from hyp-paper . The results are derived in Appendix F and presented in Table 2. The HISQ coefficients are on average an order of magnitude smaller than the ASQTAD coefficients. Consequently the residual taste-exchange one-loop corrections in the HISQ action are probably smaller than two-loop corrections and possibly also dimension-8 taste-exchange operators at current lattice spacings. The HYP action gives coefficients similar in size to HISQ.

### iii.2 Quarks

Many new taste-preserving operators would need to be added to the HISQ action in order to remove all order errors. The leading operators are listed in Table 3. None of these is relevant at the few percent level for light quarks, but terms that enter in relative order might change answers by as much as 5–10% for quarks on current lattices, where and . High-precision work requires that these errors be removed.

Nonrelativistic expansions of the various operators in Table 3 show that only the first, the Naik term, can cause errors in order . All of the others result in harmless renormalizations or are suppressed by additional powers of , and so contribute only at the level of 2–3% or less for s and less than % for s. We can remove all errors by including radiative corrections in the parameter that multiplies the Naik term (Eq. (19)).

There are two ways to compute the radiative correction to . One is nonperturbative: is adjusted until the relativistic dispersion relation,

(36) |

computed in a meson simulation is valid for all low three-momenta .

A simpler procedure is to compute the one-loop correction to using perturbation theory — by requiring, for example, the correct dispersion relation for a quark in one-loop order. The result would have the form

(37) |

where depends upon . As we will show, turns out to be negligibly small for the HISQ action (but not for ASQTAD).

Operator | Physics | Physics |

## Iv Applications: Bounding Taste Exchange

We tested our perturbative analysis of the suppression of taste exchange by computing the pion mass taste splittings for ASQTAD, HISQ, and a few other variations quentin-thesis . Some of these results are shown in Table 2 where we list the mass splitting between the 3-link pions and the 0-link (Goldstone) pion from a quenched simulation with unimproved (Wilson) glue at fm and quark mass ukqcd-w . As expected, small mass splittings are correlated with small values for the coefficients of the one-loop taste-exchange interactions (Eq. (28)). These results confirm: a) that taste-exchange interactions are perturbative in origin; b) that these interactions will be suppressed by a factor of order in the HISQ action relative to the ASQTAD action at current lattice spacings; and c) the HISQ action will be more accurate than other current competitors because it has negligible one-loop taste exchange and no tree-level errors.

We have also compared meson splittings for ASQTAD and HISQ valence quarks in full QCD simulations with light-quark (ASQTAD) vacuum polarization. We find that the pseudoscalar splittings are 3.6(5) times smaller with HISQ than with ASQTAD on lattices with fm, when both valence quarks are quarks. The splittings are 3.1(7) times smaller on our fm lattices.

Any taste-exchange effect will be 3–4 times smaller in HISQ, and so measuring the ASQTAD–HISQ difference in any quantity allows us to bound taste-exchange errors in each formalism. To illustrate how such comparisons are done, we have computed the mass of the meson and the pseudovector decay constant of the meson using ASQTAD and HISQ valence quarks for each of our two lattice spacings.

The is unstable and so its mass in a simulation is sensitive to
the volume of the lattice (it is *not* “gold-plated” in the
sense of ratio-paper ). We have not corrected for this volume
dependence, but have simulated using a fixed physical volume.
Consequently results for the mass should extrapolate to the same value
as , but this value may be a few tens of MeV larger than the
physical mass. The is an easy-to-analyze substitute for
a ; it is a meson, but without
valence-quark annihilation, and therefore hypothetical. We tuned the
mass so that the masses in each case were all identical
(696 MeV) to within MeV (statistical/fitting errors only). We
evaluated the decay constant by computing the matrix element
of the pseudoscalar density and multiplying by the bare quark mass
(see fk-fpi , for example).

Our results, shown in Table 4, suggest significant taste-exchange errors for ASQTAD, particularly in the fm simulation. Looking first at the decay constant , the ASQTAD result on the coarse lattice is 6(3)% larger than the ASQTAD result on the fine lattice, where taste-exchange effects should be at least a factor of two smaller. The uncertainty in the discrepancy is large here because of uncertainties in the values of the lattice spacings. Far more compelling is the 4.6(6)% discrepancy between ASQTAD and HISQ on the coarse lattice, which can be measured far more accurately than differences between simulations with different lattice spacings. This discrepancy is reduced to 1.9(4)% on the fine lattice, as expected since is half as large. Since errors in HISQ should be 3–4 times smaller than in ASQTAD, these discrepancies imply that taste-exchange errors in the decay constant could be as large as 5–6% for ASQTAD but only 1–2% for HISQ when fm. The errors are half that size when fm. Results from both formalisms are consistent with an extrapolated value of 180 (4) MeV.

Results for the mass are very similar. These give a final mass of 1.05(2) GeV, which is 30(20) MeV above the mass 1.019 GeV from experiment (and consistent with expectations for finite-volume effects). These analyses taken together indicate that HISQ is delivering 1–2% precision already on the coarse lattice, while a significantly smaller lattice spacing (and much more costly simulation) is needed to achieve similar precision from ASQTAD. HISQ errors should be less than 1% on the fm lattice for light quarks.

Our tests are only partial because ASQTAD quarks were used for the vacuum polarization in both the HISQ and ASQTAD analyses, but the effects of changes in the vacuum polarization are typically 3–5 times smaller than the effects caused by the same changes in the valence quarks. The agreement between our HISQ results from different lattice spacings indicates that finite- errors from the vacuum polarization are not large for these lattice spacings.

(fm) | ASQTAD | HISQ | ASQTADHISQ | |
---|---|---|---|---|

(MeV) | 1/8 | 196(4) | 187(4) | 1.046 (6) |

1/11 | 185(3) | 182(3) | 1.019 (4) | |

(GeV) | 1/8 | 1.119(24) | 1.076(23) | 1.040(9) |

1/11 | 1.057(18) | 1.052(16) | 1.005(6) |

In past work, taste-exchange errors have been estimated by comparing results from different lattice spacings, relying upon the fact that these errors vanish as . This example illustrates how the same errors can be reliably estimated by comparing ASQTAD with HISQ results from simulations using only a single lattice spacing. This approach to estimating taste-exchange errors is very efficient. It is almost certainly the simplest way to quantify uncertainties about taste exchange and the validity of the -root trick for vacuum polarization.

## V Applications: Quarks and Charmonium

As argued in Sections II.3 and III.2, we expect the ASQTAD action and, especially, the HISQ action to work well for quarks even though is typically of order 0.4 or larger on current lattices. To achieve high precision (few percent or better), we must tune the Naik term’s renormalization parameter (Eq. (19)). Here we do this nonperturbatively by computing the speed of light squared, (Eq. (18)), for the in simulations with various values of and tuning until . Our results are summarized in Table 5.

ASQTAD ( fm): | 0 | 0.38 | 2 | 0.962 ( | 9) |

0.3 | 0.38 | 2 | 1.020 ( | 11) | |

0.4 | 0.38 | 2 | 1.036 ( | 11) | |

HISQ ( fm): | 0 | 0.43 | 2 | 1.029 ( | 11) |

0.43 | 1 | 0.985 ( | 16) | ||

0.43 | 2 | 0.992 ( | 13) | ||

0.43 | 3 | 1.014 ( | 15) | ||

0.43 | 4 | 0.991 ( | 11) | ||

HISQ ( fm): | 0.67 | 1 | 1.190 ( | 20) | |

0.67 | 1 | 0.560 ( | 10) | ||

0.67 | 1 | 0.904 ( | 15) | ||

0.67 | 1 | 0.950 ( | 15) | ||

0.66 | 1 | 1.008 ( | 13) | ||

0.66 | 2 | 1.017 ( | 10) | ||

0.66 | 3 | 1.019 ( | 12) | ||

0.66 | 4 | 1.007 ( | 7) |

The ASQTAD results on the fine lattices show only small errors in even with . Values for different s are plotted in Fig. 4, together with an interpolating curve. These data indicate that the optimal choice is for this lattice spacing and mass. The tree-level prediction for , from Eq. (20), is , which indicates that radiative corrections in are of order , as expected.

The HISQ results show even smaller errors on the fine lattices. Tuning to , the tree-level value given by Eq. (20) for , removes all errors in at the level of 1%. This suggests that one-loop and higher-order radiative corrections in are negligible for HISQ compared with the tree-level corrections. The dominance of tree-level contributions is confirmed by our HISQ analysis using the coarser lattice spacing, where . Here, we found the optimal value again by tuning until for a low-momentum . (We overshot slightly and did simulations for rather than .) Our tuned compares quite well with the tree-level prediction of from Eq. (20). Consequently it is quite likely that the tree-level formula is sufficiently accurate for most practical applications today.

Note that setting cancels out the Naik term completely. Tree-level errors are then order rather than order , as in the HISQ action. Table 5 shows that these errors cause to be off by almost a factor of 2. This example underscores the importance of using -improved actions in high-precision work.

On the coarse lattice, we use the with , the smallest nonzero momentum on our lattice, to tune . It is important to verify that the tuned action gives the correct dispersion relation for other momenta as well. The data in Fig. 5 demonstrate that errors are less than a couple percent for meson momenta out to 1 GeV even on the coarse lattice where . Indeed, the errors would probably have been smaller (%) had we tuned a little more accurately.

We examined the spectrum of the meson family to further test the precision of our formalism. We used the fm lattices where , and therefore we expect errors of order 1–2% of the binding energy, or 5–10 MeV (see Section III.2). A particularly sensitive test is the hyperfine mass splitting between the and . We tuned the bare mass until the mass of the local (and lightest) in our simulation agreed with experiment. Since there are no other free parameters in our action, the simulation then predicts a mass for the . As is clear from Fig. 6, the mass in the simulation is quite accurate — certainly within the 5–10 MeV we expected.

The data shown in this last figure also bound taste-exchange errors. As shown there, different tastes of the have different masses, because of taste-exchange interactions. The maximum spread in the masses for the HISQ action, however, is only 9 MeV. This is much smaller than the spread from the ASQTAD action, where the splitting between, for example, the 0-link and 1-temporal-link pion masses is 40 MeV (compared to only 3 MeV for the same splitting with HISQ). (Uncorrected staggered quarks also show very large splittings japan-psi .) Note also that the spread in the s is three times smaller than the spread in the s. This is typical; the masses of mesons other than pseudoscalars are much less sensitive to taste-exchange effects.

The hyperfine splitting, for 0-link mesons in each case, is 109(3) MeV in our fm simulation with tuned (). The 3% uncertainty is almost entirely due to tuning uncertainties in the lattice spacings since these uncertainties enter twice: once for converting the splitting from lattice to physical units, and once through uncertainties in the mass, which are themselves controlled by uncertainties in nrqcd . Our HISQ result is somewhat smaller than the current experimental result of 117(1) MeV etac-data , but it needs three further corrections. (It is also worth noting that the current Particle Data Group average of 117 MeV is somewhat larger than the most recent experiments which find values in the range 113–115 MeV with uncertainties of 1–2 MeV; see etac-data .)

The first correction comes from the operators in Table 3 that must be added to the HISQ lagrangian in order to remove further discretization errors. Of these the most important for the hyperfine splitting is

(38) |

where . This will affect the hyperfine splitting in relative order , or at the level of MeV. The coefficient is readily computed in perturbation theory and this calculation is underway.

The second correction comes from residual taste-exchange interactions,
which from the data in Fig. 6 could be of order a few MeV.
Both this error and that corrected by are
approximately proportional to . So we can estimate them (together)
by comparing to a calculation with a different lattice spacing. We
repeated our hyperfine splitting analysis on the fm lattice
using the tuned (). The errors should be a
little more than twice as large on the coarser lattice since is
almost exactly twice as large. We obtained a splitting of 110(3) MeV
on the coarse lattice, which is essentially identical to
the 109(3) MeV we obtained on the fine lattice. Combining these
results, together with our *a priori* expectation of MeV
errors from corrections, we obtain an corrected hyperfine
splitting is 109(5) MeV.

The third correction is due to the fact that our simulation does not include effects from the annihilation of the valence quarks into two or more gluons. Such annihilations are responsible for small shifts in the and masses, as well as for the (non-electromagnetic) hadronic decay rate of each meson. The dominant contribution comes from and affects only the . The shift in the energy is proportional to the perturbative amplitude for at threshold bodwin-braaten-gpl and therefore positronium-result

(39) | ||||

This result implies that the leading correction to the mass due to annihilation can be computed from the hadronic width:

(40) | ||||

where we use experimental results from etac-data . This correction increases our theoretical value for the – splitting to 111(5) MeV, which agrees well with experiment. The sea-quark masses in our simulations are not quite correct, but both theoretical expectations and experience with previous simulations indicate that this has negligible effect on this hyperfine splitting.

This level of precision would be impossible using our ASQTAD results because the taste-exchange errors are tens of MeV — much larger than the MeV expected from other errors. This example confirms that taste-exchange errors are likely the dominant source of errors in the ASQTAD formalism. With HISQ, on the other hand, taste-exchange errors have been suppressed to a level commensurate with other errors.

Finally we also computed the masses of some radially and orbitally excited states in the family using HISQ, although not as accurately as the ground-state masses. High-precision determinations of excited-state masses require careful design of the meson sources and sinks used in the simulation. Here we did not attempt high precision, but rather used simple local sources to do a quick check on the spectrum. Our results, shown in Fig. 7, agree well with experiment to within our statistical errors.

## Vi Simulation Techniques for Highly-Smeared Operators

It would be highly desirable to create new unquenched gluon configurations using the HISQ action in place of the ASQTAD action. The use of such configurations, with lattice spacings of order fm, would significantly reduce any residual worries about errors due to taste-exchange interactions.

Such simulations are complicated, however, by the heavily smeared, reunitarized links in the action. The additional smearing and reunitarization have no effect on the cost of the quark-matrix inversions required when updating gluon configurations, but they complicate calculation of the derivative of the quark action with respect to an individual link operator . Derivative is defined by

(41) |

for any function of . We developed and tested both an analytic and a stochastic version of the derivative for this action Follana:2003fe .

The analytic version employs a unitary projection

(42) |

to reunitarize each smeared link . The main obstacle is then computation of the gauge derivative of the inverse square root. Using the product-rule identity for the matrix ,

(43) |

we can solve for directly, both iteratively by the conjugate gradient algorithm, and exactly by first diagonalizing .

Using the chain rule, we combine with standard derivatives of the base action and of the smeared links . We encode derivatives of the action and smeared links generically, allowing for run-time changes independently in either. The additional cost of computing and combining with these is minimal.

Several other analytic approaches have been developed for unitarized smearings Kamleh:2004xk ; Morningstar:2003gk . The authors of Kamleh:2004xk addressed the problem of computing the derivative of the matrix inverse square root by replacing it with a rational approximation. In Morningstar:2003gk , the smearing itself is unitary, explicitly avoiding the need to reunitarize and the inverse square root. In both, the derivatives are then computed straightforwardly.

In the stochastic approach, we define links

(44) |

where is now a field of traceless, hermitian random matrices, each with normalization

(45) |

and is a very small number. Defining to be the terms in containing , the derivative can be computed efficiently from

(46) |

averaged over a finite number of sets of random s. Our numerical experiments suggest that 10–100 sets of s are adequate for actions like HISQ. We will describe both our stochastic and analytic techniques in a later paper.

## Vii Conclusions

In this paper, we have demonstrated that taste-exchange interactions are perturbative and we have shown how to use Symanzik improvement to create a new staggered-quark action (HISQ) that has greatly reduced one-loop taste-exchange errors, no tree-level order errors, and no tree-level order errors to leading order in the quark’s velocity . The HISQ action addresses one of the fundamental issues surrounding staggered-quark simulations by allowing us to estimate taste-exchange interactions through comparisons of HISQ with ASQTAD results. We presented numerical evidence that taste-exchange interactions in HISQ contribute less than 1% to light-quark quantities, like meson masses and decay constants, at lattice spacings as large as 0.1 fm.

The suppression of all order errors by powers of makes HISQ the most accurate discretization of the quark action for simulating quarks on current lattices. We demonstrated this with a new lattice QCD determination of the mass splitting, which agrees well with experiment. This result could be improved by computing the coefficient in the correction (Eq. (38)) to the HISQ action.

Our final HISQ action is defined by Eqs. (32–35). We showed that the tree-level value (Eq. (20)) for the Naik-term parameter is adequate at the level of 1% errors for lattice spacings at least as large as fm.

The HISQ formalism will be most useful for physics and works very well for such mesons. In Fig. 8, for example, we show the – spin splitting from simulations with HISQ quarks using both our fm and fm lattices. As in the charmonium case, theory and experiment agree well here to within errors. Another very sensitive test of our simulations is to compare computed and experimental values for the mass difference , which is quite insensitive to tuning errors in the mass. We obtain 956(14) MeV using HISQ on the fm lattice, and 978(20) MeV on the fm lattice. Both values agree well with the splitting 956(1) MeV from experiment. Finally, we measured the speed of light and found for the on the coarse lattice ( fm with ), confirming that the same coupling constant values work for both and mesons. It is important to appreciate that there are no free parameters available for tuning in the extraction of any of these results; all QCD parameters were tuned using other quantities.

As we will discuss in a later paper, the HISQ formalism is particularly useful for accurate calculations of quantities like , , , and so on. These all require currents and these currents, even though they are conserved or partially conserved, have order renormalizations. Consider, for example, annihilation into a (virtual) photon. The electromagnetic current can be computed by inserting photon link operators,

(47) |

into the HISQ action and then expanding to first order in to obtain

(48) |

Current is not renormalized, because of QED’s gauge symmetry, but it is not the only operator that contributes to annihilation. In addition there are correction terms that contribute,

(49) |

just as for the gluon fields. The first correction term renormalizes the quark’s magnetic moment, the second its charge radius. Such terms are normally negligible since the extra derivatives introduce extra powers of and is small. For quark annihilations, however, the extra derivatives become quark masses instead of small momenta, and so are more important. It is easy to compute coefficients and in perturbation theory, but contributions to annihilation from these operators are indistinguishable from those coming from the leading operator . Consequently we can omit the corrections and, instead, introduce a renormalization constant for :

(50) |

where