# Massive Fermions in Lattice Gauge Theory

###### Abstract

This paper presents a formulation of lattice fermions applicable to all quark masses, large and small. We incorporate interactions from previous light-fermion and heavy-fermion methods, and thus ensure a smooth connection to these limiting cases. The couplings in improved actions are evaluated for arbitrary fermion mass , without expansions around small- or large-mass limits. We treat both the action and external currents. By interpreting on-shell improvement criteria through the lattice theory’s Hamiltonian, one finds that cutoff artifacts factorize into the form , where is a momentum characteristic of the system under study, is related to the dimension of the th interaction, and is a bounded function, numerically always or less. In heavy-quark systems is typically rather smaller than the fermion mass . Therefore, artifacts of order do not arise, even when . An important by-product of our analysis is an interpretation of the Wilson and Sheikholeslami-Wohlert actions applied to nonrelativistic fermions.

FERMILAB-PUB-96/074-T

ILL-TH-96-1

hep-lat/9604004

## 1 Introduction

The most promising avenue for a quantitative understanding of nonperturbative quantum chromodynamics—and other field theories—is via numerical (Monte Carlo) integration of functional integrals defined on a lattice [1]. Like any numerical technique this method has uncertainties that must be understood and controlled before the results are useful. In particular, although the continuum theory is defined by the limit of a sequence of lattice theories, the numerical calculations are never carried out at the limit. Because the Monte Carlo introduces statistical errors, the extrapolation to the continuum limit is imperfect. The results for physical quantities are consequently contaminated by lattice artifacts. For a practical result, this uncertainty must be smaller than, say, relevant experimental uncertainties.

The way to reduce lattice artifacts is based on the renormalization group [2]. One starts with a general action

(1.1) |

where the include all interactions with the desired field content and the appropriate symmetries. One approach to the continuum limit, which might be called brute force, is to choose the in any way that drives the lattice spacing to zero. An ideal approach would be to choose the to lie on a renormalized trajectory [2], where there are no lattice artifacts even though the lattice spacing . In the space of all possible actions specified by eq. (1.1), the renormalized trajectories lie in a subspace, whose dimension equals the number of relevant parameters. Once the relevant parameters have been fixed by physics, they and the renormalization scheme determine all the .

Unfortunately, a renormalized trajectory is mostly of abstract value, because on one infinitely many are nonzero. All practical schemes, such as blocking fields [3] or Symanzik improvement [4] use criteria such as locality [3] or the scaling dimension [4] to truncate the space of actions. (For an asymptotically free theory, such as QCD, these two criteria are not very different.) Furthermore, the calculations of the are, in practice, only approximate. For these reasons an improved action is only partially renormalized. Nevertheless, any practical action can be written

(1.2) |

where denotes (an action on) the renormalized trajectory. Usually the truncations and/or approximations used to generate will also yield estimates for the remaining cutoff effects .

This paper treats massive fermions coupled to non-Abelian gauge fields. The relevant couplings are the fermion masses and the gauge coupling. So the renormalized trajectory takes the form

(1.3) |

where denotes the fermion mass,^{1}^{1}1We use for a quark mass defined by a physical condition
and for the coupling appearing in the action.
is the scale characteristic of the gauge theory, and the
argument labels the renormalization point.
The are gauge-invariant combinations of four-component fermion and
anti-fermion fields ( and ) and the lattice gauge
field ().
For later calculational convenience we choose the bare, rather than some
physical, fermion mass and gauge coupling to
parameterize the couplings .

As the fermion mass is formally smaller than . (By asymptotic freedom as .) It is therefore tempting to expand the couplings in , as in previous analyses [5, 6, 7]. But there may be fermions satisfying ; the charm, bottom, and top quarks are examples in nature. If, in practice, is not small, perturbation theory in need not be useful, even though perturbation theory in might be. Indeed, this regime includes the charm and bottom quarks at currently accessible lattice spacings.

The static [8, 9] and nonrelativistic [10, 11, 12] effective theories address the problems of heavy fermions. Their restriction to implies that couplings of interactions between particle and antiparticle states may be chosen to vanish, and the remaining interactions in eq. (1.1) are organized according to a expansion. But for some the expansion is no longer useful. Furthermore, radiative corrections induce power-law terms, e.g. , which must be canceled by adjusting the . These terms, which diverge as , are a reminder that the effective theories are to be used at scales below (large) . Their presence implies that cutoff effects in the effective theories should be removed not by brute force, but by keeping and constructing actions systematically closer to the renormalized trajectory [11, 12].

This paper presents a way to encompass both the small and large mass
formulations.
The doubling problem is handled by Wilson’s method [13].
In addition to treating the dependence of the couplings exactly,
the central idea is to enlarge the class of interactions considered in
eq. (1.3) to include those from both the
small and the large limits.
In particular, we do not impose a symmetry between couplings of
interactions that would be related by interchanging the time axis with
a spatial axis.
For any , the four-momentum of a quark in most interesting
physical systems satisfies
whenever .
But when the characteristic four-momentum of the physics does
not respect time-space axis interchange.
Under such circumstances it is inconvenient and unnecessary to choose an
axis-interchange symmetric action.^{2}^{2}2Relinquishing axis-interchange symmetry is common in
treatments of heavy quarks with momentum-space and dimensional
regulators.
It is possible to derive deviations from heavy-quark symmetry from
the Dirac action [14, 15], while maintaining time-space axis
interchange invariance as a corollary to Euclidean invariance.
But usually the derivations are easier with a nonrelativistic
action [10, 16], the so-called the heavy-quark effective
theory [17].

In the appropriate limits, our formulation of lattice fermions shares properties of previous ones. On one hand, at dimension five or less, couplings related by axis interchange become identical in the limit : the Wilson action and the improved action of ref. [6] are recovered. But when the mass-dependent renormalization leaves lattice artifacts that are proportional to and , not . At higher dimension, however, we retain Wilson’s time derivative and incorporate “spatial-only” interactions into eq. (1.1).

On the other hand, for one can interpret the lattice theory in a nonrelativistic light. Indeed, all members of our class of actions approach a universal static limit as . For large but finite, the corrections to the static limit can be recovered systematically, provided the fermion mass is defined through the kinetic energy, and provided the general action, eq. (1.3), is truncated only at dimension (or higher). Unlike previous implementations of nonrelativistic fermions, however, our approach crosses smoothly over into the regime of tiny lattice spacings, where even for a heavy quark. Thus, after several have been tuned close to a renormalized trajectory, thereby removing the worst lattice artifacts, a little brute force can remove the rest.

Because we make no assumptions about the ratio of fermion masses to other scales, our formulation is especially well suited to fermions too heavy for small methods yet too light for large methods. With the actions given below one can test whether a given fermion is heavy enough to be treated nonrelativistically, without resorting to brute-force simulations. A practical example might be the charm quark, which has a mass only a few times , yet even on the finest lattices available today is largish, at least .

For a concrete determination of the , one must choose a renormalization group, a criterion for truncating the sum in eq. (1.3), and a strategy for determining the . For illustration we adopt here a Symanzik-like procedure [4], organizing the interactions by dimension. Carrying out this program to arbitrarily high dimension would produce a renormalized trajectory of a renormalization group generated by infinitesimal changes in . For simplicity, however, most of this paper treats interactions only up to dimension five. Although a nonperturbative determination of the couplings is possible in principle, this paper makes the further approximation of perturbation theory in , that is

(1.4) |

Except for sect. 8 we work to tree level, so we often abbreviate by . (Explicit one-loop calculations are in progress [18].) Within these approximations we determine the by insisting that on-shell quantities take their desired values, as first suggested by Lüscher and Weisz [19].

One calculational procedure is to work out -point on-shell Green functions via Feynman diagrams and tune them to the continuum limit, to the appropriate order in . An example is in sect. 4. Because this strategy is limited to a finite number of quantities, it is nontrivial to assume that other quantities are improved too [4]. An alternative is developed in sect. 5. Starting from the transfer matrix we derive an expression for the fermion Hamiltonian, valid (at tree level) for states with . Because the Hamiltonian is an operator, improving it to some accuracy guarantees the improvement to the same accuracy of infinitely many -numbers. We show that by adjusting the couplings correctly, and allowing physically unobservable redefinitions of the fermion field, one can tune the Hamiltonian to the continuum limit, i.e. to the Dirac Hamiltonian, to the appropriate order in .

In addition to equations for the , the analysis of the Hamiltonian yields two important results. One is that the Wilson time derivative needs no improvement. The other is a canonically normalized fermion field that, to the accuracy of the improvement, obeys Dirac’s (continuum) equation of motion. This field is a potent ingredient in calculations of matrix elements involving heavy-quark systems; see sect. 7.

Owing to the approximations introduced—the truncation of interactions and perturbation theory—some cutoff effects remain. If these errors are small, they may be estimated by insertions of in correlation functions. If the series for has been developed to -th order

(1.5) |

A typical term in distorts an on-shell correlation function by an amount of order , where , and . (If is omitted from the action altogether, then here .) The analysis presented below shows that the tree-level, lower-dimension are well-behaved for all masses. We also show that loop diagrams have the same or more benign behavior at large mass. In particular, as the either approach a constant or fall as , for some . These results provide evidence that the higher-dimension are well-behaved too.

In Monte Carlo programs it is customary to parameterize the action by the hopping parameter instead of the mass. In this notation the dimension-three and -four interactions are written

(1.6) |

Some terms of dimension five—to solve the doubling problem [13]—are included here too. The relation between the bare mass and the hopping parameters and is given below in sect. 2. Eq. (1.6) illustrates how our program subsumes properties of the familiar small-mass and large-mass formulations. Imposing axis-interchange invariance would set and , and then reduces to the Wilson action [13]. Rewriting and setting to zero with produces the simplest nonrelativistic action [11].

This pattern continues for dimension-five interactions. Aside from the Wilson terms in , the other dimension-five interactions are the chromomagnetic interaction

(1.7) |

and the chromoelectric interaction

(1.8) |

where and are suitable functions of the lattice gauge field , as in sect. 2. The light-quark formalism of refs. [6, 7] considers the special case , whereas the heavy-quark formalism of refs. [11, 12] sets .

The couplings , , , and are specific examples of the couplings in eq. (1.3). On the renormalized trajectory they are, therefore, all functions of . Sect. 4 shows how to adjust and so that the relativistic energy-momentum relation is obtained for all . With the correct choice, for which when , the (tree-level) lattice artifact is proportional to for and for . Similarly, results in sect. 5 include functions and that reduce lattice artifacts in the quark-gluon vertex functions to for , or (yet smaller) for .

In their on-shell improvement program refs. [19, 6] introduced changes of variables, or isospectral transformations, to expose redundant interactions. Since the coupling of a redundant interaction does not influence physical quantities, one can choose it according to theoretical or computational criteria distinct from improvement. Sect. 3 examines the isospectral transformations when time-space axis-interchange symmetry is not imposed. In our formulation many isospectral transformations are exploited to keep the time discretization, and hence the transfer matrix, as simple as possible.

The remaining redundant directions can be classified in the Hamiltonian
approach developed in sect. 5.
In a Euclidean version of the standard Dirac-matrix basis,
given in sect. 2, matrices are either block diagonal
or block off-diagonal.
Block-diagonal transformations are absorbed into a generalized field
normalization.
On the other hand, block–off-diagonal transformations (called
Foldy-Wouthuysen-Tani transformations [20] in atomic physics)
generate leeway in choosing the mass dependence of associated
couplings.^{3}^{3}3The off-diagonal interactions are precisely the ones usually
omitted from nonrelativistic formulations, yet their presence in our
formulation permits a smooth transition from large to small mass.
For example, in the action one may freely choose , as
long as the choice circumvents the doubling problem.

Our approach breaks down, just as any lattice theory does, when is too large. Fortunately, the typical momenta and mass splittings of hadronic systems usually are bounded; the energy scale around the fermion mass is dynamically unimportant. In quarkonia the typical energy-momentum scales are and , i.e. 200–800 MeV for charmonium and 200–1400 MeV for bottomonium. Similarly, in light and heavy-light hadrons the typical momentum scale is , i.e. 100–300 MeV. In some processes, such as a decaying heavy-quark system that transfers all its energy into light hadrons, a large three-momentum does arise. Then our formulation and its predecessors all require further extensions. One should appreciate, however, that the breakdown arises not from the large fermion mass per se, but from the large momentum of the decay products.

A by-product of our formalism applies to existing numerical calculations, done with axis-interchange invariant actions. For (and, furthermore, ) we derive in sect. 9 a nonrelativistic interpretation of such actions. One then sees that, with a proper definition of the fermion mass, any action described by , including the Wilson and Sheikholeslami-Wohlert fermion actions, has the lattice-spacing and/or relativistic inaccuracies of a typical nonrelativistic action. A practical bonus of the nonrelativistic regime is that it is no longer necessary to adjust differently from . In heavy-light systems, one may also set .

This paper is organized as follows: Sect. 2 introduces some notation, including a form of the action better suited to perturbation theory, and the Dirac-matrix convention used in later sections. The isospectral transformation of ref. [6] is reviewed and generalized in sect. 3, to determine which couplings are redundant. Then, to derive improvement conditions, Feynman-diagram methods are discussed in sect. 4, and the Hamiltonian method in sect. 5. With a Hamiltonian description of the lattice theory in hand, sect. 6 estimates cutoff effects in various hadronic systems. Sect. 7 considers perturbations from the electroweak interactions, needed for the phenomenology of the Standard Model [21]. Some of the issues beyond tree level are outlined in sect. 8. The relationship of our work, for , to nonrelativistic QCD is pursued in sect. 9. We discuss a few phenomenologically relevant applications more thoroughly in sect. 10. Finally, sect. 11 contains selected concluding remarks, and the appendices contain various technical details.

## 2 Notation

We shall call the form of the action in eq. (1.6) the “hopping-parameter form.” For studying the continuum limit and developing perturbation theory it is useful to present a different form. Let us introduce some notation. The lattice spacing is and the site labels are . Rescale the fields:

(2.1) |

and similarly for . The bare mass is

(2.2) |

where is the spacetime dimension, and . With these substitutions the action reads

(2.3) |

where the integral sign abbreviates . The covariant difference operators are conveniently defined via covariant translation operators

(2.4) |

where . Then

(2.5) |

define various covariant difference operators and the three-dimensional discrete Laplacian. We shall call the form of the action in eq. (2.3) the “mass form.”

The temporal kinetic energy in eq. (2.3) is written in a way that does not make the temporal Wilson term explicit. Eq. (2.3) is more convenient, however, for constructing the transfer matrix, and for comparing with nonrelativistic QCD. The spacelike Wilson term, the one proportional to , is needed to prevent doubling. A convenient choice in computer programs is , but we keep it arbitrary, because other choices may have other advantages.

For constructing the transfer matrix and for examining the nonrelativistic limit, a useful representation of the Euclidean gamma matrices is

(2.6) |

satisfying . Another convention that we use is so that . Using eq. (2.6), and , where

(2.7) |

The following split into upper and lower two-component spinors

(2.8) |

follows from eq. (2.6). This convention is chosen so that (the operators corresponding to) and annihilate particle and anti-particle states, respectively. With these two-component fields the mass form of the action is

(2.9) |

This form of the action exhibits explicitly that particles and anti-particles are treated on the same footing. (Anti-particles transform under the complex-conjugate representation of the gauge group, however, so appears instead of in the rules (eqs. (2.4)) for covariant translations acting on .)

Writing and , the four- and two-component mass forms of the chromomagnetic and chromoelectric interactions are

(2.10) |

and

(2.11) |

respectively. Except in a technical step in sect. 5 we take the “four-leaf clover” lattice approximant to the field strength

(2.12) |

as introduced in ref. [22]. In eqs. (2.10) and (2.11), and . As defined here, , , and are anti-Hermitian; similarly we take anti-Hermitian gauge-group generators .

## 3 Redundant Couplings

Before trying to determine the mass dependence of the couplings , , , and , one should establish which combinations are physical. The fields in functional integrals are integration variables, and a change of variables cannot change the integrals. Interactions that are induced by changes of variables are redundant; their couplings can be chosen with some leeway, dictated by calculational or technical convenience, rather than by physical criteria.

A subtle example of a redundancy in the space of interactions is wavefunction (re)normalization, which multiplies the field by a constant. For fermions, for example, it is sometimes convenient to fix the kinetic energy to have coefficient unity, which is the mass form of the action, and sometimes to fix the local term to have coefficient unity, which is the hopping-parameter form. But neither interaction is redundant, even though the normalization convention drops out of physical quantities.

Otherwise redundant directions are exposed by redefinitions of the field. In the analysis of ref. [6], with axis-interchange symmetry, one considers the transformation

(3.1) |

where is chosen so that is “small.” After carrying out the transformation on the target action , one expands the transformed action to . One finds changes in the normalization of the lower-dimension terms and the additional interaction : from the two independent transformation parameters, only one combination survives. Hence, of the dimension-five interactions listed in Table 1,

dim | w/ a.i.s. | w/o a.i.s. | |
---|---|---|---|

3 | |||

4 | |||

5 | |||

one (i.e. ) is redundant, and the other is not.

On the lattice the nearest-neighbor discretization of suffers from the doubling problem. Wilson’s prescription adds a nearest-neighbor discretization of to eliminate the unwanted states. By the preceding analysis [6], using instead would not change the spectrum at . When the discretization is chosen to solve the doubling problem, however, the interaction does change the spectrum of high-momentum states. Since they communicate with the low-momentum states through virtual processes, lattice artifacts proportional to remain. They can be eliminated with the other dimension-five interaction, , with a coupling proportional to .

Thus, with axis-interchange symmetry there are four interactions up to dimension five, one of which goes with wavefunction normalization (e.g. ). One coupling is redundant, and it can be chosen to solve species doubling (). The other couplings are fixed by the fermion mass () and a physical improvement condition ().

When axis-interchange symmetry is given up, the transformation in eq. (3.1) should be generalized to

(3.2) |

Applying this transformation to the target action induces the dimension-five interactions listed in Table 1. From the four independent transformation parameters, only three combinations survive: , , and . Therefore, the coefficients of , , and can be chosen arbitrarily. The last of these has no redeeming features, so should be chosen so that it never appears.

The other two redundant interactions are again used to solve the doubling problem. The term is used to eliminate states that would make contributions to the fermion propagator proportional to ; the factors in eqs. (1.6) or (2.3) provide the unique choice. Low-energy states with are lifted by adding the interaction proportional to in eqs. (1.6) or (2.3). When the mass is nonzero, it may prove convenient choose to be a function of , so we leave it arbitrary.

As with axis-interchange symmetry, the chromomagnetic and chromoelectric interactions are not redundant. Their couplings can be used to remove cutoff effects once the doubling problem has been eliminated.

Thus, without axis-interchange symmetry there are eight interactions up to dimension five, one of which goes with wavefunction normalization (e.g. ). Three couplings are redundant; two can be used to solve species doubling ( and ), and the other to eliminate . The other couplings are fixed by the fermion mass () and three physical improvement conditions (, , and ).

Redundant combinations of higher-dimension interactions can be exposed by generalizing the transformation of eq. (3.2). In particular, after dispensing with axis-interchange symmetry, it is possible to transform away interactions with higher time derivatives of and , in favor of spatial derivatives of and , and , and time derivatives of the latter. Indeed, any action with the Wilson time difference—the first line of eq. (2.3)—has an easy-to-construct transfer matrix. (This is reviewed in sect. 5.) Consequently, it is possible to implement eq. (1.3) by adding “spatial-only” interactions to .

## 4 On-shell Correlation Functions

We now turn to the mass dependence of the tree-level couplings, generically denoted in eq. (1.4), needed to bring the action closer to the renormalized trajectory. This section uses the fermion propagator to obtain the relation between the physical mass and the coupling , the correct tuning of the coupling , and the normalization of the field . Since we are interested in the full mass dependence, we do not expand in the fermion mass. Sect. 5 uses the Hamiltonian of the lattice theory to clarify and extend the analysis to and .

A well-known procedure for determining the couplings [4] is to calculate -point correlation functions and expand in momentum . In gauge theories, however, it is not known whether lattice artifacts can be removed systematically from Green functions off mass shell. Hence, one expands “on-shell” quantities instead [19]. The (lattice) mass shell specifies the energy at given spatial momentum , so on-shell improvement amounts to an expansion in . Previous analyses [5, 6, 7] also expanded in the coupling . We simply avoid the latter expansion, and thus obtain the full mass dependence.

The simplest on-shell correlation function is the fermion propagator as a function of time and spatial momentum. It is used to relate the bare mass to a physical mass and to derive the mass dependence of . In the language of sect. 3, it probes the interactions and , relative to .

Define through

(4.1) |

where

(4.2) |

where and , but for brevity eq. (4.2) is given in lattice units. To integrate over , proceed as follows: rationalize the denominator; for let , and for let , yielding a contour integral over the circle ; apply the residue theorem to obtain

(4.3) |

for ,^{4}^{4}4To obtain from eq. (4.3),
replace by 1 on the right-hand side.
where (restoring )

(4.4) |

implicitly defines the energy of a state with momentum . The residue is given below in eq. (4.12).

Expanding the energy-momentum relation in powers of yields

(4.5) |

where the “rest mass”

(4.6) |

and the “kinetic mass”

(4.7) |

(Any axis will do to define , by spatial axis-interchange symmetry.) The relativistic mass shell has , and it terminates at . From the tree-level eq. (4.4)

(4.8) |

and

(4.9) |

Eq. (4.8) shows how to adjust so that .
Similarly, eq. (4.9) shows how to adjust and so
that .^{5}^{5}5In the - parametrization
(eliminate with eq. (2.2)) this condition
is an implicit transcendental equation. In the -
parametrization one can solve for explicitly.
Setting and solving for yields the (tree-level)
condition (setting again)

(4.10) |

The dimension-five coupling is treated here as redundant; it is determined not by physics, but to solve the doubling problem. To alleviate doubling any will do, and the most natural choice is .

For small mass the Taylor expansion of eq. (4.10) is

(4.11) |

At the redundant coupling drops out, leaving unambiguously. On the other hand, the full mass dependence of can only be prescribed hand-in-hand with . The origin of the link between the two couplings is that both the kinetic () and Wilson () terms contribute to at . This and analogous links between couplings’ mass dependence are examined further in sect. 5 and Appendix A.

Beyond tree level (in perturbation theory or in Monte Carlo calculations) one would tune according to the same physical principle that led to eq. (4.10): determine the momentum dependence of the energy of a suitable state and demand that and be equal.

When and have been adjusted so that , one can rewrite eq. (4.5) as . Expanding eq. (4.4) to , one finds the lattice artifact at small mass and at large mass. To reduce further, one must incorporate higher-dimension interactions into the analysis.

Finally, let us return to the residue in eq. (4.3). In general, the residue is a scalar function of four-momentum , evaluated on shell. With a Euclidean invariant cutoff, scalar functions can depend only on ; on shell, with , the spatial momentum drops out. With the lattice cutoff, however, the mass shell is distorted, cf. eq. (4.4), so three-momentum dependence can remain. Indeed, after integrating eq. (4.2) over one finds

(4.12) |

Normally one identifies the residue with a (re)normalization of the fermion field. Now, however, it is appropriate to expand , where

(4.13) |

Then has the canonical normalization. In the hopping-parameter notation the canonically normalized field is . This notation shows clearly that the approach to the static limit, , smooth. Indeed, eq. (4.13) captures the dominant mass dependence of the field normalization to all orders in perturbation theory, cf. sect. 8 and ref. [18].

One might ask what to make of the momentum dependence of , when the action is improved to higher dimensions. The residue itself is not observable; physical quantities are given by ratios of -point functions and the propagator. With the correct on-shell improvement, the dependence of untruncated -point functions combines with that of to yield the desired results (to the order considered).

## 5 The Hamiltonian

This section introduces another method for deriving conditions on the couplings in the action. The strategy is to obtain an expansion in the lattice spacing for the Hamiltonian. For concreteness, we focus on the action . The couplings are then adjusted so that the Hamiltonian of the lattice theory is equivalent to the Dirac Hamiltonian. The idea is conceptually the same as on-shell improvement, because the “spectral quantities” of ref. [19] are just eigenvalues of the Hamiltonian. But since the Hamiltonian is an operator, it contains the information of infinitely many quantities, rather than the finite number accessible when one computes correlation functions.

This approach reproduces the derived with on-shell correlation functions. But the analysis is explicitly relativistic, if noncovariant, so one sees clearly that the results are general. On the other hand, we have not attempted to extend the method to four-fermion operators, or to higher orders in . The calculations required by those extensions seem simpler with Feynman diagrams.

There is a further conceptual advantage to the Hamiltonian. Lattice field theories are almost always formulated in imaginary time. The interpretation of the results in real time hinges on a good Hamiltonian fixing the dynamics of the Hilbert space of states [2]. Hence the implicit, but seldom stated, goal of improvement is an improved Hamiltonian; this section merely takes direct aim on that goal. Moreover, once one accepts the central role of the Hamiltonian, one appreciates why a satisfactory Hamiltonian implies a satisfactory time evolution , no matter how large is.

In lattice field theory the Hamiltonian is defined through the time evolution operator, or “transfer matrix” [2]. Therefore, sect. 5.1 starts by reviewing and extending the construction of ref. [24] to the actions and . A by-product of this analysis is the demonstration that there is no need to improve the temporal derivative in eq. (2.3). This feature is familiar from the static and nonrelativistic formulations. It is a special blessing here, because a temporal next-nearest-neighbor interaction would introduce unphysical states [6], and at large the physical and unphysical levels cross. With the transfer matrix in hand, sects. 5.2 and 5.3 develop an expansion in for the Hamiltonian itself.

### 5.1 Construction of the transfer matrix

The transfer-matrix construction with two hopping parameters differs little from the usual case [24]. The transfer matrix acts as an integral operator in the space of gauge fields; in the axial gauge a wave functional at time evolves to

(5.1) |

at time . The wave functional is also a vector in the fermion Hilbert space. For the standard gauge action the kernel may be written

(5.2) |

The factors arising from the fermion action
are operators in the fermion Hilbert space.
The factors arising from the gauge field, and , are given
in ref. [24]; in the following, they do not play a crucial role,
so we do not discuss them further.^{6}^{6}6Different from ref. [24] is the convention for the
factors in the action (compare eq. (1.6)
with eq. (2) of ref. [24]).
With our convention it is natural for time-ordering to place later
times to the left.
Thus, the kernel transfers the field from at time
to at time .

The fermion operator for action can be written

(5.3) |

where (cf. ref. [24])

(5.4) |

(5.5) |

in a matrix notation in which and are vectors and , , and are matrices depending on gauge field . The vectors and matrices of this notation are labeled by spin, color, and space. The covariant difference operator is as in eq. (2.5) and

(5.6) |

(5.7) |

The operators and obey canonical anti-commutation relations

(5.8) |

where and label spatial sites and and are multi-indices for spin and color. The fields corresponding to these operators are related to the original fields by

(5.9) |

This discrepancy in normalization between the integration variables in the functional integral and the canonical operators in Hilbert space demonstrates again that the normalization convention for the field , cf. eq. (2.1), is arbitrary. On the other hand, the propagator of has unit residue at tree level, and a perturbative series beyond tree level.

The generalization of eqs. (5.2)–(5.9) to include the chromoelectric interactions suffers from a technical difficulty. Usually one uses the “four-leaf clovers” in eq. (2.12) as the lattice approximants to the chromomagnetic and chromoelectric fields. For the chromomagnetic interaction, this choice poses no problem, because involves link variables from one timeslice only. For the chromoelectric interaction, however, the time-space four-leaf clover involves link variables from three timeslices. In that case, the construction of the gauge-field transfer matrix is more complicated, and, if the improved gauge action is any indication, it may no longer be positive [25].

To avoid this complication one can define a chromoelectric field on only two timeslices. Consider

(5.10) |

where

(5.11) |

is defined on a two-leaf clover. The projection operators in eq. (5.10) are chosen by analogy with the Wilson time derivative, cf. eq. (2.3), and as a result the standard transfer-matrix construction goes through with minor modifications. The two-leaf version differs from the four-leaf version by an interaction of dimension six, so it should not alter the tree-level tuning of .