CHINESE JOURNAL OF PHYSICS
VOL. 38, NO. 3-II
JUNE 2000
The Regularization Problem in Chiral Gauge Theories J. ZINN-JUSTIN
CEA-Saclay, Service de Physique Theorique*, F-91191 Gif-sur-Yvette Cedex, FRANCE et Universite de Cergy{Pontoise email:
[email protected] (Received Jan. 24, 2000) In these notes we review various perturbative and non-perturbative conventional regularization techniques in quantum eld theory. We recall that most of these techniques fail in the case of gauge theories with chiral fermions. This is not surprising since in such eld theories anomalies can be present. Of particular interest is the method of lattice regularization, because it can be used, beyond perturbation theory, to determine physical properties of eld theories by other numerical techniques. There the manifestation of this diculty takes the form of a doubling of the fermion degrees of freedom. Until recently this has prevented a straightforward numerical study of chiral theories. The recently discovered solutions of the Ginsparg{Wilson relation and the method of overlap fermions seem to provide an unconventional solution to this dicult problem and to indicate that the problem was in essence technical rather than re ecting an inconsistency of chiral gauge theories beyond perturbation theory.
PACS. 11.15.Ha - Lattice gauge theory. PACS. 11.30.Rd - Chiral symmetries. PACS. 11.30.Fs - Global symmetries.
I. Introduction Quantum eld theories are in general a ected by UV divergences. Regularization is a useful intermediate step in the renormalization program, which consists in modifying the initial theory at short distance, large momentum or otherwise to render perturbation theory nite [1]. So long as one is not concerned with symmetries it is easy to nd methods, like the various forms of Pauli{Villars's regularization [2]. However sometimes one wants to preserve ∗ Laboratoire de la Direction des Sciences de la Matiere du Commissariat a l'Energie Atomique
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c 2000 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
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some formal symmetry of the unrenormalized theory. As long as only global linearly realized symmetries are concerned, with some care the same methods can be used. Problems may, however, arise when the symmetries correspond to non-linear or local transformations, like in the examples of non-linear σ models or gauge theories. Indeed there are really two types of large momentum divergences. First all eld theories are a ected by divergences resulting from the coupling of an in nite number of degrees of freedom. Their precise form varies with the number of space dimensions, but these divergences can be regularized by simple methods. The second type of divergences is not speci c to eld theory, being in fact already present at the level of perturbation theory in simple quantum mechanics. These divergences are related to the problem of quantization and order of quantum operators. They are present as soon as derivative couplings are involved, or when fermions are coupled to gauge elds (because fermions are never classical). They re ect the property that the knowledge of the classical action is not sucient to infer the form of the quantized theory. It is easy to verify that in this case Pauli{Villars's type regularizations are in general not applicable or sucient to provide a complete regularization. Other methods have to be explored. In many examples dimensional regularization [3] solves the problem because then the commutator between eld and conjugated momentum taken at the same point vanishes. However in the case of chiral fermions dimensional regularization fails because no continuation of the γ5 matrix retains all its properties [4]. One may also be interested in regularization schemes which have a meaning beyond perturbation theory, either because one is concerned about the existence of quantum eld theory, and because one wants to perform non-perturbative calculations. At present the most successful scheme is based on lattice regularization of euclidean quantum eld theory. One veri es that such regularization indeed speci es an order between quantum operators, and therefore, when applicable solves the ordering problem of non-linear σ -models or gauge theories. However, again it fails in case of chiral fermions for reasons we recall. That no conventional regularization scheme can be found in the case of gauge theories with chiral fermions, is not surprising since we know examples of theories with anomalies, i. e. theories in which the symmetry present at the classical level can not be implemented in the quantum theory. It is therefore not so surprising that this problem has for many years found no solution. Recently the situation has drastically changed because solutions of the so-called Ginsparg-Wilson relation have been found. As a consequence, it has become rather clear that the absence of a non-perturbative regularization, was simply a technical issue, and did not re ect some intrinsic non-perturbative inconsistency of chiral gauge theories, as one may have feared. The new regularization scheme is the main topic of many other contributions to the meeting. Since it has a natural implementation in ve dimensions, it again opens the door to speculations about higher space dimensions. In this contribution we mainly review the conventional regularization methods to explain the diculties which appear in the regularization of perturbation theory, in chiral
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gauge theories. We then brie y indicate how the solutions of Ginsparg{Wilson relation evade some of these problems.
II. Cut-o and Pauli{Villars's regularization We rst discuss methods which work in the continuum (compared to lattice methods) and at xed dimension (unlike dimensional regularization). The idea then is to modify the behaviour of the eld propagators beyond a large momentum , to render all Feynman diagrams more convergent.
II-1. Matter elds
A simple modi cation of the propagator improves the convergence of Feynman diagrams at large momentum. For example the inverse propagator m2 + p2 of the scalar eld can be replaced by (1) p2 + m2 + α2 p4 /2 + α3 p6 /4 + · · · + αn p2n /2n−2 , and the degree n chosen large enough to render all diagrams convergent. The parameter is the cut-o . In the large cut-o limit the original propagator is recovered. This is the spirit of Pauli{Villars's regularization. More general modi cations are possible. Schwinger's proper time method suggests: Z ∞ 1 (2) = dt ρ t2 e−t(p +m ) , 2 2 p + m reg. 0 in which ρ (t) satis es the condition lim ρ (t) = 1, (3) t→∞ and decreases fast enough when t goes to zero. However the regularization has to satisfy one important condition: the regularized propagator should remain a smooth function of the momentum p. Indeed singularities in the momentum representation generate, after Fourier transformation, contributions to the the large distance behaviour of the propagator, and we want to modify the theory only at short distance. For spin 1/2 fermions a similar method is applicable. The inverse propagator m + i6 p can be replaced by m + i6 p 1 + α1 p2 /2 + · · · + αn p2n /2n . (4) 2
2
Global linear symmetries. To implement symmetries of the classical action at the quantum level, we need a regularization scheme which preserves the symmetry. This requires some care but can always be achieved for linear global symmetries, i. e. symmetries which correspond to transformations of the elds of the form (5) φR (x) = R φ(x) ,
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where R is a constant matrix. To take an example directly relevant here, a theory with massless fermions may have a chiral symmetry θ (x) = ψ (x) eiθγ . (6) ψθ (x) = eiθγ ψ (x), ψ The substitution (4) (for m = 0) preserves chiral symmetry. Pauli{Villars's inspired regularizations have several advantages: one can work at xed dimension and in the continuum. However, in all models which already in quantum mechanics have divergences due to problem of order between quantum operators, a class of Feynman diagrams cannot be regularized by this method. Quantum eld theories where this problem occurs include models which have non-linear or gauge symmetries. 5
5
Regulator elds. Let us note that Pauli{Villars's regularization has another, sometimes equivalent, formulation based on the introduction of regulator elds. To regularize the action S (φ) for the scalar eld φ: S (φ) =
Z
dd x
1 2 φ + V (φ) , 2 φ − + m
(7)
one introduces additional elds φk , k = 1, . . . , n, and consider the modi ed action Sreg. (φ, φk ): Z X 1 1 d φk − + Mk2 φk Sreg. (φ, φk ) = d x φ − + m2 φ + 2 2 z k X i (8) +V φ + φk . With the action (8) any internal φ propagator is replaced by the sum of the φ propagator and all the φk propagators zk / (p2 + Mk2 ). An appropriate choice of the constants Mk and zk then improves the large momentum behaviour of Feynman diagrams. In next section we show how the same idea allow to regularize the fermion loops in the background of a gauge eld.
II-2. Gauge elds elds.
We consider here only covariant gauges. Power counting then is the same as for scalar
Abelian gauge theory. Because Fµν is gauge invariant and the action for the scalar ∂µ Aµ is arbitrary, we can make the gauge eld propagator decrease arbitrarily fast at large
momentum. However the situation is quite di erent for charged matter elds, because only covariant derivatives are allowed. One thus gets a regularized fermion action of the form: Z ψ, Aµ = dd x ψ (x) M + D 6 2 /2 + · · · + αsD 6 2s /2s ψ (x). 6 (9) 1 + α1D S ψ,
Note this method of regularization, unlike dimensional or lattice regularizations, preserves chiral symmetry. One-loop divergences. The propagator, as a function of the momentum k, now decreases indeed as k−2s−1 . However, at the same time, the regularization has generated new 6 2s+1 in powers of Aµ one obtains in particular a more singular interactions. Expanding D
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A 6 ψ vertex with 2s derivatives. 6 . Therefore the action contains a ψ term proportional to 6 ∂ 2sA A 6 vertices of If we now consider a one-loop diagram with only fermions in the loop and ψψ this kind, we immediately see that the power counting is independent of s because there are exactly as many fermion propagators as vertices and therefore the additional powers of momentum in the numerator cancel the additional powers in the denominator. Higher orders. This cancellation is special to one-loop matter diagrams. A multiloop diagram is formed by fermion loops joined by gauge eld lines. Since the behaviour of the gauge eld propagator at large momentum can be arbitrarily improved, all multi-loop diagrams can be rendered super cially convergent. In the case of scalar matter diagrams, scalar self-interaction vertices can be added, but then the number of matter propagators exceeds the number of gauge eld vertices and again the diagrams can be made super cially convergent. Therefore the only remaining divergent diagrams are one-loop matter eld diagrams which are generated by the determinant coming from the gaussian integration over matter eld in an external gauge eld. These one-loop diagrams can nally be regularized by adding to the action a set of regulator elds. In the case of fermion matter one introduces boson (with spin) and fermion regulator elds with masses of the order of the cut-o : (M + D 6 ) ψ 7→ ψ
r X i=1
i (MF,i + D 6 ) ψi + ϕi∗ (MB,i + D 6 ) ϕi . ψ
(10)
However, these mass terms then break a possible chiral symmetry, and this method is not applicable with chiral fermions. If one is interested in preserving chiral properties one can use the following strategy: one rst regularizes the one-loop diagrams in a way which breaks the symmetry. One then veri es whether one can choose counter-terms in such a way that the renormalized diagrams satisfy the WT identities required by the gauge symmetry. One then inserts the one-loop renormalized diagrams in the general diagrams regularized by Pauli{Villars's method. Non-abelian gauge theories. Compared with the abelian case, the new features of the non-abelian gauge action are the presence of gauge eld self-interactions and ghost terms. The gauge action is Z 1 (11) S (Aµ ) = tr dd x Fµν Fµν , 4g 2 where Fµν is the curvature tensor (12) Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ]. The ghost eld action takes the form: Z µ Dµ C , Dµ = ∂µ + [Aµ , •]. Sghost = tr dd x C∂ (13)
The ghost elds thus have a simple δab /p2 propagator and canonical dimension one in four dimensions.
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The problem of regularization in non-abelian gauge theories has many features in common with the abelian case, as well as with the non-linear σ -model. The regularized gauge action takes the form: S ( Aµ ) =
Z
dd x tr Fµν P D2 2 Fµν ,
(14)
in which P is a polynomial of arbitrary degree. In the same way the gauge function ∂µ Aµ is changed into: (15) ∂µ Aµ 7−→ Q ∂ 2 2 ∂µ Aµ , in which Q is a polynomial of same degree as P . As a consequence both the gauge eld propagator and the ghost propagator can be arbitrarily improved. However, as in the abelian case, the covariant derivatives generate new interactions which are more singular. It is easy to verify that the power counting of one-loop diagrams is unchanged while higher order diagrams can be made convergent by taking the degrees of P and Q large enough. Pauli{ Villars's type modi cation regularizes all diagrams except, as in all geometrical models, some one-loop diagrams. As with charged matter the one-loop diagrams have to be examined separately. For fermion matter it is however still possible as, in the abelian case, to add a set of regulator elds, massive fermions and bosons with spin. This procedure breaks a possible chiral symmetry. In the chiral situation the problem of the compatibility between the gauge symmetry and the quantization is reduced to an explicit veri cation of the WT identities for the one-loop diagrams. Note that the preservation of gauge symmetry is necessary for the cancellation of unphysical states in physical amplitudes, and thus essential to the physical relevance of the quantum eld theory.
III. Dimensional regularization Dimensional regularization seems to have no meaning outside perturbation theory since it involves continuation of Feynman diagrams in the parameter d (d is the space dimension) to arbitrary complex values. However this regularization very often leads to the simplest perturbative calculations [5]. Its use requires requires however some care in massless theories because this rules may lead to cancellation of UV logarithmic divergences by IR divergences. More important here, it is not applicable when some essential property of the eld theory is speci c to the initial dimension. An example is provided by theories containing fermions in which parity symmetry is violated. Gauge elds. Dimensional regularization is well suited to perturbative calculations in QED. It solves the problem of geometrical models because the divergences of quantization origin which appear in a local theory have the form ~ ~ [π (x), φ(x)] = i δ d−1 (0) = i (2π )1−d
Z
dd−1 p ,
(16)
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where π (x) is the momentum conjugated to the eld φ(x). In dimensional regularization such a quantity vanishes identically. When Feynman diagrams are calculated in the momentum representation, these divergences appear in the form of powers of the cut-o , and in dimensional regularization such divergences vanish automatically due to the rule Z
Z
dd k = 0. k2
(17) Problems however arise in the case of gauge theories with chiral fermions, because the special properties of γ5 are involved as we recall below. Fermions. For fermions belonging to the spinorial representation of O(d) the strategy is the same. The spin problem can be reduced to the calculation of traces of γ matrices. Therefore only an additional prescription for the trace of the unit matrix is needed. There is no natural continuation since odd and even dimensions behave di erently. However no algebraic manipulation depends on the explicit value of the trace. Thus any smooth continuation in the neighbourhood of the relevant dimension will be satisfactory. A convenient choice is to take the trace constant. In even dimension as long as only γµ matrices are involved no other problem arises. However no dimensional continuation which preserves all properties of γ5 can be found. This leads to serious diculties if some diagrams involve γ5 and if it becomes necessary to use the identity relating γ5 to the other γ matrices (18) 4! γ5 = −µ ...µ γµ . . . γµ , where µ ···µ is the complete antisymmetric tensor then . As we recall later, lattice regularization is equally impossible in this case. This diculty is the source of chiral anomalies. Since we have to calculate traces, one possibility is to de ne γ5 by the relation (18) in all dimensions. It is then easy to verify that, with this de nition, γ5 anticommutes with the other γµ matrices only in four dimensions. If for example we evaluate the product γν γ5 γν in d dimensions, we nd: γν γ5 γν = (d − 8)γ5 . (19) Anticommuting properties of the γ5 would have led to a factor −4 instead. d k= d
1
1
4
1
4
4
IV. Lattice regularization We have explained that Pauli{Villars's regularization does not work for eld theories in which the action has a de nite geometrical character like models on homogeneous spaces (for example the non-linear σ -model) or gauge theories. In these theories some divergences are related to the problem of quantization and order of operators, which already appear at the level of simple quantum mechanics. Other regularization methods then are needed. In many cases lattice regularization [6] may be used. The advantages are the following: (i) Lattice regularization is the only established regularization which has a meaning outside perturbation theory. For instance the regularized functional integral can be calculated by numerical methods, like stochastic methods (Monte-Carlo type calculations) or
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strong coupling expansions. (ii) It preserves most global or local symmetries with the exception of the space O(d) symmetry which is replaced by a hypercubic symmetry (but this can be shown to be not too serious) and fermion chirality, which turns out to be a more serious problem. The main disadvantage is that it leads to very complicated perturbative calculations.
IV-1. Boson eld theories
Scalar eld theories. The action S (φ) for the scalar eld φ is replaced it by a lattice action. Therefore the derivative ∂µ φ a becomes a nite di erence, for example: ∂µ φ 7→ ∇µ φ = [φ (x + anµ ) − φ(x)] /a , (20) where a is the lattice spacing, and nµ the unit vector in the µ direction. The propagator a (p) for the Fourier components of a massive eld is then given by d 2 X 2 (21) −1 (1 − cos (apµ )). a (p) = m + 2 a µ=1
It is a periodic function of the components of the momentum vector pµ with period 2π/a. In the small lattice spacing limit the continuum propagator is recovered: X 2 2 1 (22) a2 pµ4 + O p6µ . −1 a (p) = m + p − 12 µ
In particular hypercubic symmetry implies O(d) symmetry at order p2 . Gauge theories. Lattice regularization de nes unambiguously a quantum theory. Therefore, once one has realized that gauge elds should be replaced by link variables corresponding to parallel transport along links of the lattice, one can regularize a gauge theory. The link variables Uxy are group elementsR associated with the links joining the sites 2 is the product of link variables on x and y on the lattice. The regularized form of dx Fµν a closed curve on the lattice, the simplest being a square on a hypercubic lattice, leading to the well-known plaquette action, each square forming a plaquette. The typical gauge invariant lattice action corresponding to the continuum action of a gauge eld coupled to scalar bosons then has the form: X X X S (U, φ∗ , φ) = β tr Uxy Uyz Uzt Utx + κ φx∗ Uxy φy + V (φx∗ φx ), (23) plaquettes
links
sites
where x, y ,... denotes lattice sites, and β and κ coupling constants. The action (23) is invariant under independent group transformations on each lattice site, lattice equivalents of the gauge transformations of the continuum theory. The measure of integration over the gauge variables is the group invariant measure on each site. Note that on the lattice and in a nite volume the gauge invariant action leads to a well-de ned partition function because the gauge group is compact. However in the continuum limit the compact character of the group is lost. It is therefore necessary to x the gauge on the lattice in order to be able to construct a regularized perturbation theory.
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We nally note that on the lattice the diculties with regularization problem do not come from the gauge eld directly, but involve the gauge eld only through the integration over chiral fermions.
IV-2. Fermion and the doubling problem [7]
Let us review a few problems arising when fermions are present in the action. We consider the free action for a Dirac fermion: Z ψ = dd x ψ (x) (6 ∂ + m) ψ (x). S ψ, (24)
To regularize this action by a lattice and preserve chiral properties one can replace ∂µ ψ (x) by (25) ∇µ ψ (x) = [ψ (x + anµ ) − ψ (x − anµ )] /2a. ~ Then the inverse of the fermion propagator for the Fourier components ψ (p) of the eld is: X (26) −1 (p) = m + i γµ sin apµ a
µ
a periodic function of the components pµ of the momentum vector. A problem then arises: the equations relevant to the small lattice spacing limit, sin(a pµ ) = 0 (27) have each two solutions pµ = 0 and pµ = π/a within one period, i.e. within the Brillouin zone 2π/a. Therefore the propagator (26) propagates 2d fermions. To remove this degeneracy it is possible to add to the regularized action an additional scalar term δ S involving second derivatives: h i ψ = M X 2ψ (x)ψ (x) − ψ (x + anµ ) ψ (x) − ψ (x)ψ (x + anµ ) . δ S ψ, (28) 2 x,µ After Fourier transformation the modi ed Dirac operator DW reads DW (p) = m + M
X µ
The fermion propagator becomes: with:
† † (p) = DW (p) DW (p)DW (p)
"
iX
(1 − cos apµ ) + a −1
γµ sin apµ .
(29)
µ
(30)
, #2
1 X (31) (1 − cos apµ ) + a2 sin2 apµ . µ µ Therefore the degeneracy between the di erent states has been lifted. For each component pµ which takes the value π/a the mass is increased by M . If M is of order 1/a the spurious states are eliminated in the continuum limit. This is the recipe of Wilson's fermion [8]. However a serious problem arises if one wants to construct a theory with massless fermions and chiral symmetry. Then both the mass term and the term (28) are excluded. DW (p)D (p) = m + M † W
X
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It remains of course possible to add various counter-terms and to adjust them to recover chiral symmetry in the continuum limit. But then calculations are plagued by ne tuning problems and cancellations of unnecessary UV divergences. Of course one could think modifying the fermion propagator by adding terms connecting fermions separated by more than one lattice spacing. But it has been proven that this does not solve the doubling problem. In fact this doubling of the number of fermion degrees of freedom is directly related to the problem of anomalies. Since the most naive form of the propagator yields 2d fermion states, one tries in practical calculations to reduce this number to a smaller multiple of two. The idea of staggered fermions [9] introduced by Kogut and Susskind is often used: rst by modifying the action one is able to decrease the multiplication factor from 2d to 2d/2 with respect to form (26). Then the remaining degeneracy is interpreted as the re ection of an internal symmetry SU (2d/2 ) of the action. Notation. For convenience we now set the lattice spacing a = 1 and use for the elds the notation ψ (x) ≡ ψx . Ginsparg{Wilson relation. Recently a decisive advance on the problem of chiral fermions has been achieved. It had been noted, many years ago, that a potential way to avoid the doubling problem while still retaining chiral properties in the continuum limit was to construct a lattice Dirac operator D satisfying the relation [10] {D −1 , γ5 } = γ5 (32) where γ5 in the r.h.s. means γ5 δxy , i.e. the identity for lattice sites. More generally the r.h.s. can be replaced by any local operator on the lattice. However, only recently have lattice Dirac operators solutions to the Ginsparg{Wilson relation (32) been discovered, because the demands that both D and the anticommutator {D −1 , γ5 } should be local, seemed dicult to satisfy, specially in the most interesting case of gauge theories. Let us brie y explain the main idea. Using the relation, quite generally true for a Dirac operator, D† = γ5 Dγ5 , (33) one can rewrite the relation (32) † D−1 + D−1 = 1 , (34) and therefore D + D† = DD† = D† D . (35) This implies that the lattice operator D has an index, and in addition S = 1−D, (36) is unitary SS † = 1 . (37)
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The eigenvalues of S lie on the unit circle. The eigenvalue one corresponds to the pole of the Dirac propagator. An explicit solution [11] can be derived from the Wilson{Dirac operator DW of equation (29). Setting A = 1 − DW , (38) one takes −1/2 −1/2 (39) S = A A† A . ⇒ D = 1 − A A† A −1/2 † has the eigenvalue one. This With this ansatz D has a zero eigenmode when A (A A) † can happen when A and A have the same eigenvector with a positive eigenvalue. In the case of the Wilson{Dirac operator (29) a necessary condition is sin pµ = 0 . (40) The presence of doublers thus depends on the sign of the corresponding values. By choosing 2M > 1 one keeps the wanted pµ = 0 mode, but eliminates all doublers which then correspond to the eigenvalue two for D, and the doubling problem is in principle solved. It is then possible to construct lattice actions that have a chiral symmetry which corresponds to local but non point-like transformations. In the abelian example, X δψx = iθ γ5 (1 − 12 D)xy ψy , (41) y
x = iθ δψ
X y
y (1 − 1 D) γ5 . ψ 2 yx
(42)
The problem is that these transformations no longer leave the integration measure over the fermion elds, Y x , dψx dψ (43) x
automatically invariant. Indeed an in nitesimal change of variables ∂ψx0 = δxy + iθγ5 (1 − 21 D)xy ∂ψy
x0 ∂ψ 1 y = δxy + iθ (1 − 2 D)xy γ5 , ∂ψ leads to the jacobian J (tr γ5 = 0) J = 1 − iθ tr γ5
X x
Dxx .
(44) (45) (46)
This leaves the possibility of generating the expected anomalies [12], when the simple Wilson{Dirac operator of the free theory is replaced by the covariant operator in the background of a gauge eld. Finally let us stress that, if it seems that the doubling problem has been solved from the formal point of view, from the numerical point of view the calculation of the operator (A† A)−1/2 in a gauge background represents a major challenge.
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References [1] For more details and references on standard regularization techniques see, J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd ed. (Clarendon Press, Oxford, 1996). [2] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949). [3] J. Ashmore, Lett. Nuovo Cimento 4, 289 (1972); G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972); C.G. Bollini and J.J. Giambiaggi, Phys. Lett. B40, 566 (1972); Nuovo Cimento 12B, 20 (1972). [4] D.A. Akyeampong and R. Delbourgo, Nuovo Cimento 17A, 578 (1973). [5] For an early review see, G. Leibbrandt, Rev. Mod. Phys. 47, 849 (1975). [6] The consistency of the lattice regularization is proven (except for theories with chiral fermions) in T. Reisz, Commun. Math. Phys. 117, 79, 639 (1988). [7] The doubling phenomenon for lattice fermions has been proven quite generally by H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185, 20 (1981). [8] K.G. Wilson, in New Phenomena in Subnuclear Physics, Erice 1975, ed. A. Zichichi, (Plenum, New York, 1977). [9] T. Banks, L. Susskind and J. Kogut, Phys. Rev. D13, 1043 (1976). [10] P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25, 2649 (1982). [11] H. Neuberger, Phys. Lett. B417, 141 (1998); ibid, B427, 353 (1998). [12] P. Hasenfratz, V. Laliena, and F. Niedermayer, Phys. Lett. B427, 125 (1998); M. Luscher, Phys. Lett. B428, 342 (1998); Nucl. Phys. B549, 295 (1999).
