RIWS~A D~L NUOVO CIM]~TO VOL. 6, N. 1 Gennaio-Marzo 1976

Supersymmetry (Fermi-Bose of Quantum Field Theory (*).

S . FERRARA (**)

C ~ R ~ - Geneva, Switzerland

(rieevuto il 24 Settembre 1975)

Symmetry): A New Invariance

105 1. Introduction. 106 2. The supersymmetry algebra. 109 3. I~ontrivial mixing of internal symmetries with supersymmetry. 110 4. Particle supermultiplets. 114 5. Superfields and Lagrangian models. 123 6. Spontaneous symmetry breaking and Goldstone fermions. "124 7. Gauge invariance and supersymmetry. 130 8. Asymptotic freedom in non-Abelian gauge theories. 132 9. Supersymmetric extensions of the U 2 Weinberg-Salam and U1 Higgs models. 134 10. Supercurrent and the graviton multiplet. 136 l l .Action principle in superspace.

1 . - I n t r o d u c t i o n .

One of the dreams of theoretical physicists has been for a long time the

search for nontrivial generalizations of space-time symmetries mainly in order

to incorporate, not in an ad hoc way, properties tha t fundamenta l interactions

of particles exhibit in l~ature beyond relativistic invarianee. Two years ago WESS and Z~M~O [1-4], generalizing from dual-model super-

gauge symmetries, succeded to construct an algebraic structure in physical

four-dimensional space-time, now called supersymmetry, which is quite re-

markable in at least two respects:

(') This review article is an extended version of the lectures given at the Copenhagen Symposium en ~ Interactions of Fundamental Particles ~, August 1975. (**) Address after December 1975: Laboratoire de Physique Theorique, CNRS, Ecole Normalo Superieure, Paris, France. On leave of absence from Laboratori Nazionali di Frascati, Frascati, Italia.

105

106 s. F~Pa~AR+~

1) the irreducible representations of this symmetry combine fermions with bosons,

2) it is a truly relativistic spiu-containing symmetry and the strictures of known previously stated no-go theorems [5, 6] are circumvented.

The algebraic structure underlying supersymmetry is not a Lie algebra and this is the main reason why previous attempts to construct nontrivial generalizations of relativistic symmetries in the framework of conventional Lie algebras failed.

Surprisingly enough, later investigations have shown that almost all known renormalizable interactions of local quantum field theory can be arranged in such a way as to manifest this symmetry.

Moreover, this symmetry has been shown not only to be preserved by the quantum corrections but also to pro~dde the less divergent model field theories known up to now.

I t is the aim of this article to give an almost complete review of the main applications of supersymmetry invariance in the physics of particles and fields.

2. - The supersymmetry algebra.

The sypersymmetry algebra, as naturally arises generalizing from dual models [7-10], is obtained by adding to the conformal algebra of space-time two 5Iajorana spinors Q=, S~ and a ehiral charge / / .

The two spinor charges~ to be called restricted and special supersymmetry transformation generators, admit the following commutation relations with the conformal and ehiral charges:

(2.1)

i i s ¢ ' , [Q=,D] = ~ q ~ , [~'~,D] -- 2

[Q~, P~] = o , [ ~ , K , ] = o ,

[Q~, K~,] - - i (y~,S)~ , [S~, P , ] = i(y~,Q)¢, ,

and the following anticommutation relations among themselves:

(2.2)

8UPERSYMMETRY (FERMI-B08E SYMMETRY): A NEW INVA_RIANCE :ETC. 1 0 7

I t is for instance in this general form tha t the supersymmetry algebra in four dimensions was originally discovered b y W~ss and Z v m ~ o [2].

We note tha t we are using a Majorana representat ion for the ~-matrices,

i.e. they are real and satisfy the properties

(2.~)

and

(2.4)

for a Mujorana spinor. I f we introduce parameters £ s which are total ly anti-

commut ing spinors ( they can be regarded as odd elements of a Grassmann algebra)

( 2 . 5 ) a i ~ j : - - (~ j~i , g, j = 1 , . .*, 4 ,

then the an t icommuta t ion relations (2.2) become commuta t ion relations among the infinitesimal action ~ = ~Q of an element of the Mgebra. Moreover we observe tha t a general supersymmetry t ransformat ion is an eight (antieom- mut ing)-parameter t ransformat ion because we must associate a Majorana spinet :¢0 to the charge Q~ and an independent Majorana spinor al to the charge S~. They can embedded in a linear x-dependent spinor

(2.6) a(x) : O~ 0 - ~ ~5#X/~O~ 1

which is in fact the most general solution of the differential equat ion

(2.7) ( ~ + y ~ a _ lg~,7. ~)a(x) : O,

which ensures tha t the product of two such t ransformations

(2.8) ~,(x) : 2i~(x)7~,o~(x)

is indeed a conformal t ransformat ion of the space-time point x, i.e. it satisfies the differential equation

(2.9) ~ a ~ ( x ) + ~ v ~ , ( x ) - ½g~ ,v~ .~ (x ) : O.

We know tha t the general solution of (2.9) is

(2.10) ~,(x) : e~ + o.)~x~ + ex , + a~x 2 - - 2 x , a . x (w~v : - - co~,a) ;

this means tha t the parameters involved in (2.10) c~n be uniquely expressed, because of (2.8), in terms of the ant icommuting parameters a °, x~, ~o, x~ as

108 s. r~Rm4.~A

the following bihnear expressions:

(2.zi)

¢~i - 0 0

a~ = 2 i ~ 7 ~ ,

the remaining independent bilinear combination of the ~'s being the par- ameter of the chiral t ransformat ion

(2n2) 4~ ( ~ y~ z+. - - ~] ' - I 0 - I 0 s¢2 9~5 sol )

We remark tha t the above algebraic system given by eqs. (2.1), (2.2) can be wri t ten in a much more elegant form if we embed the two spinor charges in a eight-dimensional spinor Z~ which tr~msforms according to the Dirac repre- sentat ion of the spinor group S U~.2 (locally isomorphic to 04.s) and the cor- responding 15 conformal generators in a 6 x 6 skew-symmetric mat r ix J~b which corresponds to the generators of the 04.s (conformal) algebra.

Then we get [11]

(2.13)

where

[Xa, JaB] = iY, tBa b Xb '

[ X ~ , H ] = - - iiyVa~'Zt,,

{zo, L} ~- 2 ( ~ j , + _or,o~H/,

]

is the chiral t ransformation matrix. We observe tha t the previous algebraic system, which is an example of

graded Lie algebra [12-21], could be useful, with regard to possible applica- tions to physics, only for massless systems. This is of course due to the fact t ha t such a closed algebra contains the eonformal charges. However there is a nontr ivial subalgebra [22-26] of the previous s t ructure generated by the Poincar6 generators M~,,, P~ and the spiuor charge Q~

(2.14) {Q=, ~ } = - 2P , , r ;~ ,

(2.15) [Q=, P , ] = O, [O~, M,~] = i(a~,,.Q)~,

whose elements commute with the square mass operator ~ P ~ and which therefore can very well be a symmet ry of a massive system.

SUPERSY~IM]~TI~Y (F]~RMI-BOSE SYMMETRY): A NEW INVA/~IANCE ETC. 109

We will confine ourselves, in the sequel, to the smaller graded algebra defined by eqs. (2.14), (2.15) in order to avoid the usual problems connected with mass- less particles.

Finally we note that the product of two transformations generated by the Q~ give, because of eq. (2.14),

(2.16) [01, (~=] = [51Q, ~ . Q ] = - 2 ~ x y . a ~ - P . ,

a four-dimensional translation with parameter a ~ - 25~7~a2 which is a nil- potent quantity, i.e. a~ = 0 for p <8 due to the aaticommuting properties of the ~'s. As a consequence any translated quant i ty / (x + a) will depend only on a finite number of derivatives at the point x.

We shall see in the sequel that this is the fundamental property which allows one to construct representations of this relativistic symmetry with a finite number of particle fields.

3. - Nontrivial mixing of internal symmetries with supersymmetry.

The supersymmetry algebra given by eqs. (2.14), (2.15) can be generalized in a nontrivial way, assuming that the spinor charge belong to some repre- sentation of an internal-symmetry group G. This has been achieved by S~d~d~, STRAT~DEE [27-29] and WESS and Z ~ N O [30].

Imagine the previous algebra written in two-component Weyl formalism

(3.1) {Q=, q.} = o,

¢*p /z ,

{Qa, Q~} = o ,

[Q=, ~,] = o.

Then if the Q~ transform now according to some representation of G and the O~ according to the complex conjugate representation, the algebra (3.1) will become

(3.2)

= (Q~, Q~} = 0 ,

f L ~Q=, 0N = 20~o;~P.

[Q~, P,] = 0 ,

q'/Im k [B~, B, .] " ~ B ,

where S~ ~ are the Itermitian matrices of the representation to which the Q~ belong and B~ are the generators of the (compact) internal-symmetry group.

Of course we have written only the relevant commutators and antieom- mutators, the others being obvious.

110 s. F~RRARA

Actual ly the algebraic system given by (3.2) can be fur ther generalized in the sense tha t the first an t ieommuta tor does not need to vanish. Namely it can have the form

(3.3) {Q2, Q p = = ,

where Z z~ are operators which commute with any element of the graded Lie algebra

(3.4) [Z L'r, G] ~ 0 for G = Z ~t, Q~, -P~, M~, B~ .

I t has been shown by ]~AAG and collaborators [31] tha t the above graded Lie algebra is indeed the most general (nontrivial) symmet ry of the S-matr ix for massive systems. The only fur ther possible generalization consists in adding an internal symmet ry G' which comnmtes with all elements of the gTaded Lie algebra..

Note tha t the only freedom one has is the group G and the representat ion for the spinor charges.

Surprisingly enough ttAAG and collaborators [31] have fur ther shown that , if the internal symmet ry is combined with the enlarged algebra (2.1), which holds for massless particles (note tha t this is t rue modulo infra-red problems !), then the group G itself is fixed to be U~ and the spinor charges are in the fun- damenta l (2V-dimensional) representatioi1 of the group. The only remaining freedom is, in this case, just the number ~" of spinor charges one has and a complete fusion among space-tinle symmetries and internal symmetries is obtained. We observe tha t only in this ease all boson charges, namely the in te rna l -symmetry generators and the conformal charges, can be wri t ten as well-defined bilinear expressions of the Fermi charges, so this is probably a possible reason why in the zero-mass case the algebra is so stringent.

Unfor tuna t ly the more general algebra we are talking about is not useful in order to construct field theory models because it always leads to non- renormalizable interactions. Also from a phenomenological point of view, the algebra considered in this section turns out to be not very interesting because the spectrum of states is ra ther unphysical (apart f rom the problem of mass degeneracy) for any choise of the group G and of its representations. We will discuss these aspects in the next section.

4. - P a r t i c l e s u p e r m u l t i p l e t s .

According to SALAmi and STRATtlTI)EE [27, 28] the representations of the supersymmetry algebra can be studied in a ve ry simple way using the Wigner me thod of induced representat ions [27].

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC. III

Consider for instance the previous algebraic system of commutation relations

(4.1) { [ Q 2 , L%] = ~ =

= o ",vail # ,

where Q~ belong to some representation of an internal-symmetry compact group G. The construction of unitary representations of (4.1) begins with the observation that the Q's charges leave invariant the m~nifold of states with given momentum P~.

On this manifold the anticommutators of the Q's become a fixed set of numbers and in fact these commutators generate a Clifford algebra. We con- sider first the simplest case where there is no internal symmetry at all, and we restrict ourselves first to the ease of nonvanishing mass. The fom'-momentum, being timelike, can always be assumed of the form P~, = (M, 0) . The little algebra of the supersymmetry algebra is therefore generated by the Q's and by the angular momentum J, the Oa-group being the little group of a timelike vector. In terms of ~Teyl spinors the anticommutation relations in eq. (4.1) become

(4.2) {Q~, Qo} : (Q~, Q~) : o , (Q~,, Q~) : 5~ ,

having chosen a suitable normalization. The Q's satisfy the algebra of creation and destruction operators and can

be used in a familiar way to build a 4-dimensional Foek space with positive metric. Indeed one can start with a particle state IJ, J3) to be considered as the Clifford vacuum, i . e .

(4.3) ~)~lJ, J3) ---- 0 ,

and build up the states

(4.4)

where nl, n2 take values (0, 0), (0, 1), (1, 0), (1, 1). These states span a 4(2J + 1)-dimensional irreducible representation of the

little algebra. The spin-parity content of such representation is ( J - ½)n, J% J -% (J + ½)-n, where V---- -p i, 4- 1 for J integer or half-integer. Such an irre- ducible representation is therefore, when reduced to the Poiacard group, the direct sum of four inequivalent representations

(4.5) JM, J - - ½, 7) ® ]M, J, iv) ® ]M, J, -- iV) ® 1M, J, -- V>,

which are mixed together by the spinor charges Q's.

112 s. F~RA~A

We consider now some examples. The simplest representat ions are those with J = 0, ~, 1, ~. Their particle contents is respectively a scalar, a pseudo- scalar and a spin ½; a pseudoscalar (scalar), a vector (pseudovector) and two spfi~ ½; a vector, a pseudovector, a spin ½ and a spin ~; a pseudovector (vector), two spin ~ ~nd ~ tensor (pseudotensor).

In the case of zero mass it can be shown tha t there are only four physical states spanned by the Q's ( J ¢ 0) of helicities ± 2, ± (2 + ½). The second and the four th of the previously described multiplets contain massless particles of helicities respectively (½, 1) and (~, 2), and can be therefore regarded as the multiplets which contain the photon and the gravi ton in a supersymmetr ic world. The reason why, for zero mass, there are only two helicity states, is very simple : if P~ is lightlike, then it can be assumed of the f o r m / ~ = (1, 0, 0, 1). The little ,~lgebra becomes

(4.6) {Q~, Q~} = 1 , (Q_~, Q2} = {Q~, ~)~} = (Q1, Q2} = o ,

the physical states are [2) and Q~I2), while Q212>, QxQ~I).) are zero-norm states. We now consider particle supermultiplets classified according to the super-

symmet ry algebra nontr ivial ly mixed with internal symmetries. As before, the rest f rame an t ieommuta t ion relations are

(4.7) { { Q 2 , = { Q 2 , = o ,

Z - - M

I f the Q~ belong to a real n-dimensional representat ion of the in te rna l -symmetry group G, then they will give 22" independent new states when applied to a given state which acts as a ground state. In the case where the representat ion is complex they will erect 24" independent states. The dimension of the rep- resentat ion of the lit t le algebra will be 22~ times the dimension of the ground state. We are going now to consider some examples. Take the Q~ to be in the fundamenta l rappresenta t ion of SU~, then n = 2 and the algebra will give 16 new states when applied to a Clifford vacuum IJ, J3; I , / 3 ) defined by the equation

(4.8) ~)~[J, J3; I , I3) = O.

Consider the lowest representat ion, namely IJ, J3; I, I3) = IO, 0). The (J, I) ~ content of the states of the little algebra will be

(4.9) (0, 0)+@ (1, ½)~@ (1, 0 ) -@ (0, 1) -@ (½, ½)-i@ (0, O) + .

The s t ructure of any irreducible representat ion will be given by vector multi- plieation of such representat ion with an 03 @SU2 mult iplet ( J , I ) w h i c h

SUPERSYMMF, TRY (FF~RMI-BOSE S Y M M E T R Y ) : A NF.W INVARIANCF, ETC. 113

acts as a ground state. I t will be noticed that the rest frame states can be classified into SU4 multiplets, namely as

(4.10) 1 6 = 1 O d G 6 ~ 4 G x .

This is because the rest frame algebra contains the SU4 generators given by the matrices

(4.11) 1 L -- •

This is true in general, in the sense that, if we start with n spinor charges Q~ ( Z = 1, ..., n), the 2 ~- (2 4") states can be uniquely decomposed into a direct sum of irreducible representations of the SU2. (SUd,,) algebra, the spin running from 0 up to n/2 (n).

As another example consider the Q~ transforming according to the vector representation of 0a. In this case 22"=64 and the states can be classified according to SUe representations

(4.12) 6 4 = l @ 6 G X 5 @ 2 O O 1 5 e ~ O 1 .

The spin-isospin-paxity contents (J, I) ~" of these representations is

(4.13) (0, 0)+Q (½, 1)i® (0, 0)-O (1, 1)-O (0, 2)-@

® (3, o)-,® (½, 2)-,® (½, 2)-,@ (o, o)+®

® (1, 2)+® (o, 2)+® (½, 1),® (o, o)-.

As a final example we consider the case where the charges are in the quark rep- resentation, i.e. they are SUn triplets. In this case the lowest representation is of dimension 212= 4096 and these states can be classified according to ~U= as

(~.14) 4096 = 1 O 12 • 66 O 220 O 495 Q 792 G

O 924 O 792' O 455 O 220 Q 6-6 Q 1-2 O 1 .

The maximum spin is J----3 (singlet). Of course all the unitary representations of the supersymmetry algebra

axe obtained by vector multiplication of such lowest representations (where the ground state is a scalar singlet) with an arbitrary representation of the internal-symmetry group and with an arbitrary spin carried by the ground

8 - R i v i s t a d e l N u o v o Oimen to .

114

state.

(4.15)

The dimension of a given repiesentat ion will be

D = 2 : ~ × ( 2 J ÷ 1 ) × d ,

S, PER2AI~A

J being the spin and d the dimension of the representat ion which defines the ground state.

5. - Superfields and Lagrangian models .

The first no , t r iv i a l representat ion of the supersymmetry algebra given in the l i terature was obtained by WEss and Zv~g~o [2] in the following form:

~A(x) = @ ( x ) ,

~B(x) = ~r~ ~f(x) ,

(5.1) ~ ( x ) = ~ ( A ( x ) - - r o B ( x ) ) ~ + (~ (x ) + r ~ ( x ) ) ~ ,

~F(x) = i ~ . ~ , ~ , ( x ) ,

~V(x) = i~ror.~, ,w(x) ,

where A and E are scalars and B and G are pseudoscalars, ~p being a Majorana spinor. :Note that , on the mass shell, the following relations hold:

[ ( [~ - - m ~ ) A = ( D - - m 2 ) B = O, (5.2) / ( iy '~ A- m)~v = O,

which means tha t the representat ion is reducible because

(5.3) ~(mA(x) + F(x)) = ~ ( m B ( x ) + G(x)) = O,

and therefore one can take

(5.4) E(x) = -- m A ( x ) , G(x) -~ - - roB(x) .

This shows tha t the mult iplet of fields given in (5.1) describes the off-shell particle field mult iplet which corresponds to the J----0 representat ion accord- ing to the nomenclature given in the previous Section.

In order to construct field representat ions of supersymmetry , following SALA~r and STI~ATHDEE [23], one considers the action of a group element

SUPERSYlgMETEY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC. 115

exp [--5Q] over the group manifold

(5.5) exp [-- i x . t )] exp [-- t~Q] -= exp [ - i x .P - OQ].

From the ant icommutat ion relations of the Q's one gets

exp [ - ~Q] exp [ - i x .P - 0Q] --= exp [ - i(x + i~yO)._P - (0 A- ~)Q],

or in infinitesimal form, for the (( superfield ~) ¢(x, 0) = exp [-- ixP -- 0Q]¢(0~ 0),

~¢(x, O)= ( - - ~ + i, ,O~,) ¢(x, O) ,

~¢(x, O) = i~,¢(x, o), (5.7)

~¢(x, O)= - i ( x , , ~ - x , ~ + O~,,~) ¢(x, O) ,

respectively for translations and Lorentz transformations. According to FER~A]~A, WESS, ZU~ClZ~O [24] it is convenient to use Weyl

spinors 0~, 0~ related to the Dirac spinor 0~ by the relations

(5.8) 0~ = 1 (1 - - i ~ ) 0 , ~ = ½ ( 1 + i~o)0.

Then it follows tha t the superfield can also be written as

(5.9) ¢(x, 0, 0) = exp [-- ixP ~- iOQ A- i0Q]¢(0, o, 0).

The main point is tha t , due to the ant ieommuting properties of the 0's, ¢(x, 0, 0) is a finite power series, namely

(5.1o) ¢(x, 0, ~) = ¢ + oz + ~ + OOM + ~1~i +

+ Oa~ Or, + 000~ + 600], -4- O000D,

where, without loss of generality, we can assume ~ to be real. Because of the t ransformation law (5.6) one easily realizes tha t the individual

components transform into

I C -+ g , Z -+ 8A, M, M*, v~, M, M* -> ~Z, ~ , (5.1]) / ~t ---> ~M, 8M*, 8v~, D , D --+ 82,

(5.6)

analogously

116 s . FERRA-RA

under supersymmetry transformations. In terms of component fields one can

write the infinitesimM tr 'msformat ion as follows:

(5.12)

C

7.

V t

t

0 i~y 5 0 0 0 0 0

0 i57 - ~ 0 0 () i5 0

0 i~757 • ~ 0 0 0 i~y~, 0

o i~l~ 0 o 0 i~Tt~ 0

0 t} 0 0 0 i~y" ~ 0

6 t: 5

So one sees that~ as expected, the superfield notat ion is nothing but an eco-

nomic way to deal with matr ix multiplication [32].

I n order to fur ther develop the tensor calculus of supersymmetry we ob-

serve that the group manifold can be parametrized in three different bu t

equivalent ways [24]:

(5.13)

exp [-- i x P 4- iOQ + iOQ] ,

exp [ - i x P + iOQ] exp [i0Q],

exp [ - i x P 4- .iOQ] exp [iOQ] .

The three expressions in (5.13) correspond to three different (equivalent) reM-

izations of supersymmetry representations, i .e. to three different definitions

of superfields which are related by the following identities:

(5.14) ¢(x, 0, 0) ---- ¢~(x T i0(~0, O, 0) = ¢2(X -- i0(~,0, O, O).

We note that, once ~ superfield has been defined as type I or I I , it becomes

intrinsically complex so that the reality condition ca.nnot be longer imposed.

This fact ca.n be better understood from the fact tha t iOcr~O is a pure

imaginary shift on the first argument.

The group action on ¢~ and ¢~ is respectively

(5.15) )

From (5.15) it follows immediat ly tha t c~/~6, ~/~0 are invariant differentia-

tions on type- I or t ype - I I superfields. Therefore for any given superfield ¢

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC. 117

(for example in real basis) there are two invariant differentiations [24, 29]:

(5.16)

We shall call D ~ , / ) ~ = (/)~) the covariant derivatives. They obey the fol- lowing relations:

(5.17) D~,DeD~ = D ~ D ~ D ~ = 0 .

The last line being obvious. Because of the existence of the covariant derivatives a (complex) super-

field can be reduced further imposing on i t the constraints

(5.18) D~¢ = 0 or D~¢ = O.

A superfield satisfying (5.18) is called a chiral field, according to the nomen- clature of SALAlVI and STI~ATHDEE [29]. (It WaS originally called a scalar superfield.)

A general (real) superfield will be called a vector superfield. Strictly speak- ing there are superfields which are nor scalar nor vector, i .e. superfields which satisfy the relation

(5.19) D D ¢ -~ 0 or D D ¢ = O .

They are called linear superfields [24]. However, as they have not been used in any interesting application, we will omit them from our discussion.

Chiral fields are called left handed or right handed if they satisfy the equation

(5.20) D~¢L= 0 or its complex conjugate D~¢~----0.

The reason of this definition is that , when translated in I or I I basis, they do no longer depend on the r ight-handed (left-handed) par t of the Majorana spinor 0.

We note tha t the general form of ¢1~ is [24]

(5.21) ¢1L---- A + 0~o + 00~ ,

118 s. F~ARX

A, % F being complex fields. Their t ransformation law is~ in te rm of real fields,

(5.22) (i 0 0 i~7~ 0 0

\ : 0 i ~ y . ~ O 0

o ~ , ~ o o l \ o /

We have confined, so far, our discussion to superfields without an additional (external) Lorentz index {~}. Of course such an index is completely irrelevant for supersymmetry transformations so it can be added without difficulty to the previous superfields without modifications of the established results.

5Iultiplication among different superfields is also well defined. I t will be an essential ingredient in order to build up supersymmetr ic model field theories. For example one easily verifies the following relations:

¢ ~ = (@)~,

¢ , ~ = (@)~,

where ¢~ = eL, ~ L - - ~ and the subscript V stands for vector N[oreover

DDCv(X , O, 6) = (DD¢)a(x, O, 0).

superfield.

If eL has components A, B, % F, G according to (5.21), then ¢~ = / ) / 9 ¢ r has components respectively given by

2', G, y" ~F, 5 A , [~B .

In terms of component fields the covariant derivatives are nothing b u t finite-dimensional matrices [25, 26]. For example the previous operation is given by the following mat r ix in the component space:

(5.23)

(i 0 o o 1)( A , 0 0 1 o y . ~ = 0 7 .~ 0 0

[ ]A 0 0 0

[BB [] 0 0 0 G/

In order to derive Lagrangian field theories which are invar iant under super- sym m et ry transformations, we remark that , because of (5.6), the last eompo-

SUPF, RSYM~:ETRY (FERMI-BOSE SYMMETRY): -4. NEW II~VARI/k~CE ETC.

nent field of a given multiplet always transforms as a total derivative

119

(5.24) ~¢,.,(x) = ir"~,X(x),

X(x) being some Fermi field. Therefore it follows that

(5.25) fd,x ¢~.~(x)

is invuriant.

The first supersymmetric model field theory which has been studied is the self-interaction of a chh'al multiplet, namely the interaction of the A, B, W, 17 G component fields [22]. In order to derive the corresponding Lagran- gian we will use extensively the superfield techniques. In particular we wiU show that the kinetic term, the mass term and the interaction term are sep- arately invariant under supersymmetry.

Kinetic term: if we start with the chiral multiplet S ( D ~ S --- 0), the kinetic energy is the D-component of the following real (vector) superfield SS:

(5.26) 1 1

which is equal, up to a four-divergence, to the /~-component of the chiral bilinear S D D S .

Mass term: the mass term corresponds to the /7-component of the ehiral multiplet S~:

i + •

Interaction term: the interaction corresponds to the/7-component of the chiral multiplet S ~ :

(5.2s) gS~ = g ( F A 2 - - .FB ~ + 2 G A B - - i~,y~A + i~ys y~B) .

Therefore

(5.29) £(x) = Lk + L~ + L, .

If we use the equations of motion for F and G

(5.30) /7 + m A + g (A ~ - - B ~) = 0 ,

G + m B ~- 2 g A B ---- 0 ,

1 2 0 s. ~ n ~ A ~

the Lagrangian (5.29) takes the more familiar form

(5.3~) 1 1 2 i 1 L(x ) = - - ~ ( ~ A ) 2 - ~ ( ~ , B ) - - ~ ~ " ~ ~- ~ m 2 A 2 ~-

1 i g2 B~) 2 - , + ~ m ~ B ~ - - ~ m ~ - - g m A ( A ~ - B 2) -- ~ ( A ~ - igCfi(A--75.B)~ f

i.e. a combination of cubic, quartic and Yukawa interactions with a well- defined relation between masses and coupling constants.

The l~oether-supersymmetry current is given by the following expres- sion [22~ 33]:

(5.32) J~(x) = ~. ~ ( A - - r ~ B ) r , y , - - (F + m V ) ~ ,

and is conserved as a consequence of the equations of motion. I t has been shown by I~IoPo~rLOS and Z~r~II~O [34] tha t this model is re-

normalizable preserving the symmetry . This means t h a t the infinities can be absorbed in supersymmetr ic counter- terms of the form given b y eq. (5.29).

Moreover, remarkably enough, it turns out tha t there is only one diver- gence, a logarithmic infinity due to a wave function renormalizat ion of the multiplet . We observe, en passant~ t ha t the fields ~ and G are auxil iary fields and do not correspond to any asymptot ic state.

The renormalized mass and coupling constant are given by the following relation [34, 35]:

(5.33) m, ---- Zm~, g, ~ Z~gB,

where Z is the wave functio~l renormallzation. I t has been shown by FERRARA, :[LIOPOULOS and ZLrMINO [36] tha t the

renormalized one-particle irreducible n-point functions F¢,...¢~, where the external lines s tand for any field of the chiral multiplet , satisfy the following Callan-Symunzik equations, at each order of per turbat ion theory :

(5.3~) m~-~m ~ - f l ( g ) ~ g - - n ~ ( g ) /'t,...¢.(p,; mlg)---- 5(g)l"~,¢,...¢.(O, pi; m ,g )

with fl(g), 7(g) and ~(g) expressed in terms of a single function

3 ! ~ ] 1 (5.35) fl(g) = 2 9 1 ~ - ] ' Y(g) - 2 1 ~- ] ' 5(g) - 1 + ] '

](g) being defined as

(5.36) log Z g2

t (g) ~ mB ~m~ - - 4 ~ 2 ~- ' ' ' ;

SUPERSYMM~TRY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC.

note t ha t (5.35) imply the relation [36]

121

(5.37) fl(g) = 3gT(g),

which shows thu t there is not fixed point which can be re~ched in u continuous way f rom the origin.

In order to unders tand the previous result in a simple wuy we c~n use super- symmetr ic per turba t ion theory [37-44].

The previous JJagrungian (and the corresponding uetion) can be wri t ten us a density over u superspace whose point ~re lubelled by x, 0~, Oe,. Spinor integrat ion is a well-defined operation. For instance one makes use of the following fundamenta l p roper ty [45, 46]:

(5.38) f dOi O¢ = 6~ .

I f we use (5.38), the action of the previous system can be wri t ten us

(5.39) f d'x [f SDDS ÷ + gSS ÷ h.c.] ÷ f [f JS-+- h.c.] .

The free propagator is obtained for g = O

1 (5.40) S(1) - - (D, D, J(1) -- mJ (1 ) ) .

16(m 8 - D2)

Using the functional differentiation rule

(5.~)

8s(1) _ a~(1, 2) = ~,(x2 - x~ + i~2~0~) ~ ( 0 1 - 03) 8s(2)

~ ( 1 ) _ (~(1, 2 ) = ~ , ( x 2 - x~- i02r02)~8(~2- 03) ~s(2)

~s(1) ~(~) ~3(2) = ~s(2) = o ,

1 ~(02 - 08) = ¥ ( 0 2 - 03)(01- 08)

one obtains

< S ( I ) S ( 2 ) > - - 3 i J ( 2 ) - -

~S(Z) _ <S(1) S(2)> - - ~ i J ( 2 )

- - i 16(m ~ ~ _ k8 ) exp Lu2~,o2~ - - ~ , . , o '~ j ,

im 16(m 2 • k s) exp [0270~ k] 02 ~01~.

122 s. FERRARA

Consider now the one-loop d iagrams for the self-energy [39]. T h e y are

X'++(1, 2) A++(1, 2)A++(2, 1 ) ,

Z+_(1, 2) = A+_(1, 2)A_+(2, 1 ) ,

which con t r ibu te to Z++, X+_; because A~+ : 0, Z'++ : 0. F o r Z'+_ one gets

f 1 1 X+_(q, 0~0~) ~_ ddk k s .~ m s (k-~- q )S+ m s

• exp [01 70~k - - O,~aOls k ] exp [-- O,~,Os(k -~- q) -~- 0120~01S(~ --~ q)] =

= exp [-- 0170~q + O,saO, sq] ddk kS + m2 (k + q)~ + m s '

i.e. a logar i thmic divergence. This d ivergence can be absorbed in a counter -

t e r m of the t y p e SS, i.e. in a wave func t ion renormal iza t ion . Because 2:++ = 0

no mass coun t e r - t e rm is needed.

We consider now the ve r t ex correc t ion [39].

The only g raphs one can wri te are

V+++(I, 2, 3) : z]++(1, 2) A++(2, 3)z]++(3, I) ,

V++_(I, 2, 3) : A++(1, 2)A+_(2, 3)Z]_+(3, 1),

which con t r ibu te to V+++, V++_. However , because 0120120130,30saO2a= O, V + + + = 0. F o r t he V++_ ve r t ex

we get

f 1 1 1 d*k kS - - m s (k -~- q)S + m 2 (k -~- t) s -]- m s exp [01~02 k] 012012"

• exp [O~y03 k -- 02saOs3 ~:] exp [03y01 k + 031 a031 k] =

f 1 ] 1 : d~k k~ + m~ (k + q)S ÷ m s (k + t) s + m sO~O's'

i.e. a finite result . Therefore there is no independen t coup l ing-cons tan t re- normal iza t ion .

I n the same w a y one can show t h a t all the h igher n -po in t func t ions are finite. Moreover CAePE~ and LEIBBRA~])T [38] have es tabl ished the superficial

degree of d ivergence of a n y g r a p h

where

d = 4 L - - 2 I + 2# ,

tt = min n + - - 1 , n _ - - 1 ,

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC. 123

L, I, n+, n being respectively the number of loops, propagators and vertices of any graph. Using the following topological relations:

one finally gets

I+E,+-~3n+, I + E _ = 3 n _ ,

L = I - - ( n - - 1 ) , n ~ n + + n _ ,

d ~ 2 - - 2 E _ - - 4 N ,

where E~ are number of external + , - lines and N ~ (E+- E_)/3 which has been assumed to be positive (if, on the other hand, £ > E+, then d = 2 -- - - 2 E + - - 4 N where N = (E_--E+)/3). The power-counting formula shows that the only divergent graphs are those with N---O, E_----E+= 1, i.e. the self-energy graphs X+_.

Finally we would like to mention that the super Sd-theory, as shown first by LANG and WESS [47], leads to a nonrenormalizable Lagrangian. This fact can also be understood, in a very simple way, using supergr~phs and the pre- vious power-counting formulae.

6. - Spontaneous symmetry breaking and Goldstone fermions.

The notion of supersymmetry is very elegant, however, as we have seen, it leads to a complete degeneracy of masses inside a supermultiplet. As Nature does not show such multiplets, it is very compelling to look for a spontaneous symmetry breaking of this symmetry.

I t has been shown by S2a.A~, ST~ATHDEE and ILIOPOVLOS and ZU~nNo [48, 34] that, when the vacuum is not a supersymmetrie singlet, then a Goldstone phe- nomenon emerges with the immediate implication that the system must con- tain a massless spin-½ fermion because of the following general argument:

The infinitesimal change of a Dirae field ~v~ under a supersymmetry trans- formation is

(6.1)

where J~,(x) is the vector spinor current. The vacuum expectation value of (6.1) yields the expression

(6.2)

because of the noninvarianee of the vacuum

(6.3) ( s ~ ( o ) } = ~ ( ¢ } ,

124 s. FERRARA

where ¢ is some scalar field related to yJ~ through supersymmetry trans- formations.

Equat ion (6.2) gives

(6.4) ~.<1'* J.~(x) ~,Ao) > = <¢> a,(x) ~ ,

the Fourier t ransform yields

(6.5) f d'x exp [ikx] < T* J,:(x) ~ (0 ) > = M(k2) kz6~ -~ ...,

which because of (6.4) gives finally

(6.6) k~M(k 2) = (¢>,

i.e. a simple pole in M(k 2) at k~= 0 with residue <¢>. This indicates tha t an intermediate s tate which is a massless spin-½ particle contributes to M(k2).

The notion of Goldstone fermion is ve ry exciting since it might give a funda- mental explanat ion of the neutr ino [49, 50].

S ~ A ~ and ST~AT~D~E [51] have fur ther shown tha t in any spontaneously broken theory, where the multiplets described in the previous Section are involved~ i.e. chiral multiplets and vector multiplets, the expression of the Goldstone field in terms of the basic fields is

(6.7) j i k

implying tha t a necessary and sufficient condition for spontaneous break- down is tha t at least one of the auxil iary fields F, G and D admit a nonvanishing expectat ion value. The subscript of these fields refers to a given internal- symmet ry index. T h e / ) , 2 fields are therefore assumed to be matrices because the vector mult iplet can only appear as a gauge mult iplet in a given renormul- izable Lagrangian. In the nex t section we will encounter an explicit Lagran- gian model where the Goldstoue phenomenon occurs.

7. - Gauge invar iance and supersyn lmetry .

The model we have described in the previous sections consists of a scalar, a pseudoscalar and a spin-½ particle in interaction among each other. However Lagrangian fields theories which combine supersymmetry with gauge in- variance have also been constructed [52-55] ~nd shown to be renormalizable in a way consistent with supersymmetry and g~uge invariance [42, 56-58, 59-62].

SUPERSYMMETRY (F]~RMI-BOS]~ SYMMETRY)-" A N~W INVARIAIgCE ]~TC. 125

The gauge superfield is a real vector muir±pier V(x, O, 0). To define gauge transformations in a way which is consistent with supersymmetry we must en- large [52] the gauge funct ion to an entire ehir~l muir±pier A(x, O, O) (D~A ---- O) so tha t

(7.1) ~V(x, O, O) ---- i ( A - A)

under ~ gauge transformation. In terms of component fields, (7.1) reads

(7.2)

8C ----B,

~M----F,

~£V = G ,

8v~ = ~,A ,

8/) = 0 .

So one realizes tha t C, Z, M, N are not physical degrees of f reedom (they can be in fact gauged away), while ~, D are gauge-invariant quantities.

The supersymmetr ie extension of the electromagnetic s trength corresponds to the following ehiral field:

(7.3) W~ = DDD, V ,

which contains the component fields 2, D,F~,,=~,v~--~v~,, y.~2, and in fact 8 W ~ = 0 under (7.1).

The free supersymmetr ie Lagrangian [2] is given by the E-component of the chiral muir±pier W~W¢,

(7.4) w ~ w ~ = _!F~4 ~ - ~ -~y ~ + ~D~. ~

The gauge superficial can be pu t in interact ion with a complex chiral muir±pier S = $1 ÷ iS2, then providing the supersymmetr ic extension of QED. If one defines S=~----SI=L iS2 (D~S±= 0), the gauge t ransformat ion on S~ is

(7.5) ~S± = T ±gAS±,

and the corresponding supersymmetr ic minimal coupling is [52]

(7.6) ½ (&3+ exp [e V] ÷ & 3_ exp [ - e V]) ;

1 2 6

of course also the mass t e r m

(7.7) 2mS+S_

is bo th gauge invar iant and supersymmetr ic . The supersymmetr ic Lagrangian is just given b y the following expression:

(7.8) (W ~ W~)p ~- m(S+ S_)~. + ¼ (S+S+ exp [gV] + S S exp [-- gV])v.

The quite unconvent ional fo rm of the interact ion t e r m is merely due to the presence of the unphysica l field C(x) (of zero dimension), which is however essential in order to write the covar iant der ivant ive in a supersymmet r ic way. The impor tan t point is tha t , using the gauge freedom, one can fix C = Z = = M = N = 0 (Wess, Zumino gauges)[52]. In these gauges the L a ~ a n g i a n

given by (7.8) reduces to the famil iar form

1 ~ 1 ] _

(7.9) L = --J) (o~'A1)~--2 (8~A~)~--'2 1(~'B2)2--2

i i ] ~) ~)1~" ~)1--~) ~2~ )" $~v2--:~m~(A~ + A[ 4- B~ + B~) --

-~m(~o~ + ~ ) - F'. , ~,.~;.-

*-* B ~

1 ~ ~ A ~ ~ 2 1 _~) g v~'(AI + -+ BI + B~)--2 g2(A1B~--A~BJ~'

where we have used the equations of mot ion of the auxi l iary fields

I Ei~-mA~=O, G~+mB~=O~ (7.10)

[ D + g(A~B:-- A~BJ = 0.

The Abelian gauge model we have described so far provides the simplest em~mple of spontaneous break-down of s u p e r s y m m e t r y if we add to the ]ore- vious Lagrangian the t e rm [63]

g g '

which is supersymmetr ic and gauge invariant .

S. FER.RARA

SUPERSYMMETR~I " (FERMI-BOSE SYMMETRY): A NEW INVA.I~IAI~CE ~TC. 127

Upon el imination of the D-componen t field this t e r m adds to the previous Lagrangian the quan t i t y --~(A~B~--A~B~). The mass ma t r ix can be diag- onalized introducing the new fields

1 1 a~ = - - ~ (A~ -- B~), a~ = ~ (.B~ + A~),

V z

1

V ~

1

and the result ing potent ia l is

1 1 g~ ~ (7.12) ~ (m ~ - ~)(a~ ~ a~) @ ~ (m 2 + ~)(b~ @ b~) @ ~- (al + a~ - - ba 2 - - b~) 2 .

A t this poin t one sees a l ready t h a t s u p e r s y m m e t r y is spontaneously broken,

since the masses of the fields in the same mul t ip le t are no longer equal. Fo r l ~ I < m ~ the fields a~, b~ have a vanishing expecta t ion value, while

< D > - - , 3 2 = . ~ , 5 , + . . . g g

and therefore 2 is a Goldstone spinor. For ]~I > m~ one of the quadrat ic t e rms has a negat ive coefficient. Now

gauge invar iance is also spontaneously broken and the vec tor field vu acquires = 2(~ - - m 2) through the usual Higgs mechanism. I t turns out t h a t a m a s s m v

the Goldstone fermion is now a linear combinat ion of 2, 75YJ1 and ~2. SALAM~ STRATHDEE [54]7 ~FERRARA a n d ZUMIN0 [53] succeeded to const ruct

non-Abelian supersymmet r ic gauge theories. I n this case the gauge t r ans fo rmat ion for the gauge superficial looks more

complicated. F o r a finite local gauge t rans format ion one has

(7.13) exp [gV] ---> exp [-- igA +] exp [gV] exp [igA] ,

where ma t r i x mult ipl icat ion is understood. Note, in fact , t h a t V = 2 ~ V ~, A ---- 2"A ~, 2 a being the group generators in

the vec tor representat ion. The finite gauge t rans format ion (7.13) corresponds to a compl ica ted (nonlinear) infinitesimal t rans format ion for the gauge field

( 7 . 1 4 ) 3V. = A~Cb~(V) ÷ A~Ob~(V) ,

where C,~(V)-~ i C ~ ( V ) + C~(V) is an infinite power series in V~. However in the Wess~ Zumino gauges

V~ Vb Vo = 0 ,

128

so that

(7.~5) eL = ~ , c~o = - ho°v°,

and one recovers the usual Yang-Mills transformations for the fields

v J , ,~", D ~ .

The pure supersymmetric Yang-Mills I~agrangian E-component oi the following chiral multiplet:

(7.16)

with

(7.17)

S. FERRARA

is provided by the

~ ~ = "~, ;t + ig[vz, 1] ,

and D " = 0 because of the equutions of motion. The Lugrangian (7.18) implies the quite striking result that a Yang-Mills

interaction with a fermion in the adjoint representation is automatically super- symmetric. The corresponding conserved (gauge invariant) spinor cur- rent is [30, 64]

(7.19) J~ ---- -- ½ Tr (Gq~yj~ ,y~) , ) .

The Yang-SIills supersymmetric system can be coupled to a matter super- nmltiplet (A~, B~, y),) belonging to some representation of the group. The form of the Yang-Mills coupling of the matter is completely analogous to the Abetian cuse. In particulur, if ulso the matter belongs to the udjoint repre- sentation, the resulting Lagrangian is [53]

(7.20) 1 ] i

~f.~to.= Tr -- ( ~ , A ) 2 - - ~ ( 2 ~ B ) 2 - - ~ ' 2 , p -

] 1 i g2 ) - - _v) m2A2 -- 7~, m ~ B ~ - - -~ mCfy) ~- ~ [A, B] ~ + g~ [A + ~,sB, ~] •

where

W~ = DD(exp [-- gV]Da exp [gV]) .

The iuvariance of (7.16) can be checked by using the local transforma- tion of (7.17).

In terms of component fields (7.16) reduces to (Wess-Zumino gauge)

i ~ . 2 i ) , Tr ( - -1G~, - -~

(7.1s) Gl,~ = ?l,v,,-- 8,v~ -~ ig[v~, v~] ,

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): A NEW INVAI~IANCE :ETC. 129

ge rm ark ab ly enough we observe[53] tha t the supersymmetr ic Yang-Mills theory with just one ma t t e r mult iplet in the adjoint representat ion manifests an addit ional symmet ry for m = 0. In fact, because of the par t icular form of the Lagra.ngian, one can combine the two Majorana spinors ,~ and yJ into complex Dirae spinor

] (7.21) Z = ~ (2 + i~) .

This leads to the Lagrangian

(7.22) Tr 1 o 1 1

- ~ ~ - ,~ ( ~ , , A ) ~ - - 5 (~ , ,B)2 -

i o g~ 77y.2Z -- ig]~[A + ~sB, Z] + ~- [A, B] ~J .

This Lagrangian, in addit ion of being invar iant under "gang-Mills and super- symmet ry transformations, is also invariant under a , baryon n u m b e r , phase t ransformat ion Z -+ exp [i~] Z, Z* -+ exp [-- @] Z* (the other fields remaining enehanged) and therefore is invariant under a global U~-group [65, 66]. The Lagrangian under consideration, being supersymmetrie , admits a conserved spinor current . We leave, as an exercise, to verify tha t this current is given by the following expression [30]:

Jz = Tr (-- 1-Gq°veToVt~ Z + ig[A, B]ysV~Z-- i y . ~ ( A -- vsB)vzX) .

Renormalizabil i ty of supersymmetr ie gauge theories has been proved using different techniques [56, 58-61, 67]. In these theories, like in the model with- out gauge fields, the same phenomenon occurs, namely the absence of mass renormalizat ion for the ma t t e r mult iplet [58, 62].

The main problem one has to deal with, in order to prove renormalizabil i ty, is the fact t ha t one has to abandon the Wess-Zumino gauges, where the La- grangian looks renormMizable by power counting, and to consider a super- symmetr ic gauge where the Lagrangiau is intrinsically nonpolynomial.

I t is s t raightforward to show tha t the supersymmetr ic generalization of the usual Fermi- type gauge-breaking t e rm (1/~)(~avs) 2 is given b y [56, 58]

1 (7.?,3) - ( D D V D D V ) ~ ,

C~

moreover one has to introduce also Faddeev-Popov ghosts which are anticom- mut ing ehiral fields. The new feature of supersymmetr ie gauge theories is therefore the presence of spin -1 fields which obey Bose statistics. These new ghosts are decoupled in the Wess-Zumino noneovar iant gauges bu t they are needed for general gauges.

9 - R i v i s t a d e l N u o v o C i m e n t o .

130 s. FERRARA

The regularizat ion procedure is also a problem in supersymmetr ie gauge theories [58]. In fact dimensional regularization [68] cannot be longer used because of the fact tha t s u p e r s y m m e t r y t ransformat ions depend on the space-

t ime dimension [69]. t towever it turns out, f rom the works of FERRARA, PI- GUET, SCHWEDA [58, 70] t ha t the higher covar iant der ivat ive me thod and the B P H Z renormalizat ion scheme can be generalized without difficulties to supersymmetr ie theories.

Concerning the problem of spontaneous supe r symmet ry break ing[71] , we would like to ment ion tha t the mechanism used in the Abelian s i tuat ion [63] cannot be longer used in the non-Abelian case. In fact , in this ease the D-componen t of the vector mul t ip le t is no longer a scalar (it belongs to the adjoint representat ion) and would break explicit ly gauge invariance. The me thod can only be applied to a nonsemi-simple group like U: with S U~ as a residual symmet ry , but there is no obvious generalization to semi-simple groups. The same difficulty would not arise if one requires a spontaneous b reak-down of the gauge group [29, 72-75], the s u p e r s y m m e t r y remaining unbroken. Remarkab ly , as shown by O'I~AFEARTAIGIt and collaborators [76], this is sufficient in order to build up a theory which is both asympto t ica l ly free and infra-red convergent , i.e. this is sufficient to equip all bu t an Abelian

subset of the Yang-Mills fields wi th masses.

8. - Asymptot ic freedom in non-Abelian gauge theories.

A question of great physical interest, in a non-Abelian gauge theory of particles, is its ul t raviolet behaviour and in par t icular when it can be

a sympto t i ca l ly free [68, 77, 78]. We observe tha t , because of the results discussed in the previous section,

the super symmet r i c Ya.ng-Mills theory, if we do not add some ex t ra couplings like g'd ,z :S 'S~S k, turns out to be controlled by one coupling constant only,

the gauge coupling constant , so there is one Cal lan-Symanzik function fl(g) common to all fields [53, 79].

]f one defines the following group invar iants :

(8.1)

T ( R ) b ~ = Tr (RaR ~) ,

C2(R) I = R~R ~ ,

C:(G) b~'~ = / ° ~ / ~ ,

where R ~ are the generators of tile representa t ion to which the m a t t e r mult i- plet (A~, B~, y~) belongs, and C2(G) is the quadrat ic Casimir in the adjoint

representat ion, i.e. in the representa t ion of the gauge mult ip le t ()~Jl~, v,~k), then the Cal l~n-Symanzik function is given, up to the second loop, by the

8UPERSYMMETRY (FERMI-BOSE SYMMETRY): A N:EW INVARIANCE :ETC. 131

following expression [80] :

(8.2)

with

(8.3)

(8.4)

A B fl(g) = - - g. + g~ 16n 2

A = n T ( R ) - 3C~(G),

B = 4nC~(R)T(R) @ 2C2(G)T(R) - 6C22(G),

n being the number of ma t t e r multiplets. The condition for asymptot ic freedom is [53, 80]

(8.5) n T ( R ) -- 3C2(G) < 0 .

Note tha t for A = 0 the theory is never asymptot ical ly free

B ( A = 0) = 12C2(R) C2(G) > O .

because [80]

(8.6) A = N ( n - 3) , B = 6 N ~ ( n - 1) ,

therefore, for n = 0, A < 0, B < 0 (pure Yang-Mills), for n = 1, A < 0, B = 0, for n = 2, A < 0, B > 0, and we have a per turba t ive zero at

g~ 1

1 6 ~ ~ 6 N

For n • 3 tha t theory is infra-red free. A problem [81], recently solved by O'RAFEAlCTAIGI-I and collaborators [76], is

tha t of having infra-red convergence in an asymptot ical ly free supersymmetr ie theory, namely to equip all the gauge mesons with mass bu t an Abelian sub- set. These authors have shown tha t this si tuation can be achieved through a spontaneous break-down of the gauge group (supersymmetry remaining un- broken). This mechanism can be triggered if one adds to the original Yang- Mills Lagrangian

(8.7) c~e----Tr ( -1-G24 "~-- / 2 Y ' ~ ) - - 2 v~7" ~ p - 2

~ u B ~ B ---~ m2(A ~ -1- B 2) -- 2 2

i g2 m~f l ~- i g [ i ( A t R F ) %- 275(BtR~p) -~ h.e.] + -~ ( A * R B + h.e.) 2

9 ~ - R i v i s t a d e l N u o v o G i m e n t o

There is a wide class of solution of (8.5). For example, if the ma t t e r field belongs to the adjoint representat ion and

G - - - S U ~ , then (8.3), (8.4) simplify into

132 s. F~RmXRX

a supersymmetr ic m a t t e r - m a t t e r interact ion of the form

(8.8) g' d~jk SiS~S ~ ,

which is of course locally Yang-Mills invar iant .

Note, however, t ha t a t e rm like (8.8) ( 'annot be generated f rom (8.7) because it is odd in S~ while (8.7) is even.

I f the m a t t e r fiehl itself belongs to the adjoint representat ion, then, irre- spect ively of the new t e rm (8.8), the condition for having asympto t i c f reedom is unchanged, i.e. ~ < 3. However the impor tan t point is t ha t the result ing potent ia l one obtains by adding (8.8) to (8.7) has not an SU~-symmetr ic mi- n imum and in fact it originates a spontaneous break-down of the gauge group.

I t turns out tha t the simplest way to achieve infra-red convergence is t ha t of having two ma t t e r mult iplets in the adjoint representat ion of ~qUa. This model is ra ther unphysical , in par t icular it still contains massless scalar par- t ides , however it shows tha t , in supersymmetr ie theories, the problem of infra-

red convergence is less severe than in conventional theories.

9. - Supersymmetric extensions of the U~ Weinberg-Salam and U~ Higgs models.

At t emp t s have been made in order to construct realistic models for unified weak and electromagnet ic interact ions with the addi t ional propert ies of super- s y m m e t r y invariance. The main difficulty one encounters is tha t , if one wants to build up a realistic model with the neutrino as a Goldstone fermion, there is place only for one kind of neutrino in ~ supersymmetr ie theory, just because the number of Goldstone particles eamlot exceed the number of spinor charges in the theory. On the other hand, if one wants to introduce two types of neu- trino, then one necessarily mus t double the charges and therefore provide the supe r sym m et ry algebra of ~ nontrivia.1 mixing with an internal- s y m m e t r y group.

Unfor tuna te ly it has been shown up to now tha t renormMizable Lagran- gians with this richer s y m m e t r y do not exist [82-84]. A model calculation showing this fact has been carried out by CAPPER and LEIB]3RAS"I)T [85].

The mos t realistic model, restr icted to the electron sector, has been given by FAYET [86]. I t can be regarded as the supersymmetr ic extension of the S U2@U~ Weinberg-Salam model of weak and electromagnet ic interactions.

The original Lagrangian contains a doublet of complex ehiral mult iplets and a singlet in teract ing with a t r iplet and a singlet of vector mult iplets , i.e. the gauge supermult iplets of S U~@ U~.

Spontaneous s y m m e t r y breaking occurs in two steps. Firs t there is a spon- taneous b reak-down of S U2 with the gauge group U1 and s u p e r s y m m e t r y remaining unbroken. A super-t t iggs mechanism [29] is opera t ing in such a

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): A NEW INVARIANCE ETC. 133

way tha t three real chiral multiplets are eliminated and three (real) vector multiplets acquire a mass. The resulting particle multiplets are therefore

1) the massless vector mult iplet (y, v) which contains the photon and the neutr ino;

2) the massive (neutral) vector mult iplet (Z, Eo, z) which contains the Z neutra l vector boson, a neutral lepton and a scalar with the same masses;

3) the charged massive vector mult iplet (W±, E±, e±, z±) which contains the W± vector mesons, the electron and a new lepton and a charged scalar z all with the same mass;

4) a complex (neutrM) chiral mult iplet (eo, ¢o, (~) which contains a lepton and two scalars.

As a second step, spontaneous breaking of supersymmetry occurs (the gauge group U~ remaining unbroken). The v-particle becomes a Goldstone fermion. All masses inside the multiplets remain the same bu t for the electron mult iplet where the masses of the leptons E_, e_ are split on each side of the W_ mass according to the formula

m E _ = r o w _ + ~ , m ~_ = r o w _ - - ~ ,

where ~ is a parameter which measures the symmet ry breaking. Z, W± are the ordinary vector mesons of weak interactions, v, y are the

electron neutr ino and the proton, e_ is the electron and E_ a heavy lepton. eo, Eo are two heavy neutrinos.

I t can be shown tha t the present model solves the difficulty related to the vanishing of the anomalous magnetic moment in supersymmetr ic gauge theories.

In fact , it has been shown by ~FAYET tha t the graphs, which in a supersym- metr ic theory exact ly cancel the QED graph for the g - 2, are higly sup- pressed in this spontaneously broken version of the gauge theory, as previousy suggested by FE~RA~A and ]:~EMIDDI [87].

FAYET [88] succeeded also in constructing in a very elegant way the super- symmetr ic version of the U1 tIiggs model. Such a model describes the inter- action of a massless vector mult iplet with a ehiral multiplet .

The Lagrangian, in the Wess-Zumino gauge, is

(9.1) 4 ~ 2

1 - - ~uq~*,~uqJ + ie ~ / 2 ( ~ ).q~ + (p* ~4oL) -- ~ (~ + e~*q~) ~ .

If ~ < 0 supersymmetry remains intact bu t gauge invariance is broken in the vacuum.

134 s. F~lmXR~

I f

i ¢ = - - ~ ( A - - iB) and E = % 4- 2R,

we can choose ( A } = 0, (B} = b > 0, where 5 satisfies the relation ~ + + {-e~ -~ O.

As z~ consequence of a super-Higgs mechanism the ehiral mul t ip le t disap-

pears and the vector mul t ip le t (B, E, ~ ) becomes massive with mass given b y m -- eb.

The new Lagrangian is

(9.2) ( S = - - ~ I , ' ~ , ~ - - i E e . E - - ~ , B ~ B -

2

7) (m d- e B ) : v ~ - - - m B d- B 2 -- i (m 4- eB)F ,E 4- eELT~ELv, .

R e m a r k a b l y enough the Lagrangian in (9.2) is nothing but the nmst general self- interaction of a mass ive supernmlt iple t which does not contain quart ic t e rms in the Fermi fields.

This Lugrangian has in fact the mos t general form

(9.3) ( ) L = V D D D D V + ~ V + ~ - e x p [ 2 e V ] ,

where, if ~: < O, spontaneous s y m m e t r y breaking does not occur. I f one makes the following identifications:

B = b e x p [eC], E = 2 n + e d e x p [eC]z ~,

then (9.3) is just tile Higgs Lagrangian. h i other words, the supersymmetr ic

Higgs model is nothing bu t the self-interaction of a massive vector mult iplet .

10. - Supercurrent and the graviton multiplet.

I t has been sho~vn up to now tha t a lmost all renormalizable theories, includ-

ing gauge theories, can be made supersymmetr ie and in addit ion it has been realized t h a t their supersymmetr ic version has less divergent quan tum corrections.

On the other hand, it is well known tha t the classical Einstein theory of gravi ta t ion does not have a consistent quan tum counterpar t .

I t is therefore of grea t theoretical interest to invest igate the propert ies of a supersymmetr ic version of gravi ta t ion [89] and in par t icular if it leads to a renormMizable theory.

8UPF.RSYMMETRY (FERMI-BOS~I SYIVIM]~TRY): A NEW INVARIAI~C]~ ETC. 13~

As a very preliminary step to this problem, FERRARA and Z u ~ N o [64, 90] have studied in detail the transformation properties of the various conserved quantities which are present in a symmetric matter system.

Let us consider first a massless supersymmetric system, i.e. a system described by a Lagrangian field theory which is (classically) invariunt under the 24-parameter graded Lie algebra introduced iu sect. 2.

I t has been shown by the previous authors that the spinor current can always be redefined in such a way that the improved spinor current

( 1 0 . 1 ) g ~ = J~i~ ,

which satisfies 7"Z = O, is a member of a real supermultiplet having the fol-

lowing general structure:

(lO.2)

where

pod, X % v~( , o) = c,,(x) + i@~z#(x) + @~7~O(v~(x)) -

- ~ j ( ~ C~(x)- ~ Q(~)) - iOoOnr. ~z,(~) +

+ ¼@~OOr. ~z:~(x) + -~ (00) ~ [] ¢~(x),

( 1 0 . 3 ) C~ = 1 ~ 3 ~J~,, v~, = -- ~Oz~,, Z~, = -- ~J#,~,

are the conserved axial current, the traceless energy-momentum tensor and the conserved spinor current. The form (10.2), together with the conservation equations and the conditions

7 " g = O , 0 ~ = O,

is actually equivalent to the following covariant conservation equations:

(10.4) D~'V~,,, = O, D t' V~,~, = O.

The supermultiplet V~ actually contains all the local currents of a supersym- metric system. The corresponding elements (charges) of the graded Lie algebra are obtained as moments of the local currents contained in the multiplet.

If one performs an infinitesimal supersymmetry transformation ~(x) as defined in sect. 2, then the multiplet V~ undergoes the following transform ation:

~C~ = i~ysZ~ ,

~Z~' =-~7~(v~. + v.~)~--~7 s~, ~

(10.5) -- 8~ C.7~7~o~ - - C~757 x 8 ~ -- 2 C~75y ~ 8 . ~ ,

i 4- i

136 S. FERRARA

I t is therefore clear t ha t the above supermult ip le t should p lay in a super- symmet r ic theory a role analogous to the stress tensor in a usual theory.

In the case of a massive sys tem eqs. (10.4) become par t ia l conservat ion equations. They become

(10.6) D ~ V~ = / ) ~ S ,

S being a chiral mul t ip le t ( D : S = 0).

These equations imply t h a t the supermult ip le t I ~ contains now a scalar

and a pseudoscalar field beyond an axial vector, the conserved spinor current and the stress tensor.

I f one implements eqs. (10.4) with the mass shell condition

(10.7) ( ~ -- m2) V~ = 0 ,

then the mul t ip le t Ir~ can be used to describe a mul t ip le t of particles of

spin 1, 3, ~ and 2. This mnlt iplet corresponds to the J = ~ - mul t ip le t according to the classification given in sect. 4. For m = 0 (10.4) become gauge conditions and the remaining degree of f reedom are two particles of helieities 23 and 2.

This m a y very well be the gravi ton supermult iplet which should p lay a fundamen ta l role in a supersymmet r ic theory of gravi tat ion. Final ly we note t ha t the massless spin-~- particle in the supersymmetr ie gravi ta t ional theory has the exact counterpar t iu the massless spin- 1par t ic le of the photon multiplet .

11. - A c t i o n p r i n c i p l e in s u p e r s p a c c .

ARNO~,\'ITT, ~2q-ATH and Zu~II>-o [91, 92] developed an action principle in cu rved superspaee which was ~t na tura l extension of the usual geomet ry of Minkowsky space to a R iemann geometry .

They considered a point on a superspace labelled b y the co-ordinates z ~ = (x' , 02) (where x, is the usual space-t ime co-ordinate and 0 ~ is a Majorana spinor) and generalized the s upe r s ym m et ry t ransformat ions to a local gauge group of a rb i t ra ry co-ordinate t ransformat ions in superspaee

(11.1) S = za'(z).

These t ransformat ions leave the line element

(11.2)

invar iant .

d s ~ = d z ~ g ~ ( z ) d z ~

SUPEI~SYM~IF.TI~Y (FER~II-BOS~ SYMMETRY): A :NEW INVAI~IANCE ETC. 137

The metric g~--~ (g,~, g~ , g:~} are gauge supermultiplets and can be in- te rpre ted as the metric tensor of superspace.

The l~iemann geometry of this space can be constructed and leads to a curvature tensor

(11.3) n'~B~- F~,B+ (-- 1)"°P~.,~,+

and to the Christoffel affinity

(11.4) I 1,ad _ ~v F ~ a = (--1)b~½[(--1)bdg~,,Bq- ( - - ) ~l~bg~D,~--g',']g

l ~.~ = ( - 1)~+~+'~"

An action principle can be formulated in superspace start ing from the follow- ing action, which is invariant under general co-ordinate t ransformations in superspace:

(11.5)

where R is the scalar curvatt tre

(11.6) R = (-- 1 ) a g ~ R ~ ,

and R ~ is the contracted g i emann tensor. The action principle leads to the supersymmetr ic Einstein equations

(11.7) RAB = 0 ,

which could eventual ly be generalized b y adding to the original action an allowed additional invariant

f dSz ~ ~ ,

which would give a cosmological t e rm in (11.7)

(11.8) R ~ = ~g~ .

The actual weakness of this approach is due to the fact tha t usual super- symmet ry t ransformations in Minkowsky space, which one would natura l ly associate to a fiat limit of the above curved space, are not solution of (11.7) nor of (11.8). This is a consequence of the fact tha t g,~B(z) is not constant in fiat space and its actual form does not satisfy the previous equations.

138 s. rnRnA~x

W o o [93] has in f ac t shown t h a t t he usua l s u p e r s y m m c t r i c equu t ions of

f ia t space c a n n o t be iden t i f i ed w i t h t h e c o r r e s p o n d i n g c o v ~ r i a n t e q u a t i o n s

in c u r v e d space in the f lat l infi t . I n o rde r to m a k e such an i den t i f i c a t i on ~ l imi t -

i ng p r o c e d u r e has to be t a k e n in to accoun t .

Such l i m i t i n g p r o c e d u r e p r o b a b l y co r r e sponds to a c o n t r a c t i o n of t h e loca l

gauge g r o u p of gene ra 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 s ill s upe r spa c e in to ~ (geo-

m e t r i c a l l y d i f fe ren t ) loc,d ga,uge g r o u p whose f la t l im i t wou ld be i de n t i f i e d

wi th t h e f lat s u p e r s y m m e t r y .

The c o n s t r u c t i o n of th i s c o n t r a c t e d g e o m e t r y a n d the c o r r e s p o n d i n g con-

s i s t en t f o r m u l a t i o n of q, s u p e r s y m m e t r i c E i n s t e i n t h e o r y of g r a v i t a t i o n is

s t i l b m open p r o b l e m .

R E F E R E N C E S

[1] B. Zu~IINO: in Renormalization and Invariance in Quantum •ield Theory, edited by E. R. CAI=X_~'ELLO (New York, N .Y . , 1973), p. 367.

[2] J. "W]~ss ~md B. ZU.~IINO: _Vucl. Phys., 70 B. 39 (1974). [3] D . V . VOLKOV ~md V. :P. AKVLOV: Phys. Lett., 46 B, 109 (1973). [4] D . V . VOLKOV and V. A. SO~OK.~: J E T P Lett., 18, 529 (1973). [5] L. O'RAFEARTAI(;H: Phys. Rec. Lett., 14, 575 (1965); Phys. Rev., 139, B 1052

(1965). [6] S. COL]~MA~ ~ ~nd J. MANDI:L.~: Phys. Rev., 59, 1251 (1967). [7] Y. AHARO-~OV, A. CASt~En and L. SI'SSKI~'D: Phys. Lett., 35B, 512 (1971). [8] O . L . G~RW~IS al~d B. SAKITA: ~'ucl. Phys.. 34 B, 633 (1971). [9] A. NEVEU and J. SCHWARZ: Nncl. Phys., 31 B, 86 (1971). [lt)] 1). RAMOND: Phys. Rev. D, 3, 2415 (1971). [11] S. F]~RRXR,~: Xucl. Phys., 77B, 73 (1974). [12] L. CORWI~-, Y. N~%MA~ and S. ST]~R~TB~RG: TAUP 448-74, Tel Aviv Repor t

(1974). [13] L. CoRw[x, Y. NE'~:.~t~ and S. ST~R~BER~: Bee. Mod. Phys., 47. 573 (1975). [14] Z. FR6LICHER and A. NIJnX1IuIS: Proc. Nat. Acad. Sci.. 43, 239 (1957). [15] K. KOD.¢IRA, L. NIICE~IBER(~ and D. C. SPn~'cER: Ann. Math., 75, 536 (1962). [16] K. KODAIRA and D. C. SPENCER: Ann. Math., 67, 328 (1958). [17] A. NIJE.',-I~['IS: Proc. K. Ned. Acad. Wet., A58 , 3 (1955). [18] A..NI,IENHUIS r~nd ~V. RICHARDSON jr.: Bull. Amer. Math. Soc.. 70, 406 (I966);

72, 1 (1966). [19] A. FAIS and V. RITTENBERG: Rockfeller Univers i ty Report Coo-3232B-74. [20] D . C . SPENCER: A?t,~l. Math., 76, 306 (1962). [21] ]). GODDAICD: Nucl. Phys.. 88B, 429 (1975). [22] J. WESS and B. Z w ~ i i o : Phys. Lett., 49B, 52 (1974). [23] A.S.~L.~t a~d J. STRAX~ID~E: Nucl. Phys., 76B, 477 (1974). [24] S. FnRRARA, J. WESS and B. Zu)~I~'O: Phys. Lett., 51 B, 239 (1974). [25] ~V. Ri2~[L and B. C. Yu.~-~: _Forts. Phys., 23, 431 (1975). [26] W. Ri/'~z ~nd B. C. ¥u~-~-: K~iserlautern prepr in t (September 1974). [27] A. SAL~)t and J. STRATI1D]~]I~: 5~01. Phys., 80 B, 499 (1974). [28] A. SALEM and J. STRATHDEE: .\'uel. Phys., 84 B, 127 (1975). [29] A. SALA~ ~nd J. ST~CA~D]~]~: Phys. t~ev. D, 11, 521 (1975).

SUPERSYMMETRY (FERMI-BOSE SYMMETRY): ~_ )TEW I:NVARIANCE ETC. 1~9

[30] B. ZUMINO: Proceedings o] the X V I I International Con]erenee on High-Energy Physics (London, 1974).

[31] R. HAAG, J. T. LOPUSZANSKI and M. SOHNIUS: ~Yuel. Phys., 88B, 257 (1975). [32] L. O'RAFEARTMGH: Nucl. Phys., 89 B, 418 (1975). [33] B. DE WIT: Phys. Rev. D, l l , 898 (1975). [34] J. ILIOPOULOS and B. ZUMINO: Nucl. Phys., 76 B, 310 (1974). [35] H . S . TSAO: Phys. Lett., 53 B, 381 (1974). [36] S. FERRARA, J. ILIOPOULOS and B. ZUMINO: ~Tuel. Phys., 77B, 413 (1974). [37] D . M . CAPPER: I.C./74/66. [38] D . M . CAPPER and G. LEIBBRANDT: Nucl. Phys., 85 B, 492 (1975). [39] R. DELBOURGO: NUOVO Cimento, 25 A, 646 (1975). [40] K. FUJIKAWA aD-d W. LANG: 5ruct. Phys., 88B, 61 (1975). [41] K. :FUJIK~WA and W. LANG: iVucl. Phys., 88 B, 77 (1975). [42] J. HONERKAMr, F. KRAUSE, M. SC~EUNERT and M. SC~dL]N])WEIN: .Phys. Lett.,

5 3 B , 60 (1974). [43] M. HuQ: Belfield preprint (Dublin, 1974). [44] A. SALAM and J. STRATHDEE: Nuel. Phys., 86B, 142 (1975). [45] F. A. BEREZIN: The Method of Second Quantization (New York, N . Y . , and

London, 1966). [46] F. A. BEREZIN and G. I . KATZ: Mathemat. Sbornik (USSR), 82, 343 (1970)

(English translation, Vol. 2). [47] W. LANG and J. WEss: JYucl. Phys., 81 B, 249 (1974). [48] A. SALAM and J. STRAT~DEE: Phys. Left., 49B, 465 (1974). [49] W . A . BARDEEN: to be published. [50] B. DE WIT and D. Z. FREEDMAN: Phys. Rev. Lett., 35, 827 (1975). [51] A. SALAM and J . STRATHDEE: IC/75/38. [52] J. WESS and B. ZuMINO: ~Vucl. Phys., 78B, 1 (1974). [53] S. FERRARA and B. ZuMINO: Nucl. Phys., 79B, 413 (1974). [54] A. SAZAM and J. STRATI~DEE: Phys. Left., 51B, 353 (1974). [55] R. DELBOURGO, A. SALAM and J. STRA~tDEE: Phys. Left., 51 B, 475 (1974). [56] B. DE WIT: Phys. Rev. D, 12, 1628 (1975). [57] B. DE WIT: Phys. Rev. D, 12, 1843 (1975). [58] S. FERR~tA and O. PIGUE~: Nncl. Phys., 93 B, 261 (1975). [59] O. PIGUET and A. ROUET: Nucl. Phys., 99B, 458 (1975). [60] A . A . SzAVNOV: Theor. Math. Phys. (USSR), 23, 3 (1975). [61] A . A . SLAV~OV: Nucl. Phys., 97B, 155 (1975). [62] R. DELBOUI~GO: ICTP/74/19 (June 1975). [63] P. FAYE~ and J. ILIOFOULOS: Phys. Lett., 51B, 961 (1974). [64] S. FER~ARA and B. ZuMI~O: Nucl. Phys., 87 B, 207 (1975). [65] A. S~LAM and J. STRATHDEE: Nucl. Phys., 87B, 85 (1975). [66] A. SALAM and J. STR~THDEE: _hrucl. Phys., 93 B, 23 (1975). [67] J. HONEtCKAMP, ]~. KRAUSE, M. SCHEUNERT and M. SCttLINDWEIN: Nucl. Phys.,

95B, 397 (1975). [68] G. 'THOOF~: Nucl. Phys., 61B, 455 (1973); 62 B, 444 (I973). [69] R. D~BOUR~O and V. B. P ~ S ~ D : ICTP/74/6 (December 1974). [70] O. PIGUET and M. SCHWED~: ~Vucl. Phys., 92 B, 334 (1975). [71] P. F~YET: P.T.E.N.S. 75-1 ENS prepr in t (January 1975). [72] L. 0'R~nAR~A~GH: Phys. Lett., 56 B, 41 (1975). [73] L. O'RAFEAETAIOH: S.B. DIAS-TP-74-46 Dublin preprint (1974). [74] L. 0'RAFEARTAIGH: J~ruel. Phys., 96B, 331 (1975). [75] L. O'RAFEARTAIGH: Lectures Notes, Comm. of the Dublin Inst. of Adv. Studies,

Series A (Th. Phys.), 22 (1975).

1 4 0 s . ~ n a A n h

[76] S. Bitowxz, l,. O'RAFI~Ar~T.~I(;H and T. S~EI~Y: Yucl. Phys., 99 I/, 150 (1975.) [77] D . J . GRoss aud F. WILCZEK: i)hy,u, l~eF. Lett., 30, 1346 (1973). [78] ]I. 11. t'OLITZ~R: Phys. 1~ev. Lett., 30, 1346 (1973). [79] 1[. S~-Z~KI: N uct. Phys., 83B, 269 (1974). [8111 T. 1~. D. Joxi~S: Y~cl. Phys., 87B, 127 ([975). [81] G. Woo: M.I.T. preprin~ (November 1974). [82] ]). 1I. DO_XDI: K~rlsruhc prcprint (January 1975). [83] P. ft. DO~-DI and M. SOIINIIS: Xttcl. Phys., 81 ]i, 317 (1974). [84[ ~R. J. FIRTH and J. D. JE:,-KLX'(;: Du,:ha~ preprint (August 1974). [85] D . M . C,~PPH~ and (~. ]n,:IBB~::X~D'~': 2~-vcl. Phys., 85B, 5(13 (19751. [86] P. FAYET: )~-,Itel. Phys.. 90 B, 104 (1975). [87] S. FERRAtCA a.nd F. ]~I:;MIDDI: Phys. Left., 55 B, 347 (]974). [88] P. FAYE'I': P.T.E.N.S. 75-7 ENS pr(~print (June 19751. [89] B. Zl-~kx-o: f uc l . Phys., 89 B, 535 (1975). [9tl] B. DE WIT and D. Z. FnEH)~IAN: Phys. ilev. D, 1.2, 2286 (19751. [91] R. Anx-owI1r and P. N-XTH: Phys. Left,, 56B, 177 (19751; "~'~ub-2259 North-

eastcrn L'nivcrsity prcprint (June 1975); Nub.-22t~i Northca~ter~l University l)reprint (July 1975).

[92~ R. ARNO\VITT, P. XAT]I and B. Z"~Hx-o: Ph~],~. Lett.. 56 B, 81 (19751. [93] G. Woo: 3[.I.T. pr(~pt'int (.~[arch i975).

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Questo fascicolo ~ stato l icenziato dai torchi i] 30-IV-1976