A Lattice Formulation of Chiral Gauge Theories

TL;DR

This paper introduces a lattice formulation for chiral gauge theories that employs a Wilson mass and a double limit process to maintain gauge invariance, applicable to anomaly-free fermion representations.

Contribution

It proposes a novel lattice approach for chiral gauge theories using a Wilson mass and a double limit to restore gauge invariance, addressing longstanding issues in lattice chiral fermion formulations.

Findings

Method effectively removes fermion doublers.

Restores gauge invariance in anomaly-free theories.

Provides a technique for computing fermion operator matrix elements.

Abstract

We present a method for formulating gauge theories of chiral fermions in lattice field theory. The method makes use of a Wilson mass to remove doublers. Gauge invariance is then restored by modifying the theory in two ways: the magnitude of the fermion determinant is replaced with the square root of the determinant for a fermion with vector-like couplings to the gauge field; a double limit is taken, in which the lattice spacing associated with the fermion field is sent to zero before the lattice spacing associated with the gauge field. The method applies only to theories whose fermions are in an anomaly-free representation of the gauge group. We also present a related technique for computing matrix elements of operators involving fermion fields. Although the analyses of these methods are couched in weak-coupling perturbation theory, it is argued that the computational prescriptions are gauge invariant in the presence of a nonperturbative gauge-field configuration.

Equations

$$S_G=\frac{1}{2g^2}\sum _x\sum _{\mu \ne \nu }\mathrm{Tr}{U}_{\mu }(x)U_{\nu }(x+a_{\mu })U_{\mu }^{†}(x+a_{\nu })U_{\nu }^{†}(x)+h.c.,$$

$$U_{\mu }(x)\equiv \exp \left[iag{A}_{\mu }(x+a_{\mu }/2)\right],$$

$$U_{\mu }^{†}(x)\equiv \exp \left[-iag{A}_{\mu }(x+a_{\mu }/2)\right]$$

$$S_N=a^d\sum _{x,\mu }\overline{\psi }(x){\gamma }_{\mu }\frac{1}{2a}[\psi (x+a_{\mu })-\psi (x-a_{\mu })],$$

$$S_{NI}=a^d\sum _{x,\mu }\overline{\psi }(x){\gamma }_{\mu }{P}_L\frac{1}{2a}\{[U_{\mu }(x)-1]\psi (x+a_{\mu })-[U_{\mu }^{†}(x-a_{\mu })-1]\psi (x-a_{\mu })\},$$

$$i{S}_F^N(p)=[(1/a)\sum _{\mu }i{\gamma }_{\mu }\sin (p_{\mu }a){]}^{-1},$$

$${\mathcal{V}}_{\mu ,a}^{(1)N}(p,l)=T_a{V}_{\mu }^{(1)N}(p,l)P_L=-ig{T}_a{\gamma }_{\mu }{P}_L\cos [(p_{\mu }+\frac{1}{/}2l_{\mu })a],$$

$${\mathcal{V}}_{\mu \nu ,ab}^{(2)N}(p,l_1,l_2)=T_a{T}_b{V}_{\mu \nu }^{(2)N}(p,l_1,l_2)P_L=ia{g}^2{T}_a{T}_b{\delta }_{\mu \nu }{\gamma }_{\mu }{P}_L\sin [(p_{\mu }+\frac{1}{/}2l_{1\mu }+\frac{1}{/}2l_{2\mu })a],$$

$$V_{{\mu }_1\dots {\mu }_{n+1}}^{(n+1)}$$

$$d_{\mu }=(2/a)\sin \frac{1}{/}2p_{\mu }a.$$

$$S_W=a^d\sum _{x,\mu }\overline{\psi }(x)\frac{1}{2a}[2\psi (x)-\psi (x+a_{\mu })-\psi (x-a_{\mu })].$$

$$S_{WI}=a^d\sum _{x,\mu }\overline{\psi }(x)\frac{1}{2a}\{[1-U_{\mu }(x)]\psi (x+a_{\mu })+[1-U_{\mu }^{†}(x-a_{\mu })]\psi (x-a_{\mu })\}.$$

$$i{S}_F^W(p)=\{(1/a)\sum _{\mu }i{\gamma }_{\mu }\sin (p_{\mu }a)+M(p){\}}^{-1},$$

$$M(p)=(1/a)\sum _{\mu }[1-\cos (p_{\mu }a)].$$

$${\mathcal{V}}_{\mu ,a}^{(1)W}(p,l)=T_a{V}_{\mu }^{(1)W}(p,l)=-g{T}_a\sin [(p_{\mu }+\frac{1}{/}2l_{\mu })a],$$

$${\mathcal{V}}_{\mu \nu ,ab}^{(2)W}(p,l_1,l_2)=T_a{T}_b{V}_{\mu \nu }^{(2)W}(p,l_1,l_2)=a{g}^2{T}_a{T}_b{\delta }_{\mu \nu }{\gamma }_{\mu }\cos [(p_{\mu }+\frac{1}{/}2l_{1\mu }+\frac{1}{/}2l_{2\mu })a],$$

$$U_{\mu }(x)$$

$$U_{\mu }^{†}(x)$$

$$\psi (x)$$

$$\overline{\psi }(x)$$

$$\delta (S_W+S_{WI})$$

$${\mathcal{M}}^{(0)}(p,k)=-i{T}_a(1-P_R)M(p)+i{T}_a(1-P_L)M(p+k),$$

$${\mathcal{M}}_{{\mu }_1\dots {\mu }_n,a_1\dots a_n}^{(n)}(p,k,l_1,\dots ,l_n)$$

$$\delta (S_W)$$

$${\tilde{\mathcal{M}}}^{(0)}(p,k)=i{T}_a{P}_{R}M(p)-i{T}_a{P}_{L}M(p+k),$$

$$D=d-N_g,$$

$$a^d\sum _x\overline{\psi }(x)\mathcal{D}\psi (x)=S_N+S_{NI}+S_W+S_{WI}$$

$$\mathcal{D}{|}_{{\gamma }_5\to -{\gamma }_5}={\gamma }_5{\mathcal{D}}^{†}{\gamma }_5.$$

$$S_{eff}=\ln (\det \mathcal{D}).$$

$$\frac{1}{/}2[S_{eff}\pm (S_{eff}{|}_{{\gamma }_5\to -{\gamma }_5})]=\frac{1}{/}2(S_{eff}\pm S_{eff}^{†}).$$

$$\underset{a\to 0}{\lim }\sum _x[{\overline{\psi }}_1(x)\mathcal{D}{\psi }_1(x)+{\overline{\psi }}_2(x)(\mathcal{D}{|}_{{\gamma }_5\to -{\gamma }_5}){\psi }_2(x)]$$

Full text

††preprint:

ANL-HEP-PR-95-59

(REVISED)

September 1996

I INTRODUCTION

The interaction of chiral spin-1/2 particles with gauge fields is a feature of many field-theoretic models, including the standard electroweak model. The implementation of chiral gauge theories in lattice field theory is, of course, a prerequisite to the numerical simulation of such theories; but it is also of importance in establishing that chiral theories can be defined outside of the domain of perturbation theory.

In recent years, a number of proposals for constructing lattice versions of chiral gauge theories have been put forward. A review of the present status of many of these lattice chiral-fermion proposals has been given by Shamir [1]. Some proposals [2, 3, 4] have not yet been studied extensively. Others, such as the Eichten-Preskill model [5], the Smit-Swift model [6], and the staggered-fermion model [7, 8], apparently fail to yield a chiral fermion spectrum because of the coupling of gauge degrees of freedom to the fermion [9, 10]. The domain-wall proposal of Kaplan [11] and the related overlap formula of Narayanan and Neuberger [12] have received a good deal of study, with encouraging results [12, 13, 14]. On the other hand, it has been suggested that both of these methods might fail along the lines of the failure of the Smit-Swift model because gauge degrees of freedom couple to the fermion at the boundaries of the regions of nonzero gauge field [15, 16]. Given the unsettled status of the proposals that are currently viable, it would seem to be worthwhile to consider alternative methods for formulating chiral gauge theories.

In this paper, we present a new method for constructing lattice versions of chiral gauge theories. Our approach makes use of a Wilson mass [17] to remove fermion species doublers. The Wilson mass breaks the chiral gauge symmetry. However, we argue that the violations of chiral symmetry that survive in the continuum limit are associated with ultraviolet- (UV-) divergent amplitudes and that the chiral symmetry can be partially restored through the addition of local renormalization counterterms to the action [18, 19, 20]. The philosophy of using local counterterms to restore the chiral symmetry has also been suggested by the Rome group [21]. (Local counterterms are required to restore the chiral symmetry in the proposal of the Zaragoza group [22], as well.) However, unlike the approach of the Rome group, our method does not entail the tuning of counterterm coefficients. Instead, we implement the counterterms by modifying the lattice definitions of the fermion determinant and operator matrix elements.

The first modification is to replace the magnitude of the fermion determinant with the square root of the determinant of a fermion with vector-like couplings to the gauge field [18, 19, 20, 23, 24]. (A related modification of matrix elements of operators involving fermion fields is also introduced.) This redefinition of the determinant implements the renormalization counterterms that are associated with UV divergences in a single fermion loop. After this modification, the fermion determinant is gauge invariant in the presence of a background gauge field, except for contributions from the Adler-Bardeen-Jackiw (ABJ) anomaly [25]. These violations of chiral symmetry cancel, as usual, when one considers a theory containing a suitable complement of physical fermions.

The presence of dynamical gauge fields leads to additional ultraviolet divergences and potentially requires the introduction of many new counterterms to restore the chiral gauge symmetry. We deal with this difficulty by introducing separate lattice cutoffs for the fermion fields and gauge fields [7, 24, 26, 27, 28, 29]. In the double limit in which the fermion cutoff is removed before the gauge-field cutoff, the violations of chiral symmetry vanish with at least one power of the ratio of cutoffs. The use of this double limit in conjunction with the modification of the magnitude of the fermion determinant has been emphasized previously in Refs. [24, 26]

Most of the analysis in this paper is couched in weak-coupling coupling perturbation theory. However, we are able to show, by exploiting the finite radius of convergence the perturbation expansion of the fermion determinant, that our method is also valid in the presence of nonperturbative gauge-field configurations.

The remainder of this paper is organized as follows. In Section II we discuss, in general terms, fermion doubling, its elimination through the use of a Wilson mass, and the breaking and restoration of chiral symmetry. In Section III we introduce a lattice implementation of a theory of left-handed fermions coupled to a non-Abelian gauge field. Although our specific analyses in subsequent sections of the paper refer to this model, our methods generalize immediately to models that contain right-handed as well as left-handed fermion fields and to models that contain scalar particles. In Section IV we discuss the nature of the violations of gauge invariance that arise from the introduction of a Wilson mass. Section V contains an analysis of the chiral-symmetry properties of the fermion determinant in the presence of a background gauge field. This analysis allows us to derive a modification of the determinant that restores the chiral symmetry in the case of an anomaly-free theory. In Section VI we discuss the difficulties that arise from dynamical gauge fields and present the double-limit procedure for dealing with them. In Section VII we indicate how the methods used in computing the fermion determinant can also be applied in computing matrix elements of operators containing fermion fields. A proof of the validity of the methods for computing the fermion determinant and operator matrix elements in the presence of nonperturbative gauge fields is sketched in Section VIII. Finally, in Section IX, we summarize our results and discuss various options for implementing our chiral-fermion method.

While this paper was in preparation, a paper by Hernández and Sundrum [30] on the same subject appeared. The methods that these authors propose for computing the fermion determinant (but not the matrix elements of fermion operators) are essentially identical to the ones proposed in the present paper. Many of the conclusions drawn in the present paper and in Ref. [30] are the same; one exception is noted at the end of Section VI B 5. However, the details of the proofs in the two papers are, in general, quite different.

II DOUBLING, WILSON MASSES, CHIRAL SYMMETRY: GENERAL

CONSIDERATIONS

It is well known that the most straightforward transcription of the Dirac operator to the lattice is problematic because of the phenomenon of fermion doubling: for each left- or right-handed particle in the continuum theory, there are $2^{d-1}$ left-handed and $2^{d-1}$ right-handed particles in the lattice theory, where $d$ is the dimensionality of space-time [31].

For the case of QCD, Wilson [17] suggested that one could remove the doublers by giving them a mass that goes to infinity as the lattice spacing $a$ goes to zero. Of course, the introduction of a mass explicitly breaks the chiral symmetry. However, this is not expected to present a serious problem in QCD, since the gauge symmetry remains intact. Consequently, the renormalization program is unaffected and one should recover the continuum theory as the lattice regulator is removed ($a\rightarrow 0$).

In the case of a chiral gauge theory, the introduction of a Wilson mass has more serious consequences. For such a theory, the Wilson mass and, hence, the UV regulator break the gauge symmetry, thereby jeopardizing the renormalization program and the decoupling of unphysical degrees of freedom. A failure of the gauge degrees of freedom to decouple may lead to an alteration of the low-energy spectrum of the theory. For example, under such circumstances, when one integrates over the gauge degrees of freedom, a chiral gauge theory can become a vector-like gauge theory [10].

In a chiral theory, one cannot completely avoid such a breaking of the gauge symmetry. There are several no-go theorems which state, under a variety of assumptions, that any gauge theory that does not exhibit fermion doubling must violate chiral symmetry [31, 32]. One can argue this very generally on the basis of the properties of the ABJ anomaly. If a lattice theory preserves a chiral symmetry, then the corresponding chiral current is conserved. In particular, the triangle anomaly is zero and remains zero in the continuum limit. But, according to the proof of Adler and Bardeen [33], there is no Lorentz-covariant Bose-symmetric counterterm that removes the anomaly in the triple-chiral-current Green’s function for a theory containing a single fermion species. That is, there is no UV regulator under which the anomaly vanishes as the regulator is removed. Hence, a lattice regulator that preserves the chiral symmetry must cancel the anomaly through the presence of multiple fermion species, i.e. doubling. Note that this argument leaves open the possibility that one might eliminate the doubling in a way such that the violations of chiral symmetry arise solely from the ABJ anomaly. Such a result is our goal.

In employing continuum perturbative UV regulators, such as dimensional regularization, one deals with violations of a chiral gauge symmetry by adding counterterms order by order in perturbation theory so as to restore the chiral symmetry in selected Green’s functions. The remaining violations of the chiral symmetry arise from the ABJ anomaly and cancel when one introduces an appropriate complement of physical fermion species. Such an order-by-order approach is, of course, incompatible with a nonperturbative regularization of the theory. However, one might still hope to effect a restoration of the chiral symmetry by introducing local counterterms with appropriate coefficients.

A heuristic argument in support of this idea is the following. Suppose that we have introduced a Wilson mass term. Then, the lattice spectrum for the non-interacting theory is identical to the continuum spectrum in the limit $a\rightarrow 0$. Suppose also that we have fixed to a renormalizable gauge. Then, the magnitude of the gauge field is much less than order $1/a$, unless a source of the gauge field has momentum of order $1/a$. Consequently, for field momenta much less than $1/a$, the interacting lattice action approaches the continuum action in the limit $a\rightarrow 0$. The conclusion is that the lattice, in this case, is simply a UV regulator. It follows that the differences between the lattice regularization and any other UV regularization must reside at loop momenta on the order of the UV cutoff of the theory. Hence, the differences must arise at short distances ($\sim 1/\hbox{cutoff}$); that is, they have the form of local interactions. Therefore, we conclude that, if there exists a satisfactory UV regularization of a chiral gauge theory (that is, one that respects the chiral gauge symmetry), then it must be equivalent to the (Wilson) lattice-regularized theory, plus local counterterms. Furthermore, if we find such a theory, it is unique, up to gauge-invariant counterterms, which merely renormalize the coupling constant.

III A LATTICE CHIRAL-FERMION MODEL

Now let us discuss the lattice implementation of a specific model: a left-handed fermion coupled to a non-Abelian gauge field. As we have already mentioned, the techniques that we present are easily generalizable to models containing right-handed fermions and/or scalar particles.

We assume that the gauge-field part of the (Euclidean) action has the standard plaquette form

[TABLE]

where, as usual,

[TABLE]

[TABLE]

are the lattice link variables, $A_{\mu}=A_{\mu}^{a}T_{a}$ is the gauge field, $T_{a}$ is a gauge-group matrix in the fundamental representation, $g$ is the gauge-field coupling, $a$ is the lattice spacing, and $a_{\mu}$ is a unit vector in the $\mu$ direction. Initially, we introduce the fermion through the “naive” lattice action for a Dirac particle:

[TABLE]

where the $\gamma$’s are Euclidean Dirac matrices satisfying $\{\gamma_{\mu},\gamma_{\nu}\}=2\delta_{\mu\nu}$. Note that, in contrast with some formulations of chiral theories, our approach retains both left- and right-handed components in the fermion field. The chiral nature of the theory arises from the coupling to gauge fields, which involves only the left-handed Dirac component:

[TABLE]

where $P_{R/L}=(1/2)(1\pm\gamma_{5})$, $\{\gamma_{5},\gamma_{\mu}\}=0$, $\gamma_{5}^{2}=1$. (In four dimensions, $\gamma_{5}=-\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}$.) The fermion propagator corresponding to the naive action is

[TABLE]

where $p$ is the incoming fermion momentum. The order $g$ and order $g^{2}$ gauge-field vertices that arise from the gauging of the naive action are

[TABLE]

[TABLE]

where the $V^{N}$’s are the vertices that arise from the gauging of the naive lattice action for a theory of fermions with vector-like couplings to an Abelian gauge field. Here $T_{a},T_{b},\ldots$ are the gauge-group matrices, $a,b,\ldots$ are the gauge-field indices, $\mu,\nu\ldots$ are the polarization indices, and $l_{1},l_{2},\ldots$ are the incoming momenta, all of which are associated respectively with the gauge fields. The incoming fermion momentum is $p$. The vertices of higher order in $g$ can be obtained conveniently from the recursion relation

[TABLE]

where

[TABLE]

In addition to the usual pole at $p=0$, the naive propagator (7) has extra poles when one or more momentum components are equal to $\pi/a$. It can be seen that half of the poles have positive chiral charge and half have negative chiral charge [31]. Thus, this doubling phenomenon leads to gauge-field couplings to both left- and right-handed species; the theory, at this stage, is not chiral.

We follow the standard approach of eliminating the doublers by including a Wilson mass term [17] in the action:

[TABLE]

We can gauge the Wilson term by adding to the action

[TABLE]

(As we shall see, it may sometimes be convenient to drop this coupling of the Wilson term to the gauge field.)

Now the fermion propagator is

[TABLE]

where $M(p)$ is the Fourier transform of the Wilson mass:

[TABLE]

The additional vertices that arise from the gauging of the Wilson term are

[TABLE]

[TABLE]

where the higher-order contributions can again be obtained from the recursion relation (12).

We see that the propagator (16) now has a pole only at $p=0$. This would seem to leave us, as desired, with a single Dirac particle with only left-handed couplings to the gauge field. Unfortunately, the Wilson terms $S_{W}$ and $S_{WI}$, having the Dirac structures of masses, lead to a nonconservation of the left-handed vector current by coupling the right-handed component of the Dirac field back into the theory. This implies that the chiral gauge invariance of the theory is broken.

Such violations of the chiral gauge symmetry cause serious difficulties. Gauge invariance is an important ingredient in the standard renormalization program. Without it, there is an explosion of new counterterms. For example, in the absence of current conservation, the vacuum polarization can generate a quadratically divergent gauge-boson mass, the light-by-light graph requires counterterms, Lorentz-noncovariant counterterms can arise on the lattice, and, in non-Abelian theories, the gauge-boson–fermion coupling can become different from the triple-gauge-boson coupling. In order to recover a satisfactory theory of chiral fermions coupled to massless gauge bosons, one would need to tune all of these counterterms in such a way as to restore the chiral current conservation. This is required, for example, to obtain a massless gauge boson and to guarantee that ghost fields decouple and that unitarity is preserved.

On the other hand, we note that the Wilson mass (17) and vertices (17) have the property that they vanish in the continuum limit $a\rightarrow 0$ for fixed momenta: they are lattice artifacts. Consequently, we expect the violations of the gauge symmetry generated by the Wilson mass to vanish, except when momenta of the order the lattice cutoff $\pi/a$ are important. That is, we expect that, in the continuum limit, the violations of the chiral gauge symmetry in the Green’s functions of the theory will persist only in UV-divergent Feynman diagrams and subdiagrams.

IV GAUGE VARIATIONS

In order to test this expectation, let us examine in more detail the nature of the violations of the gauge symmetry that result from the introduction of a Wilson mass. An infinitesimal transformation of the gauge field

[TABLE]

can be compensated, so as to leave $S_{N}+S_{NI}$ unchanged, by a transformation of the left-handed component of the fermion field:

[TABLE]

The Wilson terms, however, are not invariant under the transformation (IV). The gauge transformation results in a change in the action:

[TABLE]

By Fourier transforming (29), one can arrive at the Feynman rules for the vertices corresponding to a gauge variation. There is a $\Lambda$-fermion vertex

[TABLE]

and there are $\Lambda$-gauge-field-fermion vertices involving $n\geq 1$ gauge bosons,

[TABLE]

Here, $T_{a}$ is the gauge-group matrix associated with the gauge transformation $\Lambda$, $k$ is the incoming momentum associated with the gauge transformation, $p$ is the incoming fermion momentum, and the $l_{i}$ are the incoming gauge-field momenta. Note that the $\Lambda$ vertices (IV) contain factors of $g$ only for the gauge fields, not for the $\Lambda$ fields.

If we choose not to gauge the Wilson term, then all of the gauge variation in the action resides in $S_{W}$:

[TABLE]

In this case, there is a slightly different $\Lambda$-fermion vertex,

[TABLE]

and there are no $\Lambda$-gauge-field-fermion vertices.

In the analysis to follow, we will frequently make use of the fact that theories with vector-like couplings to the gauge field exhibit a gauge invariance, even in the presence of a Wilson mass term. A theory with vector-like couplings to the gauge field can be obtained by setting $P_{R}=P_{L}=1$ in the action (5), (6), (14), (15). Then, if one sets $P_{R}=P_{L}=1$ in the gauge transformation (IV), the gauge variation (29) and the $\Lambda$ vertices (IV) vanish, as expected. Note, however, that the gauge symmetry is violated if one drops the gauging of the Wilson term (15) from the action, as can be seen from examination of (36) and (37).

There is also a property of the $\Lambda$ vertices that will be crucial for our subsequent analysis. The $\Lambda$ vertices are linear combinations of either Wilson masses or Wilson vertices. Consequently, they all vanish in the continuum limit $a\rightarrow 0$ for fixed momenta. Thus, the gauge variations can persist in the limit $a\rightarrow 0$ only if momenta of order the lattice cutoff $\pi/a$ are important, that is, only in divergent Feynman diagrams.

V AMPLITUDES IN A BACKGROUND GAUGE FIELD

As a first step in identifying and dealing with the violations of gauge symmetry in the Green’s functions of the chiral theory, let us consider the case of fermion amplitudes in the presence of background gauge fields in which the momentum of a gauge-field quantum is limited be much less in magnitude than the lattice momentum cutoff $\pi/a$.

A Counting powers of $a$

First let us consider, in the limit $a\rightarrow 0$, the size of the contribution from a fermion loop containing zero or one gauge-variation ($\Lambda$) vertices and any number of background gauge-field vertices. We will analyze, in turn, the region of integration in which the magnitude of the loop momentum is much smaller than $\pi/a$ and the region of integration in which the magnitude of the loop momentum is of order $\pi/a$.

As we have seen, a $\Lambda$ vertex vanishes in the limit $a\rightarrow 0$ unless momenta of order $\pi/a$ are important. Thus, we expect that a loop containing a $\Lambda$ vertex will receive a vanishing contribution from the region of integration in which the magnitude of the loop momentum is much smaller than $\pi/a$. Since the external gauge-field momenta are assumed to be much smaller than $\pi/a$, one can take the $a\rightarrow 0$ limit in this region simply by taking the $a\rightarrow 0$ limits of the propagators and vertices, holding momenta fixed. In this limit, propagators and naive single-gauge-field vertices go over to continuum propagators and vertices, which are $a$ independent, while multi-gauge-field naive vertices, Wilson vertices, and $\Lambda$ vertices vanish as at least one power of $a$. Furthermore, since the trace of an odd number of $\gamma$ matrices vanishes, a $\Lambda$ vertex is always paired with a Wilson vertex or a Wilson term in a propagator numerator. The volume of integration in this region is independent of $a$. Thus, we conclude, that a loop that contains a $\Lambda$ vertex receives a contribution from this region of integration that vanishes as at least two powers of $a$ in the limit $a\rightarrow 0$.

Now we consider the region of integration in which the magnitude of the loop momentum is of order $\pi/a$. We can determine whether this is an important region of integration by examining the sizes of the propagators, vertices, and the domain of integration. (See, for example, Ref. [34] for further details.) Away from its pole at the origin, the propagator (16) is of order $a$. Here, it is crucial that we have eliminated doublers; otherwise, there would be poles in the propagator for components of the loop momentum of order $\pi/a$. An $n$-gauge-field-fermion vertex is of order $a^{n-1}$, and a $\Lambda$-$n$-gauge-field-fermion vertex is of order $a^{n-1}$. The domain of integration is of order $a^{-d}$ in $d$ dimensions. From this it follows that the region in which the magnitude of the fermion-loop momentum is of order $\pi/a$ gives a contribution of order $a^{N_{g}-d}$, where $N_{g}$ is the number of external gauge fields. Note that this result is independent of the number of $\Lambda$ vertices.***Since we are concerned only with infinitesimal gauge transformations, we need never consider the case of more than one $\Lambda$ vertex. We define the degree of divergence of a loop to be

[TABLE]

which corresponds to the expression in continuum field theory. If the loop is UV convergent, that is, if $D$ is negative, then the contribution from the region in which the magnitude of the loop momentum is of order $\pi/a$ vanishes as a power of $a$ in the limit $a\rightarrow 0$. In this case, for a loop containing no $\Lambda$ vertices, the contribution from the region in which the magnitude of the loop momentum is much less than $\pi/a$ dominates. One can obtain the $a\rightarrow 0$ limit of this contribution by replacing the integrand with the continuum expression. The resulting integral is UV convergent, and so one can extend the range of integration to infinity with negligible error. Hence, the $a\rightarrow 0$ limit of this contribution is identical to the continuum amplitude.

We conclude that a fermion loop containing a $\Lambda$ vertex gives a vanishing contribution in the limit $a\rightarrow 0$, unless the degree of divergence is non-negative. Hence, for $d=4$, the gauge variations that persist in the continuum limit arise only from loops involving a $\Lambda$ vertex and four or fewer external gauge-field vertices.

Using these same arguments, we can also conclude that a term in a loop amplitude that is proportional to a Wilson mass or vertex gives a contribution that vanishes as a power of $a$ in the limit $a\rightarrow 0$, unless the degree of divergence of the loop is non-negative. Furthermore, in the case a non-negative degree of divergence, the dominant contribution comes from the region of integration in which the loop momentum is of order $\pi/a$. That is, the contribution takes the form of a local interaction, with configuration-space size of the order of the inverse of the lattice UV cutoff $\pi/a$.

B Modifying the fermion determinant

At this point we could attempt to restore the gauge symmetry by adding renormalization counterterms to the theory. Of course, no counterterm can remove violations of the gauge symmetry that arise from the ABJ anomaly. Partly because of the absence of full rotational symmetry on the lattice, the number of possible counterterms is quite large. In addition to the usual rotationally invariant gauge-field wave-function renormalization, there are counterterms corresponding to a gauge-field mass, a rotationally non-invariant wave-function renormalization, and rotationally invariant and non-invariant gauge-field–gauge-field scattering amplitudes. The tuning of all of these counterterms in a lattice simulation would be awkward. Fortunately, there is a trick that can be used to implement the required counterterms automatically [18, 19, 20]. Motivated by the fact that a theory with vector-like couplings of the fermion to the gauge field is gauge invariant, we will attempt to rearrange the fermion-loop amplitude so that it looks like the loop amplitude for a vector-like theory.

Consider an arbitrary fermion-loop amplitude. We can write the projectors $P_{L}=(1-\gamma_{5})/2$, which appear only in the naive vertices, in terms of the unit matrix and $\gamma_{5}$ and expand the expression for the amplitude. The result is a sum, each term of which contains an even or an odd number of factors of $\gamma_{5}$.

1 The even-parity part

For those terms that contain an even number of $\gamma_{5}$’s, which we call even-parity terms, we would like to move the factors of $\gamma_{5}$ together and use the identity $\gamma_{5}^{2}=1$ to eliminate them, thereby obtaining the corresponding expression for a vector-like theory. This would amount to a simple algebraic manipulation, were it not for the fact that $\gamma_{5}$ anticommutes with the naive terms in the rationalized-propagator numerators and naive vertices, but commutes with the Wilson terms in the rationalized-propagator numerators and Wilson vertices. We would obtain a result that is proportional to the corresponding expression in a vector-like theory were we to treat $\gamma_{5}$ as if it anticommuted with the Wilson terms in the rationalized-propagator numerators and Wilson vertices. We will follow this procedure. Of course, the resulting expression will differ from the original one, and we must account for this difference. However, the difference is always proportional to a Wilson mass from a propagator numerator or a Wilson vertex. As we have demonstrated in Section V A, a loop containing a Wilson mass or vertex vanishes as at least one power of $a$ in the limit $a\rightarrow 0$, unless the degree of divergence is non-negative, and then the contribution corresponds to a local interaction. Thus, such contributions have the form of renormalization counterterms. We can drop them without affecting the nature of the theory: such a procedure amounts merely to adding renormalization counterterms to the action and choosing a particular tuning of the counterterm coefficients. Then, for the terms in the original loop amplitude that contained an even number of $\gamma_{5}$’s we obtain expressions that are proportional to the corresponding expressions in a vector-like theory. We now work out the constants of proportionality.

Consider first a contribution from a loop amplitude that contains at least one naive vertex. We are interested only in manipulating the terms containing an even number of $\gamma_{5}$’s. However, it is simplest to work out the combinatorics by moving the complete projectors $P_{L}$ until they stand next to each other, treating $\gamma_{5}$ as if it commuted with all other factors in the amplitude. Each projector is separated by $N$ propagators and $N$ vertices from another, and so, in the process of moving one projector so that it is adjacent to another, the projector flips from a $P_{L}$ to a $P_{R}$, but always winds up as a $P_{L}$ in the end. Since $P_{L}^{2}=P_{L}$, we have just one projector $P_{L}=(1-\gamma_{5})/2$ when the process is finished. The even-parity part of the amplitude corresponds to the term $1/2$. Thus the even-parity part yields a contribution that is exactly half the corresponding contribution in a vector-like theory.

Now consider a contribution from a loop amplitude that contains no naive vertices. In this case, there are no projectors $P_{L}$, the contribution is entirely even in parity, and it is equal to the corresponding contribution in a vector-like theory. In order to combine it with the even-parity parts of the contributions containing at least one naive vertex, so as to obtain a complete vector-like amplitude, we must discard half. However, since the discarded piece contains no naive vertices, it must contain at least one Wilson vertex. As we have already argued, we can safely discard such a contribution, since that act amounts to choosing a particular tuning of the coefficients of renormalization counterterms.

At the end of all of these manipulations, the even-parity part of a fermion-loop amplitude yields a contribution that is half the corresponding contribution in a vector-like theory. The effective action that one obtains by integrating over the fermion degrees of freedom is, of course, given by the loop amplitudes, weighted by $1/N_{g}$. Therefore, the effect of our manipulations is to replace the even-parity part of the contribution to the effective action by one half the effective action for a vector-like theory. Now, the lattice Dirac operator ${\cal D}$, which is defined by

[TABLE]

has the property†††This property also holds if one drops the gauging of the Wilson term $S_{WI}$ on the right side of (39). that

[TABLE]

Now, the effective action is given by

[TABLE]

Since $\det\gamma_{5}=1$, we see from (40) that

[TABLE]

That is, the even-parity (odd-parity) part of the effective action is the real (imaginary) part of the effective action. Furthermore, (41) implies that the real (imaginary) part of the effective action corresponds to the magnitude (phase) of the fermion determinant.

Therefore, we conclude that our manipulations amount to the prescription that the magnitude of the chiral fermion determinant be replaced by the square root of the fermion determinant for a vector-like theory.‡‡‡There is no ambiguity in the sign of the square root. We are identifying the square root with the magnitude of the fermion determinant, and so we always take the positive sign. The sign ambiguity associated with the Witten anomaly [35] is carried by the phase of the determinant. Since the low-energy spectrum is unchanged by our modifications of the determinant, the Witten anomaly is unaffected. In particular, the Witten anomaly is absent in this lattice implementation of the Standard Electroweak Model. This prescription has been discussed previously in the case of continuum theories [23] and in the case of lattice theories [20, 24]; an equivalent formulation involving auxiliary fermion species has also been presented [18, 19]. If one adopts this prescription, then the magnitude of the fermion determinant and, correspondingly, the real part of the effective action have an exact gauge invariance.

We note that, since these manipulations amount to the addition of renormalization counterterms to the theory, they do not affect unitarity. This is obvious at the level of the action, since, in Minkowski space, it is Hermitian even with the addition of counterterms. It is also easy to see diagrammatically: a cut of a diagram can never pass through a short-distance loop (momenta of order the UV cutoff), because the on-shell conditions and energy-momentum conservation constrain the components of the momenta of the cut lines to have magnitudes much smaller than the UV cutoff.

Of course, as we have already argued at the diagrammatic level, the manipulations that we have made do not affect the low-energy behavior of the theory. It is easy to see this directly from the action. The even-parity part of the effective action generated by a fermion with left-handed couplings to the gauge field is equal to one-half the effective action generated by two fermions, one with left-handed couplings and one with right-handed couplings. The continuum limit of the action for such a complement of fermions is given by

[TABLE]

where $\psi=P_{L}\psi_{1}+P_{R}\psi_{2}$. Here, in taking the continuum limit, we have assumed that the momenta associated with the Fourier transforms of the fields are all fixed to be much less than the UV cutoff, so that one can take the “naive” $a\rightarrow 0$ limit of operators. We conclude that the even-parity part of the effective action goes, at low momentum and in the continuum limit, to one half the effective action generated by a fermion with vector-like couplings to the gauge field, plus non-interacting degrees of freedom.

2 The odd-parity part

Now we turn to the terms in the loop amplitude that contain an odd number of $\gamma_{5}$’s, which we call the odd-parity part. The manipulations of the preceding section, which bring $\gamma_{5}$’s together and use $\gamma_{5}^{2}=1$ to eliminate them, can never succeed in converting the odd-parity parts to a vector-like amplitude: there will always be one $\gamma_{5}$ left over in the end. Thus, we must deal in another way with the violations of the gauge symmetry in the odd-parity parts that persist in the limit $a\rightarrow 0$.

Let us specialize, for the moment, to four dimensions. As we have seen in Section V A, the gauge variations that are nonvanishing as $a\rightarrow 0$ are contained in the fermion-loop amplitudes involving one $\Lambda$ field and four or fewer gauge fields. Then, one can see that the nonvanishing gauge variations correspond to the ABJ anomaly. An explicit calculation is presented in Appendix A. Here we give a general argument that the gauge variations are zero, provided that one chooses a theory in which the complement of physical fermions satisfies the anomaly-cancellation condition

[TABLE]

As we have argued in Section V A, a loop containing a $\Lambda$ vertex receives a nonvanishing contribution in the limit $a\rightarrow 0$ only from the region of integration in which the magnitude of the loop momentum is of order $\pi/a$. That means that the nonvanishing gauge variations all have the form of local interactions. In four dimensions, the odd-parity, local operators of dimension four or less that are invariant under lattice rotations and involve a $\Lambda$ field and gauge fields are of the form ${\rm Tr}\,[\Lambda\epsilon_{\mu\nu\rho\sigma}A_{\mu}A_{\nu}A_{\rho}A_{\sigma}]$ and ${\rm Tr}\,[\Lambda\epsilon_{\mu\nu\rho\sigma}(\partial_{\mu}A_{\nu})A_{\rho}A_{\sigma}]$, or ${\rm Tr}\,[\Lambda\epsilon_{\mu\nu\rho\sigma}(\partial_{\mu}A_{\nu})(\partial_{\rho}A_{\sigma})]$. These all vanish if the anomaly-cancellation condition (45) is satisfied. There remains the possibility that subleading contributions from this region of integration could give rise to violations of gauge invariance that vanish as powers of $a$. However, there are no lattice-rotationally invariant, odd-parity, local operators of dimension five involving a $\Lambda$ field and gauge fields. Hence, the violations of gauge invariance from the region of integration in which the magnitude of the loop momentum is of order $\pi/a$ vanish at least as $a^{2}$ in the limit $a\rightarrow 0$.

Similar arguments show that, in two dimensions, the gauge variations of the odd-parity part of a loop also vanish as $a^{2}$ in the limit $a\rightarrow 0$, provided that the anomaly is cancelled. In two dimensions one can achieve cancellation of the anomaly in a non-trivial theory by introducing left-handed and right-handed fermions such that the sum of ${\rm Tr}\,(T_{a}T_{b})$ for the left-handed fermions is equal to the sum of ${\rm Tr}\,(T_{a}T_{b})$ for the right-handed fermions.

We emphasize that, in contrast with the modified even-parity loop amplitudes, the odd-parity loop amplitudes do not possess an exact gauge invariance, even if (45) is satisfied. There are violations of the gauge symmetry that vanish only in the limit $a\rightarrow 0$. We have just seen that such violations can arise from the region of integration in which the fermion-loop momentum is of order $\pi/a$. In Section V A, we noted that violations of gauge invariance can also arise from the region of integration in which the magnitude of the fermion-loop momentum is much less than $\pi/a$, even in UV-convergent diagrams. In both of these cases, the violations of gauge invariance vanish as $a^{2}$ in the limit $a\rightarrow 0$.

The odd-parity amplitudes themselves are finite in the limit $a\rightarrow 0$. This follows from the fact that there are no odd-parity renormalization counterterms involving only gauge fields. In four dimensions, the lattice-rotationally invariant, odd-parity, local operators of dimension four or less involving gauge fields have the forms ${\rm Tr}\,[\epsilon_{\mu\nu\rho\sigma}A_{\mu}A_{\nu}A_{\rho}A_{\sigma}]$, ${\rm Tr}\,[\epsilon_{\mu\nu\rho\sigma}(\partial_{\mu}A_{\nu})A_{\rho}A_{\sigma}]$, and ${\rm Tr}\,[\epsilon_{\mu\nu\rho\sigma}(\partial_{\mu}A_{\nu})(\partial_{\rho}A_{\sigma})]$. When one symmetrizes under cyclic permutations of the gauge fields, the first operator vanishes, and the second and third operators are total derivatives. It can be seen in a similar fashion that corresponding operators in two-dimensional theories vanish. Since the gauge variation of an odd-parity amplitude vanishes as $a^{2}$ in the limit $a\rightarrow 0$, we can conclude that the deviation of an odd-parity amplitude from a gauge-invariant expression also vanishes as $a^{2}$.

Finally, we mention that the analysis of the gauge variations of the odd-parity parts of loops in this section does not depend on the gauging of the Wilson term. The analysis relies only on the power-counting rules and the general structure of the local interactions, neither of which are affected by the presence or absence of (15) in the action.

VI DYNAMICAL GAUGE FIELDS

We wish to generalize the discussion of Section V to include the possibility that the gauge fields are dynamical, rather than simply external background fields. The important distinction is that the gauge-field momentum can now contain a loop momentum, and so its magnitude can range up to the lattice cutoff $\pi/a$. Now we can have divergent loop integrations involving gauge-field propagators as well as fermion propagators, and the results for the counting of powers of $a$ must be generalized from those derived in Section V.

The even-parity parts of fermion loops can again be rendered exactly gauge invariant by making use of the $\gamma_{5}$ trick of Section V B 1 to replace the fermion loop by one-half the corresponding loop for a fermion with vector-like interactions with the gauge field. We have already seen that this replacement does not alter the low-energy behavior of amplitudes. Therefore, it amounts to a change of UV regulator, which is equivalent to the addition of counterterms to the theory. In the case of a background gauge field with momentum much smaller in magnitude than the UV cutoff $\pi/a$, the required counterterms were those generated by a single fermion loop. In the present case, counterterms can also be generated by multi-loop subdiagrams, including loops involving gauge fields. Fortunately, we do not need to implement these counterterms explicitly: they are provided automatically by modification of the fermion-loop amplitude.

The case of the odd-parity parts of fermion loops is more complex and requires some further analysis.

A Counting powers of $a$

We wish to study the gauge variations of the odd-parity parts of fermion loops in the limit $a\rightarrow 0$. That is, we wish to study the behavior of a diagram or a subdiagram containing exactly one $\Lambda$ vertex in that limit. As we argued in Section V, contributions involving a $\Lambda$ vertex are suppressed by at least one power of $a$ in the limit $a\rightarrow 0$ unless a momentum entering the $\Lambda$ vertex has a magnitude of order $\pi/a$. Thus, we wish to study the region of integration in which the loop momenta have magnitudes of order $\pi/a$. We might as well take all the loop momenta in a subdiagram to have magnitudes of order $\pi/a$, since we can always study the case when only a subset of the loop momenta have magnitudes of order $\pi/a$ by considering a smaller subdiagram. For purposes of the discussion in this subsection only, we assume that the gauge field has been fixed to a renormalizable gauge.

Now we use the facts that, in the region in which all momenta have magnitudes of order $\pi/a$, an $n$-gauge-field-fermion vertex is of order $a^{n-1}$, a $\Lambda$-$n$-gauge-field-fermion vertex is of order $a^{n-1}$, a fermion propagator is of order $a$, a gauge-field propagator is of order $a^{2}$, an $n$-gauge-field vertex is of order $a^{n-4}$, and each loop integration has a range of order $(1/a)^{d}$ in $d$ dimensions. From these facts, it is easy to see that a single-particle-irreducible (1PI) diagram or subdiagram with $N_{f}$ external fermion legs, $N_{g}$ external gauge-field legs, and $L$ loops is of order $a^{-D}$, where the degree of divergence $D$ is given by§§§Ghost loops, which appear with certain choices of gauge, do not affect these conclusions.

[TABLE]

Any 1PI subdiagram that contains a $\Lambda$ vertex and has a non-negative degree of divergence can potentially lead to a violation of the gauge symmetry that survives in the limit $a\rightarrow 0$. As we have already argued, the even-parity parts of fermion loops in such a subdiagram can be rendered exactly gauge-invariant by replacing the fermion loop with one-half the corresponding loop for a fermion with vector-like interactions with the gauge field. However, in the case of the odd-parity part of a loop, a $\Lambda$ vertex inside a radiative correction can give a nonvanishing contribution in four dimensions. (An example of such a contribution is shown in Fig. 1.) Hence, there are violations of the gauge symmetry in four dimensions.

One might hope that it would be possible to restore the gauge symmetry by tuning the limited number of renormalization counterterms that are associated with divergent radiative corrections[18, 19, 20]. Unfortunately, this turns out not to be the case. For example, in four dimensions, the diagram of Fig. 2 has an overall degree of divergence $D=2$. Thus, the contribution that arises from the odd-parity parts of the fermion loops yields violations of the gauge symmetry, even though the individual fermion loops have a negative degree of divergence. In particular, the diagram generates a gauge-field mass, and so would require a mass counterterm, even if gauge-field-mass generation has been eliminated at the one-loop level by modifying the even-parity parts of loops as described in Section V B 1. On examining other multi-loop diagrams, one reaches the conclusion that that all possible renormalization counterterms consistent with the cubic lattice symmetry appear.

B The double-limit procedure

We would like to restore the gauge invariance of the theory without resorting to the tuning of counterterms. If we could limit the momenta in loops involving gauge fields to be much less than the fermion-loop UV cutoff, then the arguments of Section V would apply. One way to achieve this is to introduce two different lattice spacings, $a_{g}$ for the gauge field and $a_{f}$ for the fermion field and take the limit $a_{f}\rightarrow 0$ with $a_{g}$ fixed before taking the limit $a_{g}\rightarrow 0$. Such a double-limit procedure is similar in spirit to the UV regulator employed in proving the anomaly-no-renormalization theorem [36]. A double-limit procedure has also been discussed previously in the context of lattice theories [7, 24, 26, 27, 28, 29]. The use of a double limit along with the modification of the magnitude of the fermion determinant has been discussed previously in Refs. [24, 26].

1 Interpolation of the gauge fields: general considerations

In computing the double limit, we assume that the gauge-field links that reside on the gauge-field lattice $U_{g\mu}$ are the dynamical variables, i.e., the variables over which one integrates in the path-integral expressions for amplitudes. These are the quantities that appear in the pure gauge-field action (1). The interactions of the gauge fields with fermion fields are obtained by inserting gauge-field links $U_{\mu}$, which reside on the fermion lattice, into the fermion action as in (5), (6), (14), and (15). These gauge-field links that reside on the fermion lattice are not the dynamical variables $U_{g\mu}$. We must obtain them by an interpolation of the dynamical gauge-field links.

It is often convenient to discuss the interpolation in terms of the gauge fields $A_{\mu}$, which are related to the plaquettes through (1). One can use the Hamilton-Cayley theorem to express the logarithm of an $m\times m$ group matrix (link), as a linear combination of the unit matrix and the first $m-1$ powers of the matrix. The ambiguity in the phase of the coefficients can be resolved by requiring that matrices that are close to the unit matrix have logarithms that are close to zero. This is equivalent to the requirement proposed by ’t Hooft[27] that the eigenvalues of $a_{g}A_{g\mu}$ and the eigenvalues of $a_{f}A_{f\mu}$ lie on the interval $(-\pi,\pi]$.

For simplicity, we will assume that $a_{g}/a_{f}=R$ is an integer and that the fermion lattice subdivides the gauge-field lattice, so that they coincide every $R$ sites. For each gauge-field-lattice site $y$, there are $R^{d}$ fermion-lattice sites $x=y+m$, where $m$ is vector whose components are integers satisfying

[TABLE]

We will also assume that, in the interpolation, the fermion-lattice links $U_{\mu}$ depend only on the gauge-lattice links $U_{g\nu}$ that form the edges of the surrounding hypercube. That is, we assume that the links $U_{\mu}(y+ma_{f})$ depend only on the links $U_{g\nu}(y+m_{g}(\nu)a_{g})$, where $m_{g}(\nu)$ is a vector with integer components satisfying

[TABLE]

Similarly, the fermion-lattice fields $A_{\mu}(y+m_{\mu}a_{f}+\frac{1}{/}2a_{f\mu})$ depend only on the gauge-field-lattice fields $A_{g\nu}(y+m_{g}(\nu)a_{g}+\frac{1}{/}2a_{g\mu})$.

The Fourier transform of the fermion-lattice field is given by

[TABLE]

where

[TABLE]

Note that, if $\overline{A}_{\mu}(l,y+\frac{1}{/}2a_{g\mu})$ were equal to $A_{g\mu}(y+\frac{1}{/}2a_{g\mu})$, then $\tilde{A}_{\mu}(l)$ would be equal to $\tilde{A}_{g\mu}(l)$, where

[TABLE]

is the Fourier transform of the field on the gauge-field lattice. We express the deviation of $\tilde{A}_{\mu}(l)$ from $\tilde{A}_{g\mu}(l)$ in terms of a “regulating factor” $F_{\mu}(l)$:

[TABLE]

Many different interpolations of the gauge fields are possible. However, if the interpolation is to lead to a gauge-invariant theory in the double limit, then certain minimal requirements must be met: the interpolation must lead to correct tree-level amplitudes in the continuum limit, the interpolation must provide a UV cutoff of order $\pi/a_{g}$ on gauge-field momenta, and the interpolation must relate a gauge transformation of the fields on the gauge-field lattice to a gauge transformation of the fields on the fermion lattice. We now enumerate a set of sufficient conditions for meeting these requirements.

1. a.

Locality. The interpolation must be local in the sense that gauge fields on the fermion lattice cannot depend on gauge fields on the gauge-field lattice that are separated by an arbitrarily large number of gauge-field-lattice sites. If one were to employ a nonlocal interpolation, then the gauge-field-fermion interactions would not go to the continuum (local) form in the limit $a_{g}\rightarrow 0$. The interpolation need not be strictly local; it can depend on gauge fields that are separated by a finite number of gauge-field-lattice sites. However, a dependence of the interpolation on widely separated gauge-field-lattice sites would lead to large order $a_{g}$ errors in the limit $a_{g}\rightarrow 0$. We have assumed a local form for the interpolation in (48). 2. b.

Smoothness. We take as a smoothness requirement the continuity of fields inside hypercubes on the gauge-field lattice.¶¶¶Such a criterion has been discussed in Refs. [27, 29, 37]. That is, we require that, for a given $y$, the fields $A_{\mu}(y+a_{f}m+\frac{1}{/}2a_{f})$ differ on adjacent fermion lattice sites by quantities of order $a_{f}$. There can, depending on the interpolation, be discontinuities along certain directions at the boundaries between the gauge-field hypercubes. However, the size of these discontinuities is independent of $a_{f}$.

The smoothness requirement leads to a UV cutoff on the gauge-field momentum, since it guarantees that the Fourier transform (50) vanishes as $a_{f}^{n}$ if $n$ components of $l$ are of order $\pi/a_{f}$. We can see this by making use of the elementary properties of Fourier transforms. Consider the one-dimensional Fourier transform

[TABLE]

From (54) it follows that

[TABLE]

where

[TABLE]

are the forward and backward lattice derivatives. Smoothness requires that the lattice derivative of the field $\nabla_{\nu}A_{\mu}$ be of order $a_{f}^{0}$, except possibly at gauge-field-lattice hypercube boundaries, where it may be of order $a_{f}^{-1}$. Since the number of boundaries does not grow with decreasing $a_{f}$, the left side of (55) is at most order $a_{f}^{-1}$. This implies that, on the right side of (55), $\tilde{A}_{\mu}(l_{\nu})$ can be at most of order $a_{f}$ over a range of $l_{\nu}$ that is of order $\pi/a_{f}$.

Smoother interpolations than we consider here lead to additional suppression of the Fourier transform of the interpolated field at large momentum. One can derive relations similar to (55), but involving higher derivatives. From these, it can be seen that, if, along the $\nu$ direction, the $(r-1)$st derivative of $A_{\mu}(x+\frac{1}{/}2a_{f\mu})$ is continuous and the $r$th derivative is continuous except at hypercube boundaries, then $\tilde{A}_{\mu}(l_{\nu})$ can be at most order $a_{f}^{r+1}$ over a range of $l_{\nu}$ of order $\pi/a_{f}$. It should be noted, however, that as an interpolation becomes smoother, it becomes increasingly less local, involving more widely separated sites on the gauge-field lattice. Therefore, such interpolations, in general, increase the size of the order $a_{g}$ errors in the limit $a_{g}\rightarrow 0$.

The smoothness requirement, coupled with locality, also guarantees that one recovers the correct tree-level amplitudes in the continuum limit. That is, it guarantees, that

[TABLE]

This follows immediately from the fact that, because of continuity,

[TABLE]

can differ from $A_{g\mu}(y+\frac{1}{/}2a_{g\mu})$ only by a quantity of order $Ra_{f}=a_{g}$. Here we are making use of the fact that the gauge fields associated with the tree amplitudes are continuous on the gauge-field lattice.

Therefore, we conclude that the smoothness requirement leads to the properties

[TABLE]

and

[TABLE]

As we have already mentioned, smoother interpolations result in additional suppression of $F_{\mu}(l)$ when components of $l$ are large. For example, if the interpolation of $A_{\mu}$ is “transversely continuous”, i.e., continuous along directions $\nu\neq\mu$ at the boundaries of the gauge-field-lattice plaquettes, then there is an additional power of $a_{f}$ on the right side of (60) for each component $l_{\nu}$ that is of order $\pi/a_{f}$. 3. c.

Gauge Covariance. We require that, for every gauge transformation $\Lambda^{\prime}$ of the gauge-field-lattice links $U_{g\mu}$, the interpolation of the gauge-transformed links $U_{g\mu}^{\Lambda^{\prime}}$ must yield a set of fermion-lattice links $U_{\mu}^{\Lambda}$, where $\Lambda$ denotes a gauge transformation of the fermion-lattice links $U_{\mu}$ [37]. This requirement allows one to infer, from the gauge invariance of the fermion sector of the theory on the fermion lattice, that the complete theory on the gauge-field lattice is gauge invariant.

One might imagine that one could meet this gauge-invariance requirement by fixing to a particular gauge before carrying out the interpolation. However, gauge fixing is a nonlocal procedure and, therefore, violates the requirement that the interpolation be local.

The interpolations that we will consider have the property that the gauge field $A_{\mu}$ is constant along fermion-lattice links $U_{\mu}$ that lie along gauge-field lattice links $U_{g\mu}$. For these links, $A_{\mu}$ is chosen to be equal to $A_{g\mu}$. This implies that

[TABLE]

In solving (62) for $U_{\mu}$, we choose the branch cut of the $R$th root in accordance with the definition of the gauge fields discussed earlier in this section. That is, we take the branch cut such that, if $U_{g\mu}$ is near unity, then $U_{\mu}$ is near unity.

The property (62) is compatible with the gauge-covariance requirement. In order to see that this is so, consider a gauge transformation $\Lambda^{\prime}(y)$ on the gauge-field lattice. Each link $U_{g\mu}$ is transformed according to

[TABLE]

Thus, according to (62), the fermion-field links change as follows:

[TABLE]

A gauge transformation $\Lambda$ on the fermion-lattice links that reproduces the right side of (64) can be obtained by the following procedure. First, set

[TABLE]

Then, each link $U_{\mu}(y+m_{\mu}a_{f})$ can be brought into agreement with the right side of (64) by suitable choice of $\Lambda(y+m_{\mu}a_{f}+a_{f\mu})$, where the choices can be made by a sequential algorithm, starting at the first link and working toward the last link. At the last link, the choice of $\Lambda(y+m_{\mu}a_{f}+a_{f\mu})=\Lambda(y+a_{g\mu})$ must not conflict with (65). However,

[TABLE]

and so the choice of $\Lambda(y+a_{g\mu})$ that is required by (62) is

[TABLE]

which agrees with (65).

Recently, Shamir [1] has pointed out that there is a potential difficulty in maintaining the smoothness and the gauge covariance of the interpolation procedure. He has shown that the interpolating field differs from a smooth field by a gauge transformation that is, in general, topologically nontrivial and, hence, singular. These difficulties do not appear in an Abelian theory with a non-compact gauge-field action. It is possible that they might be avoided by fixing to a suitable gauge on the gauge-field lattice. However, this issue has yet to be resolved. In the analyses to follow, we indicate those parts of the arguments that may be affected by these considerations.

2 An Abelian interpolation

As an example, let us consider an interpolation that satisfies the required properties in the case of an Abelian theory. In an Abelian theory, the gauge transformation (63) is equivalent to

[TABLE]

If the interpolation of the $\Lambda^{\prime}$-dependent part of (69) has a vanishing lattice curl, then it can be written as the lattice gradient of a potential on the fermion lattice. Then (69)) is equivalent to a gauge transformation on the fermion-lattice fields (of the same form as (69)). It is easy to see that a simple linear interpolation of the gauge field [37] has this property. Hence, it is gauge covariant under infinitesimal gauge transformations (although not under the large gauge transformations of Ref. [1]). To be explicit, one takes

[TABLE]

Clearly, this interpolation satisfies the locality and smoothness requirements. We have, for this interpolation,

[TABLE]

which implies that the regulating factor is given by

[TABLE]

We see explicitly that the properties (b) hold, as expected from our general arguments.

3 A non-Abelian interpolation

In the case of non-Abelian gauge fields, simple linear interpolations of the sort discussed in the last section do not satisfy the gauge-covariance requirement. However, ’t Hooft [27] has proposed a more intricate interpolation method that does. Here we discuss a variant of ’t Hooft’s method that was suggested by Hernández and Sundrum [30].

The first step in the method is to fix the interpolation for fermion-lattice links that lie along gauge-field-lattice links according to (62). As we have already shown, this step is consistent with the gauge-covariance requirement.

The next step is to determine the interpolation for the fields $A_{\mu}$ that lie on the two-dimensional surface of an elementary plaquette, where here $\mu$ is either one of the two directions that define the plaquette. The interpolation is given by the field configuration that minimizes the two-dimensional action for a pure gauge-field theory on the fermion lattice,∥∥∥This action is given by (1), but in two dimensions and on the fermion lattice. subject to the boundary conditions on the fields on the links bounding the plaquette. To obtain a unique solution to the minimization condition, one must fix the gauge. A convenient choice is the two-dimensional Lorentz gauge

[TABLE]

One can argue that the solution is unique as follows. The minimization condition implies that the field configurations satisfy the gauge-field equations of motion. If we neglect terms of higher order in $a_{f}$, then the equation of motion is

[TABLE]

where

[TABLE]

and we have rescaled the fields by $g$. In the Lorentz gauge, the equation of motion becomes

[TABLE]

If one sets $g=0$ in (77), then one recovers Laplace’s equation, which, with the given boundary conditions, has a unique solution. One can obtain a solution to all orders in $g$ by iteration, treating the order $g$ and order $g^{2}$ terms as source terms and using the solution to Laplace’s equation as a starting point. Hence, in the continuum limit, the interpolated field configuration that is continuously connected to the $g=0$ solutions is unique.

In order to see that the gauge fields derived through this interpolation procedure satisfy the smoothness requirement, suppose the opposite: that a gauge field has a discontinuity. Then, for at least one point $x$, the first term on the left side of (77) is of order $a_{f}^{-2}$, whereas the remaining terms are of order $a_{f}^{-1}$ or smaller. (Here we are assuming that the interpolated gauge field is bounded, which may not be true in the presence of singularities of the type discussed by Shamir [1].) Therefore, in the case of a discontinuous gauge field, the equations of motion cannot be satisfied in the continuum limit, and one concludes that the gauge field does not satisfy minimization criterion in the continuum limit.

In four dimensions, there are two more steps in the interpolation method. The third step is to determine the fields inside the cubes bounded by the elementary plaquettes. One does this by seeking a field configuration that minimizes the three-dimensional pure gauge-field action, subject to the boundary conditions along the elementary plaquettes and the three-dimensional Lorentz-gauge condition. The last step is to determine the fields inside the four-dimensional hypercubes bounded by the three-dimensional cubes. One minimizes the four-dimensional pure gauge-field action, using the fields on the cubes as boundary conditions and fixing to the four-dimensional Lorentz gauge. It is easy to see, by generalizing the preceding arguments, that these last two steps result in fields that satisfy the smoothness requirement.

Finally, there is the question of whether this interpolation method satisfies the gauge-covariance requirement. Suppose that we have obtained a field configuration on the fermion lattice by the interpolation method. Then suppose that we make a gauge transformation on the gauge-field lattice. The links bounding the elementary plaquettes will be changed in value, and a re-application of the interpolation procedure will result in a new field configuration on the fermion lattice. We wish to show that this new field configuration can be obtained by a gauge transformation on the fermion lattice of the original fermion-lattice field configuration. Here, we paraphrase the argument presented in Ref. [30].

We have already shown that there is a gauge transformation that does this for the gauge fields that lie on the links bounding the elementary plaquettes on the gauge-field lattice. Such a gauge transformation will not, in general, leave the gauge fields that lie inside the plaquettes in the two-dimensional Lorentz gauge. However, we can always make a gauge transformation on the interior of a plaquette that returns the fields to the Lorentz gauge, without changing the fields on the links that bound the plaquette. Similarly, we can find a gauge transformation on the interior of a three-dimensional cube that returns the fields inside the cube to the three-dimensional Lorentz gauge and a gauge transformation on the interior of a four-dimensional hypercube that returns the fields inside the hypercube to the four-dimensional Lorentz gauge. Since the pure gauge-field actions are invariant under these transformations, the resulting configuration still satisfies the minimization criteria. Hence, it is identical to the field obtained by applying the interpolation method to the gauge-transformed gauge-field-lattice links. Here we are assuming the uniqueness of the interpolated field configuration.

4 The Feynman rules

By considering the Fourier transform of the lattice action, one can easily derive the Feynman rules for the double-limit procedure.

The Feynman rules for the gauge-field propagators and vertices are the same as those for a theory with lattice spacing $a_{g}$. Momenta in propagators and vertices range from $-\pi/a_{g}$ to $\pi/a_{g}$, and momentum is conserved modulo $2\pi/a_{g}$. Hence, pure gauge-field loop integrations range from $-\pi/a_{g}$ to $\pi/a_{g}$.

The Feynman rules for fermion propagators, gauge-field-fermion vertices, and $\Lambda$-fermion vertices are determined by considering the Fourier transform of the fermionic part of the action. Momenta in propagators and vertices range from $-\pi/a_{f}$ to $\pi/a_{f}$ and momentum is conserved modulo $2\pi/a_{f}$. Hence, pure fermionic loop integrations range from $-\pi/a_{f}$ to $\pi/a_{f}$.

When a gauge-field line attaches to a fermion line, one must consider the effect of the interpolation in working out the Fourier transform of the gauge field on the fermion lattice, as in (50). The interpolation introduces a regulating factor $F_{\mu}(l)$ for each connection of a gauge-field line to a fermion line. The gauge-field momentum $l$, which appears in the Fourier transform of the gauge field on the fermion lattice (50), can be written as

[TABLE]

where $q$ takes on values from $(-R+1)/2$ to $(R-1)/2$ in integer steps and $-\pi/a_{g}\leq l^{\prime}<\pi/a_{g}$. We can think of the integration over $l$ from $-\pi/a_{f}$ to $\pi/a_{f}$ as an integration over $l^{\prime}$ from $-\pi/a_{g}$ to $\pi/a_{g}$ and a sum over $q$ from $(-R+1)/2$ to $(R-1)/2$. There is an integration over $l^{\prime}$ and a sum over $q$ for each attachment of a gauge-field or $\Lambda$ line to a fermion line. The quantity $l^{\prime}$ may be interpreted as the gauge-field momentum variable in the Fourier transform of the gauge field on the gauge-field lattice (52). Only $l^{\prime}$ appears in gauge-field propagators and pure gauge-field vertices; they are insensitive to the value of $q$ because, as can be seen from (52) they are periodic, with period $2\pi/a_{g}$. In a Feynman diagram, integrations over variables of the type $l^{\prime}$ are constrained by the fact that the total of the gauge-field momentum, including the variables of the type $l^{\prime}$, is conserved, modulo $2\pi/a_{g}$, in every propagator and vertex. Thus, the gauge-field-momentum variables, including those of the type $l^{\prime}$, can be re-organized, in the usual way, into independent loop momenta, which range from $-\pi/a_{g}$ to $\pi/a_{g}$, and external momenta. In general, the fermion propagators, gauge-field-fermion vertices, and $\Lambda$ vertices depend on the value of $q$, as well as on the value of $l^{\prime}$. The sums over variables of the type $q$ are constrained only by momentum conservation, modulo $2\pi/a_{f}$, along each fermion line. Aside from this constraint, there is an independent sum over $q$ for each attachment of a gauge-field line to a fermion line.

Using these Feynman rules and (61), we see that, in the limit $a_{g}\rightarrow 0$, for momenta much less than the cutoff $\pi/a_{g}$, the Feynman rules for the fermion become the continuum Feynman rules. Therefore, we recover the required low-energy behavior of the tree-level amplitudes.

5 Counting powers of $a_{f}$

In this section we will demonstrate, for an open fermion line or for the odd-parity part of a closed fermion line, that contributions that arise when gauge-field (or $\Lambda$-field) momenta of order $\pi/a_{f}$ enter the line vanish in the limit $a_{f}\rightarrow 0$ with $a_{g}$ fixed. We call momenta of order $\pi/a_{f}$ “large” momenta. In the arguments to follow, we assume that the even-parity parts of fermion loops have been modified as in Section V B 1 to render them exactly gauge invariant. One consequence of this assumption is that all of the gauge variations must arise from the odd-parity parts of loops. The argument that we present holds in two and four dimensions. We proceed by counting the powers of $a_{f}$ associated with a contribution in which large gauge-field (or $\Lambda$-field) momenta enter a fermion line.

In the initial discussion, we assume that the fermion-loop momentum associated with a closed fermion line is not large. Since momentum is conserved, modulo $2\pi/a_{f}$, along a fermion line, if one gauge-field momentum entering a fermion line is large, at least one other gauge-field momentum entering a fermion line must be large. We assume, initially, that exactly two gauge-field momenta entering a fermion line are large.

Powers of $a_{f}$ arise from the fermion propagators, gauge-field-fermion vertices, and $\Lambda$ vertices through which the large momentum flows. It is easy to see, by making use of the power-counting rules of Section V A, that the minimum number of factors of $a_{f}$ arises if the large gauge-field momentum flows through at most one fermion propagator.

Inverse powers of $a_{f}$ can arise from the sum over variables of the type $q$ in (78). If two gauge-field momenta entering a fermion line are large, there is only one independent sum, the other sum being constrained by momentum conservation. The range of the sum contributes a factor of order $R\sim a_{f}^{-1}$ for each component of the momentum that is large.

There is a regulator factor $F$ associated with each of the points at which the two large momenta enter the fermion line. From (60), we see that each regulator factor contributes a factor $a_{f}$ for each component of the momentum that is large. Hence, the minimum number of powers of $a_{f}$ is obtained by taking only one component of the momentum to be large.

By way of illustration, let us consider the case in which the large gauge-field momentum flows through exactly one fermion propagator. As we have already noted, this case gives the minimum number of powers of $a_{f}$. The fermion propagator contributes a factor of order $a_{f}^{1}$. The large momentum also flows through two gauge-field-fermion vertices or a gauge-field-fermion vertex and a $\Lambda$ vertex. The gauge-field-fermion vertices contribute factors of order $a_{f}^{0}$ and the $\Lambda$ vertex contributes a factor of order $a_{f}^{-1}$. Hence, the propagators and vertices contribute a factor of order $a_{f}^{1}$ in the amplitude and $a_{f}^{0}$ in the gauge variation. If we take one component of the gauge-field momentum to be large, the range of the sum over $q$ gives $a_{f}^{-1}$ and the regulator factors give $a_{f}^{2}$. We conclude that, in this example, the contribution to the amplitude from the factors associated with the large gauge-field momenta is of order $a_{f}^{2}$. The contribution to the gauge variation is larger—of order $a_{f}^{1}$. This is a consequence of the fact that the large momentum associated with the $\Lambda$ field contributes an additional dimensionful factor of $1/a_{f}$ to the gauge variation. Since the contributions to the amplitude itself from this momentum region vanish as $a_{f}^{2}$, we can still conclude that the amplitude differs from a gauge-invariant expression by terms of order $a_{f}^{2}$.

Now let us relax the assumption that only two of the gauge-field momenta entering the fermion line are large. For each additional large momentum, there is at least one factor $a_{f}$ for the propagators and vertices through which it flows, a factor $a_{f}^{-1}$ for the associated sum over $q$, and a factor $a_{f}$ from the associated regulator factor. Hence, contributions involving more than two large gauge-field momenta are suppressed by at least one additional power of $a_{f}$.

We can also relax the assumption that the fermion-loop momentum associated with a closed fermion line is not large. Suppose that the loop momentum is large. Then, the entire contribution of the loop, including the sums over variables of the type $q$ and the regulator factors, arises from short distances and can be expressed in terms of local operators on the gauge-field lattice.

Consider first the case of loops containing gauge variations ($\Lambda$ vertices). All of the gauge variations arise from odd-parity loops. As we have already discussed in Section V B 2, the lattice-rotationally invariant, odd-parity, local operators of dimension $d$ or less involving a $\Lambda$ field are of the form of the ABJ anomaly. (In the present case, continuum derivatives must be replaced by lattice derivatives on the gauge-field lattice, since we are really discussing the effective theory on the gauge-field lattice.) These all vanish if the anomaly-cancellation condition (45) is satisfied. There are no such operators of dimension $d+1$. Hence, the contributions to the gauge variations from the regions of integration in which both the gauge-field momenta and the fermion-loop moment are large are of order $a_{f}^{2}$, possibly times logarithms of $a_{f}$.

Now consider the odd-parity parts of loop amplitudes. Recalling our arguments of Section V B 2 (and again replacing continuum derivatives by derivatives on the gauge-field lattice), we note that the lattice-rotationally invariant, local, odd-parity operators of dimension $d$ or less involving only gauge fields all vanish under Bose symmetrization. Furthermore, there are no lattice-rotationally invariant, odd-parity, local operators of dimension $d+1$ involving only gauge fields. Hence, the contributions to the odd-parity loop amplitudes from the regions of integration in which both gauge-field momenta and the fermion-loop moment are large are of order $a_{f}^{2}$, possibly times logarithms of $a_{f}$.

Finally, we consider the even-parity parts of loop amplitudes. Because the even-parity parts of loops are exactly gauge invariant, only gauge-invariant local operators can contribute. There is a lattice-rotationally invariant, gauge-invariant, Bose-symmetric operator of dimension $d$, namely, the one that renormalizes the gauge-field wave function. Hence, there could, in principle, be contributions, in which large gauge-field momenta flow into the even-parity parts of loops, that go as $a_{f}^{0}$, possibly times logarithms of $a_{f}$. Of course, we need not show that such contributions vanish in order to establish the gauge invariance of the double-limit procedure. Furthermore, their behavior is no worse than that of the even-parity parts of fermion loops in the absence of large gauge-field momenta, which is also logarithmic in $a_{f}$.

We must also consider the possibility that, in a Feynman diagram, inverse powers of $a_{f}$ could arise from a fermion loop other than the fermion line under consideration, and thereby lead to contributions from regions of large gauge-field momenta that are nonvanishing as $a_{f}\rightarrow 0$. We have already seen that such inverse powers of $a_{f}$ cannot arise when gauge-field momenta entering the loop are large and or when both gauge-field momenta and the fermion-loop momentum are large. The local-operator argument given for the latter case also applies when only the fermion-loop momentum is large. Therefore, no inverse powers of $a_{f}$ can arise from a fermion loop.

Let us summarize these results. We have found that, in the double limit, contributions in which a large gauge-field momentum enters a fermion loop containing a gauge variation vanish as $a_{f}$ times logarithms of $a_{f}$. This result, combined with the analysis of Section V, allows us to conclude that the odd-parity parts of fermion loops can be rendered gauge invariant by taking the double limit and by requiring the fermion to be in a representation of the gauge group that satisfies the anomaly-cancellation condition. We assume that the even-parity parts of fermion loops have been rendered exactly gauge invariant by replacing them with one-half the corresponding loop for a fermion with a vector-like coupling to the gauge field. Therefore, we have achieved our goal of making all the amplitudes in the theory gauge invariant. We have also found that contributions associated with the odd-parity parts of loops are finite in the limit $a_{f}\rightarrow 0$. This implies that the phase of the fermion determinant is finite in this limit. Furthermore, we have seen that the contributions in which a large gauge-field momentum enter the odd-parity part of a fermion loop vanish as $a_{f}^{2}$, possibly times logarithms of $a_{f}$. This result, together with the analysis of Section V B 2, implies that the phase of the fermion determinant differs from a gauge-invariant expression by terms of order $a_{f}^{2}$, possibly times logarithms of $a_{f}$, in the limit $a_{f}\rightarrow 0$.

It should be noted that the detailed power-counting rules we have presented in this subsection are specific to interpolations of the gauge fields that are discontinuous in at least one direction at the boundaries of the gauge-field hypercubes. One might devise smoother interpolations in which the gauge fields (or their higher derivatives) are continuous. For such interpolations, the regulating factor $F_{\mu}(l)$ and, hence, the contributions to the amplitudes and gauge variations would be suppressed by additional factors of $a_{f}$ when gauge-field-fermion-loop momenta are of order $\pi/a_{f}$.

It may be useful to contrast our results with those of Ref. [30]. In that work, the authors make the additional assumption that the interpolation is transversely continuous. (That assumption is valid for the interpolations that we have presented.) They are then able to show that the all the contributions in which a large gauge-field momentum enters a fermion loop are suppressed by powers of $a_{f}$. Their proof applies to the even-parity parts of loops, as well as to the odd-parity parts of loops and to loops containing $\Lambda$ vertices. They conclude, as we do, that contributions in which large gauge-field momenta enter the odd-parity parts of loops vanish as $a_{f}^{2}$. However, they also conclude that gauge variations vanish as $a_{f}^{2}$. This last result seems to be at odds with our explicit example.

6 Options for computing the determinant

In the last section we demonstrated that there exists a satisfactory procedure for computing the fermion determinant. There are actually several variants of this procedure that one can employ, and some may be more efficient than others in practical calculations. We now discuss some of these computational options.

Once one has replaced the magnitude of the fermion determinant with the square root of the determinant for a fermion with vector-like couplings to the gauge field, the magnitude of the fermion determinant has an exact gauge invariance. Therefore, one can evaluate the modified magnitude of the determinant without employing the double-limit procedure, and still obtain a gauge-invariant result. That result will be equivalent to the one obtained through the doubling-limit procedure, since the effective action is unique, aside from gauge-invariant counterterms, which can always be absorbed into a redefinition of the coupling constant.

There are several advantages in calculating the magnitude of the fermion determinant without making use of the double-limit procedure. There is the obvious advantage that one would not be faced in a numerical simulation with the computational burden of taking the limit $a_{f}\rightarrow 0$ for each gauge-field configuration. Another advantage follows from the fact that, in four dimensions, the magnitude of the determinant is divergent in the limit $a_{f}\rightarrow 0$. The divergence arises from the diagram with two external gauge fields, which generates the logarithm of $a_{f}$ that is associated with the gauge-field wave-function renormalization. In the double-limit procedure, one would need to add a wave-function-renormalization counterterm, which has the effect of replacing $\ln a_{f}$ with $\ln a_{g}$, to obtain the correct renormalization of the gauge-field-fermion coupling and to obtain a finite result. This counterterm can be determined from a one-loop calculation, since radiative corrections to the fermion loop with two external gauge fields are suppressed in the limit $a_{f}\rightarrow 0$. However, it is simpler to bypass the double limit altogether in the case of the magnitude of the determinant.

One must, of course, make use of the double-limit procedure in computing the phase of the determinant. Fortunately, in two and four dimensions, the phase is finite in the limit $a_{f}\rightarrow 0$, because, as we have seen, there are no odd-parity, Bose-symmetric renormalization counterterms.

There is one advantage in using the double-limit procedure to compute the magnitude of the determinant. The vector-like gauge symmetry of the magnitude of the determinant does not preclude the generation of a mass for the fermion field. In general, the unrenormalized fermion mass will be nonzero. However, it is easy to see that fermion self-energy diagrams are suppressed in the double-limit procedure.

In the absence of the double-limit procedure, one must tune a counterterm (i.e., the hopping parameter $\kappa$) to make the renormalized mass of the fermion with vector-like couplings vanish. In practical terms, this procedure is somewhat tricky because we wish to maintain the positivity of Wilson-Dirac determinant, so that its square root is real. Of course, it is well known, from studies of theories with vector-like interactions, how to determine the critical value of the hopping parameter, $\kappa_{\rm critical}$, at which the renormalized fermion mass vanishes. There are several procedures at one’s disposal. For example, one can use the vanishing of mass corrections to the Ward-Takahashi identities, the vanishing of the Goldstone-boson (meson) mass, or the first occurrence of a zero eigenvalue of the Wilson-Dirac operator as definitions of $\kappa_{\rm critical}$. These approaches are equivalent in the infinite volume limit. In determining $\kappa_{\rm critical}$ by any of these methods, one averages over an ensemble of gauge configurations. A given gauge configuration may yield a value of $\kappa_{\rm critical}$ that differs from the ensemble average. Therefore, if one fixes $\kappa$ to be slightly below the ensemble-average value of $\kappa_{\rm critical}$, one may encounter “exceptional” gauge-field configurations, such that the lowest eigenvalue of the Dirac operator is negative and the fermion determinant is negative. On the other hand, we expect an average of the determinant over an ensemble of gauge configurations to be positive in the thermodynamic limit of an infinite number of configurations. To the extent that the thermodynamic limit is equivalent to the infinite-volume limit, the number of “exceptional” gauge-field configurations should become vanishingly small as the volume is taken to infinity.

We note that it is straightforward to reduce the size of the gauge-variant contributions that arise from the odd-parity parts of fermion loops in the region of integration in which the fermion-loop momentum and gauge-field momenta all have magnitudes much less than $\pi/a_{f}$. These contributions are a consequence of order $a_{f}$ deviations of the tree-level lattice fermion action from the tree-level continuum fermion action. Such deviations are easily removed by employing an improved tree-level action [39, 40]. To reduce the size of the gauge variations that arise from the low-momentum region, it is necessary only to improve the Wilson term in the tree-level action.

Similarly, one can eliminate the leading gauge-variant contributions that arise from the odd-parity parts of loops in the region of integration in which the gauge-field momenta entering a loop are large, but the fermion-loop momentum itself is small. As we have seen, these contributions arise from subdiagrams in which the factors along the fermion line are the same as in a one-loop fermion self-energy diagram. In particular, the leading contribution comes from the terms corresponding to a fermion-mass renormalization. Mass generation is precluded if the action is invariant under a constant shift of the fermion field [38]. If we drop the gauging of the Wilson term (15), then the action exhibits this symmetry.******It is easy to understand diagrammatically why mass generation cannot occur. If the Wilson term is not gauged, then there are no Wilson vertices, only naive vertices. Each of these contains a $\gamma$ matrix and a factor $P_{L}$. Consider a fermion-self-energy diagram. A Wilson mass from a rationalized propagator numerator vanishes when sandwiched between two naive vertices, because of the projectors $P_{L}$. The remaining terms in the propagator numerators yield contributions with an odd number of $\gamma$ matrices, and so they don’t have the form of a mass term. In the case of the odd-parity parts of loops, all of the arguments in both this section on dynamical gauge fields and in Section V on background gauge fields are independent of whether the gauging of the Wilson term (15) is retained or not. Hence, we are free to drop the gauging of the Wilson term in computing the phase of the determinant. (In computing the magnitude of the determinant, one must retain the gauging of the Wilson term in order to maintain the vector-like gauge symmetry.)

Unfortunately, the two improvement schemes that we have mentioned are of no use unless one can also reduce the size of the violations of gauge invariance that arise from the regions of integration in which both gauge-field momenta and fermion-loop momenta are of order $\pi/a_{f}$. This probably would require the use of smoother interpolations, which, as we have already argued, ultimately require nonlocality and lead to increased errors of order $a_{g}$.

Although the violations of gauge invariance vanish as powers of $a_{f}$, a sufficiently large gauge transformation could make the coefficient of the gauge variation impractically large for numerical work. Therefore, it is probably advantageous to fix the interpolating field to a smooth gauge, such as one of the renormalizable gauges. Then one would at least avoid the spurious, large, “pure gauge” contributions to the gauge field that are known to arise from UV divergences.

VII MATRIX ELEMENTS OF FERMION OPERATORS

Since a chiral-fermion action (for example, the sum of (5), (6), (14), and (15)) is not invariant under gauge transformations, if one computes matrix elements of operators involving fermion fields straightforwardly using such an action, the result is not, in general, gauge invariant. In this section, we discuss a method for computing matrix elements of fermion operators that yields a gauge-invariant result. The method that we present is related, but not identical, to the approach that we used in computing the fermion determinant.

In analyzing the matrix elements of fermion operators, we assume that any fermions in the initial and final states have been removed by the LSZ reduction. We also assume that the total number of $\psi$’s is equal to the total number of $\overline{\psi}$’s, so that the fermion operators can be Wick contracted to form interacting propagators.

A General procedure

We begin by employing the $\gamma_{5}$ trick of Section V B 1 to move all the factors $P_{L}$ to the end points of the interacting fermion propagators, treating $\gamma_{5}$ as if it anticommuted with all Wilson masses and vertices. If each interacting propagator’s end points are separated by a fixed amount in configuration space, then there is no fermion-loop UV divergence associated with the propagator. In this case, the rearrangement changes the expression by terms of order $a_{f}$ and by terms corresponding to the renormalization counterterms associated with radiative corrections to the propagators and operator vertices. If the interacting propagator’s endpoints are separated by a distance that vanishes as $a\rightarrow 0$, then there is a fermion-loop UV divergence associated with the propagator. In this case, the rearrangement also changes the expression by terms corresponding to the renormalization counterterms associated with the fermion loop. Once we have completed this rearrangement, all of the factors $P_{L}$ are associated with the fermion operators. Of course, $P_{L}^{2}=P_{L}$, and so there is at most one such factor associated with the left side and one such factor associated with the right side of each operator.

If the operators themselves are independent of the gauge field, then the modified matrix element is exactly gauge invariant, since the fermion now has only vector-like interactions with the gauge field along its propagators. Therefore, in this case, we can compute the modified matrix element without recourse to the double-limit procedure.

If an operator involves gauge fields, for example, through a gauge-covariant derivative, then, with the modification that we have described, the even-parity part of the expression associated with that operator is still exactly gauge invariant, but the odd-parity part is not. Therefore, we can compute the even-parity part without making use of the double-limit procedure. For the-odd parity part, we must invoke the double-limit procedure to ensure gauge invariance. If an interacting propagator’s endpoints are separated by a distance that vanishes as $a\rightarrow 0$, then nonvanishing gauge variations can arise from the associated fermion-loop divergence. In this case, as was discussed in Section V B 2, we must also impose the anomaly-cancellation condition (45) in order to ensure gauge invariance.††††††Since we have applied the $\gamma_{5}$ trick here to the odd-parity part as well as to the even-parity part, the anomaly takes on a somewhat different form than in Appendix A. However, the conclusion—that the gauge variations in the presence of a background field can be removed by imposing the anomaly cancellation condition (45)—is unchanged.

The power-counting arguments that we have given previously also apply to the operator matrix elements. In particular, we expect the violations of gauge invariance arising from odd-parity operator loops to vanish as $a_{f}^{1}$, and we expect the deviations of the odd-parity loops from a gauge-invariant expression to vanish as $a_{f}^{2}$.

B Example: violation of baryon-number conservation

As an example of the procedure for computing matrix elements of operators involving fermion fields, let us consider the matrix element of the baryon-number current

[TABLE]

in the presence of dynamical gauge fields plus an external source of background gauge-field quanta. We assume that $\psi^{B}$ is part of a larger column vector $\psi$ such that the gauge group of the complete field $\psi$ satisfies the anomaly-cancellation condition (45), but the subgroup associated with $\psi^{B}$ does not.

A matrix element of $J_{\mu}^{B}$ is given by a weighted average over gauge-field configurations of

[TABLE]

where $S_{chiral}^{B}(x,x^{\prime})$ is the interacting baryon propagator, with configuration-space end points $x$ and $x^{\prime}$. The subscript “chiral” indicates that the interactions of the baryons with the gauge field are left handed. Now, $F_{\mu}$ is gauge variant. However, we can modify the definition of the matrix element so as to render it gauge invariant. We apply the $\gamma_{5}$ trick of Section V B 1 to move all of the projectors $P_{L}$ in $S_{\rm chiral}^{B}$ on the right side of (80) to the factor $\gamma_{\mu}$. The terms that we discard in this procedure all vanish in the limit $a_{f}\rightarrow 0$ or have the the forms of renormalization counterterms. The result is that $F_{\mu}$ is replaced by

[TABLE]

where $S_{\rm vector}^{B}$ is the interacting propagator for baryons with vector-like couplings to the gauge field. The expression (81) has an exact (vector-like) gauge invariance. Consequently, we can compute it without recourse to the double-limit procedure.

Now $\tilde{F}_{\mu}$ corresponds to the matrix element of a left-handed baryon current

[TABLE]

in a theory in which the baryons have vector-like interactions with the gauge field. As is well known, in four dimensions, in a theory with vector-like couplings, $\tilde{J}_{\mu}^{B}$ is not conserved: its divergence is given by the ABJ anomaly, which is nonzero in the presence of background gauge fields with nonzero winding number. Thus, we have recovered the familiar result that, once one has added such renormalization counterterms as are required to render its matrix elements gauge invariant, the baryon-number current is not conserved [41].

Of course, one could also compute the violation of baryon-number conservation directly, by examining amplitudes that have unequal numbers of incoming and outgoing baryons. Such amplitudes can be computed in the standard way by considering the contributions to the path integral of the zero modes of the Dirac operator [42]. As we have argued in Section V B 1 (see, in particular, (44)), the manipulations of the fermion determinant that we advocate do not affect the low-energy modes in the continuum limit. Therefore, the lattice and continuum calculations yield the same result.

VIII BEYOND PERTURBATION THEORY

The analyses that we have presented so far have been given in terms of weak-coupling perturbation theory. In this section, we will argue that, in the presence of an arbitrary background gauge field, the perturbation expansions for the fermion determinant and interacting fermion propagators actually determine these quantities completely, except at the zero modes of the Dirac operator. This is not to imply that one can analyze the complete theory through the use of perturbative techniques. The gauge-field sector of the theory, of course, exhibits effects that are not amenable to a perturbative analysis.

Throughout this section, we will assume that the gauge-field configuration defined on the gauge-field lattice (and implicitly on the fermion lattice) is bounded. Of course, there is no universal bound that applies to all of the gauge-field configurations in the path integral. Therefore our conclusions may not hold when one sums over all configurations. Another potential loophole arises from the fact that, configuration by configuration, the gauge fields on the fermion-field lattice may become unbounded because of singularities in the interpolating field of the type discussed by Shamir [1].

A Finite volume and fixed lattice spacing

In the arguments to follow, the convergence properties of the perturbation series are crucial. Ultimately, we wish to study these properties in the case of infinite volume and in the limit $a_{f}\rightarrow 0$. However, it is illuminating to consider first the behavior of the perturbation series for the somewhat simpler case of finite volume and fixed lattice spacing.

We begin by noting that the determinant of the lattice Dirac operator ${\cal D}$ can be written as

[TABLE]

where $\partial$ is the free Dirac operator (${\cal D}$ evaluated at $g=0$). The perturbation expansion for the effective action $\ln(\det{\cal D})$ is obtained by expanding the logarithm in (84) in powers of $g$.

As an intermediate step in analyzing the perturbation series, let us examine the series in $1/\partial({\cal D}-\partial)$, introducing a parameter $\zeta$ as the coefficient of $1/\partial({\cal D}-\partial)$ in (84). At fixed lattice spacing in a finite volume, ${\cal D}$ and $\partial$ are just finite matrices. Therefore, the logarithm can be considered to be a matrix-valued function with matrix argument. Furthermore, its expansion in powers of $\zeta(1/\partial)({\cal D}-\partial)$ has a finite radius of convergence. Let $\lambda$ be an eigenvalue of $1/\partial({\cal D}-\partial)$. Then the radius of convergence of the logarithm as a matrix-valued function of $\zeta$ is $1/|\lambda_{max}|$, where $\lambda_{max}$ is the $\lambda$ with the largest magnitude. There is a branch-point singularity in the matrix-valued function whenever $\zeta\lambda=-1$.

Now, $({\cal D}-\partial)$ is an analytic function of $g$ through the link variables $U$. Since $({\cal D}-\partial)$ vanishes as at least one power of $g$ as $g\rightarrow 0$, the perturbation series has a finite radius of convergence in $g$. The branch points at $\zeta\lambda=-1$ correspond to isolated branch points in the complex $g$ plane. Consequently, one can determine $\det{\cal D}$ almost everywhere in the complex $g$ plane by analytic continuation in $g$. Of course, there are ambiguities because of the cuts that arise from the branch points. However, the ambiguity associated with a cut has no effect on the determinant, since it leads to shifts of the argument of the exponential by $2\pi in$, where $n$ is an integer.‡‡‡‡‡‡In computing the square root of the determinant of the Wilson-Dirac operator, we choose $\kappa<\kappa_{\rm critical}$. This implies that we are to the right of the cut in $\det{\cal D}=\exp({\rm Tr}\,\ln{\cal D})$, and so there is no ambiguity in the square root. The branch points themselves correspond to zero modes of the Dirac operator. As we have argued in Section V B 1, the procedure that we use to rearrange the determinant leaves the zero modes unaffected; they are given, in the limit $a_{f}\rightarrow 0$, by the zero modes of the continuum Dirac operator.

Similarly, we can write the interacting propagator as

[TABLE]

The expansion of the right side of (85) in powers of $({\cal D}-\partial)$ has a finite radius of convergence. Therefore, the perturbation expansion of ${\cal D}^{-1}$ in powers of $g$ has a finite radius of convergence. By analytic continuation, the perturbation series determines the interacting propagator everywhere except at the zero modes of the Dirac operator.

B Infinite volume and the limit $a_{f}\rightarrow 0$

Now let us take up the infinite-volume case. Here it is most convenient to examine the convergence properties of the perturbation series, using the momentum-space Feynman rules. We are ultimately interested in the limit $a_{f}\rightarrow 0$.

In order to demonstrate that our perturbative analyses hold for arbitrary $g$, we need to prove two properties: that the perturbation series for the effective action (logarithm of the fermion determinant) and the interacting fermion propagator have finite radii of convergence, and that one can take the limit $a_{f}\rightarrow 0$ term by term in the perturbation series. To prove the first property, we need to show only that the perturbation series is absolutely convergent. To prove the second property, we must show that the perturbation series is uniformly convergent as $a_{f}\rightarrow 0$. We will demonstrate this by showing that the series can be majorized. That is, we will show that for every $a_{f}$ in a neighborhood of $a_{f}=0$, the absolute value of each term in the perturbation series is bounded by an $a_{f}$-independent series that converges. Thus, the proof of the uniform convergence of the series also demonstrates the absolute convergence of the series. We will assume that the first few terms in the perturbation series of order $g^{d}$ or less have been removed, so that we do not have to deal with individual terms in the determinant that are divergent as $a_{f}\rightarrow 0$. Obviously, subtracting a finite number of terms does not affect the convergence of the series.

First we analyze the region of integration in which all the gauge-field momenta, and the fermion-loop momentum in the case of the effective action, have magnitudes much less than $\pi/a_{f}$. Consider the contribution to a term of order $g^{n}$ that contains only single-gauge-field-fermion vertices ${\cal V}^{(1)}$. The magnitude of each vertex is bounded by an $a_{f}$-independent constant times $g$. We can obtain a bound on the magnitude each fermion propagator by dropping the Wilson term and replacing $(1/a_{f})\sin(p_{\mu}a_{f})$ by a finite constant of order unity times $p_{\mu}$. Thus, the magnitude of each propagator is bounded by an $a_{f}$ independent constant times $1/|p|$. Since we are assuming, in the case of contributions to the effective action, that the fermion momentum is much less than $\pi/a_{f}$, the volume of the integration is an $a_{f}$-independent constant. Thus, each such contribution to the interacting fermion propagator is bounded by $C(gA/k)^{n}$, and each such contribution to the effective action is bounded by $(1/n)C(Ag/k)^{n}$, where $C$ is an $a_{f}$-independent constant, $A$ is the maximum magnitude of the gauge field*****Here we are assuming that the gauge-field configuration in momentum space is bounded. In fact, the gauge field may be singular in momentum space. However, if the gauge field is bounded in configuration space, then these singularities are integrable. Hence, one could eliminate any such singularities by smearing the momentum-space gauge field over a small fraction of the range of the gauge-field momentum integration., and $k$ is the minimum of the magnitudes of the gauge-field momenta. Here, we assume that the momentum of the gauge field is cut off in the infrared by physical effects or by application of an explicit infrared regulator. We also assume that one can neglect the regions of integration in which sums of gauge-field momenta nearly vanish or, in the case of the interacting propagator, sums of gauge-field momenta and the fermion momentum nearly vanish.†††Suppose that we constrain $r$ momentum integrations so that each component of momentum has a range of size $\epsilon$ relative its unconstrained range. There are $n!/[(n-r)!r!]$ ways to do this. The volume of integration of each of the $r$ momenta is reduced by a factor $\epsilon^{d}$. At most $r$ propagators are enhanced by a factor $1/\epsilon$. Therefore, the net effect of constraining momenta is to multiply the bounds we have obtained by $(\epsilon^{d-1}+1)^{n}\leq C^{n}$, where $C$ is an $a_{f}$-independent constant.

Suppose that we include the possibility of multi-gauge-field-fermion vertices. The effect of these is to replace propagator factors by powers of $a_{f}$. Therefore, we can bound any propagator factor by $[C_{1}a_{f}+(C_{2}/k)]$, where $C_{1}$ and $C_{2}$ are $a_{f}$-independent constants. This implies that the contributions to the interacting propagator are bounded by $(Ag)^{n}[C_{1}a_{f}+(C_{2}/k)]^{n}$ and the contributions to the effective action are bounded by $(1/n)(Ag)^{n}[C_{1}a_{f}+(C_{2}/k)]^{n}$. Thus, we see that, for $g$ small enough, these contributions are bounded by the terms in a convergent geometric series that is independent of $a_{f}$.

Now consider the region of integration in which some of the gauge-field momenta are of order $\pi/a_{f}$. As we have seen in Section VI B 5, such contributions are suppressed by powers of $a_{f}$. If a gauge-field momentum of order $\pi/a_{f}$ passes through a fermion propagator, then the propagator is bounded by a constant times $a_{f}$. Thus, we can again bound the propagator factors by $[C_{1}a_{f}+(C_{2}/k)]$. There are additional powers of $a_{f}$ from the regulating factors $F$ associated with the vertices. Otherwise, the bounds on vertices are unchanged. The powers of $a_{f}$ in the regulating factors more than compensate for inverse powers of $a_{f}$ associated with the ranges of the sums over the gauge-field-momentum variables $q$ in (78). Therefore, the contributions to the interacting propagator and the effective action are again bounded by $(Ag)^{n}[C_{1}a_{f}+(C_{2}/k)]^{n}$ and $(1/n)(Ag)^{n}[C_{1}a_{f}+(C_{2}/k)]^{n}$, respectively. For $g$ small enough, these quantities are, in turn, bounded by the terms in a convergent geometric series that is independent of $a_{f}$.

Finally, we consider contributions to the effective action from the region of integration in which the fermion-loop momentum is of order $\pi/a_{f}$. We see from (38) and the surrounding discussion that, for gauge-field momenta with magnitudes much less than $\pi/a_{f}$, such contributions are bounded by an $a_{f}$-independent constant times $(Ag)^{n}a_{f}^{n-d}$. The argument of the preceding paragraph shows that contributions from gauge-field momenta of order $\pi/a_{f}$ do not change this bound. Again, for $g$ small enough, the contributions are bounded by the terms in an $a_{f}$-independent, convergent geometric series.

We conclude that the perturbation series for the interacting propagator and the effective action have finite radii of convergence and are uniformly convergent in the limit $a_{f}\rightarrow 0$. Therefore, the perturbation series determine the propagator and the fermion determinant by analytic continuation, except at singularities. Furthermore, we can take the limit $a_{f}\rightarrow 0$ term by term. In this limit, the singularities correspond to the zero modes of the continuum Dirac operator.

Therefore, the conclusions that we have reached through a perturbative analysis of the fermion determinant and interacting propagator apply for arbitrary $g$. In particular, we can conclude that, in the continuum limit, the prescriptions we have given for computing the fermion determinant and the matrix elements of fermion operators give the correct low-energy amplitudes and yield gauge-invariant expressions.

IX SUMMARY AND DISCUSSION

We have presented a general procedure for constructing gauge-invariant lattice formulations of theories of chiral fermions interacting with gauge fields. The procedure involves three key ingredients: (1) the fermions must be in an anomaly-free representation of the gauge group; (2) one must replace the magnitude of the fermion determinant with the square root of the determinant for a fermion that has vector-like couplings to the gauge field, but that is otherwise identical to the original fermion; (3) one must implement the gauge-field action on a lattice with spacing $a_{g}$ and the interacting fermion-field action on a lattice with spacing $a_{f}$, define a suitable interpolation of the gauge field to the fermion-field lattice, and take the limit $a_{f}\rightarrow 0$ before taking the limit $a_{g}\rightarrow 0$.‡‡*‡A typical fermion action is given by the sum of (5), (6), (14), and (15). The corresponding action for a fermion with vector-like couplings is obtained by setting $P_{R}=P_{L}=1$. In four dimensions, all three of these conditions are required to ensure the gauge invariance of the formulation. In this procedure, the magnitude of the determinant is exactly gauge invariant. The gauge variations of the phase of the determinant vanish as $a_{f}^{1}$, and the deviations of the phase of the determinant from a gauge-invariant expression vanish as $a_{f}^{2}$, possibly times logarithms of $a_{f}$. (We note that the result of Ref. [30] for the power behavior of the gauge variations seems to differ from the one derived in this paper.)

We have also presented a closely related method for defining, in a gauge-invariant fashion, matrix elements of fermion operators in chiral theories. As was shown in Section VII B, the application of this method to the baryon-number current leads to the familiar conclusion that that current is not conserved.

The analysis of these methods is couched in weak-coupling perturbation theory. In analyzing the properties of a UV regulator, of which the lattice is an example, we are concerned with the behavior of the theory near the cutoff. Hence, one might hope, in the case of asymptotically free theories, that the perturbation expansion would be a reliable guide to that behavior.

Furthermore, as we have argued in Section VIII, in the presence of a given gauge-field configuration, the perturbation series defines the interacting fermion propagator and the fermion determinant everywhere except at zero modes of the Dirac operator. The convergence of the series is uniform in $a_{f}$, so that one can analyze the continuum limit term by term. Hence, the methods for computing the determinant and propagator are valid in the presence of a nonperturbative gauge-field configuration. We have not addressed the issue of the summation over gauge-field configurations outside of the perturbative analysis.

Shamir [1] has presented an argument that potentially undermines these analyses. He observes that, if an interpolation of the gauge field is gauge covariant, then the interpolating field is related to a smooth field by a gauge transformation that is, in general, topologically nontrivial. Hence, the interpolating field may possess singularities. Such singular fields violate the smoothness requirement for gauge fields on the fermion-field lattice that was used in the power-counting analyses of Section VI B 5 and also violate the assumption of the boundedness of the gauge fields that was made in Section VIII. It is possible that one might avoid these difficulties by fixing to a suitable gauge on the gauge-field lattice. However, this is an open question.

Putting aside questions of principle, it is not yet clear that the procedure presented will be tractable in practical numerical calculations. The obvious stumbling block is the double-limit procedure for $a_{f}$ and $a_{g}$, which could lead to computing requirements that are much greater than in the case of a single lattice-spacing limit.

In computing the magnitude of the fermion determinant, one has two distinct options. One can apply the double-limit procedure. Then one must tune a counterterm that renormalizes the gauge-field wave function in order to keep the magnitude of the determinant finite in the limit $a_{f}\rightarrow 0$ and to obtain the correct renormalization of the gauge-field-fermion coupling. The coefficient of this counterterm is readily computed in perturbation theory, since it is generated only by the diagram with a single fermion loop and two external gauge fields.

On the other hand, the magnitude of the fermion determinant is exactly gauge invariant, once one has replaced it with the square root of the determinant for a fermion with vector-like interactions. Therefore, one can compute the magnitude of the determinant by taking $a_{f}=a_{g}$. Since a vector-like gauge symmetry does not preclude the generation of a fermion mass, one must also tune a mass counterterm (hopping parameter), so as to keep the fermion massless.§§*§The diagrams that generate fermion masses are suppressed in the double-limit procedure, and so no mass counterterm is required in that case. (In practice, it may be a challenging problem to approach the critical value of the hopping parameter in such a way that the positivity of the determinant is maintained. See the discussion in Section VI B 6.) In this single-limit procedure, all other renormalization counterterms can be absorbed into a redefinition of the coupling constant. Hence, only the fermion mass and the coupling constant need be tuned in taking the continuum limit.

It seems possible that one would need to compute only the magnitude of the fermion determinant in updating gauge-field links, computing the phase of the determinant as an expectation value once equilibrated lattices had been generated. If this turns out to be the case, then the use of a single-limit procedure for the magnitude of the determinant would result in an even greater relative reduction of the computing time.

In computing the phase of the fermion determinant one must employ the double-limit procedure. This computation is mitigated somewhat in two and four dimensions by the fact that, owing to the absence of odd-parity counterterms in an anomaly-free theory, the phase is actually finite in the limit $a_{f}\rightarrow 0$. Therefore, one can carry out a straightforward extrapolation to obtain the limit.

One source of error in the extrapolation is easily reduced. As we have seen in Section VI, order-$a_{f}^{2}$ deviations of the phase of the determinant from a gauge-invariant expression arise from the region of integration in which the gauge-field momenta and the fermion-loop momentum associated with a given fermion loop are much smaller in magnitude than $\pi/a_{f}$. In this region, the deviations from the limiting result come from the deviations of the tree-level lattice action from the tree-level continuum action. The order in $a_{f}$ of these deviations can readily be increased through the use of improved actions [39, 40]. Similarly, one can eliminate the order-$a_{f}^{2}$ gauge-variant contributions to the phase of the determinant that arise from the region of integration in which gauge-field momenta are large and the associated fermion-loop momentum is small. One can accomplish this by dropping the gauging of the Wilson term (15) in computing the phase of the fermion determinant (but not the magnitude). Then there is a symmetry under constant shifts of the fermion field [38] that precludes the generation of fermion-mass terms, which give the largest gauge-variant contributions.

Unfortunately, such improvement programs are of limited utility, since errors also arise from the region of integration in both gauge-field momenta and fermion-loop momenta are of order $\pi/a_{f}$. As we showed in Section VI B 5, when one uses an interpolation in which the gauge field is discontinuous along least one direction at the boundaries of the gauge-field lattice hypercubes, these errors are of order $a_{f}^{2}$. The use of a smoother interpolation, in which the gauge fields (or higher derivatives) are continuous, would, in general, suppress these errors by additional factors of $a_{f}$. However, such interpolations are necessarily less local. In general, the order $a_{g}$ errors increase as one increases the distance on the gauge-field lattice between the sites that enter in the interpolation.

Although gauge-variant contributions ultimately vanish as $a_{f}\rightarrow 0$, the presence of large, “pure gauge” contributions in gauge-field configurations might make the approach to that limit problematic in numerical work. It is probably sensible, therefore, to fix the interpolating field to a smooth gauge, such as one of the renormalizable gauges, to ensure at least that the known, spurious, “pure gauge” contributions are absent.

In testing the ideas of this paper in numerical simulations, it would be most efficient, computationally, to consider two-dimensional theories. Then, anomaly cancellation can be achieved by introducing both left- and right-handed fermions, such that the sum of ${\rm Tr}\,(T_{a}T_{b})$ for the left-handed fermions is equal to the sum of ${\rm Tr}\,(T_{a}T_{b})$ for the right-handed fermions [43]. Strictly speaking, two-dimensional theories do not require the double-limit procedure. That is because, as can be seen from (46), the only divergent subdiagram is a fermion loop with exactly two external gauge fields; there are no divergent subdiagrams containing gauge-field propagators. However, the odd-parity part of a fermion loop with two external gauge fields is zero by virtue of the anomaly-cancellation condition (45). Therefore, the violations of gauge invariance that arise from the odd-parity parts of fermion loops vanish in limit $a_{f}=a_{g}\rightarrow 0$. Nevertheless, one could use the two dimensional theories as a testing ground for methods of extrapolating to the limit $a_{f}\rightarrow 0$ with $a_{g}$ fixed. One could check the gauge invariance of the fermion determinant and also compare the results for various physical quantities, such as the mass spectrum, with analytic results.

More stringent tests of the methods presented here could be obtained in four dimensions. Again, one could test the convergence of the extrapolation to $a_{f}=0$ and the gauge invariance of the determinant. Also, in weak coupling, one could compare results for physical quantities in the standard electroweak model with calculations in weak-coupling perturbation theory.

It is clear that the fermion determinant we have described corresponds to a complex effective action. This is a general property of chiral gauge theories that would be expected to hold regardless of the lattice formulation chosen: the effective action receives imaginary contributions that are independent of the UV regularization from finite odd-parity parts of fermion loops. It remains an open question as to whether one can devise practical means for handling such complex actions in numerical simulations.

Acknowledgements.

I would like to thank Maarten Golterman, Eve Kovács, Peter Lepage, and D.K. Sinclair for a number of illuminating conversations on the topics discussed in this paper. I would also like to thank Andreas Kronfeld and Yigal Shamir for their comments on earlier versions of this paper. Special thanks are due to Maarten Golterman and Eve Kovács for their extensive and detailed comments on several versions of the manuscript. This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, under Contract W-31-109-ENG-38.

A COMPUTATION OF THE ANOMALY

In this Appendix we present a calculation of the gauge variation of the odd-parity parts of fermion loops in the presence of a background gauge field in four dimensions [44]. For simplicity, we restrict ourselves to the case in which the Wilson term has not been gauged. If one includes the gauging of the Wilson term (15), then one must consider additional contributions to the gauge variation involving $\Lambda$-gauge-field-fermion vertices.

We will use repeatedly the fact that a trace containing an odd number of $\gamma_{5}$’s is nonvanishing only if it contains four factors that are linearly independent combinations of the the matrices $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $\gamma_{4}$. These linearly independent combinations can come from three sources: the $\gamma$ matrices associated with naive vertices in the loop, the $\gamma$ matrices associated with external momenta in propagators, and the $\gamma$ matrices associated with the loop momentum in propagators.

In order to expose the external momenta, we expand the propagators and vertices in a Taylor series in the external momenta times the lattice spacing $a$. We can use the result, derived in Section V A, that a loop containing a $\Lambda$ vertex receives a nonvanishing contribution in the limit $a\rightarrow 0$ only from the region of integration in which the magnitude of the loop momentum is of order $\pi/a$. In this region, it is easy to see, from the discussion in Section V A and the fact that the external momenta are assumed to be much smaller in magnitude than the cutoff $\pi/a$, that the $n$th term in the Taylor expansion has a relative suppression factor $a^{n}$. Thus, for a loop with degree of divergence $D$, terms in the Taylor expansion containing more than $D$ factors of the external momenta do not receive a nonvanishing contribution from the region of large loop momentum in the limit $a\rightarrow 0$. Therefore, we retain only the first $D$ terms in the Taylor expansion. For these terms, it can be seen, from the discussion in Section V A, that the region of integration in which the magnitude of the loop momentum is much less than $\pi/a$ gives a negligible contribution. Thus, we can extend the range of the integration to the entire Brillouin zone.

We also note that the $\gamma$ matrices associated with the loop momentum can never contribute the required linearly independent factors: if a term contains an odd number of $\gamma$-matrix factors associated with the loop momentum, it gives a vanishing contribution because the integrand is an odd function of the loop momentum; if a term contains an even number of $\gamma$-matrix factors associated with the external momentum, these factors can be brought together by using the anti-commutation relations and eliminated by using $(\gamma\cdot a)^{2}=a^{2}$.

Armed with these facts, let us consider in turn the various contributions up to those containing four external gauge fields.

The contribution involving one $\Lambda$ vertex and no external gauge fields vanishes by Abelian charge-conjugation symmetry.

Next consider the contribution involving one $\Lambda$ vertex and one external gauge field. If the gauge-field vertex is a naive vertex, it can contribute one of the linearly independent $\gamma$-matrix factors. The one independent external momentum can contribute another. However, that is not enough to saturate a trace containing an odd number of $\gamma_{5}$’s.

In the contribution involving one $\Lambda$ vertex and two external gauge fields, we can have at most two factors of external momentum in the Taylor expansion and still obtain a nonvanishing contribution in the limit $a\rightarrow 0$. Then, in order to obtain a nonvanishing trace, we must take all of the gauge-field-fermion vertices to be of the type ${\cal V}^{(1)}$, which involves a single gauge field, and we must retain terms proportional to the external momentum only in the Taylor expansions of the propagators. The nonvanishing contribution then comes from the diagram of Fig. 3(a), whose amplitude we denote by $A^{(2)}$, plus the diagrams obtained by permuting the gauge fields. That contribution is given by

[TABLE]

Here, sums over repeated indices are understood. The subscript “odd” on the trace means that we retain only those terms that contain an odd number of $\gamma_{5}$’s, and “perm” means permutations of the symbols separated by semicolons, i.e., permutations of the gauge fields. In the last line we have used the fact, which follows from the computation of the trace, that $I_{\mu\nu\rho\sigma}^{(2)}$ is proportional to $\epsilon_{\mu\nu\rho\sigma}$.

A similar analysis shows that the nonvanishing contribution involving one $\Lambda$ vertex and three external gauge fields is given in the limit $a\rightarrow 0$ by the diagram of Fig. 3(b), whose amplitude we denote by $A^{(3)}$, plus the diagrams obtained by permuting the gauge fields. In this case, $D=1$, and so we retain only one power of the external momentum in the Taylor expansion. The result is

[TABLE]

Here, we have used the facts that only the first Dirac trace is nonzero and that it is proportional to $\epsilon_{\mu\nu\rho\sigma}$.

It is easily seen that the contribution involving one $\Lambda$ vertex and four external gauge fields is given in the limit $a\rightarrow 0$ by the diagram of Fig. 3(c), whose amplitude we denote by $A^{(4)}$, plus the diagrams obtained by permuting the gauge fields. In this case, $D=0$, and so we set the external momenta equal to zero. The result is

[TABLE]

Again we have used the fact that the Dirac trace is proportional to $\epsilon_{\mu\nu\rho\sigma}$. In fact, direct computation of the trace shows that

[TABLE]

We see that the odd-parity contributions from the fermion loops all vanish in the limit $a\rightarrow 0$ if the anomaly-cancellation condition (45) is satisfied.

Now let us sketch a method by which the calculation of $I^{(2)}$ and $I^{(3)}$ can be completed. If we drop the color factors in (A), then the resulting expressions correspond to the calculation of the gauge variations in an Abelian theory. Since $I^{(2)}$ and $I^{(3)}$ are symmetric under cyclic permutations of the gauge fields, we can compute them by considering cyclic permutations of the Abelian expressions for the gauge variations.

Consider the quantity $\tilde{\Gamma}_{\alpha\mu\nu\ldots}^{(n)}$, which is the Abelian amplitude associated with the odd-parity part of a particular set of diagrams involving a fermion loop, $n$ gauge fields (with indices $\alpha\mu\nu\ldots$), and no $\Lambda$ vertices. We include in $\tilde{\Gamma}^{(n)}$ the diagram with no multi-gauge-field-fermion vertices and the diagram with a single two-gauge-field-fermion vertex involving the gauge fields with indices $\alpha$ and $\mu$. We note the following relation between the Abelian gauge variation and $\tilde{\Gamma}^{(n)}$:

[TABLE]

where the tildes denote the Abelian case,

[TABLE]

and $d_{\alpha}$ is defined in (13). This relation follows from the fact that the left side of (A26) is the gauge variation that one obtains by taking

[TABLE]

which is equivalent to (23) in an Abelian theory, and absorbing the transformation of the fermion fields (25) into a change of variables in the path integral. (One Fourier transforms (A28) with respect to the coordinate of the gauge field to obtain the left side of (A26).) At a graphical level, the relation (A26) is obtained by applying repeatedly the Feynman identity

[TABLE]

as in textbook demonstrations of gauge invariance at the Feynman-graph level. The $M$ terms, of course, give the $\Lambda$ vertices on the right side of (A26). For the inverse propagator terms, one does not find the simple pairwise cancellation that occurs in the continuum theory because the lattice vertices are momentum dependent. It follows from the recursion relation (12) that this momentum dependence is compensated by the contributions that one obtains by contracting $d_{\mu}(k)$ with the two-gauge-field vertices. The result is a complete cancellation of the inverse propagator terms.¶¶*¶If we had gauged the Wilson term in the action, then there would be Wilson vertices in the amplitudes, as well as naive vertices. The cancellation of the inverse propagator terms would fail in the presence of the Wilson vertices because they commute rather than anticommute with the $\gamma_{5}$’s in the inverse-propagator terms in (A30). Consequently, a more complicated identity than (A26), involving $\Lambda$-gauge-field-fermion vertices, would be obtained.

Now, $\tilde{\Gamma}^{(n)}$ receives no contributions from the region of integration in which the magnitude of the loop momentum is of order $\pi/a$. This follows from the fact, discussed in Section VI B 5, that the odd-parity parts of loops have no renormalization counterterms that are invariant under cyclic permutations of the gauge fields. It can also be seen by expanding $\tilde{\Gamma}^{(n)}$ in a Taylor series in the external momenta. The first $5-n$ terms in the expansion have a vanishing trace under cyclic permutations of the gauge fields; the remainder in the expansion is suppressed by powers of $a$ when the magnitude of the loop momentum is of order $\pi/a$. We conclude that we can evaluate $\tilde{\Gamma}^{(n)}$ (including all permutations of the gauge fields) by taking the limit $a\rightarrow 0$ in the propagators and vertices. The result is just the continuum expression. Thus,

[TABLE]

The right side of (A32) is just the continuum expression for the ABJ anomaly. We can evaluate it by considering the gauge variation of the continuum action in the presence of a UV regulator. If we impose a Pauli-Villars regulator, then we obtain expressions that are identical to those in (A5) and (A17), except that there are no color factors, the Wilson mass $M(p)$ is replaced everywhere by the Pauli-Villars mass, the limit $a\rightarrow 0$ is taken in the remaining terms in the propagators and vertices, and there is a minus sign because one subtracts the massive Pauli-Villars-regulator contribution. The results are

[TABLE]

which, upon continuation to Minkowski space, can be seen to be in agreement with previous calculations of the gauge (consistent) anomaly [45].

This result is actually independent of the choice of UV regulator. As we have already mentioned, if one assumes symmetry under cyclic permutations of the gauge fields, then there are no renormalization counterterms for the odd-parity parts of the ordinary fermion-loop amplitudes (those associated with diagrams that do not contain $\Lambda$ vertices). The absence of counterterms guarantees that the amplitudes themselves are regulator independent. Furthermore, the anomaly can be obtained from the amplitudes by varying the gauge fields according to (23) and absorbing the transformation of the fermion fields (25) into a change of variables in the path integral, as was discussed explicitly for the Abelian case in reference to (A26). Therefore, the anomaly is also regulator independent. In particular, we would have obtained the result (A) had we chosen to retain the gauging of the Wilson term in the action.