The Smit-Swift-Aoki formulation of a lattice chiral gauge theory is presented. In this formulation the Wilson and other non invariant terms in the action are made gauge invariant by the coupling with a nonlinear auxilary scalar field, omega. It is shown that omega decouples from the physical states only if appropriate parameters are tuned so as to satisfy a set of BRST identities. In addition, explicit ghost fields are necessary to ensure decoupling. These theories can give rise to the correct continuum limit. Similar considerations apply to schemes with mirror fermions. Simpler cases with a global chiral symmetry are discussed and it is shown that the theory becomes free at decoupling. Recent numerical simulations agree with those considerations

Maiani, L.

Rossi, G. C.

Testa, M.

English

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Preprint n.794
March 12,1991
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L.Maiani, G.C.Rossi and M.Testa
0% LATTICE CHIRAL GAUGE THEORIES
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ON LATTICE CHIRAL GAUGE THEORIESM
L.Maiani
Dipartimento di Fisica, Universiti di Roma "La Sapienza"
Sezione INFN di Roma
P.le A.Moro 2.00185 Roma, Italy
G.C.Rossi
Dipartimento di Fisica, Universitd di Roma II, Tor Vergata
Via F. di Carcaricola, Roma, Italy
M.Testa
Dipartimento di Fisica, Universitlt di Roma "La Sapienza"
Sezione INFN di Roma
P.le A.Moro 2,00185 Roma, Italy
Abstract
We consider the Smit-Swift-Aoki formulation of a lattice chiral gauge theory. in
which the Wilson and other non-invariant terms in the action are made gauge-invanant by the
coupling with a non-linear auxiliary scalar field, A. We show that n decouples from the
physical states only if appropriate parameters are tuned so as to satisfy a set of BRST
identities. In addition, barring unforeseen cancellations, explicit ghost fields are necessar y to
ensure decoupling. Thus, for these theories to give rise to the correct continuum limit. the
requirements discussed in our previous work have to be satisfied. Similar considerations
apply to schemes with mirror fermions. Finally, we discuss simpler case, with a global chiral
symmetry and show that the theory becomes free at decoupling. Recent numerical
simulations agree with our considerations.
M Partially supported by Ministero Pubblica lstruzione.
7bs research was supported in pan by the National Science Foundation under Grant No. PHY89.04035.
supplemented by funds from the Nauonal Aeronautics and Space Administration.
1. The lattice formulation of chiral gauge theories has received considerable attention in
recent years. As well known. the difficulty lies in the chiral symmetry breaking caused by the
so-called Wilson term, which is needed to remove unwanted fermion doublers present in the
naive discredtation of the Dirac action.
A starting point was provided by Smit 1 1 1 . To cope with the above problem, he
introduces an auxiliary field, n, with values in the gauge group. A plays the role of the
angular part of a Higgs field and it is used to make the action, Wilson term included,
invariant under simultaneous gauge transformations of the link variables U, of the fermion
fields y and of n itself ! :
Uo(x) — U O (x)h = h(x+p)U,(x)h(x)+
V(X) _y(x)h
 = h(x)W(x)
Q(X) —42(x) h = f2(x)h(x) + 	(1)
The general structure of the action, in this case, is:
S(U, y, n) = S `i1( U0 , yn )	 (2)
where S--1 is any local action constructed with U and V. In particular, a gauge field mass-
term in S\1:
(sNl)mass= 2 x	 Tr(n(x+µ)Uµ(x)f2(x)+ -h.c.] 	 (3)
gives rise to the kinetic energy tern for n.
From a different, but completely equivalent, point of view 1 2), one starts from the
non-invariant action, SN1, and obtains a gauge invariant partition function as an average over
the gauge group:
Z = f AL' zvz.	 J AS2 e'S^1 Nn ,yf2 )	 (4)
Of course, the field n is a pure artifact t ` the formulation, and we are interested in the
limit where A decouples from the physical degrees of freedom. To this aim. the parameters
We arc using a condensed notation in whi;h we do not separate explicitly the left and right-handed
components of y, which, in a chiral theor%. transform differently.
appearing in SNI should be tuned appropriately, in a way which, however. was never
completely specified, in reff.l l] and 121, and which we are going to study in detail in this
paper.
The physically interesting gauge transformations that we want to recover in the
continuum limit of our theory arc not those given in (1), but rather:
Uµjx) --Ouµ'Wg = g(x+µ)Uµ(x)g(x)+
W(x ) —►W(x )g = g(x)W(x)
OW -40(x)	 (5)
Simultaneous invariance under (l) and (5), with independent g and h, is equivalent to
the decoupling of n. In fact, the combined transformations with:
g(x)=h(x)-t=h(x)+	 (6)
leave U and W untouched and can be used to set A=1 everywhere.
Thus, the continuum action which the lattice theory is supposed to mimic (called the
target theory in ref.13)) not only has u decoupled from the other fields, but is completely Q-
independent. This leads to another necessary criterion to ensure that we have reached the
correct gauge theory. Since the target action is 0-independent, in the continuum limit the
partition, function in (4) must become proportional to V 2 , where V is the gauge group
volume:
V =	 J Ag(x)	 (7)
that is, Z must develop one factor of V for each gauge symmetry group.
In summary, we can identify three equivalent conditions, either of which can be used
to tune the parameters in Srj, in order to reach the target chiral gauge theon
i) decoupling of fl;
ii) inv ariance under both set of transformations, (1) and (5):
iii) proportionality of Z to 112.
An independent approach was proposed in ref.13). n is not introduced at all, the set
of dynamical variables is enlarged so as to include ghost fields and the parameters in Sr1 are
3.
tuned so as to satisfy the BRST identities of the target theory, in a given, but otherwise
arbitrary, gauge2 .
In the following we prove that the validity of the BRST identities corresponding to the
symmetry under (S) is a necessary condition to decouple the auxiliary field A. We will also
see that, in general, the addition of local counterterms containing only U and V. to eq.(1) is
not sufficient to guarantee the decoupling of 0. Inclusion of the ghost fields, with the
corresponding enlargement of the counterterm structure, is the minimum requimment for this
to happen.
All together, these results show that the approach based on (4) may succed only when
it becomes equivalent to the approach of ref.[3). The symmetry (1) we started with is
inessential. It cannot reduce the difficulties inherent to the construction of chiral gauge
theories outlined in 131, but can only make them worse.
Another approach to this problem has been presented in ref.[4], the so called mirror
fermion method. It consists in introducing new fermionic degrees of freedom (the mirror
ferrnions) with chiral transformation properties opposite to those of the ordinary matter. With
mirror fields it is possible to eliminate the doubling through a gauge invariant Wilson term.
However, in order to remove the degeneracy between ordinary and mirror fermions, an 0-
like field has to be introduced. This means that, after the integration over the mirror fermions.
we are left in a completely analogous situation as the one considered above, so that we need
not discuss mirror fermions anv further.
2. To spell out the decoupling conditions, we first extract 'from Z the volume facto:
corresponding to the unphysical transformations (1), through the usual Faddev-Popo%
procedure. This requires fixing the gauge. For definiteness, we specialize to the Lorentz
condition 3 and, with standard manipulations, we obtain:
(^ Aµ)2 +cdZ=t' ^^f21JUAWJJiyJ9clJc c xp^-S^1(Ln, WSJ - jd4x[ 2au	 µ
=1' JJrJi2^^J9W-0Wl^cltJc exp[- SNX0 , Wfl ) +Sg . f(A, C. c)) =1'	 (R^
2 We stress that the tuning conditions can be enforced in a completel y non•perturbauve M a> and at the
same ume per,urbauon theory can be recovered to the weak coupling limit, as verified up to one loop to
ref.13).
3	 To simplify '.he notation, we shall use here and there the continuum notation, putting. u ; = a go AO;
denoung finste differences with derivatives, etc..
4.
The vacuum expectation value of an arbitrary operator, O(U, t►, Q), is defined
i
JBQJ9UJ9tyJ94fj9cJ9c exp(-SNI (Un, A +Sg .t(A, c, c)) ON. v, A)
In the target theory, <O> would remain unchanged if we make the substitutions (S) in
the action. Of course, this is not true in eq. (9), since the action is not invariant. To enforce
this invariance, we have to by tune the parameters of SNI, and this gives us the appropriate
conditions to obtain the correct continuum theory. In formulae, this implies that I(O. g),
defined according to:
1(0. g) = f JJnAI,'AtyAt^-0c^c exp(-SNI(U go, V90) +Sg.t A. C. c)] O(U, v. A)
(10)
has to be independent of SW, in the continuum limit.
Following standard techniques, it is now straightforward to prove that the g-
independence of (10) is equivalent to the validity of the BRST identities:
jdy < bBRST(O( li , W, 0) c(y) c5µ 15µ) > = 0
Thus, by the considerations made in 1., we see that the validity of the BRST reladon^
is a necessary condition for the decoupling of Q. in the theory described by the partition
Note that it is necessary to consider operators O which are general functions of U, v
and 0. in order to reach all sectors of the Hilbert space of the theory. For 0- independent
operators, 1(0. g) is trivially g-independent. In this case, in fact, the particular A dependence
of the action required by invariance under (1) makes it possible to reabsorb g in the invariant
integration over U.
3. To make a closer contact with ref. ( 3], it is useful to consider the vacuum expectation
value of operators invariant under both transformations (1) and (S). Such operators are Q-
independent, as discussed in I-, and can be generally written as:
M. w - ISS&IF(US)l det[M(U`)l P(US. wt)	 (12)
where F is a gauge-fixing condition, det[M(U)l the corresponding Faddev-Popov
detetminant4 and P is any function of its arguments. The vacuum expectation value of 0 is:
<a(U, W) >=
= 'f.0g.0Q.OUZW.O^ exp[-Srj(U fI. WA)lb[F(U9)l det[M(Ug)l P(Ug. W9) (13)
After an obvious change of variables, the numerator of eq. ( 13), is rewritten as:
1' f -M j-OUDYJJ exp[-SN1(Un, Wn)lb[F(U)] det[M(U)] P(U, W)
	
(14)
The explicit volume factor corresponds to the exact symmetry (1).
As before, we specialize to the Lorentz gauge:
F(U) = aµAµ 	 (1S)
By standard techniques, the delta - function in ( 14) is replaced by the exponential of
(aµAµ )= , the Faddeev-Popov determinant is written as the exponential of a ghost action, and,
in conclusion, (14) is replaced by the equivalent expression:
V j.00 J(P, w .
all JJ90 JJ9c-OcJ9 µ ^L'J9W-09 exp(-SNXn, Wfl ) - Jds x( (az M /- +i aµ Dµc])P(L, W)
.1' JAf2JJ9cJ9cJ9UA^►Aiy exp(•S^l(U n, Wfl ) +Sg .f(A. c, i)1P(U, WI	 (16)
At decoupling, A disappears from the physical intermediate states in the Green's
functions (16) and it contributes only to (finite or infinite) renotmalizations of the terms in
SNl. The latter contributions survive because divergences arising from the functional
integration over A may compensate the vanishing of the A couplings. Neglecting these
4	 The arguments which follow arc valid even in the presence of Gnbo% copies, as we comment below.
6.
renormnalization effects at first, i.e. treating A as an external field, the deeoupling of A (and
the presence of a further volume factor in (l6)) is equivalent to the condition:
)(P, n)=
= jDcJ9iBUJ9YJ9* expl
-SNOO, A +Sg.gA , c, c)]P(U. W) s
= independent from n.	 (17)
It is again straightforward to see that the 0-independence of (17) gives rise to a set of
identities which are precisely the BRST identities considered in ref.13], namely:
Jdy < 6BRST(P( U- W) C(y) ^µ 679 ) > = 0	 (18)
The renormalizations implied by the A integration do not change this picture. Their
presence simply means that the coefficients of the counterterms determined by (18) will be
different from those required to decouple A in the full functional integral, eq.(16) (we shall
see belov% a simple example of this phenomenon ).
The actual values of the coefficients are, of course, physically irrelevant. In fact the
resulting Green's functions, when restricted to U and W external states, obey in the end to the
same identities and we will get to the same continuum theory either by enforcing (18) or bN
requiring the decoupling of Q in the intermediate states of eq. (16), or by enforcing eq.(11).
This result is very satisfactory. It shows that if a meaningful continuum limit can be
obtained, the resulting chiral gauge theory will be unique.
A final remark concerns the so-called Gribov problem, which represents a potential
obstruction to a program which contains the gauge fixing as a basic feature. It was shown in
ref.15 1 that to overcome the problem of Gribov copies one has to use the Faddev-Popov
determinant, instead of its absolute value. However, it is precisely with this prescription that
the Faddev-Popov factor admits interpretation in terms of a ghost field action, and leads to
the formulation of the BRST identities.
4. It is quite obvious that the identities (11) or (18) cannot be satisfied for a finite lattice
spacing, a, but only after neglecting terms which vanish like positive powers of a 13).
Even with this proviso, the limitation to local counterterms, dependent only upon U
and W. which is implicit in the scheme ( 4), may prove to be too restrictive for the identities
(11) to be satisfied. This is the case if the identities (11) turn out to require cots erum is in
the ghost sector, which destroy the geometrical nature of the gauge-fixing action.
In the approach of ref.13], counterterms are required, by renorrrtalirabdity, to have
dimension less than five and are further restricted by whatever exact syrntnecry of the target
theory can be mantained at the lattice level. For example, there is only one possible
countenerm in the ghost sector, of the form:
bSPost= C f dx Tr1 caµ(Aµ c)]	 (19)
other renormalizable counterterms being excluded by ghost number conservation and
invariance under a constant shift of c.
In the sche-ne based on (4) the requirements on counterterms are less clear, since the
theory is not renormalizable. We expect all terms which are dimension less or equal to 4, in
the limit n(x)•1, to be relevant. One may ask if the exact symmetry (1) can be used to obtain
additional limitations to the form of the counterterms, in particular if it can forbid the presence
of the dangerous one, eq.(19). The answer is negative. Using fl, one can construct the new
connection:
which transforms like Up(x) under (1). With V W one can make any possible countenerm to
be invariant under the BRST transformations associated with (1). For example, the ghost
countenerm (19) is obtained as the Q=1 limit of the BRST-invariant term (b'BRST = 0):
bBRSTTrI caµAµ - CaµAN ' caµ(R'aµA)'
and is therefore allowed.
Within the approach of ref.13], the issue of counterterms can be decided with
perturbation theory in the gauge coupling. go. The decisive countenerm (19) may arise on1%
at two-loop level, or higher. An explicit calculation is not available. at present but, in the
absence of a protecting symmetrv,we should remain on the safe side, and assume that such
countenerm is indeed required.
Another remark concerns the relation with perturbation theory. The scheme based on
(4) contains non-polynomial Yukawa-couplings with large coefficients, and is therefore not
amenable to perturbation theory. In a way, the situation recalls that found in the 4 4 theory.
(20)
b.
Due to the triviality of the theory, the remmnaliaed coupling constant becoom stall for mull
lattice spacing and renormalized perturbation theory is applicable. However, the relation
between renormalized and bare couplings is fully non-peruntative.
In contrast, the relations ( 1 g) can be implemented without problems in perturbation
theory, in the scheme of ref.[3).
S. The non-linear global Yukawa coupling studied n6merically in refs.16). [7]. see also
ref.18j, can be discussed along lines similar to the gauge case, and it provides a simple
example of counterterm non-perturbative renormalitation.
-The action can be obtained from that of the gauge case by setting:
Vµ(x) =1
In continuum notation, it is given b%
S(y, f2)= S Fkln(y ) + SYW(Y. f2) + Skin(0)
SFkinly)= jd x l WLOvL +WROVR I
Syw(y. ft) =fdx ISY WLQLQR + VR + WLULWDR'yR + h.c.]
Sk,n (ft )= ^i KL jdxTrI&0 QLDiJJL'j + 1i Kk jdx Tr(d,; ftRD f2R']=
where.
f2= (f2L, DR)	 (22)
and W is a differential operator which represents the Wilson term. The model is invariant
under:
D=(QL, RR) -+ D h -(hLOL. hRQR )
WL— hLWL
WR_ hRWR
	 (23)
with global hLR.
9
An obvious way to impose the decoupling of n from v. is to raquiree A to obey the
same equations of motion that would arise iron Ski„ alone. This implies that, for a local
transformation. hah(x):
1BOD Biy exp-ISFkin(W) +SYW(W. n) +Skin(nh)1O(W. Ah)
a independent of h(x)	 (24)
B; the change of variables:
it -. t2h+
W W h+	 (2S)
eq.(24) becomes:
jJ9AJ9YJDW ex p- ISFkin ( W h+ ) +Sy V► , (W, R) +Skin(n ) 1 O( W h+ • n) a
a independent of h(x)	 (26)
The set of equations generated by eq.(26) coincides with the Ward identities of a
massless. free fermion, continuum theory. As observed before, the 0-integration gives rise
only to non-penurbadve renormaiization of gy.
In general, the decoupling may take place only if fermionic matter is anomaly free
Otherwise. Wess-Zumino self interactions of A will arise, which prevent the decoupling.
Such interactions, however, require a non abelian structure of the symmetry group. The L k, ,
and U(I)xV(I) abelian models are special in that the continuum limit can be obtained even in
presence of chiral anomalies.
Numerical investigations have been carried out for two U(1) cases, namcly:
A=ML. OR) = (W. 1) •	 (27.a)
A=(i2L. DR) = (w2 , w)	 (27.b)
with w an x-dependent phase.
The numerical results show that it is indeed possible to tune Sy so as to obtain a
vanishing renormalized ferrnion mass. In addition, the critical value of Sy found in 171 is
I 
compatible with zero in case (27.0. as expected on the basis of the shift-symmetry studied in
ref.(9), while it is definitely non-zero and non -perturbative for case (27.b), a non trivial
renornalization due to the functional integration over fl.
We add two comments.
i. In the presence of further physical interactions (Yukaws or gauge interactions) the
masslessness of the fermion cannot be used as a tuning condition; rasher. the farnion mass is
a ph> sical quantity to be computed after the appropriate (Ward or BRST) identities are
imposed.
ii. The critical value of Sy found in case (27.b) cannot be assumed as a starting point for a
numerical simulation of the fully interacting gauge theory, since it is renortmalized non-
perturbatively by the additional interactions.
We thank M.Golterman and 1.Smit for interesting discussions. We would like to
acknowledge the warm hospitality of the Institute for Theore:;.al Physics at Santa Barbara.
»here this work was initiated and ended.
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P.Hirschfeld. Nucl. Phys. IM (1978) 37;
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