Fortachritte der Physik %,57--142
(1978)
Supersymmetry and Superfields A B D S~
u
International Centre for Theoretical Phyyics, Trie.de, Italy, and
Imperial College, London,England and
J. STRATHDEE International Centre for Theoretical Physics, Trieste, Italy. Introduction
Without doubt the invention of supenymmetryl), a possible fundamental symmetry between fermions and bosons (their contrasting statistics notwithstsnding) is one of the most elegant creations in theoretical physics. The recognition of this type of symmetry has enriched the mathematics of relativistic particle theory in a number of basic ways. 1. I n supersymmetric theories it is possible to group specified numbers of mesons and ferniions (e.g. spins zero and 112, or 0, 112 and 1, or 312 and 2) in supermzcltiplcfs
of the same mass. The existence of one member of such a superniultiplet must imply the existence of all the other members if supersyninietry is a law of nature. Before the discovery of supersymmetry it had erroneously been assumed that basic tenets of quantum field theory forbade associations of different spins in the same multiplet. 2. One can associate multi-component “superfields” with such supermultiplets and write supersymmetric Lagrangians for these [2]. Such Lagrangians are extremely restrictive in form. In particular, if renormalizability is required then they are very special indeed. The most exciting property of these Lagrangians is their “softness” in that many of the ultraviolet infinities of conventionally renormalizable theories do not appear (as a rule only wave function renormalizations are divergent). 3. Internal symnietries like isospin, SU(3), etc., can be incorporated without any difficulty into the structure of the supermultiplets. These internal symmetries can be gauged, i.e. made into local symmetries. For supersymnietric and gauge invariant Lagrangians the vector gauge bosons are necessarily accompanied by spin-113 supersymmetry partners [3]. These gauge fermions with their universal gauge coupling To our hornledge the first appenrance of supersymmetry occurs in the work of Gol’fand and Likhtman. It was rediscovered independently by AKULOVand VOLKOV nnd by Wsss and ZUMINO[I].
1)
6 ZeitschriIt ,,E’ortscliritte der Physik”, Heft 2
58
-4BDUS
SALAY and J.
STRATHDEE
are a new phenomenon in particle theory. (The universal couplings manifest theniselves not only in the interactions of gauge fermions with their vector partners, the gauge bosons, but also in their interactions with what one may call “matter” supermultiplets.) 4. Supersymmetry can be broken spontaneously in exactly the same way as internal symmetries are broken when the appropriate scalar fields develop a vacuum es-
pectation value [a]. In the lowest approximation this phenomenon is governed by the scalar ficld potential. In supersymmetric theories these potentials are highly restricted in form, unlike the case of non-supersymmetric Lagrangian theories where there is usually an embarrassment of possibilities. This circumstance is a great but austere virtue - not easily can one set up a model which is flexible enough to meet one’s requirements. Among the suprising properties of supersymmetric theories one finds that if the supersynimetry is not broken in zeroth order then it will not break in any finite order and, moreover, the scalar field expectation values - which may describe spontaneously broken internal symmetries - are given exactly by the zeroth order result (i.e. are not renornialized).
5. A basic aspect of supersyiniiietry - and one which may be associated with developments of physical significance - concerns its relation to the curved spacetime of gravity theory. This would be the generalization of spacetime to a nianifold which includes anticommuting c-numbers 0“ ( a = 1. 2, 3>4)along with the familiar z p . The notion of such a n extension of flat spacetime was used to define superfields O(z, 0). But the curved space generalization in which the metric field qu,(x)becomes one part of a superfield remains to be made convincingly. The problein of finding a supcrsymnictric extension of Einstein’s gravity theory is being tackled now by many authors and much progress has been made. However, the picture has not yet become quite clear and we shall not attempt to ’discuss the generalization of Einstein’s theory. I t is our purpose in this paper to elucidate the developments 1.-4. using the superfield notation throughout. Some of the results arc published already; others are new. We feel, however, that it is worthwhile to cover the entire development from this unified viewpoint.2) One feature of our presentation is that, from the outset, a fermionic: quantum number is defined in the superniultiplets - often at the expense of space reflection symmetry. I n the past this has been done more or less as an afterthought. I n view of the intrinsic importance of this quantum number in particle physics we’ believe the present treatment may be more apposite. One should ask the question: is this very elegant development made use of by nature? At present the answer seems to be in the negative. Now the photon and a neutrino could have shared a superniultiljlet - this would be especially plausible after the recent developments concerning unification of weak and electromagnetic interactions - but, as we shall show in Sec. V, it seems that if there is a neutrino accompanying the photon it is probably a Goldstone particle. Such a Goldstone neutrino, arising from the spontaneous breakdown of supersymmetry, is unfortunately subject to low-energy theorems which make its identification with the observed v, or vV unlikely. The neutrino which partners the photon may yet be found. The spin-312 ‘object which accompanies the graviton (spin-2) may have a fundamental role in full gravity theory - for example in circumventing singularities of spacetime. On the strong interaction side, the only indication of a symmetry among baryons and mesons appears to be the unvenality of Regge slopes but this could, no doubt, be explained in other ways. ?)
-4 number of reviews with extensive lists of references nre nvnilable [5].
Supersymmetry and Superfields
50
Section I
A . Supersymmetry as a Higher Relativistic Symmetry
The concept of a fundamental symmetry between fermions ar,d bosons is an intriguing one. It belongs to the class of so-called “higher” or relativistic symmetries in that its bnsic aim is to unite in a single multiplet, particles with different intrinsic spins. In the past such relativistic schemes have been concerned with the problem of uniting, for example, pseudoscalar wit.h vector mesons or spin 1/2,baryons with spin 312 baryons. Supersymmetry, however, is concerned with the uniting of scalar and vector bosons with spin 1/2 fermions. I n order to clarify the relationship of the new symmetry to the old proposals for uniting different spins in one multiplet, it is necessary to digress briefly on the strccture of the latter. One of the simpler such was arrived at in the attempt to make relativistic the SU(4) and SU(6) symnietries of WIONER,G ~ ~ R S ERADICATI Y, and SAKITA. The S U ( 6 ) , for example, is a phenomenological attempt to classify the hadronic multiplet structure which could be expected to emerge from the quark model if the dominant forces arc independent of spin. Thus the low-lying meson states, pseudoscalar and vector, are expected to fill out a 35-dimensional representation of S U ( 6 ) while the low-lying baryons, spins 1/2 and 312, are expected to fill a 56-fold. In predicting thcsc states as well as in several other respects the symmetry S U ( 6 ) is successful. However, and this id the main point for our discussion, the Wigner-Gursey-Radicati-Sakita symmetry is essentially non-relativktic. Its most distinctive feature is the presence ainong its infinitesimal generators of a set of operators which carry one unit of angular nionientuni. These are the operators which connect the states, such as x + and p+, whose spins differ by one unit. These generators behave in a well defined way under space rotations but not under Lorentz transformations. I n a relativistic theory such operators make no scnse; if the symmetry is to be taken seriously, they must be supplemented by new operators so as to fill out Lorentz multiplets. A number of attempts were made in this direction. One of the simplest was to embed the 3-vectors of Giirsey, Radicati ar.d Sakita in nntisymmetric 4-tensors and this was achieved by enlarging S U ( 6 ) to SL(6, C ) . The latter is in fact the smallest relativistic symmetry which can contain SU(G). Chfortunately for SL(6, C ) ,i t is a non-compact group. This means that its irrcducible unitary representations are all infinite dimensional. If the particle spectrum is to be cl.tsaified in unitary representations of this group, the z and p mesons must then he accompanied by an infinite sequence of particles, making the scheme (at tlic least) unattractive. All attempts at relativistic versions of the spin-containing synnietries met this phenomenon which came to be known as a no-go theorem. Like many no-go theorems in physics, this one apparently was also based on certain hidden assumptions and we shall explain shortly how it is circumvented in the supersymmetric framework. Briefly, the difficulties with SL(6, C ) were concerned with extra intcrnal Ylimemions” implicit in the symmetry. A rather pictorial understanding of the infinite-dimensionality of SL(6, C)’s unitary representations can be obtained by the method of induced representations. Thus, if SL(6, C ) is resolved into cosets with respect to its maximal compact subgroup S U ( 6 ) then one obtains a 35-diniensional coset space on which the full group can act. Now what can these 35 real variables signify? The answer is that they must constitute an “internal” space with 35 degrees of freedom. It is these internal degrees of freedom which manifest theniselves as an infinite degeneracy in the particle spectrum: they can contribute any amount of spin but no energy. I n another version of SL(6, C ) symmetry, one of the Lorentz tensors - an SU(3) singkt - among the infinitesimal opcrators is identified with the gencrator 5*
60
ABDUSSALAM and J. STRATHDEE
of Lorentz transformations on space-time itself. For this to make sense it is necessary to embed the space-tim,e co-ordinates, x,, in a 72-dimensional manifold. This time the new degrees of freedom contribute to the energy and one finds that, in the rest frame, energy ranges over a continuum. It is difficult to imagine how all the new dimensions could be accommodated in a physically meaningful theory. The difficulties just outlined can all be avoided by the simple expedient of associating the new dimensions with anticommuting variables. Indeed, it is now rather hard to understand why such a simple solution was not thought of long ago. The idea is to consider fielb, O(x,e), which are defined over the product of the usual space-time, labelled by four coordinates xp, with a new space labelled by a set of anticommuting c-numbers, Bi. In order to see how the new degrees of freedom are controlled one has only to expand the field O(z, 0 ) in powers of 0 and notice that the series must terminate after a finite number of terms:
where N denotes the number of independent components Oi. Because of the anticommutativity among the Bi, the coefficient fields 4i,i,..,(z)must be completely antisymmetric and there can therefore be only a finite number of them. It remains to be shown, of course, that a consistent group theory can be established on the space of z and 0. Such extensions of the Poincard group do exist and a minimal example will be discussed in detail in the following sections. A t this stage we remark only that the new dimensions must have a spinorial character in keeping with their anticomniutativity as demanded by the TCP theorem. Thus, if in the above expansion 4(z) is a commuting scalar field, we should like the next member, di(z), to be an anticommuting spinor field. For this reason the new relativistic symmetry must combine bosons with fermions and its infinitesimal generators must include some which carry a half unit of angular momentum.
B. The Minimal Fermionic Extension The smallest spinorial representation of the homogeneous, proper Lorentz group is the two-component spinor. It follows that the minimal ferniionic extension of fourdimensional space-time which can be conceived is made with the two complex (or four real) components of such spinors. This is the manifold on which is based the relativistic symmetry known as supenymmetry. I n order to establish a notation, we discuss now some of the elementary features of this manifold and of the functions defined over it. Since the concept of the four-component Dirac spinor and the associated apparatus of Dirac matrices is more familiar to physicists than the two-component (dotted and undotted) spinors and their associated matrices, we choose to work with the former. This is purely a matter of notation: the two-component spinor is identifiable as a chi& projection of the Dirac spinor. Kow, a Dirac spinor has two chiral projections (for the definition see Eq. (1.10) and in general these are independent. However, if the Dirac spinor is subject to a reality condition (the Majorana condition) then the two chiral projections are related by complex conjugation. This means, in effect, that the four real components of a complex 2-spinor are ideritifiablc with the four real components of a Majorana spinor. It is possible to set up a one-one correspondence between 2-spinors and Majorana spinors. Our main conventions are as follows. For the nlctric of space-time we take q p v= diag ( + l , -1, -1, -1)
(1.2)
Supersymmetry and Superfields
61
and write the scalar product of two 4-vectors
A B = A$,
= AJ3o - A,B,
- AZBZ - A3B3.
(1.3)
-
(There is usually no need for the more exact but cumbersome notation A B = qF*AuB, = A,Bp = APB,. The only point where confusion could arise is in the use of the gradient a, = a/axo, -a/az,, - a / a x z , -alas,.) The Dirac matrices, y,, aatisfy the antieommutation rule b p , y.1 = %,,.. (1.4) The frequently employed tensor and pseudoscalar combinations are defined by
With these definitions it is always possible to choose a representation in‘ which, the matrices yo, u , ~uZ3, , u3, are hermitian while y,, y z , y 3 , y5 and uo,,uoz,a03 are antihermitian. Another important matrix in the considerations which follow is the charge conjugation matrix C, which relates y, to the transposed matrix y,T,
The matrix C is necessarily antisymmetric. (It can be fixed, apart from a sign, by imposing the further requirements C-I = Ct = C.) The infinitesimal Lorentz transformations are defined by
where w,, is real and antisymmetric. The 4-vectors A,, and B,, are required to transform like x,, and this ensures the invariance of the product A,B,,. The Dirac spinor, is required to transforin according to
+,
The adjoint spinor $ = +tyo then obeys the transformation rule
i sg, = ~,,,*U,,,. 4 (This conies about because the matrices you,,,, like yo, yay,,, yoys and iy,y,,y,, are all hermitian.) The transformations (1.8) are reducible. From (1.4) and the definitions (1.5) it follows that y 5 commutes with uu,.The subspaces on which y5 = -i and +i are invariant and this means that the c h i d projections (1.10)
separately obey the rule (1.8).These chiral projections are of course the two-component spinors of which the Dirac spinor is comprised. From (1.9) and the properties of the charge conjugation matrix one can show t h a t the “conjugate” spinor, $ c , defined by p = CIp (1.11)
62
ABDUSSALAM und J. STRALTHDEE
transforms exactly like itself. The relevant property of C is expressed in the equation Ca:, = -aa,,C. This identity is o ie of a number involving C which may be summarized in the statement;
y,C and a,.C are symmetric C , y,C and iypy5Care antisymmetric. A Majorana spinor is one for which
$c
1
(I.12)
= $ or, in terms of chiral projections,
$7
= C$,T.
(1.13)
The Majorana spinor therefore contains the same information as a two-component s p h o r (and its complex Conjugate). Let $, and $2 be a pair of anticonimuting Majorana spinors, ($1,
$at = 0.
Then the symmetries (1.12) are expressed in a statement about bilinear forms: $Iy,,$z and $ , G ~ , $ J are ~ antisymmetric while and $liya,y5t/Jz !re symmetric under the interchange of and &. (Note in particiilar that $lya,$I and $J,u,,Jtl vanish identically for Majorana spinors.) X o w consider some of the detailed properties of the expansion (1.1) for the case of a Majorana spinor 0. Because of the anticommutativity among the four components of 8 , there are altogether sixteen independent terms in the expansion. These can be conveniently grouped as follows: @(z,0 ) = A ( s )
+ 8$(c) +
1 1 +8y50G(z)+ 40iy,y50V.(+) 4
OOF(z)
(I.14) where A , F , (7, V , and D are Bose fields, $ and x are Fermi. It is clear that, with respect to the proper Lorentz group. A , F , G and D are scalars, V , is a vector while $ and y, are 4-spinors. We shall defer discussion of the improper transformations. The product of two such expansions must again have the same form and this can be verified. Thus if e) e j = 0 3 ( x , 0) (I.15) then the components of
A , = AIAz $3
= Al+Z
+
$142
cB3 are given by
Supersymmetry and Superfields
63
I n deriving these expressions we have assumed that 0 commutes with Bose fields and anticommutes with Ferini fields. A. further assumption we shall make is that the order of anticommuting factors is reversed by complex conjugation. With this understanding the bilinears 80, 8y,O and 8iyMy50are “real” while, for example,
(G$)*
= $0 =
fiJ,C,
and this bilinear is real if Jt is a Majorana spinor. Derivat,ives with respect to the Majorana co-ordinates 8 are defined by
to leading order in 80. Since f3 and 68 are anticommuting quantities it is important to remark that the infinitesimal d e is placed to the left of a@/@ in (1.17). This is a convention we shall adhere to. The derivative defined here has the usual properties of a differential operator with two important exceptions.. Firstly, when applied to the product of two functions its effects are distributed according to the rule
a (aJ1aJ2) = a01 aJ* f a J I ag 28
8% 2e
-2
(1.18)
where the +(-) sign applies when aJ1 is hosonic (ferrnionic), i.e. when 88 commutes (nnt.icommutes) with Secondly, the product of two such differential operators is ant isymmetric, (1.19)
Applied to a general function like (1.14) the derivative takes the explicit form:
(1.20) This formula will prove useful when we come to define the action on component fields of the extcnded Poincar6 group (See. (C)). Finally, it is necessary to make some decisions about the discrete transformations P and CP. There are two possibilities. The first is to refuse to assign any distinctive quantum number to the spinorial components J, and x. I n this case, if the scalar and vector components of @(x, 0 ) are real, then the spinors should also be real (in the Majorann. sense), i.e. a~(r, e) = e)* (1.21) and there is no concept of fermion number. With such a n interpretation one can associate the transformation e --f iroe (1.22)
with space reflection. The component fields A , F and D are scalars, G a pseudoscalar, T’, an axial vector. The alternative possibility is that we arrange that the spinor components Jt and x are distinguished from the Bose components by a fermionic quantum number. There is. only one way to do this and it means sacrificing the space reflection. Suppose the positive chirality combination of co-ordinates 0, carries unit fermion-number (as well as one-
64
ABDUSSALAM and J. STRATHDEE
half unit of spin). It follows that 0- carries minus one unit. Clearly the transformation (1.23) reverses the fermion number,
e* + iyoer.
This transformation can no longer be rtesociated with a simple space reflection. Rather it must be associated with combined space reflection and antiparticle conjugation. The phase transformtttions associated with fermion-number,
e* + e f %
e
or
--f e - ’ y 4 ,
(1.23)
can now be appliedto the function @(x, 0). Suppose this function transforms according to
@(x, 0 ) + @(z,eayd).
(1.24)
Then the component fields transform according to
A
*4
i
(1.25)
+
To summarize, the boson fields A , V , and D carry no fermion-number while F iG carries two units and F - iQ carries minus two units; the chiral spinors $L and x+ carry one unit while $+ and x- carry niinus one unit. These assignments are of course compatible with the reality condition (1.21). We shall return later to the problem of defining both parity as well as fermion-number in this framework. C. Supertranslations Having established a notation acd described some of the properties of the ext,ended space-time and the functions @(x, 0 ) defined over it, we are now in a position to discuss the Poincarti group and its extension. Firstly, the action of an infinitesimal PoincarO transformation on x and 0 is given by 6zp = b,,
+ o,,,x,
i 60 = -- wp.a,,.o 4
1 I
(1.26)
where bp and w,,, = -w,,, are infinitesimal real parameters. The new transformations, which we shall call super&ransZations,are given by
(1.97)
J
6e = where E is a n infinitesimal anticommuting Majorana spinor. We shall assume that tho cornponents of E anticommute among themselves, with 0 and with all spinor fields. The peculiar feature, expressed in (I.27), that a “translation” in the Majorana coordinates 0 is associated with a &dependent translation in the space-time co-ordinate x,,
Supersymmetry and Superfields
65
is highly significant and gives rise t o the many unusual characteristics of supersymmetric theory. But first we must verify that the infinitesimal transformations (1.26) and (1.27) form a n algebra. Apply to x and 0 two successive infinitesimal transformations characterized, respectively, by the parameters b 1 , wl,d and ba, wa,9: s16,zP= w ; . ~ 2 Z ,
+ -2i z'ypfi,e
froin which it is clear that the commutator [S,, S,] is equivalent to a third infinitesimal transformation d3, characterized by the parameters
(1.28) &3=
--4i u,,"(W:.&2
-c u p ) .
The infinitesimal transformations do indeed form an algebra and, what is more, they can he integrated to yield finite transforniations. A pure supertranslation integrates quite trivially to the finite form xP
-+ x,u
+ -2i E y p e
e+o+&.
I
(1.29)
It is interesting to note the form taken by the product of two successive supertranslations.
ABDUSSALAM and J. STRATHDEE
GG
which indicates that a space-time translation, i/261y,,E2 emerges when two supertranslations are compounded. I n order to treat the representations of the symmetry defined by (1.26) and (1.27), we shall set up an abstract algebra of the infinitesimal generators. Firstly, notice that the application to xp and 0 of the first-order differential operator,
reproduces the formulae (1.26) and (1.27). This differential operator can be looked upon as a particular realization of the abstract operator, (1.32)
which defines the infinitesimal generators P, J acd S. Moreover, the conlmutator [S,, S , ] = d3, derived above, can be looked upon 8 s a part+ular realization - in terms of differential operators - of the algebraic statement,
[VS,), F(dd1 = W,)
(1.33)
*
To deduce conimutation rules among P , J and S is now straightforward. On the lefthand side of (1.33)substitute for F(8,) and E(6,) the linear expressions (1.32) with paraiiicters bl, cul, and b2, (2,~ 2 respectively. , On the right-hand side substitute for J'(d3) using the values for b3, d, given in (1.28). The resulting formula is an identity in the parameters b', wl, and ,'b w 2 , E?. By comparing the coefficients of these parameters one arrives a t the following set of rules: [Pp,
P.1 = 0
1
i [ p p j J , I ]= ~ J ' A
- vpiPv
I (1.35) (1.36)
These rules are of course obeyed by the particular realization (1.31), in terms of differential, operators, from which we started:
P,,= i a,,
1
J p , = i(zp2, - x. Zu) -
&=i($
(1.37)
+_a). i
The subalgebra (1.34) is, naturally, the familiar set of Poinear6 rules. The pair (1.35) merely indicates that the supertranslation generator S transforms as a Dirac spinor under Lorentz transforniations and is unaffected by space translations. The anticommu-
Supersymmetry and Superfields
67
tator (1.36) which closes the system reflects the truly novel aspect of this symmetry. That it must be an anticommutator rather than a coinmutator can be seen in several ways. Sotice firstly that the matrix y,,C is symmetric. Secondly, the differential operator expression for S involves the anticomniuting operator ?/@. This differential operator anticommutes with E , which means that, in general, S should anticommute with E . Going back to the commutators (1.33) and keeping only and c2 one finds
[E’S, EZS] = E’y,&aPcl
(1.38)
and, on using the anticoinmutativity of E with S, this yields the anticomniutation rule (1.36). The appearance of an anticommutator in the algebraic structure is a new phenomenon in particle symmetry. In the mathematical terminology the system (1.34)-(1.36) is called a graded algebra. The linear operators F(6) given by (1.32) belong ,to what is called an “extended Lie algebra” in that some of the co-ordinates E belong to a Grassiiiann algebra. (Mathemntical precision in such niatters/will not be emphasised here. Most of the mathematical operations needed here are quite elementary and nre familiar to particle theoreticians.) We conclude with a remark on the fermionic quantum number. Corresponding t o the phase transformations (1.23) which serve to assign one unit of feriiiion-number to the positive chirality coinponent O+, one can dcfine the operator
F
=ioy,
a eij ’
(1.39)
which commutes with P,, and J,,,and with S gives the commutator [S, F] = i y 5 S .
(1.40)
This shows that S- carries one unit of fermion-number while S+ carries minus one unit.
D. Unitary Representations I n order to generate a unitary representation, the operators P , J and S must be real. More precisely, the linear form 1 iF(6) = b,,P, - - w,.J,,. ES 2
+
must be self-adjoint. Since the parameters b and w are real numbers while E is a Majorana spinor, this reality means simply P, = Pf?,
Jp, = J t , ,
S =CST.
(1.41)
Now the unitary representations of the Poincard group are well known, SO our problem is to find out how these must be combined so as to form representations of the extended group. In solving this problem the c h i d projections S%play a crucial role. The anticommutator (1.36) resolves into three distinct pieces when chiral projections are taken, [S-, s-}= 0, [S+,S+}= 0, (1.42)
ABDIJSSALAMand J. STRATHDEE
68
That this must happen is clear from the fermion-number assignments: since there are no operators in the algebra which carry two units of fermion-number, the anticommutator (S-, S-Jmust equal zero, and likewise for {S+, S+]. Since the chirid operator S- has only two independent components and since these anticommute with each other, it follows that the product of three or more of them must vanish, S.-S,-S,- * * = 0. (1.43) The same is true of products with three or more adjacent S, factors. The next important remark is that supertranslations must leave invariant the manifold of states with fixed 4-momentum since S , commute with P,. On such manifold the anticommutator {S+,S-) becomes equal to a. fixed set of numbers, and the operators S+ end S- are aeen to generate a Clifford algebra. Since this algebra has just sixteen independent numbers - the products with up t o four factors of S, and S- - its one and only finite-dimensional irreducible representation is in terms of 4 x 4 matrices. The manifold of states with fixed 4-momentum is therefore reduced hy the action of supertranslations into four-dimensional invariant subspaces. On the other hand, the manifold with fixed 4-nlonlentum is reduced by the Wigner rotations into (2j 1)-dimensional subspaces where j , the intrinsic spin, is integer or half-integer. Taken together, the combined action of Wigner rotations and supertranslations must give rise t o invariant subspaces with 4(2j 1) dimensions. We now show in detail how this comes about. Following the method of Wigner we start with a state a t rest,
+
+
P,=1M,
p=o.
(1.44)
Tn this frame the Wigner rotations coincide with the space rotations generated by J = ( J Z 3J, 3 , ,J1,)and one can assume that the chosen state carries a definite angular momentum j and z-component j:. The basis states for an irreduzible representation of the Poincar6 group are obtained by applying to the chosen state firstly the space rotations which serve to generate the 2 j 1 values of jz and, secondly, the Lorentz transformations which generate states with a n arbitrary 2 and Po = To generate the basis states for an irreducible representation of the extended Poincar6 group we can apply the operators S,, S- and their products to the 2j 1 rest states and then apply Lorentz transformations to the 4(2j 1) states which result. We shall suppose that the original 2j 1 states are annihilated by S,. (This is not a restrictive assumption since, if the original states were not annihilated by S,, we could have multiplied them either once or twice by S+and picked out from among the resulting states a set of 2j’ 1 states which certainly are annihilated by S+.)Now multiply each of the 2j 1 states by S- to generate Z ( 2 j 1) new states. Since S- carries one-half unit of angular momentum, these new states will comprise a pair of multiplets with spins j - 1/2 and j 112. Next apply the operator S-S- to the original states. Since, in view of the anticommutativity, the product S-S- has only one independent component, this operation will generate another multiplet of spin j . The rest frame basis is now complete since products with three or more factors of S- must vanish. If the original states carried fermion-number f then, since S- carries one unit of this number, the states generated in this way carry f 1and f 2. I n summary, the rest frame states have the spin-fermion number content
+
+
+
+
+
I-. +
+
+
+
+
(1.45) (For the case j = 0 the multiplet j
- 1/2 is of course absent.)
Supersymmetry and Superfields
69
Starting with the states Ij J r which satisfy
~0IL)r= lid1 Pljz)r = 0
i. = -3,
= IiJr i19
J121iz)J
-i
+ 1, ..., i
(1.46)
FIiJr = IiJr f ,
s+\i:)J = the rest frame basis is completed by adjoining the states Ij:, l.}J+l,II = & 1/2 and I jJr+? defined by
(1.47)
(J.48)
(1.49)
The normalizations me chosen so that all states have the same (positive) norin. The positivity of the xiorni is besic to the unitarity of the representation and it is e consequence of the reality condition s+= Ci3-T (1.50) and the anticornmutation rules (1.42).The last of these rules reduces on the rest states to the form (As+,
s-}= --I +2 iys y0C11.l,
or, on taking account of (1.50)
-
1 - iy5
IS-, s-}= 2 YOM. By nieans of these rules it is straightforward to construct the action of S on the states defined by (1.46) and (1.47). Oric finds 1 SijJJ = -z Ij:,?)J+I
i
Cf+T(i.),
A
(131)
The states \jl, I.) niay be resolved into multiplets with angular momentum j j - 1/2 but this will only comp1icat.e the forinulae (1.51).
+ 1/2 and
70
ABDUSS.xLa>r and J. STRATHDEE
To coiiiplete the basis system it reniains only to apply the Lorentz boosts. There are many ways to do this, but perhaps the most commonly used conventional schenie is the one due to Jacob and Wick. The rest frame 4-momentum P p
=
is boosted to the general forni p,, = N(cosh a, sh iy sin 0
w, o,o, 0) COB
4, sh a sin f3 sin 4, sh a eos 0 )
(1.52)
(1.53) where the angles a , 0 , 4 are restricted to the ranges
o5o5n,
Oln
--x
<4 5 R.
Under this convention the so-called helicity states are defined by IP, i:>f= W,)lA)f
(1.54)
etc. The action of the supertranslation operator on these boosted states is obtained fronr (1.51) by applying the operator (1.53) to both sidcs. From the fact that S is a Dirac spillor it follows that U(L,) SU(L,)-1 = n(L,)-1 s,
(1.55)
On multiplying both sidcs of the boosted form of (1.51) by the matrix a(L,) one obtains the formulae:
-1i SlpjJ,
=
-CA
lpj~~),+l f i + T (4 ~ ,,
(1.56)
where the spinor coefficients U&J,
A) arc defined,
(1.57) The spinors u(p, 2 ) are positive energy solutions of the Dirac equation. ( p - ill)u = 0, and are normalized by WP, A) U-(P,2') = M a w , i.e. 21(p,>.) U(p,?,') = 2 M 6 1 ~ ~ .
The formulae (1.56) which express the action of S on the st.ates of an irreducible'representation of the extended symmetry take the form given regardless of the boosting conventions. Such Conventions affect only the functional forin of the spinors u(p,2).
Supersymmetry and Superfields
71
This completes the discussion of unitary irreducible representations with a finite mass. These representations are labelled by three parameters, &I, j and f, and generally incorporate four distinct irreducible representations of the Poincar6 group (as indicated in (1.45)).In the remainder of this paper, which is concerned mainly with renormalizable Lngrangian models, we shall meet only two of these. (1) The fundamental representation,
f = j = 0, with Poincar6 content
(Oh@
(i), 0
(Oh
and of course the antiparticles
(2) The “vector” representation, f = - 1, j = 1/2, with Poincard content
which is self-conjugate.
E. Generalizations To concludc this chapter on the purely group theoretic aspects of the relativistic symnictry we shall consider very briefly some possible generalizntions. The feriiiionic estension of the Poincard group dealt with above was a inininial one in that it employed (t single Xajorana spinor. The most significant sacrifice imposed by the mininiality requirement - combined with the idea that the new spinor co-ordinates should carry a fermion-number is the loss space reflection symmetry. If this is to be restored then a generalization is needed. The siniplest generalization is from Majorana to Dirac spinors which can be subjected to complex supertranslations. The foriiiulae (1.27) are simply replaced by
-
i
ax, = - Ey,e
2
2 -
- - ey,E, 2
(1.58)
so = E ,
where 8 and E are anticommuting Dirac spinors. It is a simple matter to prove that these forni a n algebra. The anticommutator (1.36) is here replaced by the set
(1.59) where S and are independent generators. (In unitary representations they are related by hermitian conjugation, = Styo.) The complex extension characterized by (1.59) is certainly compatible with reflection symmetry and fermion number. Thus, for example, all four components of S can carry one unit of fermion-number and this is clearly consistent with the parity transformation
s
-
Yos.
The fundamental representation has sixteen independent rest franie states. These can be defined by successive applications of S,, to a “lowest” state lo>-, (carrying zero spin and ferniion number -2) which is annihilated by The successive states (all of the
sa.
72
ABDUSSALAMand J. STRATHDEP:
same mass) are:
(LGO)
I.Br),
= &qd,lO)-,
I.Brd),
= ~,S,S,SblO)-,
-
They comprise: a scalar difermion and its antiparticle; two spin 1/2 particles of opposite parities and their antiparticles ; two scalard, one pseudoscalar and one polar vector, all fermionically neutral. I n Section I V we shall present a renormalizable Lagrangian model for this multiplet. It is arrived a t in a rather indirect way as a special example of a class of niodels which, although based on the Majorana supersymmetry, are set up in such a way that parity conservation emerges. Further generalizations of the supertranslations are easily invented. For example,
(1.6 I)
where e k and &k, k = 1 , . .., N , are Majorana spinors. This scheme admits an O ( N ) symmetry with a consequent enrichment of the multiplet structure. This niultiplet appears to exhibit an SU(2n) structure in the rest frame. Section 11
A. Local Superfields I n Section I a new symmetry was defined in abstract terms. Through their action on the co-ordinates of an enlarged “space-time” the supertranslations were seen to constitute an extension of the group of inhomogeneous proper Lorentz transformations, the Poincar6 group. Consideration of the algebra of infinitesimal transformations led to the formulation of commutation and anticommutation rules among the generators. On the basi~,of these rules the unitary irreducible representations, or “supermultiplets”, were analysed. By means of such considerations a general understanding of the new symmetry is achieved and it becomes possible to formulate dynamical statements in the form of selection rules, identities among scattering amplitudes, etc. To reach a deeper understanding it is necessary to formulate a local field theory for the supermultiplets and to set up Lagrangian models for its realization. A necessary first step in this programme is the concept of the local mper/iekE which we now conpider. The prototypical local superfield is the real scalar function @(x, e), whose expansion in powers of 0 was given previously,
(11.1)
73
Supereymmetry and Superfields
This field is local if it commutes with itself a t spacelike separations,
This definition of locality is compatible with the usual requirement that ordinary Bose fields commute among themselves and with fermionic fields while fermionic fields anticommute aiiiong themselves (always at relatively spacelike points). To ensure this compatibility it is necessary only to require that the spinor co-ordinates 0 and 0’ anticommute with each other and with the spinor field components +($) and ~(z), on the one hand, and commute with all boson field components on the other. The actiori on @(z,0) of an infinitesinial transformation from the algebra of the extended Poincar6 group is given, according to (1.31), by
(11.3) This formula defines the “scalar” nature of the superfield @(z,0). From it can be deduced (by substituting the expansion (11.1)) the transformation properties of ‘the various cotnponent fields. Thus, with respect to the Poincar6 group, A , F , G and D are scalars, V , is a vector, t,b and x are spinors. The behaviour of these components under a pure supertranslation is given by:
SA = E $ , 1 @J
= 1
(F
+ y5G + iypy5V‘/, - i3.A)
E,
(11.4)
=
1 (D- i 3 F + y,iOQ 2
- iy,y,i2V,)
E,
6D = -Ei2-x. Ferniion-nuniber is assigned according to the convention of Section I, i.e. 0, carries unit fermion-nutuber, F = 1. If the supcrfield O(z, 0) as a whole is required to carry F = 0, then the following distribution of F values is implied: F=
2 :F+iG,
F = ‘1.: F=
+-, x + ,
0 : A , V,,D,
F = -1
:‘++,x-,
F = -2
;F
6 Zeitschrift ,.Fortsclirltte der Physlk”, Heft 2
- iG.
ABDUSSALAMand J. STRATIIDEE
74
These assignments are compatible with the reality condition
qz, e)*
= ayz, e)
(11.6)
which may be imposed. This condition would imply that Bose components A , F,Q, V. and D are real, while spinor components are subject to the Majorana condition,
q
x
= CglT,
= CZT.
(In taking the complex or herniitian conjugate of a superfield it is always necessary to reverse the order to anticommuting factors; = $0 = eC$., etc.) In general the superfield O(z, 6) ia locally irreducible. By this we mean that is cannot be decomposed into a sum of distinct, independently transforming components without the intervention of non-local operations. (In this it is analogous to the ordinary vector field V,(z) which can be expressed as the sum of transverse and longitudinal parts, V,” = V,C a&, only with the help of the non-local operator l/P;i.e. 4 = ( l / P ) a,V,. However, such a non-local decomposition is of considerable help in understanding the structure of the representation (11.4) and its derivation will be our first task.) An important role is played in the derivation by the operation of cocuriunt di#erentiution. The covariant derivative of Q, is defined by
(a$)*
+
(11.7)
It is covariant in the sense of transforming like a spinor,
LD,
I
=2
JpV1
up.D
J
(11.8)
with respect to homogeneous Lorentz transformations and an invariant with respect to translations and supertranslations,
[ D , P,] = 0 ,
(D, 8)= 0 .
(11.9)
These results are esxily verified with the help of the differential operator expressions for the generatom (1.37). From (11.7) if follows that the covariant derivatives generate a “supertranslation” algebra of their own,
Dpl = -(YpC)4i 4 .
(11.10)
Again it is useful to distinguish the chiral coniponents D+ since, for instance, positive chirality components anticomniute and the successive application of three such differentiations must annihilate any superfield. The covariant derivative is basically a Majoran8 spinor and for this reason it is useful to define the conjugate from B by
Da= (C-1).@D,
-
or D , = -DrTC-l.
(11.11)
The usefulness of these operators lies in the fact that a - D + and B+D- are invariants of the extended group. Applied to one superfield @(x, 0 ) they yield another whose cornponents transform in exactly the same way. Out of these operators and their products we can construct invariant projections. We begin with some simple identities. The most useful identities are those which spring immediately from the anticommutativity among components of the same chirality,
D+(D-D+)= 0
and
D-(D+D-)= 0,
(11.12)
75
Supersymmetry and Superfields
which imply the identities
(D+D-) D*(B-D+)= 0 .
(11.13)
Repeated use of these identities together with the anticommutation rules
*
1 iY5 i 2 , (D*, D&}= 2
enables one to demonstrate
((D-D+)(D+D-))2= -4a2(D-D+) (D+D-),
((D+D-) (D-D+))2= -422(D+D-) (ED+). These formulae then imply that the non-local operators 1
-
E+=-- 4a2 (D+D-) (6-0,), (11.14) 1 E- = -- (D-D+)(B+R-), 4 6%
are idempotent and orthogonal, i.e.
E+a = E+,
E-2
= E-,
E+E- = 0.
(11.15)
The operators (11.14) are two of the projections needed for the deconlposition of @(x, 0). There exists only one more and it is given, of course, by
=
1
=1
1 +(D-D+,U+D-), 4a2
(11.16)
1 +(OD)'. 4a2
The decomposition of @ ( x , 0) corresponding to this non-local resolution,
+ E- + = @+(x,e ) +
@ ( x , 0 ) = (E+
is given explicit,ly by
A
= A+
El)
@(x> 0)
e)
9
+ WZ,
+ Jt- +
F = F+
+ F-,
Jt1J
G = iF+- iF-,
(11.18)
+V,~, x = -i2Jt+- i 2 $ - + i2Jtl,
V , = ia,A+ - ia,A-
G*
=
(11.17)
+ A- + A , ,
Jt = Jt+
D
el,
-PA+ - PA-
+ a2A,.
ABDUSSALAMand J. STRATHDEE
76
where Vl,, is transverse and $* are chiral. The expreqsions (11.18) can be solved for the irreducible components,
(11.19)
(11.20)
some of which are seen to be non-local. Only if the components 1 and D arc theinselves first and second derivatives (ax' a i d PO' of some localfields x' and D' can the original superfield be considered locally reducible; otherwise it is not. The transformation behaviour of the reduced components (11.19) and (11.20) can be dcduced from (11.4). One finds,
(11.21)
(11.22)
dVlp = -Su,.iy5
.
i&$,
J
The so-called chiral scalar representations (11.21)and the transverse vector representation (11.22)are of fundamental importance in the development of renormalizable Lagrangian models. For this rcason it is necessary to dicuss their properties in some detail and, to gain a deeper understanding, from other points of view. We begin with the chiral scalars. Since the superfield @(x, 0) from which @+ and 0-are obtained could have been subject to thc reality condition (11.6), it is clear that the chiral representations are conjugate, @+*. We shall therefore restrict our attention to the positive chirality @+ The existence of this kind of superfield rests on a very simple fact. The behaviour of x and 8 under supertranslations is such that the complex 4-vector N
(11.23) transforms according to d.zp = iE+y,0+.
(11.24)
Supersymmetry and Superfields
77
That is, 6z depends on O+ (or 8-) but not on 8- (or a+). This means that the space of complex-valued functions, +(z, O+), must be invariant under the action of the extended group. Since 0, has only two independent components, the expansion of 4 in powen of 8, must terminate in the sec0r.d order rather than the fourth. Indeed, one can write
= @+(z,e),
where, in the second line, we have effected the translation z --t 2 by means of a displacement operator. If the displacement operator ie expanded in powers of 0 and the terms in (11.25) are arranged in the standard form (11.1)then one will find precisely the coefficients which are indicated for @+ in the expressions (11.18). Another way to characterize the chiral superfield @+ is by means of a differential condition. Thus, the covariant derivative of @+ is:
indicating positive chirality with respect to thc spinor index. In ot.her words, @+ is annihilated by the negative chiral part of the covariant derivative,
D-@+= 0.
(11.26)
This set of covariant constraints has as its general solution the expansion (11.25). The positive chiral scalar is therefore defined by (11.26). A property of the chiral scalars which is of crucial importance in the construction of Lagrangians is their closure with respect to multiplication,
o) @+yx,e)
= @+yz, 0).
(11.27)
That the prcduct of two positive chirality scalars (at the same point) should yield a third becomes evident when tey arc expressed as functions of z and 0,. Alternatively, the linearity of the differential operator D-implies that it will annihilate the product if it annihilates the separate factors. The component fields of @&” defintd by (11.27) are given in terms of the components of @+’ and @+ as follows: ‘
A+”(X) = A+@)A d ( x ) ,
A B DSALAM ~ and J. STRATHDEE
78
Generalization of (11.27) and (11.28) to products with any number of factors is clearly possible. Indeed, it ,is a simple matter to show that the local function v(@+(z,0 ) ) has the component expansion
UP+) =
e-1/4a3y1e
+ J-$+v'(A+) +
v(A+)
(11.29) where v' and v" denote the first and second derivatives with respect to the complex variable A+ of the function v(A+).This function must be analytic in general but, in practice, will be restricted to polynomials. The above properties of the positive chiral scalar @+ are easily translated for the conjugate representation @-. Thus, the negative chiral scalar @- is expressible as a function of z* and 0- and one can write for it the expansion (11.25') Such superfield9 are characterized by the differential constraint
D+@-= 0.
(11.36')
Finally, the product of two negative chiral scalars a t the same point yields a third: negative chiral scalars are closed with respect to local multiplication. The local product of a negative chiral scalar with a positive one does not yield a chirnl result. This can be seen from the rather trivial observation that the product of two functions, +(z, 0,) and +'(z*, O-), must depend on both z and z* as well as 0, and 0-. The result niust therefore be a superfield @"(z. 0 ) of the general type with which we b e p n this section. Its components are given, in the standard notation, by
A" = A+A-',
+" = A+$-' + $+A_',
+ F+A-', G" = -iA+F-' + iF+A-', Y," = (i8,A+)A_' + A+(-ia,A-') + ++TC-ly,$-', 2'' = (ia,A+)7 . t ' + y.$+(i&A-')- (i2$+)A_' - A+(i2+-')+ B++F-' + SF+$-', IY = (-8 2A+)A_' + 2(a,A+)( 8 . L ' ) + A+(-Z2A-') + 4F+F-' + 2$+TC-1(i2- ik) #-'. F t= A+F-'
(11.30) This superfield is clearly not locally reducible. The fermion-number content of the chiral superfields is easily established. Since 0, and 0- carry F = 1 and F = -1, respectively, it follows from an examination of the expansion (11.25) that A+, $+ and F+ must carry F = f , f - 1 and f - 2, respectively. The value f , associated with A+ is a n invariant of the representation. (We shall normally take f = 0 so that $+ carries F = -1, i.e. ++ serves to annihilate one fermion and $+ creales one.) The negative chiral components A _ , F- carry the values -1, -f 1, -f 2 if @- is identified with @+*. (As a general rule we shall not identify @- with @.+* but treat it ns an independent field with f = 2. With this notational convention, $carries F = - 1 and annihilates a ferniion. Note that A- annihilates a difermion.)
+
+-,
+
79
Supersymmetry and Superfields
New categories of local representations are obtained by generalizing the scalar superfieJds to spinors and tensors of arbitrary rank. That is, one regards 0(z, 0 ) not as a single scalar superfield but as a set of superfields belonging to one of the finite-dimensional representations of the proper Lorentz group. With respect to translations and supertranslations, these fields transform as scalars, but under the homogeneous (proper) group they transform according to
a+, e)
--f
~
(
5
1
el) = q n )qz,ej,
(11.31)
,
+
+
1) (2jz 1)dimensional where the matrices D ( A ) belong to one of the familiar (2j, representations, 9 ( j , , j 2 ) - or a direct sum of such representations. This kind of generalization applies equally to the chiral (0*) or non-chiral (0, 0,) types. Two examples will illustrate the main features. The first example employs a chial spinor superfild, (Ya-)+, which has negative chirality with respect to its external spinor index, (1
+ i y d Y-+
(11.32)
= 0,
and positive chirality with respect to its internal structure, D-Y+ = 0.
(11.33)
The expansion in component fields has the general form (11.25)
where U- and V- are negative chiral spinors and H,, is a 4-vector. Under an infinitesimal supertranslation these comp0nent.s transform according to
(11.35)
6 V- = y,,iOE-M,. This represcntation is basically complex. Thus, even with the most favourable assignment of fermion-numbers F = 1, 0, -1 t o U-,M,, and V-, respectively, it is not possible to treat M, as a real vector. However, if certain constraints are applied, this representation will turn out to be equivalent to the real O1.We shall return to this question after setting out the second example. The second example eiiiploys a chiral spinor, Y++,which has positive chirality both internally and externally, i.e. (1 - iy5) !P++= 0
and
D-!P++= 0.
(11.36)
-4gain the expansion in component fields has the general form (II.%),
(
1
+ (D ( Z ) + 1 u,,F,,(z)) e+ + -1 o-o+r+(‘~) , (11.37)
Y++@e) = e - l l ~ T * y ~ ~u+(z)
where U+ and V+ are positive chiral spinors, D is a scalar and F,, is a self-dual antisymmetric tensor. (It is of course theopposite external chiralities that govern the appearance of a vector M,, in !P-+ and of a scalar-tensor combination D,F#, in !P++.)The
80
ABDUSSALAM and J. STRATHDEE
component fields transform according to
1 1 -B- V , - - B+ii+U+, 2 2
(11.38)
This representation also is basically complex but capable of being made real by means of certain constraints. Sow consider tho representation (11.35) to see if it can be made self-conjugate. Gincc Uand V- carry F = 1 and - 1, respectively, we should identify the conjugate of one with the other. For the correct balance of dimensions and also to have the correct chiralities, the associat,ion inust take the form
V- = w i X V - T ,
(11.39’)
where o is a number to be determined. If the identification (11.39) is to be coinpatiblc with supertranslations we niust have 6V- = wiiWGi7-T) i.e. according to (11.3.5), ~ , , i i k M= U oiXyPT~,*MP*, = -tfJi2yP&-JI,”*
for arbitrary E-. This gives constraints on M , viz.
o = a,(~,
- wiv,*) -
ap,,
- wJr!,*).
(11.40) ,-
Finally, it is necessary to verify the compatibility of (11.39) and (11.40) with tho espression (11.35)for GiM,. This requires !(uI = 1. The phase factor can always be absorbed and so we shall take (0 = 1. To summarize, taking
where V I Pis real and transverse, A l is real and I)l is hlajorana, we find that the transformations (11.35) reduce to the form (11.22},i.e. a real transverse vector representation. Similar considerations applied to the representation (11.38) show that it too can he real.
Supersymmetry and Superfields
81
In this case the reality conditions are
V+= iaCV+T, 2Fp. = apv,-
+ -2
(11.42)
i
a,vp
&p.lQ(alVQ
- apvl),
with V , and D real. The transformation rules (11.38) then take the form
6 v; = S+ypE+ 4-a+y,u+,
BD
=
(11.43)
1 - 1 2 U+i3E+-2 z+i2U+.
This representation also is related to O1 though it is not locally equivalent. The conncction is made through the non-local relations,
or, more compactly, YJ++= i 2 D - 0 , .
B. Green’s Functions and Invariant Amplitudes One of the main objectives of local field theory is the computation of Green’s functions. The structure of these many-point functions is restricted by symmetry considerations and, in the present case, by the requirement of invariance under the action of supertranslations. Such structural constraints are best exhibited by expressing the Green functions in teriiis of invariant amplitudes and we shall examine here some applications of this idea. Since it is our purpose to reveal the implications of the new transformations in the extended relativistic symmetry, we shall pay little attention to the subgroup of Poincar6 transforniationd, which can be handled in the usual way. It will therefore be sufficient to deal with a system of fields comprising only chiral scalars #= and their complex conjugates. The appending of Lorentz indices to such fields so as to include nonscalar representations is a relatively trivial modification and would not affect the main argument. (Sotice also that the non-chiral real field O1 can always be respresented by a chiral spinor, !P+-= D+O1,and so included in the general scheme.) To begin with we shall ignore the fermionic quantum number which distinguishes 0from 0+*. Insofar as supertranslations only are concerned, these fields are equivalent. It will therefore be sufficient to analyse the structure of innny-point Green’s functions referring to products of the fields #+ and O-, only. Afterwards we can introduce the ferniion-nunibcr and quite easily pick out the amplitudes which conserve it.
ABDUSSALAX and J. STRATHDEE
82
A considerable simplification in notation in achieved by amputating the displacement operators, exp (F1/4 8 8 y 5 8 ) , which always appear as factors in the chiral superfields. Thus, we define the amputated fields,
(11.44) which depend on only one chiral component, either 0, or 8ordinate. This property is expressed nrialytically by
a&+ 1 -iy5 -=-a6+
2
a&+ a8
(8- or a+), of the spinor co-
= 0,
(11.45)
Under supertranslations the amputated chiral fields transform according t.0
= zr
ad&+ z+i30+@5. +
(11.46)
88 'f
On occasion in thefollowing we shall use the abbreviatednotation d,(l) = d5(zl,Ol). The n-point Green's functions 6 are expressed aa vacuum expectation values, &( 1,
...,n) = (01 T*b+(I ) .-.&+(r)d-(r + 1)
d-(n)
lo),
(11.47)
in which the T*-ordering is used so as to simplify the transformation properties (i.e. avoid the comp!ications due to non-covariant surface-dependent terms). Our main purpose here is to consider the implications of the requirement that G! be invariant with respect to the transformations (11.46).This requirement is embodied in the single equation
(11.48) or, equivalently, in the chiral pair,
(11.49)
Fortunately i t is quite easy to find the general solution to these equations. One starts by looking for solutions of the form 8 = e-WM, (11.50)
Supersymmetry and Superfields where
83
W is chosen to satisfy the inhomogeneow equations
(11.51)
There are many ways to satisfy these inhomogeneous equations and we shall return to this question. For the moment assume that one solution has been found. On using this W in (11.60)and substituting it ivto the conditions (11.49)one finds that the amplitude M inudt satisfy the simplified equations
’ aM
O=C-,
68,-
(11.53)
aM
O=C-. r+i
88,
.
Thus, quite generally, the amplitude M can depend on the chiral coordinates 0,+, . ., 8 , and Or+,-, ., On- only through their differences. For example,
..
Jf = M(Olr+,
- -, *
or- l r + ;
Br+ln-,
* * *>
,
On-ln-)
(11.53)
where Ojr+ = 8,+ - Or+ and 01,- = 0 , - On-, There are altogether n - 2 such differences. Further restrictions must arise for disconnected contributions to the Green functions: only differences between co-ordinates referring to a connected piece are admissible. To simplify the discussion we shall tacitly assume that only connected amplitudes are concerned. The expression (11.53)incorporates the supertranslation invariance requirement for pll chiral Green’s functicns with only two exceptions. The exceptional cases involve the products of fields all of which have the same chirality, i.e. r = 0 or r = n, and these must be treated separately. The reason for this is the inipossibility of constructing a W satisfyin,o (11.51)when all 0’s have the same chirality. We shall return to the exceptional cases after considering the form of W. It is a simple matter to verify that the linear operator
n
is a solution of (11.51). (One needs only to draw on the fact
2 aj = 0 resulting from 1
PoincarB invariance). The expression (11.54) meets all the requirements but it is not very symmetric in appearance. ?Vhile regretting this asymmetry, we do not feel that it is a very serious shortcoming. We remark only that alternative solutions to (11.51)must take the form
w = wr, + W’,
where W’ is any linear operator constructed entirely from co-ordinate differences and hence satisfying the homogeneous form of (11.51). By exploiting this arbitrariness one could construct a more symmetrical solution but we shall not pursue the point. . Finitlly, we must deal with the exceptional, or what might be called “monochiral” cases r = 0 and r = n. Suppose, then, that r = n and the constraints (11.49)take the form
(11.55)
84
ABDCSSALAM and J. STRATHDHE
The first of these equations indicated that 6 must be a function of the IL - 1 co-ordinate differences Oj,-. The second is solved by partitioning the n - 1 terms of the sum n-1
2 iajejn+= 2 Pk+
(11.56)
k
1
in all possible ways. The partitions involve from one up to n - 1 pieces,
pk+= i a ?I. 8iln+ .
+ ... + iajrelrn+.
(11.57)
For each partition form the monomial k
which clearly is annihilated by each member of the partition and hence by the sum. The equation$ (11.55) are therefore satisfied hy
(11.58) where the sun1 runs over all icdependent partitions to each of which there corresponds n scalar amplitude M p ) . The other iiionochiral amplitudes, r = 0, are decoiiiposetl in the sanie manner. Soiiie examples of these decompositions arc given in Appendix B, where they play an important role in the softening of ultraviolet divergences. The supertrnnslation invariance requirement is fully met by the expression (11.58)in the iiionochiral case and by the expression (11.501with Jf and IY given by (11.53) and (11.54) in the niised cases. If we had bcen dealing with chiral spinor and tensor superfields, these expressions would still be valid, the appropriate Lorentz indices being tacit in the amplitiides 111 and J1lPJ.These aniplitudcs must of course respect the Poinoilr6 group. Their decomposition into scalar invariants would follow by standard methcds which we need not go into. (Even for the case of scalar superfields the decomposition of 1l-Iinto invariant amplitudes, which is just the problem of resolving multispinors of rank I ;n - 2, can be quite complicated.) Having denlt with the supertranslations it is necessary to consider now the phase transformations generated by fcrmion-number. On the chiral scalar superfields these transformations trtlie the form
(11.59) and so d- must be dist,inguished from d+*.The implications are clear. In the general case the amplitude 31 must. satisfy an identity of t.he forni
,V(e-iaO+; eiaO-)
= e2iadv-M(/3+, 0-),
(11.60)
where N - is a n integer defined as the number of times 0-occurs in e! niinus the number of times @-* occurs, (II.Gl) LV- = N ( 0 J - N ( @ - * ) . The condition (11.60) plainly constitutes a fairly strong restriction on the forni of the amplitude M .
85
Supersymmetry and Superfields
To illustrate the application of the symmetries discussed here we list a few simple examples. Firstly, from among the 2-point functions we have
(T*d+(l)&*(2))
=
1 ;z 812-0,2+u,
(11.62)
(T*&+(I)&+*(2)) = exp (-ijl+ialel+) N , where u and ill are scalar functions of the invariant zi2 = (q - z,)~.The only other nonvanishing 2-point functions are those which result from (11.62)by reversing the chirality assignments. We shall have more to say about expressions (11.62) when we come t o the insertion of intermediate states. From among the %point functions we have, for example,
(11.63)
-
where the scalar amplitudes M,, M,and M 2 depend on the invariants .&, xi3 and q 3 z2,. The iiiiportance of Green's functions in local field theories lies in their close relation to the qunntities of more immediate interest, the scattering amplitudes or S-matrix ele~iiaits.The latter are extracted froin the former by means of the limiting procedure known as tlic LSZ reduction technique. It is not our intention to derive and justify this procedure as applied to superfields but only to point out some of the novel features and eshibit a final formula. To this end we begin with a discussion of the 1-particlc contribution to the 2-point functions: the so-called free propagators. .2 scalar supermultiplet of mass DI comprises the particle states Ip),,, lp, i),,Ip)p defined in Section 1. Now we must allow also for the antiparticles $),, \ihich belong to a distinct supermultiplet of mass M . The action on these states of the supertranslation gmerator S is given by 1
S\p), = i 1
Ipl), &+T(pj.), .(11.64)
1
Sip), = i A
Ipi), Cn-T(p,I),
(11.65)
Ir
A
ABDUSSALAM and J. STRATHDEE
86
where u and v denote the usual positive a r d negative energy Dirac spinors. From (11.64) and (11.65) and the field transformation laws (11.46) expressed in the form
one can derive a number of relations among the matrix elements of the component fields between the vacuum and the 1-particle states. On suitably normalizing the components A+ and A- one finds the following non-vanishing matrix elements:
o(Fl A* 10) = 1, - 1 m ++ 10) = V+(P, A)>
(01 A+ IP>o= 1* (01 ++ I P h = U+(PA),
(11.66)
-*(PI F+ 10) = --M>
(01 F+ IP% = --M,
(01 A- ~P)z= 1,
-2@1
A- 10) = 1,
-&Zi
(01 t I P h = U-(PA),
*- (0)
(11.67)
= ?L(p#?),
F- 10) = - N .
(01 F- Ip)o = --M,
With a standard covariant nornialization for these states we can compute their contribution to the absorptive parts of the 2-point functions. Firstly, with zl0> x20,
(T*&+P)d-*(2)) =
J’g3
=
J
O(P0) 4 P 2
3&
1.Ha)(01 &+(I){ IP>O
0@1
+ c IPl)l l(P4 + 1P)z ,@I} 1
o(po)d(p2 - Ma) e-iP(zl--za) ((01 A+ IP>oo@l
+
dJ-*m 10)
&-W-* 10;
1
= --MLI+(~,
-
t
MZ) 2
e12-e,2+.
Similarly, when zl0< zz0the antiparticle niultiplet gives the negative frequency part. Taken together, these expressions clearly take the form (11.62) with the amplitude, a, given by a(z12)= - & A p ( z , , ; Ma). (11.68) For the other 2-point function one finds a different complexion. Taking xl0 > x,, one fin&
tqP0)d(pa - M Z) e- ip(=+a) 1 =
J o2 dP q p , ) d ( p 2 -
~
2 e-ip(zt-za) )
= exp (-ijz+ii+181+)d+(z, - 2,; Ma).
{
1
M a e+ ij,-(p + M ) e2- - 4
1-
e 8, e If
-+
1-
87
Supersymmetry and Superfields
Similarly when q 0< q0.This again conforms with (11.62), the amplitude M being just the Feynman propagator dF. Free propagators for the various component fields are easily picked out from these expressions. Finally, we give the supersymmetric version of the LSZ reduction formulae. Consider one of the Green functions 1'( T6+(5J n(d) lo), (11.69) where n(6) denotes a product of superfields the details of which do not concern us. According to well-known arguments, the asymptotic behaviour (zo+ f o o ) of (11.69) is dominated by the l-particle contributions. Thus, formally, 89 xo + 00,
More exactly, the 1-particle matrix elements of 17 are given by
obl n l o ) + c fi-u+W lb4 n 10) 1 =i
J dzeiP* [ ( a 2
+
~
2
M
0-8, f(Pl
10)
(01 ) ~ d + ( e) z ,17 lo)]
(11.70)
evaluated on the mass shell, p 2 = M2, with po > 0. The various matrix elements are associated with distinct &coefficientsand so can easily be picked out from (11.70). Likewise , the antiparticle matrix elements are given by the integral
= i J dre-ipz
[(a2
+W )
(01 ~ * d + (e)z ZT , lo)],
(11.71)
again with p 3 = M2 and p , > 0. The equations (11.70) and (11.71) constitute the reduction formulae d+.The same consideratiom applied to d- yield the reduction formulae
&I 17 10) + c 8+U-(P4 1b4 17 10) - y e+e1 M -
= i J dxe'Pz[(Sa
(01 17 1 9 - 2
+
~
+ c(01 I? a
= i J dre-iP+[(aa
+
2
(01
o(Pl
I? 10)
T ~ - ( ze), 17 lo)],
)
(11.72)
M -
i a - I fi+V-(P4
~
2
10(
)
-2 e+e-(ol I? 1 3 0
~ d - ( ze ,) IZ lo)].
(11.73)
It will be remarked that the two formulae (11.70) and (11.72) give precisely the same matrix elements although with different coefficients. This is a reflection of the fact that, in the asymptotic region, the superfields 4j+ and 0- are related in 8 simple way. They
ABDUSSALAX’andJ. STRATHDEE
88
behave like free fields and satisfy a pair of linear differential equations,
I1 -ODD, - MD- = 0, 2 --21 DD@+ M@+= 0,
or, in component form,
-PAT
+MF,
= 0.
Such differential equations, and their generalizations to the interacting regions whci e .they become non-linear, are properly the subject of Lagrangian field theory and it is to this subject that the following section is devoted.
Section III. Supersymmetric Lagrangiana
A. Kinetic Energy Terms and Renormalizable Interactions In this section we shall be concerned with the setting up of Lagrangian models to illustrate various aspccts of the theory. Only through a study of such models can one see how the new symmetry interacts with the standard considerations of unitarity, locality, causality and renormalizability. Now in ordinary Lagrangian field theories the locality requirement is met by starting with a local action which is a space-time integral over a local density function of the fields and their first derivatives, the Lagrangian. We shall adopt thiv rule also for the superfields. However, with fields which are defined over the eight-dimensional space of x and 0, it is necessary to decide what “density” means. A new concept seems to be needed, that of integration on the anticoiiiniuting 0. Such a concept has in fact heen described in mathematical literature. Xevertheless we shall avoid any explicit reference to this concept in the following. One reason for avoiding the f3 integral is t,hat, so far as we are aware, techniques for dealing with surface terms have not yet been developed and, while no doubt they will be, we suspect that they will turn out to be fairly elaborate. But our main reason for avoiding these integrals, however, is simply that we find we can do without them. All we need to do is make sure that any 0 dependence in the action functional is confined to surface terms of the usual (Space-time)variety. That is, we must ensure that the &dependent terms in the action density always take the form of Space-time gradients. Such surface terms do not influence the Euler-Lagrange equations which are governed entirely by the volume part of the action. These equations will be supersymmetric (i.e. covariant with respect to the extended relativistic symmetry) provided the volume part of the action is an invariant. The volume part will, in its turn, be invariant if the Lagrangian is a scalar superfield all of whose Odependent terms are space-time gradients. Therefore, the action associated with a region of space-time must take the form
S = J dzY(r,0 ) = J dx-Eo(x, 0)+ surface integral, volume
volume
89
Supersymmetry and Superfields
where Y transforms like a scalar superfield,
The corresponding 65 will clearly be represented by a surface integral. This asymmetrical treatment of z and 8 does not do full justice to the underlying extended space-time structure and our only excuse is that it is expedient. The &independent part of the action density may be characterized very simply. It must transform like a niixture of a t most three local representations, viz. the D component of the real (non-chiral) scalar @(z,0 ) and a real combination of the F components of the c h i d scalars @- and @-*. Recall that these components transform under the action of a n infinitesimal supertranslation according to
SD(z) = -si2x(z)
Because of the gradients in these formulae it follows that the volume integrals of D and 3'- (F-*) ore invariant. Another candidate for the action density, a real combination of F+ and F+*, must be rejected because of the fernlion-number which these components carry. There are no other candidates. Now the D coniponent of @ is simply the 0-independent part of ( E D ) ? @and the F component of @- is the &independent part of
[email protected], the &containing parts of these expressions are all space-time gradients. This can be understood by noting that every application of the covariant derivative D = (a138 - i / 3 26)either removes one power of 0 or adds one power along with a space-time derivative. Such are the ternis of which the action density is to be made, Y
-
(DD)'@ + DD(@-+ @-*).
(111.1)
Our use of the covariant derivative here rather than the ordinary 3/38 has no significance for the equations of motion. Given a single positive chirality scalar @+ and its conjugate @+*, what possible Lagrangians can one make? Naturally one must start with hilinears in order to generate linear, free field, equations of motion. It is not difficult to convince oneself that only one bilinear is admissible, 1 (111.2) 9, = 8 (DD)' I@+('. In terms of component fields the &independent part of -Yo is given by
~
1
1
1
1
A+*Z'A+ + ~= - l4 A+*PA+ ~ =+ laA+l2 ~ -2 4 2
$+(a - i3)I)++ IF+\*,
+ $+GI)+ + lF+12 + gradient term.
= laA+[2
(111.3)
The gradient term can be gathered into the &dependent part and forgotten. The active part of Y ois just what is shown explicit,l;yin (111.3).It has the form of a kinetic term and, taken by itself, would yield the equations of motion
PA, = 0 , 7
Zeltschrift ..Fortachritte der Physlk", Heft 2
i&+ = 0 ,
P, = 0 ,
ABDUSSALAMand J. STRATHDEE
90
and their complex conjugates. The main interest of the Lagrangian (111.3) is that it determines the dimensionality of the fields. Thus, in natural units,
[A+]= M ,
[++I = M3'2.
which is canonical for scalar and spinor fields with the kinetic term always required to have dimension 4, [ g o ] = M'
.
With @+ having dimension 1 i t is clear that no interaction terms can be constructed from thth single fieM without doing violence to the requirement of renormalizability if fermion-number is conserved. To see this, note that simple powers of @+ are forbidden since
Do(@++
@+%
+
@+S
+ -..)
'
carries two units of fermion-numbkr. The only terms which conserve fermion-number are of the form
( B W (1@+1*")
+
and these have dimension 2n 2. On a pcwer-counting basis, these terms conflict with renorinahability for n > 1. (Recall that DD has dimension 1 and for renormalizability a Lagrangian mu& not possess dimensions in excess of four.) Similar arguments applied to the case of a single negative chirality scalar 0-and its conjugate 0-* lead to the conclusion that the only possible fermion-number-conserving renormalizable Lagrangian is given by
49,
+ -8
(111.4)
where 9' is a parameter of dimension 2. This expression reduces to
plus gradient terms. The equations of motion are
and their complex conjugates. Thus, for a single chiral field of either type there th no psibzZity of setting u p a non-triz-ial theory without violating either fermbn-number conservdion M ~enormulizubility.However, for a system containing two distinct fields @+ and 0-, one can construct a non-trivial interaction term. This makes essential use of the closure property discussed earlier, viz. that the product of two scalar superfields of the eame chiral type is itself a c h i d scalar of that type. Thus, since @-* and O.+ have positive chirality, so too do their products @-*a+ and In both of these products the F components carry zero fermion-number and can be used in the Lagrangian. Thus, for the @+, @- system, the renormalizable fermion-number-conserving Lagrangian takes the form :
where h.c. etands for the (hermitian) conjugate terms. The parameters 9, M and g have dimension 2, 1 and 0, respectively. No other terms are admissible. h'otice, however, the lack of symmetry of (111.6)between @+ and 0-' type fields. Later, when we consider
91
Supersymmetry and Superfields
the question of parity conservation for the Lagrangian, this asymmetry will be relevant. Of the three parameters in (111.6) only two independent combinations can appear in physically significant quantities such as scattering amplitudes. This fact results from the invariance of the kinetic terms under the c-number shift, @+ --f @+
f
(111.7)
cJ
where c is independent of x and 0 . Such transformations affect only the scalar component A+ and it is clear from formula (111.3) that they do not alter the volume terms in (1/8) (DD)2I@+la. The remaining terms in (111.6), however, change as follows:
9'
--f
+ Mc + gca
Y
+ 2gc
bl --f M
(111.5)
9 +s. Otherwise expressed, the field transformation (111.7) is equivalent to the parameter transformation (111.8). Bnt quantities of physical interest (the quantum field theoretic analogoues of canonical invariants) should not be affected by a field tramformation. Therefore, they should not be affected by the parameter change (111.8); thcy must depend on only two invariants,
Ma
g and Y - - ,
4s
for example. We may conclude that no generality is lost by taking one of the parameters, M or 9,equal to zero. It will be instructive to examine the component structure of the Lagrangian (111.6). By means of the formulae (11.28) and (11.30) one obtains the result
Y = laA+I2
+ $+ii+$+ +
- $-(M
+ IaA-I2 + $-i2J,-+ IF-Ia + [F-*(9' + MA+ + gA+2)
+ %A+) Jt+ + A-*((M + %A+) F+ + gJ,+'C-'$+) + h.c.1
(111.9)
plus surface terms which include all the 0 dependence. The expression (111.9) is a Lagrangian of the usual sort and one can deal with it by standard methods. First of all one notices that the complex scalars F+ ar.d F- are not independent dynamical variables since their derivatives do not appear in 9 (i.e. no canonical momenta can be associated with them). These auxiliary variables can therefore be eliminated from the Lagrangian by means of the Euler-Tagrange equation which give
-3'-
=Y
-F+ =
+ MA+ + gA+a H A - + 2gA+*A-. 9
(111.10)
Elimination of the auxiliary fields leads to a Lagrangian which describes the interaction of two complex scalars, A+ and A- (the latter of which carries fermion-number F = -2), and a Dirac fermion I$= ++ Jt-,
+
+ MA+ + gAc21a+ laA-12 - I(M + 2gA+*) -4-1' + $(G+- M ) J, (111.11) - [@A+$-$+ - gA-*++'C-'Jt+ + h.c.1.
2' = ]EA+I2- 1 9
7*
92
ABDW SAUM and J. SRIATHDEE
The behaviour of this Lagrangian with respect to supertranslations has been obscured by the elimingtion of the auxiliary fields. These transformations are non-linear, viz.
dA+ = .EL+,
& = -(dl -1 %A+*) A_&+- (9’+ MA+ + gA+a)E- - i A + c - +A_&+, BA- = B+$, and the corresponding change in 9’ is a space-time gradient,. However, the physical content of (111.11)is fairly clear. The free field part is most easily read in the parametrization with Y = 0 where the bilinear terms take the form .Y(p)
=
laA+I2- INA+l2
+ $(ia- M ) + laA-I2 - (MA-1’. $J
(111.12)
Thus we are clearly dealing with a supermultiplet of particles of mass M (and, of course, the corresponding supermultiplet of antiparticles). The renormalizable interactions of these supermulbiplets are characterized by the dimensionless coupling constant 8. The generalization of (111.6) t o systems containing several chiral scalars is immediate. With the set of fields Gj+, j = 1, 2, ..., m , and Qa-, a = 1, 2, ...,n (where m and n are not necessarily equal), the Lagrangia.n must be
9=
1 -
C
[ i
+C
I@j+12
a
l@a-Iz
I
where Yo, Maj.and gajk = gakj are parameters of dimension 3, 1 and 0, respectively. No generality IS lost by assuming a diagonal form for the kinetic terms since i t can always be arranged by choosing appropriate linear combinations. Again it will be possible to eliminate some of the parameters by making a n appropriate shift Qj+3 Qj+ Cj. However, there can now be symmetry or algebraic considerations which restrict this freedom. To formulate a general rule concerning this would be coniplicated and, in the long run, futile since i t is always simpler to deal with specific cases as they arise. An obvious but nevertheless important remark concerns the field count. If the numbers, rn and n, of positive and negative chirality fermionu, $Jj+ and $Ja-, are not equal then one of two things must happen. Either some particles remain without mass, or else there must occur a spontaneous violation of fermion-number conservation (reflected, for example, by ( A _ )=/= 0). We shall poetpone further a discussion on Lagrangians of the general form (111.13) to Section IV. For the present we shall consider the simpler form (111.6). First of all, what are the equations of motion for the superfields Q+ and @-? A direct if not very illuminating answer to this question could be obtained by first setting up Euler-Lagrange equations of the usual sort for the component fields. One can deduce these equations from the expression (111.9). Then, knowing that the component equations must be compatible with supenynimetry, one should be able to arrange them in such a way that the supersymmetry is manifest, i.e. convert them back into the notation of superfields. However, such a n approach to the problem is not very satisfying. One is left with the uneasy feeling that an unnecessary detour has been made - that the underlying geometrical structure has not been allowed its due role. A more aesthetically pleasing formulation should be possible, and this is indeed the case. It is based on the notion of variational derivatives with respect to superfields.
+
Supersymmetry and Superfields
93
Consider the question of functional derivatives. Suppose we have a functional M which depends on the chiral fields @+ acd @+*. This is equivalent to saying that it depends on the component fields A+, $+, F+ and their conjugates. Derivatives with respect to these coniponents can be defined in the usual way by expressing the infinitesimal 6M as a linear form in the component variations,
Likewise, we must be able to express 6M as a linear form in the superfield variations 6@+(x,0) and 64+*(z, 0). A convenient way to write this is
The coefficients BM/B@+(z, 8 ) ) SM/B4+*(x,8 ) are the required functional derivatives. They are superfields in their own right and their component structure is determined by comparing (111.15) with (111.14). One finds that the linear forms are identical if the superfield derivatives are given by
The superfield derivatives are thus seen to have the structure of cliiral superfields with components given by ordinary functional derivatives. For example, on substituting M = @(x’, 0‘) in (111.16), one finds
which defines a supersymmetric analogue of the Dirac delta function. It is characterized by the integral property (111.18)
94
ABDUSS ~ L A X and J. STRATHDEE
for an arbitrary positive chirality supedield @+, Similarly, the negative chirality delta function is defined by
= 6 ~ 2e,;x', e').
\
(111.19)
Let us now compute the derivatives of the action functional corresponding to the Lagrangian (111.6))
8=
J Ldx
-(~90 ([@+Iz ) ~+
1
+
- 2 B D (@-*(Y M@+ + @+2)
+ h.c.) .
I
(111.20)
Only the kinetic term cause9 any difficulty and to deal with this we need to express the opcrator (4D)ain a different form. Two identities are needed for this,
( B D ) a= (B-D+ and
+ D+D-)2= (D-D+,D+D-)
0 = [D.D+, D+D-] - 2BBy,D
whence 1 2
-
= D-D+D+D-
- D-9y5D = D+D-D-D+ + Dd.y,D.
(111.21)
Using (111.21) together with the chiral property
D-@+= 0,
*D+@+* = 0,
one can put the variation of the kinetic term into the following form: 1 8( D D ) l ([@+I2 8
+ 1@-12)
=
1
+
( B D ) 2(do+@+* 'B@-*@-
+ 11.c.)
(111.221
95
Supersymmetry and Superfields
Wit.h the help of this formula we can put the variation of the action into the form:
6s =
dz
p:*{-T
(-T1).5
1
DO@+ + ( M + 2g@+*)@-}
DO@- + (9’+ H@++ da,,
1 Dy,,y,D[B@+*O+- 6O-*@-
- h.c.1.
(111.23)
Thus the variation 8s reduces to a surface term when the functional derivatives of S are set equal to zero,
6S
1 -
--DO@+ =* = 2
+ ( M + w+*) 0- = 0 , (IIT.24)
-= -DD@- f (9’+ M@+ + @+=) = 0 . 2 6S
1 -
6@-*
These are the equations of motion. They are manifestly supersymmetric.
B. Local Symmetry and Gauge Lagrangians We have just seen in Section 1II.A. that the inclusion of global internal symmetries into a supersymmetric scheme gives rise to no difficulty. Superinultiplets are combined into sets which support representations of the internal symmetry and this is effected by arranging the corresponding superfields into multiplets of the internal symmetry. Thus, expressing the positive and negative chiral fields as two independent columns, @+ = {Qi+)and 0-= I@,,-], the action of a global transformation is given by @* -+ exp (iAkQ&’)@&,
(111.25)
where the parameters Ak are real numbers and the matrices Q+ and Q- are hermitian. (In a later section expressions for the corresponding conserved currents,
(111.26) J k = O+‘Q+k@+- @-?Q-k@-, will be derived.) Generalization of the transformations (111.25)to local form is straightforward. Indeed, if the parameters Ak are allowed to vary with x,, then they must necessarily depend on 0 us well and in such a way as to preserve the c h i d character of the transformed fields. I n view of the closure property of chiral products noted before, it is clear that chirality preservation requires that the local parameters A#(z,0 ) be themselvcs chiral, aiz. (111.27) These transformations have a n unusual structure. As with any cliiral field, these inatrices can be expanded in powers of 0 , r s p (i.i+k(z, e) Q+k) = esp
ABDUSSALAXand J. STRATHDEE
96
where, of course, V+(z)carries a spinor index. The product of two transformations yields a third with u3,.
=
u,+u,+,
u,+v*+ + v,+u2+, w, = u,+w,++ V?+C-lV,+ + I.v,+u,+.
v3+
=
(111.29)
(In the last formula V,T, denotes the transpose with respect to the spinor index of V,+.) The matrices U+(z)which represent a subgroup are complex but not, in general, unitary. One might expect that such transformations could not be compatible with the usual requirements of a physically acceptable t,heory. It is remarkable that no conflict arises : all negative metric excitations are suppressed by the gauge niechanisni. As in all gauge theories, the principal vehicle of the local syinnietry is a set of potentials. In this case they take the forni of real (non-chiral) superfields, !Pk(z, 0). Their'role here is to provide a "metricyyfor the other fields in the system. Thus, the kinetic terms are to he expressed in the form
S?k =
1
(BD)2[@+te2'J'kQ+*O+ +.C"e-z'J''Q-'@-l.
(111.30)
This foriii is gauge invariant provided that the potentials transforni according to e2YtQk
+ efA+k'Qke2'l'kQke-iA+kQt,
(111.31)
where Qk = &*.'. (Thechoiceof a n exponential parametrization for the metric in (111.30) is merely a convenience. It is conceivable that other types of co-ordinates might prove useful in specific problems.) For some purposes it is useful to represent the potentials !P in the form of a herniitian matrix. For example, in the case of S U ( 3 ) one may write
y=
1k
y k -
2'
!Pk= Tr ( A V ) ,
(111.32)
where Rk, I,! = 1, 2, ..., 8, denote the Cell-Mann niatrices. A superfield analogue of the field strengths can be defined. It takes the form of a chiral spinor i !Pa++ = -- - n+D-(e-2yDa+e2y), (111.33) 2 f2 where the nunierical coefficient is chosen for later convenience. This superfield is chiral with respect to both its 0 structure and its spinor index,
D-Y++= 0
and
(1 - iy5) Y4+= 0 ,
and so belongs to the transverse vector representations discussed in Section 11, Eq. (11.36) ff. Under the gauge transformation (111.31)one finds Y++-+ ei*4+!P++e-i,'+.
(I1134)
The complex conjugate of !P++is a negative chirality field Y--. Thus or
(111.35)
Supersymmetry and Superfields
97
This field can also be defined by a formula analogous to (111.33),viz. (111.36)
It transforms according to Y-- -+
eiA+tp--e-iA+t.
(111.37)
A gauge-invariant kinetic term for the potentials is given by (111.38) where Q is a dimensionless coupling constant. I n general there can be one independent constant for each simple component in the local symmetry group. If this symmetry contains a commutative abelian subgroup, i.e. if the infinitesimal algebra contains certain linear combinations, f&Qk, which commute with all & I , then the expression (111.39) is gauge invariant (up to surface terms) and can be included in the Lagrangian. The parameters f have the dimension of (mass)2 and are very important in the niodels of massive vector aupermultiplets. On combining the expressions (111.30), (111.38) and (111.39) one obtains a gaugeinvariant and supersyninietric Lngrangian for the interacting system. The corresponding equations of motion are necessarily degenerate to a degree which reflects the gauge freedom. I n other words, those degrees of freedom which are associated with gouge transformations will not be governed by the equations of motion. They must be dealt .with by a n external mechanism. This problem is familiar from electrodynamics and Yang-Mills theory. I t s treatment in the supersymmetry context is of course more complicated in detail than for these well known cases but will be seen to follow essentially the same lines. We begin with an abelian local symmetry. For local U(1) symmetry there is just one potential, Y , a real superfield. The transformation law (111.31) reduces in this case to the linear inhomogeneous form Y -+ Y
i i -A+ + -A+*, 2 2
(IIT.40)
and the implications are clear. Thus, since Y can always be reduced, i.e. represented uniquely as the sum of chiral and transverse vector pieces, I Y = N+
+ N,* + !PI,
(111.41)
it follows that the transverse vector part, 'Y,, is gauge invariant while the positive chiral part, N+, transfornis according to
i N++N+--A+. . 2
(111.42)
(These formulae are analogous to the following, drawn from electrodynamics,
A,+A,
+ a,A, +
A, = a,A1 A:, A'-tA'+A,
(111.40) (111.41') (111.42')
98
ABDUSSALAXand J. STRATHDEE
where A:, the transvcrse part, satisfies a,,A,,' = 0. The supersymmetric analogue of this transversality condition is, of course, Eip-Y, = BD+Y,= 0.) The c h i d part, N+, is therefore completely arbitrary. It cannot be determined by m y ghuge covariant equations of motion and 80 must instead be fixed by convention. At the classical level this convention or supplementary condition may be taken in the convenient form
D 9 - Y = B+,
D-D+Y = B+*,
(111.43)
where B+ is an arbitrarily chosen chiral field. The supplementaq conditions (111.43) are manifestly supersymmetric and so are the most useful when it is desired to set up the Feynman rules in an explicitly supersymmetric way. However, it is sometimes more convenient to sacrifice this visible supersymmetry wit,h its attendant unphysical modes in favour of a description in terms of ' physical components. (Analogous to the lose of explicit Lorentz invariance in going from, say, the Landau gauge to the Coulomb gauge.) Indeed, one can choose' the components of N+ in such a way as to compensate some of the components of !PI.In this type of gauge the potential y/ may be presented in the form (111.44) where A and D are unconstrained but V , is subject to the supplementary condition
a,,V,,C4 = B ( 4,
(111.45)
with B ( s ) an arbitrary real ~ c a l a r . ~One ) very considerable virtue of this gauge is that it implies the reduction of exp (BY)to polynoinial form since Yn = 0 for n 2 3. The Lagrangian then assumes a manifestly renormalizable form. Similar considerations apply to the non-abelian symmetries where a set of real potentials Ykare involved. The linear transformation law (111.40)is no longer relevant and the transverse vector pieces Y,&are therefore not gauge invariant (just as in the ordinary non-abelian gauge theories) but it is still possible to remove the gauge degrees of freedom by means of subsidiary conditions like (111.43),
B+Ll-Yk= B+k,
D-D+?P = B+k*
(111.46)
or, alternatively, to represent each !Pkin the form (111.44) with a,V,! = Bk. On quantizing, the usual Faddeev-Popov complications are encountered. These will be discussed below. In manifestly renormalizable gauges of the type (111.44)the field strengths are given
(111.47) where the Yang-Mills covariant derivatives appear, (111.48) The normalimtione in (111.44)are chosen for later convcnience. The y5 which multiplies A(z)is an unfortunate relio of the early development of the subject when the role of space reflections waa not understood. Its removal would probably cnii3e too much confusion nt this stage, however.
3,
99
Supersymmetry and Superfields
(Note that 2, is related to 1- by conjugation, ' ,1 = CX-'T, since Y kis real.) If such a gauge is used, then it is best to discard the superfield notation and express the Lagrangian entirely in terms of component fields. The contributions (111.30), (111.38) and (111.39)t,hen become, respectively, 2 k
+ $+iV@++ F+tF+ + i@(A+tl-kQ+k$+- $+;,t-kQ+kA+) + A,tDkQ+kA++ V,,A,tV,A- + $-iV$- + F-tF- + iy'2 (A-tX+kQ-k~-
= V,A+tV,A+
- IJ-I.+kQ-kA-)- A-kDkQ-kA-,
(111.49) (111.50) (111.51)
r
where the covariant derivatives in (111.49) are given by
VpA* = (8, - iV,kQ.Q+k) A,, Vp$* = (8, - iV,kQ*') $*
(111.52)
*
There may be, in addition, a gauge-invariant self-interaction (or generalized mass term) among the matter fields, Y,,,=
-f
D+D-[@-t(lP+ MO+
= F-f(.Y
+ h@+O+)]+ h.c.
+ MA+ + AA+A+)- &(M$+ + ?LA+$++ ?@+A+)
+ A-t(lCrF+ + hA+F+ + hF+A+ + h$+TC-I$+) + h.c.
(111.53)
provided it is compatible with the global symmetry. The Lagrangian defined by the expressions (111.49)-(111.53) is manifestly renormalizable (apart, possibly, from anomalies) and respects both fermion-number and gauge symnietry. It is not manifestly supenymmetric owing to the choice of gauge (111.44). An important feature is that the left-handed gauge spinor 1- carries fermion-number F = -1, like $+ and $I-.The right-handed spinor ,I+ therefore carries F = + 1 like $+ and $-. This fact will prove significant when we come to consider the problem of parity conservat,ion. Although the gauge (111.44) appears to conflict with the requirements of supersymmetry, the contradiction is a superficial one since only gauge-dependent quantities such as Green's functions are affected. Gauge-invariant quantities such as scattering amplitudes and matrix elements (between physical states) of the energy momentufn tensor should not be sensitive to the choice of subpidiary condition and should therefore respect supersymmetry (unless of course it is spontaneously violated). A formal way to see this is by a (non-linear) transformation law for the fields, which reflects the action of supertranslations on them and which does leave the Lagrangian invariant. The method, although somewhat tortuous, can be carried through without difficulty. The idea is to follow an infinitesimal supertranslation - which destroys the form (111.44) - by an appropriate gauge transformation (111.31) which restores it. The combined effect will then leave the Lagrangian invariant (up to surface t e r m ) .
ABDUSSALAXarid J. STRATHDEE
100
Firstly, the infinitesimal supertranslation applied to (111.44)gives
Next, consider an infinitesinial gauge transformation, d,,eZy = iA+tezvr- eZyiA+,
(111.55)
where A+ has no &independent term, i.e. =
ex+ +
1 -
1 -
e-e+/+ + e+e-G+(--iat+). (111.56) 4
With this special form for A+ it is possible to solve (111.55) for 8,,Y by one iteration,
where we have defined 5'- such that e+C- = t+O-.The gauge paranieters 5, and f+ must now be chosen such that
dS!P
+ 8 , , =~
1 1 +8 8 8 ~ ~ 6+) .- (coy S D . 2 f2 16
1 -
ei~.y,06v,
(111.58)
This is achieved by taking
5,
= 2V.Y.E-
and
f+ = @ a+).-
(111.59)
and the resulting variations of V,, I. and D are given by
dV,
i
=
--ByJ,
1Tz
i 62 = fi (iy,D
+2
(111.60)
101
Supersymmetry and Superfields
Applied to matter fields @*,the transformations &Ort
+ d,@*
take the form
(111.6I
where the covariant derivatives are defined by (111.48)and (111.52).The transformations (111.60)and (111.61)necessarily preserve the form of the Lagrangian (111.49)-(111.51).
C. Parity Conservation Tn this section we wish to prove the important result thot a supersymmetric, renormalizable, fermion-number-conserving Lagrangian theory which also preserves parity must be a gauge theory of a very special form. First consider the fermion-number-conserving, renormalizable non-gauge type Lagrangian (111.13). For a parity operation to be defined, the supermultiplets @+ and 0must be put into correspondence. Firstly the spinor components can transform only according to a rule of the form JI+ +- WYOJI(111.62) 9
where w is some unitary matrix subject to the constraint wa = f l . Secondly, the fernlion-number F = 2 spin-zero components A- must transform among themselves,
A- + w’A-, where w’ is another matrix like w. Now consider the particular interaction term in (111.13)
(111.63)
Application of the transformations (111.62) and (111.63) to this term changes it to a form which is clearly not present in (111.13). This shows that interactions of the form (111.13)cannot support space reflections. We shall prove now that if a local eymnietry is present it is possible to set up a parityconserving interaction. For definiteness consider the case of local SU(n)symmetry. Since the gauge field !P - an n x n hermitian traceless matrix - contains a set of negative chirality fermions, A_, in the adjoint representation, it is necessary that the matter system should contain a t least a set of positive chirality fermions, c+, also in the adjoint representation. These fermions niust belong te a (traceless) supermultiplet S+ which we shall call the “supplementary gauge fields”. Under the action of local SU(n) the augmented gauge field system transforms according to
4
e 2 9 ~+ eiA+teapPciA+
S+ -+ e’A+S+e-’A+,
(111.64)
where A+ is a traceless n x n matrix of positive chirality and g is a dimensionless coupling constant.
102
and J. STRATHDEE
ABDUS&LAX
Other matter fields may also be present in the form of supermultiplets @+ and @-. It is necessary that @+ and 0-*should transform contragrediently. For simplicity we take just a pair of n-component columns transforming according to @+ -+ eiA+@+
@- + eiATt@-.
(111.65)
The most general supersymmetric, fermion-number-conservingand renornializable Lagrangian for this system is given by
-+. I +
y = - BD [L Tr (a-!P+ - !P+!P-) 8 2 1
+s
-
l
;[
-
1 ~D-te-~g'''@-]- - D D [@-t(M 2
[@+te2gy@+
I+
Tr (S+te2QyS+e-20'f')
+ hS,) @+
h.c.1,
(III.6G)
where h is a dimensionless coupling constant and M is a mass. In order to deal with the parity question it is necessary to express (111.66) in terms of the component fields. For 0, and Y we adopt the expansions (11.25) and (111.44), respectively, and for 8, the expansion S+ = e-l/4~+3
+ ~ + ( z+) -41 e(i + iy,)o
[
(c+(z)
/+(.)I
.
(111.67)
For convenience of writing we introduce the negative chirality antifermion 5- defined as the conjugate of c+, i.e. (111.68) 85- = (ec+)t = f+e.
+
This means that the sun], 5 = c+ 5-, is a Majorana spinor like 1. Applying the formulae (111.49), (111.50), (111.53) to the Lagrangian (111.66) one finds
2' = - Tr 2
+i
1 2
[--:U:, + X-igL + - D2 + +
V,a+W,,a+
+ f+i!#c++ j+tf+
1
g([a+t, 1-1C+ - f JLa ] ) ga+t[D, a+]
+ V,,A,tV,A, + $+i!#$+ + F+tF+ + i g(A+tI-$+ - $+LA+)+ gA+tDA+ + V,A-tV,A- + $-iJ?$- + F-tF.- + i/2q(A_tI+$- - $-l+A-) - CJAtDA+ M(A-+F++ F-tA+ - $-$+ + h.c.) + h[F-fa+A+ + A-tf+A+ + A-ta.F+ (111.69) - A-tf-J1+ - $-a+$+ - $-f+A+ + h.c.1,
(111.70)
,
103
Supersymmetry and Superfields
The appearance of (111.69) can be greatly simplified if the coupling parameter h is given the specific value h=gfi. (111.71)
Indeed, with this restriction the system is parity conserving. This can be seen more easily by eliminating the auxiliary fields, F, , f+ and D, resolving a+ into hermitian and antihermitian parts, (111.72) and introducing 4-component spinon,
(111.73) The Lagrangian (111.69)then reduces to = 2 Tr
[-;
u:.
I
+ airox + 31 (V,a)2 + a1 ( V N + B a t % XI + [b, Y d ,
+ VpA+tV,,A++ V,A-tV,,A- + $(i$ - 31)$ - g$(a - y5b) $ - g
[$XA+ - i & , f A -
+ h.c.1 - V ,
(111.74)
where the covariant derivatives are easily deducible from (111.70), (111.72)and (111.73). The conjugate spinor 2 is defined in the usuakfashion
8f
=8
=& ($
The potential V has the form:
+
V = F+tF+ F-tF-
( ~- ik).
+ -21 Tr (/+tf+ + $ D 2 ) ,
(111.75) (111.76)
where the auxiliary fields are given by
F+ = - ( M
F- = - ( M f+
=
-g 2
+ g(a - ib)) A+ g(n + ib)) A+
(111.77)
fi (A-A+t -
A+A+t - A-A-t (Recall that f+ and D arc traceless n x 7~ matrices.) On substituting the expressions (111.77) into (111.76) one finds
ABDUSSALAM and J. STRATHDEE
104
Parity conservation in the Lagrangian (111.74) with the potential (111.78) is now manifeat. For the spinois we take Fc, -+ YoQ (111.79) x +Yo% and xc -+ -YoX". Among the bosons: U,is a vector; a , A+ and A- are scalars ; b ib a pseudoscalar. (The odd relative parity of x andzc results from the definition (111.75).) Alternative parity assigntments are equally feasitrle. Thus, instead of (111.79) one could take
rcI
x
3
WYOJ,
~ y ~ x w and - ~
xc
(111.80)
-+
-~y~xcw-~,
where w is a unitary matrix satisfying w2 =; 1. The corresponding rules for the bosons are then : Uo + WUOW-~, Ui + -WUiw-', cz -+
b 3 -wbw-',
(OUOJ-',
(111.81)
A*+wwA*. This completes the discussion of the parity-conserving local S U ( n ) model. Other local synimetries are treated inexactly the same way and any number of mattersupermultiplets itlay be introduced in the form of contragredient pairs @+, @-*. I n every case the couplings are determined entirely by tAe gauge principle.
D. Conserved Currents We conclude this section on Lagrangians by considering the local conserved currents which can be constructed corresponding to a given Lagrangian. A Lagrangian of the general form (111.13) may admit a global symmetry,
SOD,= ioQ,@+,
(111.82)
where the iriatrices Q+ and Q- are hermitian. In such cases one expects to find a conserved 4-vector. This current vector must of course belong to a supermultiplet and one would like to know what significance the other members can have. We now consider this question. Notice firstly that the Lagrangian can be expressed quite generally in the forin
9' = -A D D ( 9 + 2
+ h.c.),
(111.88)
where the chiral part 9'+is given by 1 -
p+=
-zo+D-(I@i+Ia f
I@a-12)
@-:
( y a
H5i@i+
An arbitrary infinitesimal variation of the fields Di+and S9+ =
1
-- D+D-(@+tS@+ 4
f
S@z-(A5
-
(111.84)
2 g 5 i j @ j + ) S@i+
(111.85)
f goij@i+@j+)
a5-gives
+ @-* d@- + h.c.)
Jfai@s+
f gai,@i+@j+)4- @ z - ( M a i
+
and this expression can be arranged in a more appropriate form with the help of the variational derivatives of the action functional, S. The derivativcs of S are given by
S tipersymmetry and Superfields
105
formulae like (111.24), viz.
+ +
-6S- - - ’ ~ t o - @ a -
( ~ a Hai@i+
2
S@,x-
+
gaijQi,+@j+),
(111.86)
SS
%+
1 -
+
U+D-@& q - ( N d - -2 8
+ 2g,ij@j+).
Using t,hese in (111.88) one finds
W += -A4 b+D-(@+t 60,
+ @-t’s@-+ h.c.)
and for variations of the type (111.82) which correspond to a syiiiinetry this must vanish. The identity (111.87) then assunies the forni
where the “current” J(Q) is a real (non-chiral) superfield defined by J(Q) = @+?&+a+ - @-‘Q-@-.
(111.89)
When tlic equations of niotion are satisfied,
-6S- - 0,
-SS-
6@*t - O ,
then the right-hand side of Eq. (111.88) vanishes and it becomes a supersymmetric conservation law. More generally, it can be used to generate Ward-Taknhashi identities. The co~nponentsof the current J(Q) are defined by the usual expansion 1 + O$(Q) + 71 eOF(Q) f 4ey&G(Q) 1 1 - 1 +4 8 i y v y 8 VAQ) i- 7 OOOx(Q) + 5 (Be)’ D(Q)
J(Q)= A(&)
(111.90)
where the bosonic coiriponents A , F, G, V and D are real and the fermionic components and x are Majorana spinors. They are given explicitly by
+
A(Q) = A+tQ+A+- A-tQ-A-, $(&) = A+tQ+$+ - A-tQ-$-
+ C$+TQ+A+- CqT-Q-A-, F(Q) = A+tQ+F+ - A-tQ-F- + F+tQ,A+. - F-tQ-A-, + i-4-tQ-F-
Q(Q) = iA++Q+F+
+
- iF+tQ+A+- iF-tQ-A-,
+
(111.91)
+
VJQ) = A+tiz.Q+A+ A-tiT’.Q-A$+y,&+$+ $-y.Q-$-, x(Q) = SF+tQ+$+ - 2F_tQ-$- 2C$+TQ+F+- ~CIJ-~Q-F- A+tigQ+$+ + A-ti2Q-qy,C~+TiZ,Q,A+- y,C$-Tiz,Q-A-, D(Q)
+
+
- 4F-tQ-F- + 2$+i3Q+$+ - 2i-iTQ-1)-
= 4F+fQ+F+
Zeitsrliriit ,,Foitachritte der Physik“, Heft 2
- A+tz’Q+A+ + A-t’ZQ-A-,
ABDUSS A L Mand J. STRATHDEE
106
where ‘5 =
8 - 8. The component structure of the “divergence” of J ( Q )is given by
(111.92) When the equations of motion are satisfied, the identity (111.88)gives the “conservation law” D+D-J(Q)= 0 (111.93) and, since J ( Q )is real, this implies D-D, J ( Q )= 0 as well. I n terms of components,
m?) + iw?)= 0, (111.94)
of which the last is the fainiliar conservation law. In suinninry, the current 4-vcctor, V,(Q), belongs to a real non-chiral supcrmultiplet J ( Q )whose components (111.91) are listed. The divergence, a,V,(Q), belongs t o a chiral superniultiplct z + D - J ( Q )all of whose coniponents must vanish when the equations of motion arc satisfied. The corresponding Ward-Taltahashi identities could be set up in a nianifestly supersymnietric forni with thc help of the classical identity (111.88). Since the chiral part of J ( Q ) must vanish when the equations of niotion arc used, the remaining part niust take the forin of a transverse vector superrnultiplet (0,).I n these circumstances J reduces to
The matrix elements of J ( Q ) between physical states belonging to the fundaiiicntal represcntation are characterized by two invariant amplitudes. We shall pursue this excrcise in some detail now since it provides a good illustration of the methods developed in section 11. As R first step in treating the matrix elements of J(&) i t is necessary to replace it by a n equivalent chiral supcrfield to which the decompositions of Section 11.B. can be applied directly. Therefore we shall deal with the spinor superfield
y-+= D-J(&)
(111.96) which has positive chirality in its &structure and negative chirality in its external spinor index i.e. D-Y’-+ = 0, (1 iy5) Y-+ = 0,
+
107
Supersymmetry and Superfields
and it carries one unit of fermion-number, The required matrix elements can be extracted from various 3-point functiobs. A convenient one is.
iyap + M2) (822
+ Ma) (T* dJ1) d+*(2)!P,(3))
- ~ ~ - i l - ( i ~ , B ; - + i ~ . e : - ) ( ~- bla,,a,]) o12-,
(111.97)
where a and b are scalar invariants. This expression, the most general one compatible with supersymmetry and fermion-number conservation, is of the form (11.63). The LSZ reduction formulae (11.70)-(11.72) can be applied to (111.97). Thus
i2 dxl
d r z e i P l f l - i P ~[(a12 ~~
= ll.i exp libl
+ B2) (Za2 + N 2 )(T*d-(l) d+*(2) @+(3))]
- p 2 )z3 - B 3 - ( ~ 1 0 1- p202-)] . (a
+ b [ p l ,p 2 ] e12-. )
(111.9s)
The various matrix elements are obtained by expanding the right-hand side in powers of 0 and comparing coefficients. Thus, with O3 and z3 set equal to zero,
Z 2 b l l $-(a 11)2&)1 1,
Z+(PzA?)62-
+ c &+u-(PJl)
+ b[181, 1821) (4- w.
= Jf(a
$-(aIPdO
1 ~ 1 4 1
2,
(111.99)
-
Tlic terms linear in O,, give
[B(PLI
+ -.I
( kTp(Q) - ii?A(Q))[IPdO + **.I
=-W1+P,
- %+p2)y,(a
+ mz,,PZI)
(01-
- 4-1.
(IJI.100)
The ternid quadratic in O,, give no further information. To extract particular matrix elcnients from (111.99)and (111.100) is straightforward. One finds,
(111.101) where cc and b are real functions of t = ( p I - P , ) ~ .Matrix elements of tained from these by complex conjugation. 8*
$+(a)can be ob-
108
ABDUSSALAM and J. STRATHDEE
TheNoether-like technique used here to treat a global symmetry is not readilyapplicable to the geometrical aymmetries : Poincar6 and supertranslations. For example, although i t is possible to construct a current T,= J(ia,) corresponding to space-time translation, which satisfies B,D-T, = 0,it is not real and it does not satisfy B-D+T,,= 0. It does not contain the canonical energy momentum tensor among its components. At present it is not clear whethcr such currents will be a t all useful. To deal with the geometrical symmetries we shall adopt an indirect approach based on the requirement of fermionnumber conservation. Ths conserved current 4-vector corresponding to the fermion-number symmetry is given by (111.102) j,(z) = $+ye++ tj-y,$~ 2A-+iTFA-.
+
+
This is the standard Noether current obtainable by the usual considerations applied to a Lagrangian made out of the fields A+, ++, A- which carry the fermion-numbers 0, -1, -1, -2, respectively. It satisfies the identity
+-,
(111.103) from which t,he Ward-Takahashi identities can be derived. The supersyiiimetric versions of (111.102) and (111.103) can be obtained directly by application of the “boost” exp fie&),where fi denotes the supertranslation generator (to be distinguished here from the action functional, 8).The supertranslations act on the component fields according to the rules, 1 [ A , , 091 = el)* i
J
(III.104)
It is a simple matter to integrate these formulac to obtain
@&(z,0 ) = eii3Ai(z)
D@&, 0) = eiiS$+(x) 1 -D D L D ~ (e) X ,= e i i 3 F 5 ( x ) 2
rig$
,
-* cieS,
(111.105)
e-isi.
and, in similar fashion,
[(III.106)
Supersymmetry and Superfields
109
[Recall that the derivatives dS/d@* are chiral superfields with the component structure
Define the supercurrent JJx, 0) by applying the boost to the fermion-number current (111.102), i.e. J J z , 0) = eji$J,(z, 0 ) dad, (111.107)
- eii3j,(x)
e-ii3.
It is R simplc matter to show that this current can be expressed in the manifestly supersymmetric form
since this expression has no explicit 0-depexdence axd satisfies the boundary condition J,(x, 0) = j&). The current (111.108)is a real vector superfield ar,d it is clearly conserved when both fermion-number and supertranslations are valid symmetries. Indeed, the identity which incorporates the conservation law for J, is obtained by boosting (111.103), 1 - 6s ~ , J , , ( X ,0 ) = 7 z B@+ D@+
(.-)
+
(ITI.100) When the equations of motion are used it' is possible to show that J,, reduces (like the global J ( Q ) )to a transverse vector superniultiplet,
n+D-J, = B-D+J,, = 0.
(111.110)
The proof is as follows. Firstly, there is a simple operator identity
Next, in order to shorn that this vanishes, an argument similar to that used in the global case can be used. Thus, corresponding to an infinitesimal translation, 1 ?pY+ = 4 l)+D-(@+cQD+- @J2,,@-) ,
-
(111.112)
when the equations of motion are used. Further, the equations of motion imply that 9+ takes the simple form 1 (111.113) P+= -- D+D-(@,f@+- @-to-). 4
110
ABDUSSALAMand J. STRATHDEE
Subtraction of the gradient of (111.113) from (111.112) gives the desired result,
and (111.111) vanishes. Since J,,is real, it follows that D-D+J,, also vanishes and the formulae (111.110) are proved. The transverse vector part of J,, can be expanded in the usual fashion. J P k
0) = i&)
+ q-y5
S,,(4)
1 - -1 + 1 Oiy,y,Q( --2T,,(x)) + eoeia(--ySs,,(z))+ 32 - (8e)z azj,,(z),
(111.115)
where S,, is a Majorana spinor whose left-handed part is given explicitly by
iS,-(x)= A-t(y,ig - i&J = F+ty,,$+
t,h-
+ yJ3jLTF- + y.y,,Ctj+Tia.A+
'
+ y,,C$..TF- + (i2.A-t) y.y& + ~ . y , , C $ + ~ i a ~+Ai+ W -t[y,u,y.1 $-I (111.116)
(where equations of motion have been used to effect the rcarrangetnent). This is a conserved current whose time-coniponcnt So- is the density of the supertranslation generator S-. Tlie tensor T,,, is transverse with respect to both indices,
i?,,T,,,= 2,T,, = 0, although it is not symmetric. Its explicit form is given, with the help of equations of iiiotion, by 1 1 .y .t. T,,= A+fi2,,i7,a4+ - ,4_ta d .a G ,A2 2
+-
+; $+(r,, + 7. i +
~j,,.
{
+
F+tF+ F-tF-
n $+
.-+ +; $-(y,, 2.7,+ y. ?,,I $-
1 +(A+tS2A++ A - T I ~ ~ f A -h.c.) 2
1
1
- - (rl,, 2 2 - a,,a,) ( A + ~ A + A-~A-) 2
+1
P+t2.7,A+
~ . ~ 8, , l ~
'
+ A-t izJ- + 21 $+Y& + 23 JI-y t,h . (111.117) -I
The antisymmetric part makes no contribution to the integrated 4-momentnm operator. By its construction the current J,,transforms according to
1 [J,+,, ES]= B i
111
Supersymmetry and Superfields
or in terms of components,
(111.118)
The integrated quantities
can be shown to satisfy the supersymmetry algebra of Section I. (In J,. the symmetrical part of T,,, is employed.) The coniponents of J,, can be arranged into two distinct superinultiplets. Notice firstly that the covariant derivative
is n cliirnl superfield. Multiply this by yp to obtain
iy,S,,+
+ TppO-+ (111.120)
which shows that the trace, Tpp,and the self-dual part of Tl,,,- i / 2 2,jP belong with y,Sp in one supermultiplet. The remaining components make up another supermultiplet. Finally, a few words about Ward-Takahashi identities. These are based on the identities (111.88)for J(Q) and (IJJ.109) for Jp. A typical Green's function involving J ( Q )can be ivpresentcd by a path integral,
( T J ( Q )W@)) =
(do)J ( Q )n(@) exp
[+ SW]
where n(@) represents a product of superfields. The identity (111.88) can be applied directly,
ADDUSSALAXand J. STBATHDEE
112
after integrating by parts. (A singular term has been excluded by assuming 6+(1 1) = 0, see below.) Thus, for example, ~
1 -
2 D + D - ( T J ( lQ) , @+(2)@-t(3)) tt
= 7 d+(L 2) Q+(T@+(Z) @-'(3))
i
- 7 (T@+(2) @-+(WQ-6+(3,11,
1 D + n ( T J ( l ,&) 0 4 2 ) 0+?(3))= 0, 2
where 6+(1 , 2 ) denotes the supersymmetric delta function
This delta function has t,he iniyortnnt feature that it vanishes when 0 , = 02, a yropcrty used in the above derivation. I n similar fashion one can derive Ward-Takahashi identities for the supercurrent J,. Only the details differ. Section IV. Model Building
A. Spontaneous Symmetry Breaking So far we have tacitly assumed that supexsymnietry is unbroken. The discussion of representations in Section I and invariant amplitudes in Section 11, for example, wcre based on the presuriied existence of a supersymmetric vacuuni. Our purpose now is to discard this assumption and investigate some of the consequences of a degeneracy in the vacuuni. I n other words, we shall suppose that the underlying dynamics (as expressed in the Lagrangian) is invariant and that a conserved supercurrent exists, but that the ground state is not invariant. In this case the space-time symmetry is reduced to the Poinrar6 group and a Goldstone phenomenon occurs. On the one hand, the most distinctive aspect of supersymmetry - the grouping of fermions into supermultiplets with bosons - is lost while, on the other, a new and equally distinctive feature emerges, the Goldstone "neutrino". The mechanism is analogous to the familiar chiral dynamics of the pion where chiral SU(2)LxSU(2)R symmetry is realized by means of a triplet of massless pseudoscctlar Goldstone boeons. When the chiral symmetry breaks spontaneously to SC'(2) there necessarily appears a set of massless pmticles with the quantum numbers of the lost symmetries. The associated currents are then dominated at low energies by the eschange of these massless particles, and varioun low-energy theorems result. Exactly the same thing happens when supersymmetry is broken spontaneously. Let the spinors $*(z) belong to scalar superfield? of the kind introduced in Section I1 so that, under a n infinitesimal supertranslation, d$+ =
*2 (P*- i 3 A & ) iv5
E.
(IV.11
If thc vacuum is at least translation-invariant then (3.4)= 0 and we have !&/+' = (F,e
E
-
(11.2)
113
Supersymmetry and Superfields
If the vacuum respects'fermion-number conservation, then (F+) = 0 but we shall not require this. If either (S$+) or (B+-) is non-vanishing,' then the vacuum cannot be supenymmetric. Formally, (IV.3) where So is the time component of the current S, introduced previously. The vacuum expectation value of this equation can be expressed in the compact form
(W*(O))
S,(4
= ia,(T*ES,(4
$*(OD
(IV.4)
provided that the vacuum is translation-invariant and the current SJz) is local and conserved.8ubstitute the expressions (IV.2) for s++and remove the parameter E from both sides of (VI.4). One finds
(IV.5) In moiiicntum space,
where the aiiiplitudes M , ( ~ .) .,. depend on k2. The identit.ies (IV.5) t,ake the form
(IV.7) The second of these serves only t.0 eliminate M,(*' from the decoiiiposition (IV.6). The first gives the explicit form of M,(*), vi;. a simple zeroinass pole with residue (F*). "his indicates that thc intermediate states which contribute to the two-point function (1V.G) must include a massless particle of spin 1/2: a Goldstone fermion. A similar argument can be repeated with the spinor components, f - , in the gauge fields. According to the formula (TILGO), i (1V.S) ( 8 L ) = -- (0) E- ,
liz
which may be non-vanishing if supersyninietry is broken. The Goldstone spinor will in general turn out to be a mixture of the spinor fields present in the system. The relative proportions in the mixture are governed by the expectation values of the scalar fields, F, and D. Perhaps the siniplcst way to see this is through the expression for the conserved supercurrent S,(z),
(IV.9)
114
ABDUSSALAM and J. STRATHDEE
On linearizing this about the vacuum solution, one finds
wehere the Goldstone spinor, v-, is defined by fv- = (FA+)$-
i + CIj+T(F+)+ = (Dk) L'
(IV.11)
lie
and f is a normalizing factor. If ferniion-number is conserved, then (F,) seen to be a pure negative chirality ferinion field. Low-energy theorenis can be derived for v- by writing
=0
and v- is
Current conservation then implies that the neutrino source Bv- = - ( l / f ) a,R,+ is a soft operator at low energies.
13. Mass Generation Owing to the very strict controls that supersymmetry iniposcs on the form of the action if often happens that one or more particlcs are found to he massless. Of these orily the Goldstone spinor, associated with the spontaneous breakdown of supersymmetry can be understood in group-t,lieoretic ternis. Other instances of this phencnienon can usually be explained by a particular representation content in a given systein, or by the exigencies of renormalizabilitywhich scverly h i i t the variety of interactions. Here we revicw the mass problem in general ternis and list the various ways in which ~ R S can S be generrrted. We conclude with two niechanisiiis due to Slavnov wherein the supersyrnnietry is broken explicitly albeit softly.
B. 1. Mass generation for scalar suyeriiiultiplets To begin, consider the free spin-l/2 particle. It is usual to represent the massive spin-1/2 states by iiieans of a pair of chiral spinors $+ and $- which belong to independent rcpresentations of the proper Lorentz group. They satisfy the Dirao equation
-ia$++ M & = 0 , --a$- + M$+ = 0 ,
(IV. 12)
and, in the Lagrangian, the inass term takes the forni -M$-$+
+ h.c.
(IV.13)
The fields $+ and $- may be related by hermitian conjugation,
Jt-
= C$+T,
(IV. 14)
but this is possible only if the states carry no internal quantum number such as baryon or lepton number. Using (IV.14) the mass term (IV.13) takes the form
IM$+~C-'$++ h.c.
(IV.15)
115
SuFersymmetry and Superfields
and this term clearly violates such quantum numbers. (Of course, if 31 = 0 one can discard either $+ or 9- and allow the remaining one to carry the quantum number.) The massive spin-1/2 fermion is now considered to belong to a niassive scalafsuperniultiplet of the extended Poincard symmetry. It has scalar partners which carry fermion number F = 0 and F = 2. This multiplet can be represented by 8 pair of chiral scalar superfields @+ and @-, which are just the supersymnietric extensions of $+ and $-. They satisfy an extended version of (IV.l2),
(IV. 16) 1 -DDO2
+ MO+ = 0,
and, in the Lagrangian density, the mass term takes the form 1 2
-- DD(M@-*@++ h.c.)
= M(F-*A+ - $-$+
+ A-*F+) + h.c.,
(IV.17)
which includes the spinor term (TV.13). The fields @+ and 0-may be related by complex conjugation if there is no feriiiionic quantum number present. The supersymmetric extension of (IV.14) is simply @- = @+*. (IV.18) However, we shall continue to assume that a ferniionic number is involved and so will not identify 0-with a+*.To conclude, in general, mass generation for scalar superniultiplets requires the co-operation of mukiplets @+ a n 3 @- in a Lagrangicrn term of the type (IV. 17). The massive supermultiplet (represented by @+ and @-) has the content: one spin-l/2 massive particle carrying fermion-number F = 1, one spin-zero particle with F = 0 and one spin-zero particle with F = 2. All these particles have the same iiiass M. The iiiost general renormalizable and fermion-number-conserving interactions of a inassive scalar niultiplet are described by the Lagrangian (111.6). B.2. Mass generation for a vector superniultiplet
So much for the scalar superniultiplet. The vector supermultiplet is much more complicated. First of all, the only known way to set up a renormalizable interaction for massive vector particles is through the agency of a spontaneously broken gauge synimetry. Hence the vector supermidtiplet must be associated with a gauge superfield Y(z, 0). For the spontaneous breaking'of this syninietry a set of Higgs scalars is needed and they must be carried in scalar superfields 0,.The two superfields, Y and @+, co-operate to make a vector superrtjultiplet of massive stcites - rather as @- and @+ co-operate to make a scular supermultiplct. In general, one may consider a non-Abelian local symmetry which involves a number of generators Qkand associated gauge fields !P.The symmetries are spontaneously broken if
* 0.
&"(A+)
S o w , it is well known that the vector components acquire in this way the mass term,
where the mass niatrix ( L Wis) ~ given ~ by
and the gauge coupling strengths are implicit in the charge matrices Qk. The new feature of the supersymmetric gauge theory is the generation of fermion masses in the fusion of left-handed gauge fermions Lkwith right-handed matter fermions $+ in the term
i p ( A + t )I?Qk++
+ h.c.
(IV.21)
There may of coursc be additional left-handed fermion components, $L, in matter superfields @-. T h e full mass matrix then takes the form,
(IV.22) where M is, in general, a rectangular matrix. The non-vanishing fermion masses are includcd anlong the eigenvalues of the matrix
(IV.23) If the off-diagonal elements (A,?) Qk m are non-vanishing, then there is mixing betwccn the gauge fermions It- and the matter fermions +-.The fermion spectrum will then differ substantially from the vector spectrum. (If there is no such mising then N J l t includes a submatrix which is effectively the same a5 the vector mass matrix (IV.20).) I n other words, this mixing term (A+t)Qkm is a signal of supersymmetry breaking - although such breaking can happen even when there is no mixing.
B.3. Minimization of effective potential to generate scalar masses Scalar masses are obtained by expanding the potential
V = F+tF+
+ F-tF- + -1 (Dk)’ 2
(IV.24)
about its minimum with F+, F- and Dkexpressed in terms of A+ and A- as discussed in Section 111. If supersymmetry is not broken then all F’s and D’s must vanish a t the minimum and the masses are obtained by retaining only their linear parts,
(IV.25)
.
where the notation (. .) means “evaluated a t the minimum”, and we have assumed that fermion-number is not violated, i.e. ( A _ )= 0, (Iffermion-number is violated, then we should have more terms in the linearization,
Supersymmetry and Superfiolde
117
etc. We shall continue to assume that fermion-number is preserved since it means a great simplification.) If the supersymmetry is broken spontaneously, then some components among the F’s and D’s will fail to vanish a t the minimum. In these components it will be necessary to retain quadratic terms. Such terms act to separate the degenerate terms of a supermultiplet. For instance, if (09=/= 0, then the scalar mass terms will contain
(Dk) (A+t&+’A+- A-t&-’A-), which separates states with F = 2 from those with F = 0 and both from those with F = 1. To find out whether such a term will appear in a given model is a matter of detailed investigation. One must minimize the potential and see if any F or D is nonvanishing there. However, if the model is not a very elaborate one it willusually be apparent when there are no values of A+ and A- for mhicn all F’s and D’s vanish simultaneously. A problem which often arises is that of degeneracy in the classical minimum. The potential attains its minimum value not a t a single point or even on a set of points which are connected by symmetries of the system, but on a larger manifold. This phenomenon indicates the presence of niassless (pseudo-Goldstone) scalar states. I n ordinary gauge theories when this kind of instability appears in the classical approximation it is renioved by the quantum corrections. However, a peculiar feature of supersymmetric Oieories is that such instabilities can persist to all orders. Indeed, it can be proved that the pseudoGoldstone particles remain without mass to all orders i n perturbation theory if the supersymmetry is not broken i n zeroth order. On the other hand, if the supersymmetry is broken in zeroth order, then either the instability is removed or it is amplified, i.e. the pseudoGoldstone particle acquires either a real or an imaginary mass. Again, it requires detailed investigation to decide which alternative is chosen in any given system.
B.4. Slavnov’s mass generating terms The appearance of vacuum instabilities seriously limits the utility of strictly supersymmetric schemes. If such a pathology arises in a particular model one can only conclude that it is not amenable to perturbative treatment. Whether such a model is sick in some even more fundamental way cannot be answered a t present. In order to circumvent these difficulties we consider now the introduction of pseudo-explicit symmetry breaking terms which serve t o lift the vacuum degeneracy in zeroth order. Two mechanisms for breaking supersymmetry in a particularly soft way have been invented by Slavnov. The first (Type I) mechanism is applicable to Lagrangians which admit a global U(1)symmetry. This symmetry is then gauged in the usual supersymmetric way and then a singular limit is taken in which the gauge coupling vanishes while the associated auxiliary field D becomes infinitely large. At no stage is the supersynimetry of the extend:d system broken explicitly and, in fact, a Goldstone spinor is present. However, this Goldstone spinor, along with the U(1)gauge vector, is decoupled in the limit, and the end result is a scalar mass term which, so far as the original system is concerned, acts like an explicit symmetry breaker. The derivation goes as follows. The supposed global symmetry acts on the matter fields 0-through “charge” matrices Qi and it is required that this symmetry remain unbroken, &*(@*> = 0 . It is made local through the introduction of a gauge superfield !Powhich couples to @& through the kinetic terms,
(IV.26)
118
ABDUSSALAXand J. STRATUDEE
where go denotes the new coupling constant and the dots indicate all other gauge Couplings. With a U(1) local symmetry it is possible t o intrcduce a linear tern1 (IV.27) where p is a parameter with the dimensions of mass. The new gauge field gives rise to a new term in the classical potential, viz. (IV.28) and thie is the only term in the entire Lagrangian which becomes singular in the limit go --f 0 (keeping p fixed). In this limit the vector and spinor components of Yo will decouple and the only relic of this field will be the terms
(Iv.29) of which the first, a n infinite c-number, is irrelevant and may be discarded. Only a n effective mass term for A+ and A- remains. As remarked above, this mechanism is not an ordinary sort of synimetry breaking. Although the term (IV.29) acts, in effect, as an explicit breaker of supersymnietry, it has a symmetry respecting origin in the gauge field Y,, of which it is a sort of tadpole. I n other words, one can look upon the mass insertion, p2, as the contribution of a zerofrcquency external Do line, P2 = go(D0). (IV.30) I n thG view the qmmetry breaking is spontaneous ruther than explicit. This viewpoint is inis portant when one inquires into the matter of counter-terms. I n supersynimetric theorieonly wave-function counter-terms are necded and this means that only one new divergence is introduced into the theory by this mechanism: the renormalization of Do, which will. manifest itself as a lqurithmic divergence in pz. By virtue of this paucity of independent parameters, Slavnov was able to set up a iiiodel which is both ultraviolet free arid infrared stable. Slavnov’s second (Type 11)mechanism is applicable to systems where it is possible to construct chiral bilineara which are invariant under any local symmetries of the system. Let @+Tv@+ be one such invariant, where is a symmetric numerical matrix. Intrcduce the pair of auxiliary negative chirality singlets S- and S-’ whose kinetic terms and interactions with @+ are governed by a-new term in the Lagrangian, 1 a (L)D2)(S-*S-’ + h.c.) -
2
The non-positive definite kinetic term here indicates that one of the new states must be associated with a negative metric,
119
Supersymmetry nnd Superfields
However, it is not difficult to show that these states do not couple to the physically significant ones. To see this, reduce (IV.31) to component form,
[Zu-* aa-‘
+ t-iO[-’ + I-*/-’ + ~ - * ( 2 h A + ~ @ ’++ h$+TC-l q++)
+ ru2 f-’]
- f-(21~4+~q$ -+f-*(hA+Tp4+) )
= [u-*(-c?~u-’
+ (I-* +):
4- h.c.
+ 2hA+T?$i’++ h$+TC-lq$+)+ f-(iOC-’ - 2 h ~ I + ~ q $ + )
+ hA+TqA+)- , U ~ A + ~ I I A+, ]h.c.
(I-’
(IV.33)
with the neglect of surface terms. h’ow make a field transformation, i.c. substitute in (IV.32) the expressions 1 U-‘ = 67 (2hA+TqF+ h$+TC-’j7$+),
+
f-’
=
f-
+
62
- hrl+Ttld,..
Thc resulting Lagrangian,
[
au-* 26..
+ f-iat-+
(
f-*
7b2)f - - , 2 A A T , ~1A ++
+-
h.~.
describes a pair of non-interacting superfields S- and 8-, tcgether with a scalar iiiass tern is, -/13A+TY)A+ 11.c. (IV.31)
+
which is rclcvant to the original systcni. Again this is, in effect, an explicit supersyninietry breaker although it has a symmetry respecting origin. Just as in (IV.30), one can write = -I(/-*), (IV.35) and this implies that p2 must diverge logarithmically as in the Type I case. (If the physical systeni contrtins a singlet @-then the parameter Y - in the term Y F - - nmst develop a logarithmic divergence due to contributions from (IV.34).)
C. Models To illustrate some general features we list three examples. The first two are parityconserving models in which an internal synimetry is spontaneously broken but supersyiiinietry is preserved. The third is a very simple model with a discrete symmetry wherein the supersymmetry is spontaneously broken and a Goldstone fermion appears. c.1. Local V(1)
This parity conserving nicdel involves, in addition to the gauge potential Y’, a neutral singlet S+and a pair of charged fields @+ and @-. The Lagrangian is given by
9 = f (DD)2([@+I2 8
--D D 2
c2gv
+
[@-\2
e-%v
+ IS+12)
(-f4 F-Y.+ + -8E D- D Y + pq @-*S+@+
f hi.
120
ABDUSS u m and J. STRATBDEE
which reduccs in the Wcss-Zumino gauges to
Y
=
1 4
1 2
1 2
-- u;, + 2i-a-n+ y (8,a)Z - - (a,b)2
+ IVrA+12 + IV,A_l2 + $i$$
- g$(u - y5b) $ - g fi [$(xA+ - i y 5 f A - ) + h.c.1 - V ,
(IV.36)
where the potential is given by
with
(IV.38) On substitution of these expressions into (IV.37) the potent.id reduces to1' = g2(a2
+ b2)(IA+12+
IA-12)
+ 2gZIA+12IA-I2 + -1 02 (7€ - IA+I2+
2
IA-12)
.
(Iv .39) I n (IV.36) the covariant derivatives take the form appropriate to a n abelian symmetry, 26%.
Up,= 2,U. - E.U,,
V,$
=
(a,, - iqU,,) I),
V,A+ = (?,
- igU,) A , .
(IV.40)
Minimization of the classical potential (IV.37) is trivial. Indeed one can see that all the auxiliary iields fIV.38) vanish a t the point
(a) = (a) = ( A - ) = 0,
(A,) =
(IV.4 1)
provided E/g > 0. This p i n t is non-degenerate and, since ( V ) = 0, it is clearly an absolute niinimum. The vacuum respects supersymmetry as well as parity and fermionnuinber. The local symmetry is broken, however. One can show that all particles in this model have a common mass. The free Itgrangian is easily obtained by substituting into (IV.36) and (IV.39) the shifted field ..(IV.42) and retaining only bilinear terms. One finds
where the common mass is given by
M = W .
(IV.44)
121
Supersymmetry and Superfields
The scalar field A , can be removed by a gauge transformation. There remain two real scalars, A , and a ; a real pseudoscalar, b ; a scalar diferniion, A _ ;a real vector, U,,; two Dirac spinom, (I) x ) and 1/1/2y5($ - x), of opposite parity. Interactions are characterized by a single dimensionless coupling constant g. The model is presumably renormalizable but we have not examined the quantum corrections. Although this model can have no broader interest we have included it here as the simplest representative of a supersymmetric system in which both parity and fermion-number are conserved. I n fact, it describes the renormalizable interactions of a single multiplet of the complex supersymmetry discussed in Section I.
+
c.2. U(3)local x SUl3)global
A model of the same type but with more structure is constructed from the gauge potertials Yo,Yk(k= l, 2, .., €9,their supplementary pieceb S+O, S+k and a pair of matter fields @+ and 0-in the representat,ion (3,g). The parity-conserving Lagrangian is
.
Y = - (DD)2 Tr (@+texp (2gO!P + %!IJ) @+ + 0-t esp (--2goYo - 2gY) 0-) + @+o(2
[
8 l -
+ Ti-1 Tr (S+texp ( 3 g Y )8, exp (-2gY)) A
.I
+ P T r @-t(g&+O + gS+)@+ + h.c.
I
1
,
(IV.45)
where !P = Y k A k and S+ = S+k,ikare traceless 3 x 3 matrices. I n this case one can show that the potential is mininiized a t a point which respects both supersymmetry and global SU(3),vb. (A,) = fTo x unit matrix, (A-) = 0 , (S+O)= (As,) = 0.
(IV.46)
There are no massless particles. I t turns out that the system comprises two multiplets of the complex supersymmetry, an SU(3) singlet with mass f - 2 g o E and an octet with niass
1-4gzl/3g0.
(2.3. O'Ra ifeartaigh'e model This is the simplest mcdel in which spontaneous breakdown is realized. It contains three chiral superfields, a0-,@,+ and Oz-, and admits a discrete syninietry @o-
+ @&,
+@ -,
@,+ -+ -@,+,
.
(IV.47)
The Lagrangian is
3
1 -
= - (DDp
8
(1@O-l2
+
1@1+12
+
1@Z-l2)
(IV.48) and the potential is given by
V = iF0-1' Zeitvclirift ,,Fortschritte dcr Phyaik''. Heft 2
+
+ IF2-]',
(1x49)
ABDUSSALAJI and J. STRATHDEE
122
where the auxiliary fields are
-Fo-
=
h
-Fl+
3 + - A2 : + ,
= mA,-
+ hA:+Ao-,
-F2- = mA,,.
Of the three extremal conditions aV/aA:- = 0, etc., oiily two are independent:
0 = mA2-
+ hA:+Ao-,
(
O = m2+ There are two solutions, but one is deeper, viz.
(AI+) = [ ( - 2 / h s ) (m2
+ h3)]1'2.
Since only the ratio &/Ao- is fixed here the minimum is degenerate. The system contains a massless di-lepton in the zeroth order. To compute (Ao-) and A(?-) one must take account of quanturii effect^.^) Notice nt2
(Po-) = h, (Fl+)= 0,
m[(-2/h2) ( (F2-) = -
7 d
+ kq)]l'r,
which certifies the supersymnietry breakdown.
D. Unification of Weak and Electromagnetic Interactions in a Supersymniet ric Approach To conclude this discussion, we turn to the problem of incorporating the weak and electromagnetic interactions of leptons and hadrons in a supenymnietric framework. As yet there exists no satisfactory resolution of this problem and the following paragraphs are intended to point up the problems as they appear to us now. In the absence of any more compelling alternative, we shall begin with a iiiiniiiid extension of the S U ( 2 ) x V (1) iiiodel of weak and electromagnetic interactions where leftThere is a standard technique for obtaining the one-loop contribution to the effective potential. Make the c-number shifts
4,
A+
+ ao-
+ A+,
A,+, A,+ -+a,+ i
A,-
+ 8%-
+~ 2 -
nnd pick o u t the terms which are bilinear in the q-numbers. This shiftedfree Lagrnngian determines t i set of masses which are functions of uQ-,etc. Thc one-loop contribution is then given by
(IV.50) On restricting the variables a,, uI+and a,- to the subspnce on which the claasicnl potential is minimized (keeping only a,,-, say, ns a n independent variable) the counter-terms, like V"), are constants. One then minimizes V(" with regard to a,,-. It is found that this minimizes a t a,- = a,- = 0. Leptonnumber is conserved. One finds, in addition to the Goldstone lepton (left-handed),a lepton and a dilepton bf mass Jf = ) I - ( n ~ z ? a h ) , a pair of ordinary bosons of mass nnd ViW f n ~ z . Finally, the pseudo-Goldstone di-lepton is found to have the (mass)?
+
fw
123
Supersymmetry and Superfields
handed (matter) leptons and quarks are classified into doublets and the right-handed are (matter) singlets. The supersymmetric extension involves introducing doublets and singlets of scalars to go with the fermions and a (gauge) triplet plus Ringlet of fermions to go with the gauge vectors of S U ( 2 )x U(1). This much is the bare niinimuni but in practice many more fields are needed. The new left-handed gauge fermions ( S U ( 2 )x U(1) triplet and singlet) in the supersymmetric approach can have nothing to do with the matter fermions (leptons and quarks) which are supposed to be doublets. They must therefore be very heavy (or, in soine other way effectively decoupled from present-day physics). They must therefore be given right-handed partners from among new matter fields. These new matter fields may then be thought of as part of an “exteLded gauge system”. No members of the extended gauge system - apart from the photon - have yet been seen; presumably they are all very heavy. We shall continue to assume the existence of the ferniionic quantum number F associated with the chiral components of the supertranslations. This number will not be identified with any of the known ones such as electron-or muon-number. If these nunibers are all separately conaerved, as we shall assume, then there will be no mixing between the gauge fermions and the matter fermions. This is an important requirement if the universality of the weak interaction (equality of weak (ev,) and (pvJ couplings, is to be maintained. (It follows also that the Goldstone fermion, arising through spontaneous breakdown of supersymnietry, will not be identified with either of the known neutrinos.) To ensure the universal strength of the charged weak current one assigns left-handed fermions or right-handed antifermions of doublets of S U ( 2 ) 1x V (l ) y e.g.
while right-handed ferniions or left-handed antifermions are singlets, en-
with
I =‘O,
Y = -2,
pz+
with
I = 0,
Y = +2.
Since vL has no right-handed partner, it must remain massless (at least if electronnumber is conserved). Likewise for Y R ’ . The quarks are similarly assigned, e.g.
(”n), with
I = 112,
Y
PR
I = 0,
Y = 413,
nR
I=O,
Y
=
=
1/3,
-213.
A major difficulty concerns the scalar partners of these fermions. These scalar ter.d to be degenerate with the fermions. Sotiee firstly that if these scalars carry globally conserved quantum numbers, like electron- or baryon-numbers, this will reqhire their vacuum expectation values to vanish. I n turn, this will mean that the masses of these supermultiplets arise entirely from their interactions with the extended gauge systems. This extended gauge system has two parts: the vectors and left-handed fermions contained in gauge superfields Y k ( zO), , and the scalars and right-handed ferrnions containcd in a set of positive chirality scalars Di+.It
o*
124
ABDUV SALAXand J. STRATHDEE
is the scalar components of the latter which will play the role of Higgs fields and generate all masses. Now there is some latitude in the assignment of the scalars but, unfortunately, all the versions we have examined suffer similar defects. The main defect is the persistence of a mass formula by which the average (mass)aof fermions in any charged supermultiplet is equal to the average (mass)2 of bosons. This means, for example, that if the electron is Bssigned to a scalar supermultiplet, then even after the breaking of supenymmetry, its two bosonic partners must have masses -1/2 MeV. This rule, for which we have no general proof, perists have when Slavnov's mechanism is used. This will be illustrated in the example which follow^.^) TOillustrate the main features of a supersymmetric version of weak and electroniagnetic interactions, we present a model based on the local symmetry L!~U(~)~X U(l)y ,x U(l)y, wherein the electric charge is identified by
(IV.51) The other four symmetries must he broken spontaneously. The gzuge superfields !P, !Po, and !Po, include three electrically neutral and two charged ones comprising the photon, two neutral vectors, the charged vectors and the accompanying left-handed fermions. Right-handed fermions are contained in a pair of positive chirality doblets @,+ and !D2+ with the (I,Y,, Y a )assignments @I+ (1/2, - - 1 , O ) , (IV.52) (1/2,0, 1).
-
-
These fields, together with the gauge fields Y , !Po,,Yo,make up what we designated as the extended gauge systems. Matter leptons will take one of two possible forms, either lefthanded doublet paired with right-handed singlet, 03(1/2,0, - 1) , (IV.53) s3+ (O, -')
-
' 9
J
or right-handed doublet paired with left-handed singlet
Fractionally charged quarks (to which, for simplicity, we shall confine our considerations) are assigned in a similar way except that they carry fractional hypercharges and each doublet is paired with two singlets instead of one so that all quark states are massive. (See Eq. (IV.91) for the assignments of the second singlet.) 6) An alternative one might considcr is the use of vector supermultiplets. These contain two fermions along with a scalar an d a vector. One fermion could be light and the other heavy like the bosons. However, such a n approach brings its own difficulties. Firstly, if the theory is to be renormalizable then the vector will have to belong t o a gauge system and one will not be able to make do with SU(2)x U(1). Rather, one woulcf have to associate a n independent symmetry generator with each lepton and quark. This very large symmetry would then have to break in such a way as to yield one relatively light charged vector W+ and perhaps a neutral Zo. The leptons and quarks would belong to thc adjoint representation an d i t is hard to imagine how they could all resemble doublets of some relatively weakly broken SU(2) subgroup. The problem of setting up a universal weak interaction would secm to be insurmountable in such a framework.
Supersymmetry and Superfields
125
The matter interactions are severely limited with these assignments. Only two types of coupling can be made, 1 -
+
-hS4._O;-it2@,+ ,+ + h.c.). 2 J~~Y@J-S~~%
(IV.55)
Xotice the global symmetries, 03--+
SB+ -+ eiaS3+,
eia03-,
G4++ eiB04,, %9+
S4- --f e W 4 + , --f
(IV.56)
@I#*+
which may be interpreted as electron and muon number symmetry. The gauge interactions are given by the kinetic terms,
plus the supersyiiiiiietric analogue of the Yang-Mills Lagrangian (111.38). Wit.h two commutative U(1) groups in the local synimetry we can have two dimensional parsmeters, ;, and t2,in the term 1
(my (ElYOl 8
+
E?u62).
(IV.58)
l'he scalar potential is the standard one,
+
1 E - A f + A l + Ai+Al+ - la4-I2, -D 01 - -1 91
- - D1 o , Q2
91
=
52 -+ AJ+A2++ Al-A3- - 3/a3+la-
92
(TV.OO)
ABDUSSALAM and J. STRATHDEE
126
where the notation for component fields is exactly as in previous sections, wiz.
If the various quantum numbers, fermion, electron, n u o n etc., and electric charge are all . conserved then, in the vacuum only two fields can develop a n expectation value, ( A ; + )= vl
and
(A;+) = v2,
(IV.61)
where the values vl end vz are fixed by minimizing the potential (IV.59). They can be expressed in terms of parameters in the Lagrangian by solving the equations
(IV.63) On expanding V a t the minimum, one finds the following mass terms:
V
M
2g21v,A,-*
+ vlA2+IZ
- 2g2(Vi2- v2')) 1%-12 + (h2V22 - 2g2(v12- v z 2 ) ) luf12 f
(fLVa2
+ 2(g2 + gI2)v,2(Re
AI0)2
+ (f?vz2 + 2g2(vi2- vz2))IA3-12 +
(/12v,2
+ 2#(v,2
- v 2 ) ) lA,+12
- 4g2v,v2(ReAle) (Re Azo)i2(g2 + qz2)vZ2(ReA,0)2, (IV.63)
(where superscripts refer to electric charge). All non-Goldstone components, except for A:- and A:+, have maw and these masses are real if fizz
> 2g21v12 - vz21
and
h2vZ2> 2g21vI2- Va21.
(IV.64)
The pseudo-Goldstone states A:- and A:+ could receive masses from the radiative corrections. They are without mass in the tree approximation because of their association in superrnultiplets with the neutrinos $&, &. Couplings to the extended gauge systems provide only a modestly effective means for breaking the supersymmetry : charged multiplets are split but neutral ones are not. Fortunately, it is not necessary to rely on radiative effects for lifting this degeneracy. Instead we can exploit the Slavnov Type I mechanism. The necessary global V (1) symmetries are simply, electron-number, muonnumber, etc., and so we can immediately adjoin to (IV.63) supplementary mass terms of the type (IV.29), ,UI~(~A,O~' f IA3-12 - 1%-12)
+ ,U,2(IAd012+ IA,+I2 -
la,f12)*
(IV.65)
For the positivity of a3- and a,+ masses we must have
(IV.66)
127
Supersymmetry and Superfields
Finally the masses of the two real neutral components Re Alo and Re AZ0are determined hy the matrix 2(g2 g?) V12 --g2v1v, (IV.67) 2(g2 9z2) vz2 292vlvz
(-
+
)
+
whose eigenvalues are certainly positive. Thus, subject to the above inequalities, all scalar masses are real and the minimum is established. In particular, the assumed conservation of charge and the various fermionic quantum numbers is justified. Consider now the spinor and vector masses. The spinor mass terms are contained in the various Yukawa couplings of A, and A,. They are
-
-jVz$&+
- )%5,+_$,+,
+i2gvlF$;+
9
+ i2gv,I-+$,+,
,
+ i l / ~ v l ( g & -- glx;-) $:+ - i f i ~ ~ ( g 1 ~ -qzl!-) $;+
+ h.c.
(IV.68)
The masses of the charged particles can be read iminediately from this. For definiteness, suppose that the electron is contained in @ ,, S,, while the muon is in #j4,S4,j.e. that ebclongs to a left-handed doublet while p+ belongs to a right-handed one. (It would be equallypossible ifp-were assigned to a left-handed doublet. There is no problem of electron muon mixing if none is introduced in the bare Lagrangian.) I n addition to the two charged leptons m(e-1 = f U 2 , e- = $i53+, (IV.69) p+ = 5;-
+ + $+:,
= hv,,
++)
there are two charged fermions in the gauge system,
f?+ = $ + ; Q- =+6;1,
+ a_+, m(Q+)= 2gv,, + Q--, n@2-) = 2gV1.
(IV.70)
Associated with the charged leptons (IV.69) are the two neutrinos, V L = $!-
and
vR’
= $:+,
(IV.7 1)
which are without mass. Finally there is a set of mass terms for the neutral fermions in the extended gauge system. These can be arranged in the form of a rectangular mass matrix,
with
(IV.72)
-QVZ The canonical represent,ation of this matrix would take the form
).,
m1 0
Y,
(IV.73)
ABDUSSALAXand J. STRATHDEE
128
whcre U and V are orthogonal matrices (3x3 and 2x2 respectively) and the "eigenvalues" m, and mzare reel and positive. These eigenvalues together with the angles in U and V are obtained in a straightforward fashion by diagonalizing the symmetric niatrices M M T and M T M . (It is worth remarking that MTM is presisely the scalar mass matrix (IV.67) while, as will be seen, MMT is the vector mass matrix. This elegant relationship is a manifestation of the underlying supersymmetry : the massive scalars, spinors and vectors belong to a pair of vector supermultiplets). There are four angles involved in the representation ('IV.73), three in U and one in V . They can all be expressed in terms of the three independent ratios, g,/g, g2/g and v1/v2. Let. us write
v=
'.
(IV.74)
(i:,")*
where C1 = cos O,, S, = in 0,: etc. [These angles can be used to define two massive neutral spinon :
Rl' = ( c 4 $ 1 ° 02'
= (s4$10
- s4$2')+
+
c4$Z0)i
+ +
i((c3cZcl
- s3s1)
+
i((s3c2c1
c3s1)
'?I' j.1'
- (c3cZs1 - (s3c2s1
+
&3Cl)112'
- c3cl)
- c3sZ1.3)-
- s38ZA3)(IV.75)
and one massless left-handed spinor: v L ~ =
(s2cla,o - S,S,A~O + c24)-.
(IV.76)
The coefficients in (IV.76) can be expressed entirely in terms of coupling constant ratios,
(IV.77) where e denotes the electromagnetic combination, 1
1
1
1
2 =s+S+?.
(IV.78)
The other angles and the masses m,, m, are given by straightforward but rather lesq interesting formulae. We pass on now to the vector masses. The vector mass terms are obtained in the usual way from the Eggs fields A,, and .tizi. They are (IV.79) %J2(.12 vz2) IW+? VI210JV, - 91F,0)2 z.'2z(gW, - gzVz0)',
+
+
+
where (W+, W,,W-) denote the gauge vectors of ~ 9 U ( 2 )and ~ , V,O, V,O those of U ( l ) y , x U(l)ya. The neutral ternis can be arranged in the form ( V1"V~'W3)
i
9I2Vl2
0
-S91"1'
0
!7*2"2a
-0S*"z2
-sg1Vl2
- _ ( Vl'~VZ'~3)M M T 2
-g922.'z2
q2(vla
+ %a)
) (3
Supersymmetry and Superfields
129
with M given by (IV.72) and (IV.73). The two massive vectors acd the photon nre, therefore, 21 =
(C3C2CI - S3SI) PIo - (C3C#l
2, = (S3CzCl
+ C$,)
+ S3CI) VZo - C,S,W3,
VIO - (S,CZS, - C3C1) Vz" - S3SzW3,
A = S@1Vlo - SZS, V20
+ C2
(IV.81)
W 3 .
In summary, the system comprises the following states: (1) A charged vector supermult,iplet
1 spin 3
R+ = &+
+ iL+,
m(@) = 2gv2,
(lV.82)
spin I 2
-
-f2- = - v;+ - ZL-, ..
and its conjugate. ( 2 ) Two neut.ral vector superinultiplcts
??@-)
=
2gv,,
ABDUSSALAJI and J. STRATHDEE
130
These niultiplets are unbroken in zeroth order. Their respective masses are given by mi,? = (g2
+ g*2) + (g2 + 9z2)vz2 F l/((92 + V12
912Y V12
- (g2
+ 9z2)
%2)2
+
ag.vlzv22.
(W.85)
(3) The photon supermultiplet
(IV.86) 1 spin 2
vLy
+
= S2C1A7- - S z S 1 l ~ - C.J3- (the Goldstone neutrino).
The vector supertnultiplets (1)-(3) exhaust the gauge system. Leptonic matter is classified in scalar superrnultiplets: the Q3-, S3+and 04+, S,- type combinations of which there may be duplicates. For definiteness we consider one of each and label them “electronic” and “muonic” respectively. (4) The electron supermultiplats, one charged,
spin 0
a;+,
1 spin 2
e- = ti+
spin 0
A;-,
+ q3-,
m(e-) = fv2,
+
(IV.87)
(note m2(a3-) m2(A3-)= 2?n2(e-)) and one neutral, spin 0
A;-,
”(A30)
= jil,
(IV.88) 1
spin 2
m(vs) = 0 .
vL = $:-,
( 5 ) The muon supermultiplets, charged,
spin 0
A:+,
1 spin 2
p+ = -$ :
m(A*+)= l/hZv,2
+ ,-:C
+ 2qY”(v12-
v22)
+
p*2,
m(p+) = iiv,,
m(a4+)= I/h2v23 - 2g2(v12- v22) - p*2, spin 0 a,+_, (again m2(a4+) m2(A4+)= 2m2(p+)
+
(IV .89)
and neutral,
(IV.90) 1 spin 2
vR’
= $:+,
m(v8’) = 0 .
Finally, since the baryonic supermultipIets should not contain any neutrinos, it is necessary to adjoin extra singlets, S;+ or &-. Moreover, if the baryonic multiplets are
131
Supersymmetry and Superfields
fractionally charged, then their hypercharge assignments and, consequently, their Inasses will be different from (3) and (4) above. Consider the (I,Y,, Y 2 )assignments, @3-
83,
-
( I / % Y , Y - I),
@4+
(0, y , Y - 2 )
s4-
-
s3+ (0, Y + 1, Y - 1)
-
-
+ 1, Y ) , (0, y + 1, y + 11, (I/% Y
-
Sh-
(IV.91)
(0, Y , Y ) ,
under which the following interaction terms are permitted:
-L2 ~D[!ZJ-(f”Qj2+S3+ + f’@l+S;,)+ (h”L’:-@,T + h’S;+_@T+) izz!ZJ,+ + h.c.1.
lIV.92)
One finds the following spectrum of baryonic states. (6) Supermultiplets with charges Q = Y 1, Y and masses given by
+
AY+l Q+
fh’%22
+ ( Y + I) 2g2(v12-
v22),
(IV.93)
(IV.94) To sunimarize, what we have shown above is that the demand of universality of couplings can be niet, if the extended gauge sector fermions do not mix with matter (lepton and quark) fermions. (One important consequence of this is that gauge fermions exist which are about as massive as W and 2 particles.) The chief drawback of the nicdeI presented lies in that there also exist in the model (charged) spin zero particles - companions of electrons and muons - with light masses. The hope is that radiative effects may cure this defect of the model. ~
~
As is well known, there are no anomalies in the theory, provided that there are two (integer charged) leptons and four quarks among the matter fields with their conventional assignments of Y quantum numbers. (The extended gauge fermions do not produce anomalies.) R,
ABDUSSALAM and J. STRATHDEE
132
To obtain an effective Lngrangian useful for phenomenology a t relatively low energies (< 6 GeV) it would be reasonable to integrate out all massive components of the extended gauge systems, i.e. the supermultiplets (1) and (2) which are very heavy (> 50 GeV). The photonic supermultiplet (3) must of course be retained in the effective Lagrangian. That part of the interaction Lagrangian which is linear in the gauge field defines a set of currents,
3 (QjoJ(.cdjO)+ qpJ(zYp) + +jOJ(+jo)
j-1.2
+
QjO)
J(ajO)
+ (a-J ( P ) + Wp+J(Wp,-)+ 4+J(+-)+ J ( a + )L?++ h.c.).
_ .
Elimination of these gauge fields yields the effective intraction Lagrangian,
We shall not pursue this any further in view of the unrealistic character of this incdel. All of the currents in this effective Lagrangian receive contributions from lepton- or baryon-number carrying scalar fields which cannot be ignored. Their masses must, according to formulae like (IV.87), (IV.89), (IV.93), (IV.94) be of the same older as those of their fermionic partners. (This being so, there is even a long-range pseudoelectromagnetic interaction between scalars and spinors due to the exchange of the photonic neutrino (IV.86).) Appendix A
Feyniiian Rules
A number of apparently renormalizablc Lagrangian models has been given. In this appendix rules are developed for computing terms in the perturbation series. To take full advantage of the supersynimetry, these rules are presented in a manifestly supersymmetric fashion. The formal derivation of Feynnian rules proceeds very much as in any ordinary quantized field theory. Consider firstly the most general renormalizable system made entirely from scalar superfields (i.e. without any local symmetries). The action is given by
Introduce external sources Ja+,J i - and define the connected vacuum amplitude Z(J) by the usual path integral,
i
exp- Z ( J ) = b
s
(a@do)*)
(J-t@++ @-tJ+
1
+ h.c.} . (A.2)
133
Supersymmetry and Superfields
Connected Green's functions are given by the functional derivations of Z evaluated a t J = 0. For example,
For inany purposes it is convenient to have a special notation for the functional derivatives of Z a t arbitrary values of J. Thus we write
(A.3)'
defining the field averages as functionals of the external currents. In principle these expressions can be solved to give the external currents as functionals of the field averages to which they give rise. We denote field averages by the sanie symbols as are used for the field operators since no confusion can possibly arise. Higher derivatives of the connected vacuum amplitude are denoted quite generally 8s follows : B 6 B ... Z ( J ) = G(o+(l) @+(2) @-t(n)).
dJ-t(l) BJ-t(2)
dJ+(n)
* * a
As an alternative to the connected vacuum amplitude Z ( J )one niay choose to work with the effective action r(@) defined by
The effective equations of motion then read
and they are solved by the expressions (A.3). The effective action is characterized by oneparticle irreducible graphs and it may be obtained, in principle, by solving the DysonSchwinger equations:
(A.7)
In these equations the fields @ are the independent variables and one is to solve for the In order to do t h i s it is necessary to havd the 2-point functions G(1, 2), functional r(@). which appear in (A.7) expressed as functionals of @ rather than J. This is done by defining G(1, 2) as a functional inverse of the matrix of second derivatives of P(@).We shall not pursue this matter here. Although the perturbation series for F(@)is simpler than that for Z(J) in that it contains only irreducible graphs, the latter is easier to derive rules for. The deriv at'ion proceeds as follows.
ABDUSSALAM and J. STRATHDEE
134
Firstly, separate the classical action into two pieces,
a(@) = So(@)+ f l u ( @ ) ,
(A.8)
where S , contains all the trilinear ternis (i.e. Sois obtained from (A.l) by setting qaii = 0). Then the connccted vacuum amplitude is given by 2
exp
f
Z ( J ) = exp - 8,
(t-&)
exp
f z,(J), 2
where 2, denotes the amplitude corresponding to S,. It is easily evaluated. To evaluate Z o ( J ) notice , firstly that the corresponding effective action, ro, is just So(@). This is implied by Eqs. (A.7). Turn next to Eqs. (A.6). With replaced by So they become linear inhomogeneous equations, viz.
r
1 -680 - -- DDOa- + 3,+ Mai@i+= -Ja+,
ma-
2
(z4.10)
which can be solved with the help of the identity
One findg,
(A.11) @+)(I.
=
-(a?
+ iwMj-i(+
BDJ-
1
+ M ~ ( J+ + -9) .
Finally, substituting these solutions into the right-hand sides of Eqs. (.4.3),a quadrature gives
Z o ( J )= [ d z ”
[+ (BD)a( J+t( P- MMt)-’J+ +
1 -
+
J-+(S2 J!tM)-l J - )
1
- - DD( -J-t(aa + M+M)-’ Mt(J+ + 3) + h.c.) , 2
(A.12)
where t.he Feynman inverse is understood. This functional can also be expressed in the somewhat more cumbersome form,
which serves to define the zeroth order expectation value,
(11.14)
Supersymmetry and Superfields
135
and the zeroth order propagators, Go(@+(l)@+t(2))= (aI2
+ MtM)-l
GO(@-(l) @-t(2)) = (al'
+ MMt)-'
Go(@-(l) @+t(2))= -(a12 GO(@+(l) @-t(2)) = -(a12
(-4.15)
+ MMt)-l M &(l, Z), + JItM)-' Mt d+(I, 2).
The chiral delta functions are given by
where OIz = 0, - Oz. It is practical to work with the truncated fields d+ introduced in Section ILB., (A.16)
For these fields the propagators are considerably more simple, G o ( d + ( l )d++(2)) = (a,2
+ MtM)-' exp (81-ii3102-) d(r, - z2),
q,(d+(i) 6-+(2))= --(a12
+ M + M ) -~t~
2
e12-e,2+qxl-
22).
The perturbation series for the connected vacuuin amplitude Z ( J )is now given by substituting the expression (A.13) (or (A.12)) into (A.9) and expanding in powers of s,, (A.18)
This prescription may be suniniarized in a set of nionientuni space Feynman rules as follows : - (i) Draw diagrams in the usual way with 3-leg vertices corresponding to the interaction terms gaiid:-di+&i+ and g&da-d;*@+ -
+
(There may he 1-leg vertices corresponding to the linear term, h.c., but in practice these can always be eliminated by a suitable shift in the fields ai+. Such a shift would modify the inass matrix but not the couplings.) The vertices each carry a definite chirality, positive or negative.
136
ABDCSSUAM and J. STRATHDEE
(ii) With each vertex associate the factor
+ + p 3 ) i poii
(positivc chirality)
+ p , + p 3 ) i gzij
(negative chirality).
( 3 7 ~ ) ~ S ( pp, z or
(2.2)' B(p,
n
These factors are presented graphically as in Fig. 1 10
I I
1 I
1 I
I
-A
_I_
i
+
j
I
Fig. 1 positive chirnlity
-
i
negative chlrality
(iii) With each line (joining B p9ir of positive vertices associate the propagator
d i
+~
-- [(.+
1 2
t ~ )~ -t ]i , , el2-oI2+.
Wit.h each line joining negative vertices associates the propagator
w
-- [ ( - p z i
1 + ~ l l r l t ) - nilai i - e,z+e12-. 2
With each line joining vertices of opposite chirality associate as appropriate one of the propagators 1
( -p2
+ M + N ) Gexp ~ (i3,-pO2-),
where the 4-momentum is directed froin the positive vertex (1) to the negative vertex (2). (Sote the asymmetry, 8,-p02-= -8z+p81-.) These propagaton are represented graphically in Fig. 2,
+ 1
+
-----.a
i
a
i
i
2
137
Supersymmetry and Superfields
(iv) With each external line associate the wave function d p t
+
= ~ p ( p )& + p ( p )
+ 1 eFe+F,eXt(p).
, i.e. extract the coefficient of the
(v) At each vertcx apply the operator term
n
+,e,
vertices
and integrate over the momenta,
(The rule (iv) applies to incoming our outgoing particles. The modifications necerrsary to characterize lines which terminate on the external currents J + are easily arrived at.) To conclude this appendix we list the component stnxture of the propagators (A.17). These are conveniently separate into three sectors according to value of the fermionic number F in the intermediate states. The non-vanishing components are as follows.
1. F
= 0 sector
Go(A+(1) A+t(2))= (az
+ MtM)-' d(x, - z2), (A.201
Qo(++W 3.
F = 2 sector
$-(a)= (az + M
+ M )+ - iy5 ' 2 Mt d(2, - 2J.
Go(F+(l)F+?d(2))= -ayaa
+ MfM)-'
+ MMt)-'d(z, GO(L(1)27,712)) = -(a2 + MMt)-'M Qo(A-(l) AJ(2)) = ( a 2
Qo(F+(l) A J ( 2 ) )= - ( a 2 10 Zeitschrift ,Fortschritte der Physik", Heft 2
d(2,
- 22),
- 221, (A.22) d(z, - z2),
+ MtM)-' Mt d(z, - x 2 ) .
138
ABDUSSALAYand J. STRATHDEE
Vertices are governed by the interaction Lagrangian, p g
+ A:-(gAi+Fj+ + rCIiT,C-'rCIj+)l + h*c-
= gatj[C-Ai+Aj+ - 2$di+rCIj+
(-4.23)
The formulae (A.20)-(A.23) give a good indication of the complexity of the model described by the Lagrangian (A.l) whose Feynman rules have been listed above. However, they give no hint of the remarkable cancellations coming from the supersymmetry. To see these effects it is definitely worthwhile to employ the superfield notation and associated Feynman rules. Appendix B Regularization and Renormalization
In order t o regularize the model of Appendix A in a manifestly supersymmetric fashion i t is sufficient to replace the kinetic terms by the expression
whcre R* (a) are polynomials in a/ax,
R*(a) = 1
+ E*a2 + q * ( W 2 +
(B.2)
The exact form of these operators is not important. They need only be of sufficiently high order to dampen the short distance behaviour of propagators so as to make finite the radiative corrections to the Green's functions. The parameters t ,v , .. are to be taken to zero a t the and of any computation. (In terms of the familiar momentum space 1/A2,7 1 1 ~ 4 ,..) cut-off, A , we have E Since the regularizing factors R+ are introduced only in the bilinear terms they will cause changes in the propagators but not in the vertices. It is a simple matter t o construct the regularized propagators. One has only to modify the equations (A.lO) by intrcducing the factors R+ and R-. They become
.
.
N
1 -
-DDR-(a) 0-+ 2
+ M@+= - J + ,
and they are solved by
Differentiation with respect to J , gives now the regularized propagators,
Supersymmetry and Superfields
139
where a = alax,. It is clear that the light-cone behaviour of these propagators can be made as soft as required by choosing the polynomials R, (a) of sufficiently high order. Since the regularized kinetic terms (B.l) are manifestly supersymmetric, i t follows that the regularized amplitudes constructed from the propagators (B.5) will be compatible with supersymmetry as well as finite. This is important for the arguments which follow. The renormalization of supersymmetric models is greatly simplified by the occurrence of zeros in certain amplitudes whose canonical dimensions would, on the face of it, seem to indicate divergence. A good analogy is given by the electrcdynamic vacuum polarization, q m ( 4 = (a, a2 - a,,&, W Z 2 ) . The kinematical factor reduces the "effective" dimensionality of l?,,,(k) from 2 to 0 and so the divergence is logarithmic rather than quadratic. This analogy is not perfect, however, in that the softening is due entirely to current conservation - i.e. gauge synimctry - whereas, in the supersymmetric system t,here is, as we shall see, a subtle cooperation between the symmetry and the constraints due to renormalizability. In order to reduce the discussion to essentials, we shall henceforth ignore the internal degrees of freedom (including the fermionic number, F) and deal with the prototypical system comprised of a single scalar superfield, @+, in self-interaction, 1 -
9' = - (BD)']@+I2 8
-
M 9 DD a@+ +@+2 + - @+3 + h . ~ . 2l - ( 2 3
The simplification is only notational. It will become apparent as the argument proceeds that it is applicable to the general case as well. Assume now that a supersymmetric propagator regularization of the kind discussed above has been made and consider what sorts of divergence might be expected in the limit when the regularization is removed. The m a 1 dimensional argument would indicate the presence of divergences in the folIowing derivatives of the effective action,
evaluated a t @+ = @+* = 0. Typical components are shown on the right-hand sides in order to reveal the canonical dimensions in a familiar way. Thus the quantity (a) has dimension 2, and one is therefore prepared to find that it diverge quadratically. The amplitudes (b), (c) and (a) have dimensions 1 , O and 0 respectively, and could therefore 1 o*
140
ABDUSSALAM and J. STRATWEE
diverge logarithmically. These are the only quantities which even potentially diverge') and, as it happens, only (d)actually diverges. Fortunately there is a simple explanation for this suppreaeion of divergences and there is no need to enquire too deeply into the graphical structure. Of particular importance are the formulae (II.38) which express supersymmetry for the nunwchiral Green's functions, f3*(1,
..a,
...d*(n)).
n ) = (Td*(l)
The amplitude 8, is a function of el+,ON,
...,fl,,+t
(Be71
which must satisfy the constraints
and can therefore depend only on the n - 1 differences Oin+ and in a very restricted way, moreover. Only those combinations of the Bin+ and momenta which are annihilated by 91-1
the sum, 2 piei.+, can appear in 4., For example, . 1
where, to save space, we use the abbreviated notation, ea =
-21 e-e+.
.
The scalar amplitudes Q1, 4, .. depend only on scalar combinations of the momenta. We shall now prove that there can be no radiative corrections to the monochiml vertex functions (a), (b) and (c) evaluated a t the origin of momentuni space. All we need in addition to the group theoretical results indicated in the structure (B.9) is the remark that the required amplitudes (a), (b) and (c) can be obtained as coincidence limits on amplitudes like &+(l, 2), &+(l, 2 , 3 ) and &+(1,2,3,4),respectively. Thus, for example,
=
a + g 5 0R + (. 1 , 2 ) )
9
e.=e,
=
3,
since, according to (B.9), &+(l,2) must va&
(B.10) in the coincidence limit OI2 = 0.
The absence of a quadratic divergence from the mass term (b) is to be expected since the supermultiplet includes a fermion whose mass can, at worst, diverge logarithmically. Quadratic divergenm in the scalar masses must therefore cancel. A third derivative, like (c) but involving one differentiation with respect to $+*, neednot 5 considered: its finiteness is assured by supersymmetry since thcre is no corresponding term, (1/8) (DD)*(@+*@+*) in a renormalizable Lsgrangian.
141
Supersymmetry and Superfielde
I%e second formula of (B.9) gives &+(1,2>3)1e,-e,= (PI
+
~
2
~ )A ~G
(B.ll)
2
+
which vanishes at p 1 pr = 0. If we understand this monochiral amplitude not as the 3-point Green's function but, rather, as the hybrid quantity
then the limit (B.ll) representee the self-energy operator, i.e. the radiative correctionR to the amplitude (b) (see Fig. 3) Fig. 3 Self-energy operator
+
The crucial result here is the emergence of the kinematical factor ( p , p J a which causes the self-energy to vanish at the origin of momentum space. Equivalently, (B.12)
Since the potential divergence was only logarithmic we may conclude that 17 is strictly finite. An entirely analogous argument is applied to the monochiral function
Its structure is exhibited in (B.9), which shows the vanishing of this quantity in the limit ol,=
+
OJ
p1
= p3 = p4 = O.
This means the absence of radiative corrections to the coupling constant. Tlie aiiiplitude (c) is finite. Evidently the arguments given here will go through in exactly the same way when fermion number and other internal degrees of freedom are restored. Only the trilinearity and monochirality of the interaction was used (along with the supersymmetry displayed in the formulae (B.9)) and these features are common to all the renormalizable models made from chiral scalars. Finally we may mention that the amplitude (d) is truly divergent and must be compensated by a wave function counter-term of the form
(2- 1)
1
(BD)2I@+l2.
(B.13)
This wave function rescaling will of course affect the parameters 3,M and g in the usual way
$ + a , = Dl23J M -+ M , = ZM, fJ 3 Or
=m g .
(B.14)
142
ABDUSSALAM and J. STBATHDEE
Referencea
[Z] Yu.A. GOL'TAND, E.P. h K R T b I A N , JETP Letten, 13,323 (1971); D. V. VOLKOV,V. A. SOROKA, Phys. Letters 46B, 109 (1973); J. WESS, B. Zu~mo,Nuclear Phys. B 70,39 (1974); Phys. Letters 49 B, 62 (1974). [Z] ABDUSS w , J. STBATEDEE. Nuclear Phys. B 76,477 (1974); S. FERR~BA, J. WESS,B. Z m o , Phys. Letters 61 B,239 (1974). [3] J. WESS. B. Zomo, Nuclear Phys. B 78, 1 (1974) ; ABDUSSaLaar. J. STBATHDEE, Phys. Letten, 61 B, 353 (1974); 5. FERBARA, B. ZUMINO,Nuclear Phys. B 70,413 (1974). [a] P. FAYET.J. I.x.xopoa~s,Phys. Letters 61B, 461 (1974); L. ~'RAIFEAIITAIOH, Nuclear Phys. B 97, 331 (1976). [5] B. ZUMINO,Proceedings of the 17* Internstionnl Conferance on High Encrgy Physics, London 1974; S. FEBRARA. Rivista del Nuovo Cimento 0, 105 (1975); P. FAYET,S. FEBRABA, Phys. Rep. 88 C, 249 (1977); Abdus Salam and J. STRATHDEB, Phys. Rev. D 11, 1521 (1975).
