FR3looo?2. CEA-N-2175 Note CEA-N-2175 Centre d'Etudes Nucléaires de Saclay Division de la Physique Service de Physique Théorique INTRODUCTION TO UNIFIED THEORIES OF WEAK, ELECTROMAGNETIC AND STRONG INTERACTIONS - SU(5) - par Alain BILLOIRE, André MOREL -Novembre 1980- Note CEA':-2175 DESCRIPTION-MATIERE (mois clifs extraits du thesaurus SI DON MIS) an français an anglais MODELES DE JAUGE UNIFIES UNIFIED GAUGE MODELS INTERACTIONS FORTES STRONG INTERACTIONS INTERACTIONS FAIBLES WEAK INTERACTIONS INTERACTIONS ELECTROMAGNETIQUES ELECTROMAGNETIC INTERACTIONS GROUPES SU-5 SU-5 GROUPS COURS LECTURES H Note CEA-N-2175- Centre d'Etudes Nucléaires de Saclay Division de la Physique Service d? Physique Théorique INTRODUCTION TO UNIFIED THEORIES OF WEAK. ELECTROMAGNETIt AND STRONG INTERACTIONS - SU(5) - par Alain BILLOIRE. André MOREL ri CEA-N-Z175 - Alain BILLOIRE, André MOREL INTRODUCTION AUX THEORIES UNIFIEES DES INTERACTIONS FAIBLE, ELECTROMAGNETIQUE ET FORTE - SU(5) So—aire.- Ces notes constituent la version écrite d'une série de cours donnés au DPh-T a Saclay pendant l'hiver 1ï'9-19S0. L'intention est de fournir une introduction aux théories dites unifiées des interactions faible, électromagnétique et forte. Il ne s'agit en aucune façon d'une revue des travaux de recherche récents ou en cours ; nous avons plutôt cherché a présenter in» exposé aussi élémentaire que possible et renvoyant au minimum 5 d'autres ouvrages. En particulier, le* connaissances en théorie des champs e' en théorie des groupes nécessaires a la compréhension du texte ont été ramenées au minimum et nous nous sommes efforcés d'en rappeler les principaux éléments au moment d'en faire usage. La liste de références, loin d'être exhaustive, devrait être suffisante pour guider le lecteur souhaitant approfondir les bases de la théorie r.u trouver des détails concernant les progrès récents. Dan? une premier, partie, nous rap­ pelons le modèle SU(2) • U(1), en insistant sur les pro:.limes qui, laissés ouverts dans ce cadre, trouvent une solution dans celui des théories unifiées. Dans la deuxième partie, nous expliquons de façon systématique comment on peut construire des théories unifiées, et 1 CEA-N-2175 - Alain BILLOIRE, André MCREL INTRODUCTION TO UNIFIED THEORIES OF WEAK, ELECTROMAGNETIC AND STRONG INTERACTIONS - SU(5) Summary.- These notes correspond to a series of lectures given at Saclay during winter 1979-1980. They are meant to be an introduction to the so-called grand unified theories of weak, electromagnetic and strong interactions. By no means do they constitute s review of the current research work on the subject. We tried to be as elementary and self-contained as possible. In particular, tho required knowled­ ge in field and Lie group theory has been kept at the minimal level, and we made an effort to recall the basic theoretical tocls when they were needed. Our list of references, far from exhaustive, should be sufficient to guide the reader who is interested cither in lear­ ning more about fundamental topics or in finding d.-t.uls about recent progress reported in the literature. In a first pa^t, we recall in a very elementary way the standard SU(2) S U(11 mod.l of electroweak interactions, putting the emphasis on the questions «hich. arc left open by this model and which unified theoriss help to answer. In part II, we explain in a systematic way how unified theoi;rs can ht constructed, and develop the SU(5) model in great detail. Oth^r arc not models, like SO(10) and F0, presented, net bfnuse of any développons le modèle SU(5) en détails. D'autres modèles, comme S0(10) ou E0, ne sont pas présentés, non par préjugé mais parce que SU(5) est le plus simple et le plus étudié. 11 se trouve d'ailleurs que la plupart des concepts et résultat; généraux ne sont pas spéci­ fiques à un groupe de symétrie particulier. La préparation de ces notes doit beaucoup à d'utiles discussions avec nos collègues du DPh-T de Saclay ; nous remercions en particulier F. Hayot et R. Lacaze pour leur lecture critique d'une grande part du manuscrit. Nous sommes très reconnaissants envers D. Bunel et F, Lefèvre pour leur très grande efficacité et le soin apporté à la présentation de ces notes. 1980 130 p. Commissariat à l'Energie Atomique - France prejudice, but essentially because SU(5) is the simplest one and has been subject to the deepest investigations up to now. Also it appears that most concepts and general results are not specific to any parti­ cular symmetry group. We wish to acknowledge many helpful discussions with our Colleagues in Saclay during the preparation of these notes, and especially thank F. Hayot and R, Lacaze for their critical reading of a substantial part of the manuscript. Finally, we are grateful to D. Bunel and F. Lcfevre for their very efficient typing. 1980 130 p. Commissariat 3 l'Energie Atomique - France TABLE OF CONTENTS Part I 1. INTRODUCTION 2. THE ALGEBRAIC STRUCTURE OF THE SU(2) 9 U(l) MODEL 3. SPONTANEOUS SYMMETRY BREAKING AND THE HIGGS MECHANISM FOR SU(2) 8U(1) 4. ANOMALIES Part II 5. IN SEARCH OF THE UNIFICATION GROUP 6. DESCRIPTION OF THE SU(5) MOOEL 7. RELATIONS BETWEEN THE SU(3), SU(2), U(l) COUPLING CONSTANTS 3. HIGGS BOSONS, SU(5) BREAKING, MASSES a) .i.rsc stage breaking : SU(5) » SU(3) B SU(2) 9 U(l) b) Second stage breaking : SU(3) 9 SU(2) 9 U(l) - SU(3) 9 'J (1) e.n. c) Generation mixing d) 3-L invariance 9. RENORMALIZATION OF THE SU(5) PREDICTIONS a) Coupling constant renoraalization b) Fermion mass renormalization 10. THE PROTON LIFETWE APPENDIX A : YANG MILLS AND GROUP THEG.IY APPENDIX B : YOUNG TABLEAUX FOR SU(N) REFERENCES - 3 - 1. INTRODUCTION Our understanding of elementary interactions is mainly based on the observation chat they obey a number of symmetry laws. Some of these laws seem Co be exact. Such are Poincari (Lorentz + translational) or CPT inva­ riances. Also, in the field theoretical formulation of the interactions, exact local gauge symmetries are realized, such as che U (I) symmetry c.a. which generates electromagnetic phenomena (QED), or as far as we know, the SU(3) colour symmetry, responsible for strong interactions (QCD). Such symmetries, when exact, imply the existence of aassless, spin 1, gauge bosons. Particles can then be classified according to representations of the corres­ ponding (Lie) groups, their quantum numbers being the eigenvalues of the com­ muting generators. For example, che OED Lagrangian is invariant under the local transformations ? •» 0 '? , e.a. U (x) - exp {-iea(x)Q} , (I.I) e.a. of che feraion fields. The charge operator Q is the generator of the transfor­ mation, a(x) an arbitrary real function of space time. Local invariance implies the existence of a aassless gauge boson, the photon. Clearly only the produce eO is physically relevant. If Q is normalised so that its eigenvalue in the electron state 0 is -1, then e is the usual electron charge (e*/(4Ti) «a is che fine structure constant). But at this level, nothing guarantees chat another particle has, in units of e, a charge equal to a fraction or to a multiple of Q , as we chink it is the case. It is so because U(I), being abeiian, contains only I-dimensional irreducible representations, so that different particles, belonging co different irreducible represencations of U(I), have unrelated quantum numbers (here charges). Kere is che first question which any unification aims to answer : Question I : Why axt JUtt. chaxqte comneruuAate 1 In order to go ahead, we need to classify particles in representations of some larger group g, in such a way that non-trivial commutation relations between che generators imply relations between the O.N. of che particles(even if chese particles belong Co different represencations). SU(3) is another, apparently exact, symmetry of nature. The elementary fermions appear either as members of che fundamental representation (quarks q., i"l,2,3) or as singlets of SU(3) (leptons, e,.>e ,u,u v etc.). Therefore, Ue.a . (1) cannot 'e identified co a U(l) subgroup of SU(3)^. Rather, we have as a symmetry group Che direct produce - 4 - gQ - SU(3)c« U<B (1) . (1.2) From this examole, we see chat the needed group g should not contain U (1) as a factor, if we are to explain that charges are commensurate, but ocmf cours. e it should contain g as a (non invariant) subgroup in order to accomodate in particular the colour and charge quantum numbers of the known elementary fields. Weak interactions (W.I.) are also considered as deriving from a gauge sym­ metry group, namely SU(2). But this symmetry cannot be an exact symmetry : at present energies, W.I. appear as short range interactions, (Fermi type interac­ tions), which cannot be described by zero mass vector boson exchange. Both their smallness and their short range nature may be accounted tor at the same cime if the gauge fields associated with the SU(2) symmetry have masses M large as com­ pared to the relevant energy scale (the proton mass). Their exchange in a 2 fermion •* 2 fermion transition leads co an effective Fermi interaction of 2 2 strength G_ of order g /M , where g is the dimensionless SU(2) coupling constant. gauge boson (M) | 2 <SCM2 Fig. f. 1 - Gauge, oo-ion zxchange. at ixani^eA. iouaxtd Q} and for a. baon ireu,i ,'.f <Kvej "u-Je to an z^icXiv?. fouA. Fzmi inttiaition oj i£re.ngth Gr~g2/$ io\ ]02\ « M*.