PROC. 19th INT. CONF. HIGH ENERGY PHYSICS TOKYO, 1978
P 9b Unification, Superunification and New Theoretical Ideas
A. SALAM
ICTP, Trieste
ceeds as follows: §1. Introduction a) Find an internal symmetry group G, The Organizers have kindly allotted to me simple or semi-simple (with discrete symmet five parallel sessions—nine hours of the Con ries), which includes SU(2)xU(l)xSU(3)c ference—to report to the plenary session on. and whose fermionic representations describe These include sessions on 1) unification of the known (and predicted) quarks and léptons. strong, weak and electromagnetic interactions, b) Write a local theory of this internal 2) supersymmetry and supergravity, 3) quantum symmetry group G, with one coupling param electrodynamics, 4) formal field theory and 5) eter g and with Yang-Mills spin-one gauge new theoretical ideas. With my deepest particles. (For a semi-simple group G, dis apologies to those physicists whose beautiful crete symmetries can ensure that there is only work I will not be able to report, Ï have one gauge coupling constant.) The gauge reluctantly decided, in view of the shortness particles include octet of colour gluons cor
± of time, and the experimental interests of the responding to SU(3)C and W 9 Z° and y cor majority of the audience, to structure my responding to SU(2)xU(l). Assuming that report around the theme of unification ideas the gauge theory is asymptotically free, the for the four basic forces: strong, weak, elec unifying constant g would manifest itself at tromagnetic as well as gravitational. For low energies, through renormalization group 2 there is no question that after the unification considerations in the form as=g J47r for strong implied by SU(2)xU(l), the unification of the SU(3)C symmetry and the fine structure con other two forces with what may call the "elec stant a for the SU(2)xU(l) "electroweak" tro weak force" is likely to be one of the most force. motivating concerns in particle physics and c) The descent from G to SU(3)C on the will affect the development of the subject. colour and SU(2)xU(l) on the flavour side
New theoretical ideas often take some five determines the ratio a/as in terms of the high to ten years to mature. My concern will be unifying mass M (expressed in units of a low not so much with the shortrange but rather mass ju~few GeV) in accordance with the with the long-range aspects of unification standard ideas of Georgi, Quinn and Weinberg. over a perspective of ten to twenty years ; In most of the unification models considered, in terms, if you like, of not just the LEP it is usually assumed that the unification mass and the ISABELLE accelerators, projected M is realized physically as governing the for the late 1980's but their successors of 10 masses of ultra-heavy gauge mesons. TeV centre of mass, projected for the later d) There is spontaneous symmetry break 1990's. The weak force will become com ing (SSB) which is accomplished by introduc parable to the electromagnetic above 100 ing a set of Higgs-Kibble spin-zero representa GeV. The question we shall be posing is tions of the group G, and a Higgs potential this; is the unification with the strong likely which gives rise to observed masses and other to manifest itself directly at a relatively low broken symmetry phenomena. With the energy like 10 TeV, or does it manifest itself Higgs multiplets, one introduces two sets of only at the inaccessible energy in excess of new coupling parameters: f's coupling fer 1012 TeV? mions and scalars and X9s describing the Higgs self-couplings. The Higgs sector is not Standard unification ideas as tightly controlled as the basic fermionic To unify weak, electromagnetic and strong, and the gauge sectors. Later we shall see the standard renormalizable gauge model pro that supersymmetry (Bose-Fermi symmetry) is 934 A. SALAM one way of controlling the arbitrariness of this through supersymmetry is exceedingly con sector. stricting. It would have been pleasing (in van e) To include gravity—a spin-2 gauge Nieuwenhuizen's phrase) if nature had "indeed theory—a compact internal symmetry G can been aware of our efforts". It does not not suffice. Einstein's gravity can be for seem to be on present evidence—or at least on mulated as the gauge theory of Weyl's SL(2,C) present evidence as we interpret it now. symmetry. This group structure must then be Perhaps we need to re-examine, ab initio, the included in the basic group together with the charge concept, questions like how many compact G. flavours, how many colours, how many quarks f) To include gravity, a radically new and and leptons, why internal symmetry and why fascinating approach has been favoured. non-Abelian internal symmetry at all? In There are many ways to describe this—and a this respect, I shall finally touch upon some of part of this talk will be concerned with these the deepest ideas reported at this Conference, diverse approaches—but one of the simplest connected with space-time topology and inter is to say that one brings gravity into a unified nal symmetries. Thus there will be four parts gauge scheme not just through SL(2,C)~ to this talk. 0(3, 1), but instead through the larger structure 1) The standard unification models utiliz Sp(4)^0(3,2), which includes and reduces ing "simple" or "semi-simple" gauge groups. to SL(2,C)^0(3, 1) after a Wigner-Innonue 2) Global supersymmetry, unification of contraction. Fermi and Bose objects into one multiplet— g) This structure Sp(4) permits of a rather the general aspects of unification through special inclusion of internal symmetries of the grading; extended supersymmetries; and their type O(JV), and the corresponding Yang-Mills possible embedding in higher dimensional particles, through the procedure of "grading". space-times; use of extended supersymmetries Grading is the adjoining of fermionic anti- to unify gravity with matter. commuting generators to Sp(4) to give what 3) Local supersymmetry ; gauging of super- are known as ortho-symplectic structures symmetry itself, leading to a theory of self- OSp(4|A0 which include Sp(4)xSOQV) "bo- interactions of Einstein's gravitation, and of sonic" generators. "Grading"—the Fermi- these with the spin 3/2 gravitinos as well as Bose supersymmetry—implies that all multi their supersymmetric interactions with other plets of the graded groups contain equal types of supersymmetric matter. numbers of fermionic as well as bosonic com 4) The search for internal symmetries within ponents. For the particular case of OSp(4|8) space-time topological ideas. (with 0(8) internal symmetry), the basic multiplet contains one graviton (spin 2), eight gravitinos (spin 3/2), twenty-eight Yang-Mills PART I (spin 1) particles, fifty-six spin 1/2 fermions and §11. The Standard Models of Strong plus seventy Higgs scalar s. There are two distinct Electro-Weak Unification via Exchanges coupling parameters, the gravitational K^=An of Spin-One Gauge Particles ^Newtonian and the Yang-Mills g. Note that it is the grading, the Fermi-Bose super- As indicated in the introduction, one starts symmetry, which makes the basic fermions with a group G (simple or semi-simple)1
(spin 1/2) come to belong to the same multiplet 1) G includes SU(2) x U(l) X SU(3)C ; as the conventional gauge particles (spin-2 2) Its fermionic representations—and pre graviton, plus the Yang-Mills spin one objects) ferably the fundamental—should describe together with the spin-zero Higgs particles. known and predicted quarks and leptons2; This is a radically new type of unification 3) We shall assume that the local (gauge) where it is not so much the "uni-constant" version of G spontaneously breaks into SU(2) x aspect of gauge unification which is emphasiz U(l) X SU(3)C with essentially one heavy mass ed but the "uni-multiplet" aspect which is scale M; all gauge bosons not contained more to the fore. within SU(2)xU(l)xSU(3)c are ultra-heavy, This ambitious superunified approach their masses being of order M. This is follow- Unification, Superunification and New Theoretical Ideas 935
ed by a second, more familiar breaking of an electro-weakly broken global SU(3)C admitt
SU(2) x U(l) x SU(3)C, with a medium mass ing of integer-charge (vector and axial) gluons scale. and quarks. Such particles do not need Surprisingly, there are not very many absolute but only partial confinement. candidates for the unification3 models. If G To motivate partial confinement, we remark is "simple" the current choice is between that in the parton model language, the effective
4 5 6 G=SU(5) or SO(10) or E6 . For the "semi- masses of quarks and gluons (inside the known simple" case1 (with discrete left-right-colour- hadronic bags) is rather small. There is no flavour symmetries guaranteeing one basic contradiction with medium or heavy physical constant g) the only offer is masses, outside the bags for integer (or even
4 fractional) charges. G=[SU(4)] = SU(4)Z x SU(4)*|flavour The "Archimedes effect"—light quarks and xSU(4) xSU(4)*| x cololir gluons inside—and heavy outside—is well (or more generally perhaps [SU(n)]4). known in other branches of physics. (Elec Except for quark and lepton spectra, the trons in a metal, nucléons in shell-model
"simple" groups SU(5), SO(10) and E6 offer calculations, are examples of situations where fairly similar dynamical predictions, so far as the effective masses are different from physical the high mass unification signals are con liberated masses. The distinction between cerned. these examples and the case of the quarks and a) For "simple" groups the unification vector gluons is only quantitative. The Archi mass M is as a rule of the order of 1012 TeV, medes effect for quarks and gluons appears as are the masses of all gauge particles not to be much stronger.)
contained within SU(2)xU(l)xSU(3)c. In In order to give an estimate of the ratio of this sense these groups predict the end of the the inside to the outside mass, we must have directly experimentally accessible particle phy a theory of partial confinement. (For example, sics within a forseeable future. For the semi- is it vector gluons which are responsible for simple case, the situation is very different. particle confinement or is it tensor gauge Here M and therefore the ultra-heavy gauge particles—see below.) A formula based on particles have masses which can be as low possible partial confinement through vector as 10 TeV. [This is connected with the possi gluons has been suggested by A. de Rujula, bility of chiral colour—see below.] Thus R. C. Giles and R. L. Jaffe (MIT preprint, direct tests of strong force unification with CTP 637, June 1977). These authors surmise the "electroweak" force may be envisaged / that mout—min^C/27ramvt Here mout is the already at ISABELLE (c.o.m. energy 1 TeV) outside mass of a colour multiplet with the and possibly even at PETRA and PEP. Casimir operator C, a is the Regge slope
b) The semi-simple G offers a choice be parameter and mv is the vector gluon mass tween liberated integer-charge quarks and con inside the bag. For mv->0, mout^oo, i.e., fined fractional charge quarks. one recovers exact confinement. For mv~\0
The two tables summarize the salient fea MeV one would obtain mout for quarks^ 10~ tures of these models. Several remarks are 15 GeV. in order. 2) The semi-simple G=[SU(4)]4 with left- 1) In the models based on the simple right flavour colour symmetry must contain groups, SU(5), SO(10) and E6, colour SU(3) colour axial SU(3) gluons, in addition to vector as a good global symmetry appears to preclude SU(3) gluons. Let us assume that these axial integer-charge quarks. With fractional char objects are relatively light (m^< 100 GeV): ges for quarks, SU(3)colour gluons are neutral. so that SUcL(3)xSUci2(3) is the low medium
With exact SU(3)C they would remain massless. mass colour symmetry. It is this larger
Since no massless gluons have been observed, structure (SUfli(3)xSUU(3) vs SU(3)C) which there should be an exact confinement of these provides the essential element for the semi- and possibly of all colour. On the other simple [SU(4)]4 to exhibit a unification mass hand, the semi-simple [SU(4)]4 (or more gener M as low as 10 TeV. ally [SU(#)]4) model permits the alternative of The reason for this is simple; the grand 936 A. SALAM unification mass scale M is essentially con a lepto-quark X mass of the order of 10M05 trolled by a relation like7 GeV, one obtains a lifetime for a 10 GeV quark of the order of 10~13-10"15 sees or shorter. The proton decay in this model is a sixth order (g6) process; proton=qqq->three neutrinos+mesons, the most favoured channel being three neutrinos plus one, two or three where j8 « 11 /3 X 1 /1 6TT2 X Casimir operator of mesons. the low-energy residual symmetry group. The The problem of the semi-simple [SU(4)]4 Casimir operator for a low-energy strong unification model is the converse of the one group like SUcL(3)xSUCjR(3) is twice that for for the simple groups. Here we must guarantee
SU(3)C. This means that for the same left- that the proton lives long enough. (The hand side, the unification mass M required present lower limit estimates of proton lifetime for [SU(4)]4 descending into chiral colour is (1030 years) are essentially for the two-body many orders of magnitude smaller than the mode, protons-*or n°) (F. Reines and unifying mass needed for the simple groups M. F. Crouch, Phys. Rev. Letters 32 (1975)
SU(5), SO(10) or E6, which descend into 493).) It is important to note that this parti vectorial colour SU(3). cular decay mode is forbidden in the basic 3) Together with the unifying mass scale, model using [SU(4)]4. sin2# can be determined by the standard techni It is good to remind ourselves that proton ques of ref. 7. The surviving candidate groups stability is an empirical law, unmotivated by for unification are not sharply distinguished any fundamental theoretical reasons. This by the predicted values of sin2#, except that realization has been hammered into theoretical
E7—a favourite at one time—seems implausible consciousness by the fine work of the group now. led by Professor Reines. Since the value of 4) There is, again, a sharp distinction the proton lifetime parameter is now so deci between simple and semi-simple groups for the sive for distinguishing between simple and semi- predicted proton lifetime. The distinction is simple unifying groups—and between high important, not only in that the semi-simple (1012 TeV) and low mass (10-100 TeV) unifica [SU(4)]4 gives a lifetime near to the present tion, it seems important that this parameter experimental lower limit while the "simple" is measured afresh, at least once more. groups give substantially longer life scales. It In a contribution to the Conference, M. is also important because the mechanism for Yoshimura had argued that the dominance of proton decays are completely different. matter over antimatter in the present Universe For the "simple" groups this decay is a is a consequence of baryon number non-con second-order process in the gauge coupling serving reactions in the very early fireball. His P->g+(#+?)->#+X-* anti-lepton. Since the computations give a small ratio of baryon (so-called lepto-quark) gauge meson X respon to photon number density of the same order sible for this decay must have a mass ~M~ as observed. This work has been extended by 1015 GeV for the "simple" groups, the proton S. Dimpoulous and L. Susskind (SLAC pre lifetime is generally in excess of 1038 years. print, 1978). The (fractionally charged) quarks are them selves of course stable against decays into The semi-simple option integer-charge leptons. A. If the semi-simple [SU(4)]4 (or [SU(rc)]4) For the integer-charge semi-simple [SU(4)]4 is indeed the unifying group, one may list a model, the quarks themselves are unstable number of further signatures for this theory. (the predominant decay mode is q->neutrinos The rather low grand unification energy of +mesons). In contrast to the case of the around 10-100 TeV is the energy at which the "simple" groups, this decay, however, is a quark-lepton distinction would disappear, and consequence of spontaneous symmetry break when the so-called strong interactions acquire ing. (The decay mode quark-»anti-leptons+ their basic O(a) strength. This is the energy mesons is forbidden by a selection rule.) With where leptoquarks (X) of this theory, of masses Unification, Superunification and New Theoretical Ideas 937
around 10-100 TeV, may begin to make their neutrinos over anti-neutrinos in a beam-dump appearance. For example, for ISABELLE experiment would signal nucléon dissociation with 1 TeV in the c.o.m., we may expect a mechanism into quarks and their subsequent substantial contribution to pp-* p+[i~ + • • • decays. There is also an expected asymmetry
through the Drell-Yan pp-^qq+ • • * ->#+(X+ for ve versus /OH >ju+-\-ju~ + • • - with a lepto-quark Before closing this section, two general X-^q+ju of 10 TeV mass. Likewise, in the remarks are in order. converse reaction e+ +e~ -»(#H-X)+e~ ~^q+q 1) Some of us (Abdus Salam and J. Strath- one will begin to feel the effects of X-particles dee, ICTP, Trieste, preprint IC/77/153, to be
at energies ^mx/30. published in Phys. Rev., P. Caldirola and E. B. It can be shown that in an integer- Recami, M. Pavsic, Phys. Letters 66A (1978) charge liberated quark version of this theory 9 and F. W. Hehl, Y. Neeman, J. Nitsch and the neutral axial SU(3) colour gluons (being P. Von der Heyde, submission to this Con companion particles to vector gluons) pos ference) have argued that confinement—partial sess a direct effective interaction with e+e~~ or exact—may have as its origin the interaction and ju+p~ systems. Assuming that their mas of quarks and gluons with spin-2 gauge par ses lie in the PETRA-PEP energy range, one ticles (strong gravitons). These are described would expect asymmetries in the forward- for example by the Einstein equation for a backward e+e~-^ju+ju~ around these axial strong tensor field with the Newtonian
gluon masses. Then asymmetries will be constant GN replaced by a strong constant
2 sharply pronounced, provided the masses of G#m nilcieoii~ 1. If there is, in addition, a cos-
the axial gluons and the vector gluons are mological term with a parameter Xs (replacing
propitiously related (see J. C. Pati & Abdus the conventional cosmological parameter Xg),
40 Salam ICTP preprint, August 1978). The where As/Ag^Gs/GN?zl0 , the spin-2 Einstein- main decay modes of axial gluons are into de-Sitter equation possesses a classical solution vector gluons plus <^(or ai). Thus one may describing a "closed" de-Sitter micro-Universe
1/2 13 expect characteristic signals at PETRA and of radius i?micro-(^^/6)- ^10- cms.
PEP of the type: [The f00 component of the strong tensor equals
J 2 + f =l rG À /6 r +2ju /r. Here ju is the mass e +e~-> axial gluons-^vector gluons+(^)-> 00 s s s s parameter for the source quark. Note the r2 confining term in the potential.] A test quark
C. Regarding integer versus fractional of mass jut in the strong gravity field is charges for quarks, G. Rajasekharan and described by a Klein-Gordon equation (~f)~1/2
P. Roy (Pramana 6 (1976) 303) and J. C. dfi^ff^ du 0)+fA 0=0. This equation Pati and Abdus Salam (Phys. Rev. Letters 36 possesses 0(3, 2) symmetry and gives an (1976) 11) showed that for deep inelastic pro energy spectrum, co=l/R-(2n+1+3/2+ cesses, a colour suppression mechanism is V9/4+ju2). The system exhibits discrete operative, such that the colour component of levels only and no continuum. There is exact quark charges does not shine forth as a rule confinement of the quarks inside a hadronic for deep inelastic eN. However, such sup bag of radius i?^10~13 cms. pression is not expected for Compton scatter One can introduce SU(3) of colour and show ing. The data of D. O. Caldwell et al (Phys. that with an appropriate choice of parameters Rev. Letters 33 (1974) 868) for deep inelastic the resulting octet of spin-2 gauge mesons Compton has recently been analysed by H. K. produce confinement for colour singlet states Lee and J. K. Kim (Phys. Rev. Letters 40 only. From this point of view, a theory8 (1978) 485) who remark that the integer charges unifying the strong and the electro-weak forces, for quarks are favoured over fractional charges, without taking into account strong gravitons in spite of uncertainties of the parton models (spin-2 gauge particles) is incomplete. for such a comparison. The appropriate group which describes eight D. Since in the [SU(4)]4 integer-charge coloured spin-2 gravitons together with nine model, quarks decay into neutrinos+mesons, Yang-Mills spin-one objects is SL(6, C) which
but not into anti-neutrinos, an excess of contains SL(2, C)xSU(3)c. If strong as well 938 A. SALAM
Table I. Simple "Unifying" Groups.
The major distinction between SU(5) and SO(10) lies in the four-component neutrino in the latter and left-right symmetry. None of the groups above permit of chiral colour.
Table II. Semi-simple [SU(4)]4 with discrete left-right-flavour-colour symmetry.
as Einstein gravity must be included together, fundamental entities of which all other particles for example, with the unifying compact group are composed, one would like to introduce (r=SU(5), one should be considering the only the fundamental representation of the structure SL(2, C)xSL(10, C). unifying group. As can be seen from Tables 2) The second general remark concerns the I and II this is already no longer true, for any proliferation of quarks and leptons. This is of the groups proposed. At the very least, the most serious problem which model builders there is a replication of the representations- must face. If one is thinking in terms of which implies that the group structure is not Unification, Superunification and New Theoretical Ideas 939
iii) there must be an infinity of Higgs par ticles as well. The restriction that neutrinos should be mas sive comes from cosmological considerations which limit the number of massless neutrinos to something like three or four, this number depending on the figure for helium abundance. The limitations on mass increase comes from (g—2) limitations for electrons and muons. There are also limitations from asymptotic freedom—e.g., the celebrated limitation of 16(1/2) quark flavours from asymptotic free describing those quantum numbers which dif dom of SU(3) of colour. ferentiate one basic multiplet from another. 3) A final remark; as mentioned before, Assuming that the numbers of flavours and besides the basic gauge coupling parameter, colours, etc. is not infinite—increasing with there are the set of the Higgs-fermion coupling the energy range experimentally explored—the parameters (/) and the Higgs self-coupling only answer to this situation would be to set parameters (X) in the theory. It is important up gauge theories in terms of pre-quarks that one should find theoretical reasons to ("preons") or sub-quarks or sub-stratons9'10— decrease the arbitrariness implied by these basic entities which might carry the individual constants. Based on a demand for asymptotic basic quantum numbers. (Quarks carry two freedom, two suggestions have been made to quantum numbers: colour and flavour. "Pre evaluate these in terms of the gauge parameter ons" would carry only one.) At the present g- time all known quarks and leptons may be a) Abdus Salam and J. Strathdee (IC/78/ considered as made of nine (hypothetical) 44, Phys. Rev. to be published) have used an entities—five carrying flavour and four car improved perturbation theory (where Dyson's rying colour (the fourth colour representing irreducible graphs for gauge Higgs interactions lepton number). This number, nine, of in are modified with line and vertex insertions— dependent quantum numbers, may possibly the effect of these insertions being estimated be reduced, but not very much. It is con by using the running coupling parameter ceivable that for energies above 10-100 TeV g(k2) in the irreducible graphs). They show or so, quarks and leptons can no longer be that the effective ^'s thus computed are finite, considered as point particles and their form (provided the bare /i's vanish) and equal /(= factors must be taken seriously. The "preon" g2lb times a group-theoretic factor. Here b is notion will then come into its own. the parameter which appears in the renor- Concluding the discussion of how many malization group formula quarks and how many leptons there may be, one may remark the following. If the flavour and colour quantum numbers 2 4 really are manifestations of topological struc Since /teSectiYe^g (rather than ^g ) the Higgs ture of space-time, as I shall discuss later, it is masses are thus of the same order of magnitude fully conceivable that the flavour and colour as the gauge masses. Salam and Strathdee quantum numbers increase with the energies also show that the theory is asymptotically considered when we probe deeper into the free. structure of the space-time manifold. Gla- b) In a communication to this Conference, show (Report to the Oxford Conference, July N. P. Chang and J. Perez Mercader have 1978) has studied such a situation and he con studied a class of SO(iV) grand unified theories cludes that in such a situation where the Yukawa set of couplings and Higgs i) neutrinos would in general be massive, self-couplings are fixed by the eigenvalue con
ii) the masses of new quarks mN increases ditions for asymptotic freedom. (The eigen faster than Np, any j?, value conditions require the existence of con- 940 A. SALAM stants /, 1, A, where the running parameters of such a multiplet inevitably (J. C. Pati and Abdus Salam, Phys. Rev. Letters 31 (1973) 661) gave rise to are all expressed as functions of the running lepto-quarks (superheavy gauge particles) and a fun gauge constant g(t); for example f(t)=fg(t), damental unification around the superheavy mass. X(t)=lg\t) and A(t)=Ag\t). Here t=\og k2). These three ideas of Pati and Salam, ie,
The technique used was originally studied by a) gauging SU(3)C for colour and thereby generat N. P. Chang, Phys. Rev. D10 (1974) 2706; ing strong interactions (in addition to SU(2)x U(l) for the electroweak), in the contest of grand M. Suzuki, Nucl. Phys. B83 (1978) 269; E. Ma, unification Phys. Rev. Dll (1975) 322 and E. S. Fradkin b) grouping quarks and leptons in the same multi and O.K. Kalshnikov, J. Phys. A8 (1975) plet of a basic symmetry group 1814. But the major new result is that, c) and the inevitability of lepto-quarks through the though replications of the basic Fermi families gauging of this group, are the cornerstones of all current unified gauge models of strong and are needed, the number is limited and depends electroweak forces. on the particular group SO(iV). Whereas the model of Pati and Salam was based on a
c) Finally, before concluding this section I "semi-simple" non-Abelian SU(4)|FxSU(4)]c group must mention some nice ideas of H. Terazawa, structure, with discrete symmetry F<-»C in contrast K. Akama, Y. Chikashige and T. Matsuki, the next unifying model—that of H. Georgi and S. L. Glashow (Phy. Rev. Letters 32 (1974) 438) which who have considered unified models of the incorporated the features listed above, was "simple." Nambu-Jona-Lasinio type for all elementary Another distinction between these prototype models particles (including gravity).10 They start with was left-right symmetry which was a part of the Pati- a non-linear Heisenberg type Lagrangian and Salam model, but abandoned in the Georgi-Glashow model. SO(10) and E however can admit of left- from this construct an effective Lagrangian 6 right symmetry. For complete references see J. C. which combines SU(2)xU(l) with SU(3) of Pati, submissions to this Conference 971, 972 and 973. ± colour. The photon, W , Z° and physical 2. Ideally it should be the fundamental represen Higgs appear as collective excitations of lepton- tation of the group, since other representations can be antilepton pairs or quark-anti-quark pairs. constructed out of it and particles corresponding to They obtain mass formulae for W and Z these other representations are composites. In prac particles. Extending their work to gravity tice, as we shall see, this is seldom the case for the groups now being considered. This is because we theory they obtain a relation like a=3TT/ appear to have been caught almost unawares by the (HQ2) ln 4ir/tc N Grn2 where N is the number Q Q Q problem of quark and lepton explosion after the b of quark flavours. They predict on this basis quarks and r leptons have been discovered. that there exist a dozen leptons (six neutrinos 3. M. Gell-Mann, P. Ramond and R. Slansky: and six charged leptons) and a dozen flavours Rev. mod. Phys. (to be published) have attempted to limit the choice of permissible groups G by imposing and three colours of quarks. additional requirements. For example, by requiring that colour conserves parity exactly, they are limited to References and Footnotes to Part I groups which are vector-like in colour and permit There is a number of fine reviews of the material chirality in the flavour sector only. By making the covered in this lecture. I shall in general use these standard requirement that colour be an absolutely for purposes of referencing and not refer to individual exact symetry, they motivate fractional charges for papers unless a particular paper needs identification, quarks and thus confinement; by requiring further that or has not been included in the reviews. the group structure relate the lepton and quark charges, 1. I believe one of the first models attempting to they come to limit themselves to SU(5) or the three ex unify stong with electro-weak forces was that of ceptional groups F4 D SU(3) x SU(3)C, E6 D SU(3) X I. Bars, M. B. Halpern and M. Yoshimura (BHY), SU(3) x SU(3)C, E7 =)SU(6) x SU(3)C. Lawrence Lab. Report LBL-990 (1972). Unhappily 4. H. Georgi and S. L. Glashow: Phys. Rev. at the time the model was proposed, the role of gauged Letters 32 (1974) 438; A.J. Buras, J. Ellis, M. K. colour SU(3)C QCD, mediating strong interactions Gaillard and D. V. Nanopoulos: Nucl. Phys. through the operation of a group structure and com 5. H. Fritzsch and P. Minkowski: Ann. Phys. muting with SU(2) x U(l) was not sufficiently recognized. (NY) 93 (1975) 193; Nucl. Phys. B103 (1976) 61; H.
This was achieved in the SU(2) xU(l) xSU(3)c model Georgi: Particles and Fields (APS/DPF Williams of strong and electroweak unification by J. C. Pati and burg), Ed. C. E. Carlson (AIP, New York 1975), p. 575. Abdus Salam (reported in the Proceedings of the 6. F. Gursey, P. Ramond and P. Sikivie: Phys. Batavia Conference, Vol. 2, Sept. (1972) p .304, Review Letters 60B (1976) 177; F. Gûrsey and M. Serdareglu, talk by J. D. Bjorken and Phys. Rev. D8 (1973) 1240). Yale preprint COO-3075-180 (1978); Y. Achiman Pati and Salam also introduced the idea of grouping and B. Stech, Heidelberg preprint, HD-THEP 78-6; leptons and quarks in the same multiplet. A gauging Q. Shaft, University of Freiburg preprint (1978). Unification, Supemnification and New Theoretical Ideas 941
7. H. Georgi, H. R. Quinn and S. Weinberg : Phys. space-time. Its algebra contains new genera Rev. Letters 33 (1974) 451. tors Qa which are on the same footing as the 8. C.J. Isham, Abdus Salam and J. Strathdee: Poincaré generators and J^. In partic Phys. Rev. D3 (1971) 867; ibid., D8 (1973) 2600 and 1.1. Rabi 70th Birthday Volume (American Academy ular, there is the fundamental relation of Sciences, New York 1977), Series II, Vol. 38, p. 77; and references therein. 9. J. C. Pati, Abdus Salam and J. Strathdee: It is natural to attempt to see this symmetry as Phys. Letters 59B (1975) 265-268 and references therein, acting in an enlarged space-time. In fact, by in particular, the work of W. Greenberg. adjoining the anti-commuting Majorana spinor 10. H. Terazawa, Y. Chikashige and K. Akama: co-ordinates da to the Minkowski , one can Phys. Rev. D15 (1977) 480; H. Terazawa, Y. Chikashige, realize the "supertranslations" which are K. Akama and T. Matsuki: ibid., D15 (1977) 1181; generated by Q H. Terazawa: ibid., D16 (1977) 2373. a
PART II A non-trivial feature of this flat super-space- Supersymmetry and Supergravity time on which the extended Poincaré symmetry acts is its non-vanishing torsion (see below). §111. Global Supersymmetry Supersymmetric systems are characterized The unification we have discussed above has by a balancing of fermionic and bosonic no arbitrariness in respect of the gauge multi states. The spin content of the irreducible plet, once the group structure is specified. It unitary representations is is unique. There is also no arbitrariness in respect of the fermionic multiplet, if it is stipulated that we shall only use the fundam ental representations—all other representations referring to composite particles. There is however arbitrariness regarding Higgs scalars. This can be cured if, for example, the Higgs where / is some integer or half-integer. scalars were tied to the fermions. This is The balancing of Fermi and Bose com what supersymmetry does. Supersymmetry ponents in supermultiplets is reflected in the is Fermi-Bose symmetry introduced into parti suppression of quantum fluctuation effects. cle physics by Y. A. Golfand and E. P. This is brought about by cancellations between Lichtman (JETP Letters 13 (1971) 323) and Fermi and Bose contributions. One of the rediscovered by J. Wess and B. Zumino (Nucl. most striking features of supersymmetric Phys. B70 (1974) 39). In its simplest global dynamics is the absence of certain ultraviolet form, it forces the Higgs scalars to belong to divergences: for example, (1) absence of the same representation of the internal sym coupling constant and mass renormalizations, metry group as the fermionic multiplet. In (2) vacuum energy density, (3) the absence— the so-called extended super symmetries ^ the at least at the one-loop level, of quantum next unification is achieved; the Yang-Mills corrections to the classically computed energy as well as the basic fermions, as well as the of a monopole solution in a supersymmetric Higgs, could all belong to the same multiplet version of the Georgi-Glashow model, (4) the of the internal symmetry. In its final form, finiteness of supergravity and extended super- extended local supersymmetry the same multi gravity for one- and two-loop diagrams and plet contains the spin-2 graviton, the Yang- (5) the impossibility of breaking supersym Mills fields, the basic fermions as well as the metry spontaneously through quantum loops, Higgs scalars, and there is a unification of if it has not been broken at the tree diagram gravity with matter. level. It would appear that, owing to can cellations among quantum fluctuations, super- Global aspects symmetric theories are particularly well suited In its most direct form supersymmetry should to a semi-classical treatment. Paradoxically be viewed as an extension of the symmetry of then, balancing of bosons with fermions in a 942 A. SALAM supersymmetric manner appears to bring presentation of the internal symmetry which quantum theories nearer to the classical. commutes with supersymmetry. The infini Examples of renormalizable supersymmetric tesimal supertransformations are field theories (1) First, the self-interacting chiral scalar multiplet comprising the complex: scalars
A+9 F+ and the left-handed spinor a Again an auxiliary field D 3 is needed to close the algebra. The gauge supermultiplet may be coupled to a matter supermultiplet by a where m denotes the common mass and h is generalized version of the minimal principle. a dimensionless coupling. This Lagrangian is For example, invariant (up to a 4-divergence) under the infinitesimal supertransformations where g denotes the Yang-Mills coupling constant. Note once again, that the elimina tion of the auxiliary field D, results in a quartic (Higgs) self-coupling of the field A+ The complex scalar F+ is an auxiliary field: its equation of motion is purely algebraic and with a coefficient which is tied to the gauge it can easily be eliminated from the Lagrangian. coupling g as well as to the Yukawa coupling A result of this elimination is the appearance of constant in the term ig A/2 A\.X^ 6) It has been advocated that for realistic spinor generators Qa-+QaU i=l,2, • -, N. models, supersymmetry may be broken by (This may or may not be associated with an addition of soft non-supersymmetric mass increase in the number of Bose dimensions: terms (e.g., for fermions alone). This proce see below.) The basic anticommutator be dure is not only aesthetically non-pleasing; it comes is also likely to destroy the fine features of cancellations of quantum fluctuations noted above. Application of globally supersymmetric ideas where and Vtj are central charges. (As to weak and electromagnetic phenomena has pointed out by Haag, Lopuszanski and Sohnius been made but is not promising. A super- the algebra will admit a set of 0(N) generators, symmetric version of SU(2)xU(l) theory Its which have dimension of mass. The would require a large number of new and as central charges are in a sense commuting con yet unseen states. First, the leptons acquire tractions of the 0(N) non-Abelian charges.) scalar partners and the gauge vectors acquire The point about extended supersymmetries spinor partners. Unless the symmetry is very is that the spin content of the particles com badly broken the leptonic scalars would have prised in a multiplet possesses a wider range. to be light (~1 MeV) and the gauge spinors For example, for N=4, and for lightlike heavy (~100 GeV) except, of course, for the one of the multiplets comprises one state of photonic spinor which should be light or even helicity ±1, four of helicity ±1/2 and six of massless. Second, in order to generate masses helicity zero. Thus there is the possibility of there must be a Higgs doublet of scalars and a unimultiplet of combined N=4 extended this will acquire spinor partners. However, supersymmetry (and if desired an internal if supersymmetry is to break spontaneously symmetry SU(«)) which contains Yang-Mills as well as SU(2)xU(l) then the Higgs system particles plus the basic fermions plus the will have to be even larger. The supersym Higgs scalars, all in one multiplet. metry will have to break if only to make heavy Renormalizable Lagrangian models have the electron's scalar partner. One might been devised which realize some of these prefer to do this explicitly rather than spon extended symmetries. In particular N=2 taneously although on aesthetic grounds that (complex supersymmetry) and N=4. The 944 A. SALAM latter example also illustrates a deeper mecha by E. Poggio, H. N. Pendleton and by D. R. T. nism at work in that it finds its most natural Jones: another example of the cancellations formulation as a supersymmetric pure Yang- between fluctuations in supersymmetric theo Mills theory in a ten-dimensional space-time ries. Since all other possible infinities in the just as the N=2 extended supersymmetry may model refer to wave-function renormalization be realized in a six-dimensional space-time. and since these presumably can be eliminated For N=4, the ten-dimensional space-time by suitable choices of gauge parameters, this symmetry is broken down explicitly by model (up to the two-loop level) can claim to restricting all momenta to the physical four- be the only known finite field theory of the space so that the extra dimensions contribute conventional type. only to the "internal" quantum numbers. The N=2 supersymmetric model (a super- But there survives a global internal 0(6)^ symmetric version of the Georgi-Glashow SU(4) corresponding to rotations in the extra model) is also very intriguing as has been dimensions. shown by A. D'Adda, R. Horsley and P. Di The supersymmetry generators Qai comprise Vecchia, Phys. Letters 76B (1978) 298; A. a sixteen-component Weyl-Majorana spinor D'Adda and P. Di Vecchia, Phys. Letters with respect to the underlying space-time sym 73B (1978) 162 and E. Witten and D. Olive metry 0(9, 1). They decompose into four (HUTP 78/A013). The two central charges four-component Majorana spinors of 0(3, 1). in this theory can be identified with electric The Yang-Mills Lagrangian is (0 and the magnetic charges (M) defined as appropriate surface integrals. Witten and Olive show that Q and M may be regarded as fifth and sixth spatial components of a light where Fa is made out of the ten vectors A% fic like six-momentum—a refinement of the a and X is a sixteen spinor. (The index a Kaluza-Klein theory. refers to any internal local symmetry, e.g., SU(n) of the conventional type which can be §V. Local Supersymmetry and Supergravity operative indepedently.) The Lagrangian is Local supersymmetry is obviously the next invariant under the transformations extension. Before one considers approaches based on a geometrization of the superspace manifold, one may mention the graded algeb The field content with respect to the unbroken raic approach due to Chamseddine and West, subgroup 0(3, l)xO(6), or, rather, its cover MacDowell and Mansouri and Neemann and ing, SL(2, C)xSU(4)is Regge. Here Weyl's SL(2, C) internal sym metry which on gauging gives rise to the theory of a spin-2 field is replaced by the orthosymplectic OSp(4, 1),which after gauging all of which belong to the adjoint representa gives rise to a field theory of spin-2 and spin tion of whatever additional conventional local 3/2 fields. Just as for the case of Weyl's internal symmetry SU(rc) which is being gauged. SL(2, C), there is no compulsion to introduce Among the Bose components are six space- general covariance in the context of graded time scalars and a four vector. The fermions OSp(4, 1). However, this can be done, and by comprise a left-handed 4 together with a right- a further process of constraint and contraction, handed 4*. a generally covariant supersymmetric theory It is possible to break the global internal of spin-2 and spin-3/2 fields is constructed symmetry from 0(6) to 0(2) (complex super- which agrees with the Stony Brook-CERN symmetry) by introducing a mass term. With supergravity (see below). respect to complex supersymmetry the multi To go back to the alternative superspace plet is yet irreducible. It contains only one approach to local supersymmetry, just as the coupling constant—the constant pertaining to bosonic co-ordinates xm undergo general co the local SU(#). The model is remarkable in ordinate transformations, one expects the that this coupling is two-loop finite (as shown combined superspace co-ordinates (xm, dtl)= Unification, Superunification and New Theoretical Ideas 945 ZM to do likewise. The natural extension of supertranslations in flat superspace is the group of general co-ordinate transformations in eight-dimensional curved superspace. One then tries to set up a graded Einstein-Cartan theory using frames and connections. The basic superfields are the one-form The main problem is to impose sufficient constraints so as to reduce the set of inde A where the 8x8 matrix EM (z) includes the pendent components to a realistic number. a original vierbein em (x) together with many This has been solved by Wess and Zumino by other components and 0MB°(Z) includes the restricting the torsion c spin connection o)mb (x). in addition to me superspace general co ordinate transformations—which are inevitable r T —one must choose an appropriate generali while leaving free the components Tab and Tab . zation of the local Lorentz transformation, Three remarks are in order. First the local or frame rotations, of Cartan and Weyl. Two group is SL(2, C) and, with respect to this suggestions have been made. These are OSp(4, group, these constraints are covariant. Se c 3+1) advocated by Arnowitt and Nath and cond, the constraint Tab =0 is a direct generali SL(2, C) suggested by Wess and Zumino. zation of the one usually imposed in Einstein- There is a third approach due to Brink, Gell- Cartan gravity: it solves to give the spin c Mann, Ramond and Schwarz, which, using connection 0ab in terms of E and dE. Third, e 126 auxiliary fields of Breitenlohner, relies on the non-vanishing (but fixed) values of Tafi the local group being OSp(4, 1). Of these, are what is needed to reproduce global super- the first corresponds to super-Riemannian symmetry as the flat space limit. geometry—the claim is that nothing is lost in Global supersymmetry corresponds to the A replacing the super-vierbein EM by the super- one forms metric where rj denotes the OSp(4, 3 + 1) invariant metric. The second alternative allows more which should emerge as solutions of the independent components in the super-vierbein supergravity equations of motion. and corresponds to a non-Riemannian geo The constraints on T reduce the number metry. of independent superfields among E and 0 to Out of the primary objects E and 0 of Wess just give, va and u (of which the latter may be and Zumino, it is possible to construct cur gauged away completely along with certain vature and torsion two-forms components of the former). The well-defined A G linear transformation rules for EM and 0AB corresponding to general co-ordinate and local Lorentz transformations should be realized which contain all possible covanant combi through a non-linear group action on the fields nations involving one derivative of E or 0. va and u. The Lagrangian for the independent In detail fields is given by the extremely elegant ex pression The corresponding equations of motion are covariant 946 A. SALAM where Ga and R are covariant objects con tained among the surviving components of R D and T C. For example, they can be ABC AB (Here k enters in the vacuum metric arising extracted from T \ since ah in the spontaneous breakdown.) These field equations are then seen to produce the super- gravity equations in the limit &->0. Two ad ditional results have been stated (see talk by P. Nath and NUB 2361, June 1978): First, when the Wess-Zumino or CALT supervier- Since u can be removed by a gauge trans bein E A are combined to form the metric formation, it follows that the equation R=0 M g , this metric is identical to the above is simply a consequence of G =0. The latter MN a gauge complete metric when field equations must contain all the independent dynamical are imposed, i.e., modulo the supergravity information. field equations, the vierbein and metric spaces In their communication to the Conference are identical. Second, Arnowitt and Nath Wess and Zumino show how the constraints have also gauge completed the vierbein to they have imposed, solve to give an expression 0(#2) without auxiliary fields by expanding for E (after gauge fixing) in terms of two the tangent group to include elements of (#-space) derivatives of the field v . Thus the a OSp(4, 3 + 1). These vierbein then reproduce Lagrangian expression det E is expressed their metric without the limit k~*0. directly in terms of the field va. Since later However, Arnowitt and Nath suggest that we shall see that the super-field va contains the limit k-+0 should not be taken as e (the nothing but the field components of Stony vector gauge coupling constant) is propor Brook-CERN formulation of supergravity as tional to k in their theory. Thus k^O allows modified by the groups working at Imperial for the existence of minimal couplings in gauge College, CERN and Lebedev Institute, we supersymmetry without the difficulty found in obtain a direct connection between the super- supergravity of a concomitant cosmological space geometrical approach and the "compo constant. The authors have examined the nent" approach of the next section §VI. k^O theory under the hypothesis that the A somewhat different approach is followed spontaneous breaking preserves global super- by Arnowitt and Nath who choose OSp(4, 3 + symmetry and have obtained the following 1) as the local group. They eliminate the con results (see talk by P. Nath and NUB 2343, nections corrections are included. It is a direct pre ponents: the fields of Stony Brook-CERN diction of the k^O gauge invariance since supergravity. supersymmetry leaves no arbitrariness in the The principal dynamical fields are the vier- Higgs potential. An additional analysis shows bein and the Rarita-Schwinger The product rule for "same" chirality scalars is exactly the same in supergravity as for global supersymmetry—no dependence on the gravitational constant K enters this rule. The product rule for opposite chirality scalars is modified to the extent that ordinary derivatives are replaced by the following "covariant" forms : The auxiliary fields M, N and all vanish when the action of pure supergravity is ex- tremized. In this they resemble the fields D of pure Yang-Mills and also in that they play an essential role in closing the algebra, greatly simplifying the coupling of the gauge system to matter supermultiplets. The anticommu- Finally, the supercovariantization of the F- tator of two local supertransformations el9 e2 type and D-type densities, whereby the pro takes the form blem of giving global F's and D's and how to write supercovariant Lagrangians, is solved by the following expressions : l x where t=2ë2y e1 and ôG9 8L9 3S denote general co-ordinate, local Lorentz and local super- symmetry transformation, respectively. Any globally supersymmetric matter system can be coupled to supergravity with the help of auxiliary fields. This is based on a "tensor calculus" of matter multiplets which includes the following operators : i) multiplication of same chirality scalars ii) multiplication of opposite chirality sca lars 0+X0'L=0", iii) construction of F-type density for 0+9 iv) construction of D-type density for 09 where 0 denotes the scalar multiplet (A B + 9 9 respectively. Once again note that for /c=0 X, F G) and 0 is the vector multiplet (C, C, 9 these expressions reduce to those for global H K V 1 D). The global behaviour of 9 9 a9 9 supersymmetry. With the help of these rules these components is fixed by the superfield one can make any globally supersymmetric expressions system into a locally supersymmetric one. These rules are seen therefore as the graded extension of the well-known minimal principle, whereby flat space systems are "covariantized" by coupling to the gravitational field. Unification, Super unification and New Theoretical Ideas 949 One simple application of these rules would geometry. For example the positive chirality be to make a "cosmological" density by field 0+ satisfies inserting in the F-type density the constant chiral field and it can be represented in the form where U is defined up to a kind of gauge thereby obtaining the term transformation (Note that the field strength W is chiral in Another application would make use of a a self-coupled scalar multiplet to generate spon this sense.) The rule for constructing a taneous breakdown of the local supersym scalar density out of @+ is then obtained as metry. The associated Higgs effect has been follows : shown to eliminate the Goldstone spinor from the physical spectrum while giving mass to the spin 3/2 field. This was shown in the presentation to the Conference in the lecture of J. Scherk. This clarification of the structure of Stony Brook supergravity by the introduction of auxiliary fields is one of the most important achievements in the supersymmetry field in the last year. The theory thus arrived at where E denotes a chiral density which in a provides the irreducible basis upon which all special (chiral) gauge is given by the geometrical formulations based on super- metrics and super-vierbeins, etc. must be founded. In particular the field va of Wess and Zumino must after gauge-fixing comprise The Lagrangian given above for the U(l) the set e/, (p^ Ap9 S and P. gauge superfield can be given also in the The coupling of matter, including systems chiral form with local internal symmetries, has been given an elegant formulation in the non-Riemannian The Lagrangian for a self-interacting scalar supergeometry of Wess and Zumino. First, supermultiplet coupled to supergravity is given the Lagrangian density for a U(l) gauge bv Zumino in the form superfield V is given (in the notation of 2- component spinors) by A where E=det EM and the superfield strength Wa is defined by with the chirality of C where @AB denotes the superconnection. (In a suitable gauge the components of Sfv agree Corresponding to this the equations of super- with those defined by Ferrara and van Nieu- gravity acquire right-hand sides wenhuizen and Stelle and West.) The concept of chiral superfield can be given a meaning in this non-Riemannian where J is a generalized supercurrent 950 A. SALAM expressed in terms of Ha(x, 9) (=va in the notation of Wess and Zumino). The Gôteborg-Caltech approach uses the Breiten- lohner fields Ba(x, 6), Blahl(x, 6) and Ba{x, 6), It satisfies the conservation law among which are included the components A AB A of Ha(x, 6). They express E , @ and T in terms of the Breitenlohner fields, which with themselves are shown to depend on com ponents of Ha(x, 6). Arnowitt and Nath work with metrical quantities g (x, 6). They show It contains the energy momentum tensor, the MN that the Gôteborg-Caltech formulation of spinor supercurrent and an axial vector cur supergravity equations is equivalent to their rent. metric tensor formulation on physical mass- An interesting new contribution to the shell. Their formulation contains one extra understanding of supergravity is due to Ogievet- parameter k; the limit k-^0 is necessary to sky and Sokatchev. They consider the group obtain Stony Brook supergravity (and the flat of volume preserving general co-ordinate trans superspace when the Newtonian constant formations in four complex Bose+two vanishes). complex Fermi dimensions (z'u, 6a). In some Wess and Zumino give a Lagrangian. When sense this corresponds to a subgroup in the their additional covariant constraints on E general co-ordinate transformations in eight and 0 are solved, and the result is substituted real boson+four real Fermi dimensions. The with their Lagrangian, the final Lagrangian idea of Ogievetsky and Sokatchev is to pick coincides with that of the Imperial College- out a submanifold in this space by specifying CERN-Lebedev workers. The approach of four of the Bose co-ordinates in terms of the Gôteborg-Caltech and also of MacDowell remaining Bose and Fermi variables. reported at the Conference relies on formulat ing equations of motion and working with with xp and £P real. They show that by these for the E and 0 fields. The approach imposing suitable co-ordinate conditions, i.e., can be used for extended supergravity also 11 fixing the gauge the components of H can be for the cases N=l9 2, 3 (see below). set into correspondence with the variables of simple supergravity: e/, b**, M and N. §VIL Prospects for a Physical Theory Moreover, a consistent algebraic structure is To construct a physical theory, one may ensured by this construction. (It is interesting take the view that all we need from the spin-2 to note that the group of general co-ordinate gravi ton and the spin 3/2 gravitino (with the transformations from which they start contains Newtonian coupling parameter) is to super- only twenty real Bose and twenty real Fermi covariantize a given globally supersymmetric functions of z^. This is more restricted than Lagrangian. In this approach the graviton the sixty-four real Bose+sixty-four real Fermi and the gravitino stand apart from the matter functions of which enter the general co (including Yang-Mills gauge) supermultiplets. ordinate transformations of real superspace.) One may be more ambitious and consider In spite of the brilliance of this idea, as extended supergravity theories in which the yet the geometry of this manifold is obscure graviton is accompanied by a number N<& and we need rules for constructing covariant of spin-3/2 gravitinos and the system admits objects, Lagrangian densities, etc. a global internal 0(N) symmetry. Also in To summarize, the Ogievetsky and Sokatchev cluded in the graviton supermultiplet are superfield Ha(x, 0) is the object nearest to the states with successively lower helicities which I.C.-CERN-Lebedev set of fields, in terms of transform as antisymmetric tensors of 0(N). which the Stony Brook supergravity is Physical states—assumed massless—are con recovered. Wess and Zumino show how veniently tabulated according to helicity À. their constraints can lead the supergeometric For a set of values of N their multiplicities quantities used by them (EA, 0AB, TA) to be are as follows: Unification, Superunification and New Theoretical Ideas 951 Explicit expressions for Lagrangians for N< near to physical reality theories like ex 4 have been given (see D. Z. Freedman's tended 0(N) supergravities are likely to be. report, this Conference) but only partial re Starting with one (Einstein) graviton, we are sults are available for the larger values of perhaps dealing for the 0(8) case with 8 spin N. (The global symmetry in these cases can 3/2 gravitinos, which for SU(3) of colour be extended from 0(N) to U(iV), by separat (embedded in 0(8)) split into (3+3* +1 + 1), ing the spinors into chiral pieces. For ex 28 spin-1 particles (a colour octet of gluons+ ample, in the case JV=3, the three Majorana- two electrically neutral singlets (photon+Z°) Rarita-Schwinger spinors of spin 3/2 decom and 9+9 fractionally charged (±1/3, ±1/3, pose into a left-handed 3 and a right-handed +2/3) perhaps superheavy bosons; plus 56 3*. There are three vectors, A% out of which spin-1/2 objects, comprising 3+3 + 3+3 quarks three field strengths can be made. The of charges 2/3, —1/3, —1/3, 2/3, a neutral self-dual and anti-self-dual pieces of these octet (of gluinos) a singlet of charge—1 (the transform as 3 and 3* under SU(3).) electron), a colour six-fold of charge —1/3 For a general JV<8—this restriction being and two Majorana neutrinos. There is no needed in order that no helicities higher than place for any W± (since 0(8) cannot accom 2 occur—there are N(N—l)/2 spin-1 vector modate SU(3)C as well as SU(2) X U(l)) nor for fields ALjn in the same multiplet with the the muon, nor for the r lepton. There are graviton field, but they do not gauge 0(N). two coupling constants, the Yang-Mills g~e, Rather, they gauge (non-minimally) a set of and the Newtonian K, together with the inevi central charges ZLin with respect to which the table cosmological term. (The cosmological graviton is neutral. Hence there are only constant 3e2/*c2 is 10118 times larger than the magnetic moment type couplings of A? to this astrophysical limit—in Freedman's words multiplet. It is possible, however, to introduce "the record for the worst comparison of matter multiplets with non-vanishing central theory and experiment"). There is hope from charges. To these the Ap couple in the usual some of S. Hawking's recent work that this minimal fashion with strength g•=tern, where large cosmological constant may turn into a ra=mass of matter multiplet. virtue of the theory; such a constant ap It is possible to turn the global 0(N) into parently implies that space-time is made up of a local symmetry with the ALjn as gauge fields extremely tiny granules of de-Sitter black- by adjoining various terms to the Lagrangian holes. including a cosmological term and a spin-3/2 An elegant way of writing the 0(8) extended mass term. The 0(N) gauge coupling is given supergravity has been suggested by E. Cremmer by g~tc\/À, where À denotes the cosmological and J. Scherk, who show that if one extends constant. These models can perhaps best be space-time to eleven dimensions, a dimensional approached by a process of gauging the graded compactification from eleven to four dimen de-Sitter group OSp(4, N) and then Wigner- sions gives a theory with eight spins or charges Innonue contracting it. There are thus two (gravitinos). independent couplings in extended supergravi- Another avenue for realizing a physical ty Yang-Mills g and Newtonian /c, or fc and theory may lie along conformai supergravity. the cosmological constant A. In this report I have not mentioned conformai Finally, we must raise the question of how supersymmetry, or conformai supergravity. 952 A. SALAM If the Poincaré group is extended to the 15- 1/p4) [through the Lagrangian (WR^-lfiR2)] component conformai group, the grading together with the spin 3/2 gravitinos (two of needs eight rather than four fermionic gene positive and one with negative metric) plus a rators. To close the algebra, one also needs spin-1 axial particle. There is no cosmological one further pseudoscalar bosonic generator term. (connected with chiral transformations), mak On account of the soft propagator (1/p4 for ing for a total set of 24 generators constituting gravitons), theories of this variety—even when a graded SU(2, 2; 1), instead of the 14 conformai symmetry is broken (spontaneously generators in the Poincaré case with its con or otherwise) by the addition of the Poincare tracted de-Sitter ancestry through OSp(4, 1). supergravity Lagrangian—are known to be In addition to the simple conformai super- renormalizable. One can set up a renormali- symmetry, extended conformai supersymmet- zation-group scheme—this has been done by ries can also be constructed corresponding Salam and Strathdee (IC/78/12, to be publi to a graded SU(2, 2, N). These structures shed in Phys. Rev.) and independently by J. have the merit of incorporating an internal Juive and M. Tonin (Padua preprint, IFPD SU(A0 instead of the internal 0(N) as is the 2/78). These authors claim to show that case for the graded OSp(4, N). In this sense the massive unphysical spin-2 ghost in theories (of incorporating SXJ(N)) this theory is an with Lagrangians (l//c2)iî+(l/g2)iî^iî^+(l/ advance over the Poincaré case. g'2)R2, is innocuous so far as unitarity dif So much for the global conformai super- ficulties are concerned, provided the constants symmetry theory. The local version of this g and g' have values for which the gravity theory (conformai supergravity) has been propagator shows an anomalous dimension. constructed—the Lagrangian having been They give criteria for this to happen. written down by the groups working at Stony Such explorations are important, even with Brook, City College New York, and CERN. out supersymmetry, particularly in view of the The Lagrangian describes the propagation of feeling that supergravities are not likely, in spin-2 dipole objects (quartic propagators three or higher loop diagrams, to prove finite Unification, Superunification and New Theoretical Ideas 953 on shell, as had once been fondly hoped for matter. from the example of one or two loops (for details of these calculations, see the report References to Part II to the Conference by D. Z. Freedman). Even Most of this section of the report was written in if supergravities had proved finite, the fact that collaboration with Dr. J. Strathdee. the high-energy behaviour increases like KEtcA For supersymmetry a number of reviews exist : P. Fayet and S. Ferrara: Phys. Reports 32 (1977) {KtcY'1 for an #-loop graph with E external C5. gravitons, implies that unless a summation Abdus Salam and J. Strathdee: Fortschr. Phys. 26 technique is devised, there is no hope of (1978) 57. gravity theory exhibiting Froissart boundedness. For various aspects of supergravity see the excellent In this connection Salam and Strathdee have reports to this Conference by D. Z. Freedman (Stony Brook), J. Schwartz (Gôteborg-CALTECH), surmised that if gravity theory is asymptoti B. Zumino (CERN and Karlsruhe) and P.Nath (Northe cally free—as suggested by Fradkin and astern). The groups not represented directly at the Vilkovisky—the running parameter K(tc) may Conference are the Imperial College group, K. S. fall as l/fc. This implies, not only that the Stelle and P. C. West, Imperial College, London, pre high-energy behaviour is Froissart (^tce~E) print ICTP/77-78/6, S. Ferrara, P. van Nieuwenhuizen, CERN preprint TH. 2463, E. S. Fradkin and M. A. but also a possible renormalization technique Vasiliev, IAS-788-3PP (1978) and V. Ogievetsky and for loop calculations, when one starts with L. Sokatchev, Dubna preprint E-2-11702 (1978). Dyson's irreducible graphs in a first-order Palatini formulation of Einstein's gravity and PART III uses the running constant K(tc)~l/fc in place of A:, to obtain an estimate of infinity behaviour §VIII. Topological Ideas and the Origin of the of the insertions in lines and vertices. Internal Symmetries To summarize the prospects of supersym The central problem—made the more seri metric and supergravity theories, my personal ous by the quark and lepton explosion situa feeling is this. Spontaneously broken global tion—is the problem of understanding the supersymmetry may prove to be an important charge concept in a fundamental manner. To principle for constructing physical Lagrangians. make "fundamental understanding" more This will find confirmation or otherwise if the precise one may use the analogy of the gravita ± W and Z°'s are accompanied by correspond tional charge (mass). Einstein succeeded in ingly heavy leptons. If confirmed, one will linking it with the curvature of space-time. need simple supercovariantization of such a Can we hope that a similar fundamental under unified globally supersymmetric theory and a standing will be achieved for the other possible Higgs effect, whereby the Goldstinos charges? Perhaps this way we shall under give mass to the gravitinos. The prospects stand the secret of how many charges (flavours of renormalizability for such a theory are and colours) there are, of the symmetries they unlikely to depend on its supersymmetric are associated with, and why these sym character. These will probably be settled by metries appear to give rise to non-Abelian incorporating techniques like the use of runn group structures. ing coupling constants or the asymptotic There is the conjecture made already in the safety ideas of Weinberg. 1950's by Wheeler, Shônberg and others that It would be good to have a complete super- it is the topological structure of space-time— space formulation for such supercovarianti both in the large and the small—which gives zation; superspace is an important extension rise to the charge concept. If this conjecture of space-time, one where no problems of direct is correct, one may ask, what is the origin of measurability arise for the extra fermionic the Lie group symmetries (even of the humblest dimensions of space-time. Regarding extend SU(2) X U(l)) within the context of topological ed supergravities, my own feeling is that they ideas? will prove their relevance for strong super- Topological ideas have recently played a gravity, perhaps in a spontaneously broken crucial and a central role in Yang-Mills theory, extended superconformai form, before one with the discovery of the 't Hooft-Polyakov uses them for unifying Einstein's gravity with monopole and the instanton-meron solutions. 954 A. SALAM In fact a conjecture has been made by D. Olive projective two-space CP2 can be given a posi based on the celebrated equivalence of Sine- tive definite Riemannian metric for which Gordon theory in 1 + 1 dimensions with the The index theorem would Thirring model. What looks like as a topolo gical charge in one formalism, may appear as then seem to imply that nR—nL=~l/8, which an internal symmetry Noether charge in the is absurd. The reason that one appears to other. Could this be the prototype for an get a non-integer contribution to the index is understanding of all Noether carges? that CP2 does not have a spin structure; one My own feeling is that this is unlikely to cannot consistently define spinors on CP2. be the case. Normal instanton or soliton How does one correct this situation ? Hawk physics is done on S4 or S3. This is obtained ing and Pope in a beautiful paper in Phys. as the one point compactification of Euclidean Letters have answered this question. They space. Technically, for example, the different point out that there exists a covariant con instanton sectors for a gauge group G cor stant two form which can be taken as respond to different principal fibre bundles associated with the U(l) symmetry and with with fibre G over S4. Now the classification the inclusion of this F^, the Atiyah-Singer of bundles over spheres is rather easy. In theorem reads : fact the set of all G bundles over Sn is in (1, 1) correspondence with and in particular 4 on S the bundles are classified by 7r3(G). How ever, for every simple, connected Lie group The contribution of the first term depending and hence bundles are always labelled by an on F^ can be computed. This contribution integer (the "winding" number) which can be is precisely ra(ra+l)+l/8. Peace is restored related to the Yang-Mills fields (connections) if both terms are added; n ~n is indeed an by the well-known Pontryagin-Chern formula L B integer. Now what is the moral of this. The moral is that a compact space-time structure CP2, We started from a gauge group G and have would be unacceptable physically unless it progressed from G to classes of integers (Z) was supplemented with an "internal" U(l) for compactified space-time manifolds M. symmetry. The U(l) is motivated by space- Can the rather humble Z's lead one back to time topology. From this observation, Hawk the non-Abelian group G itself? ing and Pope argue that there may be a con In this respect an important suggestion has nection between the topology of space-time recently been made by Hawking and Pope and the spectrum of elementary particles. and Back and Freund. Let us go back to This theme has been taken further by Allen the characteristic classes—and in particular to Back and P. G. O. Freund in a contribution the second Pontryagin and Euler classes of a to the Conference. They rebel against the general space-time with no gauge field except starting point of Hawking and Pope. Why 2 the gravitational. These classes are represent the compact CP from which Hawking and ed by the curvature tensor in a well-known Pope start? Freund himself was one of the 2 manner, with the celebrated Atiyah-Singer first to suggest consideration of CP in the index theorem stating that for such a situa context of gravitational anomalies. However, tion : there has to be some dynamical reason for this. Back and Freund instead start with general (pseudo) Riemannian space-times and show that all matter fields must appear in suitable where nR and nL are the numbers of zero mass multiplets of a gauged symmetry in order to spin states of right and left helicities. permit the definition of a generalized spin Consider now a compact space without structure. (They argue that one cannot a boundary, for example CP2. The complex priori restrict oneself to manifolds that have a Unification, Superunification and New Theoretical Ideas 955 spin structure; this would be an unacceptable of W2. Hence the Stieffel-Whitney class cor restriction for example in a standard func responds to some new type of quantum number tional integral formalism for quantum gravi whose physical significance has not yet been ty.) The simplest choice, they discover, for elucidated. (For details see S. Avis and C. J. the gauge structure for a general Riemann Isham, Cargese lectures 1978, IC preprint manifold is SU(2)—the first indication of a ICTP/77-78/23) Another example is the mob- non-Abelian structure from space-time topolo iosity or twistedness quantum number (C. J. gy. With a gauged internal SU(2) symmetry Isham, Proc. Roy. Soc, in press) which can be one can define spinors of any four-dimensional assigned to scalar or spinor fields which cor Riemann manifold. Physically one may at respond to cross sections of G-bundles when tempt to identify this SU(2) with the universal G=Z2. (The number of inequivalent Z2 weak isospin SU(2) factor of the unified elec bundles is equal to the number of elements troweak group. The corresponding weak in the group H\^t, Z2).) isospin statistics connection they discover J. Kiskis has tried to relate these ideas to requires bosons (fermions) to possess integer the basic concept of how to define the signs of (1/2 odd integer) weak isospin. This un charges (including that of the electric charge) fortunately will not accommodate the simplest when a gauge group is enlarged to include Higgs structure currently utilized in SU(2)x certain discrete symmetries, such as charge U(l). However, one might generalize weak conjugation in a space-time which is not isospin to SUL(2) X SU^) with two quantum simply connected. He concludes that in a numbers (kL, kR), such that fermions carry space-time with a handle, for example, such a half odd values of kL-±kR and bosons carry definition is not possible and there exists a integer kL+kR. mechanism for global violation of charge con This is powerful stuff—space-time Rieman- servation. Kiskis makes the remark that we nian topology dictating (at least partially) know nothing about global topology of real internal gauge symmetries, plus an internal space-times on a cosmological scale, nor do charge-statistics relation restricting the types of we know anything about space-time topology multiplets one may contemplate. It is sug at distances shorter than 10~16cms. How gestive; it is deep, could this be the whole ever, the problem about exotic topologies will story? always be, what is the dynamical mechanism Though not directly relevant to the question which produces "handles" for example in of where the internal symmetries come from, space-time? there are still other directions into which the There is yet one more approach to the "ex delicate interplay of space-time and group planation" of internal symmetries. I have topologies can lead us into. For example, always been very impressed by a remark which remark that the classification of G bundles in ' Res Jost made in his lecture on New Theoretical general space-times is different for different Ideas at the Sienna Confernece in 1963. Jost Lie groups. While for a four-dimensional said; given a number of choices, he believed manifold ^£ and for any simply connected nature always made the brashest, the least groups G (e.g., SU(«), Sp(w), G2, etc.) subtle and from one point of view the "least {G bundles}^H4(^, Z)=Z if is com imaginative" choice. To solve the r, Q puzzle of 1956, nature did not resort to a use of pact and oriented parity-conserving density matrices between ^0 if ^ is non-compact, accidentally nearly degenerate states. It chose the situation is richer for U(#)'s or SO(«)'s. instead, the brash, bold, expedient of parity For example, for SO(n), with a non-compact violation. Likewise in our times for neutral ^ {SO(a) bundles} ^H\^,Z2) so that the currents, in the low-energy regime, nature classifying element is the second Stieffel— has chosen the simplest structure it could, Whitney class (W2). It is noteworthy that the SU(2)xU(l). although the Pontryagin class of an SO(n) The least imaginative—though perhaps the bundle can be expressed in terms of the brashest—resolution to the problem of internal Yang-Mills fields this is apparently not true symmetries is to assume that what we are 956 A. SALAM witnessing is the structure of extra space-time dimensions. We really do live in an extended §IX. Spontaneous Compactification space-time. If space-time does possess extra The idea seems to be due originally to dimensions (perhaps all spacelike) and if the Cremmer and Scherk (Nucl. Phys. B108 (1976) curvature pertaining to these happens to be 409) who were looking for an interpretation of so large that these fold upon themselves, with the extra dimensions in dual models. As a dimensions much smaller than 10"16 cms (and bonus this mechanism may explain the super perhaps with a hierarchy even approaching heavy scale in grand unified schemes. Since Planck length 10"33 cms), we would not the dual models are supposed to be renor- apprehend them, except indirectly. The so- malizable, one is able to incorporate gravity called internal symmetry quantum numbers as well as all other forces in a renormalizable are then the only (indirect) window we possess framework. For the compactification dis for apprehending these extra dimensions. cussion, however, the dual model is replaced by The idea is not new: it started with the a "low-energy approximation" in the form of Kaluza-Klein introduction of the 5tlx dimen a classical Lagrangian containing local fields sion and its connection with electric charge. and satisfying general covariance, gauge in It was pursued in the 1950's by Abraham variance, etc. Pais and in the 1960's by T. Takabayashi (see The Lagrangian contains a 4+N dimensional Proceedings of the 8th Nobel Symposium, metric tensor g and a set of gauge vectors Ed. N. Svartholm, 1978 (Almqvist and Wick- fic for an internal symmetry group, which to sell), p. 157) following an earlier work of start with at least has a non-geometrical origin. H. Yukawa, Phys. Rev. 91 (1953) 415, 416. These fields couple with a strength e. The What is however new is a recent attack on Lagrangian may or may not have scalars. It this problem by Cremmer, Scherk, Schwarz is conjectured that the ground state solution and others (reviewed in this Conference by Scherk), taking their cue from dual models may have non-trivial topology—in both metric and group spaces, i.e., (g^y^tyj^ and (Aa )^0. in ten space-time dimensions (Nucl. Phys. fi B108 (1976) 409, where the "compactification" (Cremmer and Scherk in treating the N=2 of the extra dimensions is seen as a spontaneous case assumed a monopole form for Ap. on 2 symmetry-breaking phenomena, through which the sphere S .) the highest unification mass (Planck mass) A non-vanishing Chern class for (Af) im enters particle physics). Although topological plies obviously that some components of a ideas were already introduced by Cremmer (A fiy are non-zero and so must contribute to and Scherk in order to motivate stability of Tfc on the right-hand side of Einstein's equa structures in extra-dimensions, a new twist tions. Hence one expects (g^} not to be has been given to these ideas by Horvath and flat. The non-trivial topology is supposed to Palla who seek the masslessness of the neutrino stabilize this situation. through a use of the Atiyah-Singer theorem as The simplest non-trivial topology that can applied to extended space-time. Thus the be imagined in [4+N] is iî4 X SN and the most notions of curvature and topology of extra symmetrical (and therefore lowest) solution space-time dimensions motivate the charge with this topology would be the product of concept, the non-Abelian symmetries, the Minkowski 4-space with an iV-sphere. The highest unification energy and the relatively group of motions would then be Poincaréx low masses of physical objects we are dealing SO(Ar+1). Corresponding to this, the 4-space directly with, including the masslessness of the components of the gauge vector must vanish, neutrinos. The major unsolved problem then but the remaining N components need not— is: how many extra dimensions and why that at least if the gauge group contains SO(JV+1). many? If the N/2 Chern class is non-vanishing then In the next section we examine these ideas of the gauge vector cannot vanish and the situa spontaneous compactification in some more tion is self-sustaining. (According to Horvath detail. and Palla, ICTP, Trieste, preprint IC/78/37, submitted to this Conference, the gauge group Unification, Superunification and New Theoretical Ideas 957 G need not be so large as SO(iV+l). The groups. main need is that the groups allow gauge field After the spontaneous compactification, all configurations on SN with non-vanishing N/ fields can be expanded in series of hyper- 2'th Chern class and for this TT^_I(G) should be spherical functions defined on the internal nontriviaL). To close the cycle, it has been space with coefficients dependent on the first shown by G. F. Luciani that the S0(A4-1) four co-ordinates. In effect the effective ac symmetry is itself local and may possibly be tion has the form of an infinite component identified with the non-geometrical gauge field theory, but the masses of the components group we started with. are different and depend both on the indices The main check on the consistency of this of the hyperspherical functions and the re idea is a computation of the excitation presentation the components belong to. All spectrum; i.e., one must expand the back the non-zero masses turn out to be of the ground values showing that the first-order order of Planck's mass. Thus at low energies terms cancel while the second-order terms are only the zero mass components have physical positive definite. relevance in the sense that either they remain At the strictly classical level one could exactly massless or they obtain their non-zero perhaps verify the stability against small physical masses by some other mechanism. perturbations by computing the mass spectrum It is in the question of zero mass fermions but one does not know if the possibility of that topologically non-trivial compactification tunnelling can be excluded. of gauge fields plays an important role. This In the papers of Cremmer and Scherk, Higgs' is with the use of the Atiyah-Singer index fields are included and a monopole solution of theorem which relates the zero-energy eigen- the 't Hooft type is conjectured in the extra space of the Dirac operator to the topological dimensions. In their paper with Horvath invariants characterizing the global gauges, and Palla (Nucl. Phys. B127 (1977) 57) the among which are the gauges referring to the Higgs is absent and a Wu-Yang type monopole compactified space. Thus the number of zero is assumed. mass objects (neutrinos) may be connected with For N=2 the ansatz takes the form (with the topology of the internal space. G=SO(3)): where i?0 and p are constants. They are obtained rather trivially since the field equa tions reduce to an algebraic form. There is also a cosmological term since the Higgs potential is taken in the form By a careful (and therefore possibly an unnatural) adjustment of K0, there is the possibility of the model giving a possible ex planation of the large unification masses of the order of Planck mass for the "simple" unified P10: ICFA Chairman: Y. YAMAGUCHI Speaker: E. L. GOLDWASSER Scientific Secretaries: Y. KIMURA Y. ASANO (Wednesday, August 30, 1978; 14; 00-14: 30)