CONFORMAL INVARIANT QUANTUM FIELD THEORY

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CONFORMAL INVARIANT QUANTUM FIELD THEORY

G. Mack

To cite this version:

G. Mack. CONFORMAL INVARIANT QUANTUM FIELD THEORY. Journal de Physique Collo-

ques, 1973, 34 (C1), pp.C1-99-C1-106. �10.1051/jphyscol:1973108�. �jpa-00215188�

CONFORMAL INVARIANT QUANTUM FIELD THEORY

CONFORMAL INVARIANT QUANTUM FIELD THEORY G. MACK

Institut fiir Theoretische Physik, Universitst Bern, Switzerland

1. - INTRODUCTION.

-

In this report we will address ourselves to the question of the construction and properties of a non-trivial exactly conformal invariant quantum field theory (QFT). Such a theory, if it exists, has a good chance of being relevant to the description of the real world in certain high energy limits

-

or possibly at intermediate energies

-

^{as will}

be stated below. Emphasis in this report will be put on non-perturbation techniques.

We shall study the theory by looking at its connected Green functions

1 2

A graphical notation will be used as indicated, in which an n-point Green function will be represented by a bubble with n long legs. The dressed 2-point function (propagator) will be represented by a line

and sawing off a leg from a bubble means amputation with the dressed inverse propagator G-' (xlx2).

Because of the spectrum condition the Green functions can be continued to the Euclidean domain

x4 = ixO real

Later we shall consider such Euclidian Green functions.

This simplifies the group theoretical analysis.

To build a theory one must first of all have a set of equations from which the Green functions should be determined. It is well known that the Green functions in Lagrangian QFT satisfy an infinite set of coupled non linear integral equations. They can be derived e.g. from a cut off Lagrangian with subsequent removal of the cut off. For simplicity I will write down the equations for $3-theory (in D

2

6 space time dimensions) as they were obtained by Symanzik long ago [l] ; they are given in figure 1 below. Similar

(i) Bethe-Salpeter equations -+ define kernels

defines "2i-kernel"

(ii) Dynamical equations (for 4 3 -theory)

ren. SD-Eq. for Propagator

FIG.l.- Renormalized integral equations for n-point functions.

Some words of explanation may be needed. The equations involve not only the Green functions them-- selves but also certain auxiliary amplitudes.

Firstly one defines amplitudes which are 1-particle irreducible in one channel, they are constructed out of Green functions proper according to _\r(

There and below the dotted lines in the bubble may serve to remind one that the amplitude is not symmetric in all its arguments but only in those attached to the upper and lower legs separately.

The other auxiliary amplitudes are the Bethe-Salpeter kernel and the so-called "2 particle irreducible kernels",

they are distinguished by a symbol "B" resp. "2i".

For our purposes they should be thought of as deter- mined in terms of the Green functions by the

equations exist for other theories.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973108

C 1 - 1 0 0 G. MACK

Bethe-Salpeter equations, figure l(i). In the case of nor any other external parameters, but only the the 2i--kernels this is a straight definition, while

for the Bethe-Salpeter kernel one must solve an integral equation.

If the BS- and 2i-kernels are thought of as so expressed in terms of the Green functions, then the dynamical integral equations, figure l(ii), are equations relating in general the n-point function to an n + 1 - point function and lower ones, so they are coupled. There are infinitely many, one for every n;

they are integral equations since there is always a term which involves an integration over loop momentum.

Two of them are in fact integro-differential equations, they are the renormalized Schwinger Dyson (SD)-equations for vertex and propagator. The latter has been written here in a peculiar form [21 which differs from ref. 1 but is equally correct. The primes

'

^{stand for}

differentiation with respect to momentum flowing between two external legs, or multiplication with a coordinate difference.

because we have integro-differential equations, boundary conditions xust be supplied also. They are the usual renormalization conditions which determine mass and coupling constant. In momentum space

a

-1

G-'(~,-~) = 0 and

-

^{G (p,-p) = -i at p2 = m2 (1.4)}

a

^P 2

.. ,. ..

V(p _{1 2 3} p p ) = ig for pipj = 4m2 (36.. _{1 J}

-

¹⁾

The vertex V is the amputated 3-point function.

In canonical perturbation theory, all the equations of figure 1 are solved by iteration, at each iteration step boundary conditions (1.4) are used ro recover the vertex function V and the propagator from their derivatives.

In the following we shall inquire into the possi- bility of attacking the dynamical equations of figure 1 non perturbatively, i.e. without recourse to iteration.

2

: - DILATATION AKD CONFORMAL IN\JARLANCE

. -

^Our

starting point will be the observation that the dynamical equations(Fig. 1) have a lot of symmetry.

The crucial point is this: the dynamical equations figure 1 do not involve mass m or coupling constant g

Green functions themselves. Mass and coupling constants enter only in the boundary conditions (1.4).

Thus in particular there is no dimensional parameter in this whole system of equations, therefore one will naturally expect thatthey are exactly dilatation

invariant. And indeed they are that and more.

Let us be more precise. Dilatations are coordinate transformations

2 2 xu + pxu

,

p > 0 so that ds2 + P ds

for the line element. The field transformation law under infinitesimal dilatations can be found from the theory of induced representations :3], viz

64 = -icD@ with

The "dimension" d of 4 will be taken to be a real number (we thus exclude the possibility of a non- diagonalizable matrix) it is a new quantum number analogous to spin. Through (2.1) also the transfor- mation law of the Green functions is fixed, and one can check that the whole system of equations (Fig. 1) is indeed exactly invariant, for any d. Following Wilson we shall allow non canonica1,i.e. non-integer dimensions d.

Noreover the equations (Fig. 1: allow for an even larger symmetry: they are also exactly invariant under conformal transformations

xu + o(x)-'(xu

-

cux2) where o ( x ) = 1

-

^2c.x + c x 2 2

where cL characterizes the transformation. Conformal transformations are a kind of space time dependent dilatations 141, as is evident from the transformation law of the line element

The field transformation law is

where Z = 0 for scalar fields 4 . uv

I n Minkowski space one must restrict oneself to infinitesimal transformations of Green functions, but in the Euclidian domain also global ones can be allowed. Sumning up we emphasize once more the fact: The dynamical equations (Fig. 1) are exactly

-

dilatation - and conformal invariant.

CONFORMAL INVABIANT QUANTUM FIELD THEORY C1-101

3. - FIXED POINTS AND ASYMPTOTIC LIMITS.

-

The equations being invariant, the transform of any solution must evidently be another solution. In general this new solution must be expected to satisfy different boundary conditions. Let us consider dilatations by a factor p ? A -1

.

In this case the transformed solution satisfies boundary conditions of the same form (1.4) characterized by parameters mr2, g', apart from a finite field strength renorma- lization:

o ld transformed

m2,g +- m'2, g'

-

¹

dilatation by X

These are exactly the facts expressed by the renor- malization group [5] (m 2 = 0) resp. callan-~ymanzik equations [6] (m 2 # 0). It is easy to show that

m v 2 = l2m2 while g' = g' (X,g)

is a function of A and g which satisfies a certain composition law that follows from the group property of dilatations. As a consequence it is uniquely determined by B(g) Z *'(A,g)A

a

With the equations being invariant it is natural to ask whether they also have some invariant solutions.

A trivial one is found right away: For a massless free field all the equations are of the form 0 0 and the solution is invariant if d = l(resp.%(D-2)

as shown in the transformation law. We shall make the HYPOTHESIS: There exists a nontrivial dilatation and conformal invariant solution, with the coupling constant gm # 0.

Evidently this requires that gf(A,gm) = gm, i.e.

B(g-) = 0. The Green functions of the conformal invariant solution will temporarily be denoted by GGML(x l...~n), their physical significance depends on the slope B1(gm). The well known result of the Callan Symanaik analysis [5,6] is that

A + if Bt(ga) < 0

(3.1) X -t 0 if B1(gm) > O and m2 = 0 except for overall normalization. gm is then called an UV-stable resp. IR-stable fixed point. The general case of several coupling constants has been discussed by Wilson [5]. Assuming the coupling constants are sufficiently close to the eigenvalues there will be a range of A for which approximate equality (3.1)

holds. This range may or may not extend to ("intermediate energy scaling")

We shall now address ourselves to the problem of constructing the exactly conformal invariant theory.

There are to date two approaches, the second of which is still in development:

1. Skeleton graph theory (Migdal-Polyakov) 2. Group theoretical approach

The skeleton graph approach was started by Polyakov [ 6'1 and Migdal [7] and further developed by Parisi, Peliti [8]

,

Todorov, Symanzik and the author [9,2]

.

For completeness a short Lccount of its main ideas will be given in the next section. A more extensive review has been given in reference [lo]. The presenta- tion of the group theoretical approach in the following sections will be based on the authozk work ill]. Some related ideas can also be found in the work of

Polyakov, Migdal [12] and Ferrara et al., [13,14]

although they do not consider the infinite set of integral equations of figure 1.

4.

-

SKELETON GRAPH THEORY.

-

Here one still relies on iterative solution of those integral equations for the n 2 4 point function, i.e. all of the equations of figure 1 except the SD-equations for vertex and propagator (last two equations of figure l(ii)).

Thus one transcends perturbation theory by one step.

The starting point is Po1yakov:s observation [6]

that in an exactly conformal invariant theory the 3-point vertex (and the dressed propagator) are completely determined by invariance, except for the coupling constant g and the a priori also unknown dimension d of the field. In the Euclidean domain

G(X,O) =

r

(d) (x')-~ (4.1)

where x. = x.

-

^{x and = 62 = 63 =} t(D-d) in D

i j 1 j

space time dimensions. The normalization of the propagator is physically irrelevant and may be chosen according to convenience.

Out of the dressed vertex and propagator, the n 5 4-point functions are constructed by skeleton graph expansions [15]

.

For example, for the Bethe-Salpeter kernel

C1-102 G. MACK

In formulae G(Ax l...A xn) = G(x l...~ ) where AeSO(5,l) acts

It remains to solve the renormalized SD-equations for the vertex and propagator. Consider first the equation for the vertex. Inserting the unique conformal inva- riant Ansatz for vertex and propagator together with skeleton graph expansion (4.3), the RFS will be an infinite series, each term of which is again a conformal invariant 3-point function and must there- fore be proportional to the unique expression (4.2).

(The derivatives are unnecessary in conformal theory, the requirement of conformal invariance plays the r81eof boundary condition which replaces (1.4).) Factoring out the common x-dependence we are thus left with an algebraic (as opposed to integral

-

⁾

equations involvingthe unknown parameters g, d,

where functions f. are determined by working out the individual skeleton graphs. Similarly the propagator equations results in another equation

1 = g2h (d) + g h2(d) 4 ⁺

.

^(4.5)

1

Thus there are two equations for two unknown parame- ters, so one may hope for a unique solution. If so the theory will have no free parameters, in particular the coupling constant will be fixed in agreement with the discussion in section 3.

One might further speculate that for realistic models a solution (with g real and d > k(D-2) as required by positivity) may exist for some internal symmetries and not for others, thus providing a dynamical explanation of internal symmetry. Unfortu- nately such ideas hinge on the question whether and how a meaning can be given to the infinite sums in equations (4.4) and (4.5).

naturally on light-like 6-vectors.

Consider then G(AxlAx2; x3...x ) as a function on the group {A}. It can be harmonically analyzed [I?]

,

i.e. expanded into unitary irreducible representations of the group SO(5.1). It may be helpful for the reader to compare with the conventional partial wave expansion, i.e. expansion in unitary representations of the rotation group.

Irreducible representations of SO(5,l) [resp.

SO(D+I,I)] are characterized by

x

= [k,~], where II specifies a representation of the Lorentz group SD(4) [resp

.

^SO(D)]

,

and 6 is a complex number. We will only need completely symmetric traceless tensor representations of the Lorentz-group, their rank will also be denoted by R.

The partial waves of an n-point function G(x l...X ) will be denoted by

G

:

(XIX

^{3...~n) E (5.1)}

x3 Xn

Here a = (al,,.a ) is a Lorentz multi-index, with the a.

same II as in

x

= [ ~ , 6 ] .

They are conformal invariant and transform with respect to x 3...~n in the same way as the Green functions, while they transform with respect to x according to equations (2.9,(2.2) with d replaced by 6, and C the completely symmetric traceless R-th

FN

rank tensor representation of the Lorentz group, ac- ting on indices a .aR.

To write down the conformal partial wave expansion one needs in addition the representation functions, or rather Clebsch -Gordan kernels

They are uniquely determined conformal invariant 2 3-point functions similar to expression (4.2), again a= (a1.. .aR), and -X

-

^{,D-61 Q[}

^.

5.

-

CONFORMAL PARTIAL WAVE EXPANSION.

-

We will now The expansion is then

turn our attention to the group theoretical approach.

pi ⁼

^{f d x}

&

⁼ {dx~dxri?xlx2

1

^XI X

Consider Green functions for Euclidian arguments x.

-

^*-. G ~ ( X ~ X ~ . . . X ~ ) (5.3) There exists a manifestly conformal covariant forma-

n-

2

-

^{n- 2}

lism [3,9] in which Euclidian vectors x are identified

In comparison with ordinary partial wave expansion X with points on a projective "light cone" in 6(resp.

plays the role of angular momentum 11; x,a are "magnetic D+2)dimensional space, and Green functions become quantum numbers" replacing m, and

r-'

are spherical homogeneous functions on that cone. In this formu-

functions like x Integration is over the principal 11m '

lation conformal symmetry takes the simple form

series of unitary representations

CONFORMAL INVARIANT QUANTUM FIELD THEORY C1-103

with integration running from &D-im to 4 ~ + i m , and the

~lancherel weight c(x) is some polynomial in 6 for even D.

A special case of interest is the expansion of the 4 point function G(x .x4). In this case the partial wave amplitude is a conformal invariant 3- point function and therefore its x-dependence is completely fixed

All the dynamical information is thus in the factor g(x) (analog to partial wave amplitudes a ) . Our

e

graphical notation uses the group theoretical fact that

l"

can be obtained from

r

-X by amputation with a suitable propagator.

The expansion for 4-point function and Bethe-Salpeter kernel can then be written as [12] (5.6)

, . ,

.

with partial wave amplitudes g(x) resp. b(x) depending only on X. The li-prescription is necessary because the last term on RHS of equation (1.3) is too singular to be expandable in representations of the principal series. There is however also an expansion for the full amplitude, it has the same g(x) and differs only in the path of the X-integration. This follows from the dynamical integral equations.

Lastly there is a Plancherel-theorem which expresses completeness and orthogonality of the expansion functions. It reads

where 6(x,xt) is essentially a Kronecker-6 in 9, and a &-function in dimensions 6.

6. - SOLUTION OF INTEGRAL EQUATIONS.

-

If the partial wave expansion for all the n z 4 point Green functions and kernels is inserted into the integral equations

(Fig.11, it turns out that they are thereby diagona- lized due to the Plancherel theorem (5.7), and thus reduce to algebraic (as opposed to integral) equations.

These algebraic relations amount to simple factori- zation properties. Let us for a moment leave the

SD-equationrfor the propagator out of consideration, and consider all the others. The result is then that the integrands of (5.3).

have poles in 6 at

x=x0

= [Il.d] (6.1) for all n 2 2, with d = dimension of the field$.

Moreover the residue must factorize

residue = (g2bo)-l

H )

^(6.2)

i.e. is expressible in terms of the ordinary Green functions. The constant bil= res g(x) at

x

=

x0.

Thus the integral equations amount to simple FACTORIZATION PROPERTIES of the partial waves.

Finally, the propagator's SD-equation requires that for n = 2 the LHS of (6.1) has a pole in 6 also at

for D space time dimensions. The Bethe-Salpeter equations suggest then that such a pole should be expected for all n, barring cancellations. As we shall see, the pole at

x

=

x2

is related to the existence of a stress tensor with dimension D , and the pole at

x

=

x0

to the existence of the fundamental scalar field $(x) with dimension d.

Let us show for an example how these results obtain.

Consider e.g. the SD-equation for the 3-point vertex.

Because conformal invariant 3-point functions are unique

for

x

= [0,d]

4 - 4 5

O

The proportionality factor g will be called the coup- ling constant. Inserting this and the partial wave expansion (5.6b) of the BS-kernel gives

The Plancherel theorem (5.7) cannot be applied directly, since

xo

is not in the principal series, but it can be used after a process of analytic continuation. As a result equation (6.4) simplifies t 0

S(xo) = 1

From the Bethe Salpeter equation defining the BS-kernel one finds in a similar way [12] that g(~) = b(x) [1-b(x)]-'. Thus the partial wave ampli- tude for the 4-point function has a pole at

x

=

x

-

see equation (5.5) and (5.6a) to compare with (6.1).

7.

-

TENSOR FIELDS. - In general the partial wave amplitudes are expected to have poles not only at

x

=

x0

and

x

=

x

₂ but at many other points

x

=

xa.

^In

fact the poles come in pairs at

xa

and -X due to some symmetry property of partial wave amplitudes.

The physical significance of these poles is as follows:

To every pole at some x a = L$a,6a] with 6 _> 4D, there corresponds a symmetr'ic traceless tensor field

oa

al.. .alla (x) of rank La and with dimension 6

.

^Their

Green functions are given by

This definition is justified by observing that it will satisfy the standard integral equations

I&\

etc. (7.2)

The physical significance of the poles of the partial waves suggest the natural POSTULATE:

g(x) is a meromorphic function of

x

(i.e. i n 6 for all L) and so are the partial wave amplitudes of all higher n-point functions.

This analyticity property cannot be derived from the integral equation alone but is a necessary require- ment for the validity of operator produced expansions B la Wilson [15] and Ferrara et al. [13,141. Similar analyticity properties hold in each order of pertur- bation theory as a consequence of Weinberg's power counting theorem.

8.

-

OPERATOR PRODUCT EWANSION.

-

From the preceding discussion it is now clear how one should proceed to derive operator product expansions. Essentially one will do this by shifting the path of the

&-integration in (5.3) to the right, picking up the contri1)ution from any pole in

x

in the partial wave amplitudes which the path crosses. Because of (7.1) their residues are related to matrix elements of tensor field operators.

Before any path shifting can be done, one must however go over to representation functions of the second kind which have good asymptotic properties as x1 + x3. This is well known in Regge theory where one splits Legendre functions P, into Q, and QQ-k-l.

Here an analogous split is possible. It is convenient to carry out a Fourier transform with respect to the last argument x in (5.2). One can write

where we have dropped some factors. We will use a

Using symmetry of partial waves in

x

-+ -X one can rewrite the partial wave expansion as

Now we are ready to shift the path of the 6-integration to the right. This will produce an asymptotic

expansion because the QX have good asymptotic properties.

Contributions come from all poles of the integrand.

Using definition (7 .l) of matrix elements of tensor fields, the result reproduces the conformal invariant operator product expansions of Ferrara et al. Rewritten in terms of operators in Minkowski space they are

If qx is expanded in powers of iV one recovers the usual Wilson expansions 1151, with coefficients that respect scale invariance as they should.

We:have oversimplified matters in some points.

The most important qualification is as follows.

Among the poles which the path crosses is one at -xo, i.e.!d = 0 , 6 = D-d. Its contribution looks like coming from a field 0 shadow (x) with dimension D-d, its ²

D-3d

coefficient scales with x

.

However it appears that this is not a physical local field, at least in summed up perturbation theory it does not exist. Thus this unwanted contribution must be cancelled when the Born term (last term on RHS of (1.3)) is added to (8.3). It turns out that such a cancellation can be achieved for all the n-pcint amplitudes precisely because of the factorization property (6.2) which, as we saw, is equivalent to the validity of all but one of the dynamical integral equation. Cancellation occurs if the constant of proportionality g b takes the special value 2, viz 2

tg2 = res x=x0 g(x)

It is mosc interesting that the argument can be turned

CONFORMAL INVARIANT QUANTUM FIELD THEORY C1-105 around: If the Green functions satisfy operator

product expansions without a shadow contribution of D-3d

the above type (i.e. scaling like x ) then this guarantees factorization property (6.2) and therefore validity of all but one of the integral equations.

The remaining SD-equation for the propagator holds also if in the operator product expansion there is a term which sc.ales as the contribution from the stress energy tensor should.

In conclusion one may say (simplifying slightly) that operator product expansions are the solution of the dynamical integral equations, figure 1.

9.

-

CROSSING SYMMETRY AND DUALITY. - The problem of solving the infinite set of coupled non linear integral equations, figure 1, has been reduced to the much simpler algebraic one of finding meromorphic partial wave amplitudes which satisfy factorization

(6.2) and (6.3). However there is a price to pay.

Not every solution will be acceptable, we must remember that the Green functions G(x l...x ) must be symetric in their arguments xl...xn. We shall call this crossing synunetry. Note that in a local theory the Green functions for different ordering of arguments can be obtained from each others by analytic continua- tion in the differences x.-x.. Thus crossing synunetry

1 J

is a non trivial requirement related to locality [16!.

Consider for instance the 4-point function. We can expand in 3 different ways

.

^{. (9.1)}

= X X

I",

^{= ffdxg(x'}

^>;j

^=.

^{. .}

with the same g(x) because of crossing symmetry. We have made use of the remark after equation (5.6) to write the expansion for the full connected Green function.

Now the integrand in (9.1) can itself be expanded in the crossed channel

with some crossing kernel C(x,xl) that can be explicitly determined from group theory. Inserting this in (9.1) we see that crossing symmetry holds if g(x) satisfies the homogeneous linear integral equation

also imposes restrictions. They are again non-coupled, linear integral equations, (one for each n), as opposed to the system of coupled non-linear integral equations.

from which we started. The problem of constructing a theory has thus been reformulated. It requires that one finds meromorphic (in X) partial wave amplitude which satisfy factorization (6.2) and (6.3) and in addition solve the linear integral equation (9.2) and higher ones. One hope which one may have is that this new formulation helps in curing the convergence problems of skeleton graph expansions. Another one is that the techniques used in dual resonance models may prove helpful.

The requirement of crossing symmetry can be sharpened to a kind of duality requirement as follows. Let us make the very strong assumption that the path of the 6-integration in equation (8.3) can be closed to the right; this means that operator produced expansions are convergent and not just asymptotic ones. Then the crossing relation for the 4-point function becomes

# =

^{poies X in}

^x

^{res g(x)}

R

^{poles res g(~)}

^w

^=.

^.

^.etc

This duality type relation can o f course be obtained directly from operator product expansion, assuming they

converge, and has been known to several authors (Ferrara et al, Polyakov). It is not known whether the just mentioned convergence assumption is consistent with axiomatic positivity.

10. - CONCLUSION.

-

Attacked by new non-perturbation techniques, (conformal) field theory reveals new faces.

i) Emphasis on operator algebras. Roughly speaking operator product expansions are

the

solution of the dynamical integral equations of Lagrangean field theory.

ii) Bootstrap aspect reminiscent of the old bootstrap idea that particles are bound together by exchange of the same particles in the crossed channel. Here the fundamental objects are of course fields, not particles, cf. e.g. equation (9.3).

iii) There are remarkable similarities in mathematical structure to dual resonance models

-

like emphasis on meromorphy, factorization and crossing symmetry.

Crossing synrmetry for all the higher n-point functions

References [I] SYMANZIK (K.), Lectures in High Energy Physics,

ed. Jaksic, Zagreb: 1961, New York: Plenum press 1965.

/2] MACK (G.) and SYMANZIK (K.), Commun. Math. Phys.

27 (1972) 247.

[3] MACK (G.) and S A M (Abdus), Ann. Phys. (N.Y.) 53 (1969) 174 and references therein.

4 KASTRUP (H.A.), Nucl. Phys. 58 (1964) 561.

5 GELL-MANN (M.) and LOW (F.E.), Phys. Rev. 95 (1954)

11

1300.

WILSON (K.), Phys. Rev. D3 (1971) 1818.

[6] CALLAN (C.G.) Phys. Rev. D2 (1970) 1451.

SYMANZIK (K.), Comun. Math. Phys. 23 (1971) 49.

I~'~POLYAKOV

^(A.M.), JETP Lett. 12 (1970) 381.

7 MIGDAL (A.A.), Phys. Lett. 37B (1971) 98, 386.

8 PARISI (G.) and PELITI (L.), Lett. Nuovo Cimento

t i

2 (1971) 627.

191 MACK (G.) and TODOROV (I.), Phys. Rev. (to be published).

[lo] MACK ( C . ) , Lecture notes in physics 17, W. Riihl and A. Vancura eds., Springer Verlag, Heidelberg 1973.

[ll] MACK (G.), Group theoretical approach to con- formal invariant quantum field theory, preprint Bern 1973.

[12] MIGDAL (A.A.) and POLYAKOV (A.M,), as cited in: A.A. Migdal, to appear in Nucl. Phys.

[I31 FERRARA (S.)

,

GATT0 (R.)

.

^GRILLO ( A . F . ) and PARISI (G.), Nucl. Phys. B49 (1972) 77;

Lett. Nuovo Cimento 4 (1972) 115.

1141 FERRARA (S.), GATT0 (R.) and GRILLO (A.F.)

,

Lett. Nuovo Cimento 2 (1971) 1363;

A12 (1972) 952; Nucl. Phys. B34 (1971) 349.

[15] WILSON (K.), Phys. Rev. 179 (1969) 1499.

BJORKEN (J.D.) and DRELL (S.D.), Relativistic Quantum Fields, McGraw-Hill, New York 1965.

1161 JOST (R.), The general theory of quantized fields, Amer. Math. Soc. Publ. Providence R.I., 1965.

[17 ] WARNER (G.)

,

Harmonic Analysis on Semi-simple Liegroups, I, 11, Springer Verlag, Heildelberg 1972.