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 SIDON 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
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 Fig. f. 1 - Gauge, oo-ion zxchange. at ixani^eA. iouaxtd Q} and for a. baon ireu,i ,'.f Both che range, M~ , and G_ are small for large M. However a mass cerra cannot be put by hand in Che Yang-Mills part L„ „ of che Lagrangian. The trouble wich such a mass term for che gauge boson is chat it spoils renormalizibilicy, che renormalizable theories being che only ones in which we know how Co compuce (in percurbacion). Forcunacely, symmetry breaking (mass generation) can be achieved wichout spoiling renormalizibility (spontaneously broken symmetry - Higgs - 5 - mechanism). The historical evolution of the ideas about the application of Y.M. theory to ueak interactions may be fruitfully followed by consulting Veltman's lecture uotes . In the first part of these lectures, we describe the standard aodel of electroweak interactions known as the Salaa-Weinberg model . We first exaaine the algebraic structure of the model for an exact SU(2) 0 0(1) symmetry (Section 2), then the Higgs mechanism in its simplest version (Section 3), and we finally discuss the problem of the so- called Adler anomalies (Section 4). - 6 - 2. THE ALGEBRAIC STRUCTURE OF THE SU(2) 8 U(l) «DEL We use f as a generic symbol for a Oirac spinor, f for its Dirac conjugate and f. _ its left (right) components. Throughout these notes, we adopt the con- L's [ 3 ] ventions of Bjorken and Drell for the metric and y matrices f f L,R- (-r^) • «•» In che Fermi model of neutron 3-decay, the 4-fermion interaction is described by °F u — H, IT + h.c. (2.2) /I U H is the hadronic charged weak current, and the leptonic charged weak current is given by which can be written, using 2*2 matrix notations, as L VlT2 Here t is the linear combination of Pauli matrices acting on a two /I component spinor ( j formed with the left handed neutrino and electron. The hermitian conjugate of I/\ which is present in che complete interaction (2.2), contains "". This structure is suggestive of an SU(2) structure of the weak currents where che charged ones behave as che £ components of a weak isospin 1 object T . Hence one guesses che existence of a neutral comrjnent T... and thus of a neutral current, represented by 'J2 in che space of 2 component (isospin 1/2) spinors. In a Yang-Mills realisation of the SU(2) symmetry, one introduces 3 vector fields W. , i-1,2,3 also noted W , which couple to fermions according co an SU(2) invariant Lagrangian, which can be written in general as U -g FL YU W .T FL+ (l*»î) (2.4) F stands for a set of fermions. Its left (right) handed components F^ (F_) belong co some representation of the SU(2) group, g is the dimensionless weak coupling constant. In che relevant fermion representation, T represents the 3 generators of 5U(2), whose commucacion relations are - 7 - tTi-V"i£ijkTk (2-5) s... is the completely antisymmetric tensor (e. . .* I) . We recover the standard form (2.2*) of L if left (right) handed feraions transform like SU(2) doublets (singlets). The representation of T is T/2 in the former case (weak isospin 1/2) and 0 in the latter. The neutral boson WT couples to the feraion current J3 " \ f T3 FL (2'6) Written in terms of the charge eigenstaces of the W bosons, W" - — (W *\t), WJ u ^ y y ' y the piece (2.4) of the Lagrangian (vith 1*0 for R-fermions) is One could chink of identifying this neutral current with the e.a. current. This is not possible for at least two reasons i) e.m. interactions conserve parity, while Che current (2.6) distinguishes between left and right fermions. ii) the neutrino "charge" defined by the coupling of J^ to W , does not vanish, but is opposite to Che electron charge since At least one other field B must be added. The simplest way to do so is Co consider ic as generated by a new (J(I) local symmetry, characterized by a coupling g' and a generator Y. The symmetry (here taken to be exact) is thus SU(2) 0 U(l). Consider a pair (a,b) of fermions whose left handed components form an 5U(2) doublet ( J . The part I «'- of cheir Lagrangian corresponding co che total neutral current reads : - » - t( (: 5> r l , ,,?iTI ï:?.--H - L » « *>*« (ï)I te * «• h \ »y T -R * *'5 RY j ^ T bt } (2'8) This way of writing underlines the fact that the L and 1 coaponent* are in dif ferent representations (isospin 1/2 and 0 respectively) of SU(2). In the first tera the generator T is represented by a anltiple of l.fg1, «j,since «,,h» belong to the saae representation of SU(2), whereas the eigenvai-tt.s T. and Tv_ of T (a b' • * are different. He now want to identify in I -*-' * pi*c* which describes the photon interaction with particles a and b, oaaely ^a! •-•«•.*) \**{il - <2-» with a given charge aacrix, diagonal for the physical states. This atans that Q, the generator of (J (I) aust be a linear coobination of T, and Y. At the saae tiae, we exchange the two fields w tad S for two new fields, chc eleceroaagnetic field A, and another one, Z, , to be identified with the field of the weak neutral boson. This transformation of fields aust be orthogonal in order co leave unchanged the purely kinetic part of the Yang-Mills Lagrangian (while preserving the heraiticity of the fields) : Here for anv field C we use the notation y KG - i G - 5 G . (2.12) Thus the aost general field aixing we can perfora is paraartrized by a rotation of angle 3 (the Weinberg angle) U> sind A • cota to 4 (2.13) 3 • cosd A - sind Z° u J. u Inserting these expressions Into chc Lagrangian of let. (2.8), we rewrite Le as - 9 - L(a,b) _ L(a,b) L(a b) with l( b) (i 5) Y [g Sin9 T + cose A' * " { ' L u 3 8' ll(j) y • (i,b)R Yy [g sine T3 + g'cos0 |](*) } A (2.15) R The 2 term is a short hand notation for the two distinct terms concerning a_ and b_. It is consistent with Eq. (2.8) if we recall that T, is represented by T./2 and 0 respectively in the left and right fermion spaces. Later on we shall come back to the part L *' , proportional to Z . For now we express that L ** is identical with L * ; in Eq. (2.9). This identity gives e.m. eQ - J g sin9 T3 + g'cos6 | - fg sin6 T3 + g'cos6 |] (2.16) Given the Lagrangian, the second equality determines the Weinberg angle in terms of g and g'. The first one connects the physical charges to the parameters and to the SU(2) 9 U(l) quantum numbers. Note that Eq. 2.16 is general whatever representations of SU(2) are used for the left and right fermions. It has to be always understood as a relation between (p *p) matrices in a p fermion space (a,b,c,...) whose left and righ components are in twc different p-dimensional representations of SU(2) 9 U(l). Consequences i) The second equality in (2.16) has no solution for generic U(l) quantum numbers of the fermions under consideration. Consistency between the several scalar equations contained in this matrix equation implies strong constraints on the weak hypercharges. In the case where a pair (a,b)L form a doublet while a_, b_ are single's, the following relation must hold 2 Y- - Y„ + Y. (2.17) L 3R bR If there are several such "families" of particles, the corresponding hypercharges (n) (n) n) (YT , Y , Y^ , n-1,2,...) must satisfy Eq. (2.17) tzpcuuvtzlij. There is aR R another constraint for the solution of Eq. (2.16) to exist. One has (Y* "Yh } e(Q -0.) - g sin© - g'cost ^—— (2.18) a • 4 r - 10 - so that for anv ti»l,2,... t e -f ' ** , R' (2.i9) 8 g and hence *R °R *R bR Remark At this point any SU(2) transformation on the fields leaves the physics invariant. The direction chosen for the third axis in SU(2), and thus for the electric charge, is still arbitrary. When the symmetry is broken (see next section), the charge generator is by definition that one which is left unbroken. ii) For a pair (a,b) fulfilling Eq. (2.17), let us consider the charge difference «Q - 0. . From Eqs. (2.18,19), is universal for all pairs \ ao a 'D &a o. classified in the standard way. Specializing to the particular pair (v ,e), we have A * 1 and hence ve Ya ~Yb e*g sind • g'cosô s (2.20) We may now use the fact that, as emphasized in Section 1, only the product g'Y is physicallv meaningful, and choose the normalisation of Y such that (Y -Y. )/2-l, aR R Eqs. (2.16) are then replaced by the relations 3 m T + Y/2 e m g sinô (2.21) tge M g' /g In usual presentation of this model, the first of Eqs. (2.21) is imposed and the two others are derived from the requirements that OED conserves parity and that the model has the right charge content. Here these two requirements are first expressed through Eqs. (2.16) under their most general form. This approach allows us to emphasize that a) implementing QED in *:he SU(2) 8 U(l) original symmetry implies strong constraints on the charges and weak hypercharges of the fermions(Eqs. (2.17-19)). - Il - b) once these constraints are verified, Eqs. (2.21) result on one hand from the phvsical fact that 0 » 0, on the other from a proper choice of normalisations of the generators T (£.-_» 1 in the commutation relation (2.5)) and Y . Of course this choice, which is arbitrary, has no physical consequences. iii) It happens that fundamental fermions actually seem to appear in nature in pairs with Q -0. - 1, for example the quark pairs (u,d),(c,s) ,(t,b),... 3 D (?) and the lepton pairs (v ,e),(\> ,y),(v ,x),...(?). One may consider this fact as predicted by the SU(2) 9 U(l) model. The prediction is that, in a given n-plet, the charges are Q, Q+l,...,Q+n-l, if Q is the smallest one in this n-plet. But note chat it is not a step towards the answer to question I of Section 1, "why are charges commensurate ?". For example, the model can accomodate a fermion pair (A,B) with Q.*^ and Q-T-l, that is charges which are neither commensurate between themselves, nor with the electron charge. iv) Coming back to the weak neutral current and making use of Eqs. (2.21), we obtain Che following expression for L _' as defined through Eqs. (2.8,13,14): Zo -«zy L( b> 2 Z0 " TloTë {û,b-)[Yu(l-Y5)T3-2sin 9 ^ q](j)} (2.22) Me see that the weak N.C. contains two terms. The first one is the neutral coun terpart of the weak charged current, proportional to I-y- (V-A interaction), the second is proportional to Che e.m. current, with a coefficient proportional to 2 sin 9, which is chus measurable for example by comparing the cross-sections for 2 neutral and charge currents. The present experimental value of sin 9 is 2 [4] sin 9 - 0.229 i 0.014 From equations (2.21), we know that Cgô * a— , but we have only one relation between g and g', namely W-? (2-24> egg Hence Che value of che Weinberg angle is not predicted. It is so because che symmetry group is not simple : each of its 2 invariant subgroups SU(2) and U(l) comes with an independent coupling. As a consequence, we may ask a second question which a unified cheory has Co answer : - 12 - 1 Quzition II : tilku lb 4-tn 9 what it li ? Let us now summarize the content of this section. The fermion world we know is compatible with a classification of fermions according to the symmetry group SU(2) 8 0(1) ; the algibfuUc i&iuctune. of weak and electromagretic interactions is well described if all left handed fermions are classified into SU(2) doublets, the right handed fermions being singlets. The couplings characterizing the strength of both interactions are derived from the electron charge e and the Weinberg angle 9 whose values are not predicted by the theory. Question I about commensurability of charges remains unsolved. The model is not yet suitable for weak and electromagnetic phenomznolagij. As announced, SU(2) 9 0(1) is strongly broken, at least at present energies. On one hand the gauge vector bosons W~»Z must acquire masses (large as compared with the present energy scale) in order to reproduce both the weakness and the short range nature of weak interactions (Section !). On the other hand, fermions (but the neutrinos ?) have non zero masses, which break the symmetry of the Lagrangian. A fermion mass term is -mzr\H + h*i) (2-25) (We recall that a(I±Y5)- 0+Y5)a and that (1+Y-) (1-Y5) - 0). Under an SU(2) transformation, a_ and a. remain unchanged whereas % and L are transformed into combinations of a_,bT, and aT,bT according to the reoresentations D and *i/2 : D of SU(2). Thus the 'lass tern of the pair (a,b) is not invariant. A last ingredient for weak interaction phenomenology is Cabibbo mixing. The quark pairs to be considered in weak doublets are not (u,d),(c,s) etc... but some linear combinations of them (respecting of course charge conservation). Neglecting heavier pairs, one has to replace (u,d),(c,s) by (u,d ), (c,s ) wiLh d • d cos 9 + s sin 9 C (2.26) -d sin 6 + s cos 9 , i c * c c where 9 is the Cabibbo angle. Generalisations tc -nore pairs lead to mixing matrices which are no more orthogonal, but unitary, allowing for phases between states and thus "natural" C? violations. This very important subject will be treated in Section (8c) below. In all the rest of these lectures, the - 13 - mixing matrix will be taken equal Co unity. The next step Co be achieved is to break SU(2) • U(l) symmetry in such a way that a) che U (1) symmetry is preserved, that is Che photon remains mass less, e.m. • b) the fermion and weak vector bosons W~,Z acquire masses, c) the theory remains renormalizable. The last requirement is the most constraining. The standard model which fulfills all the above conditions is obtained by spontaneous symmetry breaking, or Higgs mechanism. Its description is Che subject of the next section. - 14 - 3. SPONTANEOUS SYMMETRY BREAKING AND THE HIGGS[51 MECHANISM IN SU(2) 8 U(l) Spontaneous symmetry breaking (SSB), also cal led "hidden synmetry", refers to the situation where, though the Lagrangian one starts with has some exact symmetry, the corresponding physical world has not. In quantum theory, this feature say occur when there are several (in fact infinitely many) degenerate possible groundscates : this set of degenerate groundstates is symmetric as a whole, but the particular one which is chosen by nature is not. It must be stressed that, once chosen, the vacuum is unique, as it is separated from Che ocher possible ones by infinite potential barriers. The vacuum, however, say preserve part of the original symmetry. Since in general symmetry of the vacuum implies current conservation and synmetry of the physics, Che study of the vacuum symmetry properties is of fundamental importance. a) SSB of a global U(l) symmetry The simplest field theory example of SSB is the following. Let us consider che Lagrangian for a complex se >*r field ip(x) » dp. (x)+ûP2(x))//2~, interacting through a potential V(cp) L(x) - (3yw)0'V) - V(ip) (3.1) 7(«p) - j ( À is che coupling constant, a looks like a mass term (with an unspecified sign for che moment). X must be positive in order for the hamiltonian Co be bounded from below. The Lagrangian (3.1) is invariant under che global phase transfor mation (*) quancum correccions can be compuced in a systematic way using che generating functional of the one oarticle irreducible sraohs^b] # - 15 - of zero kinetic energy), which minimizes the potential V((p). From Eq. (3.2) the pJtential is extremum for 3V - aa • 0 (3.4) 5a 2 For a<0, the minimum is obtained for a«0 : the classical vacuum is tp=0 ; it is U(l) invariant and there is no symmetry breaking. For a>0, the minimum is obtained for UP » -— • being a maximum of V (the "mass" squared A^S^ is negative at tp « 0). As a function of Re I -V.J . The. minvmm o{ the. pottrvtiat i* xeackexi {OK j Among all the possible degenerate vacua, nature chooses a particular one ,Jo i'3„ 'P*-= e ° , say, and the U(l) svmmetrv, reoresented by the invariance of the v2 figure 3.1 under any rotation around the axis of the bottle, is broken. Now we can always redefine Che field tp by the U(l) transformation (p -» e io.°i p in 3uch a way that Che new field (asgain noted 0) has the real vacuum expectation value J /»"!. o - 16 - In order co construct a perturbation theory around the classical solution, we aust use a translated field X.U) + i \Ax) a X(X) = _! £ « (o(x) - -4 , (3.6) whose vacuus expectation value vanishes. In terms of x» tn< Lagrangian (3.1) reads (up to an uniaportanc constant) . I ,. ,2 12 2 * i (VX2)2 \ , 2 2.2 Xao ,2 2, x x J ~ï* l* 2 ""J-Xj •? - ,\ J* (3.3) Lee us now consent upon this new form of che original Lagrangian, in terms of che two fields Xi *nd Xi* i) &4.vz field, with mass proportional co che non vanishing vacuum expectation value J //2~ of che original field iii) as a function of che new fields, che original Lagrangian is no more symmetric. However, che form (3.7) of che Lagrangian is not the most gerjral one for cwo real fields x> and \- with quarcic interaccions; not only because m? » 0, buc also because Che mass tn. and che 4 and 3 degree couplings of che fields are relaced co each other. The use of SSB is chus a way of restricting che number of parameters of a non symmetric theory. To summarize : scarting with a Lagrangian for a complex field The fact that m-* 0 is a direcc consequence of the original symmetry. For small fluccuacions of <2(x) around its classical value 0, one may write che original field '?(x) as z • ix7(x) » ^(x) - -2 = = 4 «xp [ix, Thus to the order \~ , this field configuration leads to the sane potential n energy as Che vacuum J /*T does. If moreover we choose x2 i- (3.9) with no oscillations in x (long wave length limit), The existence of a mass less scalar field ''Goldstone boson), associated with a global symmetry spontaneously broken by an asymmetric vacuum, is generic and is the object of the famous Goldstone theorem. The Goldstone boson survives and remains massless to all orders in perturbation theory. For non abelian global symmetries, one finds as many oassless Goldstone bosons as there are group generators which do not leave the vacuum invariant. Neither such (*) particles are observed in nature , nor Che long range forces chey would give rise co. Hence global symmecry breaking is not yet the mechanism we look for to break SU(2) 8 U(l) down to U (1). As we shall see now, in Che case of e.m. loccui symmetry, che GoIdscone degrees of freedom are used Co give masses Co che gauge fields, and associated massless bosons disappear. This is what is known as che Higgs mechanism. b) SSB for a local SU(2) 8 U(l) gauge symmetry. The Higgs mechanism One introduces a doublet * of comolex scalar fields G) V - ÇV* tp*) , (3.10) interacting through a potential v($) - £ (***)2 -a *T* . (3.1D which is manifestly invariant under both SU(2) and U(l) local transformations (*) In a completely differenc concext, the pions are considered as the Goldstone bosons of che spontaneously broken SU(2) chiral invariance ((1)-»e~l^ "2~" (j) of massiess nCD. The pion mass is not strictly zero, however, because of che non .rero masses of the light quarks. - 18 - î(x) * exp 1-i a(r).T/2} exp {-i a'(x) ji $ . (3.12) The kinetic part O $ )(3 4) is not invariant because 3, i(x) and 3 a'(x) are u u u not zero. As usual the symmetry is restored by introducing the gauge fields W. and B already considered in the previous chapter, replacing 3, by tha covariant derivative D, 3 - D « 3 +±4 W .T •• i£-B Y . (3.13) y y u 2 u 2 u The part of the Lagran^ian involving $ then reads L* • L. , as well as the pure Yang-Mills Lagrangian and the part containing fermions (see below), is locally gauge invariant. SS3 occurs in a way very similar to that encountered in the previous example. The vacuum is obtained for W (x) * B (x) » 0 , and $(x) equal to a constant SU(2) spinor minimizing V($) : i) * - 0 for a > 0 ii) $ • -—; x(arbitrary constant unit SU(2) spinor) for a< 0 The latter case is the one we are interested in. There is now a privileged direction in SU(2) space, which, after a global SU(2) transformation on Che fields (a redefinition of che fields), allows one to write the $ vacuum as 0 /i ^ V The symmetry associated with the transformations induced by T and T is now 2 broken but one combination of T and Y is conserved. This combination coincides * (*) with the charge operator Q-T.. + Y/2, provided Y is taken to be Y»- 1 . (The field which acquires a non vanishing expectation value has zero charge). With this choice for Y, SU(2) B U(l) correctly breaks down Co U (I). Before e.m. investigating Che physical content of our theory, we may use Che SU(2) invariance co perform in each space cime point x,c a particular SU(2) transformation which Had we defined Che fields so chat Î • -s L , we would obtain Y.*-I, Che hypercharge of Che complex conjugate Hi^gs field. transforms the *U)5Uo) • ".16) where For the time being, we keep $(x) under the form (3.16) and translate the field ip(x) by its vacuum value in order to do perturbation theory around small fields. We set X&L. tp(x) - 2l (3.17) ,1 JÎ We next identify the spectrum of the theory by looking at the quadratic part 1 2 L- of L. (Eq. 3.i'0. Apart from the kinetic term y O, x) » U contains mass terms which are the terms proportional to 0 after the replacement (3.17) in °1 2 2 L.. As before V($) generates a mass term - j (\a )\ which tells us that the neutral Higgs boson x we are left with is massive, its mass being m^' a A (3.18) Let us collect the a te Lin in (D f. W . !• g'B I 1 3 u 2 s u 2 (3.20) It is Chen straightforward to find - 20 - 2 (W +W } (gW (3>21) i-2 " — -§ H» 1 2 * 3 "«^V tp *• y y y J + From this expression, we innediatly see that W. and W,, and hence W", the charged weak bosons, have acquired non zero masses, namely a2 2 m2+ - -2 (3.22) H" 4 2 2 (x+^ > g r , 12 The second term in Eq. (3.21) can be rewritten as 0 s cosSW, - * cosôB Y , 8cos26 L 3y 8 y J and we recognize inside che square bracket the combination Z° - cos6W, - sin6B (3.23) U 3 y hi which we had identified in section 2, as the neutral weak vector boson (Eq.(2.13). Eq. (3.23) is true provided g'Y/g » tg6, which, together with g'/g - tgô (Eq. 2.21) requires that the eigenvalue of Y for the Higgs doublet is 1. This is just a check as we had already obtained Y»I by noting above that Che part of che Higgs doublée $ which acquires a non vanishing vacuum expectation value must correspond co a neutral field, not to destroy exact U (1) invariance. The orthogonal combination, che phocon A = sin6W, +cosdB does not appear in 2 V Jy U Lip and chus remains massless. This has co be so since A is associaced wich che u unbroken symmetry. A second consequence of this SSB scheme is che celebrated relation between che charged and neutral weak boson masses 2 2 2 J 2 0 8 "w °z -—irm • -r (3-24) o 4cos 9 cos 3 As to che physical Higgs boson x» we note(Eqs. (3.11,16 and 17)) chat it has cubic and quartic self couplings. 1c also couples co che gauge boson (Eq. 3.21) : one or cwo Higgs bosons couple Co a pair of Z 's and Co a pair W W wich predicced screngchs. A few numbers By comparison of che Salam-Weinberg model for S decay co che Ferai incer- action (see Fig. 1.1), one gees GF/^7- g*/(8mj) - -^ , (3.25) independenc of che gauge coupling g. o From unificadon of weak and electromagnetic interactions g"/47T » a/sin 9 - 21 - From low energy weak interaction phenomenology ,-5 -2 1.05 * 10 proton From high energy weak interaction phenomenology (charged and neutral weak currents, ed polarization experiments etc..) sin29 « 0.229 ± 0.014 Eqs. (3.24,25) then allow to compute (with errors essentially due to that on sin 9) ' g /4IT » 3.11.10- 2 By - 81.1 GeV » 90.1 GeV (3.26) "Zo a - 260 GeV - 260 fk GeV V HlggS As X is undetermined, the Higgs mass is unknown. c) Couplings of Higgs to fermions. The fermion masses. Fermions nave been left aside in the above description of the Higgs SSB mechanism. We recall that in the unbroken theory, all fermions are massless since a mass term of the form -f (fL £R * fR V is not invariant under SU(2) when L and R fermions belong to different repre sentations of SU(2). For the standard model where L-fermions (Resp. Refermions) are doublets (Resp. singlets), the above term transform under SU(2) like a doublet. With a Higgs doublet $ at our disposal we can build a new piece L._, by coupling directly the $ field to f.f in. an SU(2) invariant way. In fact, there are two independent singlet fermions associated with each SU(2) doublet, u_ and d_ for example for the (u,d) pair (with the possible exception of the (l,j) pair3 if the J'S are indeed massless - no right neutrino). In the unitary gauge, i "(iflfxw » am* ^ aay C0UPle c" a T- - —1/2 fermion, while the charge c X> conjugate field * - it, (jjx))* - ("°J ) ••» inetapaneantly couple to a T3«+I/2 fermiou. As an example, a general Yukawa coupling of * to (u,d) quarks is -gd iOix)Z[d7i - g^ 3 J ~Sf Jl Vl " *f /2 IT. By a phase transformation on f_ (and/or f_), ara can always make gf real positiva, so that one finally gats •f " 7f «f <3-28> For the general problem of fermion massas, see section 8c. Fermion masses are thus "naturally" generated by the same SSI mechanism which gives the gauge bosons masses. However, there are no predictions for the fermion masses, *% they are proportional to arbitrary Yukawa coupling constants. That neutrino are mass less in the present model has bean put in bv hand in (*) deciding chat there is ao right handed neutrino . Nevertheless, Eq. (3.28) is interesting if read as giving %f as a function of m- and a which are known expe rimentally : che x fi«ld is predicted to couple to a feraion proportionally to che fermion mass. Higgs production in particle collisions is thus most probably accompanied by heavy quark production. The phenomenology, and references to che relevant literature, of the Higgs boson production and decay may be found in Ref.f 7 }, Various bounds on the Higgs and feraion sasses have been discussed (see Kef.(8 ]). They will not be reproduced here. Let us just say that che general strategy for deriving them is co infer from che success of percurbative calculations in electrowaak interactions chat chc various new couplings introduced (*) Using a Higgs triplet, che left handed neutrino by itself could be given a so-called Majorana mass [6 ]. A Hajorana particle is a spin 1/2 particle (massive or noc) of definite charge conjugation. Such a particle has no electromagnetic inceraccion at all. In chc zero mass case, there is no difference between Majorana and Dirac ^articles. in the model, (A for Che self couplings of the Higgs and g, for its Yukawa coupling to the fermions) are small enough. From Eqs. (3.18) and (3.28), bounds on these couplings implv bounds on the corresponding masses. d) The number of parameters in the standard SU(2) 8 U(l) model Restricting ourselves to the 6 quark model (3 qiiark pairs (u,d), (c,s), (t,b) and 3 lepton pairs (v ,e), (v ,y), (v ,T), with massless neutrinos), we e 4 T have 9 fermion masses (or cheir 9 couplings to the Higgs boson). Then come the 4 parameters (3 angles and one CP violating phase) of the unitary matrix which allows us to build the hadronic weak charged currents starting with the mass eigenstate quarks (generalisation of Cabibbo mixing (see section 8c)). Associated with the gauge fields are the coupling a and the Weinberg mixing angle 9, with the Higgs doublet the self coupling À and the vacuum expectation value a , that is 4 more parameters. The minimum scheme then with 1 Higgs doublet and 3 fermion families involves 17 free parameters. This number increases very fast with the number F of families. As we will see in section 8c, the unitary generalized 2 Cabibbo matrix contains (F-l) physically relevant parameters. In addition there are 3F fermion masses, so that the total number of parameters depends on F as F(F+I) : 8 more parameters if F changes from 3 Co 4 1 Here we might ask a chird question. Qutitum III : how can ant reduce tkz wmb&x o{ £tee pasiamvtvu ? We close this section with a co&ment about the possibility of introducing more Chan 1 Higgs doublet or larger SU(2) Higgs multiplets. First we observe chat tnulti-Higgs systems cause some theoretical problems. (i) in order for charge co be conserved, all vacuum expectation values have co point in the same SU(2) direction, which impose constraints on the parameters of the Lagrangian. (ii) unless unnatural conditions are fulfilled [ 9 ] , models with multi-Higgs systems involve flavour changing neutral interactions induced by Higgs exchange. This is in contradiction with experimental data on, say, L +uu . That a single Hig3» doublet U-J.".•!•• not lead to such a disease is shown in section 3c : che Hig^s interact!;.-s ;«ornes diagonal in flavour space at the same cine as che mass matrix is diagonalized. - 24 - Assuming that these difficulties are overcome, let us examine the consequences of introducing an arbitrary number of arbitrary Higgs multiplets. In Eq. (3.13), -+W• T-*• Y —à— and •=• B are replaced respectively by -*• -*• I W.Th and {[ YhB , (3.29) h h where T., Y. are the representations of the SU(2) 9 0(1) generators for a Higgs multiplet labelled by h. For each h, the charges inside the multiplet are given by % " \ + V2 • where T- runs from -L to +L. , L bein» the weak isospin of h. If Y. is such that one of the charges Q. is zero, the vacuum expectation value of the corres ponding field <& can be used to generate masses for the gauge bosons. For this field, Y./2 - -T, so that its contribution to the mass term is n \ V i g2a2 fw2 T2 • W? T2 1 • 5- (cos6 W - sine B )2 T2 2 h >• * lh 2 2h-l 2cos28 3 U 3h where T.. , i»l,2,3 here represents the matrix element of TT in the neutral state o Lh 2 2 2 Lh ta . The ratio My/(cos 9 MI ) can be read off directly from this expression : -TIT " Y i / (3'30) cos 8 ^ I a T o h h Given T. , the eigenvalue of T-, for 2 12 2 1 •*! 2 (T T (T T \ ' 7 Ih * 2h> * 7 h - 3h> The value predicted for o then is in general I <2 «w,, ,„ K..x ,>2 0Ch - i -Ji y—-, îi— (3.31) T hK n 3Jh - 25 - The value of p can be experimentally measured by comparing the strength of the charged weak current relative to the neutral weak current. Indeed, in the same way as charged current interactions are measured by GF 8°W we get from Eq. (2.22) a Fermi like neutral interaction measured by so that p is given as well by » -i s The preseut experimental value determined by comparison of v,v induced charged and neutral current interactions is o - 0.98 ± 0.05 (3.32) exp If there are several different values of L in the Higgs sector, Eqs. (3.31, 32) are not very constraining. In fact any a priori value of p can be reached. On the contrary an arbitrary number of replications of the same Higgs pattern with the same I and T. leads to a simpler equation , KI+1) " T2 °th " I 3 ~ ' (3'33) T3 independent of the continuous variables J. . If we admit that experiment indicates 0*1, the equation becomes 3T2 - 1(1+1) A little bit of arithmetics shows that the lowest solution for I,T. integers is I - J T3 - £ 2 (The next one is 1-48, T. - i 281) - 26 - It is a quite exotic configuration, not very attractive to accomodate the known fermions. For 1/2-integer isospin, the lowest solution is the standard one 1-1/2 T3 » ± 1/2 (The next one is I » 25/2, T3 - t 15/2 '.). The conclusion is : there is a remarkable agreement between the standard theory with I- 1/2 and the experimental value of p. What we have learnt here above is that an arbitrary number of Higgs doublets would work equally well, up to the above mentioned problems i) and ii). 4. ANOMALIES. Unitartty of the S-matrix and renornalization In the unitary gauge we have used Co describe the Higgs mechanism, the particle content of the theory is made apparent at the very beginning since 3 out of che 4 Higgs fields disappear from the Lagrangian, the corresponding degrees of freedom being carried by the helicity zero components of the weak + bosons W",Z . In particular, there are no unwanted Goldstone nodes. However the unitary gauge is not suited for the study of the renormalizability of che theory . In this gauge, the propagator of a massive gauge boson tf is in momentum space K k. ,unit.fTr^ _ -I f u vl which is of order I for large k components, leading to badly behaved Feynaan graph integrands. This fact makes the «normalization program difficult to pursue. It is better to come back to the original Lagrangian which involves the Goldstone bosons. The «normalization Chen is simpler to achieve,but in turn one has to show chat the Goldstone bosons have no physical effects. This happens Co be true only in the absence of the so-called anomalies, as we are Co show now. For simplicity, we consider che case of SSB for all(l) gauge symmetry, where only one complex Higgs field ip(x) is introduced. The non-vanishing vacuum expectation value a I/I of tp can be made real by a global (i.e. x-independenc) phase transformation. So we can write as in seccion 3.a, but we do not ieX x2 - 0 by a local gauge transformation. Proceeding next as in seccion 3.b, but with D, * 5 • igW , we gee che Lagrangian L,ïW ""2* Xao*l •î * g % % *\ (4.3) + terms of degrees 3 and 4 in che fields. - 28 - The first two lines are the same as in Eq. (3.7). The third one contains the I _iw usual - -r F F^ and the W mass tern generated by SSB. Lastly coœs a term which is new with respect to the case of the unitary gauge Xo - 0» namely u * u g a a 3 ^,, coming from (D by -g 3 (3. \Tjx2 (integration by part), which can be eliminated by choosing the (Landau-*t Hooft) gauge y The quadratic part of the Lagrangian is diagonalized and the W-propagator now is (Landau) „ _-! f _ Vjw] 2 -1 to be compared with Eq. (4.1) in the preceding case. It behaves like (k ) 2 at large k values, which leads to the dimensional counting of a renormalizable theory. As a slight disgression, we mention that the Landau gauge is a special case of :ne general 't Hooft gauge (or Rr gauge). In the Rr gauge, one adds, to the Lagrangian (4.3), the "gauge fixing" term - £ (3 r-g -7- x2) > where C is the gauge parameter. The Landau gauge is recovered in the limit Ç -» ». In all cases, the crossed term -g a 3 tr x? *•• exactly cancelled out. The It-propagator is r . k k • -1 u v P (k) II8I ((1 h uv 2 I,* k-Oç, l ^ ^ 2 »8- (pendent: mass 2 .4 X2 ^ Ic is actually nassiess in the Landau gauge. In the absence of anomalies, the theory will be gauge invariant in the sense chat all C-dependent quantities cancel in the S-matrix. Conversely, anomalies are manifested by poles ic ;- dependent positions in che S-matrix. We close che disgression on Rr-gauges, and go on with ; infinite (Landau) r - 29 - The x2 field, which here is nassless, is present in Che Lagrangian. Is che unwanted Goldstone boson back ? In fact, Che coaparison of Eqs. (4.1) and (4.4) leads co k k .(Landau) -(Unit.) u v .. .. BW k which expresses that apart froa the physical W pole at k • au , P u con- 2 tains a pole at k «0, with furthermore the wrong sign (ghost). There are thus 2 two origins for poles at k *0. It has been shown that, in the absence of fermions, or if che fermions have no axial coupling Co Che gauge bosons, che poles coming from che x? field and from chc gauge field cancel each other, leading to an S-matrix without any Goldstone boson present [l2l(*) (hence the word unitary used to qualify the particular gauge where x2 *•* absent from the beginning). Let us remark that in the Landau gauge, the Higgs phenomenon is not the absence of Goldstone boson - the Goldscone cheorem remains valid in the pre sence of gauge fields, but rather the presence of a ghost pole, which precisely cancels the observable effects of the Goldstone boson. The anomaly problem The only exception to che above cancellation cheorem is provided by che [131 existence of che so-called triangle anomaly . We present che proble!mm in che form ic has been discussed (and solved) by Bouchiac, Iliopoulos, Meyer.[14 ] One considers a simplified model with a gauge symmetry U (I) 6 U.,(l). The model contains one fermion f which has usual E.M. interactions with the phocon A , and cransforms under che "weak" gauge group U„ as fL * ."I*™ f, (4.6) The f-W interaction is thus described by ifLY^O^igWy)fT^ifRY^(3u.igWy)fR w - i!Ou + ig wyY5)Y f (4.7) 'in che case of a non Abelian theory, che quantization procedure introduces additional fields (the Fadeev-Popov ghosts) vhich also participate in che cancellation of the unwanted Goldscone bosons and finally disappear from che S-macrix. - 30 - One next incroduces a coaplcx neutral Higxs field •Cfj^'» h.c. (4.8) This coupling is U„ invariant provided under GyO) « * * r2i«*<*> (4.9) U„(I) is subsequently broken as tp acquires a non vanishing expeccacion value 3 fil. He set o '7ïIfh'l7ïlY5£^ (4-n) ant' the feraion aass is a • ••I 6. (4.12) Finally, according Co Che ipW inceraccion induced by (4.9) , Che U boson has che mass a., , «J « 4g2 a2 (4.13) In che Landau gauge, che W propagator is given bv Eq. (4.5), its residue at 2 *ukv 1 k - 0 being j- • "hila Che ;<« propagator —y has residue *l. We are now °W k 2 co show chac chere is a pole at 1c » 0 in che forward T( elastic amplitude, wich a non vanishing residue, che cancellation between che GoIdstone pole and chac of che W propagator being incomplets. At che lowest order in a and g, these poles occur through che following Feynman graphs (che notations are 2 indicated directly on che drawings, together with the residues at k* • 0 of the W and ;<2 propagators). The residues corresponding to each of che graphs are R^ -4^CWV UH'V' (4.15) - 31 - eï< ktt ktt' m2w v k = k, + k2 P A T* Fig. 4.1 - Cont/UbutionA to the. polz At k *0 in thz ^ofwrnxd YY AcatteAing amplitudz: W In the. ditzct channel la) ; Gold&tonz boAon \„ in thz dinzct channel ib]. Using Chç expressions (4.12) and (4.13) of che f and W masses, che cocal residue can be written as "vv-^{("^) W)--* ^ •"'"'} •(*••« It is understood in these expressions chat T^ , U represenc che sum of che cwo cerns where che cwo photon lines have been exchanged. Hence chey are sym- •x-yv meeri c in che change (k.^-k,, 'J -*v). It is clear on (4.16) chat if k T , the - 32 - divergence of the axial current carried by W, is equal to 2mU , then R*0. One could expect that it is so : recall that the Noether theorem states that to any exact continuous symmetry of the Lagrangian is associated a conserved current, so that in the limit where the U„(l) symmetry is restored, m~0 and k,:xTWV should vanish. In fact, the so-called "normal" Ward identity kVv 2m Uy1 v (4.17) a can be derived, as is the Noether theorem, from the classical equations of motion. It can also be obtained from formal (but illegal) manipulations on [13] Feynman graph integrands. However, Adler has shown by actual calculation of T and U that yv 2 WVp (4.18) &? 2m U + Sir £ ° k a relation which is known as the anomalous Ward identity : in the case of the axial current, the "normal" identity does not survive the one loop correction. We do not want to reproduce the calculation which leads to the result (4.17), but just draw the attention of the reader to a few points which are useful for what follows. The contribution of the graph uv . to the tensor T is (4.19) let Reg. t 4 where the loop integral j dp means that some régularisation is needed, or Reg which we will come back lacer on. The trace is made on Dirac indices, and yv u I ) 1 1 (4.20) S " i TFm) ' tpHt--ni) (5*-JL (jHt,-ra'1 ) - 33 - The graph with the two photon being interchanged, namely Yv happens to give the same contribution as t . In order to show that, it is convenient to use -p as integration variable, so that the second contribution involves Tr t^ with 1 1 \> \ \i (4.21) 2a Y ^FSY -PHI,-*Y 5Ya -^^-m Using that Tr t_ is invariant under transposition, and its invariance under cyclic permutations of the factors, its contribution to T is equal to that of tl uv i 2ct -p" -a -rfT*l£-a Vs -i -kj-m " -* yU Fi yJ FÇ5 Vs F7-=S c+ where C is the charge conjugation matrix, such that C Y3 C+ - -Y3' Cï;C » Yc Since Y Y„, , we find tllf = t.y v -yv.'a' 5 ' '5'e a 2a Hence iaf 2/ d% TrUf ] (4.22) Reg The same doubling occurs for I•iMTV . Note here for future reference that if y. is replaced by 1 (vector coupling), there is one - sign less in the manipula tions made for t« above, so that Tr t and Tr t~ exactly cancel each other (special case of the Furry theorem : a fermion loop with an odd number of photon legs attached gives no contribution). The integral involved in (4.22) is a priori linearly divergent (of the form / —£• at large p's). As a consequence, given a prescription for computing - 34 - (4.22), one might get a different result by making the change of variable p - p • Xjkj • X2k2 . (4.23) That it is possible to produce an unambiguous contribution from the triangle graph comes from the requirement of gauge invariance with respect to the photon indices u,v, which states : klu TaV "k 2vT ÏV « ° • (4-24) These two relations are used to define two a priori divergent integrals appearing in (4.21) as a combination of the other integrals, which converge. The rest of the calculation is straightforward. One computes 2m U and compares with (k,+k.) T^ to find the result announced in Eq. (4.17). The ÎIYJTYA anomalous term eUvP°" j, u comes from a term proportional to e (k.-k_ ) in T , the coefficient of which, formally given by a divergent integral, being fixed by the condition (4.24). This condition actually forbids arbitrary translations of the form (4.23), which would generate additional terms linear in the 4-momenta k.,k«. The above results can be understood in the following way. Feynman graphs must be regulated prior to any manipulation. Anomalies happen when there is no regularization which preserves all the Ward identities of the formal theory (the normal ones) for both types of vertices, vector (Eq. (4.24)) and axial (Eq. (4.17)). When it is so, some of the gauge symmetries of the Lagrangian are broken by radiative corrections (in the case of SSB, by "gauge symmetry" we mean independence of the S-maerix on the gauge fixing parameter 5). The anomalous Ward identity precisely expresses this lack of symmetry of Che theory wich respect to that of the Lagrangian. The result (4.18) is obtained if the electromagnetic current conservation (Eq. (4.24)) is imposed. One regularizes the theory in a U (1) invariant way (e.g. Pauli-Villars regulators). Then 6 «is» the regulated T and U obey normal vector and anomalous axial Ward identities. As che regulators are removed, T and U are finite (no renormalization) and thus obey the same identities. In Che discussion of Che anomalies, we have emphasized che point of view of che unicary of che S-tnatrix. The anomaly problem is at the same time intimately connected wich che renortnalizability of che theory : che counter- teras in che Lagrangian which would cancel che anomalous piece of Eq. (17) - 35 - is non renormalizable and gauge dependent. The solution to the anomaly problem Suppose now we have not I, but n fermions f. in the Lagrangian, with charges e.,g. and arbitrary masses m. (given by arbitrary couplings G. to Higgs fields 2 / ? 2 V a Therefore, the condition for the triangle anomaly to disappear is a«0. In the present simplified model, this condition can be realized, e.g., with two fer mions of opposite weak charges g, "-g«, and electric charges equal or opposite. It has been shown that once the 3 boson vertex has no anomaly at the one loop level, there is no anomaly at all for any diagram at any order in perturbation theory . Hence a«0 is ththie only condition to be fulfilled for the renormali- zation program to be pursued The condition a • 0 clearly is a constraint on the quantum numbers of the fermions. As a consequence, if these quantum numbers are related to each other by symmetry properties induced by some gauge group, the absence of anomaly appears as a condition either on the group itself or on the group representations to which the fermions belong. If this condition is fulfilled the theory is said to be anomaly free. We actually have to consider anomaly free theories only, since otherwise we are faced with all the problems of gauge dependent quantities in the S-matrix or of non renormalizable counterterms in the Lagrangian. Note that these problems are present in any gauge theory, abelian or not, spontaneously broken or not. It follows that the absence of anomaly is a constraint of funda mental importance for grand unified models, as we shall see at length. We consider the triangle graph in a general cas» of gauge symmetry, for 3 arbitrary external gauge fields with group indices a,b,c. They couple to the feroion loop through representations Ta, T , Tc of the associated generators, abc with strengths aeasured by the gauge coupling constants. Mote that T , T , T need not be the generators of the same simple Lie group : for example the W W 3 vertex involves the generators T , T~ of SU(2) and Y of U(l), the coupling cons tants being respectively ; and %'. Finally, the couplings of the gauge fields - 3» - Co che feraiea leap any involve y, or not. la all casas, since y~« l, say odd (even) nantir of y,'s caa always be reduced to only oae (tare) at a given place. Therefore, as far aa the aaoaaly is concerne* „ «a have to coasider T^Tlb k2c r k,b v_c c^& T Tkt i rt ki Due co che equality of che cvo Fa yen graph contributions co the »iy. the absence of anoanly is expressed by abC a , C D (R) 5 TrR [T (r ,T >] - 0 (4.25) The curly bracket oaaas anticoaawtator, aad cha trace runs over cha indices of cha feraion reprasencacion %. In order co draw useful consequences froa this equation, we first recall a few properties of traces over {roup repre sentations . i) Tr _ [T*] • 0 for any representation R (irreducible or not) of any iimplt Lie group. Sut Tr[T*] i 0 for (1(1) which is not siapla. ii) If chc group G is a direct produce of Lie groups G , C ,..., then any T is che direct sum of it* representations in each group 7 (I) ,(2) and Tr'Tj - i*r_ (T(l>) * Tr, (T(2V (4.26) T;r_r i„(!)-l2 k i ) • • • ,i TCl) < Tr, T^:> « (4.26)' - 37 - iii) If the representation R is reducible, R«R. • R_ + ... TrR[T] « I TrR [T] . (4.27) i i iv) D vanishes for any real representation of a Lie group. The proof goes as follows. If T is the matrix representing a generator in a unitary represer T ... representation R, -I represents it m the conjugate representation R : T«T implies (.-*•*)*-.-i(î-(--T*)Tt ) If R is real, by definition R and R are equivalent, i.e. they differ by a unitary transformation U. Under U, D is invariant (trace property). Hence DabC --Tr(TaT{TbTTcT}) - - Tr(Ta{Tb Tc}) « - Dabc . v) SU(N), S + 2 and S0(6) (which is locally identical to SU(4)) are the only simple compact Lie groups which have representations with non zero D (Recall that U(l) is not simple. All irreducible representations of U(l) have D # 0, but the trivial one Y-0). In a first step, the demonstration uses the fact that the only simple compact Lie groups which have non real representations t173 are SU(N) Ml, SO(4N+2) N ^ I and E6. [ 18] vi) It can U« shown that for any STT(N) irreducible representation R Dabc(R) - dabC A(R) , (4.28) dbc where only A(R) depends on Che representation* Therefore, if D (R)«0 for one particular set a,b,c of indices, then (provided d is not zero for this set) A(R) «0 and 0 (R) vanishes for any set of indices. The values cf A(R) for all R's of SU(N) are given in Ref.[l9]. Coming b3ck to SU(2) « U(l), we exanr'n th* anomaly problem for the repre sentation 6or.stjd by the following 7 fermic::» • (i 29 (ï)t • H • <* • C-)L • •; • - > - 38 - The only triangle diagrams which may cause anomalies are those which contain at least one W, otherwise there is no Ye in the loop. If there are 3 W's, âbc then D (R) » 0 because of the property (v) (all representations of SU(2) are real). If there are one W and two B's, the anomaly vanishes because of properties (i) and (ii). Finally, if there are two Ws and one B, one may replace the generator Y/2 associated with B by (Q-T-). The term with T_ vanishes (3 SU(2) generators in D ) and we are left with only two cases corresponding to W W y and W W y vertices, o o These vertices are proportional respectively to D*"Y » Tr({T+T~}0) and D33Y - Tr(Tj Q) + - 2 2 2 {T T } * 2(T. +T-) gives the same contribution as T, and therefore the anomalous term is weighted by the quantity 2 ASU(2) 8U(1) - Tr(T3 Q) The contribution of the quarks comes from the left handed doublet (u,d)L The contribution of the leptons is .leptons 1 . 1 A " 4 Qe * ~ 4 so that the anomaly does not vanish. Including other similar families cannot help. A way out is to postulate the existence of new quarks and leptons with right handed SU(2) couplings in order to cancel separately Aqu s and A p (vector like theories). There is however no experimental evidence for such new fermions. A far simpler mechanism is the one where the quark and lepton contri butions co che anomaly cancel each other. This happens if the quarks appear with 3 colours. Then : Uarks Upton9 A - Tr(T* Q) - 3 A« • A - 0 We have 3hown chat no criangle graph with external W's or y (or 3) gives rise - 39 - to any anomaly if we consider a 15 member family with quarks being triplets (3) (antiquarks antitriplets 3) and leptons singlet representations of SU(3) . We are thus naturally led to the gauge group SU(3) 8 SU(2) 8 U(l), and we have to check that there is no new anomaly associated with triangle graphs with one or several external gluons. That it is so follows from the following considerations. If there is one external gluon, D vanishes from properties (i) zeW° and (ii) above. For 2 external gluons, we have to consider D66 which vanishes for the same reason, whereas D**^ does not appear (no y. involved in the corres ponding triangle graph). The above arguments work for all similar 15 dimensional families (c., s., v, , u), (t., b., v , T) etc., or linear combinations of them. So the anomaly cancels family by family. Why it is so is a question vhich has to do with the general problem of flavours : why successive generations, how many of them, etc... . We conclude that the gauge group SU(3) 9 SU(2) 8 U(l) has no anomaly at all for replicated 15 dimensional representations, the colour degree of freedom being essential for the argument. This symmetry group is spontaneously broken through a Higgs mechanism involving a colour singlet, SU(2) doublet of complex Higgs fields as before (section 3). Therefore the symmetry is broken down to SU(3) 8 U (1) at low energy, which is exactly the remaining exact symmetry 6 «in* we are looking for. - 40 - PART II GRAND UNIFIED THE0RIES-SU(5) Before going into the main subject of this second part, we summarize the results we have obtained with the standard model of electroweak interactions. This model - gives rise to a renormalizable theory of weak interactions, - unifies charged and neutral weak currents, - leads to the right phenomenology of all electroweak processes which have been measured up to now. The colour degree of freedom has been useful to prevent from anomalies which otherwise would spoil the renoroalizability of the theory. For that purpose, a global SU(3) symmetry is sufficient, but we know that when extended into a local gauge symmetry (QCD), SU(3) is a good candidate to be a theory of strong interactions. In particular it has the property of being asymptotically free, allowing perturbative expansions for the so-called hard phenomena. There are also indications that it gives rise to confinement of quarks and gluons. Therefore the SU(3) S SU(2) 9 U(l) local gauge group provides us with a very successful model of elementary interactions (but gravity). However the model has some serious defects which we recall here by collecting the various questions we have been led Co ask in Part.I. I. Why is electric charge quantized ? (Q • T- + Y/2 and Y, the U(l) generator may take any a priori value). II. How to predict the value of the Weinberg angle ? At the present level, we still have in fact 3 independent couplings i, i and sin8. III. How can we reduce the number of free paramecers ? In addition to the couplings, we would also like to predict the feroion masses. - 41 - Grand unified models answer these questions in a definite (questions I and II) or a partial way (question III). It was outlined in the general intro duction (section 1), especially in connection with the problem of Che quanti' zation of charges, chat a unified model should be built on a symmetry group § containing g - SU(3) , 0 17(1) as an unbroken, non invariant subgroup. This group also has Co contain SU(2), the weak group, in order to reproduce the successful Salam-Weinberg model of electroweak interactions. Finally, the theory should be free of anomalies (section 4) in order to be a renornalizable theory. However the large group g cannot be an exact symmetry group since only g is observed as an exact symmetry at low energies. It happens that the above consi derations lead to rather stringent restrictions in oodel building : it is shown in section 5 how SU(5), vhich fulfills all Che conditions required, appears as a very natural (and unique in a sense Co be made precise) candidate for g. The SU(5) model is described in section 6, and its predictions on SU(3), SU(2), U(l) couplings shown in section 7. Th* Higgs mechanism is introduced in section 8 in order to get the breaking of SU(5) into SU(3) 9 SU(2) 9 U(l) in a first stage, and into SU(3) 0 U (1) in a second one. It is a very nice property fi •ID* of SU(5) chat these breakings can actually be achieved in a rather elegant way chrough the Higgs mechanism. The great weakness of the model however is undoub- cely chat Che coupling constants of che model have Co be adjusted with a fantastic precision in order to yield che required low energy physics, since Che fundamental scale of che theory (responsible for the first stage breaking), is as large as 14 about 10 GeV (section 9 - Renormalization of couplings and masses). The possi bility of matter instability (nucléon decay) being the most spectacular predic tion of grand unified models, we finally present the main results obtained in che recenc past years on che procon lifetime (section 10). Apart from the hierarchy problem, another question for unified theories is that of family replications. One would like to be able at least to classify all families in one irreducible representation of che unifying group. In SU(S), che number of families is somewhat restricted by phenomenological implications of che model (value of siaQ, fermion mass ratios), but there is no theoretical argument in favour of any particular family number. Other groups than SU(S) are of interest, in particular S0(10), especially if neutrinos happen to be massive. We will however limit ourselves to SU(5) in the present lectures. We recall chat gravity is left aside in all these models. Superunification, based on extended supergravity, wich all particles (from spin zero bosons Co H - 42 - the gravi ton) in the sane irreducible multiplet, may give an answer to the problems left opened. The corresponding models are however in their early infancy and will not be considered here. h - 43 - 5. IN SEARCH OF THE UNIFICATION GROUP In the above introduction, we have given arguments for embedding the SU(3) 9 SD(2) 9 U(l) group into a simple Lie group G . This will give us relations between various previously unrelated parameters of the theory. In particular the strong, weak and U(l) coupling constants will be equal (up to well defined numerical factors due to different normalization conventions). As a consequence, sin 9„ will no more be a free parameter. Such a theory (*) predicts the existence of exotic bosons. After breakdown of G to SU(3) 9 SU(2) 9 U(l) , the resultant exotic interactions are suppressed by the extremely high mass acquired by these bosons, so that the SU(3)§SU(2) vU(l) pattern "observed" at present energies (up to a few hundred GeV) is recovered. The relations between coupling constants implied by G are no more exact after breaking, but are still good approximations at very high energies. To be more precise, if the theory is renormalized at a scale very large compared to that where the breaking occurs, its symmetry (with all its consequences) is prac tically restored. Conversely, in order to allow for perturbative calculations, at present energies, the theory has to be renormalized using a substraction point at a scale u comparable with these energies. There the symmetry G is strongly broken, but che mass and coupling parameters at the scale u can be computed from what they are in the symmetric case by the so-called renormali- zation group equations (RGE). Since strong and electroweak interactions have very different strengths at present energies, and since couplings vary only logarithmically with che scale u , one finds a theory containing exotic bosons of fantastically high mass (of che order of 10 GeV). Such high masses are independently required in order co push che rate of proton decay (induced by che exotic interactions) below present experimental bounds (lifetime ~ 10 years). However, on* finds chat che predicted unification scale is smaller than che Plank mass scale 19 (10 GeV) where gravity can no more be neglected. In what follows, we assume chat chere are no other fermions than che presently known (but che cop quark c) fermions, grouped into "replicated" By "exotic", we mean new (with respect co che known y , gluons and weak bosons) and wich unfamiliar charges. Also, chey have characteristics of quarks and leptons at che same cime (leptoquarks). - 44 - families : (u,d,-j ,e) , (c.s.v ,u) , (t.b.v ,r) e ii T which in the simplest unification scheme (the only one to be considered here) are classified in equivalent multiplets. One thus gives up explaining the existence oi these families as well as their number. In this respect, the situation will not be better than in the SU(2) #U(1) model. In a maximal uni fication, one would like to be able to classify all particles in a single irreducible multiplet. We assume that all neutrinos are strictly massless and for definiteness consider the first family. There are 15 spinors of given chirality. 7 of them have chirality +1 (right handed in the zero mass limit), namely («~»ai»di}R The 8 other ones have chirality -1 ; they are (•" . v€ , i" 1,2,3 is the colour index. Since the gauge transformations are y inde pendent, fermions belonging to the same irreducible representation of the gauge group must have the same chirality. Rather than che above right handed fermions, we thus consider their 7 charge conjugates, which are left handed. In order co be more precise about our definitions concerning fermions and their representation in Oirac space, we recall a few properties of Dirac spinors. Let f be a fermion spinor. From follows fR - f I -j The charge conjugate of t is represented by che spinor C - T r - c(f)T where C is che charge conjugation nacrix. It is unitary and verifies - 45 - C Y C - -Y 4 Froa chese definicions, we obtain che following property of che charge coa- Q jugate (f_) of a right handed fermion : I -Y 1-Y C(f )T 5fT 5CfT One should be careful about the place of che index R,L with respecc Co che parenthesis : che conjugate of a light handed f«ration is Che corresponding Zz$£ handed antifermion (f ) . In the representation used throughout these notes for che y matrices, C is given by 0 -io„ •iff. We have C - C The following relation holds for two fermions f and f' ,C -C -,T ;T _ -z f'" ff • - f f • •»• ff f The second equality is due co che face that fermion fields ancicommute. For the same reason ,C u -C „ , u , f*w Y* f f YH f (5.1) and f,C t fC - f i £' (5.2) The last equation is valid up to a cocal derivative which does not contri- r 4 buce m che action j L(x) d x . When che fermions have group indices, we denote chera f. . If f. trans form according co a represencacion D - 46 - f. - D.. f. then its conjugate (f.) transforms according to D* . We denote it as f and thus have 1 J f - (fL) * (D^) f The place of the indices will then be very important in all subsequent manipulations. With these properties and conventions, our 15 basic left-handed fields transform according to the following SU(3)9SU(2) representations \ 'eL : (1,2) d : (3,2) V iL (1,1) (5.3) •L i (3,1) \ 4 •• (3,1) / The bracket (n.,n.) means chat the corresponding fermions belong Co the representation of dimension n. of SU(3) and to the representation of dimen sion n~ of SU(2). n means Che conjugate representation of n . (*)[20] If the unifying group is to be a simple compact Lie group , it must be (*) The definition of "compact", together wich useful properties of Lie groups which are noc all explained in Che cexc, can be found in Appendix A. The role of compactness for che commensurability of Che charges is under lined chere. - 47 - t) either one of the classical groups listed below Number of Name Definition Rank generators SU(n) n*n complex unimodular unitary matrices U n2-l n-i U1" » U"1 , dec U - I SO(n) n'n real unimodular orthogonal matrices • n(n-l) [n/2] 2 the integer T -1 a - a , det o - I part of n/2 US (2n) 2n*2n complex unitary symplectic matrices S n(2n+!) n P The following equivalences between some of these groups hold : S0(2) **# U(I) S0(3) »* SU(2) SO (4) ~ SU(2) x SU(2) (SO(4) is thus not simple) S0(6) ^# SU(4) •Li) or one of the 5 exceptional groups listed below Name •f Generators Rank G7 14 2 F 52 4 4 h 78 6 E 133 7 7 E 248 3 8 In order Co choose among chese candidates, let us first remark chat Che massless fermion free Lagrangian - 48 - l i • e*3e* • l (û). 3u£ • I Wjldj} l has global U(15) - SU(15)#U(1) symmetry. Hence provided one admits (it will always be the case in practice) that G is a symmetry separately of the free and of the interaction Lagrangians, 6 oust be a subgroup of SU(15), or SU(IS) itself. The choice is further strongly reduced by the following considerations. The 3 and 3 representations of SU(3) being not equivalent to each other, the basic 15-plet representation, which has the decomposition (1,2) + (3,2) • (1,1) + 2(3,1) under SU(3)«SU(2) (Eq. (5.3)), is complex (not equivalent to its conjugate). We recall that the only simple Lie groups which admit complex representations ar«H7] SU(N) , DM S0(4N+2) , N # 1 and E, o Next we remark that SU(3) 9 SU(2)0U(1) has rank 4 (4 commuting generators). G thus has at least rank 4 . From the above tables, the only admissible rank 4 group is SU (5) (*\ Let us now show that, irrespective of the rank (>4), SU(5) is in fact the only admissible group which provides us with a 15-dimensional represen tation where the 15 particle family can be arranged. From the above cables (*) If one allows G to be not simple, but a power of simple groups, all of them associated wich Che same coupling conscant, SU(3)9SU(3) is another rank 4 candidate. One SU(3) must be identified with SU(3) colour, so chat Che other one should be Che eleccroweak group. In such a case, Che charge operator Q is craceless in any representation. Since Che quarks (SU(3)C triplets) and che lepcons (SU(3)C singlets) muse be in separate representa tions of che second SU(3), one predicts that the sum of the quark charges should be zero, which is wrong. - 49 - and previous comments, we are left with U! SU(N) , 5 - the antisymmetric tensor, wich 3 » 3(S-l)/2 . D«I5 for N»6 , buc this represencation must be rejected because its reduction co SU(3) is not real. - che symmetric censor, with D « !f(N-H)/2 . D-15 for N-5 . S0(5) will be discussed at length below This represencation will be seen to be unacceptable. - the other irreducible representations all have too large distensions. Finally, the only reducible 15-diaensional representation is the sua of che fundaaental and antisymmetric tentor represencacions of SU(5). Ic will happen co be che correct one. {-U.} E, : che smallest represencation has 27 dimensions. [-ccc] S0(4N+2) , N>2 . The lowest dimension represencacions are : - the vector representation, with D»4N*2 . D cannot be 15, furthermore, this representation is real. 2M - the spinor representation, with 0«2 (>16) . Hence there is no 15- dimensional representation. Here one should nocice chat if neutrinos are massive, then che righc handed v has co fit into che considered representa tion of the unifying group, enlarging che family to 16 members. This fact makes che study of S0(10) very important, although it is not made here. Also of course one should underline the fundamental interest of all expe riments a bouc neutrino masses and oscillation phenomena. - 50 - 6. 9CSCRIPTI0H OF THE SU(S) HQOa^1 ] 'Jo have just sbown chat all eta* siaple ceapacc Lie troops ocHer than SC(S) coacaia ao suitable 15-diswnsional representation. Ic n—ini to be show ttaat aojoag ch* IS-diaensiooal representations (irreducible or act) of SC(S) Cher* is at lease one with the required properties. All SU GO represeatacioos caa be obcained fro* proper IT syeaetrized ceasorial powers of che fundaaeucal (H dimensions) represeatacioa. Lee * be elrs fiinii—ncal (or vector) representation, (*)" = ?* ics coaplex conjugate. V traosforas uoder SO (S) infinitesiaal transformations according co a Va * T»x -i 5a* ?3 (6.1) or in aatriz notations. v •* -a -a act u i - - *» « 5a* and !» •* J»" *i 1> 5a BY censorial produce of cwo vectors 1/ and which both transform according co M - M -i(<5oM • M&xTl . (6.2) The cwo conjugate representations are obtained by che similar produces of J and The cracelcss pare of J 9 ^a 3 .i 31M , a3 - y*' which cransforms according co U - M -L(;.xK - M<5a ) -y-itîafMl . (6.3) 51 - For completeness, we give Che lise of all SU(5) representations of dimension less or equal to 50, together with their Young tableaux (see appendix B) . 10 15 24 35 40 45 50 • a m EP p Their conjugate representations are 10 15 24 35 40 45 50 Note the identity of the 24 and ~2k tableaux, which reflects that the 24 is equivalent Co Che "24" : it is real. In order Co assign Che 15-family particles Co a representation of SU(5), we need Che SU(3) 9 SU(2) decomposition of Che representations of interest. The SU(3) and SU(2) subalgebras of SU(5) can be realized as A. - che 8 matrices where che X.'s (i-1,8) are Che 8 SU(3) 3*3 matrices. the 3 matrices where che T.'s ar«> che 2*2 Pauli matrices. l Accordingly, tb/? representation 5 can be decomp:.-'d into and which transtorr..- 'ike \3,1) and (1,2) respecciv. -, In che \ same way, (3\D + (1,1) - 52 - The 10 (resp. 15) representation is obtained from the antisymmetric (resp. symmetric) part of the product [(3,1) + (1,2)] 8 [(3,1) + (1,2)] (6.4) We have (3.1) § (3,1) - (1,1) + (6,1) (1.2) 8 (1,2) - (1,1) + (1,3) In the right hand side of these two equations, the first reps. (3,1) and (1,1) are symmetric, the other ones antisymmetric. Next we write the sum of the 2 crossed term (3,1) 0 (1,2), as the sum of the symmetric aad antisymmetric products, to be respectively attributed to the 15 and 10 representations. To summarize, we get the following decompositions : 5 - (3,1) + (1,2) (6.5) 5 - (3,1) + (1,2) 10 - (1,1) + (3,2)A + (3,1) (6.6) 15 - (1,3) + (3,2)s f (6,1) . (6.7) By computing the decomposition of 5 9 5, one gets for the adjoint representation 24 « (8,1) * (1,3) + (1,1) + (3,2) + (3,2) . (6.8) We see from Eq. (6.7) that the 15 has not the right SU(3) 8 SU(2) content (ic contains an SU(2) triplet and a colour sextet), whereas the reducible representation 5 + 10 is exactly suited to accomodate the 15 left-handed fer mions of Eq. (5.3), higher representations being too large. So, we are led to arrange v. and e. and the 3 anti-down quarks d^ into a 5 representation. That we put d and not u in 5 follows from Tr?Q« 0 since Q is a traceless generator in any representation (SU(5) is a simple Lie group). Turning the argument the other way around, we see that Q . » ^r- Q- , i.e. Che fact that quark and electron charges are commensurate (answer to question I) is required by the postulate of symmetry under the simple group SU(5). The 5 is thus the vector 53 - The 3 first components transform like the complex conjugate of the SU(3) fondamental representation. In writing the last two fermions, we have used the standard form I _) for the left handed spinor of the representation 2 of SU(2). Since 2 (which is equivalent to 2) appears in the 5 rep., we used 2 = i-c„ Put in a different way, if f. 6 rep.2, f «e ^f., where £ ^ is the antisymmetric tensor with two indices, transforms according to 2. By charge conjugation of the 5 representation, we obtain the 5 rep. which contains the right-handed d. quarks, positron and antineutrino, (6.9) so chat the charge operator Q in this representation is -1/3 -1/3 ^5- - 1/3 (6.10) It generates Che U (I) transformations. Let us explicitely construct the 10 as the antisymmetric product of two 5's, V and i>'. Denoting 4* as \ c•.1 (6.11) ID. where c stands for colour and JJ for weak isospin, the 10 I.. - -= ('Ji.*! - •!).•,•!) - 54 - is represented by C.uJ clVc2cl clC3"c3Ci -vi 'l'VVi c c'-c c- C,l*) c c u> -* c- 2 3 ^l 2 2 2 2 C^Ul c c «J-« c' /2 ^I 3 3 2 12 2 1 M is antisymmetric and only the elements above the diagonal are written. Using the fact that in SU(3), the 3 representation c transforms as E1^ c.c., — J which expresses that the 3 is the antisymmetric part of 3 9 3, we see (i) that the upper left 3x3 matrix of M is of the form 0 c3 -c2 0 c1 0 It is invariant under SU(2), and hence corresponds to SU(2) singlets. (ii) that,Q, being a multiple of unity in this subspace (Eq. (6.10)), all c^s have the same charge, which is twice the c. charge. Hence Q(cL) (6.12) More generally, if Q5g,(ij) is the charge of the element ij of the product + bein 595, Q5e5(i-j) "(Q^ii (Qgtii' ^5 8 given by (6.10). So, the three states c , i*l,2,3 are to be identified with the three left handed anti-u-quarks u , i-1,2,3. The lower right 2x2 matrix of M is SU(3) singlet. Its non-vanishing element, ut.uii-'jjyi)! is the antisymmetric product of 2 SU(2) doublets and thus is SU(2) singlet. Its charge is Q(e )+0(vc)-+l. Hence it has to be identified with the positron. Coming to che elements M.., i»1,2,3 and j*4,5, chey obviously transform - 33 - according co the representation 3 of SU(3) at fixed j, and the representation 2 of SU(2) at fixed i : they form the part (3,2) of the representation 10 of SU(5). + c Their charges are chose of the states (d.e ) and (d.v ) respectively for columns 4 and 5, that is 2/3 and -1/3. Hence Che 4 column is u.ç and Che 5 is d: . L T. In cerms of particles, we may thus finally write 0 u -u 1 -u3 0 u d2 2 1 '••' - it u -u 0 d. (6.13) -u. "ul "u2 "dl -d2 -d. In writing chis matrix we implicitely made some phase conventions (without any physical consequences) for the various states appearing. For example, we could have written -u., -d. instead of u.,d.. The l/r'T factor is also for convenience. Remark That ic is possible to arrange Che known quarks in Che fundamental • antisymmetric censor representations of SU(5) is not at all automatic. In order Co see chat, let us assume chat there are M colours, and that we are to build * c ' an SU(N +2) unified theory of SU(N ) 9 SU(2), where SU(2) is Che standard one. c c Then in the fundamental we set N quarks d , the e and the v , From Tr Q » 0 1 e we get Q.» - -s- 0 . In che representation generalizing Che expression (6.11) a sc of che antisymmetri e c censor, Che charges of che anci-up quarks would now be -2/N , whereas che up quarks appearing in che N +1 column would have charge 1-1/N. Me see chat only for Nc»3, Q ^ - -Q^ . By the way (N H ) • ^ only for N^ * 3. The 24 gauge fields, belonging Co che adjoint représentâtiou of SU(5), can be classified according Co che decomposition given in Eq. (6.3). We easily recognize che 3 gluons (rep. (8,1)), che 3 W's (rep.(1,3)) and che B boson (rep.(1,1)). In addicion Co these bosons of SU(3) 9 SU(2) 9 U(l), we next have 12 "exotic" gauge bosons. Six of chem, X., Y. (i-1,2,3), belong co Che (3,2) - 56 - representation, and their charge conjugatesX , Y to the (3,2) representation. These particles are coloured and thus presumably confined. Each of Che pairs (X. ,Y.) or (X^Y1) form an SU(2) doublet. The charge of any of the 24 gauge bosons is easily derived by building the 24 rep. from Che product 5 9 5, so that with Q, given by Eq. (6.10). We verify that the gluons (i,j«l,2,3) have no charge. The X., defined as the j*4, i»l,2,3 objects, have charge --T--1--4/3, and che Y. (j»5;i*I,2,3) have Q — -1/3. Of course the anti X,Y have the opposite charges. Anomalies We have seen (section 4) chat the 15 member family has no anomaly in the context of SU(3) 0 SU(2) 9 U(l), that is when Che external legs of the triangle diagram are gluons, W~ ,2 or y bosons. The absence of anomaly was guarantee by che relation 2 Tr[Q T3 ] « 0 , (6.15) where che trace runs over the 15 fermions. If we restrict che trace co che fermions belonging co che 5 or co the 10, we find 2 Tr5[QT3 l-iQe--i 2 rr10CQT3 I-3*^ *<|-}>-•! : che anomaly disappears by cancellation of anomalies in che 5 and che 10. It curns out chat chis is sufficient co deduce that the 5 +• 10 representation has no anomaly at all, as shown by application of property vi) of 0a c»Tr[Ta{T Tc}] abc (section 4) ; since D »0 when a,b,c correspond to W or y bosons, it vanishes for any set of indices. 7. RELATIONS BETWEEN THE SU(3), SU(2), U(l) COUPLING CONSTANTS Lee us explicitly build che generators in Che representation 5. As it was Che case for che charge operator Q, chis allows Co build then in any representation. The SU(3) 8 SU(2) 9 U(l) generators are very simple. The eight gluons are coupled Co Che SU(3) generators V*/2 11 •.. 18 (7.1) a where che À-. »' s are che usual Gell-Man matrices normalized according to Tr(X U« 2 3 . The W bosons are coupled to the SU(2) generators 1,2,3 (7.2) TV7 where che t ' s are che Pauli matrices. The B boson is coupled to che U(l) generator M/3 /3 Y (7.3) vf 1/2 where Y is che conventional hypercharge. These 12 generators are (orcho) normalized according co Tr, .(**) " î •M The 12 ocher heroician generators are constructed from Che requirement of orthonormalicy. They are similar to che Pauli matrices T./2 and T2/2 , which couple Co W. and W. in SU(2). We have line j / 0 J ! column 4 —-( — ail ocher elements vanish (7.4) ine 4 \ '-^-acolum n j 1 line j column 4 i f ° i" ^—{ all other elements vanish (7.5) XT \ i [ 0 I column J \ ^ 1 liine 4 - 58 - and similar matrices T • , T ? with line or coluan 4 being replaced by line v! v. "J J 12 1 " or coluan 5. X.* and Y.*~ are the corresponding 12 gauge bosons. The 24 SU(5) generators of Eqs.(7.2-5) are hcraitian and they are associated vith 24 real gauge fields. As stated in Section 2, one often uses complex coabina- tions of fields with definite electric charges, such as the W and W weak vector bosons 1 1 i * •• — (W1 • ilT) , (W ) ,1 coupled with -1" (T, t it,) Jl i —v according Co 1 2 W*T* • h.c W,t • ÎÏ2T* In che saae way the charged X's X. • -i- (x! - iX?) , Xj - (X.)T J ft J J i are coupled to che complex combinations of generacors j j (TY) - J- (T . +iT -) , (T_) - (TY )' (7.6) X ,'T X1 XZ X . X according Co T x! + T , X* - (T„)(T )J j X. + h.c x: J xX:T J * J J Wich similar notations one also has T . Y1. • T , Y? - (T_)j Y. • h.c i j We will use che following nocacion ,j\ (Tx>- (7.7) - 59 - 0 I Z,J (TY)- (7.8) where all the elements of Z:L (Zr,) are zero but the one in the line j, column 4(5) which is equal to 1//7 . It is sometimes convenient to use a matrix A which represents the 24 gauge bosons as a whole. By definition (see appendix A) 24 V T* (7.9) a-1 a The factor fl is chosen in such a way that Tr(AyAv) - 2 I AJA^ Tr(T*Tb) a,b U V - LI Aa Aa a Under a global infinitesimal SU(5) transformation defined by the 5*5 matrix 5a, A^ transforms as Au - Au -i[5a,AU] It has the following expression in terms of the gauge fields X.Z Y.z\ '\< T x+ il S B^^ • (7.10) (X.Z *Y.Z ) X Y *•! / where use has been made of the complex X and Y for convenience. In terms of the representations * and M , che free Lagrangian is Lfree " l > (V *\ + * > «*U> ' M We now define by convention, for k,Z • I,...,5 : - 60 - and 1*7, («)'klv so chat <1> and M exist as respectively a vector and a (antisymmetric) matrix, whose elements are conjugate Dirac spinors. Note the index transposition in the last equation, which is convenient in order to rewrite the free Lagrangian as Lfree » ii??i|) + iTrM3fM The couplings of the fermion to the gauge fields are the most easily deduced from this equation by replacing the derivative 1 by the covariant derivative 0 » i + igA (see Appendix À). In the 5 representation, one directly obtains a L - - g A^ ^T YU V • (7.11) Recalling chat M transforms like the product t|; 9 (p of two fundamentals one gets the following 5*5 matrix representation of AM (see Eq.(6.2)) : ^ M>kZ ' I < {[ T a a T - I A; (T M + M T \ 'It I Hence M 0 M generates the coupling term 24 _ - 2g [ AJ Tr(M Ta y M) . (7.12) a«l y The factor 2 cancels that which occurs from the I//2* in the definition of M in terms of the individual spinor fields. Eqs.(7.11) and (7.12) contain all che information about any particular coupling. In particular, the gluon, W and B couplings of Che SU(3) 9 SU(2) 9 U(l) model are recovered under Che form G Y q -* a quarkI s « uT •3»'J I VuifL"g/TBy ^ 't\\t (7<13) fermions fermions - 61 - By comparing the above expression with Eq.(2.8) we see that the SU(5) model implies g' * As a consequence, we get a prediction for the Weinberg angle defined by Eq.(2.21) cg 9 . Si . ^| (7.15) so that sin29 • | . (7.16) As a second consequence, the model also predicts that not only weak and electromagnetic interactions are governed by the same coupling constant g, but also that g is the strong interaction coupling constant 2 <* e . 2, 3 ,-, .,. — « -j - sm 9 - j . (7.17) s g We will show later on how these predictions, which are exact in the exact SU(5) symmetry limit, are modified in a predicted way at present energies where SU(5) is strongly broken. Before going on it is useful to notice that the predictions we have obtained in SU(5) for sinô and ct/a are valid in much more general situations. Indeed they only rely on the existence of a multiplet, irreducible or not, made of one or several families ({u,d,e,v } , {c,s,u,v } , ...) , and on the e U fact that the unifying group is a simple group. Let T, and Y/2 be the matrix representations, for the quoted multiplet, of the third component of the weak isospin and of the weak hypercharge (with their usual eigenvalues). In gene ral, they are not normalized in the standard way, that is one has 2 Y2 TrT^ f Tr-y The hypercharge generator T_ is suitably normalized if written o 4 Tr T2 with C • s- Tr Y - 62 - T and T are then coupled to the corresponding gauge fields with the same 3 o strength g . Thus it is quite general to have 7 /„'\2 ? 4 Tr A 2 tgH « (L) . c j2 2 5 in all cases where the same 15 family (or several of them) forms a represen tation of a simple group. From the same argument Tr 2)2 A \ ' TrQ2 " 8 is general under the same conditions. Apart from the SU(3) 9 SU(2) 9 U(l) interactions, which are now related with each other, there are new ones mediated by the X and Y gauge bosons. From Che general expressions of Eqs.(7.ll and 12), using the explicit contents of the 'p and M representations, and the matrices T^ , T„ of Eqs.(7.7 and 8), the new interaction terms are for the first family A i \ u j'uk,L/ x ljk - -S- Y? fvY <£ - 5 y t • e Û-Y d. T) * h.c . (7.18) The experimental implications (proton decay) of these exotic interactions will be discussed later on. Here we only remark that these new interactions violate baryon (S) and lepton (L) number conservation. This is an unavoidable consequence of any unification scheme where quarks and lepcons appear in Che same irreducible representations. As we will see however (Section 8d), there is a global U(l) invariance left which implies that B-L is conserved. - 63 - 8. HIGGS BOSONS, SU(5) BREAKING, MASSES SU(5) has Co be broken ; if it was not, we would not have cine Co discuss ic before all our procoos have decayed. This scans chat Che breaking of SU(S) down co SU(2) •SUf2^#U(1) oust occur at an extremely high energy scale, in such a way chat che gauge bosons responsible for che exotic interactions acquire aasses of che saae order of magnitude and lead Co essentially no observable effect (with the eventual exception of proton decay at an extremely low rate). After such a (first step) breaking, we cone back co che situation of che SU(3) »SU(2) §U(1) model, wich however the bonus of relations between che couplings. The latter symmetry gets itself broken down co SC(3) tU (I) cm. (second seep), like in che Salam-Weinberg aodel. This breaking occurs ac a considerably lower energy Chan che first one. In che SU(5) model, symmetry breaking is supposed Co be spontaneous, chrough che Higgs mechanism. We recall che aain lines of this scheae. The Higgs boson potential is supposed Co be minimum for a non zero vacuum expec tation value of some of che Higgs fields. Through their minimal couplings Co che gauge bosons, chey give some of these gauge field masses, and, chrough Yukawa couplings, chey also give feraion aasses. There are as many generacors broken by che non vanishing expectation values as there are real Higgs bosons which disappear, cransfering cheir degrees of freedom co che gauge bosons. If the symmetry of che Higgs potencial is not larger Chan che symmetry of che whole Lagrangian, the ocher Higgs bosons survive as massive physical particles. If this symmetry is larger, some of these Higgs bosons remain sassless (pseudo- [221 GoIdstone bosons ), ac che lowesc order in perturbation. As che two stages of breaking occur at very different energy scales, ic is convenient co treat Chen successively. a) The first stage of spontaneous symmetry breaking (SSB) SU(5) - SU(3) a SU(2) 9 U(l) Our first problem is to determine the nature of che Higgs aulciplet re quired. As one wanes co give masses co che 12 exocic bosons X and Y, one needs ac least 12 Higgs bosons. On an ocher hand, given a Higgs represencacion, there are only a few possible patterns of unbroken subgroups. The remaining - 64 - symmetries afcer SSI of SU(X) have been studied in the literature for the Higgs representations of low dimensions. The results are just listed below, illustrations being given in the forthcoming detailed discussions about SC(S). P Remitting symmetry after ! Higgs boson representation spontaneous breaking j Fund—ental D"M SU(S-l) I Ancisi rtric rank tensor SU(h-2) o-^ or S0(21*t) ttric 2 rank tensor SU(S-l) or SO(JI) Adjoint D • S"-1 SU(i) • SU(N-i) • 0(1) or SU(îl-l) «7(1) Tapit S.l SamttXtf patZenn a^ttx SSB by a, given faggà tcptejeitt&tûm. In tkt itcond column, l Hand* iox tkt integer pant oi S/2 . "OK" meou that ont ci tkt àeo poAbLbLLLtLu OK tkt otktx ii ckoitn accoxding to tkt value* oj tkt pOAamttvu in tkt Higg* potential. This table shows chat 24 Higgs fields in che adjoint representation offer che simplest possibility for obtaining the required first stage breaking of SU(5). As an illustration of how these results on SU(X) breaking are obtained, we examine che two cases of the fundamental and che adjoint Higgs representations, che former one being of interest for the second stage breaking of SU(5). i) Higgs fields in che fundamental representation of SU (3) The Higgs fields form a vector of dimension S, {h ; , which transforms (*) Ont can show ehac only che real rcprescncacions of SL'(5) may contain a singlet of SU(3) • SU(2)#U(I) , and chus have co be considered. The next real irreducible representations bevond che adjoint have dimensions 75 and 200 ! - 65 - according to Eq.(6.1) h - (1 -iôct) h The most general SU(N) invariant renormalizable Higgs potential can be written as V(h) - -u2 h\ + | 0»\)2 • (8.1) 2 U and X are real numbers. X must be > 0 in order for V to be bounded from below. The ground state is obtained for 3V gjp - 0 , at - I.... ,N a that is for 2 a a (-U +Xh hct) h » 0 (8.2) One solution of (8.2) is {h} * 0 , for which V»0 . This is the minimum for negative / . For positive y , the minimum of V is reached for the other solution 2 hX« - — (8-3) o oa x 4 of energy V " ~ 2T 2 Thus for u >0 , there is spontaneous symmetry breaking, {h } is a vecco of fixed length and of arbitrary orientation in the representation spac . One can always transform the h fields into a new vector (which we relabel h ), by a global SU(N) transformation, in such a way that the new h points along the N coordinate axis. Then / 0 h o JF7\, This vector is invariant under the SU(N-l) subgroup acting upon the (N-l) first coordinates. This is the result announced in the above table. There 2 2 are (N -1)- ((N-l) - 1) • 2N-1 broken generators. According to che general - 66 - theorem, 2N-1 out of the 2N real Higgs bosons of {h} disappear, so that there is only one real physical Higgs particle left. ii) Higgs fields in che adjoint representation of SU(N) They can be represented as a second rank traceless tensor H. , which transforms under SU(N) according to Eq.(6.3) H * H - i[<5o , H] The most general invariant potential is (up to a cubic term, which is absent if we impose the symmetry M •* -H) u2 2 *! 2 2 X2 4 V - -^-TrHS-i- (Tr^) • — Tr H . (8.4) ^4 4 2 There is SSB if u is positive. We want Co find out the possible ground [23] states for this potential . Due to Che SU(N) invariance of V, che various possible ground states differ from each other by SU(N) transformations. We may thus look for the minimum of V after a transformation of the fields which diagonalize the (hermician) matrix H . The new fields (relabelled tf) form che matrix i l a. I a. i-l l so chat che potential reads We first look for a minimum of V ac £a. " R* fixed. One has Co minimize \_ /, a. ac fixed R and for £ a. • 0 . Using 2 Lagrange multipliers p. and o. for these constraints, the a.'s must verify Vi + 2P2ai + Pl " ° So, all the a.'s are solutions of che same 3* degree equation, and chus cake ac most 3 different values. If they are all equal, chey all vanish due - 67 - Co [a.»0 (V=0). We now specialize co SU(5) and enumerate all possible configurations with at least 2 different values for Che 5 a.'s . (A) 2 different values. The solution is (1) either -^ (1,1,1,1,-4) /20 (2) or y^R (-1,-1,-1,2,2; (B) 3 different values. Their sum is 0 as there is no second degree term in Che 3 degree equation Chey verify. One finds (3) either -7: (0,0,0,1,-I) /2 (4) or -y (1,1,-1,-1,0) For these 4 solutions, the values of V are of the form U2 2 R4 Vn * "TR •T /13 7 I l\ * TX. • with s » i-TT , -TQ , •=• , -rj respectively. There is a minimum with respect Co R for X.+A.s > 0 (otherwise V is not bounded from below). The minimum, l L n obtained for •» 2 R2 - U Wn 4 is vn 4(X.=y + _X- S ) i i. n For a given point in Che X.,X2 plane, Che absoluce minimum is Che smallest of che 4 values V n For X. <0 , che minimum solution is solution (1). H - a I ' 1 ] , a2 - -^ o o| J o 20Xj • 13X2 -4/ - 68 - For \ >0 , the minimum solution is solution (2). -I -1 -1 (8.6) 3/2 A 3/2 2 l 4*2 In order to determine the remaining symmetry, we remember that under a transformation of SU(5) represented by 5a, H transforms according to H * H - i[5a , H 1 o o L oJ so that W is invariant for all 5a's such char o [6a , HQ] -0 For A >0 and 30Aj + 7A > 0 , 5a is any matrix of SU(3) «SU(2) %U(l) , while for X <0 and 20A + 13A > 0 , 5a is any matrix of SU(4) »U(1) . This property illustrates the last result of Table 8.1. The allowed regions in che X.,A? plane and che remaining symmetry pattern after SSB are sketched in Fig.8.1, SU(3)« SU(2) 0 U(1) SU(A)« U(1) Fcg.l. ? - SU(5) bwa.ki.nQ by Higg* baton* in the adjoint xepxeientaJtion. In the plane o$ the two coupLLnq* \x and X2 oi the Hiqa* potential (Eq.[8.5)1, the. \on.bidden domcUn (unbounded potential] -ci ihaded. In the. allowed domain, the. xegiom tbJtiij axe. pfiuexved axe hepaxated by \2 «o. - 69 - From now on, we assume that we are indeed in the situation where X. is positive and larger than -30A./7 . This situation is stable against radiative [24] corrections to the Higgs potential In the sane way as the gauge fields nay be considered as members of Che algebra, and represented in the representation 5 by Eq.(7.9), we write Che Higgs fields in the adjoint representation as the 5X5 matrix, 24 H - /i y H Ta a-l a which satisfies 24 Tr(H H) - I H,2 a-l a Using the explicit form of the T 's given in Eqs.(7.1,2,3,7,8), we write VW*! (8.7) «vwv VT In this equation, all H fields have vanishing vacuum expectation values, che constant /ITa T_ being separated out. The Higgs bosons have 0 D che same charge as che corresponding gauge bosons, which also belong to the adjoint representation : a 3 (i) che 8 H 's (coloured and chus presumably confined), Hy and H_ have zero charge. <«•> - £ (»i - ,$ - (H"/ has charge +1. (iii) che 3 complex H„ and che 3 complex H^. (colour criplecs) have charge -4/3 and -1/3 respeccively. The gauge fields which are coupled co chose Higgs fields which have non vanishing expeccacion values acquire masses. The Higgs-gauge couplings are obtained from che tern y i Tr [(DyH)(D H)] - 70 - of Che Higgs Lagrangian, where D is the covariant derivative DWH » S'^H • ig AJ[ [T*,H] Ue have used hemician fields so chac D is hermician. The gauge boson mass term is idencified as P a b Ji^.A . - -VA A Tr([TL ,H ]J L[l ,H J]) 2 a ab y,b 2 a ykb \ * o ' o / wich H given by Eq.(8.6). The mass matrix thus reads 2 2 a 1 rf1 h. - g Tr ([T ,H ] [^.HJ ") . (8.3) a,b o The 12 generators associated wich the gluons, the W's and the B obviously coanute with H . These gauge bosons chus remain massless, as they must do. The masses of the ocher 12 X and Y bosons are easily computed from Eq.(8.8) and Che explicic representations of the corresponding generators (Eqs.(7.4) and (7.5)). One finds 2 2 25 2 2 ,a Q, °X * aY " TS ao • (8*9) The reason why all these aiasses are equal is Chac, ac chis stage, SU(3)9SU(2) is exact. The 3 H„ and che 3 IL remain massless in che Landau gauge. They can be gauged away (unicary gauge - see che beginning of Section 4), and cheir 12 degrees of freedom are transferred Co che 12 gauge bosons which become mas sive. The 12 physical Higga bosons H_ , IL. , H_ acquire masses, which we com pute by caking che pare of che Higgs potential (8.4) quadratic in these fields, using tha representation (8.7) of H . One finds : 2 5 , 2 \ ' 1*2 *o ml » 5L al (8.10) Hw 2 o •L - (,5viVo 3 - 71 - The 12 physical Higgs bosons have extremely high masses of the same order as the X and Y masses also proportional to a (see Section 9 for orders o magnitude). The choice \. very small seems unnatural, and anyway such an artificial adjustment would be destroyed by higher loop corrections to the Higgs potential. b) Second stage of breaking SU(3) 0 SU(2) 0 U(l) -» SU(3) 0 Ugm (l) The second stage of SSB breaks SU(2) 0U(1) down to U (1) , and also e.m. gives masses to the fermions, which are still mzssless after the first stage exactly as in the unbroken Salam-Weinberg EO«" '. By the way, it happens that there is no Yukawa coupling at all between L i fermions of the S and 10 and the Higgs fields in the adjoint representation (see below), so that not only H , but all Higgs bosons of the adjoint decouple from fermions. We recall that the form of a Yukawa coupling of fermions to a generic Higgs field $ is fRfL*+h.c The problem is to determine which Higgs representation can b' involved. The 5 multiplet '\> is ip • {dj^ , e t -v }R and Che conjugate \pc - {d , e , -v} The 10 multiplet is l + M - {u ,uitdit« }L and its conjugate is • iC r i ,i -, M - [u^ , u , d , e }R 72 - Terns like f_f. are thus to be found in (*) •it 9 .U or M 0 ,Vi We recall that the charge conjugate of a spinor f is defined as c - T f - C(f)1 so that ,MC - (C(M)T) o o -M Yo Yo. C - MTC* He have used the fact that in any gasaa matrix representation Y *y and T o o C « -C . The only Higgs representations giving rise to fermion mass terms are Chen Co be found in (i) 5 • 10 - 5 • 45 whose Young tableaux (see Appendix) are • (8.11) :*B + (ii) or 10 § 10 - 5 • 45 * 50 with Young tableaux B«B •F- (8.12) We firsc see chat the adjoint is absent, the candidaCe Higgs representations being 5, 45 and 50. Moreover the representation must contain at least the (*) — — ~c 'u*'l) , ,V|*M and \', • ]/ Yukawa couplings are forbidden by chirality. c v 9 '•j can couple to reps 10 and 15. In the latter case Che neutrino could acquire a Majorana mass. - 73 - (1,2) represencacion of SU(3)»SU(2) in order Co give SU(3) invarianc masses Co che fermions. This leads Co rejecC che SO represencacion uhich decomposes under SU(3)»SU(2) according Co 50 » (8,2) +(6,3)+ (o",l) + (3,2; +(3,1) + (1,1) The 5 and 45 represencacions have Che following SU (3) • SU(2) decomposicions 5 - (3,1) + (1,2) 45 - (8,2) +(6,1) + (7,3) + (3,2)+ (3,1) + (3,1) + (1,2) . The simplesc choice is Co assume che exiscence of one mulciplec {h} of Higgs bosons in che fundamencal represencacion. The Higgs pocencial is minimum for a non vanishing value u> of che Higgs fields {h}, chac is (see § 8.a)i)) < [ hn >» a» . The value of ui a-l a vill be far smaller Chan chac of a characterizing che SU(5) + SU(3)»SU(2) »U(1) breaking, in order for che first stage breaking Co be far scronger Chan che second one. As ve have seen in Seccion 3, che order of magnitude of u> is 200 GeV, whereas, as will be shown in Section 9, a is of order 10 14 GeV. The complete Higgs pocencial wich boch che adjoint H and the fundamencal {h} represencacions present, and coupled to each other, is (assuming again h * -h and H — -H symmetries) : 2 \ 'X V - -H- Tr H2 + — (lr H2) + —• Tr H4 - m2 hfh + ^ (hrh)2 + 2 (tfh) TrH2 +| hVh . (8.13) Suppose first a * 6 * 0 . The situation has been studied in pare a) of this 2 seccion. For posicive o , che minimum is obtained for 2 m2 -1 | . (8.14) 3/2 Al 4 A2 " "#VI 3/2) \ It is clear that in order for SD(3) symmetry to be preserved by the 2 stage breaking, the vector h has to point in a direction contained in the plane of the two last coordinates. Hence h has to be h - 0 . (8.15) o ' • Once it is of this form, an SU(2) transformation which leaves H unchanged, o allows to write it hQ - | 0 | , (8.16) fm2 u> •• so chat che SU(2)9U(1) SSB generated by {h} is identical to that of the Salam-Weinberg model, as described in Section 3. Therefore, the crucial point is co insure that the minimum of the complete potential V of Eq.(8.13) is actually obtained for h in the form (8.15), when the couplings ct and S of the h and H fields are non zero. Rzmank One should notice that even in che case o • S • 0 , h and H get coupled by radiative corrections (e.g. gauge boson exchanges). In the absence of such couplings, the part of V (Eq.(8.13)) which contains only {h} has SU(5) symmetry, a larger symmetry than chac of che rest of the Lagrangian after che first stage breaking. As stated at the beginning of the present section, we might have to face a situation where pseudo-Goldstone bosons (massless physical particles not associated wich a broken symmetry) remain in the [221 spectrum . This is easy to understand in our particular case. Suppose h has che form (8.15) in che absence of h-tf couplings. Among che 4 real fields contained in h, and h. , 3 fields are "eaten" by the 3 gauge bosons W - 75 - and one field remains as a massive Higgs boson, as in the Salam-Weinberg model. But the 6 real fields of h. , h. , h. remain massless. There is no mystery about what happens. In fact, in the absence of coupling, the generic situation for h after the first stage breaking is not (8.15), but rather, 0 \ say I 113 , since the h fields know nothing about which axis among the 5 0 / SU(5) axis are the 3 SU(3) axis and the 2 SU(2) axis. In this case, both SU(3) and SU(2) are separately broken. 5 real Higgs fields among the 6 of h. , h. , h. disappear, and I remains as a massive particle, while as usual 3 disappear among the 4 of h, , h, , the last one being massive. So, in a generic, uncoupled situation, one in fact does not expect pseudo-Goldstone bosons to be present. If the couplings are turned on (by radiative correc tions or explicitly by non zero values of a and S), Che h fields which do + 2 not disappear anyway get large masses ( - a ) due to Che terms h h H and hf H2 h . o A complete treatment of the coupled potential (8.13) can be found in Ref.[25]. There it is shown that for 8 positive, the couplings of {h} to H actually provide ones with the orientation required for {h } (Eq.(8.15)). The desired breaking SU(5)-&- SU(3) • SU(2) «U(D -^ SU(3) *U (1) is (*) e*m* thus obtained in a natural way . The minimum of the potential is finally realized for the following Higgs configuration h o (8.17) H o-o a I -1 3, f(l*e) f(l-s)i Here "natural" means that Che desired breaking does not require any fine tuning of the parameters, so that it survives small variations and presumably to higher order effects. In our case, only inequalities on Che Higgs potential couplings have to be fulfilled. What will be seen Co be unnatural is Che value of m/ff (hierarchy problem, see Section 9). - 76 - The terms proportional to e in H transform like an adjoint representation of SU(2). They reflect the SU(2) breaking induced by h .As shown in Section 3 (value of p » G„/G_, in excellent agreement with a breaking by an SU(2) Higgs doublet), the weak interaction phenomenology requires a e to be much smaller than Fermion masses We come to che predictions of SU(S) for the fermion masses, generated by the second stage SSB. The most general Yukawa couplings of the Higgs bosons in the 5 rep. to the fermions of the (5 + 10) rep. take the form (Eqs.(8.11,12)) : LHi8gs(5) „ Cha-B l2ea*Yonh ,T CV • „.c . (8.18) aô 4 at SY on In order co understand che structure of che second term (Che normalization is a matter of convenience), one remarks chat 5 can be obcained as che totally ancisynmecric produce of four reps.5 (i.e. of two reps.10). So, e being che fully antisymmetric tensor with 5 indices, the second term is of the form \ h *>* , che singlet piece of 5 9 5 . Since only hj has a non vanishing vacuum expectation value u> , che first term is readily seen to yield the following mass term : M. - Gw ~JS M + h.c I 30 Giii d +h C (8 19) .J [\ '4iL * < \) ' ' - 77 - The coupling constant G «ay be complex, G* i;G! - eifO . Any fcrmiou f appears in M under the fora —iip so that each chiral ferra ion tield f , f may be redefined, f -• e f . L R L L or f •*• e f in such a way that M. reads w _ JGJM ( r Ti, ^ • •\ /o -,n\ M. • - -—— ! i i d. • e e ) . (8.20) /2~ ^ i ' The second mass tern is 5SY5n T M2 « - i-Gu,c [M3Y] C* Jl5n • h.c . (8.21) The right hand side of this equation is obviously symmetric under 3-—»y and 5*-n . Ue remark that it is also symmetric under che exchange of the two pairs (8y) and (an). So is the £ tensor, and moreover we have : T T T ( C) C*MM - [(.( C*.UC U ] VM Vr * 5n5n » l 'V*V * 5nSn] " - • Ue have made use successively of the facts that che l.h.s., a scalar in Oirac space, is invariant under transposition, that two femion fields X anticotnnute and chat in any gamma matrix representation C »-C . Hence the four terms in (8.21) where one index among 3,7,5 i is equal to 4 are equal. Ue nay thus write m ijf1 c m ^r M2 » -ïse ^ (M^) C M^ • h.c where i,j,k- 1,2,3 . Using the explicit particle concent of the 10, namely Mk4 - u^/,7 and \ ^ M.. - £.. -i- • £.. C —- ij iJm /j ljm -n C C • 1 , we obtain „ _ G.jj ijk — m . M: ' '-•- Hj.U R ukL * h'c Gwu_ u , *• h.c - 78 - By phase transformations of the fields w. and u_ . node in order to absorb chc phase of C . as above for M. , we finally fet - !C! a ) u u • ' - a This neans term S, concerns only u-quarks and thus leads to no prediction at all since G is arbitrary. On the contrary, th* equation (8.20) predicts chat che down quark and the electron have the san* aass. This degeneracy is a consequence of the SU(4) invariance of {h }. It is however broken by radia- o civc corrections (sec Section 9), the «narks being subject to stronger inter actions Chan cbc electrons, below the scale a where SU(5) is broken. o In what follows, we outline what happens if one aultiplet of Higgs in the 45 (instead of the 5) is used. This 45 can be constructed froa the product 5 •TO" * 45 • 5 . Since 75 » [5 • 5] . , the 45 is represented iy antisyau by a chird rank censor T* , antisyanetric with respect to the upper indices. Furthermore /_ T , which would clearly represent the 5 , has tc vanish. l * Let us now assuaw that SSB occurs (with the 45) in such a way chat SU(3) #SU(2) »0(1) is broken down co SU(3) «U (1) . The Higgs fields which gee non vanishing vacuum expectation values aust have zero charge. The only zero charge coaponents of T are : T. - -T7k (i,k - 1,2,3) and T, » -T as one verifies fron che general relation Q(TxV) " (V» *( VSB *(< VYY ' •vith .}- being che charge matrix in che representation 5 'Sq.f6.I0)). The Higgs expectation values oust be SU(3) syaaetric, so chat we set : -T^ - 5* - - and che trace condition on T finally imposes Since che <*5 represencation appears both in 5 • 10 and 10*10 , there are - 79 - again two possible Yukawa couplings, namely SY G -jF Mfl T + h.c and 0 T G e *^ (M fl) C*M T^ + h.c ctB py on Substituting the expectation values for T, one obtains from the first cerm 2Gu(-3 7f H + I J1 M ) + h.c » •? Gu>(-3 e* e* • J d£ d^) • h.c The second term yields L:ik T a6Yk5 T -6Gu» £ GM..) CM.. + 2GOJ £ (M _) CM. + h.c - 0 IJ 4k a6 ky ' as one verifies by manipulations similar to those made for the breaking by rep.5 . After suitable phase transformations on the fields, the mass term becomes M - -»T |G| id (3 e* e+ • T d1 dj . (8.23) The prediction of SU(5) for a rep.45 (alone) of Higgs bosons is thus that the electron should have a mass three times larger than the down quark mass, while the up quark should be massless. Such a situation before mass renor mal izat ion does not appear as very appealing. Of course one may mix both r»ps.5 and 45 of Higgs bosons, and play with the parameters, but very little of .:e predictions is left. c) Generation mixing So far we have considered the case of o«ly one family. In the SU(5) model, nature is supposed to contain at least 3 such families. Let {>];'} and a M' be the corresponding representations 5 and 10 where the subscript a dis tinguishes between various generations. The prime on •]) and ,U means chat, in general, y and M' are linear combinations of the li and M containing the a a mass eigenstates, as we shall see now. In Che presence of several families, the most general f ermion Lagrsngian is - 80 - L - iV 0 V + iTrOi' t> M')+ hi1 G . ,«• + h M' TC G , f + h.c.,(8.24) aa aa a ab b a ab 5 where sunnation over the generation index is understood. The term involving the covariant derivative is by definition proportional to unity in genera tion space. The Higgs part, where all SU(5) indices have been omitted in the couplings to the Higgs bosons {h}, contains matrix couplings (in genera tion space) G . , G , , which generalize the couplings 6 and G of the pre vious section. They have no reason to be diagonal, and are in general arbi trary complex matrices. When h is replaced by its vacuum expectation value, one obtains two fermion mass matrices, and as announced, the fermions of <|/' and M' are not mass eigenstates. Explicitly, the mass matrices are G.u / . —-, .N , G* . +G. . ab l-Tri ,, + + \ —r1 k , ab ba _,_ . ,„ «> L * Id' d!. -»• e e. I - u u' u • h.c (8.25) mass nr \ a lb. a b^/ a kb^ - u' ,d' , e' here stand for charge 2/3, -1/3 quarks and charged lepton respec- a a a tively. They are linear combinations (to be determined from G and G) of, resp., the (u,c,t,...), (d,s,b,...) quarks and (e,y,T,...) leptons, which are the physical mass eigenstates fermions. The two non diagonal mass ma trices in (8.25) are : Mab - Gab % (8'26) (8 27) and Mab » (Gab*Gba)f ' We have to show that they can be diagonalized through global unitary trans formations acting only on the generation indices. Being global, they do not affect the "kinetic" terms f 0f . Since the two matrices in (8.25) con cern d*3 and e's on one hand, u's on the other, we first diagonalize M, and then diagonalize M . We first recall that any regular matrix A can be put into diagonal form a by a bi-unitary transformation (V,U) V A u" - a (8.28) a. is furthermore real and positive definite. The proof is as follows. The matrix A A , being heriuitian, can be diagonalized by a unitary matrix U : - 81 - •"• t 2 L'A A'J - a. where a. is defined as Che square root of a whose all elements are positive. A being regular, det(a) = |det A| + 0 , and a exists. We define a second -1 unitary matrix V«a UA * and Eq. (8.28) follows. LeC now (V,U) be che pair of matrices in the generation space which diagonalizes M of Eq.(8.26). The new fields ty" - v H>' (8.29) if - 0 M' are such chat d" and e are mass eigenstates. We drop Che double prime for them and write n m m m m d e A a a The new u-quarks, u" are still unphysical since (V,U) in general does noc diagonalize M of Eq.(8.27). From (8.29) and (8.24), the mass matrix between u" states is U* M if and we diagonalize it by a new bi-unitary transformation (V,U) which aces on charge 2/3 quarks only, not to destroy what is already achieved. The new u-cype quarks are mass eigenstates, with masses m In order to construct explicitly Che physical u-quarks, we remark chac the symmetrymmetry of M . (Eq.(8(Eq.(8.27). ) allows a simplificacion. Ic implies chat >5 also is symmetric, so chac :J M - sT N* In terms of che diagonal form ,'f of .V , which by definition verifies N * V" M U , this equality means ~ ~T ~2 ~2 ~ T V U M - ,VI V U — - 82 - ~ ~T 2 It shows that the matrix S • V U , which commutes with the diagonal M , the elements of which are by assumption all different (no degeneracy in flavour space), is also diagonal. Since it is unitary, e 5 ab ab We then replace V by S U , and get for the mass matrix between u-type quarks URMu{ ^»»l û" (U1 S7) ÏÏ 0 u" which shows that the left handed and right handed mass eigenstates are "kL ÏÏU*L "kR sïï*ukK Here we have restored the colour index k. Finally, since all neutrinos are massless (in the model), any linear combination of neutrino fields is a mass eigenstate. It is a matter of convention to set v a * v"a According to their definition in terms of the original fields, the physical fermions fall in either of the three classes (d,e,v) or u._ or *kL Since the interactions with the gauge bosons were written in terms of the original fields (with primes), these interactions which involve differenc class physical fermions contain Cabbibo type mixing angles and (or) phases. Let us make this point more explicit. The fermion-gauge boson interactions, in terms of the original fields, are (Eqs.(7.13,18) : a g G Y + f* YY - f ~ < a £. ^' u-T«A ' ** I L U 2 L quarks fermions fermions • i 1 ijk -,. r- * t (u') . y J u "it.) Y? +' ijk ,-,, l e - -: (a'). , •f h.c ', <[ L •j kL - 83 - A sum over all generations is understood everywhere. The first transforma tion (V,U) which acts over all types of fermions leads to the same inter actions in terms of the d, e and u" fields. But the second one, defined by Ô7.ÏÏ), or equivalently (SU* , ÏÏ) , distinguishes the u" from the d and e fields, and Che ul' from the ul' . As a consequence the interactions with the W~ and X,Y gauge bosons, which involve ul' and \i!l linearly, are modified. Recalling that d»d" and u • Uul' are physical particles, we obtain for the charged weak current : + U + -i- W Zl y dT' + h.c - — W Z? v^d" + h.c - — W+ •! y^U d_ + h.c . (8.30) n u ^ L The unitary matrix U is the usual generalized Cabibbo mixing matrix. The charged weak current couples che charge 2/3 physical quarks (u,c,t,...)T to che linear (charge -1/3) quark combinations tf(d,s,b,—) . U depends 2 T on F real parameters if there are F families. However, the 2F quark fields have 2F-1 arbitrary relative phases which can be used to absorb 2F-1 phases of the U matrix elements in che quark field definitions. Hence che number of physically relevanc parameters in U is F~-(2F-1) • (F-l) . It is 1 for F-2 (the Cabibbo angle), 4 for F » 3 (3 Cabibbo-like angles and 1 phase, [•77] responsible for CP violations in weak decays ). In performing the relevant unitary transformations in che piece of the interaction Lagrangian which violates barycn number conservation, one has co be careful about che notations. Consider for example the quantity (u'). y u/, which couples to X^ . From our conventions (see che beginning of Section 5), (u1). • u,J is the Dirac conjugate of the spinor u,J • (u!) c c — Since for any fermion (f ) «(f) , the current (u'). y u/ can be written HI. J u &*i u,IVk • ("'\ \ "Û. ;\ C (U' ) r U,'f jR •; kL As a consequence, when we transform che u'-fields into che physical u-fields, we have Co cake care of che different transformations occurring for che lefc - 84 - and right u" UL " U UZ " U'U "L but u^ - UT u£ - UT UT S~ so that c T T c (u^) - tr ïï s (uRy and (u^)C » u£ S* ÏÏ U From these identities, we find : «•'^^"kL " SV'V The complete interaction with the X,Y bosons finally reads in terms of the physical fermion fields : X? -g__ j^dVy . +£-> (u)jY;jS uj -«•^ (^«"i-^VV^ (S)JV'KL) +h'C (8'3I) Recall chac in this formula, the symbols u,d,e,v are generic for respecti vely charge 2/3 ,-1/3 quarks, charged leptons and neutrinos. We see that in addition to the ÏÏ matrix which already appeared in the weak charged cur rents, we have a new matrix S in the currents. In the present case, as we have shown, S is diagonal and its phases are not observable at the tree diagram level. Since furthermore, we know from experiment chat ïï is closed to unity (small Cabibbo angles and C? violating phase), the interaction described by (8.31) is nearly identical to the original one (7.18). This simplicity disappears for more complicated Higgs systems : 5 is no more diagonal because the original mass matrix M is no more symmetric. Even with U - 1 , (8.31) is no more diagonal in generation space. Let us make a final remark : we have stated chat che first family was composed of - one 5 concaining che physical dT , e. and \> ; Lu 6 - 85 - - one 10 containing the physical d^ , e , a combination of the various left handed charge 2/3 quarks (u,c,t(u,c,t,...), T and another combination of the corresponding left handed antiquarks. — c ™ In the same way, the second family is composed of a 5 made of sT , uT and M , and of a 10 with s, , u and 2 combinations of respectively quarks and antiquarks, and so on. Concerning these statements, the following comments are in order. (i) As already said above, any linear combination of massless v's is a mass eigenstate, which allows ones Co define the neutrino states in such a way chat weak interactions induce only v •* e , v? * u , transitions. e |J The pairing (e,v ) , (U,\J ) , ... is thus only a labelling convention for / ue+vu\ ue~vu\ Che v's . The pairing le ,—sr*1) and (lu , ———I , say, involves exactly the same physics although in a rather awkward way. (ii) The mixing between the u's is severely constrained by existing data on weak interactions. For example the matrix U restricted to the two first families is nearly unitary (success of the original Cabibbo model with only one mixing angle •? ). So, Che other mixing angles contained in U must be small. (iii) In principle, one may associate che pair (e,v ) wi'h any quark family. Such a choice would lead Co Che same present energy phenomenology except for Che fermion masses . But this unusual association would however clear ly show up in X and Y interactions. With Che choice (e,s) , (u,d) for example, + o • +o che proton decays p — u TT and p - e K rather than p -» e T and + , p -» 'j K become Cabibbo allowed. (iv) There is however no possibility, in the simplest SU(5) model for, say, che first multiplet co be made of che down quarks and of some linear combina tion of che physical lepcons. In Section 9 below, it is shown chat afcer continjation down to present energies, the (asymptotic) prediction mr-m. compares veil wich experiment (under the assuraotion of 3 families only). But the other mass predictions, in particular m^»^ , have problems. Ic is thus not impossible, chough a little bit bizarre, to make the associations (e,s) and (u,d). - 36 - d) 3-L invariance As we have seen (Eq.(8.31)), the X and Y gauge bosons connect quarks and leptons and thus the baryon and lepton numbers 8 and L are not conser ved. However, B-L is conserved in the simplest SU(5) model as we shall see right now. The unbroken theory exhibits an additional global abelian inva- —iBx riance which we call U (I) » e . U commutes with the SU(5) transfor- x x nations. The gauge bosons and the Higgs bosons in the 24 are real (the 24 is a real representation) so they must be U singlets. Let x. r and x.n — be the eigenvalues of X for the fermions of the 3,3 and 10,173 respectively, x„ that of the Higgs bosons in the 3. The inva riance of àc D ii. and '.$/,„ D 'Jjtn requires x »-x_ and x, „"-x . - u 3 10 u 10 35 10 JQ In order for the couplings of the fermions to Che Higgs bosons of the 5, namely for terms of the form (Eq.(8.18)) : •5*5 M1Q and h5M^0CM10 co be invariant, one has -x. - x. • xt- » x_ • 2x._ • 0 0 3 10 3 10 Hence setting arbitrarily x10 one obcains x. - 3 (8.32) x.h » -2. 3 The breaking of S'J(5) by Che adjoint Higgs representation preserves the U (I) symmetry because chese Higgs are U invariant. This is not so for che second scage breaking since Che non vanishing value J of h is shifted -i'3x, * by e 3 under a U transformation. Now che hypercharge Y of SU(5) also gets broken at che second stage. Using che explicic form of Y in a rep.5, - 87 - (8.33) and x- * -2 (Eq.(8.32)), we see that, for the Higgs bosons in the rep.5, (X+2Y) has the eigenvalues -10/3 h I ",0/3 (Xj*2Y) - -10/3 Thus X+2Y remains unbroken when the h fields acquire a non vanishing expec tation value in the last two directions * . Lee us now compute X+2Y for the fermions. Calling B and L the usual baryon and lepton numbers, we obtain m. and (8.34) -1 - -L _ L,J For the fermions in the representation 10, we get fX+2Y\ I {—) c " " 3 - B c m . - \ • »..« VL m e e One may wonder where is the Goidston* boson associated with the spon taneous breaking of the global U symmetry. In fact, as X+2Y is unbroken, there is only one. Goldstone boson, associated with X-2Y, say (and not two associated with X and Y), and chis boson is "eaten" by the 3 boson, which is massive. This miraculous escape from the Goldstone theorem is known as che 't Hooft mechanism. - 88 - X+2Y /x+2Y\ 2 Hence B-L » —=3 — is conserved. Moreover, since xadj ,.. «0 , 5—=—\ 5 J/ ..«-r- a Yad j adj. from which we learn chat (B-L)„, , - (B-L)- - (B-L) giuonlons y B whereas (B L) (B L) (8 36) ~ X,Y " ! " " " Y* T. ' - X ,Y The same is true for the corresponding H_ Higgs bosons. Of course the non A, X vanishing of (B-L)T v is connnecced co the fact that the X,Y gauge bosons (as well as the Higgs H_ „) couple on one hand to 2 quarks (B-2/3 , L • 0) and on the other to one antilepcon (B»0 , L«-l) and one anciquark (B * -1/3 , L«0) . For example che transition "L T UR * Y * \ * dR is allowed by the Lagrangian of Eq.(8.31) and indeed leads Co che possibi lity of proton decay. Remote B-L is not conserved for any Higgs system. The addition of a rep.45 or 50 is innocent in chis respect, buc suppose instead chat we have an addicional rep.15 of Higgs bosons. The U invariancs of che Yukawa coupling -c h x j, j. HI. requires X.,*6 . Assume now chat che zero charge, colour $m- glec, SU(2) criplec component of H., , (H.,).. , acquires a non vanishing exoeccacion value. This has several consequences : 2 •w l) che relation , "•. - 1 is violated ; cos~-* tn_ ii) che neutrino acquires a Majorîna mass from C a8 C 55 <(5 \*3(HI5) > - \ (vL) <(H15) > ; iii) B-L is violaced as X+2Y is equal to 6*4 for (H.,),- . This point is related co point ii): 3-L invariance forbids a Hajorana mass at all orders. In such a case there is a physical Goldscone boson associaced with (B-L) SSB. The only way co gee rid of ic is Co make che original Lagrangian noc U invariant (e.g. by adding che crilinear coupling h.tuH., ). - 89 - 9. RENORMALIZATION OF THE SU(5) PREDICTIONS This chancer intends Co show how ouch che predictions of exact SU(5) are altered at present energies by S0(5) breaking. The coaputational cool is che , - . [28 ] renomalization group a} Coupling constant renormalization 6 L* Sit S? md g3 be defined fro» che part of che Lagrangiai. containing the 3, the W's and che gluons : quarks fermions -g, /TB fermions Similarly g is che X or Y coupling co che fermions. In absence of SSB, SU(5) symmetry means g, » g- » g, » g. In the case of SSB, these equalities can be crue only ac very high energies, above che mass scale J at which breaking occurs. To be more precise, let us recall chac coupling constants are defined for a given scale u(the value of the external momenta at which the «normali zation conditions are imposed). If u, which is arbitrary, is varied, the cou plings and masses have co be varied in order for che physics co remain unchanged. This invariance of physics wich respecc co u leads co che so-called renornaliza- cion group equations (R.G.E). g., g.,, g, and g are thus functions of u ("running coupling constants"), and they all become equal at very large u's. The SCE for the couplings have the form : u -«— g (y) • 3 ([g.(u)l, masses, Higgs self couplings and couplings co fermions). The 3 functions are computable in perturbation. For u such larger than a'l particle masses (y >> 7 ), the situation is identical to che symmetric one (where all particles are massless) : che four 3 's are thus equal, and accor- dingly che g's remain consistently equal in this region. As u decreases, che masses of che X and Y become non negligible, so chac che values of che various i'3 become different. >s a consequence, che corresponding %'s have different - 90 - évolutions between che region u «a and present energies. Before discussing these evolutions, we recall how che renormalization program is achieved. The (*) Green's functions ? coaputed with Feynaan rules for given bare mass • and coupling g are in general represented by divergent integrals. These integrals are made convergent by soae régularisation procedure, such as for example che introduction of a cutc-off paraaecer A in the loop integrals over internal aoaenta. A renormalized Green's function !*, depending on renormalized mass m and coupling g, can be defines as 7 (**) Z is a product of renormalization constants . Ic contains I/? - one factor Z " for each external gauge boson leg, - one factor Zj^~ for *acn external fermion leg, - other factors associated with other external legs we have not to consider here. The limit A — -H» is taken at fixed g and a, g and m being redefined as func- o o tions of g, o and A according to g « lia Z g A— * ° m * lia Z m The renornaiization constants (Z , Z , Z., Z_,...) are functions of g, m and A g 9 A F (or g , m and A). They diverge as A goes to infinity and are determined by che requirement chac che Limic A-« of Eq. (9.2) exists for all I.?.I. Green's functions. Proving renormalizability consists in showing that it is actually possible co compute these renormalization constants order by order in g free the requireaenc chac all I.P.I. Green's functions are finite. The renornaiization constants are determined up to parts which remain finite as A-*». Accordingly, che Green's funccions are obtained up co soae constants (with respect co excernal Ic is sufficienc to consider che so-called "1-particle irreducible" (i.P.I.) Green's functions (no way to decompose the diagrams into two disconnecced pieces on cutting I internal line only). All Green's funccions can Chen be built without introducing any new problem of convergence. v They are constants wich respect co excemal momenta, but funccions of che parameters (see below). - 91 - tnca). These consCanes are fixed by Che "renonalizacion conditions", i.e. by specifying Che value of a finite nuefaer of l.P.I. Green's functions at given values of the external taoaenca, usually defined as euclidien values such that pT«-•» y 2 . By Che renorsalizatiou conditions, g and • beco"— functions of Che particular aass scale u chosen. Hence, they are "runn jpling and mess g(u) and a(u). One has some freedom in choosing which set of I.P.I. Green's functions is used Co impose the renoimalizacion conditions. (It is sufficient to fix che values of all the propagators and of one vertex). Since Che coupling cons- cants (except g.) appear as the coefficient of the Cera of che Lagrangian which is cubic in che gauge fields A , che 3 A vercex nay be conveniencly used in che following way. In che Lagrangian, cbe cubic Cera has che form «a fady ?uvc <^y <9'3> where f is che group structure constanc and, in momentum space, P._o is a Lorencz censor linear in che boson aoaenta. The 3 A vertex (che 3 A's are all either gluons or W's or X,Y) contains a Cem of chis type, which at the renormalizacion point nay be written V^—j2)f3aYpuvo • <9-4> plus other terms with different Lorencz censors in the p.'s, which are finite and do not require renormalization. We then sec che renornalization condition 2 ga(y) - ??_(ua(u ) , (9.5) and from Eq. (9.2) tor che 3 A vercex (where group and Lorencz indices where omitted), we gee g(u; - l\n F° , (9.6) where Z, and F° have to be calculaced in rpercurbacion as functions of i , i A a °o o and A, Ac lowest order, g (u) « g and che coupling constants are SU(5) sym metric. The -difference, by one order of magnitude, between che strong and electro-.(^necic coupling constancs observed at present energies uacj is actually due to radiative corrections. The point is chat che usual percurbative - 92 - m X Y expansion is not valid due to the presence of large logarithms In * . The role of the R.G. is precisely to sum up all these logs, leading °to a final result quite different from the lowest order. At next order (one loop), F° is given by the graphs (computed at p. «-u ) 0 ~ + I +1 f A'H 9^ /a» gJ J&' A'H A A A'n A and Z. is computed from the A boson inverse propagator fA A AvWV^I^WWW = wvww + f 9oy9o A',H ^wJ* ) A',H 2 "> and a renormalization condition imposed to it at p »-u~. In both cases, the third graph involves all the (Higgs and gauge) bosons which couple co che particular A considered. For example, if A is a gluon, the loop may involve gluons, X or Y bosons, or Higgs bosons. But it never involves W's or B's. 3 Since che fermion and boson propagators occur only in graphs of order g , at this order we may use che renormalized masses as well as che bare ones in che graph compucation. Finally, as far as che dependence on A is concerned, all loop incegrals happen Co be logarithmically divergenc, leading Co terms \2 2 Log —j , where M is some typical scale. For each graph, Che only scales are M che masses m. appearing in che loop, and u, so chat all che results obcained chrough equacions (9.6) co (9.8) can be gathered under the form : 3 + / b In \ A g, T * °%) (9.9) s0 1? fOTL Noce Chat, as chere is only one bare coupling g , che four g (y) are not independent. Once che value of one of them has been fixed, che values of che others are determined, f has a limit ?L(—— ) as A*» It can be shown that mfrVj-u- "L fT is moreover regular in bcch limics -0 and — •* » . x is a conscanc of order coming from che loop integrations. The sum runs over all possible - S3 - (*) loops, m. being the corresponding (bare or «normalized) nass . One sees from Eq. (9.9) that for u ouch larger than all the nasses of the problem, the relation between g (y) and & is that of the massless (unbroken) theory, the 9 O sane for all a's : E-[ Ï H.* y4 *L(J. •)]•"$ This is the result announced : g.(u), g2(w). g3(U) *nd a (y) are equal for very high y's. Suppose now that we change the «normalization scale from y to y '. From equation (9.9) the relation between g (y) and g (y') reads for y close to y', •ft • after taking the limit A •*• » : 2 2 5 ga(y)-ga(y')^ ^g^L)[bLinï-^.IL(^) -?L(£)] .o(g ) . (9.io) 3 In this equation, for each loop L g (L) has to be interpreted as a cubic monomial 31 2 of the renormalized coupling constants, that is g (L) • g'(y') or g (y')g. (y') 2 3 a a or g (y')gv(y') or g. (y'), according to which vertices appear in L. For example, if the graph involves chree-gluon vertices, g (L) is to be reinterpreted as g-(y'). The evolutions of the four coupling constants are thus coupled to each other. However, as stated above, the 1 loop graphs for F . (F0.) never involve W bosons (gluons) or B bosons as internal lines. Hence, at this order, the evolutions of g-(y) and g«(y) decouple from each other and only depend on themselves and on g (y). For y' and y very different, Eq. (9.10) becomes useless (lar03 ln(y'/u)), and we must use the R.G.E. Recalling the definition of the S function 8(g(y)) - u^j. g(y) , (9.11) we are going to use (9.10) in order co differentiate g(y) with respect to y, which gives-the 1 loop contribution to 3, and Chen to integrate the differential equation to get g(y). We remark that since ~L is regular in both limits m./y * 0 and m./y •* », y--— fI —-\ vanishes in both cases and gives no contribution to S provided y is chosen far away from all masses. So, the contribution to 3 from a given loop is : ^Strictly speaking, some loops involve particles of different masses. Eq. (9.9) is thus somewhat symbolic. This complication however does not modify the conclusions. - 94 - i) - ^" b^ for u » ». , as in a oassless theory ii) 0 fc) for u2 « mjf •L Hence extremely heavy particles decouple, which is a special case of Che Appelquist 129] and Carrazzone theorem . That means that for u far below M_, M_ the X and Y contributions Co the evolutions of «u and g. are negligible. If at the same time U is well above My and M_, these masses can be neglected, and g~ évoluâtes as an unbroken SU(2) coupling constant. In order co compute the evolution of g.(u), we have to use another Green's function. We choose the gauge boson-fermion vertex (more precisely that part of it which couples to the fermion via ly ! coupling). Here, in addition to Z_, U » we need the fermion «normalization constant Z_, which can be computed from Che renormalized fermion inverse propagator. That the evolution of g. decouples from the others (namely g~ and g.) is not as simple as it was for g« end g~. Never theless it holds (in Che one loop a proximation) due to Che Ward identities which re lace Che fermion propagator to the fennion-B field vertex. They have Che consequence that the renormal ized gauge coupling g. is rela'ad to tiie bare one simply chrough «1 " ^B* *o (Mute chat any reference co the external fermion has disappeared). So we have co consider onl; che B propagator (to compute Z_)and the decoupling holds as for g2 and g-. Summarizing : for My - « u « M_ _, the X and Y contributions Co che 3*s vanish and che W,Z masses can be neglected. The chree coupling conscauts g-(u), g-(u) and g.(y) "run" as independent SU(3), SU(2) and U(l) coupling conscants (SU(2) and U(l) are still unbroken). As a side remark, let us point out chat a great variety of «normalization conditions can be used. They all lead to equivalent results (in perturbation), che resulcs given by one renormalization scheme being obtained from chose given by anocher one chrough a change of che coupling conscants and masses. This is one of che aspects of che R.G. invariance. However this does not mean chat che 3 functions obtained in different schemes are equal (8 is even gauge dependent) . T - 95 - Only when all particles have zero mass, the first two coefficients of the expansion of 3(g) are universal . The saae "universality" holds in Che so-called minimal schemes (MS) where the renormalization is done in a way completely independent of the. masses. Note ChaC Chis makes Che MS not directly suitable for our purpose, as it leads to mass independent 8-functions, which apparently violates the decoupling theorem. In fact, in such schemes, the perturbaciv* expansion is not *ralid at low energies, even after the R.G. Mj y "improvement", due to Che presence of large logarithms. In -—*— . The problem of the large logs and that of the lack of decoupling are related to each other : [3i] if all Che logs are summed up, the decoupling is restored For any pure SU(N) gauge group (and for 'J(l) setting N-0 in Che formula [321 below), b„ is found Co be If 2 2 bN - -j^ ( IIM-2 Tr(Tj) - Tr (TjJ) ) . (9.12) The first term in the bracket is the gauge boson contribution in Che loops, a is coc summed over. The Tt's (Tg's) are Che gauge group generators in the representation of the fixed chirality massless fermions (complex «assless Higgs), The trace of (T*.)2 has che sane value for SU(3), SU(2) and U(l), since the TJ'S are generators of SU(5), normalized as such. Taking SU(3) for définiteness one has T*-i£ , Tr(TV-! ' for each fixed chirality SU(3) triplet. There are 4 such triplets per family : {u.1} , {d.1} , {u. } , {d. .} . So, for F SU(5) families of negligible mass fermions, the fermion contribution to bN is The Higgs boson contribution is intrinsically small. We have to work in a manifestly renormalizable gauge, such as che 'c Hoofc, or R, gauge (see section 4). In Chis gauge, there are two classes of Higgs bosons. A first class has /ery large masses, of the order of M_ „. This is che case first for che 12 physical Higgs bosons of che adjoint representation. It is also the case for che 12 unphysical ones because in chis gauge (seccion 4), 1/2 che unphysical bosons have masses proporcional to nu „/Ç (we keep Ç finite, - 96 - for example the Feynman-'t Hooft gauge £• I). Finally, the 6 coloured fields of the fundamental representation {h} have nasses of order m_ _ because, as explained in section 8, they couple to the adjoint H. All these very massive Higgs bosons decouple and give no sizable contribution to 8 for u « m_ _. We are left with 4 real fields in the fundamental Higgs representation. One is the physical Higgs, m ~ BL, _, the 3 others are unphysical and have masses 1/iit 2 w»^ of order "Vz'« So, only 4 out of the (24+ 10) real Higgs bosons may give contributions to the 3 functions (*) . For y » OL, _, their masses can be neglected. In this limit, their contributions are given by l 3 K2, I 3 f-s u rrhZ^ t for U(l) T.\2 1 I for SU(2) (ll) b - - -^ Tr (£)" 12tr 2 (iii) b - 0 for SU(3) They are indeed small as compared to those of the fermions and gauge bosons. Collecting all the contributions to the coefficients b„, we finally obtain r 3 To" bN " 12TT UN- 4F-< (9.13) ^0 Me note that F cancels in differences b„-b„f , and, neglecting the Higgs boson contributions, chat the b's increase linearly with N. We are now ready to study the evolutions with u of the couplings. Setting 8i(^> ct.(u) - , , the solution of the R.G. equation (9.11) for a. is at the 1 loop order the celebrated relation srfcr * rW+ biln h (9.14) 11 ÏI We recall that any way Che so-called unphysical Higgs bosons (together with the Fadeev-Popov ghosts), which have £-dependent masses, have only the effect of compensating unwanted poles at k^-m^/Ç in the transverse p_rt of che gauge boson propagators. For % finite, the contributions of this transverse part and of che unphysical Higgs separately vanish. îjt £ infinite (Landau-1t Hooft), che poles are k2«0, che decoupling does noc hold, but the two contributions cancel each other. L. - 97 - This result is valid in any region for u and M , !t^ « — « !L, provided a. (u) and a. (M) are both sawll as coapare to I. a~l(u) - a~l(u) *|a7l(u) 1 3a,(u) tg29(u) (9.15) 5o2(u) Running coupling œnstants 15 m 2 mass seated w-10 nyY -TO (GEV) F-ûi 9 7 - Sketch oi the. evocation* o$ the tunning coupling constants ct.(u). liTBWWeen m„ z and mx y, the. a'* otf SU(3), SIM), U(J] evo&utfe ^idepeâderttty. A4 u ~ Tu» v <£ appacacTted, -the X,/ ba*cn contrUbution& become impoKtant and ttnd ta xeAtoil the. complUi iymetty. Above m^ y, aJUL coupling* ate clou, to a;!ul. The evolutions of the three couplings are sketched on Fig. 9.1. AC a mass scale a little above the H and Z masses, one has ou > a, > a.. (The latter inequality 2 3 is due co eg 9 < T) . From b. > b, > 0 and b. < 0, we get that cu decreases faster than a- , and chat a. increases. These 3 coupling constants are thus in a good position to meet"(* )' . As one deduces from the decoupling properties, the Conversely, grand unification (like SU(5)) "explains" why the strong inter actions associated with a gauge group SU(3) are stronger than the e.m. interactions associated with a gauge group U (1) c SU(2) 9 U(l). « «ID * - 99 - three curves cone close to each other in the region u ~ •_ _. A precise descrip tion of this region requires taking the effect of the X and T mass carefully into account. For still larger u's the three curves have essentially aerged and there is only one rapidly decreasing (b. »td SU(5) coupling constant left. Froa the experimental data on two of the couplings, one can now compute the X and 7 sus, and the value of the S0(5) coupling constant in the asymptotic region. This allows one to then predict the value of the third coupling constant. In other words, one can coapute sinB, say, in terns of a and the presently known (?) *s. (*) As a first rough estimate, one does as if the three approximate expressions (9.14) for a., a, and a. net at soae unification point M_. at ML _. Let a„ be this cCOS9 Hn value. One has 1 a (y) s GU 2 I -1,5 y a(y) a,(y) 3a. (y) (!b i * bz) In GO GU Thus 2 18 - 1 ^>b. •b.-fb. y (9.16) ÎÏÏIT " ISrîûT " TH \J i ) ln The (negligible mass) fermion contribution cancels out in this combination (as it has to) and we obtain the simple formula 1 8 11 50 m" (9.17) ci(y) 3a (y) M2 219 n ( TQ? la ~Y-GU had we included Che Higgs contribution). We vill apply this for- y aula for u a little above the W and Z masses. The value for a is roughly known thanks to deep inelastic scattering experiments, whose results are compatible with (*) In this approximation we do not commit ourselves wi'h any assumption about the nature of the (simple) unifying group. The only assumption made is chat the breaking of the unifying group down co SU(3) § SU(2) 9 U(I) occurs in one step only. - 99 - a (u) - V-Tm "* i i («-I») * In y /A (33-4F)ln y /A* and A — 500 MeV. In this formula F is the number of faailies which can be actually excited at the energy scale y. In order to coapute a at y « M», we rhus have to take into account all quark thresholds opened below M„. As a first approximation, one considers a new quark as infinitely heavy below the corresponding threshold and as aassless above it. This means that F in formula (9.18) changes by half a unit at each threshold. If a. is still represented by Eq. (9.18), A also has to change at each threshold, faking the associated non perturbative phenomena, i.i order to keep a (u) continuous. a(y) is known on shell with an extreme precision 2 2 1 a(y --me) -T3T . Ue have to extrapolate this value in the euclidean region, up to y of the order of OL,, a- where we start our analysis. One finds : a(!V " m This value, together with that of a (My) obtained from Cq. (9.18) and the relation (9.17) leadsco : and one deduces from any of the evolution equations More refined computations have been performed which treat correctly the mass [331 effects . A detailed discuss,<«j is given in R«f.[34] . the result for M~ „ is 14 Mg y - (6 x 3) 10 GeV (for A - 0.5 GeV) , i.e. one order of magnitude below Che previous result. The error quoted is the one estimated in Ref.[34], With the latter value of M„ „ and Eq. (9.15), one oredicts - 100 - a. 0.206 • 0.006 This theoretical estiaacion is to be compared with the present world average [4] experimental value sin29„ - 0.229 • 0.014 . (9.19) It is not clear whether or nor there is a discrepancy between these two values, which by the way differ only by about 1.5 standard deviation. In particular the present experimental determinations neglect radiative corrections. T351 b) Fermi on mass renormal izati on *• The predictions for fermion mass ratios, like m »m. for SSB by a 5 of Higgs bosons, are also strongly altered by rediative corrections. As a matter of fact, only fermion masses relative to some superhigh energy scale may satisfy the lowest order predictions. This is certainly not the case for the mass definer. as the position (computed perturbatively) of the fermion propagator pole (whatever it means for a confined particle, especially if it is light). The scale then involved is the physical mass of the particle, which is small. On the contrary, fermion masses defined from the value of seme Green function evaluated at a scale u much larger than m_ „ verify the naive SU(5) predictions properties (at lease for the simplest Higgs systems). Such definitions are the followings : (i) the fetsnion mass is defined from the Yukawa Higgs-feraion-fermion 2 2 2 vertex G(p.) at the point p. • -u » m_ _. As shown in section 8,b the fer mion masses appear, at the tree level, as products of a Yukawa coupling constant G by the vacuum expectation value ui of some Higgs field m - A us G . (9.20) In the «normalized theory, this relation becomes m(u2) - A J (y2) G (p2 - -u2) . (9.21) - 10! - The vacuus expectation value is evaluated fro» a potential renoraalixed at che point $ « u, and the Yukawa vertex is evaluated at che point p.2 «-y 2 »•_2 _. The ratio of, say, the electron aass to the down ouark mass is Chen • (U2) A G (u2) ——5--T1 x h (9.22) A Z •d(u ) d Gd(u ) 2 A «A, in the case of SSB by a S of Higgs bosons. For y -»• » the Yukawa couplings are S0(5) symmetric (exactly as the gauge couplings) sc we recover che lowest order SU(5) result for the mass ratio. This would not be true however in a moi general case where several Higgs nultiplets give aasses to the fermions . (ii) che fermion mass is defined from che fermion propagator A Sp(p) iBjl (9.23) f-m(-p ) by the value of the function m(-p 2) at p 2 »-y 2 , y 2 » m_2 _. This definition is useless in che involved cases above mentioned where the fermion masses do not 2 obey the naive broken SU(5) prediction as y +»• In subsequent uses of masses in perturbative computations, one sees jo*m(y ) if che typical scale involved is of order y . If we were to use o(y), / \ y » y , che perturbative expansion would sot be valid due Co large logi—1 . ° \wo' In order co compare SU(5) predictions with "experimental" determination of fermion masses we have co scale them from their region of validity y » BL. down co che low energy region where chey are measured. Here again, che renorma- lizacion group makes che job of resuming che large logs . The renormalizacion group equacion for Che masses is 4-SiHi. m(y) Ym(a(y)) , (9.24) j in u m where y , called che mass anomalous dimension, is computed in perturbation. The solution of che RGE is lny / Y(a(U>) d(lny) n lnyo m(y) « m(yQ) e , $ - 102 - g(u) Y.(g') g(u) a{* } or m(u) - m(u ) e ° (9.25) o where use has been aade of the RGE for g, d(ln(u)).£|W dg(u) 3 2 At the l-loop level, 3 • -b £r and y •" -fc- Y .h and y° are constants if all the masses are neglected in the computation. The result of (9.25) is U »(U) - « If we choose to define the fermion nasses froa the fernion propagators, we have to compute all fermion self energies in order to obtain the Y '*• For 2 ™ U larger than the X and T masses all (fernion) y 's are equal due to the SU(5) symmetry being exact. The masses of, say, the electron and the down quarks evoluate with y ,but in the standard Higgs scheae their ratio reaains fixed at I. As the value of y is decreased well below M_ , the X and Y contributions gu co zero and the various mass parameters follow different evolutions. As for che coupling constant renormal zation, we make an approximate treatment, neglecting all masses exept the X,Y mass, taken as infinite for Li's below m_ v (in fact up to y *a. J, and assuming that che SU(5) prediction is verified at u • m_ „. At first order we have to consider only one loop diagrams like m where a can be a gauge or a Higgs boson (Higgs bosons will be however subse quently neglected). Renormalization is achieved by imposing the renormalization condition MP'P' ) 2 2 - ' 2~~ p .-y i*-m(u ) At a different point, che propagator has become - !03 - IP —u ?-« 2 2 2 with, from che definition of YB , •(U* ) « n(u )(l-Y (a(u ))ln JT-) « u'~ u The computation is simpler in the Landau gauge, where it can be shown that the femion wave function renomelixatioa Z_ is finite. If furthemore all amsses are neglecced, Z_ is a constant, fixed to one by the renoraalizacioa conditior.. 2 That amans hip ) 5 I. The coaputatioa of the diagram Chan gives : <"è l. «î>i.j (T?j.i <9-2" *tj The T?'s (T?'s) are che S0(3), S0(2), 0(1) generators in cha rapresancation of the right (left) fermons. One suns over all possible gauge bosons. The reason for che occurrence of TL, T is the following. In order to get a(u'), -I 2 one coaputes S_ (p) from the self-energy diagram, and, due co A(p ) • I ndi') -^Tr S^Cp)! % 2 IP —U* The trace is taken on Dirac indices, and cha only contribution to u(u') cones froa chc case where che incoming and outgoing fanions have opposite chiralities. Since chiralicy is conserved at the gauge boson-femioa vertex, the transition fro» L to ft in the feraion propagator goes as depicted on the oaxc figure o U L,j R,j The incarnai fermion propagator changes che chiralicy (by its mass tern) and conserves the gauge index. The concrary occurs ac each of che two vertices. The initiai gauge index i is finally conserved since I *îj TJk ' 5Lk *c, t ' Meglecting the large mass X,7 sauge boson contributions, wa onlv have co The v ' s are gauge dependent. However this dependence finally cancels in mass ratio» («.«.-—7—T-) . - nyta) - 104 - consider those of the SU(3), SU(2) and U(l) gauge bosons. The results are the followings. i) SU(3) contributes to the anomalous dimension only for quarks because the leptons are colourless. One finds : 2 .(3) 3 N-1 2TT N L K ' 2TT 2« (N-3) (9.28) IT SU(2) ii) SU(2) gives no contribution as (T_) iii) (7(1) contributes by Y<»Y > o ..i«(2)xïy2ir V 2 2 i 1 7 " Tïïiï for charge ? quarks " + (9.29) 2W for charge - •=• quarks 9 "SJTF for charged leptons Using the above results for y , the values of b„ (Eq. (9.13)) and the solution m N (Eq. (9.26)) of the RGE for m the SU(5) predictions for fermion mass ratios thus get modified into (neglecting the Higgs contributions to the y functions) 12 3 4F mj(y) m (^) "s^' /us(,ki''N g (9.30) m e(u- ) m,,(u"u)- SZTÏÏ"T-T "^"^GtT/ This result is rather sensitive to the number F of SU(5) families. For A«0.5 GeV, one obtains 3.6 md(10 GeV) 4 5.3 (9.31) m (10 GeV) " for e 8.5 The present experimental situation implies that F is at least equal to 3 (though the t quark has not been discovered yet). In this case, one obtains at u • 10 GeV, from the known values of the e,u,T masses ; - 105 - r °b 6 GeV m =* 0.4 GeV s m, ~ 1.6 MeV 1d [24] This prediction, which was made before the discovery of the y , is in good agreement with what is thought to be the bottom quark mass. Conversely, if m. is correctly evaluated from the y data to be around 5 or 6 GeV, the SU(5) prediction indi^-trer that tiiere are only three SU(5) families. The top quark, then, would be the last quark to be ever discovered. The modifications due to RGE of the exact SU(5) predictions for strange and down quark masses clearly go in the right direction. Quantitative analysis is however extremely hard to perform, due to the lack of an operative definition of what a light quark mass is. - 106 - 10. THE PROTON LIFETIME In this short section we just gather the various ideas and formulae, already presented in different parts of these lectures, which are relevant to the problem of nucléon stability. Quantitative results on the proton lifetime within SU(5) models will be reported, but we also have to point out that many of the properties exhibited by SU(5) are not specific to it and belong to most grand unified models. Nucléon decay may occur only if the baryon number B is not conserved (disregarding neutron 3-decay which does not occur in most nuclei), leading to mesons and leptons. At the end of section 7, we noted that B (and L) conservation is unavoidably violated in models where quarks and leptons are in the same irre ducible representation. Such is the case in SU(5) where the right handed d-quarks, • c • e , v are in the rep.5, the left-handed u, d-quarks, e in the representation 10, and their charge conjugates in the corresponding conjugate representations (with similar assignments for the other families). Whenever such a situation is reali zed, there are gauge bosons coupled to currents build from a quark and a lepcon. In SU(5), tnese gauge bosons are the X and 7 bosons, and the general couplings of Eqs. (7.11) and (7.12) led us to the explicit formula (7.18) which exhibits all the B,L violating interactions induced by gauge invariance. Besides :he currents of the form q y I which mix quarks and leptons, Eq. (7.18) also con tains terras like qcY..q', involving a quark and an antiquark, which allow for c c qq' or q q' annihilation. Again this property is not specific to SU(5), but rather belong to any model in which at least some of the quarks and antiquarks are classified in the same irreducible representation (like the 10 of SU(5)). c Therefore, in models where both cuM-Zntb q y I and q Yuq' are coupled Co at least one of the gauge bosons G, there is the possibility of AB » I processes generated by che elementary interaction qq'-^q"0* • (10.1) AC presenc energies, Che relevant symmetries are SU(3), SU(2), U(l) which all conserve separately B and L. However, if these symmetries are che symmetries remaining afcer SSB of a larger symmetry group g, then che process (10.1) may become possible, although its rate is considerably lowered by symmetry breaking. The situation is exactly the same as in the Salam-Weinberg model which leads co an effective 4 Fermion-interaction, weak at low energies as its strength is measured by G_, - 107 - °F g2 -= - -^ . (10.2) /2 8 ^ In SU(5), the strength of the B,L violating interactions is -^« -K- , (10.3) ^ 84,Y and, as we have seen in section 9, m_ _ is of order 10 GeV. So, hopefully, the corresponding rate is extremely small. In the broken situation, there is in general other interactions leading co B,L violation, which are mediated by some of the Higgs bosons which couple to fermions. In SD(5), broken by an adjoint and a fundamental representation of Higgs bosons, only the latter couple to fermions (section 8b). We recall chat in che standard model, among Che 10 real Higgs fields (h), 3 are "eaten" Co give masses Co che W~, Z bosons, and 1 is che low-mass physical Salam-Weinberg boson which does not lead to B,L violating processes (it is colourless). The 6 other ones (colour triplets, SU(2) singlets) induce B,L violation via the inter action described by Eq. (8.18) for one generation, and Eq. (8.24) for che general case of any number of generations. The effective strength of che four fermion interaction mediated by che Yukawa coupling G (or G) is h 2 GB L ° « ~ °7h The coupling G is related co che fermion masses (Eqs. (8.20), (8.21)), so chat one has g mf G ~ m./u) e* f °W where m, is a typical fermion mass and u> che vacuum expectation value of the h Higgs field. On che other hand, che mass m. of Che 6 Higgs bosons under consideration, generated by the couplings to the Higgs H of che adjoint is [24] of order ra„ „ (g m^ „ if che h-tf coupling is generated by radiative corrections ) Therefore one has ,h raf . „ 1 GB,L~-7I * GB,L ~ • (1°'4) - 108 - with o at aost of order 1 and presumably of order g . Biggs contributions to nucléon decay are found to be saall, especially because it involves only light fermions. In order to compute the proton and neutron decay rates in SU(5), various steps have to be achieved. First one has to take into account generation nixing. The X,Y interactions of Eq. (7.18) are replaced by the general one of Eq. (8.31) in section 8c. As argued there, the nixing matrix U is phenomenologically close to unity and S, a diagonal unitary aatrix, has no visible effect at lowest order. There is no coupling dd-gauge boson, so that in a proton or a neutron, one nay have uu or ud annihilations only. There are two kinds of elementary graphs to be considered (Fig. 10.1). d d *0 u u F-cg. [0.1 - a.) an example oi annihilation qn&ph. Even $o* ÏÏ » 1, one. may get second ge.nesiation {imiont> in the. iinal itati. fa) exchange gnaph. In the limit U * 1, only $£ut geneAation pKoducti axe allowed. Graphs of type a (annihilation) allow a change of generation, leading either to a strange quark accompanied by u or v , or to fermions of the first generation. For Che exchange graph of Fig.IOb^, transitions from one generation to the other is Cabibbo suppressed. Once all graphs of type a or b_ have been listed, one has to compute the corresponding matrix element, which is easy from Eq. (8.31). One then uses some wave function in order to describe the nucléon in terms of the u and d quarks. Since, due to the high value of n^ Y, the interaction is a contact interaction, the decay rate is proportional to the wave function at the origin squared |tj/ (0)| . Finally, one needs some interpretation of the final states containing quarks and - 109 - leptous. One aay compute the inclusive decay rate by stoning over all possible qq' final states for a given outgoing lepton. This gives the total nucléon width for each lepton type and the inverse life time. Fro» the above conside rations, one expects the life tiae to be of the form T~T I I (I0-5> g*l*(0)|Z4 up to Clebsch-Gordan coefficients given by the interaction Lagrangian (8.31), and numerical constants coming from phase space. It is clear from Eq. (10.5) that the most important piece for computing T is the value of tn_ _. We refer to section 9 fox a discussion of what the problems are, and to Refs.[33,34] for a complete study of the uncertainties 4 2 involved. The value of g is (4ir (*„) at a scale of order a_ _, by definition of the unification point (a~,~T7r) < Ic nas to De further renormnlized by [24] radiative corrections and proper use of the renormalization grcuo . The [34] presently admitted result found for x is 32*2 T « 8 x 10 years in the standard SU(5) model. Since it happens that the range in between 10 (the present experimental 33 lower bound) and ~ 10 years seems to be experimentally accessible, the last question of interest is that of the branching ratios for the various allowed [37 33] decay modes. This subject has been recently studied by many people ' It is generally agreed that the most favoured channel is ir e as it contains the lightest particles and because the corresponding exchange graph is Cabibbo allowed. There is more variety in the predicted branching ratios for other experimentally interesting channels such as e p , e n, a w. The channels involving neutrinos are generally found to be suppressed. Among the properties found in SU(5) for AB • 1 processes, all of which have not been quoted here, some are more general. For example, one mav show [38 1 by inspection of general effective Lagrangians violating B and L conservation that B-L conservation should hold at a very high degree of accuracy, that AS-AB transitions are forbidden, or that relations due to the weak isospin structure of the Lagrangians exist between some inclusive rates involving given chirality - 110 - lepcons. On another hand, it oust be possible to distinguish between various unified models by more specific experimental investigations, such as the \> /e ratio or u polarization in nucléon decays. Also it has been noted that, although the simplest Higgs mechanism leads to suppressions of channels like p+ke or IT u (Cabibbo suppression), it would not be so in other schemes involving at least 2 Higgs multiplets. So the measurement of ratios of "Cabibbo forbidden" to "Cabihbo allowed" decay rates may help in distinguishing between various possibilities. This is especially worthwhile as the Higgs pattern is intimately connected to the very important problems of fermion masses and flavour mixings. We do not want to try and give any general conclusion about the topics covered in these lectures. Let us just emphasize the central role of proton decay in all grand unified theories based on SSB of simple Lie group symmetries like SU(5), 0(10) and E,. Alternatives to the theories based on spontaneous symmetry breaking have been proposed. In the so-called dynamical symmetry breaking schemes, a new m&ss scale in the range 1 to 100 TeV should show up, and specific predictions about a new, rich, spectroscopy below 1 TeV and about C? violation have been made Finally we recall that it may be that only when gravity is included a [41 1 new fundamental insight into particle physics will be obtained - Ill - APPENDIX A - YANG-MILLS AND GROUP THEORY This appendix intends to reaind the reader with a few results and useful [1U [20] concepts in gauge field and Lie group theory . Suppose we have a field theory with fermions (and scalers). The Lagraugian L is taken to be invariant under some compact (*> Lie group of transformations G. The various fields belong to (generally reducible) representations 0 of G. U may be chosen unitary (any continuous, finite dimension, representation of a compact Lie group is equiva lent to a unitary one). L is thus invariant under the field transformation tiKx) •* 0 *(x) g with U - O-1 - U _i . (A.l) 8 g r For the time being, U is independent of the point x. One says that L is invariant under a global transformation. In the vicinity of the identity e, any element g of G can be written as g « • - i -5o ta . (A.2) The <5a 's are the infinitesimal parameters of the group, the t 's its generators. By real linear conbinations of the generators, one generates the Algebra a. for the group. The inner product of two elements of a is -i times their commutator. The structure constants of a. are defined by the relation a b ab C [c , t ] - i f c t . (A.3) The structure constants are real. The appearance of a factor i in Eq. (A.3) cones from the choice made in (A.2), which is frequent in Physics. Were the generators be multiplied by -i, chere would be no i at all in Che algebraic manipulations, but chey would appear in Che relations between observaoles and generators. (*) A set M is compact if any infinies subset of M contains a sequence which converges co an element of M. For example, any closed region o' finite extension of TRn is compact. A Lie group is compact if its parameters have finice domains of variation. - 112 - To each representation of the group it corresponds a representation of its algebra. Let T be the aatrix representing t in the space of representa tion which the vector <\> belongs to. The infinitesimal transformation of ? reads J J £ a jk k or in matrix notation 41 * 4, -i 5aa T* ¥ . (A.4) Here T • T and since T is a representation of a [T3 Tb] - ifab TC f c An -ideal I is an invariant sub-algebra of a, : for all elements t of a, [I,t] c i I In «hat follows, we restrict ourselves to the so-called 4 Gauge fields are introduced to insure invariance under group transformations which now depend on Che point x where they are performed : *(x) - U (x) 'p(x) The global symmetry is enlarged to a locat symmetry. Under local transformations, the kinetic part of the free fermion (or scalar) lagrangian namely LÔ^H" (or (3 tp) O^cp)'), is no more invariant. The invariance is restored if the derivative 3 is replaced by the covaJUMit derivative. 3U * Dy - 3U + i Ay(x) Ta . (A.5) a The A (x)'s are the gauge fields. Their transformation properties are designed - U3 - to compensate the terns generated by 3 acting on the x-dependent transformation. If one considers the gauge field as a aeaber A of the representation of a, defining Ay = I A^ Ta , (A.6) a A transforms according to Ap(x) £ AuU(x) « U(x)Au(x)u",(x)-iU(x)(3UU_l(x)) . (A.7) Then oi»e verifies chat DU(A) * Dy(AU) • U DW(A) if1 which insures that ty 0 ¥ is actually invariant. Intrinsically, one defines the gauge field as a member of the algebra itself by Aw(x) - A^(x) ta so chat A (x) is a particular representation of £ (x). From (A.7), we obtain chac under an infinitesimal transformation, Aw transforms according to Ay(x) * Ap(x) - i 5o [Ta,AuOc)] + 3y(5a (x)) Ta . (A.8) a a For 5a constant, the last term drops out, and the transformation of A is that of the. adjoint xe.pn.e.ie.rvÙltlon of the algebra. With each element c of che algebra, Che adjoint representation associates Che linear transformation of t tb * -i[ta, cb] . The adjoint representation of a simple algebra is irreducible (there is no subsp-'.e of che representation space which is invariant under all group trans formations) . Note chac che matrix T associated with t in the adjoint repre sentation verifies by definition b a b c a b c - (T ) c c . -i[t , e ] - 114 - Hence (T ) » f c c Ue finally have to add the pure gauge field term in the Lagrangian. To our disposal, we have the gauge invariant quantity Tr(F F ), where FWV - -i[Du,DV] - 3V-3VAU -i[A^,AV] A mass term Tr(A A ) for the gauge fields would spoil the gauge invariance. In taking traces, we have to specify which matrix representation Aw of A^ we choose. A usual choice is Ay «AwTa a ' T being the matrix representing t in the adjoint representation. This particular choice has no consequence since, for any simple group, if in the adjoint repre sentation we have Tr^. (Ta TN - gab , then in any irreducible representation R a b ab TrR (T T ) - C2(R) g with C,(R) > 0. The only exception is the trivial representation which associates 0 to all elements of a. Since the gauge group is supposed to be compact and semi-simple, one can always find a basis for the algebra for which the matrix g is unity. Tr ..(Ta Tb) - 5lb . aaj One can show that in this basis the structure constants are totally antisymmetric. The complete Lagrangian Committing scalars) finally reads : 4g a uv,a where g is Che coupling constant and M Che fermion mass matrix. When the algebra is simple, one cannot separate it into pieces transforming independently. Hence there is only one coupling conscanc when we have exacc gauge symmetry under a simple group. - 115 - The above results can be generalized to groups which are (locally) direct products of simple groups and (eventually) of U(l) groups. This is in fact the most general case : we want the form Tr(F F ) to be non degenerate (all gauge fields present in the kinetic part), and positive definite (all kinetic terms of the same sign). This requirement implies that the gauge group is a direct product of a semi-simple group and of an abelian group. The algebra is the direct sum of the corresponding algebras (which commute between themselves). Since any semi-simple Lie Algebra is a direct sum of simple algebras, the algebra of a general gauge group is a direct sum of simple algebras and eventually of U(I) algebras . The most general Lagrangian to be considered thus is : Él-M^'-O•*k \ l * i iJ (* + i I L )4» - I? M ? . k * There are as many coupling constants as there are factor groups. Under the field rescaling A. •+ g. A, , L gets the more familiar form (now including eventual complex scalars) : 1 Î uv k \a, . *k *\ k ^ * -ÏM** [Oy+i I ^ AJJ)*[2 - fl+M2 $ . (A.9) As shown above, if we restrict ourselves to simple Lie groups (such as SU(5)), the theory depends on one coupling constant only, and it is unified in this sense. The charges of the'various particles are moreover seriously constrained. For example, if some representation appears several times (several families), each generator has the same representative for all these replications, and in particular the charge operators are the same. Finally the charge opera tors in different irreducible representations are also related. (*) In the U(l) case, Che algebra and all its irreducible representations are isomorphic to B (except the trivial representation by 0). Since all commutators vanish, the adjoint representation is the trivial one and cannot be used to define Tr(F'^vF ). In any representation, A^ is represented by A^(x)Y/2, where Y is the (J(l) Hypercharge (Y«0 in the adjoint). The Lagrangian is '--VF**-—y F* !F,,, , + i •i> Ci* i A Y/2)il> ^ - 3y AV - iJ Ay - 116 - A further important point is that moreover the charges are commensurate. It seems a priori very easy to build a candidate for the charge which has non commensurate eigenvalues : such are the eigenvalues of the SU(3) generator A+X in the fundamental representation : JoQ / "* \ VX8 -i-L ^7 _ _2_ I Let us explain why the charge operator Q cannot be of this kind if the theory is based on a compact Lie group g. We consider a simplified example in which the group generator associated with Q is, say, the one parameter group G. .-«•». :'< «' (A.,o, G has two non commensurate eigenvalues I and e. It is non compact since a varies from -a» to -H» (as opposed to the case Q« (5 ), p,q integers, where the maximum length of the variation domain is 2irpq, finite). Moreover, because 1 and e are non consensurate, any point of the square x£ [0,2ir], y€ [0,2ir] can be arbitrarily closely approached by a point x - a(mod.2ir), y - «a (mod.2ir). One says that the set {x ,y } is dense in the square[0,2ir] . Accordingly, any transformation 0 0 ty of the 2_ parameter group [U(l)l can be approached by an infinite sequence of elements of the J_ parameter group G. But if G is a subgroup of the unifying group g, the above sequence has a limit which belongs to g because g is compact (defi nition of compactness - see first note of this section). Hence in our example, g should contain two unbroken U(l) generators, that is two say two photons. This situation is neither that of the real world, nor the one we describe by the- suc cessive breakings of SU(5) : U (1) is the only abelian component left unbroken. Note that compactness of g was essential for the argument : charge is not quan tized in the Salam Weinberg model because SU(2) 9 U(l) is not compact. - 117 - APPENOIX 8 - YOUHS TABLEAUX FOR SU(«) All irreducible representations of SU(H) can be obtained a* properly ttrized powers of the fund—innl representation. This is why the irrede- cible representations of the pemvtation group of V objects, and hence Toung Tableaux are relevant to the study of SC(H) representations. * Toung tableau is conposed of boxes, each bos corresponding to one fundamental representation '? These boxes are arranged in rows in increasing length order. D A young -tab/con For SU(N), there is no tableau containing coluuns with sore than SI boxes. The tableaux nade of only 3 box coluuns represent a singlet. Any S box coluan in a tableau can be ommitted, unless nothing is left, in which case the tableau represents a singlet (we call it a unit tableau). The conjugate representation q of the SU(N) fundamental a is built from che totally ancisyunetric product of !f-l fundanentals q(I),q(2),...q(îl_l) : i J*\'"h-\ (i) (2) „(»-!) q ~ e q* q9 Ic is represented by a N-I box column, More generally, each Toung tableau cor responds to a given symmetry pattern with respect to box permutations (in fact Co one irreducible representation of the permutation group), obtained as follows - one first symmetrises each row - one Chen antisyamecrizes each column (this destroys row symmetry, unless all rows have che same length). Examtjiu D means - 118 means v means (ù to_ + 'i)„e> ) to - Ob Example fax SU(5) (unit t-any number of 5 box columns) t(R) t(R) B R«I0 lira CD 15 IT 45 Û5 24 24-24 When necessar/ (last two examples), we have to rotate upside down the tableau obtained by complement of t(R) :o a unit tableau, in order to obtain a proper Young tableau (rows in non increasing length order). The representations are labelled by their dimensions in the above examples. The dimension of a repre sentation of SU(N) given by its Young tableau can actually be computed directly from the tableau, as explained now. A simple recipe is : (i) make t>.< ;v iuct of the factors indicated below for each box of the considered tableau - 119 - N N+l N+2 N+3 • • * • N-1 N N+l N+2 • • • • N-2 N-1 N N+l • • * • • ; • • (ii) divide the result obtained in (i) by the product on all boxes of factors r.. For the box j, r. * {1+ the number of boxes to the right of box j J. j 6 j + the number of boxes below it} . ExamplzA EE N(N+1) (N-1)N*(N+1) 2 Î1 gga (N-1)N(N+1) S3 3 The nature and multiplicities of the irreducible representations entering the decomposition of the product R. 8 R- of two irreducible representations can also be found from Young tableau considerations. For each operation we give the example of the product •B-B (i) One first labels each row of the R- tableau : all boxes of the first row get label a, those of the second one get label b, etc... *B <9 (ii) One enlarges the Young tableau of R. in all possible ways by adding one box labelled a to R. on the right or below each of its own boxes - 120 - R1 - ff3 Hôi (iii) One excludes the tableaux which are not Young tableaux (rows not in non increasing order or columns with more than N boxes). In the example, one throws the last tableau away. (iv) One starts again by adding the b boxes to the allowed tableaux, again excluding illegal tableaux B3- nus The other possible additions of b do not lead to Young tableaux. One goes on with boxes c etc., and stops when all the boxes of R- have been used. (v) AiLong the tableaux which are left, one further reject those - which have several a's, or b's, ... in the same column (no such case in the example) - which are not a "lattice permutation". A tableau is a lattice permutation under the following condition. One starts counting the numbers N(a), N(b),... of boxes a,b,... occurring in the tableaus obtained, from xlqht to It^t and from up to down. At each stage of the counting these numbers must be found in non increasing order : N(a) > N(b) > In our example, chis rule leads us to reject the 3 and 5 tableaux obtained - 121 - after the step (iv). The result for the product of two antisymmetric tensor representations of 5U(N) is thus B • B ® w The equality of the dimensions on both sides of the equation can easily be checked. The first non trivial application is to SU(3) (for Stf{2) H * unit tableau), in which case the last tableau does not exist (4 box column) and the second one can be reduced to Q (a 3 box column can be omitted). The result is then to be read : 3 B 3 • 6 • 3. Once all the steps from (i) to (v) have been made, one has all the irreducible representations of SU(N) contained in R. 9 R,, with furthermore their right multiplicities, obtained as follows. A given representation occur ring in the decomposition has a multiplicity equal to Che number of times its Young tableau appears with not all labels a,b,... at Che same place. If a tableau appears several times with all the labels at the sa'e place, then it is counted only once. We end up with a second example, where only the allowed tableaux are tteqc. We compute F « We first add the a boxes, and obtain 4 distinguishable tableaux to which we add the b box M a JL e rru ET 2 -H E e a lEl - 122 - __a a © a b mmi—fflâEi • HE El We see that the representation appears twice, all others appear only once. 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