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A method for simulating chiral fermions on the lattice
Physics Letters B 288 (1992) 342-347 North-Holland
PHYSICS LETTERS B
A method for simulating chiral fermions on the lattice D a v i d B. K a p l a n 1,2 Department of Physics, University of California at San Diego, 9500 Gilman Drive 0319, La Jolla, CA 92093-0319, USA
Received 6 June 1992
I show that a lattice theory of massive interacting fermions in 2n+ 1 dimensions may be used to simulate the behavior of massless chiral fermions in 2n dimensions if the fermion mass has a step function shape in the extra dimension. The massless states arise as zero modes bound to the mass defect, and all doublers can be given large gauge invariant masses. The manner in which the anomalies are realized is transparent: apparent chiral anomalies in the 2n-dimensional subspace correspond to charge flow into the extra dimension.
A successful chiral regulator for gauge theories has p r o v e n elusive. D i m e n s i o n a l regularization explicitly violates the chiral gauge symmetry, as does the conventional P a u l i - V i l l a r s technique; zeta function regularization fails because the chiral Dirac o p e r a t o r does not p r o v i d e an eigenvalue problem. A lattice regularization technique would be desirable, allowing one to simulate the standard model as well as strongly coupled chiral gauge theories [ 1 ]. However, the naive a p p r o a c h suffers from the existence o f doublers that render the theory vector-like instead o f chiral [ 2 ]. Some m o r e sophisticated approaches a p p e a r to suffer other obstructions [ 3 - 7 ], although some have yet to be thoroughly tested [ 8 ]. A n y such regulator must correctly reproduce the a n o m a l o u s W a r d identities o f the c o n t i n u u m theory. Taking the c o n t i n u u m limit must fail in theories where the gauge currents have a n o m a l o u s divergences; on the other hand, nonconservation o f global currents must be allowed in the continuum. As Banks has e m p h a s i z e d recently, a necessary criterion for chiral lattice theories is that they must correctly describe p r o t o n decay in a simulation o f the s t a n d a r d m o d e l [ 6 ]. The point is that the s t a n d a r d m o d e l has a n o m a l y free chiral gauge currents, but a n o m a l o u s global currents, such as baryon n u m b e r [ 9 ]. A lattice regulator that achieves this 1 S1oanFellow, NSF Presidential Young Investigator, and DOE Outstanding Junior Investigator. 2 E-mail address: [email protected]. 342
while preserving gauge invariance will have to be devious, as "no-go" theorems imply [ 10 ]. In this letter I show that chiral fermions in 2n dimensions m a y be simulated by Dirac fermions in 2n + 1 dimensions with a space dependent mass term. In particular, when there is a d o m a i n wall defect in the mass parameter, the lattice theory will exhibit massless chiral zero modes b o u n d to the 2n-dimensional d o m a i n wall, as well as doublers with the opposite chirality. Since the original ( 2 n + 1 ) - d i m e n sional theory is vector-like, the doublers m a y be r e m o v e d by means o f a gauge invariant Wilson term. This system can be used to simulate a 2n-dimensional chiral gauge theory (for n = 1, 2) provided that the massive fermion modes which are not b o u n d to the mass defect decouple, and provided that the gauge fields can be constrained to propagate along the defect. I argue that whether or not these conditions can be met depends on an algebraic relation which is just the one that ensures that the 2n-dimensional gauge currents are exactly divergenceless. Global currents, however, m a y have a n o m a l o u s divergences even in the regulated theory before taking the c o n t i n u u m limit, due to the curious role played by the extra dimension. I begin by recapitulating some well known facts about chiral zero m o d e s o f a free Dirac fermion in 2n + 1 dimensions coupled to a d o m a i n wall. I work in euclidian space a n d take the 2 n + 1 coordinates to be z~= {x, s} w h e r e / t runs over 1, ..., 2 n + 1 while x is
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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the coordinate in the 2n-dimensional subspace. I use hermitian gamma matrices satisfying {~'u, 7~} = 2Ou~ and define 7:,+,-/5.
(1)
Note that/'5 is the chiral operator in the 2n-dimensional theory. The fermion mass is assumed to have the form of a domain wall - a monotonic function of s with the asymptotic form
m(s)
, +m+_ ,
m.+ >O.
(2)
s~+oo
27 August 1992
dimensional lattice with lattice spacing a and sites labelled by z = nza. The action is given by (7)
SD = Z ~zI~D ~-Jz, z
where the lattice Dirac operator is given by 1 2n+ l
RDT~=~a E u=l
7u(Tz+;,--Tz-;,)+m(s)Tz,
(8)
a n d / i corresponds to a displacement by a in the z u direction. For simplicity I take the fermion mass to be a step function,
The Dirac operator is then given by
I~D=[~+m(s)]=[y.V+FsOs+m(s)]
.
(3)
Since the mass m ( s ) vanishes at s = 0 one might expect massless states bound to the defect. Such "zero modes" would have to satisfy/£D ~0 = ~" V kUo in order to describe massless propagation along the defect. Specifically, the zero modes must have the form ~
= e x p ( i p . x ) (9. (s)u+ ,
(4)
where u+ are constant 2" component chiral spinors /~Sb/_+ = ~ / _ +
,
m ( s ) =moO(s) -- sinh(a/to) O(s) , a
O(s)=-l,
s~-a,
=0,
s=0,
=+1,
s~>+a.
(9)
Then there are two chiral zero mode solutions ~ g given by 7J~ (p, z) = exp(ip.x) 0.+ (s,p)u+_ ,
(10)
where the transverse wavefunctions are given by
and the functions (9+ (s) satisfy
[ +Os + r e ( s ) 1(9_+(s) = 0 .
(9+ (s,p) = e x p ( - / t o [s]) , (5)
The solutions to the above equation are
¢_+(s)=exp(-T-im(s')ds'),
(9_ (s,p) = ( - 1 )'(9+ (s,p) .
( 11 )
These solutions satisfy (6)
~ff (p, z)
~,-sin(pa) ~ud-(p, z)
0
however, with the asymptotic form for m (s) given in eq. (2), only (9+ (s) is normalizable. Thus there is a single, positive chirality massless fermion bound to the mass defect. What makes this model of interest for a lattice approach to chiral fermions is that the ( 2 n + 1 )-dimensional theory has neither chirality nor anomalies, yet possesses a low energy effective theory (well below the asymptotic mass scales m± ) which is a theory of a massless chiral fermion in 2n dimensions. As I will discuss below, the anomalies in the effective 2n-dimensional theory are realized through finite Feynman diagrams and therefore have their canonical form even in the regularized theory. A similar model can be formulated on a ( 2 n + 1 )-
/'5 ~ff =_+ 7tff,
(12)
and, unlike the continuum example, are both normalizable and localized along the mass defect at s = 0. For Ipal << 1, iy.sin(pa)/a-,il~, the inverse propagator for a massless mode travelling in 2n dimensions. Therefore the mode ~d- for small p corresponds to the positive chirality state (4) found in the continuum. However, the lattice model is seen to actually describe 2 2"+~ massless modes in the continuum, rather than one, corresponding to ~d- and ~6- withp near the 2 2n corners of the 2n-dimensional Brillouin zone. The conventional analysis of doubler modes [2] reveals that these 2 2"+1 modes correspond to 2 n positive chirality states and 2" negative chirality states. The resultant lattice spectrum is 343
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therefore completely vector-like in the continuum, unlike what I wish to describe. Nevertheless, since the (2n + 1 )-dimensional theory (7) is itself vector-like, it is possible to add a Wilson term to eliminate the unwanted doublers. Thus I add to SD the action
Sw=aw ~.. ~PzI~w~z, z
2n+ 1
gw 'z= / z2= l 2n+ 1
=
~
(~u=+~-2~Uz+~=_a).
(13)
/t=l
Zero modes will now correspond to solutions satisfying eq. (12) with Ro replaced by/
( +-Os+moO(s) +wa A~ +--
[ c o s ( p , a ) - 1 ] ~+_(s)=O. a
(14)
i=1
I will analyze this equation for the particularly convenient choice of w=0.5, though there is actually a broad range of values that will have similar solutions. Defining 2n
F~)=
~ [1-cos(ap~)l~>O, i=1
eq. ( 14 ) becomes (for w = O.5 )
~+_(s+ a) = -merf(s)(9+_ (s) , mere(s) =amoO(s) - 1 - F ( p ) =+amo-F(p)-l,
s>0,
=-amo-F(p)-l,
s<0.
(15)
In order for there to exist normalizable solutions for ~+_(s,p), Imeffl must satisfy the following conditions: ~+:lmeffl>l
fors<0,
Imeffl < 1 f o r s > 0 , ~-:lmefrl
fors<0,
Imeefl > 1 f o r s > 0 .
(16)
Since amo and F ( p ) are positive, the conditions for 344
27 August 1992
a normalizable ~_ solution can never be met, and so all 2n of the ~6- modes have been eliminated. On the other hand, there are normalizable ~ + (s, p ) solutions for allp satisfying
O
(17)
Thus, for 0 < amo < 2 there is always a normalizable solution for small p, since F (0) = 0. Furthermore, on the Brillouin boundary F>~ 2, and so all of the doubler solutions for ~+ have been removed for the same range ofamo. This shows that the Wilson action ( 13 ) has eliminated all but one of the 2 2n÷l zero modes bound to the domain wall in the continuum limit. Only a single positive chirality mode remains. It may seem surprising that the Wilson term could remove an odd number of chiral modes, since one expects them to be removed in pairs with opposite chirality. In this case though, the chiral doubler modes are paired with modes "at infinity", rather than with each other. The effect is more easily understood in finite volume. Consider a lattice of length L in the s direction, with periodic boundary conditions. Now if there is a domain wall at s = 0 , there is necessarily an "anti-domain wall" at s = + ½L. For every zero mode at s = 0, there is a zero mode of opposite chirality at s = + ½L. The vector-like nature of the ( 2 n + 1 )-dimensional theory is now manifest: every mode is chirally paired, though the s dependent mass function has allowed one to separate some chiral pairs by a distance ½L with exponentially small overlap. At finite volume one can also see how the peculiar discontinuity in the spectrum implied by eq. (17) comes about. At finite L with periodic boundary conditions, only the p = 0 mode is exactly chiral; for p > 0 satisfying the condition ( 17 ), the modes have masses exponentially small in L, corresponding to a finite probability that the mode can tunnel to the anti-domain wall and flip chirality. As L-+oo, this probability goes rapidly to zero and the modes are exactly chiral as seen above. For values ofp not satisfying eq. (17) the modes are massive for any L, finite or infinite. I conclude that at infinite volume the spectrum of the free theory contains (i) doublers with mass O(w/ a); (ii) Dirac fermions with mass O(mo) that propagate in the full 2 n + 1 dimensions; (iii) an exactly massless chiral zero mode bound with F5 = + 1 bound to the 2n-dimensional defect. For momentum
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IPl << 1/a the mode has the conventional continuum propagator. The theory is vector-like at every step: both the doublers with mass at the cutoff, as well as the mass mo scattering states, are all Dirac fermions. The zero modes are also paired: in finite volume one sees that there are zero modes of opposite chirality bound to an anti-domain wall, which are carried off to spatial infinity in the infinite volume limit. When background gauge fields are coupled to the (2n + 1 )-dimensional Dirac theory, there appears to be a paradox. One would expect that at energies well below rno all massive fermion degrees would decoupie, yielding an effective 2n-dimensional theory of zero modes at one of the domain walls. However, such a theory is in general anomalous and cannot be simultaneously gauge invariant and chirally invariant. In the continuum, the zero mode gauge current jzm should have a divergence equal to [ 11,12 ] D i J zm = iI'5 Cn c il...i2,Fiti2 ...Fi2,_ ,2, ,
( 18 )
where (7.= ( - 1 ) n 22.zc.n!, Fs = + 1 is the chirality of the zero mode, and the indices run over the 2n dimensions. On the other hand, the lattice theory is manifestly finite and gauge invariant. So what is going on? The resolution of the paradox is that the massive fermion states that live off the domain walls do not entirely decouple - even in the low energy limit - and so the effective theory is not truly 2n-dimensional. Callan and Harvey first described this effect some years ago [ 13 ] while studying the properties of zero modes on topological defects and their relation to the anomaly descent relations [ 14,15 ]. They showed that the combined effect of the background field strength F~a and the space dependent mass term LPm=+m(s)~I
j
(19)
was to distort the heavy fermion vacuum, producing a Goldstone-Wilczek current [ 16 ] far from the domain wall, given by
-
i m(s) - 2 I m(s) I C , ~ ...... 2,F~,~2...F~2 . . . . 2..
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Note that the factor m ( s ) / I m ( s ) l = O ( s ) for a domain wall mass, and that this legacy from the heavy modes does not vanish in the limit I mol ~oo. Because O(s) has opposite signs on the two sides of the domain wall, the current has a nonzero divergence: in the presence of nonzero E and B fields, the current has a net flux onto or off the domain wall. Callan and Harvey pointed out that this flux precisely accounts for the anomalous divergence (18) of the zero mode current. Therefore, what appears as an anomalous divergence in the effective 2n-dimensional theory is simply charge flowing on or off the mass defect from the extra dimension. Note that the sign of the mass coupling in eq. (19) determines both the sign of the Goldstone-Wilczek current (20) and the chirality F5 of the zero mode on a given mass defect, which figures in the anomaly equation (18). Callan and Harvey also showed that the current (20) may be directly computed from an effective action, which is just the Chern-Simons term [ 17 ] proportional to m (s) / Im (s) I that was induced in the effective theory by integrating out the massive fermion modes [18] ~tl The extra dimension is the loophole in the Nielsen-Ninomiya theorem [ 10 ] through which the fermions have wriggled. The currents of the ( 2 n + 1 )dimensional lattice are in fact exactly conserved. Nevertheless, there is anomalous current violation in the 2n-dimensional effective theory due to charge flowing in from the extra dimension. As the ChernSimons action is the only new local operator that can be induced in the effective theory that does not vanish as the fermion mass increases, the only effect the massive fermions have on the low energy theory (aside from renormalizing couplings) is to correctly reproduce the anomalous Ward identities. As the Goldstone-Wilczek current (20) does not vanish in a regulated theory, the 2n-dimensional anomalous Ward identities may for the first time be directly measured on the lattice. Such a numerical computation has recently been performed by Jansen [20 ], whose results show that starting from 2 + 1 dimensions, one can measure the 1 + 1 anomalous divergence ofa chiral current. He finds that without the Wilson term ( 13 ) the theory is vector-like and all 2ndimensional flavor currents are conserved. However,
(20) ~ See Naculich [ 19] for a discussion of the subtleties involved. 345
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for nonzero Wilson coupling w, he finds a nonzero divergence for the currents, and the rate of charge violation on the domain wall is proportional to the applied electric field, as prescribed by the anomaly equation ( 18 ). Chiral theories with nontrivial anomaly cancellation can also be simulated in background gauge fields. The simplest example of such a theory is the 3-4-5 chiral Schwinger model: one starts with a (2 + l )-dimensional theory with three Dirac fermions ~'/3, ~'/4 and g/5 coupled to a U ( 1 ) gauge field with charges q = 3, 4 and 5 respectively. The mass term for these fermions is given by
..C~'m=m(s)[~P3tII3÷~4tIJ4--!Ps~[15] ,
(21)
SO that the effective ( 1 + 1 )-dimensional theory on the s = 0 defect consists of charge q=3, 4 fermions with F s = + 1, and a q = 5 fermion with F s = - 1. As the two-dimensional anomaly in the gauge current is proportional to Trq2Fs, it vanishes; the ChernSimons term induced by the heavy fermions is proportional to Tr q 2(m/iml ), and so it and the resultant Goldstone-Wilczek current also vanish. As has been confirmed numerically by Jansen [20], the gauge current is anomaly free, even though the individual flavor currents are anomalously divergent. The next step is to introduce dynamical gauge fields. Including the usual ( 2 n + 1 )-dimensional Yang-Mills action is not appropriate for describing a 2n-dimensional gauge on the domain wall. There would exist an additional polarization for the gauge bosons, corresponding to the (2n + 1 )-component of the vector potential and appearing as an adjoint pseudoscalar in the 2n-dimensional world. Another problem is that the gauge bosons would propagate off the defect, giving rise to the wrong force law between the zero modes. The conventional ( 2 n + 1 )-dimensional gauge action is not required though, since the action need only be invariant under gauge and 2n-dimensional Lorentz transformations. Thus I consider a gauge action of the form 2n
Sg=fl, ~ TrU41D,j+2/32~TrU41D,.,. i,j= 1
(22)
i
In general, r, and r2 are independent and can be functions of s. It appears that simply setting to zero the (renormalized) coupling r2 would suffice for simulating a 2n-dimensional gauge theory at each s slice. However, in the interacting theory, this is clearly 346
27 August 1992
not always possible. It is instructive to consider the continuum Yang-Mills equations for these gauge bosons in 2n + 1 dimensions,
r, DjFij+Dsfl2Fi,=Ji, fl2DiFsi=Js.
(23)
From the second equation one sees that Fis~Js/p2. Therefore, taking the limit f12--+0 does not eliminate the unwanted Fi, term from the first equation - unless Js vanishes. There are two possible contributions to Js, from the zero modes and from the Goldstone-Wilczek current (20). In fact in the infinite volume limit there is no zero mode contribution to Js since gi72n+'~= #F5 V= 0, where V(x) is any of the zero modes on the mass defect ~2. Therefore, for J, to vanish, the component of the Goldstone-Wilczek current orthogonal to the domain wall must vanish. As we have seen, this
occurs if, and only if, the gauge currents in the 2n-dimensional theory are anomaly free. I conclude that if the 2n-dimensional gauge currents are anomalous, there is nothing one can do to make the theory look like a 2n-dimensional gauge theory - it looks inherently ( 2 n + 1 )-dimensional, where it makes sense. However, I speculate that for anomaly free chiral gauge currents, the f12-'0 limit decouples the unwanted gauge degrees of freedom, and the lattice model looks like a 2n-dimensional chiral gauge theory. In that case, three- and five-dimensional models of this nature are expected to possess second order phase transitions, corresponding to continuum chiral gauge theories in two and four dimensions. Note that the effective 2n-dimensional gauge coupling will be related to fll by a factor of the normalization of the transverse wavefunction ~+ (s), correctly accounting for the change in dimensionality. Suppose the mechanism described here works and one can simulate the standard model - how does the proton decay? As I argued above, the only requirement for recovering a four-dimensional gauge symmetry was that the chiral gauge current be anomaly free. The global currents, such as B and L will still be anomalous. Given an SU (2) instanton event in the s = 0 mass defect, a proton will flow off the domain wall and its baryon number will reside in the vacuum ~2 At finite volume there is an exponentially small probability for the zero mode to tunnel to the anti-domain wall, contributing to Js.
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p o l a r i z a t i o n o f the e x t r a d i m e n s i o n . S i n c e B - L is a n o m a l y free, the v a c u u m will disgorge a p o s i t r o n into o u r f o u r - d i m e n s i o n a l w o r l d (I a m a s s u m i n g a single f a m i l y ) . As far as we are c o n c e r n e d , the p r o t o n has decayed; little do we suspect that it is h o v e r i n g n e a r b y b u t out o f sight in the extra d i m e n s i o n . I w o u l d like to t h a n k T. Banks, A. C o h e n , K. J a n sen, J. Kuti, A. M a n o h a r , M. Peskin, J. Preskill a n d S. S h a r p e for useful c o n v e r s a t i o n s . A f t e r h e a r i n g a b o u t this work, J. Preskill p o i n t e d o u t to m e t h a t the a u t h o r s o f ref. [21 ] p r e v i o u s l y d i s c u s s e d d o m a i n walls a n d t h e C a l l a n - H a r v e y effect on the lattice. T h e y m e n t i o n e d the possibility o f using f e r m i o n z e r o m o d e s o n d o m a i n walls to s i m u l a t e chiral lattice theories, b u t r e j e c t e d it. T h i s r e s e a r c h was s u p p o r t e d in p a r t b y the D e p a r t m e n t o f Energy u n d e r c o n t r a c t # D E - F G O 3 9 0 E R 4 0 5 4 6 , by the N S F u n d e r c o n t r a c t P H Y 9057135, a n d by a fellowship f r o m the Alfred P. Sloan Foundation.
References [ 1 ] See, e.g., H. Georgi, Nucl. Phys. B 156 (1979) 126; S. Dimopoulos, S. Raby and L. Susskind, Nucl. Phys. B 173 (1980) 208. [ 2 ] L.H. Karsten and J. Smit, Nucl. Phys. B 183 ( 1981 ) 103. [3] G.T. Bodwin and E.V. Kovacs, Phys. Rev. D 35 (1987) 3198; D 37 ( 1988 ) 1008; D 41 (1990) 2026. [4] M.F.L. Goltermann, D.N. Petcher and J. Smit, Nucl. Phys. B 370 (1992) 51. [5]M.F.L Golterman and D.N. Petcher, On the EichtenPreskill proposal..., Washington University preprint WASHU-HEP-91-61, [6] T. Banks, Phys. Lett. B 272 ( 1991 ) 75; T. Banks and A. Dabholkar, Decoupling a massive fermion..., Rutgers preprint RU-92-09 [hep-lat/9204017 ]. [ 7 ] M.J. Dugan and L. Randall, On the decoupling of doubler fermions..., MIT preprint MIT-CTP-2050, Nucl. Phys. B, to be published.
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[8] L. Maiani, G.C. Rossi and M. Testa, Phys. Lett. B 261 (1991) 479; A. Borrelli, L. Maiani, R. Sisto, G.C. Rossi and M. Testa, Nucl. Phys. B 333 (1990) 335; M. Bochicchio, L. Maiani, G. Martinelli, G.C. Rossi and M. Testa, Nucl. Phys. B 262 (1985) 331; G.T. Bodwin and E.V. Kovacs, Nucl. Phys. B (Proc. Suppl. ) 20 (1991) 546; C. Pryor, Phys. Rev. D 43 (1991) 2669. [9] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [ 10] H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 185 ( 1981 ) 20; B 195 (1982) 541(E); B 193 (1981) 173; Phys. Lett. B 105 (1981)219; L.H. Karsten, Phys. Len. B 104 ( 1981 ) 315. [ 11 ] S. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 60 ( 1969 ) 47. [12]P.H. Frampton and T.W. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; 51 (1983) 232(E); Phys. Rev. D28 (1983) 1010; L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 ( 1983 ) 269; B. Zumino, Lectures Les Houches Summer School ( 1983 ); R. Stora, Cargrse lectures ( 1983 ). [ 13 ] C.G. Callan Jr. and J.A. Harvey, Nucl. Phys. B 250 ( 1985 ) 427. [ 14 ] B. Zumino, Y.-S. Wu and A. Zee, Nucl. Phys. B 239 (1984) 477. [ 15 ] L. Alvarez-Gaum6 and P. Ginsparg, Ann. Phys. 161 ( 1985 ) 423; 171 (1986) 233(E); O. Alvarez, I. Singer and B. Zumino, Commun. Math. Phys. 96 (1984) 409. [16] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 986. [ 17 ] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975;Ann. Phys. (NY) 140 (1982) 372. [ 18 ] N. Redlich, Phys. Rev. Lett. 52 (1984) 18; A.J. Niemi and G. Semenoff, Phys. Rev. Lett. 51 (1984) 2077. [19 ] S.G. Naculich, Nucl. Phys. B 296 (1988 ) 837. [20] K. Jansen, Phys. Lett. B 288 (1992) 348. [21 ] D. Boyanovsky, E. Dagotto and E. Fradkin, Nucl. Phys. B 285 (1987) 340; E. Fradkin, Nucl. Phys. B (Proc. Suppl.) IA (1987) 175; E. Dagotto, E. Fradkin and A. Moreo, Phys. Lett. B 172 (1986) 383.
347
PHYSICS LETTERS B
A method for simulating chiral fermions on the lattice D a v i d B. K a p l a n 1,2 Department of Physics, University of California at San Diego, 9500 Gilman Drive 0319, La Jolla, CA 92093-0319, USA
Received 6 June 1992
I show that a lattice theory of massive interacting fermions in 2n+ 1 dimensions may be used to simulate the behavior of massless chiral fermions in 2n dimensions if the fermion mass has a step function shape in the extra dimension. The massless states arise as zero modes bound to the mass defect, and all doublers can be given large gauge invariant masses. The manner in which the anomalies are realized is transparent: apparent chiral anomalies in the 2n-dimensional subspace correspond to charge flow into the extra dimension.
A successful chiral regulator for gauge theories has p r o v e n elusive. D i m e n s i o n a l regularization explicitly violates the chiral gauge symmetry, as does the conventional P a u l i - V i l l a r s technique; zeta function regularization fails because the chiral Dirac o p e r a t o r does not p r o v i d e an eigenvalue problem. A lattice regularization technique would be desirable, allowing one to simulate the standard model as well as strongly coupled chiral gauge theories [ 1 ]. However, the naive a p p r o a c h suffers from the existence o f doublers that render the theory vector-like instead o f chiral [ 2 ]. Some m o r e sophisticated approaches a p p e a r to suffer other obstructions [ 3 - 7 ], although some have yet to be thoroughly tested [ 8 ]. A n y such regulator must correctly reproduce the a n o m a l o u s W a r d identities o f the c o n t i n u u m theory. Taking the c o n t i n u u m limit must fail in theories where the gauge currents have a n o m a l o u s divergences; on the other hand, nonconservation o f global currents must be allowed in the continuum. As Banks has e m p h a s i z e d recently, a necessary criterion for chiral lattice theories is that they must correctly describe p r o t o n decay in a simulation o f the s t a n d a r d m o d e l [ 6 ]. The point is that the s t a n d a r d m o d e l has a n o m a l y free chiral gauge currents, but a n o m a l o u s global currents, such as baryon n u m b e r [ 9 ]. A lattice regulator that achieves this 1 S1oanFellow, NSF Presidential Young Investigator, and DOE Outstanding Junior Investigator. 2 E-mail address: [email protected]. 342
while preserving gauge invariance will have to be devious, as "no-go" theorems imply [ 10 ]. In this letter I show that chiral fermions in 2n dimensions m a y be simulated by Dirac fermions in 2n + 1 dimensions with a space dependent mass term. In particular, when there is a d o m a i n wall defect in the mass parameter, the lattice theory will exhibit massless chiral zero modes b o u n d to the 2n-dimensional d o m a i n wall, as well as doublers with the opposite chirality. Since the original ( 2 n + 1 ) - d i m e n sional theory is vector-like, the doublers m a y be r e m o v e d by means o f a gauge invariant Wilson term. This system can be used to simulate a 2n-dimensional chiral gauge theory (for n = 1, 2) provided that the massive fermion modes which are not b o u n d to the mass defect decouple, and provided that the gauge fields can be constrained to propagate along the defect. I argue that whether or not these conditions can be met depends on an algebraic relation which is just the one that ensures that the 2n-dimensional gauge currents are exactly divergenceless. Global currents, however, m a y have a n o m a l o u s divergences even in the regulated theory before taking the c o n t i n u u m limit, due to the curious role played by the extra dimension. I begin by recapitulating some well known facts about chiral zero m o d e s o f a free Dirac fermion in 2n + 1 dimensions coupled to a d o m a i n wall. I work in euclidian space a n d take the 2 n + 1 coordinates to be z~= {x, s} w h e r e / t runs over 1, ..., 2 n + 1 while x is
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 288, number 3,4
PHYSICS LETTERSB
the coordinate in the 2n-dimensional subspace. I use hermitian gamma matrices satisfying {~'u, 7~} = 2Ou~ and define 7:,+,-/5.
(1)
Note that/'5 is the chiral operator in the 2n-dimensional theory. The fermion mass is assumed to have the form of a domain wall - a monotonic function of s with the asymptotic form
m(s)
, +m+_ ,
m.+ >O.
(2)
s~+oo
27 August 1992
dimensional lattice with lattice spacing a and sites labelled by z = nza. The action is given by (7)
SD = Z ~zI~D ~-Jz, z
where the lattice Dirac operator is given by 1 2n+ l
RDT~=~a E u=l
7u(Tz+;,--Tz-;,)+m(s)Tz,
(8)
a n d / i corresponds to a displacement by a in the z u direction. For simplicity I take the fermion mass to be a step function,
The Dirac operator is then given by
I~D=[~+m(s)]=[y.V+FsOs+m(s)]
.
(3)
Since the mass m ( s ) vanishes at s = 0 one might expect massless states bound to the defect. Such "zero modes" would have to satisfy/£D ~0 = ~" V kUo in order to describe massless propagation along the defect. Specifically, the zero modes must have the form ~
= e x p ( i p . x ) (9. (s)u+ ,
(4)
where u+ are constant 2" component chiral spinors /~Sb/_+ = ~ / _ +
,
m ( s ) =moO(s) -- sinh(a/to) O(s) , a
O(s)=-l,
s~-a,
=0,
s=0,
=+1,
s~>+a.
(9)
Then there are two chiral zero mode solutions ~ g given by 7J~ (p, z) = exp(ip.x) 0.+ (s,p)u+_ ,
(10)
where the transverse wavefunctions are given by
and the functions (9+ (s) satisfy
[ +Os + r e ( s ) 1(9_+(s) = 0 .
(9+ (s,p) = e x p ( - / t o [s]) , (5)
The solutions to the above equation are
¢_+(s)=exp(-T-im(s')ds'),
(9_ (s,p) = ( - 1 )'(9+ (s,p) .
( 11 )
These solutions satisfy (6)
~ff (p, z)
~,-sin(pa) ~ud-(p, z)
0
however, with the asymptotic form for m (s) given in eq. (2), only (9+ (s) is normalizable. Thus there is a single, positive chirality massless fermion bound to the mass defect. What makes this model of interest for a lattice approach to chiral fermions is that the ( 2 n + 1 )-dimensional theory has neither chirality nor anomalies, yet possesses a low energy effective theory (well below the asymptotic mass scales m± ) which is a theory of a massless chiral fermion in 2n dimensions. As I will discuss below, the anomalies in the effective 2n-dimensional theory are realized through finite Feynman diagrams and therefore have their canonical form even in the regularized theory. A similar model can be formulated on a ( 2 n + 1 )-
/'5 ~ff =_+ 7tff,
(12)
and, unlike the continuum example, are both normalizable and localized along the mass defect at s = 0. For Ipal << 1, iy.sin(pa)/a-,il~, the inverse propagator for a massless mode travelling in 2n dimensions. Therefore the mode ~d- for small p corresponds to the positive chirality state (4) found in the continuum. However, the lattice model is seen to actually describe 2 2"+~ massless modes in the continuum, rather than one, corresponding to ~d- and ~6- withp near the 2 2n corners of the 2n-dimensional Brillouin zone. The conventional analysis of doubler modes [2] reveals that these 2 2"+1 modes correspond to 2 n positive chirality states and 2" negative chirality states. The resultant lattice spectrum is 343
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therefore completely vector-like in the continuum, unlike what I wish to describe. Nevertheless, since the (2n + 1 )-dimensional theory (7) is itself vector-like, it is possible to add a Wilson term to eliminate the unwanted doublers. Thus I add to SD the action
Sw=aw ~.. ~PzI~w~z, z
2n+ 1
gw 'z= / z2= l 2n+ 1
=
~
(~u=+~-2~Uz+~=_a).
(13)
/t=l
Zero modes will now correspond to solutions satisfying eq. (12) with Ro replaced by/
( +-Os+moO(s) +wa A~ +--
[ c o s ( p , a ) - 1 ] ~+_(s)=O. a
(14)
i=1
I will analyze this equation for the particularly convenient choice of w=0.5, though there is actually a broad range of values that will have similar solutions. Defining 2n
F~)=
~ [1-cos(ap~)l~>O, i=1
eq. ( 14 ) becomes (for w = O.5 )
~+_(s+ a) = -merf(s)(9+_ (s) , mere(s) =amoO(s) - 1 - F ( p ) =+amo-F(p)-l,
s>0,
=-amo-F(p)-l,
s<0.
(15)
In order for there to exist normalizable solutions for ~+_(s,p), Imeffl must satisfy the following conditions: ~+:lmeffl>l
fors<0,
Imeffl < 1 f o r s > 0 , ~-:lmefrl
fors<0,
Imeefl > 1 f o r s > 0 .
(16)
Since amo and F ( p ) are positive, the conditions for 344
27 August 1992
a normalizable ~_ solution can never be met, and so all 2n of the ~6- modes have been eliminated. On the other hand, there are normalizable ~ + (s, p ) solutions for allp satisfying
O
(17)
Thus, for 0 < amo < 2 there is always a normalizable solution for small p, since F (0) = 0. Furthermore, on the Brillouin boundary F>~ 2, and so all of the doubler solutions for ~+ have been removed for the same range ofamo. This shows that the Wilson action ( 13 ) has eliminated all but one of the 2 2n÷l zero modes bound to the domain wall in the continuum limit. Only a single positive chirality mode remains. It may seem surprising that the Wilson term could remove an odd number of chiral modes, since one expects them to be removed in pairs with opposite chirality. In this case though, the chiral doubler modes are paired with modes "at infinity", rather than with each other. The effect is more easily understood in finite volume. Consider a lattice of length L in the s direction, with periodic boundary conditions. Now if there is a domain wall at s = 0 , there is necessarily an "anti-domain wall" at s = + ½L. For every zero mode at s = 0, there is a zero mode of opposite chirality at s = + ½L. The vector-like nature of the ( 2 n + 1 )-dimensional theory is now manifest: every mode is chirally paired, though the s dependent mass function has allowed one to separate some chiral pairs by a distance ½L with exponentially small overlap. At finite volume one can also see how the peculiar discontinuity in the spectrum implied by eq. (17) comes about. At finite L with periodic boundary conditions, only the p = 0 mode is exactly chiral; for p > 0 satisfying the condition ( 17 ), the modes have masses exponentially small in L, corresponding to a finite probability that the mode can tunnel to the anti-domain wall and flip chirality. As L-+oo, this probability goes rapidly to zero and the modes are exactly chiral as seen above. For values ofp not satisfying eq. (17) the modes are massive for any L, finite or infinite. I conclude that at infinite volume the spectrum of the free theory contains (i) doublers with mass O(w/ a); (ii) Dirac fermions with mass O(mo) that propagate in the full 2 n + 1 dimensions; (iii) an exactly massless chiral zero mode bound with F5 = + 1 bound to the 2n-dimensional defect. For momentum
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IPl << 1/a the mode has the conventional continuum propagator. The theory is vector-like at every step: both the doublers with mass at the cutoff, as well as the mass mo scattering states, are all Dirac fermions. The zero modes are also paired: in finite volume one sees that there are zero modes of opposite chirality bound to an anti-domain wall, which are carried off to spatial infinity in the infinite volume limit. When background gauge fields are coupled to the (2n + 1 )-dimensional Dirac theory, there appears to be a paradox. One would expect that at energies well below rno all massive fermion degrees would decoupie, yielding an effective 2n-dimensional theory of zero modes at one of the domain walls. However, such a theory is in general anomalous and cannot be simultaneously gauge invariant and chirally invariant. In the continuum, the zero mode gauge current jzm should have a divergence equal to [ 11,12 ] D i J zm = iI'5 Cn c il...i2,Fiti2 ...Fi2,_ ,2, ,
( 18 )
where (7.= ( - 1 ) n 22.zc.n!, Fs = + 1 is the chirality of the zero mode, and the indices run over the 2n dimensions. On the other hand, the lattice theory is manifestly finite and gauge invariant. So what is going on? The resolution of the paradox is that the massive fermion states that live off the domain walls do not entirely decouple - even in the low energy limit - and so the effective theory is not truly 2n-dimensional. Callan and Harvey first described this effect some years ago [ 13 ] while studying the properties of zero modes on topological defects and their relation to the anomaly descent relations [ 14,15 ]. They showed that the combined effect of the background field strength F~a and the space dependent mass term LPm=+m(s)~I
j
(19)
was to distort the heavy fermion vacuum, producing a Goldstone-Wilczek current [ 16 ] far from the domain wall, given by
-
i m(s) - 2 I m(s) I C , ~ ...... 2,F~,~2...F~2 . . . . 2..
27 August 1992
Note that the factor m ( s ) / I m ( s ) l = O ( s ) for a domain wall mass, and that this legacy from the heavy modes does not vanish in the limit I mol ~oo. Because O(s) has opposite signs on the two sides of the domain wall, the current has a nonzero divergence: in the presence of nonzero E and B fields, the current has a net flux onto or off the domain wall. Callan and Harvey pointed out that this flux precisely accounts for the anomalous divergence (18) of the zero mode current. Therefore, what appears as an anomalous divergence in the effective 2n-dimensional theory is simply charge flowing on or off the mass defect from the extra dimension. Note that the sign of the mass coupling in eq. (19) determines both the sign of the Goldstone-Wilczek current (20) and the chirality F5 of the zero mode on a given mass defect, which figures in the anomaly equation (18). Callan and Harvey also showed that the current (20) may be directly computed from an effective action, which is just the Chern-Simons term [ 17 ] proportional to m (s) / Im (s) I that was induced in the effective theory by integrating out the massive fermion modes [18] ~tl The extra dimension is the loophole in the Nielsen-Ninomiya theorem [ 10 ] through which the fermions have wriggled. The currents of the ( 2 n + 1 )dimensional lattice are in fact exactly conserved. Nevertheless, there is anomalous current violation in the 2n-dimensional effective theory due to charge flowing in from the extra dimension. As the ChernSimons action is the only new local operator that can be induced in the effective theory that does not vanish as the fermion mass increases, the only effect the massive fermions have on the low energy theory (aside from renormalizing couplings) is to correctly reproduce the anomalous Ward identities. As the Goldstone-Wilczek current (20) does not vanish in a regulated theory, the 2n-dimensional anomalous Ward identities may for the first time be directly measured on the lattice. Such a numerical computation has recently been performed by Jansen [20 ], whose results show that starting from 2 + 1 dimensions, one can measure the 1 + 1 anomalous divergence ofa chiral current. He finds that without the Wilson term ( 13 ) the theory is vector-like and all 2ndimensional flavor currents are conserved. However,
(20) ~ See Naculich [ 19] for a discussion of the subtleties involved. 345
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for nonzero Wilson coupling w, he finds a nonzero divergence for the currents, and the rate of charge violation on the domain wall is proportional to the applied electric field, as prescribed by the anomaly equation ( 18 ). Chiral theories with nontrivial anomaly cancellation can also be simulated in background gauge fields. The simplest example of such a theory is the 3-4-5 chiral Schwinger model: one starts with a (2 + l )-dimensional theory with three Dirac fermions ~'/3, ~'/4 and g/5 coupled to a U ( 1 ) gauge field with charges q = 3, 4 and 5 respectively. The mass term for these fermions is given by
..C~'m=m(s)[~P3tII3÷~4tIJ4--!Ps~[15] ,
(21)
SO that the effective ( 1 + 1 )-dimensional theory on the s = 0 defect consists of charge q=3, 4 fermions with F s = + 1, and a q = 5 fermion with F s = - 1. As the two-dimensional anomaly in the gauge current is proportional to Trq2Fs, it vanishes; the ChernSimons term induced by the heavy fermions is proportional to Tr q 2(m/iml ), and so it and the resultant Goldstone-Wilczek current also vanish. As has been confirmed numerically by Jansen [20], the gauge current is anomaly free, even though the individual flavor currents are anomalously divergent. The next step is to introduce dynamical gauge fields. Including the usual ( 2 n + 1 )-dimensional Yang-Mills action is not appropriate for describing a 2n-dimensional gauge on the domain wall. There would exist an additional polarization for the gauge bosons, corresponding to the (2n + 1 )-component of the vector potential and appearing as an adjoint pseudoscalar in the 2n-dimensional world. Another problem is that the gauge bosons would propagate off the defect, giving rise to the wrong force law between the zero modes. The conventional ( 2 n + 1 )-dimensional gauge action is not required though, since the action need only be invariant under gauge and 2n-dimensional Lorentz transformations. Thus I consider a gauge action of the form 2n
Sg=fl, ~ TrU41D,j+2/32~TrU41D,.,. i,j= 1
(22)
i
In general, r, and r2 are independent and can be functions of s. It appears that simply setting to zero the (renormalized) coupling r2 would suffice for simulating a 2n-dimensional gauge theory at each s slice. However, in the interacting theory, this is clearly 346
27 August 1992
not always possible. It is instructive to consider the continuum Yang-Mills equations for these gauge bosons in 2n + 1 dimensions,
r, DjFij+Dsfl2Fi,=Ji, fl2DiFsi=Js.
(23)
From the second equation one sees that Fis~Js/p2. Therefore, taking the limit f12--+0 does not eliminate the unwanted Fi, term from the first equation - unless Js vanishes. There are two possible contributions to Js, from the zero modes and from the Goldstone-Wilczek current (20). In fact in the infinite volume limit there is no zero mode contribution to Js since gi72n+'~= #F5 V= 0, where V(x) is any of the zero modes on the mass defect ~2. Therefore, for J, to vanish, the component of the Goldstone-Wilczek current orthogonal to the domain wall must vanish. As we have seen, this
occurs if, and only if, the gauge currents in the 2n-dimensional theory are anomaly free. I conclude that if the 2n-dimensional gauge currents are anomalous, there is nothing one can do to make the theory look like a 2n-dimensional gauge theory - it looks inherently ( 2 n + 1 )-dimensional, where it makes sense. However, I speculate that for anomaly free chiral gauge currents, the f12-'0 limit decouples the unwanted gauge degrees of freedom, and the lattice model looks like a 2n-dimensional chiral gauge theory. In that case, three- and five-dimensional models of this nature are expected to possess second order phase transitions, corresponding to continuum chiral gauge theories in two and four dimensions. Note that the effective 2n-dimensional gauge coupling will be related to fll by a factor of the normalization of the transverse wavefunction ~+ (s), correctly accounting for the change in dimensionality. Suppose the mechanism described here works and one can simulate the standard model - how does the proton decay? As I argued above, the only requirement for recovering a four-dimensional gauge symmetry was that the chiral gauge current be anomaly free. The global currents, such as B and L will still be anomalous. Given an SU (2) instanton event in the s = 0 mass defect, a proton will flow off the domain wall and its baryon number will reside in the vacuum ~2 At finite volume there is an exponentially small probability for the zero mode to tunnel to the anti-domain wall, contributing to Js.
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p o l a r i z a t i o n o f the e x t r a d i m e n s i o n . S i n c e B - L is a n o m a l y free, the v a c u u m will disgorge a p o s i t r o n into o u r f o u r - d i m e n s i o n a l w o r l d (I a m a s s u m i n g a single f a m i l y ) . As far as we are c o n c e r n e d , the p r o t o n has decayed; little do we suspect that it is h o v e r i n g n e a r b y b u t out o f sight in the extra d i m e n s i o n . I w o u l d like to t h a n k T. Banks, A. C o h e n , K. J a n sen, J. Kuti, A. M a n o h a r , M. Peskin, J. Preskill a n d S. S h a r p e for useful c o n v e r s a t i o n s . A f t e r h e a r i n g a b o u t this work, J. Preskill p o i n t e d o u t to m e t h a t the a u t h o r s o f ref. [21 ] p r e v i o u s l y d i s c u s s e d d o m a i n walls a n d t h e C a l l a n - H a r v e y effect on the lattice. T h e y m e n t i o n e d the possibility o f using f e r m i o n z e r o m o d e s o n d o m a i n walls to s i m u l a t e chiral lattice theories, b u t r e j e c t e d it. T h i s r e s e a r c h was s u p p o r t e d in p a r t b y the D e p a r t m e n t o f Energy u n d e r c o n t r a c t # D E - F G O 3 9 0 E R 4 0 5 4 6 , by the N S F u n d e r c o n t r a c t P H Y 9057135, a n d by a fellowship f r o m the Alfred P. Sloan Foundation.
References [ 1 ] See, e.g., H. Georgi, Nucl. Phys. B 156 (1979) 126; S. Dimopoulos, S. Raby and L. Susskind, Nucl. Phys. B 173 (1980) 208. [ 2 ] L.H. Karsten and J. Smit, Nucl. Phys. B 183 ( 1981 ) 103. [3] G.T. Bodwin and E.V. Kovacs, Phys. Rev. D 35 (1987) 3198; D 37 ( 1988 ) 1008; D 41 (1990) 2026. [4] M.F.L. Goltermann, D.N. Petcher and J. Smit, Nucl. Phys. B 370 (1992) 51. [5]M.F.L Golterman and D.N. Petcher, On the EichtenPreskill proposal..., Washington University preprint WASHU-HEP-91-61, [6] T. Banks, Phys. Lett. B 272 ( 1991 ) 75; T. Banks and A. Dabholkar, Decoupling a massive fermion..., Rutgers preprint RU-92-09 [hep-lat/9204017 ]. [ 7 ] M.J. Dugan and L. Randall, On the decoupling of doubler fermions..., MIT preprint MIT-CTP-2050, Nucl. Phys. B, to be published.
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[8] L. Maiani, G.C. Rossi and M. Testa, Phys. Lett. B 261 (1991) 479; A. Borrelli, L. Maiani, R. Sisto, G.C. Rossi and M. Testa, Nucl. Phys. B 333 (1990) 335; M. Bochicchio, L. Maiani, G. Martinelli, G.C. Rossi and M. Testa, Nucl. Phys. B 262 (1985) 331; G.T. Bodwin and E.V. Kovacs, Nucl. Phys. B (Proc. Suppl. ) 20 (1991) 546; C. Pryor, Phys. Rev. D 43 (1991) 2669. [9] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [ 10] H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 185 ( 1981 ) 20; B 195 (1982) 541(E); B 193 (1981) 173; Phys. Lett. B 105 (1981)219; L.H. Karsten, Phys. Len. B 104 ( 1981 ) 315. [ 11 ] S. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 60 ( 1969 ) 47. [12]P.H. Frampton and T.W. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; 51 (1983) 232(E); Phys. Rev. D28 (1983) 1010; L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 ( 1983 ) 269; B. Zumino, Lectures Les Houches Summer School ( 1983 ); R. Stora, Cargrse lectures ( 1983 ). [ 13 ] C.G. Callan Jr. and J.A. Harvey, Nucl. Phys. B 250 ( 1985 ) 427. [ 14 ] B. Zumino, Y.-S. Wu and A. Zee, Nucl. Phys. B 239 (1984) 477. [ 15 ] L. Alvarez-Gaum6 and P. Ginsparg, Ann. Phys. 161 ( 1985 ) 423; 171 (1986) 233(E); O. Alvarez, I. Singer and B. Zumino, Commun. Math. Phys. 96 (1984) 409. [16] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 986. [ 17 ] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975;Ann. Phys. (NY) 140 (1982) 372. [ 18 ] N. Redlich, Phys. Rev. Lett. 52 (1984) 18; A.J. Niemi and G. Semenoff, Phys. Rev. Lett. 51 (1984) 2077. [19 ] S.G. Naculich, Nucl. Phys. B 296 (1988 ) 837. [20] K. Jansen, Phys. Lett. B 288 (1992) 348. [21 ] D. Boyanovsky, E. Dagotto and E. Fradkin, Nucl. Phys. B 285 (1987) 340; E. Fradkin, Nucl. Phys. B (Proc. Suppl.) IA (1987) 175; E. Dagotto, E. Fradkin and A. Moreo, Phys. Lett. B 172 (1986) 383.
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