Local approach to dilatation invariance

Nuclear Physics B197 (1982) 347-364 O North-Holland Publishing Company

LOCAL A P P R O A C H TO D I L A T A T I O N I N V A R I A N C E G. BANDELLONI, C. BECCHI, A. BLASI and R. C O L L I N A Istituto di Scienze Fisiche dell' Unit, er.~it& di Genot'a e Istituto Naziona/e di Fisica Nucleare, Sezione di Genot, a, Italia

Received 12 October 1981 Scale invariance is analyzed locally by coupling the energy-momentum tensor to a source which is the metric field of curved space-time. The resulting theory at the classical level has no mass parameters only if the general coordinate transformation group can be represented in Weyl's scheme. We further discuss the quantum extension of the theory; the Ward identities become anomalous under radiative corrections and the anomaly is shown to be connected with the instability of the classical metric field representation. The anomalies, recognized as the well-known trace anomalies for the energy-momentum tensor, are then reabsorbed by a perturbative alteration of the original metric field transformation law and we prove the modified Ward identities to be renormalizable in the fiat limit. Finally we show that our approach is equivalent to the well-known parametric equations of the Callan-Symanzik type only if the dilatation invariance is not spontaneously broken. In the presence of spontaneous scale breaking we derive a functional equation which will be applied to cases of physical interest in a forthcoming paper.

1. Introduction In this p a p e r we p r o p o s e a n e w description of d i l a t a t i o n i n v a r i a n c e in q u a n t u m field t h e o r y [ 1 - 3 ] b a s e d o n a system of local W a r d identities (WI). O u r a p p r o a c h should be p a r t i c u l a r l y useful in the analysis of r e n o r m a l i z e d q u a n t u m field m o d e l s where the d i l a t a t i o n i n v a r i a n c e is s p o n t a n e o u s l y b r o k e n , a l r e a d y at the classical level, by the n o n - v a n i s h i n g v a c u u m e x p e c t a t i o n value of a scalar field with n o n - t r i v i a l c a n o n i c a l d i m e n s i o n [4]. B e y o n d the classical a p p r o x i m a t i o n , it is well k n o w n that scale i n v a r i a n c e is modified by a n o m a l o u s terms (trace a n o m a l i e s of the e n e r g y - m o m e n t u m tensor) i n d u c e d by the radiative corrections [1-3, 5]. In the a b s e n c e of constraints, such as scalar field r e m a i n i n g massless or a y~ s y m m e t r y for fermions, the a n o m a l i e s c o m p l e t e l y spoil the naive scale i n v a r i a n c e since g e n u i n e mass p a r a m e t e r s could e n t e r into the picture h i d d e n in the a n o m a l o u s terms. T h e only way to o v e r c o m e this difficulty a n d m a i n t a i n a w e a k e r but still m e a n i n g f u l scale i n v a r i a n c e criterion at the full q u a n t u m level is to have a m i n i m a l i t y c o n d i t i o n for the a n o m a l i e s . T h e first step a l o n g this line is an analysis of the d i l a t a t i o n p r o p e r t i e s of the b r e a k i n g s which a m o u n t s to s t u d y i n g their c o m m u t a t o r with the e n e r g y - m o m e n t u m tensor. T h e w o r k i n g s c h e m e is b o r r o w e d from the usual c u r r e n t algebra models w h e r e the r e n o r m a l i z e d c o m m u t a t o r s are discussed by i n c l u d i n g in the theory a set of classical gauge fields a n d by asking the i n v a r i a n c e of the v a c u u m to v a c u u m 347

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transition functional under gauge transformations of the classical fields [6]. Accordingly we couple the e n e r g y - m o m e n t u m tensor to a metric field and remark that, if this field belongs to Weyl's representation [7] and the theory is invariant under general coordinate transformations, then scale invariance is insured at the classical level. This scheme becomes critical at the quantum level due to an instability of Weyl's representation of general coordinate transformations. By this we mean that there are perturbations of the transformations laws of the metric field, preserving the algebraic structure of the group which cannot be brought back to the original form by a redefinition of the metric field as a function of itself. In this light the dilatation anomalies correspond to a perturbation, induced by the radiative corrections, exactly along the direction of the instability of Weyl's representation. This fact immediately suggests that we may completely reabsorb these anomalies by a suitable quantum alteration (hence proportional to h) of the metric field transformation law, and therefore by a modification of the original WI. The logical and conclusive step would now be to prove the renormalizability of the altered WI, thereby completely characterizing the anomalies. This last point requires an amount of mathematical work which can be avoided by the following shortcut. We study the theory in the flat limit where the instability of the metric field can be transferred to an ad hoc external scalar field (spurion field) directly coupled to the trace of the e n e r g y - m o m e n t u m tensor. By this trick we shall be able to complete our analysis of the trace anomalies and prove the renormalizability to all orders of the altered WI In a successive paper we will apply this scheme to physically relevant models with spontaneous breakdown of dilatation invariance. In sect. 2 we analyze the scale invariance properties of scalar field models e m b e d d e d in a metric field and describe the Weyl representation. Sect. 3 contains some general comments about the fate of Ward identities under perturbations induced by radiative corrections and the proof of the instability of Weyl's representation for the metric field. In sect. 4 we study the possible anomalies to the classical dilatation Ward identities and characterize them as the minimal trace anomalies in the flat limit. We also introduce the "spurion field" and write a set of altered Ward identities. In sect. 5 we prove the renormalizability of the new Ward identities by a double expansion in powers of h and of the spurion field. Sect. 6 is devoted to a comparison of the functional equation obtained from our Ward identities with the better known parametric equations of the Callan-Symanzik type. In appendix A we summarize the necessary ingredients to extend the treatment to quantized gauge fields and to the case of spontaneous breaking of the dilatation symmetry. Some technical aspects of the proof of renormalizability are given in appendix B.

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2. Scale invariance and the Weyl representation We begin along the lines sketched in the introduction where it is suggested to analyze the dilatation behaviour of a theory by considering its extension to a curved space-time defined by a classical metric field. The invariance under general coordinate transformations of the enlarged theory should imply that the original model is scale invariant. In the following we limit ourselves to a theory with only scalar quantized fields; the case of vector fields is treated briefly in appendix A while the analogous procedure for fermion fields requires the use of the vierbein formalism [8] and is left to the reader. For the sake of clarity we first outline the well-known case of enlarging a model by adding a metric field belonging to the Einstein-Riemann representation. Suppose we have a single scalar field ~o(x) with transformation law 6~o(x) = A" (x)a~,q~(x), x " ~ x ~' - A ~ ' ( x ) ,

(A ~'(x) arbitrary).

(la) (lb)

We introduce the metric tensor g"~(x) assigned the transformation law 6g ~ (x ) = A ° (x )a,g"~ (x ) - Oo~ " (x )g"~ (x) - aoA ~(x)g~" (x).

(2)

Now the action of the model, which in the flat limit is F ~ ' = I d ' / & ° ( x ) = I d4g[lcg~'q~(g)egu~°(x) + V(q~(g))] ,

(3)

can be transformed into /-'el = I d 4 x ~ / - d e t l g ° ' ~ l [ g~'''(x)O~'q~(x)O~p(x)~-2 V(~(x))] ,

(4)

thereby showing explicitly its invariance under the transformations in eqs. (1), (2)*. Let us however stress an important point; the invariance properties of the action in eq. (4) derive only from the Lorentz invariance and locality of the action in eq. (3); in particular no hypothesis of scale invariance was needed, hence the model may well contain mass parameters. This procedure is therefore of no help in identifying scale-invariant theories. On the other hand, if the theory is scale invariant, we have an alternative and more promising way. Indeed, suppose V(q~) = c~o4 and define the new field variable ~o'(x) =- ( - d e t

Ig..I)'/%(x),

(5a)

the modified metric tensor ~ ' " (x ) =- g"" (x ) ( - d e t Ig~l) 1/4 ,

(5b)

* We are, of course, free to add to the above action more complex terms [1-3, 7] which become trivial in the flat limit, such as I d4x ~/-det Igo,,~(R + R~o~+. • •), where R is the scalar curvature.

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350

and the auxiliary vector field

w~,(x)=---~0~, In ( - d e t [go,,[).

(5c)

Substituted into eq. (4) they give

F d = I d 4x[sr/.t~,(0~,+to~,)q~t(c3.+w.)~ot -c'~,'4],

(6)

which does not depend explicitly on det ]g~,.l. The relevant observation is that this does not happen if the original action contains terms of the type ~o2 or q6. In other words if our classical action depends only on the fields ~0, sr~'" and to,. to which the following (Weyl's*) general coordinate transformation rules are assigned:

6~(x ) = h " (x )O.q~(x ) + ~,.lt ~'(x )~o(x ) ,

(7a)

6~'"(x) = h"(x)Oo~'"(x)+ ½0~°(x)~'"(x)-OoA~'(x)~°"(x) -OoA "(x)~'"" (x), (7b) 6to~, (x ) = A o (x )Ooto. (x ) + ~.A o (x )to,, (x ) - ~0~,O.A o (x ) ,

(7c)

then the same action is necessarily scale invariant. Accordingly we can characterize scale-invariant" m6dels by a set of functional differential equations obeyed by the classical action of the theory, i.e. d

W,~(x)F =-~,; 1

o,.

/ o,, 6 +20,,k~" ( x ) ~ F

~FCl

(x)6ff,,,,(x) /

o,,

6

,Sto.(x)

el\

/

el\

)

6

cl

t

,Sw(x)

6____~1.c~

&p(x)

(8) The operators W , ( x ) defined in eq. (8) describe the effect of a general coordinate transformation [eqs. (7)] of the fields ~, (~'~, to,, and satisfy the algebraic identity

[ W . ( x ) , W.(y)] = -

0

~(x-y)W.(x)-~x.8(x-y)W.(y).

(9)

The classical functional F c~ is the starting point to generate a quantum theory for the scalar field ~ in curved space-time so that the correlation functions of the quantized theory are naively expected to preserve the invariance properties of the classical model. As usual these properties are easily exhibited with the functional technique [9] i.e. by introducing into the action a term [d4xJ(x)q:, (x) where the source J(x) is assigned the transformation law

6J(x) = A O(x)Oj(x)+ 30.A °(x)J(x) . * W e w o u l d like to t h a n k D. Z w a n z i g e r f o r p o i n t i n g o u t this a s p e c t to us.

(10)

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351

The expected invariance under general coordinate transformations of the connected G r e e n functions of the theory is now written for their generator Zc[J ] as

8

/ ,,,

8

Z

(r,,,,(x)

8

+ a~,w.(x)~ z c

8w~(x)

8 1

/

\

Z

8w,(x)

) -J(x)O"8-)--~x)x)Z~

8

+ ~O,~J(x)6-~x)Zc ) = 0 .

(11)

The original WI (8) for the functional F generating the proper vertices of the theory are recovered from eq. (11) through a Legendre transformation. Now the classical sources may be interpreted in terms of the operators their functional derivatives define. The metric field generates a symmetric tensor operator which combines with the vector defined by 8Zc/Sw, (x) to yield a tensor

O,~(x)=

x) ~8~, 6(~-(~) Z~-~O~"

6~Z~ '

8oJ,,(x)

(12)

which is directly interpretable as the e n e r g y - m o m e n t u m tensor of the theory. The other terms vanishing in the flat limit which can be added to the classical action are automatically transferred into the generating functional and contribute to the definition of the time ordered products of field q~ and e n e r g y - m o m e n t u m tensor operators. The feature which is non-trivially connected with scale invariance is that the trace of the e n e r g y - m o m e n t u m tensor in eq. (12) is given by the divergence of the current 8Z~/Sw~. We notice that, if no gauge fields are present, even this contribution to the trace can be made to vanish [1, 3] by introducing in the action a term d4xR~o 2 with a suitably chosen coupling and we obtain the improved energym o m e n t u m tensor of C a l l a n - C o l e m a n - J a c k i w [1 ].

3. Instability of Weyl's representation Before analyzing the detailed behaviour of the WI under radiative corrections it is useful to have a qualitative picture of the possible alterations of the WI. Let us assume that the vertex functional of the renormalized theory be invariant under field transformations which are perturbations of eqs. (7), i.e. O

W , (h, x ) F [ ¢ ] = 0 ,

(13)

O where the W~, (h, x) operators satisfy the commutation rules in eq. (9) and coincide

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G. Bandelloni et aL / Dilatation inuariance

at h = 0 with eq. (8). Notice, first of all, that the W ° ( h , x) operators are not unique since they are not invariant under a local redefinition of the sources such as J(x) ~ J(x) + eZ(~)J(x) ,

(14a)

~""(x)-* ,;'~"(x) + e Z " ~ ( ~ ) ,

(14b)

to, (x) -* to, (x) + e z , v(~r)to~(x),

(14c)

which, when substituted into the functional Z [ J ] (or F[~]) amount to a new choice of the time-ordered products and therefore have no effect on the physical interpretation of the model. The question, however, remains if all the possible quantized versions of our theory which are invariant under general coordinate transformations (if any) correspond to W~,o (h, x) operators which are related by source transformations as in eqs. (14). In perturbation theory [6, 10] this is certainly true provided any perturbation of the classical representations in eqs. (7) corresponds to a source redefinition. In this case we shall call stable the original representation. This analysis clarifies three possible alternatives for the fate of classical WI in a full quantum theory. First and worst, the radiative corrections definitely destroy the symmetry of the theory; thus the WI become meaningless at the higher orders". Second, the representation of the symmetry group carried by the fields is unstable and the radiative corrections alter, but don't spoil, the WI. Third, the representations are stable and the renormalized WI coincide after a time-ordered product redefinition with the original ones. Concerning our WI, we now show that the representation in eq. (7b), i.e. Weyl's representation for the metric field, is indeed unstable. We can therefore conclude that the quantum corrections could alter the classical WI [eqs. (8), (11)] due to the instability of the representation (second case) and that there might also be true anomalies (first case). We then proceed to check the instability of Weyl's representation. Let us consider a generic infinitesimal deformation of the representation in eqs. (7), which restricted for simplicity to the sole metric field, can be written as 6h"~ ( x ) = A °(x)Ooh"~ (x)-(O"A " (x) + O"A~(x)- ½rl""OoA°(x)) -(OoA"(x)h°"(x)+O,,A~(x)h°"(x)-½h""(x)O~°(x))+eX"V(x),

(15)

where X "~ (x) = X " ( x ) =- A ~ ° ( h ; x)OoA ~'(x)+ B ~ ( h + C~ ~° (h ; x)wo(x)A ~ (x), hU~(x) = ~'""(x) - 7/"~

; x)Ooh"T(x)A n(x) (16) (17)

rt "~ the flat metric. The tensor X " " ( x ) given above is the most general one which * As an example we mention Adler's anomaly in current algebra.

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preserves the dimensionality of the 1.h.s. of eq. (7b). To first order in e the algebraic c o m m u t a t o r conditions [eq. (9)] yield for X""(x) the set of equations

f dax [a~(x)Wo(x)X""(X,; y ) - A ~ (x ) Wo(x )X~'" (A z; y)]=X~'"(,~.l OA2-)t 20h.1; y). (18) We do not discuss the general solution of eq. (18) but consider the term X""(A ; x) = (77"~ + h~'"(x))O~°(x),

(19)

which is easily seen to satisfy eq. (18), and compare it with the expression X"V(A ; x) = I daYA°(Y)W°(Y)Z~'"(h(x))'

(20)

which exhibits the effect of the transformation in eq. (14b). The comparison is carried out in the flat limit ( h " " ~ 0) where eq. (19) reads X~'~(A ; X)IF.L = rl"~aoa°(x),

(21a)

and eq. (20) becomes

X~V(t~.x)ll..l..:2Id4y/~O(y)[aA( '

¢~

6h~(y)

"Ao

r~ .~qz~,,(h(x))

4 6h ,(y)]3

= a ( - 4 a " a "(x) + n"~a2°(x)).

,=0 (21b)

It is apparent that the r.h.s, of eqs. (21a) and (21b) cannot coincide; therefore we conclude that the term (rl"" +h""(x))O~,AP(x) is a real instability for it is already such at the fiat limit. In much the same way one can prove that an alteration of the ~ field transformation law such as 6q~ = a PÙ,,~ + ~ a , ~ P~, + eaoh°~

(22)

is also an instability. We shall see in the following how all the breakings introduced by radiative corrections into the original WI are reabsorbed through instabilities of this kind.

4. T h e n e w Ward identities

We discuss the connection between the possible quantum anomalies of the dilatation Ward identities and the representation instabilities described in sect. 3. This study will be carried out only in the flat limit which from eq. (8) reads

a o, , __L8 r~,~ W.(x)r°%.~..=-a.~(x)8-~)F -aa.(,~(x)a,p(x) :

+2a,,[~F-a+. , a °' '

rc'= " 6h ~ (x) I'+' )-}a.a, ~aoJo(x)

0 " (23)

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G. Bandelloni et al. / Dilatation invariance

By applying the quantum action principle [11 ] we can analyze the possible breakings affecting the WI; we find that the radiative corrections will produce terms corresponding to the insertion in the G r e e n functions of local operators of canonical dimension up to five. Such terms appear in the WI for the vertex functional at their first non-vanishing order as local polynomials in the field q~ and their derivatives. To be able to characterize them we shall discuss the WI at this first non-trivial order and use the fact that the symmetry corresponding to the Poincar6 group is known to be a priori fulfilled. Thus the translation invariance of the theory insures that the integrated WI are obviously satisfied, yielding W~,(x)F[F.L = 8 " I , , , ( x ) ,

(24)

where I , , ( x ) is a local functional with canonical dimension up to four. Furthermore, the Lorentz invariance of the theory insures that I,,, is a symmetric tensor. To prove this statement, choose A ° ( x ) = A ° " x , , with , 1 " " = - A "° [i.e. the p a r a m e t e r h"(x) corresponds to a Lorentz transformation] and obtain, after multiplication of eq. (24) by A"(x) and integration, the relation f I~u~](x)d4x = 0 ,

I[~,~l=(I~-l~,),

"

(25)

whose general solution is It~,~l(x) = OoI~,,~jo(x).

(26)

We now go back to the r.h.s, of eq. (24); separate it into its symmetric (curly brackets) and antisymmetric (square brackets) parts and substitute from eq. (26) to get =~8 I(,~r+ 2o o Jfu,,]o,

(27a)

which can be rewritten as 8 ~ I ~ = 201-"I~".) + !20~0 o (Itu,; ° + Iro~l~) ,

(27b)

since the last term on the r.h.s, identically vanishes. With a little rearrangement of the indices we find 8~i~,~ = ~0 ~ ,"(I(~,~l+8°[Iro~l~, + I i , M ~ ] )_= a . I~,,~ , I.

(27c)

This is as far as we can go by using the unbroken Poincar6 group symmetry. A further reduction of the anomaly is obtained by introducing a counterterm I ~zv t ~,~ ( x ) I ~ , , 4 in the lagrangian. Recalling that we are working in the fiat limit we obtain, instead of eq. (24), the expression IV, (X)FIF.,. = a ~ , I ( x ) ,

(28)

where l ( x ) is Lorentz invariant and has dimension up to four. The next step in the characterization of l ( x ) involves the reverse process; we substitute the usual derivatives with the covariant ones, introduce the necessary

G. Bandelloni et al. / Dilatation invariance

355

metric tensors £"~ (x) and decompose the polynomial I(x) into a sum of polynomials lax), each of which transforms under a general coordinate transformation as

6L (x ) = .~ ° (x )8oI~(x ) + ¼p,8o~ O(x )L(x ) ,

(29)

with 0 <~p~ ~<4. All the terms with p~ < 4 are compensable by introducing in the lagrangian the counterterms [4/(4 -p~)]Ii(x)*; the remaining term with p~ = 4 corresponds to a true anomaly only if, in the flat limit, it cannot be written as a divergence, since contributions like I = cg,k" can be compensated by a counterterm -4w,,k'. We conclude, therefore, that the minimal anomalies to our WI, in the flat limit, are given by those operators of canonical dimension equal to four and which cannot be written as a divergence; these we immediately recognize to be the well known Callan-Symanzik or trace anomalies. This shows that in general the original program of characterizing a scale-invariant theory to all orders by means of a set of Ward identities cannot be carried out with the operators W , ( x ) as they stand. On the other hand, it is now easy to prove that the above anomalies correspond to a deformation of the WI along the direction of the instability discussed in sect. 3. Indeed we are free to add to the classical lagrangian the coupling h a x I 4 , i.e. the first order in h"'" of the term (det Isr"~] - 1 ) I 4 . This freedom can be usefully exploited since if we write the WI corresponding to the altered representation of the metric field

6h"~(x) = A °(x)~oh""(x) + ~(1 + h)O~a "(x)(o"" + h"~(x)) -O"A ~ ( x ) - a ~ A " ( x ) - O o A " ( x ) h ° " ( x ) - a o A ~ ( x ) h ° " ( x ) ,

(30)

we obtain at the lowest order in h and in the flat limit the anomaly free expression

W°(x'h)rr~"'-~

W"(x)-}~°" ShC,(xl F.,..=O"

(31)

Before being too happy with these new WI, notice that the addition to the classical lagrangian of a term h**I(~) does not spoil eq. (31) in the flat limit regardless of the scaling properties of I(~o). This means that the description of the old anomalies by the term - ~hO, (6/6h*A (X))F]F.L. does not guarantee its minimality. Hence, to be sure that only scale-invariant l(~) are admissible, we are forced to analyze the WI outside of the flat limit, which corresponds, as mentioned in the introduction, to studying the algebra of the energy-momentum tensor. This program, which is the natural approach*", turns out to be algebraically untractable. Therefore, we shall adopt and illustrate an "unnatural" alternative but sufficient to solve the problem without going out of the flat limit. * Such terms are, in fact, absent since the theory is scale invariant. t . We may mention that a possible solution of this problem may be sought by giving a regularization method which respects the WI after the alteration; we know that such a method cannot be found for the original identities which indeed develop the anomalies even in the flat limit.

G. Bandelloni et al. / Dilatation invariance

356

To characterize the operators which are scalar densities of weight four we introduce a source for them, i.e. a classical field ~r(x) with, variation 6o-(x ) = A ~ (x )Ooo.(x ) .

(32)

Now the classical WI assume the form (o')

W,,

cl

( x ) F [V.L.----W,,(X)FCtPV.L.+O,,O'(X)

FC'IF.L. = 0 .

(33)

Our strategy is that of trying to reabsorb the trace anomalies by transferring the instability of the metric field representation to the o- field. Thus we alter the cr field representation in eq. (32) to 6o'(x ) = A ° (x )Ooo'(x ) - hOoA ° (x )

(34)

and we analyze the validity, in the flat limit but to all orders in the o" field, of the deformed WI

w', Ix)rlF,. + h a ,

7--v-z,, rl .L. = 0 . oo'~x )

135)

5. Renormalizability of the new Ward identities The proof of the renormalizability of the WI in eq. (35) amounts to verifying that the breakings induced by the radiative corrections can be compensated order by order within a theory possessing a finite number of parameters. It is easily seen that the presence of the tr field does not alter the properties of the breakings as deduced in sect. 4; hence, they can be reabsorbed, order by order in the WI of eq. (35). The problems connected with the vanishing dimension of the o- field remain, however, open; indeed both the classical action and the breaking terms now depend upon arbitrary powers of tr. It is precisely the necessity to identify this infinite number of o" couplings which differentiates the present analysis from the well-known ones on the perturbative renormalizability of WI [6]. In our case we are naturally led to consider the generating functional F and the breaking terms as double power series both in the parameter h and in the ~r field itself. The proof then proceeds by showing that the anomalies at order h n and o"m can be compensated by a suitable choice of the couplings at order h "-1 and ¢r"÷1 At the classical level (h = 0) eq. (35) implies that the action be constructed with invariant couplings to all orders in the o" field. Suppose that, given an arbitrary subtraction procedure and a definite choice of the o- couplings, we have been able to satisfy the WI in eq. (35) to all orders h"o-" with n + m < N + M and 0 ~ 1 except the terms with n = N and O<~M'<~m <~M. Let us denote the minimal anomaly at order hncr M' by hN a~, Y~N'M'~(~, 0"; X ) .

(36)

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357

Our analysis says that y(N.M'~ is a local, classically scale invariant functional and it is defined modulo terms of the type hN & I ~(N'M'~ (~, (r; x) ,

(37)

where the local functional I v(NM'~(~, o-; x) has dimension equal to 3. Assume, for the moment, that we have at our disposal a unique integrated local functional X CN-l'M'÷l~ (~, or)

(38)

which is classically scale invariant, homogeneous of degree M ' + 1 in the or field and satisfies (3~"~&r(x) X

(r'-l'M'~l) (~, tr) = 3~,Y ~N'M'~(~, ( r ; x ) + a, 3,I ~(N'M'~ (~, or; x ) .

(39)

Granted this, we introduce in the action the counterterm h '~ aX(N ~.M'+1)(¢, Or) which implements our WI up to the order hNtr M' thereby completely identifying the invariant couplings at the order hN-~tr M'÷~ which were left free up to now. The proof of the solvability of eq. (39) for arbitrary M', N and its uniqueness for M ' > 2 is given in appendix B. By repeating this procedure at fixed N and increasing M ' we reach the point M ' = M where it stops (to start again at hN+~o'°), since the minimality of the anomaly at order hNcr~*x is not insured if the WI do not hold at order h N -1cr'~'2. We have therefore given an inductive process which allows one to maintain the WI in eq. (35) to an arbitrary order in h and (r and defines our theory in terms of a finite number of free parameters, which are formal power series in h to be fixed by the normalization conditions. The free parameters are the ones appearing in the or field-independent, scaleinvariant lagrangian, i.e. the coupling constants of the scale-invariant original theory, plus the coefficient of the terms -

I

day 3~r(y)IH

'I

g

()),

I

½ d4yo'(y)[a~E]20"(y)+3.(A{..~}(y)3~(y))],

d4y o'(y)d~[(D~o. + Dorl.~ + D.rl.o)3~3oo'](y)

(see appendix B), all of which, however, contain derivative couplings for the ofield and therefore have no effect on the physical properties of our theory.

6. Local scale invariance and parametric equations We would like to end this paper by pointing out the relations between our approach to local scale invariance, characterized completely by the WI eq. (35), and the well-known parametric equations of the Callan-Symanzik type [2].

358

G. Bandelloni et al. / Dilatation in variance

The analysis proceeds by first Fourier transforming to m o m e n t u m p the W I in eq. (35) and then selecting the contribution to particular p r o p e r G r e e n functions by a suitable n u m b e r of functional derivatives with respect to ~ and or fields. We thus obtain the equation (k.)

/'-(..,.i k 1. . . . , k " - p . . . . .

k " ; q 1. . . .

,q ,,1 ,-

k"-

+p

a~l

_~,....)

k l,...,k,,.ql,

,...,qm

1,_

k"-

Y.

t~=l

+ m ~ ' . ~ qt~,[,~,,,,, ul[t " ) ( 1k , . . . . . k,,. q i . . . . .

,(3=1

+p

/3=1

q¢ - p . . . . , q,,

1,

~=1

_ 1 ~`.... ,(k 1. . . . .

......

k,;ql,...,q,,

,

1 , _ y. k " -

Y. q t ~ + p

a- 1

+p,.,. lr*tn.t,n) k 1 . . . . k n ; q 1. . . . .

....

oF--1

q m- 1., -

p ......

k,,_"~lqO+p) ~;1

)]

~=1

k() _ ,,=1

-~

i a=l

q° +p /3=1

q .....

q a=l

r r%~'Dt.m)

p.tt.

1 - - . ,~A

-zo.1

x

(n.m)'t[

J~p; k

1

...

,

k";

ql ,...)q

B=I

i

-

.,--1

k"-

cx=l

+ -~p,.p i

,./~(,~.... )[~ p ; k

1. . . . , k . ; q l

.....

q,.

1,_

£

k ~_

~,=1

"(n r e ÷ l ) (

+ hp, F

"

i

. . . k l, . . . k. ";. ql

q ,,-1 , ,,=1

= - ( W. ~ , ( p ) I a

,,,..,)

k 1 , . . . . . k"" q 1. . . . .

q m l,-

Y. q O + p

)

~=1

"'z' q O + p ) B= 1

m-I

k'" -

m 1

qa

y.

) +p,p

/3=1

k" ~=1

Y.

+p

=0,

(40)

13-1

where the notation is as follows:/"( .... )(k 1 , . . . , k" ; q ~ , . . . , q'" ) is the p r o p e r G r e e n function with n ~,-field and m or-field (amputated) legs of m o m e n t a k 1' . . , k. , ,., q l . . . , q ,,,,respectively. ~(,.m). 1. . . . , q , , ) i s t h e s a m e l, t p ; k l, . . . , k " ; q G r e e n function with an extra vertex corresponding to the a m p u t a t e d leg a3,(p); analogously/a~(,.,, ) (p ; k 1, . . . , k" ; q 1, . . . , q " ) has the insertion of a vertex coupled to the external field / ~ ( p ) . F u r t h e r m o r e , in eq. (40) we have taken explicitly into account the overall m o m e n t u m conservation. We now c o m p u t e lim3 o ( v.r,e , t, p ),,~,,,,.,,) l) k 1, . . . , k n., q 1. . . . q ,~ 1, -

p~(1

"

k"¢,=1

~ B=I

+p

(41)

G. BandeUoni et al. / Dilatation invariance

359

which explicitly reads* ( "~ kXOk : + ' "Y.- ' q ,a~ o q x ) l\.~,..m,( k t . . . . . . k"" q ' . . . . . +(n

-

4 ) F ~.... I(kl , . . . ,

k"

;q

1. . . .

,q

q ~-, ,-

,n-I , -

k .

i

k" - .~..t q~ )

-

,x-- I

/3~1

/

+ 4 h F ' . . . . . 1 ~ ( k l , . . . . . k"" q I . . . . . k

q ., 1,

qe, 0 = 0 . ~l

(42)

13=1

At this point we consider two alternatives. Suppose first that the dilatation invariance is not spontaneously broken; in this case the operator 0,,--

d4x So.(x) ,

(43)

which corresponds to the insertion of the vertex coupled to a cr field at zero m o m e n t u m , is well defined. Furthermore it commutes with the W,,(x) operator in eq. (35); it follows that the functional O,~F satisfies the same WI as the functional F and we have at our disposal a, by now, classical argument to deduce that the operator 0,. is given by a parametric derivative at cr = 0. Thus, for example, in a gqr4 theory we immediately get f.i,,.a,

k 1, . . . . k ~-1 , -

k " ; 0 =(0,~/~) I'''1~ k 1, . . . .

=/3~a~F° ( . m"

(

kl,

+ n y f ~'''°1 k I

.

•

.

•

. ,

.

1, -

k"

.

k

~

n

k"

l,

~ k"; 0

- ~ k"

1, -

k" t l = l

(44) which substituted into the l.h.s, of eq. (42) yields the parametric equation for the model. Hence in a model with no infrared problems the WI in eq. (35) can be reduced to the better known parametric equation of the Callan-Symanzik type. Of course, the same procedure cannot be followed in the second possibility, where the dilatation symmetry is spontaneously broken and we remain with the functional non-parametric expression in eq. (42). The physical relevance of this last equation will be the subject of a forthcoming paper. * Notice that the Green functions involving the external vertices coupled to o3,,(p) and t ~ l pl do not contribute to eq. ~42); the first one vanishes in the limit p ~ 0 and the second one appears only with its traceless transverse part in eq. 140).

360

G. Bandelloni et al. / Dilatation in variance

A large part of this work was developed during the visit of one of us (C.B.) to the Laboratoire de Physique Teorique et Hautes Energies de l'Universit6 Pierre et Marie Curie in Paris whose hospitality is gratefully acknowledged. The other authors would like to thank the Istituto Nazionale di Fisica Nucleare for granting them the opportunity to continue the collaboration.

Appendix A In this appendix we describe how the WI in eqs. (8), (11) modify when gauge fields are present a n d / o r there is a spontaneous breaking of the dilatation symmetry. For reasons of economy, but without any lack of generality concerning the substantial aspects of the problem, we do not consider fermion fields in our discussion. It is quite easy to introduce into our identities [eq. (8) or (11)] the effects of a spontaneous breaking of the symmetry, a n d / o r to introduce quantized gauge fields associated with an internal gauge group. In the presence of spontaneous symmetry breaking we rewrite eq. (8) for a multiplet *i of scalar fields after the replacement ~oi--~ + v, with v~ the vacuum expectation value. The gauge fields a ~ ( x ) and the corresponding Fadeev-Popov ghost fields C~(x), G ° ( x ) will be assigned the variations, under general coordinate transformations, 8a~,(x)--A (x)O~a~,(x)+~,A ( x ) a p ( x ) ,

(A.la)

8t~" (x) = A~(x)0ot~ ~ (x),

(A. 1b)

8 C ~ ( x ) = a ° ( x ) O . C " ( x ) + ½~oa° ( x ) C ~ ( x ) .

(A.lc)

It is well known that, when quantized gauge fields are present, in order to be able to discuss the renormalizability of the BRS identity we need the auxiliary classical fields y~(x), y~"(x), ( ~ ( x ) coupled to the BRS [10, 12] variations of the fields ~o~,a ~., ( ~ , respectively. For the sake of completeness we shall give here their behaviour under a general coordinate transformation: 8yi (x) = Ao(x)0oy~(x) + 3aoX° (x)y~ (x),

(A.2a)

8y o. (x) = A o (x)Ooy "~ (x) - aoA " ( x ) y"° (x) + 0,3 o (x) y"~" (x),

8(" (x) = 0o (A "(x )~'" (x)).

(A. 2b) (A.2c)

It follows that the classical action obeys the functional identities [corresponding to eq. (8)] W . ( x ) l ' ~ ' = - a . a ~ ( x ) BaZ(x)

-a.(a.(x)~r

)+a.c (x) 8-Kv~x)

8 ~, ,.. 8 1.~, aC"(x) F~']J + a . C ( x ) ~-~x)F-v (x)a,.a~.O(x)

-½a.(C"(x)----~8

G. Bandelloni et al. / Dilatation invariance

-a"(T°°(x) sT""(x) +auy~(x) 6T~(x) +a,X

o,.

361

J

-

-~ (x)a.~l"

8___S__I.d+2Oo(,O~(x)~F~, ) ~ (x)

(x)af.,.Cx)

+ a.q~Ax) ~ F

-~a~.[(q~(x)+ v,) 6w,(x)

J = 0.

(A.3)

The above W. (x) operators commute with the corresponding B RS identity provided the gauge is fixed by a function g'~(x) whose variation under a general coordinate transformation is*

8g ~ (x) = A "(x )dog" (x) + ½a~"(x )g~ (x) .

(A.4)

The commutation property of the BRS identity with the W,,(x) operators in eq. (A.3) allows a separate discussion of gauge invariance and of scale invariance without any interference between them.

Appendix B In this appendix we discuss the solution of the equation a,, 8--~x) X

= a,Y'~(x)+

(x),

(B.1)

where the upper index denotes the homogeneity degree in the tr field. The unknown X ~"+~ is an integrated local functional of canonical dimension equal to four, a, YU'~(x) is the given minimal anomaly to our WI and is therefore defined modulo the local functional O~,a~l~t~(x) with the canonical dimension of I ~"~ equal to three. Clearly the solution of eq. (B.1) coincides with that of 6_ X ~''+I~= Y~"~(x)+O~l~l"~x'V~(x)

So'(x)

(B.2)

* N o t i c e that a g e n e r a l l i n e a r g a u g e of the 't H o o f t [ 13] type d o e s not h a v e the w a n t e d t r a n s f o r m a t i o n p r o p e r t i e s ; s h o u l d o n e c h o o s e a n o n l i n e a r g a u g e function g", it is m o r e c o n v e n i e n t to i n t r o d u c e an auxiliary, L a g r a n g e m u l t i p l i e r , classical field A" with the B R S t r a n s f o r m a t i o n c " --, A ~ ~ 0 a n d with v a r i a t i o n for g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n & l " = A c'a,,,l~" + ½aoA"A" and specify the g a u g e - f i x i n g t e r m as A ° g ~ - ~A~A ~,

G. Bandelloni et al. / Dilatation invariance

362

Now a particular solution of the above equation is easily found by setting

Y~)(x) = ~ y~k)(~, &r; x)crk(x), k

(B.3)

=0

and verifying that d 4 y Ytk)(q~, 0or; y ) o rk" 1

p

(B.4)

k=O

satisfies eq. (B.2) with the Ojp(x) term given by . 1 I • OJp(x)= k~=0k - ~ d4Ytrk+l(Y)ff~-~

y(k)

(q~,0tr;y) ;

(8.5)

this particular solution is uniquely determined from the minimal anomaly. We proceed by analyzing the homogeneous equation ¢~

&r~x)

x(n.1)

rx

~ I-u(n)

= ,,v-H

•

(B.6)

Notice that the r.h.s, of eq. (B.6) cannot be chosen in a completely arbitrary way but that it obeys a consistency condition, the equivalent of the mixed derivative condition for the integrability of an ordinary partial differential equation. Hence v(n) we first derive this consistency equation and find the class of admissible O,,IH . Multiplying both sides of eq. (B.6) by tr(x) and integrating, we obtain, by the homogeneity properties of X (" +'),

X,,;+l,=n +Il

f~, ..... ,,,,(X), o XO'~X)O~tH

(B.7)

which substituted back into eq. (B.6) yields the wanted expression

8 .(,i)(y). nO.Ir"~')(x)=f d4y~(y)8--~-~x)0.1H

(B.8)

To study this equation we expand the term c9,1[~''~ on a local basis of divergences as

O.IH

(X) = a,l~2tr"(x) + a2,'-q(cr"-'(x)~tr(x)) + a3cT~,O,.(cr ~ l(X)tg"cT"or(X)) + a 4 0 . (Or"-l(x)O"Vqo'(x))

+OSS](A"(x)o-"(x))+O"O"(At,..)(x)o'"(x))+O':(B,.(x)o'"(x)) + 0,~ (o'"-l(x)C,.(x)O':~"o'(x)) + O'~(D,=(x)cr '' l(x)Do'(x)),

(B.9)

where the power counting constraint on l~t '') implies that A", C~, D . are local functionals of canonical dimension equal to 1, A{..) is symmetric, of canonical dimension equal to 2 and B,. has canonical dimension equal to 3. The analysis is

G. Bandelloni et al. / Dilatation ineariance

363

now a matter of substituting eq. (B.9) into eq. (B.8) and isolating the coefficients of the independent monomials in or(x) and its derivatives. The computation is tedious and straightforward; we here report the results with a few comments. There are three particular cases which must be considered separately: n = 0 is obvious since eq. (B.8) holds identically for an arbitrary I[~ °~ which, in the fiat limit has a finite n u m b e r of independent terms due to the power counting constraint. Accordingly we find

X~' = I d4yor(y)0,,/~'~°'(y)= - f d4y O,.o-(y)~I~'°' 8 (y) "

(B.10)

The case n = 1; the general basis given in eq. (B.9) collapses to only four terms and we can expand ~,~,.~" " ~ l ~ ( x ) = a l l - ] Z o ' ( x ) + O " ~ ( A ~ , ( x ) o ' ( x ) ) + O

,9 (Al~,.,l(x)tr(x))+O~'(B~,(x)tr(x)). (B.11)

The same analysis now yields the conditions A , , ( x ) = O,

B . ( x ) = -a"Al,~,4(x),

(B.12)

and we obtain the solution

X~ I = ½f day o-(y)[al[]2o-(y) + a"(Atu~l(y)a%r(y))],

(B.13)

which also depends on a finite number of parameters since the canonical dimension of At~,~I is two. An analogous procedure must be followed for n = 2; indeed here we have the identity 2 (tr(x)Do" (x)) + 4a '~a" (o"(x)3.3.or (x))

- ~-]2(¥ 2 (x)

-

2a" (tr (x)a. (-]cr (x)) = 0 (B.14)

so the basis in eq. (B.9) becomes redundant. Deleting the al coefficient we find the conditions a2=a3=a4=O,

(B.15a)

A , ( x ) = - ~D~,(x),

(B.15b)

C , ( x ) = 2D~,(x),

(B.15c)

A~..~(x ) = rh,~OADx (x ) + cg,.D,,(x ) + O~D. (x ) , B,,(x) = - ~7-]D~(x)-O~a 1 D.(x),

(B.15d) (B.15e)

G. Bandelloni et al. / Dilatation invariance

364

and therefore the solution

X~ ) =

~

I d4X O'(X)8"[(D,(x)n,T + D,(x)rl,, + D,(x)n,o)O'tr(x)d°tr(x)], (B.16)

which again depends on a finite number of parameters for the canonical dimension of D,(x) is one and we are in the flat limit. The same analysis repeated for the generic case n > 2 yields a system of h o m o g e n e o u s equations for the coefficients a~, a2, a3, an, A,,, B,, C,,, D,,, A(,~,,~ with only the vanishing solution. Therefore we have X ~ +1~ = 0 ,

n >2.

(B.17)

Let us stress again that the h o m o g e n e o u s solutions -~H, v t ~ X ~ , X ~ I given explicitly in eqs. (B.10), (B.13), (B.16) depend on a finite number of parameters and all involve derivative couplings for the tr(x) field. As mentioned in the text, these have no effect on the physical properties of our theory when they are introduced as counterterms in the action.

References [1] S. Coleman, Dilatation, in Properties of fundamental interactions, ed. A. Zichichi IEditrice Compositori Bologna, 1973) p. 359; C.G. Callan, J.S. Coleman and R. Jackiw, Ann. of Phys. 59 (1970) 42 [2] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227; S. Weinberg, Phys. Rev. D8 (1973) 3497 [3] D.Z. Freedman, l.J. Muzinich and E.J. Weinberg, Ann. of Phys. 87 (1974) 95; D.Z. Freedman and E.J. Weinberg, Ann. of Phys. 87 (1974) 354; R. Stora, Meeting on renormalization theory, CNRS Report (June, 1971) [4] J.S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888 [5] J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D16 (1977) 438; L.S. Brown, Institute for advanced study preprint (1980) [6] K. Symanzik, in Renormalization of theories with broken symmetry, Cargese lectures in physics, vol. 5, p. 179, ed. D. Bessis (Gordon and Breach, 1972), and references therein; C. Becchi, A. Rouet and R. Stora, Renormalizable theories with symmetry breaking, in Field theory, quantization and statistical physics, ed. E. Tirapegui (Reidel, 1981) [7] H. Weyl, Ann. der Phys. 59 (1919) 101; Space-time matter, translated by H.L. Brose IMethuen, London, 19221; L. Smolin, Nucl. Phys. BI60 11979) 253 [8] H. Weyl, Ann. der Phy. 54 (1917) 117 [9] C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, 1980) [I0] G. Bandelloni, C. Becchi, A. Blasi and R. Collina, Ann. Inst. H. Poincar6 XXVIII (1978) 522; XXVIII (1978) 255 [11] J.H. Lowenstein, BPHZ. renormalization, in Renormalization theory, ed. G. Velo and A.S. Wightman (Reidel, 1976) and references therein [12] C. Becchi, A. Rouet and R. Stora, Ann. of Phys. 98 (1976) 287 [13] G.'t Hooft, Nucl. Phys. B50 (1972) 318