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Finiteness of the Chern-Simons model in perturbation theory
Nuclear Physics B345 (1990) 472—492 North-Holland
FINITENESS OF THE CHERN-SIMONS MODEL IN PERTURBATION THEORY A. BLASI and R. COLLINA Dipartimento di Fisica, Universitd di Genova, Istituto Nazionale di Fisica Nucleare, sez. di Genova, Via Dodecaneso, 33-16146 Genoa, Italy Received 30 November 1989 (Revised 25 May 1990)
The Chern—Simons action in the Landau gauge is characterized by a BRS symmetry and a local dilatation invariance Ward identity whose quantum extension controls the trace anomaly in the flat limit. We show that the trace anomaly corresponds to all orders to operators which are BRS variations and therefore it cannot have contributions from the Chern—Simons gaugeinvariant term. This is sufficient to insure the vanishing of the /3-function. We further show that, in a renormalization scheme which preserves scale invariance, there are no finite one-loop corrections to the parameters of the model.
1. Introduction In the panorama of topological field models [1] the Chern—Simons theory in three dimensions seems to occupy a prominent place. The nonperturbative analysis of this model has contributed to clarify its relation to two-dimensional, conformal field theories and to knot theory [2, 3]. The model has also interesting features from the perturbative point of view, where the numerical computations have proven the vanishing of the /3-function up to two loops and have succeeded in identifying new knot invariant polynomials [4—61.When looking at the Chern—Simons theory from a strictly perturbative point of view, we have to deal with the following problems: dimensional regularization cannot be used, so either we look for alternative gauge-invariant regularization procedures [61,or we discuss the quantum corrections of the model in a regularization independent way. Secondly, the coupling constant of the theory, whose quantization is deducible with non-perturbative arguments, appears here to be a free real parameter; consequently the vanishing of the /3-function does not follow directly*. Finally, the *
Indeed, even the characterization of the model as in ref. [7] by means of global extra symmetries, while identifying the gauge fixing (Landau gauge) uniquely, still leaves room for a renormalization of the gauge-invariant Chern—Simons term.
0550-3213/90/$03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
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topological character of the Chern—Simons gauge-invariant action, i.e. its independence from the choice of a metric, appears to be spoiled by the necessary introduction of a gauge-fixing, metric dependent, term; that this is not the case was argued by Witten in ref. [2]. Here we would like to address the question of the vanishing of the /3-function to all perturbative orders in the Chern—Simons theory in the case of flat euclidean space-time. The suggestion which we shall follow is that the metric tensor appears only in the gauge-fixing part of the classical action, and therefore, as noticed in ref. [4], the classical energy—momentum tensor is a BRS variation. To implement this idea at the quantum level we need a symmetry which characterizes the algebra of the energy—momentum tensor; in the Landau gauge, where no dimensional parameters are present, this is provided by Weyl’s representation of general coordinate transformations. This representation is unstable under radiative corrections and its flat-limit version acquires a correction term which, as shown in ref. [81, controls the “minimal” trace anomaly of the model. The next step is to extend the original “true” BRS symmetry, which commutes with the local dilatation transformations, as to include the spurion fields needed to characterize the trace anomaly in such a way as to guarantee that the trace anomaly itself remains a “true” BRS variation. The conclusion can then be stated as follows: since the Chern—Simons theory in the Landau gauge has no infrared problems [9], there is a standard connection between the coefficients of the “minimal” trace anomaly and those of the differential operators appearing in the Callan—Symanzik equation. On the other hand the trace anomaly turns out to be a true BRS variation and thus the differential operators appearing in the Callan—Symanzik equation must all correspond to insertions which are themselves BRS variations; this excludes the presence of the invariant Chern—Simons term and insures the vanishing of the corresponding /3-function to all perturbative orders. We further analyze the radiative corrections of the theory and show that there are no finite one-loop renormalizations provided the renormalization scheme does not violate scale invariance. The paper is organized as follows: in sect. 2 we introduce the BRS and local scale invariances which can be used to characterize the model at the classical level. Sect. 3 illustrates the algebra of the energy—momentum tensor and the strategy employed to characterize the minimal trace anomaly in the flat limit as a BRS variation. Sect. 4 contains the results on the renormalizability of both symmetries: the extended BRS invariance, which is discussed by the usual cohomological methods (the details are in appendix A), and the local dilatation Ward identity, whose detailed analysis has already been performed in ref. [81.In sect. 5 we collect the results on the vanishing of the /3-function and the absence of finite one-loop corrections to the model. The technical details concerning this last point are treated in appendix B.
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2. Classical action and symmetries In this section we shall characterize the Chern—Simons action in the Landau gauge by its invariance under the corresponding BRS operator and the additional, compatible, local dilatation transformations. These two symmetries are sufficient to guarantee, at the classical level, that the energy—momentum tensor is the BRS variation of a local operator. The possibility of extending this scheme beyond the classical level and in particular the instability of the local dilatation Ward identity, will be discussed only in the flat limit; we introduce here also the tools needed for this analysis. The Chern—Simons gauge-invariant action is
‘Cs
=
—
~k1~”PJ d~x[A~a~A°~ + lfabcAaAb,4c}
,
(2.1)
where f~t~Care the real completely antisymmetric structure constants of a simple Lie algebra. In a perturbative approach, the coupling constant k is not quantized and therefore it appears as a free real parameter. We choose the gauge-fixing term to be covariant and with no dimensional parameters, i.e. the Landau gauge with action
‘GF
~
(2.2)
,
where c~,b’~are the Faddeev—Popov ghosts and antighosts and d~the corresponding Lagrange multipliers. The nilpotent BRS transformations are
(2.3a)
sA~=—(V~c)”=_(aca+fabcAcch) sca
=
-~f~c’~c~, sb” =d~,
sda
=
0.
(2.3b—d)
Notice that ‘GF does not contain any free parameter which can be reabsorbed in a multiplicative redefinition of the b’~,d’~fields, compatible with (2.3c)*. In order to be able to discuss the BRS symmetry at the quantum level we also add invariant external fields y~(x),~a(X) and the couplings
IEF=fdx[Y~(~)~
2f~}.
*
(2.4)
The most general choice + fora the gauge-fixing term compatible with eq. (2.3) is, up to an integration 3xb~[d~A~ by parts, sf d 1d” + ~ both a1 = 0 (Landau gauge) and 02 = 0 are enforced by the local dilatation invariance.
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The classical action F~=ICS+IGF+IEF
(2.5)
satisfies the functional identity 5T’~ 6[~Cl
(sF’~)=fd3x[_~~~..~. +
6F’~~f -~--~-~-~-
c~
3pc!
+da_~~j.
(2.6)
The absence of dimensional parameters in (2.5) denotes that the theory is scale invariant. Wishing to realize this symmetry at the local level, and hence to discuss the invariance under general coordinate transformations x~ x° AU(x), we must abandon flat space-time; as pointed out in ref. [8], the suitable framework is provided by Weyl’s representation of general coordinate transformations where the fields transform as —
+
=
=
Aa3ca
=
~
—
3~A°~4~,
(2.7a) (2.7b)
—a~A~~ —0
1~” + 40~A~”
)I~4T
6d~ AUo~d~~ + ~0ffA~d~
(2 7d)
=
=
i5yaP.
A~ab~ +~
=
=
—
(2.7e)
3AJLya~lr +
(2.7f)
0~A”~’,
)t~r0~a + 0aX~~Ta,
=
—
(2 7c)
0,TA~Wa+ ~0,,X’w~’
(2.7g) —
‘~
0~0~A”;
(2.7h)
~T’~” is a unimodular metric tensor and w~an auxiliary connection. Notice that the fields A~and C’1 transform as a true vector and scalar respectively, while the rest are tensorial densities. The corresponding invariant action
IC!
=
f d3x{ —
—
~ke~P(A~0
[(~“o~+w~)b’1
A~+ lfabcAaAbAc)
—
y~~’1](Vc)’1 (~a~+ —
does not contain any dimensional parameter.
+
~a1fabCCbCC}
(2.8)
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To discuss the compatibility of the BRS transformations (2.3) and the local dilatations (2.7) we perform a Legendre transformation to the connected functional ZC; denoting with jai.~,K’~,J’1 Ha the currents coupled to the quantized fields A~,~a, c” b’1 respectively, the BRS operator becomes SZ C
[d~x
6
6
6
J’1~+Ja+Ha ~ 6~
8K”
zC
(2.9)
while the transformations in eq. (2.7) yield the local operator
_o~_~)~
W~(x)ZC~[~a~r(~.
+
_Jao.....~...
20~(~(r~ .)
6
—
~
8
+~0 K”— ~ 6K”
—
—
8
~H’10 + ~0 H’1— +0 ya(r ~ “6H” ~‘ 6H” ~
6 6ya~L
(2.10) The operators (2.9) and (2.10) obey [S,W~(x)] =0,
(2.11)
so that the BRS and the local dilatation invariance can be analyzed separately. At the classical level we have the identities
SZ~=0,
H’~(x)Z~=0.
(2.12a,b)
The canonical dimension and the Faddeev—Popov charges of the fields A~, d”, b”, ~a 7a~. ~a
(2,
—
1),(3,
—
~
w’~ are respectively (1, 0), (1, 0), (1,
—
1), (0, 1),
2),(0,0),(1,0).
3. The energy—momentum tensor~strategy The action F’~ in eq. (2.8) contains the metric tensor ~ only in the gauge-fixing part; this fact signals that, at least at the classical level, the
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energy—momentum tensor of the model is the BRS variation of a local quantity. In order to write explicitly this property, which will play an essential role in the proof of the vanishing of the /3-function, we first identify the energy—momentum tensor in terms of functional derivatives. Interpreting the classical sources w~’ in terms of the operators that their functional derivatives define, we see that ~
8
6
~
6 )zC_~o~
3’i~v~A(~)
6~(X)zC
(3.1)
corresponds to the energy—momentum tensor of the theory, whose trace is proportional to the divergence of the vector 6Z~/&o”(x). Let us introduce the external fields G~ and (2’~,with Faddeev—Popov charge + 1, and the modified classical action
f d3x[( G’-’”0~
+ fl~) b”] A~.
=
pd +
The corresponding connected functional
ZC!
(3.2)
satisfies the new identity
+n-~2~=o,
~c1s2c!+fd3x[G.’~
(3.3)
which we shall refer to as the “extended” BRS symmetry to distinguish it from the “true” BRS transformations obtained at G~LV 0. The local dilatation invariance is preserved by assigning to G~,12~the transformation laws =
=
~0UAaG~~~,
(3.4a)
6f1~’=A~0~fl~ —0~A~11”+ 40~A”Q~ ~G’~’P0p0alt”.
(3.4b)
=
—
0~A~~GIrV 0,JAVGSLff —
+
—
The modified operator
H~(x)+O~G~ 6G~
+~a~na—~— +a~(n~~_)
~
—
~
(3.5)
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still commutes with $~.Now from the identity (3.3) we obtain
6
6
6G~”(x)SZ~’~GI~,12~.O6[l~(x) SZ~IG~,n~o0 =
(3.6)
=
or, in terms of the “true” BRS operator 5, the relations 6 6 56G~(x) Z~lG~,~o6~() =
6
6 Z~’IGI’~,Q~O6~~v()Z~t~(3.7a) =
6
6
S 6)ZCIG~W’O
6~()ZG1~.f21’=O
(3.7b)
6~()ZC.
It is now apparent that the energy—momentum tensor in eq. (3.1) is, at least at the classical level, the “true” BRS variation of a local operator. The strategy, which the above scheme naturally suggests to discuss the quantum extensions, is that of analyzing the validity at the quantum level of the equations SZC0,
l’(x)ZC=O.
(3.8a,b)
It is well known that eq. (3.8b) cannot hold true with the operator i4’~(x)as it stands due to the presence of the trace anomaly [10] which appears as an instability of Weyl’s representation. Here we shall adopt the method of ref. [8]and rather than discussing the complete algebra of the energy—momentum tensor in curved space-time we shall consider the flat limit ~ 6’~”,w~ G”” 0. To keep track of the trace anomaly in the flat limit we introduce a dimensionless spurion field o-(x) coupled to it and to insure that the trace anomaly is a BRS variation we also need a dimensionless ghost spurion .~(x).By repeating the analysis of ref. [8], we arrive at the result that, in the flat limit and at the least order, the minimal anomaly to the Ward identity for the local dilatation invariance is given by =
=
=
=
W~~,(x)ZCIFL=0,lI(x),
(3.9)
where 1(x) has canonical dimension equal to three and cannot be written as a divergence. Accordingly we shall investigate the quantum implementability of the Ward identity, 6 W~(x)Z~ W~(X)ZCIFL + 0~(x) 61(x) 6 6cT(x)
ZCIFL=0,
6
ZCIFL + 0~cr(x)6cr(x) ZCIFL
(3.10)
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together with the extended BRS identity §ZC~SZC~FL+fd3x~(x)
(3.11)
8~ZCIFL=0.
that the term proportional to h in eq. (3.10) is the one which compensates the trace anomaly and that the spurion field .~(x)is an artifact only needed to insure that the trace anomaly, which corresponds to 6ZC/6a~(x)[~~o,is a “true” BRS variation. Indeed, if we assume that (3.11) holds true we immediately find that Notice
8 6u(x) ______
=
S
6 ~ 6~(x)
(3.12)
The program is therefore that of discussing the possibility of maintaining at the quantum level (3.10) and (3.11); a positive answer to this question implies that no term proportional to the invariant Chern—Simons action contributes to the trace anomaly.
4. Perturbative renormalization In this section we analyze the quantum implementability of the two symmetries of the model in the flat limit, but to all orders in the cr(x) field. Concerning the local dilatation Ward identity, our treatment follows closely the approach of ref. [8], where it is emphasized that we must consider the possibility of having in the action arbitrary powers of the o-(x) field. In other words, the generating functional of the proper vertices is a double formal power series in h and a- and the infinite number of ci couplings are determined perturbatively in terms of the ci independent ones by the Ward identity itself. The presence of the .~(x)field, which carries one unit of Faddeev—Popov charge, does not alter this picture since its couplings are controlled by the extended BRS identity. The two symmetries can be analyzed independently since the corresponding operators commute. We begin with the extended BRS symmetry; it is well known that both the stability and the anomaly problems can be settled by analyzing the cohomology space of the linearized nilpotent operator 6 ~ 6y’1~6A~ 61d
§L
=
Jfd~x
6 SL+fdX~_,
6a
6F’~6y~ 6 6A~
6 6b”
6F’~ 6 8c” 6~”
6F’~ 6 6~,” 6c’~
6 6a-
(4.1)
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identifies the linearized “true” BRS operator. In particular, the soluwhere tions of the cocycle condition 5L
(4.2) where
f~()
=
(4.3)
is a local functional of canonical dimension equal to three and carrying p units of Faddeev—Popov charge, are relevant for analyzing the stability (p 0) and for the compensability of anomalies (p 1). To translate (4.2) to the space of forms, we follow ref. [11] and rewrite it as the descending chain of equations =
=
§Lz1~(x) +d~l~
1(x) =0,
(4.4a)
§L~i~1(x)+d~~2(x) =0,
(4.4b)
§LL1~2(x) +d~~3(x) =0,
(4.4c)
SL4~3(x) =0,
(4.4d)
where d is the exterior differential and the upper index denotes the degree of the form*. To solve the system (4.4), we first look for the general solution of (4.4d), i.e.
for the local cohomology of the ~L operator in the space of functions of the fields. The analysis in appendix A yields as solution, in the sector p 0 =
=
and, in the sector p
=
§LS~”(x) + tfa~ca(x)c~(x)cc(x)
(4.5a)
1 =
§LS~(x).
(4.5b)
By working up the ladder of relations (4.4) (see appendix A for details), we find ~~x)
=
d4~(x)+ §LS~I(x) +
L1~(x)
=
*
t(A~o~A~ + lfabCAa AbA~~ ) d.~/LA dx~A dx’~,
dS~2)(x)+
§LS~(x).
(4.6a) (4.6b)
The descending eqs. (4.4) have been analyzed in models with a different external field content in ref. [12].
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Hence we conclude that the Chern—Simons action is stable and that there are no anomalies to the extended BRS symmetry. Furthermore the term zi$2~(x)in eq. (4.6a) can be analyzed according to the 1(x) field content and dimensionality, as =
H~(x) + m(o.)1(x)f’1
b’1(x)bb(x)d~(x)e~~dx~A
dx’~A
dx’~
=H~(x) +4m(a-)SL[1(x)f’1b’1(x)bb(x)bc(x)Ie~vpdx~~~AdxvAdxh11, (4.7)
where H~(x) is I independent. Now recalling that that §LS~l(x)
=
SLH~(x) +
f d3y 1(p)
SL is also nilpotent, we find
H~(x),
(4.8)
which implies that the I couplings in the effective action are uniquely determined by those of a- and therefore they do not correspond to independent parameters. Having analyzed the extended BRS symmetry we pass to the local dilatation invariance Ward identity; the presence of the ci, I fields does not alter the minimality properties of the breakings in the flat limit, which can be reabsorbed by introducing couplings with increasing powers of ci. Accordingly we can repeat the analysis of ref. [8] and show that the minimal anomalies at the order h”, citm can be compensated by a suitable choice of the couplings ~ At the classical level we find that the action is given by the Chern—Simons term plus a contribution as in eq. (4.8) where I1~(x) contains the gauge-fixing term and all the acouplings*. This concludes the renormalizability of the symmetries of the model.
5. Conclusions In sect. 4 we have shown that the local dilatation invariance Ward identity in the flat limit and the extended BRS symmetry are implementable to all perturbative orders. This implies that the Landau gauge is a stable choice, and, more important, that the trace anomaly is identified as a true BRS variation. To make contact with the /3- and y-functions which identify the Callan—Symanzik parametric equation of the model let us recall that the Chern—Simons theory in the Landau gauge has no infrared problems [9], so that the functional ~ZCmfd3x
*
6u(x)~’
(5.1)
Notice that in this model the presence of the extra terms, finite in number, which contain derivative couplings of the a--field found in ref. [8], is forbidden by the extended BRS symmetry.
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which obeys the same Ward identity as Z~,is well defined. It follows that, at aO/Oa- is given by a parametric derivative. Thus we have 0
=
0,
0
=
/3(k) —Z~I~~o + ~ yZ,J~O, Ok
(5.2)
where the invariant counting operators are given by (5.3a)
~
N
2
=
I d3x[H
-H” +K
N3 =fd3x{J~~j
=
+~a±~j
=
{s,fd3xK~~~~},
{s,fd3x~a~}.
(5.3b)
(5.3c)
Recalling that 0
0
—Z~I~.0 S—Z~[~0, Ocr 01 =
(5.4)
we immediately arrive at the desired result that the function /3(k) in eq. (5.2) identically vanishes and that only the counting operators contribute to the r.h.s. of eq. (5.2). The presence of possible finite counterterms can also be discussed within a renormalization scheme which implements BRS and scale invariance. Let us remark that, in this particular case, where there is no “natural” normalization of the model, the issue of the finite contributions is really crucial to give meaning to the perturbative approach. Indeed, once we have chosen a renormalization scheme for the Green functions of the theory and once the lagrangian parameters are identified by arbitrary normalization conditions, the appearance of finite corrections to the Green functions defining the parameters makes the theory unpredictive since scale invariance forbids a non-trivial link between the different finite contributions obtained by changing the normalization point. The only way out is that they vanish in any renormalization scheme which preserves scale invariance. That this is the case at the one-loop level is shown in detail in appendix B. A regularization scheme which violates scale invariance may lead to different results which however are due to the breaking explicitly introduced by the regularization and which can be reabsorbed by finite counterterms [6]. Concerning the observables of the model, we notice that the exact scale invariance implies that their vacuum expectation value is given by (scale-invariant)
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c-numbers. At the local level this is a rephrasing of the well-known fact that the model has no local observables, while for globally defined quantities this suggests that the structure becomes trivial. This perturbative result confirms the non-perturbative analysis of the model [2] according to which, for flat background, the physical Hilbert space becomes one dimensional.
Appendix A The solution of the system of descending equations (4.4) is based on the of knowledge of the local cohomology of the ~L operator in the space V~3~ Lorentz-invariant polynomials built with the fields and their derivatives, with canonical dimension up to three and any value of the Faddeev—Popov charge. As a first step let us rewrite the action of ~L on V~3~ as that of an ordinary differential operator: defining the independent variables ~
for any field cP(x) (A~,b”, d”, c”; ordinary differential operator
P.i
~-a yaP.
I,
a-},
the operator
0
3pC!
n0
(A.1)
=0k.. .0~’1~(x)
~‘~‘Oy’~P.(x) 0A~P.P.(x)
...O0 ~ ~0Aa;~(x)
OF’~ +0P.1 ...0 ~“Oc”(x)
0
3~CI
~OA~(x)
“
0y~IP.(x)
Ole!
3~C!
—0
reduces to the
~L
Oy~
P.(x)
+0
~
...O
~0~’1(x)
0
P.(x)
Ocf’P.
0
O~~P.P.(x)—O‘~“...OO ~ “Oc~(x) ~
+d~P.P.(x) ba()
0
~.
+I;P.P.()
•~(x)
(A.2)
When acting on Vt3~,~L further simplifies since the summation index n, due to dimensionality and Lorentz invariance obeys the bounds n 3 dim(~)for cI~ ~
—
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A. Blasi, R. Collina
(~,b’’), and n
2 — dim(1) for ‘1
~
a
{A~,Ca, yale
(9
a
19
19
_fabCAb
cc~a
_f~Abc~~ P.,
OAP.;(,
~3
PayaP.
a
8a-;p~
a
a
+da;P.~
19
fabcybP.cc. OyllP.
ePf0bCA~AC_____
(9
—+1. ‘~
19
~k a
+{f
9
Oa-
P.”
+
19
+1—+I.
~
b,P.
+ [_fabcAbcc~
I}, or explicitly,
+keM~A” ____+y~~~’___ P;”3~aP. P.
13
ab”
a-,
a
~9 — C~
19
Chern—Simons model
—b”~-—— P.a~a _fabcybP.Ac
t9
+ ~fa cbcc____ +fabccb ~ cc____~_ 19C~ +f
a +f”~b”;
abcbb; PACP.—
a
~
19~a
c(cbcc
+ c.~c~)
Iv~
(A.3)
V(”
OyaP.}
To identify the solution of the equation
.~LX(x) 0,
(A.4)
=
3~ and with X(x) E V~ counting operator O N=A~-~-
.~L
a +AP.;V
as in (A.3), we adopt the filtering [13] induced by the
0
~
0
0 0 —+c” “OC~P. ;P~P19ca
Oc”
O o / a +y~L____+~__+2Iba__+ba
(
0y~’
O~”
ad”
+b” ;le19l~,a
a ;levada
P. ~
0
a
Oh”
a
0
+YaP.
a
~
—I
b~P.~)
a
I+u—+cr. I ~a.P.Ocr
P.
a +ci. ~P.VOO~cev
a
+1 ~
+1
a
a
+1 ;lei’ 01
(A.5)
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485
so that =
+ .~ +
~
[N,~~’L~j =
v
~
and =
{.~(o) ~(1)}
2
=
(g~(1))
=
0,1,2
(A.6)
{~) .~L2~} =0;
+
(A.7)
~ .~, LI~ correspond to the expression in the round, square and curly brackets respectively in (A.3). The space V~3~ also decomposes under N, so that X
E
=
X(a),
NX(0)
aX~
=
(A 8)
a=0
Clearly the cohomology of can be discussed separately in each finite-dimensional eigenspace X~,which can be embedded in a Fock space where the action of a creation operator is identified with the multiplication by the corresponding variable and the annihilation operator is given by the derivative with respect to the same variable [14]. In this framework the cohomology space H~°~ of ~ coincides with the kernel of the Laplace—Beltrami operator ~
=
{(~(o))t .~)}
a
a a —+A’1--——+A’1 P.OAa
=~~—+~“
;P.OCa
a
lePOCa
+ 2y’1~—~+y—~j
a
19 P.
a~i~
~
a ab~~
a
~
a
P.~9J 3a
+cia+ci;P.a+ci;leea
+I~+I;P.~+I;P.C1.
(A.9)
It is evident that H~°~ contains only arbitrary functions of the fields c’’ without space-time derivatives; hence the general solution of (A.lOa)
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Chern—Simons model
is given by y....
(A.lob)
~(0)1~+F(Ca)
We can now identify the cohomology space of the complete operator ~L; by substituting eq. (A.8) into eq. (A.4), we find the chain of equations 0,
=
+
(A.lla)
~X~t=0,
~ ~
+
indeed
(A.llb) 0,
=
a
=
1,2
(A.llc)
The solution of eq. (A.lla) yields X~1~
+
=
F~’~(c”) ,
which substituted into eq. (A.llb), recalling the expression of ~
—
~(1)~(!))
=
—
lfabcCbCcBF(!)
(A.12) yields (A.13)
Now the r.h.s. of eq. (A.13) belongs to H~°~ and therefore cannot be written as a coboundary; hence we find ~
X~2~~(0)~(2)
+
=
~/~1)~(1)
+
F~2~(c”) ,
(A.14a)
0.
(A.14b)
0
fahCchcCl(!)(Ca)
=
Oc
The semisimple character of the group further implies the vanishing of F~~(c”) T”c”. The same reasoning applies to the successive iterations and we have to discuss the solution of =
a
fabcChCCF(’1)(cQ)
=
0,
a
=
2,3,...,
(A.15)
ac where F~”~T~’1...a,]Cal =
...
(A.16)
Ca,,
and TE”1.~ is a completely antisymmetric invariant tensor in the adjoint representation; for semisimple Lie algebras we have T’1
=
T~”~’~ T~”~1 0, =
=
T[~2~~dI tf’’~ =
(A.17a, b)
A. Blasi, R. Collina
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Chern—Simons model
487
plus possible contributions with a ~ 5 depending on the particular group. Therefore the solution of eq. (A.4) is given by
x=
~
~L~+tfa~C’1CbCc+
r0T[a1a~Ical...ca~~.
(A.18)
a~5
This identifies the cohomology of the -~L operator at least as far as the discussion of the stability and anomaly are concerned; indeed we shall see that the contribution tfabccacbcc reconstructs exactly the Chern—Simons action when inserted into the descending chain of equations (4.4), while the absence of the term proportional to c’1c”c’~c’~ in eq. (A.18) insures that the theory is anomaly free. Indeed, going back to eq. (4.4d) with the choice p 0, 1, relevant for the stability and anomaly discussion we have from eq. (A.18) the solutions =
z~(x)
=
§LS~(x) +tfa~~ccacbcc
(A.19a)
=
§LA~(x).
(A.19b)
With the above substitutions eq. (4.4c) yields —
dS~(x))
—
d~1~(x))0.
=
—
4t d(f’1~cac~!~C~),
(A.20a) (A.20b)
=
After some algebra we can rewrite the r.h.s. of eq. (A.20a) as =
—
§L(f
CAa(x)cb(x)CC(
x) dx~’)
(A.21)
and obtain the solutions ~tT(x)
=
§LS~’~(x) + d~~(x)+ tfabcAa(x)cb(x)cc(x)
(A.22a)
=
§L~~(x)+ d~(x).
(A.22b)
The procedure iterates for the successive steps, to arrive at Lt~(x) §L~1(x) + d~~(x) =
+
t(A~(x)O~A~(x) + lfabCAa(x)Ab(x)AC(x))dxP.
A dx’~A dx”,
(A.23a) =
§L~~(x) +
dz~?~(x).
(A.23b)
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Chern—Simons model
The expressions (A.23) guarantee the stability of the Chern—Simons action and the absence of anomalies.
Appendix B Here we shall analyze in detail the question of the finite corrections to the Chern—Simons model in the Landau gauge. As pointed out in sect. 5 the absence of such corrections is crucial to have a sensible perturbation expansion. The discussion will be limited to the one-loop level, and the symmetries to be preserved are the BRS and local scale invariance; it is convenient to first perform the field redefinition so that the coupling constant g 1/ ~ explicitly appears only in the interaction terms; the CS classical action, in the Landau gauge, takes the form =
F
=
J d~x{
OC.A~+
—
~cP.1P(Aa
~gfabCAaAbAc)
—
[O”b” yle’1](Vc)’1 —
—
OP.d’1A~+g~~’1f’1~cbcc},
(B.1)
where (V~Cy’= (Olec” +gf”A~c’~’).
(B.2)
Since the gauge fixing is linear, we shall consider the BRS symmetry together with the constraint* 61
(B.3)
6d’1(x)
P.
Thus the free parameters at our disposal are the gauge and ghost fields renormalizations ZA, ZC and the coupling constant redefinition Zg. These parameters are identified by the Green functions
6~F
6~F
621
6~’1(p)6eb(k)6cc(o)’
6~(p)6cb(0)’
6A~(p)6A~(k)6A~(0)
(B.4)
at an arbitrary normalization point, where the “tilde” denotes the Fourier transform and all fields are set to zero after the functional derivation. Now if the ZA, Z~,Zg parameters have the h expansion
Z *
This corresponds to the relation
=
)‘i
2Z~2~ + ...,
1 + hZw + h + Y2
=
0 in the Callan—Symanzik equation (5.2).
(B.5)
A. Blasi, R. Collina
/
489
Chem—Simons model
we have, at the one-loop level
6~F
\(1)
=g(z~’~ + Z)f’1~+
(6~a(p)6Eb(k)6Cc(O))
621
/
)(1)
______________ 6~(p)6cb(O)
(B.6a)
Gabc(p, k),
=
i612bpP.(Z~0 z~1))+ G~(~),
(B.6b)
=
_geP.1~Pfhhl1~~(z~,l) + 3Z,~’))+ ~
k), (B.6c)
—
(1)
63F _____________________
(6A~(p)8A~(k)6A~(0))
and, employing the Feynman rules derived from (B.1), we compute 3
~ k)
G~(~)
G~(p,k)
=
~
6 CP.CPf’1mnfbmnfcmrpP.kPf
(2~-)
g2 9/2
=
(2~-)
C(2)6
EP.CPPJ
d q
d3q
(p—q) 2 (k—q) 22’ q
2
(B.7a)
(B.7b)
2’
q (p—q) d3q
=~_g
(21r)
famnfbrnfcmrf
2
q (p+k—q)2(p—q)2
x [~2qCqPq~~+ qP.qP(2pv + kC) + qPqCpP.
+
q”qP.(kP +p”)
+qe(kl~p~~ —k’3p~’)_qP(p~~pC+k~epe) qP.(pl’pP +kvp~))] d3q
(2w)
brnfcmrfq2(p+kq)2(pq)2
x [2q”q”q~’2pleqvql~ 2qleqP(p~+ k’) —
—
+
2p”q~(p’+ k’)], (B.7c)
where C(2)6” faedf bed. Notice that G”~(p,k) in eq. (B.7a) is finite and it indeed vanishes; since we are interested in the (eventual) finite parts in eqs. (B.7b, c), whose integrands are logarithmically divergent, we can extract them, up to an irrelevant arbitrary constant reabsorbable by the freedom of the lagrangian
490
A. Blasj, R. Collina
parameters, by the operator
FPfd3q
/
Chern—Simons model
PA O/O~
5[15].
2
2
=
Thus we have
,
~ q (p—q)
_2p5f~3q~(P
q (p—q)
(B.8)
which immediately yields FPG~(p)=0.
(B.9)
The same procedure also gives
FPG~~(p,k)
,
(2~r)
3 2 d 2 q(p+k—q)(p—q)
famnfbrnfCmrf
x [+q’1qP.kP
—
qP.qPkC +
qu(kP.pP
—
2
kPpP.)
2k
1’p’)
—
qltk”pP
—
x
(qVqP.(/~P +p~) _pleq~’qP —
+qP(pP.pv + 2kPpP.
—
kP.p”)
5(p + k
2 —q),~
(p + k —q)
+qP(2pl~kC— k
qP.qPkC + qv(kP.pP
— qle(pPpP
+ kVPP))].
—
k”p1’) (B.10)
Thus, for the finite parts, we obtain from eq. (B.6)
(Z~+ Z~1))Fp 0,
(B.lla)
(zr)
(B.llb)
=
—
(Z~’)+
3Z~)Fp
=
=
0, 1 ~ 6g NC(2)
fP.VP
FP ~
p, k),
(B.llc)
where N is the number of generators of the group. A straightforward computation
A. Blasi, R. Collina
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Chem—Simons model
491
now yields Eleep
FP G~~( ~, k)
=
6ig3 EP.vPfamnf~fcmrkP.pP (2~-) 6
xf
q~d3q
2
2kA(P+k_~)A) (p+k—q)
2(1
q (p+k—q)(p—q) so that from eq. (B.11) we find (Z~’))Fp =
(Z~)FP
=
(z~’)FP 0. =
(B.13)
It follows that there is no possibility of having finite contributions in a scheme where the scale invariance is preserved. Any regularization procedure which violates this prescription (in particular the Pauli—Villars scheme which introduces dimensional parameters) may find finite contributions which survive even in the limit when the regularization is removed. These finite contributions are an artifact of the method and they can be reabsorbed by a finite counterterm which restores scale invariance. Finally we remark that these one-loop results may be extended, by a Bethe—Salpeter—Dyson expansion [161, to the higher orders. We would like to thank C. Becchi, F. Delduc and G. Bonneau for discussions on the subject of this paper and 0. Piguet for useful comments.
References [I] E. Witten, Commun. Math. Phys. 117 (1988) 353; 118 (1988) 601 [2] E. Witten, Commun. Math. Phys. 121 (1989) 351 [3] T.P. Killingback, Phys. Lett. B219 (1989) 448; G. Moore and N. Seiberg, Phys. Lett. B229 (1989) 422; J.M.F. Labastida and A. Ramallo, CERN-TH-5934/89, 5932/89; J. Frölich and C. Ching, ETH preprint ETH-TH/89-l0; M. Bos and V.P. Nair, Phys. Lett. B223 (1989) 61; G.V. Dunne, R. Jackiw and C.A. Trugenberger, MIT preprint CPT/771; H. Murayama, University of Tokio preprint UT-542; S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, lAS preprint IASSNS-HEP89/20 [4] P. Cotta Ramusino, F. Guadagnini, M. Martellini and M. Mintchev, CERN-TH-5277/89 [5] E. Guadagnini, M. Martellini and M. Mintchev, CERN-TH-5394/89, 5419/89, 5420/89; D. Birmingham, M. Rakowsky and G. Thompson, ICTP preprint ICTP-IC/88/387 [6] L. Alvarez-Gaumé, J.M.F. Labastida and A.V. Ramallo, CERN-TH-5480/89 [7] F. Delduc, F. Gieres and S.P. Sorella, CERN-TH-5391/89 [8] G. Bandelloni, C. Becchi, A. Blasi and R. Collina, NucI. Phys. B197 (1982) 347 [9] T. Clark and J. Lowenstein, NucI. Phys. B113 (1976) 109; J.H. Lowenstein, Commun. Math. Phys. 24 (1971) 1
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Chem—Simons model
[10] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. (N.Y.) 59 (1970) 42; S. Coleman, in Properties of fundamental interactions, ed. A. Zichichi (Ed. Compositori, Bologna, 1973), p. 359 [11] B. Zumino, Chiral anomalies and differential geometry, in Relativity, groups and topology II, ed. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) [12] C. Lucchesi, 0. Piguet and K. Sibold, Int. J. Mod. Phys. A2 (1987) 385 [13] E.C. Zeeman, Ann. Math. 66 (1957) 557 [14] J. Dixon, Cohomology and renormalization of gauge fields, Imperial College preprint (1977/78); G. Bandelloni, J. Math. Phys. 27(1986)1128 115] H. Epstein and V. Glaser, Ann. Inst. Henri Poincaré 19 (1973) 211 [16] R.W. Johnson, J. Math. Phys. 11(1970) 2161
FINITENESS OF THE CHERN-SIMONS MODEL IN PERTURBATION THEORY A. BLASI and R. COLLINA Dipartimento di Fisica, Universitd di Genova, Istituto Nazionale di Fisica Nucleare, sez. di Genova, Via Dodecaneso, 33-16146 Genoa, Italy Received 30 November 1989 (Revised 25 May 1990)
The Chern—Simons action in the Landau gauge is characterized by a BRS symmetry and a local dilatation invariance Ward identity whose quantum extension controls the trace anomaly in the flat limit. We show that the trace anomaly corresponds to all orders to operators which are BRS variations and therefore it cannot have contributions from the Chern—Simons gaugeinvariant term. This is sufficient to insure the vanishing of the /3-function. We further show that, in a renormalization scheme which preserves scale invariance, there are no finite one-loop corrections to the parameters of the model.
1. Introduction In the panorama of topological field models [1] the Chern—Simons theory in three dimensions seems to occupy a prominent place. The nonperturbative analysis of this model has contributed to clarify its relation to two-dimensional, conformal field theories and to knot theory [2, 3]. The model has also interesting features from the perturbative point of view, where the numerical computations have proven the vanishing of the /3-function up to two loops and have succeeded in identifying new knot invariant polynomials [4—61.When looking at the Chern—Simons theory from a strictly perturbative point of view, we have to deal with the following problems: dimensional regularization cannot be used, so either we look for alternative gauge-invariant regularization procedures [61,or we discuss the quantum corrections of the model in a regularization independent way. Secondly, the coupling constant of the theory, whose quantization is deducible with non-perturbative arguments, appears here to be a free real parameter; consequently the vanishing of the /3-function does not follow directly*. Finally, the *
Indeed, even the characterization of the model as in ref. [7] by means of global extra symmetries, while identifying the gauge fixing (Landau gauge) uniquely, still leaves room for a renormalization of the gauge-invariant Chern—Simons term.
0550-3213/90/$03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
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topological character of the Chern—Simons gauge-invariant action, i.e. its independence from the choice of a metric, appears to be spoiled by the necessary introduction of a gauge-fixing, metric dependent, term; that this is not the case was argued by Witten in ref. [2]. Here we would like to address the question of the vanishing of the /3-function to all perturbative orders in the Chern—Simons theory in the case of flat euclidean space-time. The suggestion which we shall follow is that the metric tensor appears only in the gauge-fixing part of the classical action, and therefore, as noticed in ref. [4], the classical energy—momentum tensor is a BRS variation. To implement this idea at the quantum level we need a symmetry which characterizes the algebra of the energy—momentum tensor; in the Landau gauge, where no dimensional parameters are present, this is provided by Weyl’s representation of general coordinate transformations. This representation is unstable under radiative corrections and its flat-limit version acquires a correction term which, as shown in ref. [81, controls the “minimal” trace anomaly of the model. The next step is to extend the original “true” BRS symmetry, which commutes with the local dilatation transformations, as to include the spurion fields needed to characterize the trace anomaly in such a way as to guarantee that the trace anomaly itself remains a “true” BRS variation. The conclusion can then be stated as follows: since the Chern—Simons theory in the Landau gauge has no infrared problems [9], there is a standard connection between the coefficients of the “minimal” trace anomaly and those of the differential operators appearing in the Callan—Symanzik equation. On the other hand the trace anomaly turns out to be a true BRS variation and thus the differential operators appearing in the Callan—Symanzik equation must all correspond to insertions which are themselves BRS variations; this excludes the presence of the invariant Chern—Simons term and insures the vanishing of the corresponding /3-function to all perturbative orders. We further analyze the radiative corrections of the theory and show that there are no finite one-loop renormalizations provided the renormalization scheme does not violate scale invariance. The paper is organized as follows: in sect. 2 we introduce the BRS and local scale invariances which can be used to characterize the model at the classical level. Sect. 3 illustrates the algebra of the energy—momentum tensor and the strategy employed to characterize the minimal trace anomaly in the flat limit as a BRS variation. Sect. 4 contains the results on the renormalizability of both symmetries: the extended BRS invariance, which is discussed by the usual cohomological methods (the details are in appendix A), and the local dilatation Ward identity, whose detailed analysis has already been performed in ref. [81.In sect. 5 we collect the results on the vanishing of the /3-function and the absence of finite one-loop corrections to the model. The technical details concerning this last point are treated in appendix B.
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Chem—Simons model
2. Classical action and symmetries In this section we shall characterize the Chern—Simons action in the Landau gauge by its invariance under the corresponding BRS operator and the additional, compatible, local dilatation transformations. These two symmetries are sufficient to guarantee, at the classical level, that the energy—momentum tensor is the BRS variation of a local operator. The possibility of extending this scheme beyond the classical level and in particular the instability of the local dilatation Ward identity, will be discussed only in the flat limit; we introduce here also the tools needed for this analysis. The Chern—Simons gauge-invariant action is
‘Cs
=
—
~k1~”PJ d~x[A~a~A°~ + lfabcAaAb,4c}
,
(2.1)
where f~t~Care the real completely antisymmetric structure constants of a simple Lie algebra. In a perturbative approach, the coupling constant k is not quantized and therefore it appears as a free real parameter. We choose the gauge-fixing term to be covariant and with no dimensional parameters, i.e. the Landau gauge with action
‘GF
~
(2.2)
,
where c~,b’~are the Faddeev—Popov ghosts and antighosts and d~the corresponding Lagrange multipliers. The nilpotent BRS transformations are
(2.3a)
sA~=—(V~c)”=_(aca+fabcAcch) sca
=
-~f~c’~c~, sb” =d~,
sda
=
0.
(2.3b—d)
Notice that ‘GF does not contain any free parameter which can be reabsorbed in a multiplicative redefinition of the b’~,d’~fields, compatible with (2.3c)*. In order to be able to discuss the BRS symmetry at the quantum level we also add invariant external fields y~(x),~a(X) and the couplings
IEF=fdx[Y~(~)~
2f~}.
*
(2.4)
The most general choice + fora the gauge-fixing term compatible with eq. (2.3) is, up to an integration 3xb~[d~A~ by parts, sf d 1d” + ~ both a1 = 0 (Landau gauge) and 02 = 0 are enforced by the local dilatation invariance.
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Chern—Simons model
475
The classical action F~=ICS+IGF+IEF
(2.5)
satisfies the functional identity 5T’~ 6[~Cl
(sF’~)=fd3x[_~~~..~. +
6F’~~f -~--~-~-~-
c~
3pc!
+da_~~j.
(2.6)
The absence of dimensional parameters in (2.5) denotes that the theory is scale invariant. Wishing to realize this symmetry at the local level, and hence to discuss the invariance under general coordinate transformations x~ x° AU(x), we must abandon flat space-time; as pointed out in ref. [8], the suitable framework is provided by Weyl’s representation of general coordinate transformations where the fields transform as —
+
=
=
Aa3ca
=
~
—
3~A°~4~,
(2.7a) (2.7b)
—a~A~~ —0
1~” + 40~A~”
)I~4T
6d~ AUo~d~~ + ~0ffA~d~
(2 7d)
=
=
i5yaP.
A~ab~ +~
=
=
—
(2.7e)
3AJLya~lr +
(2.7f)
0~A”~’,
)t~r0~a + 0aX~~Ta,
=
—
(2 7c)
0,TA~Wa+ ~0,,X’w~’
(2.7g) —
‘~
0~0~A”;
(2.7h)
~T’~” is a unimodular metric tensor and w~an auxiliary connection. Notice that the fields A~and C’1 transform as a true vector and scalar respectively, while the rest are tensorial densities. The corresponding invariant action
IC!
=
f d3x{ —
—
~ke~P(A~0
[(~“o~+w~)b’1
A~+ lfabcAaAbAc)
—
y~~’1](Vc)’1 (~a~+ —
does not contain any dimensional parameter.
+
~a1fabCCbCC}
(2.8)
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Chem—Simons model
To discuss the compatibility of the BRS transformations (2.3) and the local dilatations (2.7) we perform a Legendre transformation to the connected functional ZC; denoting with jai.~,K’~,J’1 Ha the currents coupled to the quantized fields A~,~a, c” b’1 respectively, the BRS operator becomes SZ C
[d~x
6
6
6
J’1~+Ja+Ha ~ 6~
8K”
zC
(2.9)
while the transformations in eq. (2.7) yield the local operator
_o~_~)~
W~(x)ZC~[~a~r(~.
+
_Jao.....~...
20~(~(r~ .)
6
—
~
8
+~0 K”— ~ 6K”
—
—
8
~H’10 + ~0 H’1— +0 ya(r ~ “6H” ~‘ 6H” ~
6 6ya~L
(2.10) The operators (2.9) and (2.10) obey [S,W~(x)] =0,
(2.11)
so that the BRS and the local dilatation invariance can be analyzed separately. At the classical level we have the identities
SZ~=0,
H’~(x)Z~=0.
(2.12a,b)
The canonical dimension and the Faddeev—Popov charges of the fields A~, d”, b”, ~a 7a~. ~a
(2,
—
1),(3,
—
~
w’~ are respectively (1, 0), (1, 0), (1,
—
1), (0, 1),
2),(0,0),(1,0).
3. The energy—momentum tensor~strategy The action F’~ in eq. (2.8) contains the metric tensor ~ only in the gauge-fixing part; this fact signals that, at least at the classical level, the
/
A. Blasi, R. Collina
C’hern—Simons model
477
energy—momentum tensor of the model is the BRS variation of a local quantity. In order to write explicitly this property, which will play an essential role in the proof of the vanishing of the /3-function, we first identify the energy—momentum tensor in terms of functional derivatives. Interpreting the classical sources w~’ in terms of the operators that their functional derivatives define, we see that ~
8
6
~
6 )zC_~o~
3’i~v~A(~)
6~(X)zC
(3.1)
corresponds to the energy—momentum tensor of the theory, whose trace is proportional to the divergence of the vector 6Z~/&o”(x). Let us introduce the external fields G~ and (2’~,with Faddeev—Popov charge + 1, and the modified classical action
f d3x[( G’-’”0~
+ fl~) b”] A~.
=
pd +
The corresponding connected functional
ZC!
(3.2)
satisfies the new identity
+n-~2~=o,
~c1s2c!+fd3x[G.’~
(3.3)
which we shall refer to as the “extended” BRS symmetry to distinguish it from the “true” BRS transformations obtained at G~LV 0. The local dilatation invariance is preserved by assigning to G~,12~the transformation laws =
=
~0UAaG~~~,
(3.4a)
6f1~’=A~0~fl~ —0~A~11”+ 40~A”Q~ ~G’~’P0p0alt”.
(3.4b)
=
—
0~A~~GIrV 0,JAVGSLff —
+
—
The modified operator
H~(x)+O~G~ 6G~
+~a~na—~— +a~(n~~_)
~
—
~
(3.5)
478
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A. Blasi, R. Collina
Chern—Simons model
still commutes with $~.Now from the identity (3.3) we obtain
6
6
6G~”(x)SZ~’~GI~,12~.O6[l~(x) SZ~IG~,n~o0 =
(3.6)
=
or, in terms of the “true” BRS operator 5, the relations 6 6 56G~(x) Z~lG~,~o6~() =
6
6 Z~’IGI’~,Q~O6~~v()Z~t~(3.7a) =
6
6
S 6)ZCIG~W’O
6~()ZG1~.f21’=O
(3.7b)
6~()ZC.
It is now apparent that the energy—momentum tensor in eq. (3.1) is, at least at the classical level, the “true” BRS variation of a local operator. The strategy, which the above scheme naturally suggests to discuss the quantum extensions, is that of analyzing the validity at the quantum level of the equations SZC0,
l’(x)ZC=O.
(3.8a,b)
It is well known that eq. (3.8b) cannot hold true with the operator i4’~(x)as it stands due to the presence of the trace anomaly [10] which appears as an instability of Weyl’s representation. Here we shall adopt the method of ref. [8]and rather than discussing the complete algebra of the energy—momentum tensor in curved space-time we shall consider the flat limit ~ 6’~”,w~ G”” 0. To keep track of the trace anomaly in the flat limit we introduce a dimensionless spurion field o-(x) coupled to it and to insure that the trace anomaly is a BRS variation we also need a dimensionless ghost spurion .~(x).By repeating the analysis of ref. [8], we arrive at the result that, in the flat limit and at the least order, the minimal anomaly to the Ward identity for the local dilatation invariance is given by =
=
=
=
W~~,(x)ZCIFL=0,lI(x),
(3.9)
where 1(x) has canonical dimension equal to three and cannot be written as a divergence. Accordingly we shall investigate the quantum implementability of the Ward identity, 6 W~(x)Z~ W~(X)ZCIFL + 0~(x) 61(x) 6 6cT(x)
ZCIFL=0,
6
ZCIFL + 0~cr(x)6cr(x) ZCIFL
(3.10)
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A. Blasi, R. Collina
Chem—Simons model
479
together with the extended BRS identity §ZC~SZC~FL+fd3x~(x)
(3.11)
8~ZCIFL=0.
that the term proportional to h in eq. (3.10) is the one which compensates the trace anomaly and that the spurion field .~(x)is an artifact only needed to insure that the trace anomaly, which corresponds to 6ZC/6a~(x)[~~o,is a “true” BRS variation. Indeed, if we assume that (3.11) holds true we immediately find that Notice
8 6u(x) ______
=
S
6 ~ 6~(x)
(3.12)
The program is therefore that of discussing the possibility of maintaining at the quantum level (3.10) and (3.11); a positive answer to this question implies that no term proportional to the invariant Chern—Simons action contributes to the trace anomaly.
4. Perturbative renormalization In this section we analyze the quantum implementability of the two symmetries of the model in the flat limit, but to all orders in the cr(x) field. Concerning the local dilatation Ward identity, our treatment follows closely the approach of ref. [8], where it is emphasized that we must consider the possibility of having in the action arbitrary powers of the o-(x) field. In other words, the generating functional of the proper vertices is a double formal power series in h and a- and the infinite number of ci couplings are determined perturbatively in terms of the ci independent ones by the Ward identity itself. The presence of the .~(x)field, which carries one unit of Faddeev—Popov charge, does not alter this picture since its couplings are controlled by the extended BRS identity. The two symmetries can be analyzed independently since the corresponding operators commute. We begin with the extended BRS symmetry; it is well known that both the stability and the anomaly problems can be settled by analyzing the cohomology space of the linearized nilpotent operator 6 ~ 6y’1~6A~ 61d
§L
=
Jfd~x
6 SL+fdX~_,
6a
6F’~6y~ 6 6A~
6 6b”
6F’~ 6 8c” 6~”
6F’~ 6 6~,” 6c’~
6 6a-
(4.1)
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Chem—Simons model
identifies the linearized “true” BRS operator. In particular, the soluwhere tions of the cocycle condition 5L
(4.2) where
f~()
=
(4.3)
is a local functional of canonical dimension equal to three and carrying p units of Faddeev—Popov charge, are relevant for analyzing the stability (p 0) and for the compensability of anomalies (p 1). To translate (4.2) to the space of forms, we follow ref. [11] and rewrite it as the descending chain of equations =
=
§Lz1~(x) +d~l~
1(x) =0,
(4.4a)
§L~i~1(x)+d~~2(x) =0,
(4.4b)
§LL1~2(x) +d~~3(x) =0,
(4.4c)
SL4~3(x) =0,
(4.4d)
where d is the exterior differential and the upper index denotes the degree of the form*. To solve the system (4.4), we first look for the general solution of (4.4d), i.e.
for the local cohomology of the ~L operator in the space of functions of the fields. The analysis in appendix A yields as solution, in the sector p 0 =
=
and, in the sector p
=
§LS~”(x) + tfa~ca(x)c~(x)cc(x)
(4.5a)
1 =
§LS~(x).
(4.5b)
By working up the ladder of relations (4.4) (see appendix A for details), we find ~~x)
=
d4~(x)+ §LS~I(x) +
L1~(x)
=
*
t(A~o~A~ + lfabCAa AbA~~ ) d.~/LA dx~A dx’~,
dS~2)(x)+
§LS~(x).
(4.6a) (4.6b)
The descending eqs. (4.4) have been analyzed in models with a different external field content in ref. [12].
A. Blasi, R. Collina
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481
Hence we conclude that the Chern—Simons action is stable and that there are no anomalies to the extended BRS symmetry. Furthermore the term zi$2~(x)in eq. (4.6a) can be analyzed according to the 1(x) field content and dimensionality, as =
H~(x) + m(o.)1(x)f’1
b’1(x)bb(x)d~(x)e~~dx~A
dx’~A
dx’~
=H~(x) +4m(a-)SL[1(x)f’1b’1(x)bb(x)bc(x)Ie~vpdx~~~AdxvAdxh11, (4.7)
where H~(x) is I independent. Now recalling that that §LS~l(x)
=
SLH~(x) +
f d3y 1(p)
SL is also nilpotent, we find
H~(x),
(4.8)
which implies that the I couplings in the effective action are uniquely determined by those of a- and therefore they do not correspond to independent parameters. Having analyzed the extended BRS symmetry we pass to the local dilatation invariance Ward identity; the presence of the ci, I fields does not alter the minimality properties of the breakings in the flat limit, which can be reabsorbed by introducing couplings with increasing powers of ci. Accordingly we can repeat the analysis of ref. [8] and show that the minimal anomalies at the order h”, citm can be compensated by a suitable choice of the couplings ~ At the classical level we find that the action is given by the Chern—Simons term plus a contribution as in eq. (4.8) where I1~(x) contains the gauge-fixing term and all the acouplings*. This concludes the renormalizability of the symmetries of the model.
5. Conclusions In sect. 4 we have shown that the local dilatation invariance Ward identity in the flat limit and the extended BRS symmetry are implementable to all perturbative orders. This implies that the Landau gauge is a stable choice, and, more important, that the trace anomaly is identified as a true BRS variation. To make contact with the /3- and y-functions which identify the Callan—Symanzik parametric equation of the model let us recall that the Chern—Simons theory in the Landau gauge has no infrared problems [9], so that the functional ~ZCmfd3x
*
6u(x)~’
(5.1)
Notice that in this model the presence of the extra terms, finite in number, which contain derivative couplings of the a--field found in ref. [8], is forbidden by the extended BRS symmetry.
482
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A. Blasi, R. Collina
Chem—Simons model
which obeys the same Ward identity as Z~,is well defined. It follows that, at aO/Oa- is given by a parametric derivative. Thus we have 0
=
0,
0
=
/3(k) —Z~I~~o + ~ yZ,J~O, Ok
(5.2)
where the invariant counting operators are given by (5.3a)
~
N
2
=
I d3x[H
-H” +K
N3 =fd3x{J~~j
=
+~a±~j
=
{s,fd3xK~~~~},
{s,fd3x~a~}.
(5.3b)
(5.3c)
Recalling that 0
0
—Z~I~.0 S—Z~[~0, Ocr 01 =
(5.4)
we immediately arrive at the desired result that the function /3(k) in eq. (5.2) identically vanishes and that only the counting operators contribute to the r.h.s. of eq. (5.2). The presence of possible finite counterterms can also be discussed within a renormalization scheme which implements BRS and scale invariance. Let us remark that, in this particular case, where there is no “natural” normalization of the model, the issue of the finite contributions is really crucial to give meaning to the perturbative approach. Indeed, once we have chosen a renormalization scheme for the Green functions of the theory and once the lagrangian parameters are identified by arbitrary normalization conditions, the appearance of finite corrections to the Green functions defining the parameters makes the theory unpredictive since scale invariance forbids a non-trivial link between the different finite contributions obtained by changing the normalization point. The only way out is that they vanish in any renormalization scheme which preserves scale invariance. That this is the case at the one-loop level is shown in detail in appendix B. A regularization scheme which violates scale invariance may lead to different results which however are due to the breaking explicitly introduced by the regularization and which can be reabsorbed by finite counterterms [6]. Concerning the observables of the model, we notice that the exact scale invariance implies that their vacuum expectation value is given by (scale-invariant)
A. Blasi, R. Collina
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Chem—Simons model
483
c-numbers. At the local level this is a rephrasing of the well-known fact that the model has no local observables, while for globally defined quantities this suggests that the structure becomes trivial. This perturbative result confirms the non-perturbative analysis of the model [2] according to which, for flat background, the physical Hilbert space becomes one dimensional.
Appendix A The solution of the system of descending equations (4.4) is based on the of knowledge of the local cohomology of the ~L operator in the space V~3~ Lorentz-invariant polynomials built with the fields and their derivatives, with canonical dimension up to three and any value of the Faddeev—Popov charge. As a first step let us rewrite the action of ~L on V~3~ as that of an ordinary differential operator: defining the independent variables ~
for any field cP(x) (A~,b”, d”, c”; ordinary differential operator
P.i
~-a yaP.
I,
a-},
the operator
0
3pC!
n0
(A.1)
=0k.. .0~’1~(x)
~‘~‘Oy’~P.(x) 0A~P.P.(x)
...O0 ~ ~0Aa;~(x)
OF’~ +0P.1 ...0 ~“Oc”(x)
0
3~CI
~OA~(x)
“
0y~IP.(x)
Ole!
3~C!
—0
reduces to the
~L
Oy~
P.(x)
+0
~
...O
~0~’1(x)
0
P.(x)
Ocf’P.
0
O~~P.P.(x)—O‘~“...OO ~ “Oc~(x) ~
+d~P.P.(x) ba()
0
~.
+I;P.P.()
•~(x)
(A.2)
When acting on Vt3~,~L further simplifies since the summation index n, due to dimensionality and Lorentz invariance obeys the bounds n 3 dim(~)for cI~ ~
—
484
/
A. Blasi, R. Collina
(~,b’’), and n
2 — dim(1) for ‘1
~
a
{A~,Ca, yale
(9
a
19
19
_fabCAb
cc~a
_f~Abc~~ P.,
OAP.;(,
~3
PayaP.
a
8a-;p~
a
a
+da;P.~
19
fabcybP.cc. OyllP.
ePf0bCA~AC_____
(9
—+1. ‘~
19
~k a
+{f
9
Oa-
P.”
+
19
+1—+I.
~
b,P.
+ [_fabcAbcc~
I}, or explicitly,
+keM~A” ____+y~~~’___ P;”3~aP. P.
13
ab”
a-,
a
~9 — C~
19
Chern—Simons model
—b”~-—— P.a~a _fabcybP.Ac
t9
+ ~fa cbcc____ +fabccb ~ cc____~_ 19C~ +f
a +f”~b”;
abcbb; PACP.—
a
~
19~a
c(cbcc
+ c.~c~)
Iv~
(A.3)
V(”
OyaP.}
To identify the solution of the equation
.~LX(x) 0,
(A.4)
=
3~ and with X(x) E V~ counting operator O N=A~-~-
.~L
a +AP.;V
as in (A.3), we adopt the filtering [13] induced by the
0
~
0
0 0 —+c” “OC~P. ;P~P19ca
Oc”
O o / a +y~L____+~__+2Iba__+ba
(
0y~’
O~”
ad”
+b” ;le19l~,a
a ;levada
P. ~
0
a
Oh”
a
0
+YaP.
a
~
—I
b~P.~)
a
I+u—+cr. I ~a.P.Ocr
P.
a +ci. ~P.VOO~cev
a
+1 ~
+1
a
a
+1 ;lei’ 01
(A.5)
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Blasi, R. Collina
/
Chem—Simons model
485
so that =
+ .~ +
~
[N,~~’L~j =
v
~
and =
{.~(o) ~(1)}
2
=
(g~(1))
=
0,1,2
(A.6)
{~) .~L2~} =0;
+
(A.7)
~ .~, LI~ correspond to the expression in the round, square and curly brackets respectively in (A.3). The space V~3~ also decomposes under N, so that X
E
=
X(a),
NX(0)
aX~
=
(A 8)
a=0
Clearly the cohomology of can be discussed separately in each finite-dimensional eigenspace X~,which can be embedded in a Fock space where the action of a creation operator is identified with the multiplication by the corresponding variable and the annihilation operator is given by the derivative with respect to the same variable [14]. In this framework the cohomology space H~°~ of ~ coincides with the kernel of the Laplace—Beltrami operator ~
=
{(~(o))t .~)}
a
a a —+A’1--——+A’1 P.OAa
=~~—+~“
;P.OCa
a
lePOCa
+ 2y’1~—~+y—~j
a
19 P.
a~i~
~
a ab~~
a
~
a
P.~9J 3a
+cia+ci;P.a+ci;leea
+I~+I;P.~+I;P.C1.
(A.9)
It is evident that H~°~ contains only arbitrary functions of the fields c’’ without space-time derivatives; hence the general solution of (A.lOa)
486
A. Blasi, R. Collina
/
Chern—Simons model
is given by y....
(A.lob)
~(0)1~+F(Ca)
We can now identify the cohomology space of the complete operator ~L; by substituting eq. (A.8) into eq. (A.4), we find the chain of equations 0,
=
+
(A.lla)
~X~t=0,
~ ~
+
indeed
(A.llb) 0,
=
a
=
1,2
(A.llc)
The solution of eq. (A.lla) yields X~1~
+
=
F~’~(c”) ,
which substituted into eq. (A.llb), recalling the expression of ~
—
~(1)~(!))
=
—
lfabcCbCcBF(!)
(A.12) yields (A.13)
Now the r.h.s. of eq. (A.13) belongs to H~°~ and therefore cannot be written as a coboundary; hence we find ~
X~2~~(0)~(2)
+
=
~/~1)~(1)
+
F~2~(c”) ,
(A.14a)
0.
(A.14b)
0
fahCchcCl(!)(Ca)
=
Oc
The semisimple character of the group further implies the vanishing of F~~(c”) T”c”. The same reasoning applies to the successive iterations and we have to discuss the solution of =
a
fabcChCCF(’1)(cQ)
=
0,
a
=
2,3,...,
(A.15)
ac where F~”~T~’1...a,]Cal =
...
(A.16)
Ca,,
and TE”1.~ is a completely antisymmetric invariant tensor in the adjoint representation; for semisimple Lie algebras we have T’1
=
T~”~’~ T~”~1 0, =
=
T[~2~~dI tf’’~ =
(A.17a, b)
A. Blasi, R. Collina
/
Chern—Simons model
487
plus possible contributions with a ~ 5 depending on the particular group. Therefore the solution of eq. (A.4) is given by
x=
~
~L~+tfa~C’1CbCc+
r0T[a1a~Ical...ca~~.
(A.18)
a~5
This identifies the cohomology of the -~L operator at least as far as the discussion of the stability and anomaly are concerned; indeed we shall see that the contribution tfabccacbcc reconstructs exactly the Chern—Simons action when inserted into the descending chain of equations (4.4), while the absence of the term proportional to c’1c”c’~c’~ in eq. (A.18) insures that the theory is anomaly free. Indeed, going back to eq. (4.4d) with the choice p 0, 1, relevant for the stability and anomaly discussion we have from eq. (A.18) the solutions =
z~(x)
=
§LS~(x) +tfa~~ccacbcc
(A.19a)
=
§LA~(x).
(A.19b)
With the above substitutions eq. (4.4c) yields —
dS~(x))
—
d~1~(x))0.
=
—
4t d(f’1~cac~!~C~),
(A.20a) (A.20b)
=
After some algebra we can rewrite the r.h.s. of eq. (A.20a) as =
—
§L(f
CAa(x)cb(x)CC(
x) dx~’)
(A.21)
and obtain the solutions ~tT(x)
=
§LS~’~(x) + d~~(x)+ tfabcAa(x)cb(x)cc(x)
(A.22a)
=
§L~~(x)+ d~(x).
(A.22b)
The procedure iterates for the successive steps, to arrive at Lt~(x) §L~1(x) + d~~(x) =
+
t(A~(x)O~A~(x) + lfabCAa(x)Ab(x)AC(x))dxP.
A dx’~A dx”,
(A.23a) =
§L~~(x) +
dz~?~(x).
(A.23b)
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A. Blasi, R. Collina
/
Chern—Simons model
The expressions (A.23) guarantee the stability of the Chern—Simons action and the absence of anomalies.
Appendix B Here we shall analyze in detail the question of the finite corrections to the Chern—Simons model in the Landau gauge. As pointed out in sect. 5 the absence of such corrections is crucial to have a sensible perturbation expansion. The discussion will be limited to the one-loop level, and the symmetries to be preserved are the BRS and local scale invariance; it is convenient to first perform the field redefinition so that the coupling constant g 1/ ~ explicitly appears only in the interaction terms; the CS classical action, in the Landau gauge, takes the form =
F
=
J d~x{
OC.A~+
—
~cP.1P(Aa
~gfabCAaAbAc)
—
[O”b” yle’1](Vc)’1 —
—
OP.d’1A~+g~~’1f’1~cbcc},
(B.1)
where (V~Cy’= (Olec” +gf”A~c’~’).
(B.2)
Since the gauge fixing is linear, we shall consider the BRS symmetry together with the constraint* 61
(B.3)
6d’1(x)
P.
Thus the free parameters at our disposal are the gauge and ghost fields renormalizations ZA, ZC and the coupling constant redefinition Zg. These parameters are identified by the Green functions
6~F
6~F
621
6~’1(p)6eb(k)6cc(o)’
6~(p)6cb(0)’
6A~(p)6A~(k)6A~(0)
(B.4)
at an arbitrary normalization point, where the “tilde” denotes the Fourier transform and all fields are set to zero after the functional derivation. Now if the ZA, Z~,Zg parameters have the h expansion
Z *
This corresponds to the relation
=
)‘i
2Z~2~ + ...,
1 + hZw + h + Y2
=
0 in the Callan—Symanzik equation (5.2).
(B.5)
A. Blasi, R. Collina
/
489
Chem—Simons model
we have, at the one-loop level
6~F
\(1)
=g(z~’~ + Z)f’1~+
(6~a(p)6Eb(k)6Cc(O))
621
/
)(1)
______________ 6~(p)6cb(O)
(B.6a)
Gabc(p, k),
=
i612bpP.(Z~0 z~1))+ G~(~),
(B.6b)
=
_geP.1~Pfhhl1~~(z~,l) + 3Z,~’))+ ~
k), (B.6c)
—
(1)
63F _____________________
(6A~(p)8A~(k)6A~(0))
and, employing the Feynman rules derived from (B.1), we compute 3
~ k)
G~(~)
G~(p,k)
=
~
6 CP.CPf’1mnfbmnfcmrpP.kPf
(2~-)
g2 9/2
=
(2~-)
C(2)6
EP.CPPJ
d q
d3q
(p—q) 2 (k—q) 22’ q
2
(B.7a)
(B.7b)
2’
q (p—q) d3q
=~_g
(21r)
famnfbrnfcmrf
2
q (p+k—q)2(p—q)2
x [~2qCqPq~~+ qP.qP(2pv + kC) + qPqCpP.
+
q”qP.(kP +p”)
+qe(kl~p~~ —k’3p~’)_qP(p~~pC+k~epe) qP.(pl’pP +kvp~))] d3q
(2w)
brnfcmrfq2(p+kq)2(pq)2
x [2q”q”q~’2pleqvql~ 2qleqP(p~+ k’) —
—
+
2p”q~(p’+ k’)], (B.7c)
where C(2)6” faedf bed. Notice that G”~(p,k) in eq. (B.7a) is finite and it indeed vanishes; since we are interested in the (eventual) finite parts in eqs. (B.7b, c), whose integrands are logarithmically divergent, we can extract them, up to an irrelevant arbitrary constant reabsorbable by the freedom of the lagrangian
490
A. Blasj, R. Collina
parameters, by the operator
FPfd3q
/
Chern—Simons model
PA O/O~
5[15].
2
2
=
Thus we have
,
~ q (p—q)
_2p5f~3q~(P
q (p—q)
(B.8)
which immediately yields FPG~(p)=0.
(B.9)
The same procedure also gives
FPG~~(p,k)
,
(2~r)
3 2 d 2 q(p+k—q)(p—q)
famnfbrnfCmrf
x [+q’1qP.kP
—
qP.qPkC +
qu(kP.pP
—
2
kPpP.)
2k
1’p’)
—
qltk”pP
—
x
(qVqP.(/~P +p~) _pleq~’qP —
+qP(pP.pv + 2kPpP.
—
kP.p”)
5(p + k
2 —q),~
(p + k —q)
+qP(2pl~kC— k
qP.qPkC + qv(kP.pP
— qle(pPpP
+ kVPP))].
—
k”p1’) (B.10)
Thus, for the finite parts, we obtain from eq. (B.6)
(Z~+ Z~1))Fp 0,
(B.lla)
(zr)
(B.llb)
=
—
(Z~’)+
3Z~)Fp
=
=
0, 1 ~ 6g NC(2)
fP.VP
FP ~
p, k),
(B.llc)
where N is the number of generators of the group. A straightforward computation
A. Blasi, R. Collina
/
Chem—Simons model
491
now yields Eleep
FP G~~( ~, k)
=
6ig3 EP.vPfamnf~fcmrkP.pP (2~-) 6
xf
q~d3q
2
2kA(P+k_~)A) (p+k—q)
2(1
q (p+k—q)(p—q) so that from eq. (B.11) we find (Z~’))Fp =
(Z~)FP
=
(z~’)FP 0. =
(B.13)
It follows that there is no possibility of having finite contributions in a scheme where the scale invariance is preserved. Any regularization procedure which violates this prescription (in particular the Pauli—Villars scheme which introduces dimensional parameters) may find finite contributions which survive even in the limit when the regularization is removed. These finite contributions are an artifact of the method and they can be reabsorbed by a finite counterterm which restores scale invariance. Finally we remark that these one-loop results may be extended, by a Bethe—Salpeter—Dyson expansion [161, to the higher orders. We would like to thank C. Becchi, F. Delduc and G. Bonneau for discussions on the subject of this paper and 0. Piguet for useful comments.
References [I] E. Witten, Commun. Math. Phys. 117 (1988) 353; 118 (1988) 601 [2] E. Witten, Commun. Math. Phys. 121 (1989) 351 [3] T.P. Killingback, Phys. Lett. B219 (1989) 448; G. Moore and N. Seiberg, Phys. Lett. B229 (1989) 422; J.M.F. Labastida and A. Ramallo, CERN-TH-5934/89, 5932/89; J. Frölich and C. Ching, ETH preprint ETH-TH/89-l0; M. Bos and V.P. Nair, Phys. Lett. B223 (1989) 61; G.V. Dunne, R. Jackiw and C.A. Trugenberger, MIT preprint CPT/771; H. Murayama, University of Tokio preprint UT-542; S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, lAS preprint IASSNS-HEP89/20 [4] P. Cotta Ramusino, F. Guadagnini, M. Martellini and M. Mintchev, CERN-TH-5277/89 [5] E. Guadagnini, M. Martellini and M. Mintchev, CERN-TH-5394/89, 5419/89, 5420/89; D. Birmingham, M. Rakowsky and G. Thompson, ICTP preprint ICTP-IC/88/387 [6] L. Alvarez-Gaumé, J.M.F. Labastida and A.V. Ramallo, CERN-TH-5480/89 [7] F. Delduc, F. Gieres and S.P. Sorella, CERN-TH-5391/89 [8] G. Bandelloni, C. Becchi, A. Blasi and R. Collina, NucI. Phys. B197 (1982) 347 [9] T. Clark and J. Lowenstein, NucI. Phys. B113 (1976) 109; J.H. Lowenstein, Commun. Math. Phys. 24 (1971) 1
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Chem—Simons model
[10] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. (N.Y.) 59 (1970) 42; S. Coleman, in Properties of fundamental interactions, ed. A. Zichichi (Ed. Compositori, Bologna, 1973), p. 359 [11] B. Zumino, Chiral anomalies and differential geometry, in Relativity, groups and topology II, ed. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) [12] C. Lucchesi, 0. Piguet and K. Sibold, Int. J. Mod. Phys. A2 (1987) 385 [13] E.C. Zeeman, Ann. Math. 66 (1957) 557 [14] J. Dixon, Cohomology and renormalization of gauge fields, Imperial College preprint (1977/78); G. Bandelloni, J. Math. Phys. 27(1986)1128 115] H. Epstein and V. Glaser, Ann. Inst. Henri Poincaré 19 (1973) 211 [16] R.W. Johnson, J. Math. Phys. 11(1970) 2161