t/08/00061
Department of PHYSICS
UNIVERSITY OF BERGEN Bergen, Norway Scientific/Technical Deport Ro. 110
Gauge Theories C. Jarlskog
Department of Physics, University of Bergen Bergen, Norway
Lectures given at "Advanced Summer Institute 197$ on Hev Phenomena in Lepton Hadron Physics", University of Karlsruhe - Germany, September k - 16, 1978. GAUGE THEORIES
Cecilia Jarlskog Department of Physics, University of Bergen, Bergen, Korvay
TABLE OF COMTETre 1. Introductory remarks 2. Ingredients of gauge theories 3. Symmetries and conservation lavs U. Local gauge invariance and quantum electrodynamics (QED) 5. Wenk interactions, quantum flavour dynamics (QFD) 6. Construction of the standard SU(2) x U(l) model 7. Unification of veak and electromagnetic interactions in the standard model ' 8. The weak neutral current couplings in the standard model 9. Spontaneous symmetry breaking (SSB) 10. Spontaneous breaking of a local symmetry 11. Spontaneous breaking scheme for the standard SU(2) x U(l) model 12. More quarks and leptons, masses and the Cabibbo angles. 13. The four quark and six quark models; CF - violation lit. Grand unification 15. Mixing angles in the Georgi-Glashov model 16. The Higgses of the SU(5) model 1?. The proton/neutron decay in SD(5)
The metric and notations, in these lectures, are as in J.D. Bjorken and S.D. Drell, Eelativistic Quantum Fields, McGrav - Hill, 1965. s
i. uRimwcnmr UMMB ID these lecture*, an introduction to the unified gauge the- ories, of weak and «l»ctromagn*tic interactions, ia given. Strong interactions are treated only briefly; they constitute the.thea* of Professor aachtaann'* lectures.
The idea of unifying seemingly different force* of nature is not new. More than a century ago. Jams Clerk Maxwell (1831-1879) succeeded in unifying electricity and aegnetisa. Within the last decade most of the effort has gone into unifying weak force* with electroaagnetisa. Such a unification was suggested, by Schwinger, to be realized through a triplet of vector fields, whose universal coupling would generate both the weak and electromagnetic forces (J. Schwinger, Ann. ot Thy*. 2, 1*07 (1957)). Sala» and Ward (Ruovo Ciaento 11, 568 (1959)) suggested that the vector bosons should be introduced a la Yang and Mills. It is interesting to note that at that tine the muon-neutrino had not yet revealed itself; the known
leptons e, u, \>e, were supposed to fora a triplet.
The Schvinger model was extended by Glashow (Duel. Fhya. 22, 579 (196l)) who, in addition to the triplet of vector bosons, in troduced a new neutral vector boson. In the Clashow model, the two neutral vector bosons, denoted by Z3 and Z*, nix. The linear com binations Z3cos8 + Z'sin» and -Z'sinS + Z'cos6 were assumed to cor respond to the physical particles (photon and Z°, in present termi nology). Nowadays, 9 is usually referred to as the Weinberg angle. The model predicted neutral currents. Glasfaov prophesized: "For no choice of 9 is the interaction of neutral current small compared with weak interactions involving charged currents". t The model was constructed for leptons; we know that it is not easy to detect the leptonic neutral current, even when it is not small.
In the Glashov model of 1961 the vector meson masses vere put in by hand and therefore the model is not renormalizable. The next major step was taken by Weinberg (Phys. Rev. Lett. 1£, 126k (1967)) who employed a spontaneous symmetry breaking mechanism to generate masses. He introduced four Higgs particles, whereof three disap pear by giving masses to the intermediate bosons and one (denoted by IIP) remains. The model relates the masses of the neutral and charged intermediate bosons, Itø = cos6 Mz. Weinberg guessed chat the model may be renormalizable.
Salam and Ward had also been working on introducing masses via spontaneous symmetry breaking (see the talk by A. Salam in the pro ceedings of the 1968 Noble Symposium, Ed. N. Svartholm).
The extension to hadrons, for four quarks, was given by Wein- bere (Phys. Rev. DJ, 1*12 (19T2)) «be utilised the so-called OOr- • mechanism (Olaahow, Iliopoulos and Maiaai, nys. «er. K, l8j
In 1960's the unified model* and Yang-Mills type theories vere not particularly popular (however, t remember Veltmen giving • series of lectures in a Copenhagen summer school in the late 60's). Die unified theories beesas fashionable only after t'Hooft (Amsterdam International Conference 1971) demonstrated that the spontaneously broken gauge theories are renoraalisable. Two years after, in 1973, neutral currents vere discovered at CBU. Sine* then the unified theories have become mor* and more the standard framework for describing nature. nowadays, it is believed that, the weak, and electromagnetic interaction are described by a gauge theory based on the (non- abelian Lie) group 80(2) x Vil) and strong interactions follow a BU(3) in colour. One goes further and advocates a unification of all nongravitationol interactions, for example as in the SU(5) model of Ceorgi and Glashow (Fhys. Rev. Lett. JS, 1*38 (I97*)h Of course, ve have not yet seen any intermediate vector bosons {excepting the photon) nor any Higgs particles, which are the back bone of the modern unified theories; we have not had enough energy to produce any W or Z°, etc. The fact that the experiments are in an amazingly good agreement with the simplest gauge models makes us believe that ve are on the right track; with enough energy we shall be able to admire the glorious intermediate bosons. i>
s. lmmaaxm or a) The fundaaental constituent! of aatter (leptons and quarks). b) Intermediate particles. c) Higgs particles. a) The fundamental constituents of aatter are feraions vitb spin }. There are tvo types of constituents: leptons and quarks (hadrons). The lepton family contains the observed ones w, e~, v , u~, v and T~ (and of course their antiparticles). The evidence for v , although indirect, is substantial. Leptons are integrally charged. Toe quark (hadron) family contains the "observed" quarks up (u), dovn (d), strange (s), charm (c) and beauty or bottom (b). The need for an additional quark, called truth or top (t) is ur gent. '*'-•» hope that Petra vill discover the truth. In the conventional approach, each quark exists in three var ieties (colours). For example, there are three up quarks, etc. b) In gauge theories, the existence of the constituents of matter, together with the Holy Principle of the local gauge invariance (discussed below), leads to the existence of intermediate vector (spin 1) bosons. Their job is to mediate interactions. The family of mediators contains Maxwell's photon (the only member directly seen so far) vhich mediates electromagnetism. We believe that there are further members, such as W*, Z which mediate weak inter actions at lov energies; gluons C., i»l, ..., 8 vho mediate strong interactions, etc. Mediators are integrally or fractionally charged depending on the theory. c) The Higgs particles are theoretically needed betes noires. Some day, perhaps we will learn to live without them, but so far their existence is indispensable. They are believed to be the origin of all masses except their own. We shall discuss their properties in section 10. 3. SYMMETRIES AND CONSERVATION LAWS The Holy Principle in gauge theories is the Principle of Local Gauge Invariance, as we shall discuss shortly. Before ve do so, 1st us remind ourselves, that th» concept of invarianc» (or covari- ance), which is vsry essential in physics, is osatly foraulated in Lagrangian languag». For «xaapl», th» principl» that th» lavs of physics ars independent of th» particular Lorsnts fraas chosen re quires th» Lsgrangian density to b» s scalar quantity; the equa tions of action aust b» covariant, »tc. [Exercise: State the physical principle» leading to the lavs of «nergy-aoaentua conservation, angular aoaentua conserva tion and parity conservation (if it had been true). Hov are these conservation lavs guaranteed in the Lagrangian formal ism?) Hot all conservation lavs have to do vith space and time. There are a rather large number of physically estsblished conserva tion lavs, vhich pertain to "internal" properties. Let us give a fev examples: a) Conservation of charge. b) Conservation of baryon number B, and lepton numbers Le. !• and LT. c) It is commonly believed that colour is an exact symmetry, and only the colour singlet states are observable. There are hovever theories in vhich the colour leaks out. Experimentally, there is, up to nov, no evidence against the conservation lavs a - c. Approximate conservation lavs have also been observed in nature, for example d) conservation of strange-, charm-, beauty-(ness) by strong and electromagnetic interactions. These are violated by weak interactions. e) Isospin symmetry in strong interactions. f) SU(3)-symmetry, again in strong interactions. We leave it to our readers to ponder on the physical prin ciples responsible for the symmetries above. We may ask whether the conservation lavs a) and bl are exact? Are all physical states, vhich go through our bubble chambers, calorimeters, etc, colour singlets? We shall return to some of these questions later on in these lectures. Let us conclude this section by simply noting that the experimentally established conservation lavs are easily incorpor ated as symmetries into our Lagrangians, even if ve do not under stand their origin. «. LOCAL GMJGB IIYARMM» ARD OjUAIHW UKIROIKIIAMICS (QH>) Vt «hall nov construct the siaplest «xaaple of a (auge theory ••ploying the principle of local gauge invarlanc*. Suppose that ve bad only a singl* Baseless femion f. then the fres Lagr angi an, describing f is given by t\ - i •f(x)Y|i £- »f(x), (1,-1 > Dote that i -r— is the four aoaentum operator in quantim Mechanics. P We often refer to this tera ss the kinetic energy tern, in analogy with L • T - V, where the Bass tern goes into V. JL is invariant under the global gauge transformation iA »f(x) + e vf(x), (s-2) where A is a constant, viz. X> •» e-1 £ e1 «Jt . This trans formation is called global, because A is a universal constant, in dependent of space and tine. The Holy Principle in gauge theories is that the above invuri- ance should hold locally, i.e., for A » A(x,t), where A is a real function of the space and time. In other words the phase of tp_(x) is unobservable and may be chosen at will. If we accept this law, we must modify our Lagrangian, because iA(x) *f(x) * e *f(x). (1,-3) implies jt •+ JCI - * (x) Y « (x) ifl^x' whereby the Lagrangian is not invariant and the phase matters. Since in the additional term T— is a vector, we introduce a vector field A (x), and replace Xi with JL given by M 4 = i *f(x) yv [•£- - i e A (x)] *f(x) (k-1.) where c is an arbitrary constant. ITie local gauge transformation is now iA(x) -iA(x) p V *r(x) H- e *f(x), *f(x) •* e *f(x), A (x) •<• A^x) + 6/L . {k 3) Requiring «C, to be invariant gives ' "V-k^- («-5-) This simple example illustrates the major feature of gauge theories: the requirement of local gauge invariance (in our case under phase transformation U(l) implies the existence of the inter mediate (gauge) bosons (here the Av field). A" Bay be interproted as the photon field («hereby c • e Of; e f unit of charge and Of s the charge of the fermion f) if ve are alloved to ad* a kinetic energy tera, for the photon, in accordance vith the gauge principle. The tera (Y) XJ • T7 F FVU , F " 3 A - 8 A (ll-é) o It uw * uv p v v |i has the desired properties. Notice t'uat the photon mass oust vanish identically, because a mass tera * 1 a* A A violates the rule {U-%) and is therefore forbidden._ A feraion Bass term, on the other hand, may be added . •• - mf tf(x) *f(x), since it is in variant under Ct-5). Collecting all term ve have the Lagrangian of QED L* * *fW \\k ' * e V"] *TM - ¥ F„v F"V - - mf*f(x)i|>f(x) . ;it-T) Putting f * electron, Qf « -1, gives QED for electrons which has been the most successful theory ve have ever had. The fine struc- e2 1 . ture constant o * -r— « r= is fortunately small: furthermore, the *m JLJI perturbative approach makes sence (the theory is renormalizable). The basic interaction in QED is represented by the diagram The constants I e | Q and m^ are not predicted by the theory but determined from experiment. QED is an example of an abelian (named after the Norwegian mathematician Miels Henrik Abel (1802-1829) gauge theory, i.e., the A's (if ve had several of them) are just c- numbers and commute with each other. Note also that the gauge boson (Y) carries no charge and the coupling constant Qf is not universal. The latter is seen by taking several basic fermions such as u, d, e, y, ... and going through the arguments above, viz., 4 * *, = *I *ilx\ [»£- - ie(lJ A"(x)] *J(X) (I,-8) •j(x) * e * * ^(x) . AM(x) • Ap(x) • «AM(x) (k-9) Substituting (k-9) in (MM together with the requirement of in- variance yields Aj(x) - Qj A(x) , «A"(x) - i^- , (k-10) where A(x) is arbitrary. The point is that the phases A.(x) of the fermion fields are no longer unobservable and irbitrary, we mist choose them proportional to their empirical charges! Ot course, there is nothing in the theory which tell us why 0_ • 2/3, Oa • - 1/3, Q, • -1, etc. The gauge group U(l) nas this ugly arbitrariness, i.e., it allows an arbitrary coupling constant for each fermion (or group of fermions, see below). 5. WEAK INTERACTIONS, QUANTUM FLAVOUR DYNAMICS (QFD) In physics slang, each kind of particle is said to distinguish itself from other kinds by having a particular flavour. The ob served leptonic flavours are ve, e, v„, ..., and the hadronic fla vours are supposed to be u, d, s, ... . ELectromagnetism is flavour blind (viz. ^5""/'). f "Flavour dynamics"shows up in weak interactions, where the charged currents are flavour-changing. For example an electron-neutrino is turned into an electron, etc. In this section we discuss the conventional theory of weak interaction, for four quark flavours and four leptons, i.e., the theory that we believed in until 1975, before the T was discovered. The charged-current weak interactions were described by the Cabibbo-theory supplemented by the Clashow-Iliopoulos-Maiani (GIM) mechanism + h.c. + d •»1 {"« *J ™-*5>»i °ij j /(l-Ts^l Wx + h.c. (5-1) i,j=l,2 Here W^, v., etc. refer to the field operators for W , neutrino, etc.; g is a constant S2/M?L =», M„ • mass of W-, 0 « Fermi constant. Furthermore «1 "V . V2 "V u Ul •U tt2 " e i, • « *j • M dj • d d2 • B The o's are given by U) (h) /co89c "nec\ fl • Cabibbo angle. \-sin9c coe8c/ y The elegant form ($-1) had gradually emerged from several decades of research on weak interactions. Each piece has a raison d'etre and X, > whenever applicable, gives a good description of data. After the discovery of the T-lepton and the quark flavour b, it is believed that we should modify (5-1) by introducing v3 " «T, l3 • T, u3 » t, d3 » b (where t is a to be discovered quark flavour) and let the sum over i and j run from 1 to 3. Furthermore a.. » 5. - and a is a general three by three unitary matrix. We shall discuss the four and six quark models in detail later on. After the discovery of neutral currents in 1973, we know for sure that the JS° above cannot explain all weak interactions. As ve shall see in the next section, the gauge theories provide a natural scene for neutral currents. What is much more, they allow the unification of weak and electromagnetic interactions which is, of course, marvelous. Moreover, it has been known for quite some . time that the theory described by (5-1) is plagued by awful dis eases, viz., some physical cross sections and transition probabil ities are infinitely large and unceiculable. Gauge theories pro vide a framework in vhich the diseases are cured, one gets sensible expressions for cross sections, etc. Has Nature solved her prob lems by utilizing some local gauge theory? We hope so, and are en couraged by data which are in amazing agreement with the simple gauge model described in the next few sections. 6. CONSTRUCTION OF THE STANDARD SU(2) x U(l) MODEL In this section we describe the construction of the standard model of weak and electromagnetic interactions, the so-called SU(2) x U(l) model. The model is "standard" in the sence that it seems to explain all data on neutral currents (a task vhich no other model can fulfill) provided we forget the confused situation in atomic physics experiments. Let us, for simplicity, assuae that there vera only two ele mentary fermions in nature , denoted by f and f and that Qf • Qf> * 1, where Q is the electric charge. These assumptions are. based on the observation that nature exhibits several such couples. For example (f,f) could be (ve,e), (v„,|i), (u,d), etc. We now construct a gauge theory, proceeding along the main road laid out in section h but have to make appropriate detours and modifications. As in section It ve introduce the kinetic term for the (nasslesB) f and f • yttf.f) - i[f Y f; f + t'r ^ f] . f - #f(x) , f - *f,(x) (6-1) We remember (see eq. (5-1)) that the charged currents encountered in nature are of V-A form. In order to account for this fact, we decompose the fields into L = left and R • right components f = |(1 - Y5)f + |(1 + Y5)f » fL + fR , (6-2) f (1 f L,E - i * V • and similarly for f . Substituting into (6-1) yields f f + f + 4(f .f •) = i[v I; L * V h R *frk L * h 4 Again, we look at^C°c, eq. (5-1), and observe that the left-handea fermions (f. and t!) seem to be intimately related to each other; they go into each other by sending off a W. For example, in (5-1) X X *j Y (1" Y^v. = | 1.(1 + Y5)Y*(1 " Yj^ = 2ljLY viL . Therefore it is natural to write L f f X0(f,f-) = i {iiL.iih fe f ) • fR Y h R •'R y h B} • fL (6-U) Now we shall impose the Local Gauge Principle. Here, we require that neither the "direction" of (fi ]• in a weak isospin space spanned by fjj and f£, nor the phases of the three fields •l * (fl) ' *2 S fR ' *3 E fR (6"5) L See section 12 for generalization to several f's and f *s. u should be aaasureble. Ihe gang* group is, therefbra, 8U(2) x U(l> and J^tf.f) aust be •odified to ba invariant wdar ^(x) * {axj(i a(s) • D a*p(i »j(x))} »j(x) . {6-6) 8U(2) U(l) Hera atx) are the three (real but otherwise arbitrary} parameters of the group 8U(2j; 3 are the Pauli aatricss, ft-*l'«J*]o _ | represent the three generators of 8U(2); 5 *2 • 5 v3 • 0, because *2 3 are singlets. Finally 6,(x), j " 1,2,3 are arbitrary real functions. " JL . as it utand», is not invariant under (6-0. Proa section U, we Knov that ve must introduce three vector bosons (one for each generator) for the SU(2' group and one additional vector boson for the U(l) group. Hits ve replace 3 . with j j«l I ax9X " Here w" » (wJ.w(W1»11,, W*'W2»uu, Wtf3»*3' 1); g and yj are four constants, and the redundant constant g' has been introduced for convenience. Note that the Vf's couple only to the doublet e •* g' , Q, +y. , A.(x) + B.(x) j j «j j whereby we have invarianee provided 6.(x) = y.B(x). and the gauge transformation of the B field is according to"1 B" * B" • i, afial . (6-8) 12 H»re B(x) is an arbitrary real function. It mains to add the kinetic tara «£ (B) » - r BMVB , B » » B - » B (6-9) o a uv • uv uv w n and the "U(l) problem" is solved. ' Ve now fccus our attention on the 8U(2) part, vbigb only affects tj. The 8U(2) group is a nonabelian Lie group , i.e., the commutator of generators,(here represented by o's) is nonvanishing and equals a linear combination of the generators. TO repeat, the question is how should W transform so that £j is invariant under *l •* exp(i a(x) • g)t|T The answer to this question vas given by Yang and-Hills (Fhys.Rev. 26.19U1951*)). The reader aay check that for an infinitesimal transformation (small o(x)) one gets 3a w" + « oxWM (6-10) -y g 3x • k 1 «J •+ W" + i c j - 1.2,3 M g jkm a u 3x" Note, since W and o are vectors, eq. (6-10) expresses the most gen eral behaviour expected, viz., W gets translated as well as rotated. We now must supply tf with a kinetic energy tern which should remain invariant under (6-10). Again, here the analog of (6-9) does not vark. Indeed, the requirement of invariance imposes self- interaciio» for W (the reader should check this) 3 X M = - £ V «J HWU»J - - £ w . wuu , (6-ii) o 4 [_> uv « pv ' 3-1 where «iv - \ < - \ i * ««jta *X (6-i2) The self-interaction diagrams are W™ W* W* * Sophus Lie (18U2-1899) was a mathematician from the most beauti ful fjord area in Norway called Sognefjorden. I wish the inter ested reader will someday have the chance to visit that area. Let ui ausaariie: the interaction LBgraagiaa ^ - i 7 •j(x)y» [-4j - i «| . BB - i «^BJ tiU) (6-13) is invariant under the cauft transformation • ,(x) * |.xp(i S • |) .xp(i ^«(x))1 BM^BM+ 1 »|Ixi C6-1») 5" * V» + ±iJs . 5x5". •.-0- •j " fB. *, " *i Furthermore, ve learned hov to add kinetic t-rns for the B and V fields. However, these particles must be nassless, because a mass tern-r M| B BV , etc. violates the gauge principle, eq. (6-lU). T. UNIFICATION OF WEAK AMD ELECTROMAGNETIC INTERACTIONS IN THE STANDARD MODEL In the last section ve constructed an SU(S) x U(l) model which has the correct form for the charged currents (up to mixings). The charged current interactions are mediated via W . From (6-13) *- = g *j Y *-£ R *, W1-iW2\ ft' V V i-r. : | If.? (»'X^> w'+iw2 o U li " zfe {? '"(l-Tjlf W" + h.c.} , (7-1) P Æ \ v v.} Ik JCf above doaa not daaerib* tb* raal world, baeauaa Vauat ba •assies* giving long-range forces, vharaaa, va know that vaak interactions ara abort-rancad. Latar on va aball a** that va can hava both awaive V'a and tha Lagrangian daaeribad abova provided vs introduce the aaai via the lo-ealled Biggs «echaniss). Let us aaauaa that ve had done ao. Then, a ceaparivon vith the local V - A Lagrangian for «i-decay yield! -Si.-S. (7-2) 8Hj Hov t.e turn to the neutral current», which result fromJC,, eq. (6-13). These are mediated by B„ and w'. Since these objects are massless and as yet unphysical, ve may for» tvo orthogonal linear combinations of them denoted by A„ and Zp "By "BZ (7-3) *WY *WZ where R is an orthogonal matrix with elements a^ , etc. We shall show later that via the Higgs mechanism, mentioned above, A and Z could be made to represent the physical particles, where A (Z ) is a massless (heavy) particle. Vfe wish to identify A with the photon field, therefore, we must require that the terms in£,, eq. (6-13), containing A must add up to give the electromagnetic Lagrangian, A, , V JL^ = i £ ^ y (-ie^ihv <= e{tlfVf + QflrVf} Ap (7-1.) j-l Here Q. is the (matrix representation) of the electric charge, viz., /*r (Qf " I) ' VQf • W • (7'5) «,.' 15 Tha thraa ralationa io (7-5) ara auaaariaad is tba oparator fom <» - y • * . (7-6) when *j' VJ. **im VJ (7-6a) *i • «r - 5 • y2 • «r • Y3 ' «r• Thus •teB *e X *j Y" «*j \• e j,,,e" AP • (T-7) Now ve calculate the term in JL vhich contains A . Using eqs. (6-13) and (7-3), ve find 3 Comparing (7-8) vith (7-7) yields '3 e + g y (7 9) «j ' « -wy T ' "BY j • ' eff * *j) • « Vy T * «' %yj (T-9a) The relation (7-9) is valid provided 1 8'n«8 and e X. = g aBy y^ . (7-10) Remembering that the (redundant) constant g' may be choosen at vill, ve put (7-11) « "WT = 6 = E' aBY yj = Yj (7~12' 2 2 By orthogonality of the matrix , (a„ ) + (a„ ) = 1. Thus e*(£ + ^)=l. (7-13) s Putting SL, - in&w we have the famous relations « " Sine; • «* - ZZi- ' ll~ where 8„ is usually called the Weinberg angle. I*t us summarize: the SU(2) x U(l) model contains tvo neutral mediators U* and B„, from which, by a rotation, ve amy Ion the fields A. and Z^. fee model contains four "coupling constsnts", viz. £, gV. sod a para meter, 6„. Identifying A» with the photon field fixe» these constants, viz., g - jAg- , g' - 53*5- , y, - Q, - } , y2 - Q, and y, « Qfi. Thus the model introduces one nev parameter, the angle By, vhicb is empirically' determined from the neutral current data. Rote that the U(l) group is rather unpleasant; only for a very specific (and incomprehensible) choice of the phases fi:(x) " yig(x) vith y: as given above, do the particles f and f* obtain their ob served charges. The hope'is that the SU(2) x U(l) model is a sub structure in a more elaborate scheme, vhere the origin of the eigenvalues y; and the quantization of charge is explained. The SU(5) model of weak, electromagnetic and strong interactions (see section Ik) provides a good example of hov such an elaborate scheme might look like. The SU(2) x U(l) model unifies weak and electromagnetic inter actions in the sence that Av and Z„ are intimately related to each other (eg.. 7-3) and to Wp. The weak and electromagnetic couplings are of the same order, viz. g * • _ . ' Thus the apparent huge differences in the strength of these interactions are due to the differences in the masses, H - 0, y „ , (ÆjåY = ( 52y V „ 38GeV "w Uo ; \^asin\J " |sinew| MW We shall see later that M_ = — , if the Higgs mechanism is done • W a la Weinberg (Phys.Fev.Letters 19,126!i(l967). Thus W and Z a*-e beyond the reach of the present accelerators. 8. THE WEAK NEUTRAL CURRENT COUPLINGS IN THE STANDARD MODEL We go back to the interaction Lagrangian of the standard model and, using eq. (7-3), determine the term proportional to Z J o, Using g By = g' a_ = e and Q = -^ + y , obtained in section 7 IT v* rind ^ * 5 (3-2) Since ft i» an orthogonal matrix and ve had introduced »in9y • ay , ciiSy • aBY> ve nat have det(ft) • 1, and aBZ » -ainSy. Therefore, A"" " Hafsa; ftt, J •. - .i»\ C] •"• V" ' * ,f, <8"3> where • , • ( 1 ; J is the electromagnetic current, defined in eq. <7TJ), end jjj*0- is the weak neutral current. We may rewrite J as C" W+ VVm VWL + W.'+ - Thus where the left and right coupling a, „ are defined by aL - I3L - Qsin2eH (8-5) Here Q is the charge of the fermion involved and 1^ is the third component of its veak isospin (^^L = i **or **an< * *3L = "^ for **'^* Although (8-**) vas derived for a hypothetical couple of fermions (f,f ) with Q = Q-,+1, the relation (8-U) is valid also when there are several such pairs, provided they are all treated on the same footing. [We shall return to this question later on (see section 12)J. Accepting +,his Tact, we may immediately write down the neutral current couplings for any pair of elementary fermions. For exupl* (i) f.f• - v. - i£ -1 = -i^ , a9 - -i . ^ - o (6-6) di) f.r- - u,d - «^ - § , cd - - i . 15 - -1£ - i «t • 2 - § 'in\ ••*"-§ BinZew • •£ • " I • 3 BinV (8-7) Furthermore all sequential neutrinos (negative leptons) have the same couplings as v (e). Similarly, in the hadronic sector, all charged 2/3 quarks nave the same couplings as the up quark, etc. Jj We may derive a local four-fermion interaction Lagrangian from * ", eq. (8-3), provided the Z is very heavy J^.c. „ _g _G_ U f.¥ L f. + J f. T B f. > XiaL?k^Lfk + 4fk^Bfk}> <8"8> L = |(l - Y5) , R =|(1 + YS) • The relation (8-8) describes the neutral current interactions of any two fermions f; and f^ , f; * f^ . For example to get the neutrino-up quark interactions we put f; = v, fk = u, a? = ai", aL = aL* e*0- "ote that we have used the Weinberg relation My = cosøjj • Mg. For f; = fk the factor 2 in front of G must be removed. °. SPONTANEOUS SYMMETRY BREAKING (SSB) In the previous sections ve discussed tvo examples of local gauge theories, namely QED and the Sl)(2) x U(l) model. Indeed the unobservability of the local phases of the fields led in a natural way to the existence of interactions, mediated via massless spin one (gauge) bosons. This masslessness of mediators, necessitated by the gauge principle, is just fine for QED but catastrophic for 19 weak interactions. He knov froa experiment» that the weak aedia- tors (W*, Z°, etc.), if they exist at all, are certainly heavier than 10 OeV. At this point one sight take the attitude that gauge theories, in spite of their beauty, Ward-identities which ensure renormal- izability, etc. are relevant for QED but not for weak interactions. Within the last few years, only a small group of brave physicists have dared to work along this line. Unconventional models for weak interactions have been constructed, but so far no viable alterna tive to gauge theories has been found. The mainstream of activity has gone in the direction of taking for granted the gauge principle in a world where there are no masses. In the real world, the gauge world, the gauge symmetry is, by assumption "spontaneously" broken whereby the masses are created. Below, we give a short survey of the SSB as it is employed in particle physics. The phenomenon of SSB is known to occur in several branches of physics. Usually systems exhibiting SSB have an infinite number of degrees of freedom. Since field theory has also an infinite number of degrees of freedom, it might well be that the theory of constituents of matter is a spontaneously broken one. Let us consider a system, lescribed by a classical Lagrangian «£ which is invariant under a continuous group of transformations G. We examine the ground states tf the system. If the system possesses a unique ground (vacuum) .state (which must be invariant under 0) we have a situation in which the symmetry properties of «t and the vacuum are the same. The relevant theory is a normal one. However, it might happen that the system has several ground states, which transform into each other under G. Then, if by some reason, one of the ground states is singled out as the physical ground state of the system (the others being unphysical) the sym metry is lost and the relevant theory is said to be spontaneously broken. It is known that the spontaneous breaking of a global symme try leads to existence of massless particles (excitations) called Goldstone bosons. Very fortunately, our gauge theories are based on invariance under a local group of transformations. When the local invariance is broken, instead of getting additional massless bosons, we get massive gauge bosons. We shall now describe the SSB mechanism in a little bit more detail. 20 9.1 nie Spontaneous Breakdown of a Global Symmetry A nice example of the S8B phenomenon ia provided by the Gold- atone model. Let us consider the Lagrangian 2 X. . + *£- -*£ _ ,2 „,*„ . h^, = T _ v , * » 2 • V • + m2 pip + h(tpip) , (9-1) where ipj, i = 1,2 are real scalar classical fields and n2,h are constants. We shall only consider the case h •- 0, because other wise V(|ip|) •* -• as | For m2 > 0 we find V 2 0, the minimum of V corresponds to | The unique vacuum 2WZ = Thus 'the system has an infinite number of ground states, which 'lie' on a circle in the cPj - ' Suppose now that we choose on* of the (round states, for ex ample *0 f (ipj » v.ipj • 0) as the physical (round state then the syaaetry is spontaneously broken» because the state «fe is not in variant under the transformation (9-3)• To see the structure of the theory, we expand the potential hear the vacuum state, where the theory is expected to be stable. Put •I «&«.•*£ (eft).*.- l i-1,2 ° i,j«l,2 ° (9-5) where the index o indicates that the quantities are evaluated at the point 1 *" J * I'O [-M* • hOpf • «j)].y * 2^.«,., (^-) - 8h( auxaip. L ' '^i V2'J ij i j' N^i^j/ o i JVj O (9-6) Thus the mass matrix, defined by i Jo «>0 = (ip, = v , tp2 = 0) has only one non-zero element, m2 = 2h v2 = —r*— = 2ii2, while m12 = ra21 = m22 = "• Therefore the 'particle' V((P ) = v2 ipj2 + interaction terms * , V""" 3(p! 3ip! *£" ( The spontaneous breakdown of the U(l) symmetry has led to apparition of a massless (Goldstone) boson, represented by the field «Pj. The field tp! has the mass <^2\t . The reader may wonder what-happens if r.he would choose a different vacuum, say *' = Ci = -r= , V( + 0«( Thus the linear combination TjCwJ + The arguments in this section can be easily generalized. r *?i Suppose we have a Lagrangian,<&( Look for the vacuum state by using —r—— = 0 . Suppose that KPj there are several vacuua. These will transform into each other under the group. Choose a particular vacuum / <*, and disregard the rest. Under an infinitesimal n'o action «Wof tho e croup the vacuum is changed by the amount 6 T* The potential is invariant under 0 inplying «V-^ftp, -0 , .0 i.e., ipT^k^K-0 '-1 » <9-u> Take the derivative -r—- of (9.11) (9-12) *P,8»i jk k 3tp. ji At 1* • •VjX*,"0 r-1, ....» (9-13) By our assumption Tr«p * 0 for r = 1, ..., n'. Thus Trtp span a n' - dimensional subspace in the N dimensional space spanned by the IP's, and therefore, m .- has n' zero eigenvalues, or in other vords there are n' Goldstone Dosons in the theory. Let us apply this formalism to the special case considered above (P-* ( 6ij> = iAip 6(Pj + i«tf>2 = iA((Pj + i 6 Thus the generator of the U(l) group in the | ) space has the form ,0-1, T = I I and M. 0' 24 -!j VVo-° w! 4% "JU From here ve obtain »|2 • a|2 « 0 , i.e., the When a local symmetry is spontaneously broken the situation becomes quite different, as we shall discuss below. In quantized field theory the SSB is assumed to be realized by letting the field operators acquire vacuum expectation values *i " *i + vi ' <0l*il0> " ° » <0l*il0> " vi • ^ c-numbers vj are chosen such as to minimize the classical potential. 10. SPONTANEOUS BREAKING OF A LOCAL SYMMETRY We start with the same Lagrangian as in the previous section (eq. (9-l)) and require invariance under the local phase trans formation ant derivative 3 - icBu, where c is a constant Å • (6+iBB/x^ - ***) -v( ** is invariant under the joint gauge transformations Again h is taken to be positive. For m2 > 0, «t is the QED Lagran- gian for a self-interacting charged spin zero particle, if we identify Bu with the photon field. The vacuum of the As before we turn to the interesting case m2 = -u2 < 0 , and ?5 assuae that the around state is defined by the ainisnai of the po tential V(«), -T™ • 0 , «hereby ve obtain the relation (9-k). i W? • «p2 " v2. Awns these infinite nuaber of vacuua we select one, e.g. iPj • v„ - ^ B B,,IV + interactions terms involving (10-3) The Lagrangian (10-3) is indeed remarkable. It seems that we have a massive scalar tpj (with mass Æu), a massless one spin BLater, Sz •= 0, ±1. Thus there is a redundant degree of free dom in (10-3). Indeed the massless = 2 We may now choose A such that lA Jt-I B B — - i^fo, • v) '1 2 "M " T p„ ° V ' llv2 1 •ote that gauging away (p, instead of IP, lead* to an unphysical From (10-lt), Hg » jgv|, Hp, » Æu, where M refers to the mas*. A describes a aaasive vector particle interacting in a very speci fic vay with a real scalar field. In the quantized theory the pre scription is as given as the end of the last section. The particle V is usually referred to as a Higgs particle and the mechanism described above is an example of the Higgs, Kibble, etc. mechanism. It is widely believed that all masses are generated via the Higgs mechanism, the reason being two-fold. Firstly it is appealing to imagine that a beautiful massless world built upon the local gauge doctrine suddenly undergoes a spontaneous breaking whereby all masses are created. If the reader does not buy this argument the second one can be given, i.e., no alternative has been found so far. Ever since the heroic discovery, by t'Hooft, that the SSB does not spoil the renormalizability, such theories are much cher ished. For each specific gauge theory, a spontaneous breaking scheme has to be cleverly invented in order to achieve the purposes in mind. We shall give some examples later on. It is fair to say that the Higgs sector of gauge theories usually does not match at all the beauty of the rest. The Higgs sector introduces new parameters, sometimes many of them. Let us hope that some day we will understand the origin of the masses. 11. SPONTANEOUS BREAKING SCHEME FOE THE STANDARD SCJ(2) X U(l) MODEL The standard model, described in sections 6-8, has initially It gauge bosons W-, W° and B. The breaking mechanism should be devised such that W* and (the linear combination of W° and B corre sponding to) Z° become massive! whereas the photon remains massless. Thus we need to introduce at least >i real scalar fields ipj. These "Higgses" must transform as multiplets under SU(2) x U(l), so as to couple to the H's and B. We must make sure that the Higgs poten tial V( lr,t « 1,2,3t and y, where Q • T5 + y. Clearly v should be invari ant only under Q, then 3 "ayamtries" are broken and the corre sponding Goldatone bosons aay be gauged away to yield three Massive bosons. The four Higgaes are introduced elegantly, a la Weinberg, in a doublet + ( + i«P ), where ip. are real. The Higgs Lagrangian now reads •£ - |(£ - ig |. W- ig'yB^ip) - V( where the requirement of local gauge invariance under 8U(2) x U(l) has given tjje specific couplings to V and B. We nay write V(ip) - -p2(P ip + h(ip r * vacu.im * ° tp • tp1 + v v0/ 1 J (gf • 5 + c'yB J v i »hw «"U (11-3) Here m|: is the mass matrix. We check easily that the photon is massless, T = " , Vl(« V T3+ g'w) v\ * v1eQTyl = ° • lll'u) The mass of the Z is obtained from ,2 P= M Z Z,J (11-5) |(«-v z *3 • «'-B y)»| V I Z P • Using y = Q - T3 , Qv = 0 we find v e MI (11-6) Finally, the mass of the charged W's SB I O,W>*0,V2 I2 » O2 _2 O2 ^ s ' ;' v - *£- (wI+iw2)(w,-iwp - *^- w*w . Krom (11-6) and (11-7) ve obtain the Weinberg relation 2 2 Mj - a yM| - co» ø H| (11-8) Using the empirical relation M„ • y~. ?t ' we find that for the present preferred value sin28 « r- , the mediators are awfully heavy fy ~ 76 GeV, Hg ~ 88 GeV. The analysis of this section can be generalized. Clearly the photon remains massless as long as the electric charge is conserved, viz. Tu K v = 4 H. W. m?. !• a,, A + V AMA I Ye- a,, T. vf =im* AWA Here g, T and W are the coupling constants, generators and gauge bosons respectively. The sum over j involves only those terms V «J V.Y = e , and Q' = ^y T. . Thus mlm^ ~ |(J vl = 0 J j because v is electrically neutral The Higgs doublet (ll-l) is sufficient for giving masses to all fundamental fermions. Of course fermion mass terms m4n|> do not spoil renormalizability and could be added by hand. Nevertheless, it is more attractive if all masses should have the same origin, say Higgses. Suppose ve vant to give masses to up and dovn Quarks. We have the multiplets /u\ , u_, d„, gauge invariant Higge-faraion interaction reads X • «d<»-5>L (£) ** * eu<»'5>L J) «. * »•- <*«> Upon spontaneous breakdown cp° •» •£kss " (cdv°» *A + < V"* VR * h-C- If the c's are real (this can always be arranged as we shall dis cuss later) Thus •£ = m.dd + m uu+— ddV + — uu*° (11-10) d u v v° A very specific prediction of the spontaneously broken SU(2) x U{l) model is the existence of the scalar boson 12. MORE QUARKS AND LEPTONS, MASSES AND THE CABIBBO ANGLES In this section we start by reminding ourselves that Nature seems to be repeating herself and so far we have failed to under stand her reasons for doing so. In fact by now there are three distinct families of leptons and quarks, viz; the fundamental family ve, e, u,, u,, u,, d,, d,, d, the second (superfluous?) family vy, p, Cj, C2» C3, Sj, s2, s3 the third (super superfluous?) family vT, T, tj, t2, t3, bj, b2, b3. (12-1) The fnrailies are also sometimes referred to as generations, however, there does not seem to exist any mother-daughter relationship among them. In (12-1), the indices 1, 2, 3 refer to the colour degrees of freedom. As discussed before, the evidence for vT, although sub stantial, is still indirect and the truth (t) remains to be dis covered. In gauge theories, one normally follows Nature's own pattern: a repeated world including n (super)" fluous families is simply described by repeating the theory constructed for just one faaily, for exaaple the fundamental faaily. (this state of affairs is, of course, not satisfactory.) Bere ve shall asstaw that there are n families, V V V dj i"1' —" Q - 0, -1, 2/3, -1/3, (12-2) where ve have suppressed the colour indices. In (12-2), for ex ample u, - u, u » c, u, • t, etc. Furthermore, ve shall assume that the standard SU(2) x U(l) model is correct, the left multi- plets are doublets and the right-handed ones singlets. In con structing the theory, ve follow the steps in section 6, including of course all the elementary fermions in nature, (12-2). The kin etic energy term reads 4 - *£ [ViUJ + Vi* ii \v£UJ + v& dJ • j=1 (12-3) where the sum over the colour has not been explicitly exhibited. This relation is the analog of (6-1). The decomposition into left and right components may be done exactly as in (6-3). Since we have more than 2 elementary fermions, we do not know, a priori, how to choose the doublets, viz., there are infinite number of possibi lities and the physics depends on the choice made. Of course, because of the charge assignments, we cannot put leptons and quarks in the same doublets (otherwise ve would get W's with fractional charges). We now restrict ourselves to the hadronic sector. The gener alization to leptonic sector is trivial (see below). Due to our lack of knowledge, the most general choice for multiplets is to take (arbitrary) linear combinations *JL = ** f = 1,.. . n (12-lt) where (u.L) (u,B) UjL " Ajk \L u jR " V \R (12-5) (d.L) , (d,R) ajL=Ajk \t djR = Ajk «u . J-l.-..». Here the A'c are n x n aatrices. They have to be unitary in order to leavea£Q invariant {we «hall return to the origin of these Ma trices soon). Suppressing the indices, etc. j J 3 (ia-6) -1 A* » A , for each A>u, A, " *£ fa ' aj>L»£ ^ "jP * £ UjH + V-H}' (12-7) Now we "gauge" the kinetic energy term, whereby the mediators W and B are born (just as in section 6). Thus (see eq. 6-7) is sandwiched between the fields in (12-U) just as in (6-7); - - °3 . remembering o u'.R = a d!R » 0 , y = Q - -5- , Q is the charge matrix, o » Pauli matrices, viz. /u''\ £ ( 5 D X *£ { v X»x Vt*-;»TXD ^ + r.Ryy d.j, J D=^-igi.W-ig. yB <»-« So far all the (matter as well as gauge) fields are massless. According to our assumptions, in the real world, the SU(2) x U(l) symmetry is spontaneously broken, whereby the masses are generated. The breaking mechanism is assumed to be a la Weinberg, i.e., due to a doublet of Higrø;cs. Thus we may repeat the arguments in section 11. 'ltie quark - lliccs interaction (see eq. 11-9) before SSB reads ^(quark-Higgs) =£{cjk(u'. , dj^J u^ + c.,(=", d') (Jd^} + j,k +. h.c. (12-9) Here the c's and c s are arbitrary constants. The SSB is achieved by letting (fP -+ ipP + v and dropping terms involving where /a I !\ "L.B 2 ^.R* ' Mjk " v cjk • Mjk ' VCjk d' Pn/ L,R I n L,R Now we see clearly the significance of the matrices A in eq. (12-5). They originate from diagonalizing the mass matrix and getting rid of the pseudoscalar terms in (12-10). For Q » 2/3 sector, we write M' in "polar coordinates" M' * m U , (12-11) where m is hermitian and U is unitary, m is diagonalized by a transformation m = ST^D S, where S is unitary, and D is diagonal and real, "L M' uii * ll,c" = "L S~1D s U "B + h*c" We put uH = B U «J (12-12) UL " S UL , u, M' ul + h.c. =uLDuR + uRDuL = uDu Thus the elements of D are the physical masses (12-13) Evidently, the charge -1/3 lector can be diagonalized in the saae way. Comparing (12-12) with (12-5), yields A(u'L) - S+ , A(u'R) - V* S+, etc. We now return to the interaction term in (12-8). As we have just seen the primed fields are not the physical ones and therefore we should express then in terms of the physical unprimed fields, by utilizing (12-5). It seems that ve then get a very complicated expression. Fortunately, the generalized GIM - mechanism is oper ating here and there is no trace of the matrices A in the neutral current sector, viz. J 3Jh j (12-llia) "2l{"jLUjL-3jLdjL}« j _ I "jR y UJR " L % 5JH ujR * 1L »J« UjR ' <1*-lkb) 15jB' dJH--iZajRajH (l2-ll,c) j j The neutral currents are thus 'naturally' flavour conserving: there are no strangeness-changing or charm-changing, etc. neutral currents, independent of the specific form of the matrices A. All we had to do is to treat all Q = 2/3 quarks on an equal footing (and similarly for Q = -1/3 quarks). The above arguments apply just as well for leptons, i.e., there are no flavour-changing neutral currents (such as ye, ve \>v, etc. The absence of such currents in the standard model is one of itB attractive features. Strangeness-chancing neutral currents are known to be highly sup pressed (for example from K0, •»• v |T). Similarly there are good limits on charm-changing neutral currents from D° - D transition. In summary the neutral current sector of the standard model is as given in (8-l|) and (8-5), Ij^ = j for neutrinos and charge 2/3 quarks, Io^ =. -\ for V and all charge -1/3 quarks . We now focus our attention on the charged current sector in eq. 12-8, •CC'°'=^YX(l'V*)aj-W*+h-C- 5 A(1 A 1 1 u u d « u •" (9i en)»(81,..,en)u •" (ipi,.., / . i8i •(8,,..,en) = (12-16) ie* i i it*i» k The reader may now write Ujjj = |U. |e J and convince herself that, by judicial choice of (e) = (ex,.i,6n) and (to) = («],..,<%), 2n - 1 of the parameters in *(6)U •"' ( 13. THE FOUK QUARK AMD SIX QUARK MODELS; CP - VIOLATION Up to 1975 we had. only four quark flavours (u, d, c, s) and four leptons (ve, e, v„, u). The charged current interactions were described by l(1 h (13-1) •^ ^v -^k\v - where (J is a 2 x 2 general unitary matrix. According to our dis cussions in the previous section U has just one parameter, which we take to be a rotation angle, viz., fd\ cos8 sin8 (13-2) \s -sm8„ cose where 6. is the Cabiobo angle. We aay rewrite JC° dcL " cos9c *! * 8in9c *t (13_3) s , « -sine d. + eos8 S . cL c L c !•T Thus, the doublets chosen in nature are / u 1 / c (l3-»«) ldc/L \sc / i.e., the most general ones one could have. Of course the single parameter 6 could also be put in the u - c sector. If we start with the doublets in (13-U) and the right-handed singlets (uR, cR, dR, Sp) and construct the standard SU(2) x U(l) model we could for get about all rotation matrices because all their effects are in cluded in the single parameter 8. which appears in the doublets (13-"»). The relations (13-"») summarize the celebrated GIM mechanism. In the standard model the GIH mechanism (which tells us that there are no flavour-changing neutral currents) is perfectly "natural". Ve "understand" why there should be a Cabibbo angle, but cannot compute its magnitude. Why is there not a second Cabibbo-like angle in the leptonic sector? Why do we not get . cos8' sine's /e\ = -^ (» ) \=/cos8'v - sine'v physical\ (13-6) +Sine u + ^physical H ' e cosB'vJ whereby 6' is eliminated. By definition ve is the partner of the electron, etc. For quarks ve cannot do so because the d and a quarks «re distinguishable (they have different "masses"). In conclusion in the standard model, with four quarks and four leptons, the GIH mechanism is natural; the lepton numbers Le and Ly are also automatically conserved provided the neutrinos are massless. Since 197? the third family of leptons and quarks have entered into the arena. We know that the GIM mechanism (vith just one Cabibbo angle), which was so natural and attractive as well as in very good agreement with data must be at most only approximately valid. We nov examine the generalization of the standard model to the case of six quark flavours (u, c, t, d, s, b) and six leptons. Applying the results in section 12, we find that the neutral cur rent sector is flavour conserving and given by (8-l() and (8-5), where I3L = J for all Q = 2/3 quarks and neutrinos and I31, * -} for all Q - -1/3 quarks and negative leptons. The charged current sector is the only place where new parameters (generalized Cabibbos) can enter, viz., )U sJ W~ + h.c. , -tf- (u, c, t)v*(l - YS)U| s |wt + h.c. , (13-7) 2/2 where we have used (12-5). Here U is a 3 x 3 unitary matrix and (according to our discussions in section 12) may be taken to have (3-1)2 = fc parameters. These parameters are normally taken as three rotation angles and a phase a la Kobayashi and Mas kava. It is easily seen that the Kobayashi - Maskawa choice corresponds to taking u =fii fil(6) 0 0 \ c = cosB , s = sine , t(0,0,6) = | 0 1 0 I (13-9) 0 e-W Multiplying the Matrix wa have V» Vj / "' 1 c,c c -» s e * C C , +, C e (13-10) U - I -s,c2 2 3 2 3 l 2 S 2 J " 8 l C C -C el -c,s s +c c e I »1 2 - l»2 3 2'3 * 2 3 2 3 this is the famous Kobayashi - Maskawa generalization of the Cabibbo - GIH - matrix which described the world as long as we had only four quarks. Comparing (13-10) and (13-2) we find that 6. » 8 (the Cabibbo angle) but now we have two more arbitrary angles and a phase 6. Bone of these is calculable. The phase could possibly be the origin of CP - violation. Hote that this phase is removable if any of the angles 8j, 82 or 8» vanishes. For example, if 8 » 0 we have ,(8.) » 1 and then putting (d) " t(0,0,S)d, d • (sj , reooveB the phase. Similarly, using /^(BjlttO.O.O • »(0,0,6)^(6^, we see that for B2 = 0 again the phase can be absorbed into u » (c). Finally if 8j = 0, we get U »Æi(62)*(0,0,6)^(63). The reader can easily convince herself where V is a 2 x 2 unitary matrix. Removing of phases in such a 2x2 matrix is rather trivial. We now discuss very briefly the phenomenological consequences of the six quark model. How big are the angles flj, j = 1,2,3? i) Cabibbo universality. Before knowing that there are more than four quarks we used to identify the strength of u -*- d transi tion with "cos8c" and u -» s by "sin8c", see eq. (13-3). Further more the analysis of baryon decays, etc. gave always z 2 sin 8c + cos ec = 1 to a good approximation. Nowadays we know better. From U3-10) we have that transition relative coupling where, to measure u*+d c. B-decay u<-*s sjCa hyperon decays 38 c**d -»l<&. V« * M W*X is « c*+s cjC2Cj-»2S3t , «tc. Since the Cabibbo universality vorked veil with just four quarks, 2 2 i.e., "sin 8c" • "co» »c" - sfc| + cf * 1 we My conclude that a, cannot be so large. Furthermore, since the four quark model pre dicted that charmed particles should predominantly decay into strange particles and this prediction is experimentally verified « mist also be a small angle. Remembering that the Cabibbo angle is also snail we find that the Kobayashi - Maskawa matrix, eq. (13-10. is of the form 1 »1 -S] 1 s3+s2 e" J (13-11) i« -s2-s3e Now we consider the present knowledge on the angle in (13-10). The muon lifetime together with the ratio of the supperallowed 8- transitions give cos8c • cj. We find coa26 « (0.972 ± 0.00U)(1 + A - A.) , (13-12) C MP where A - A is due to radiative corrections. Theoretically U p 2 A - A„ -0.021 for oin 6H » J . Thus sin26 = sin28i • 1 - cos28 « c * c * 0.028 ± O.OOlt for A - AD • 0 u P • 0.01(7 ± ff.OOU " A - A = -0.021 u p (13-13) However, from the semileptonic decays of the baryon octet we obtain 2 2 2 "sin 6 " = Gin e! cos 83 = 0.052 ± 0.002 , (13-1>») 2 where "sin ee." emphasizes that this quantity was improperly identi 2 fied as sin flc. Putting together (13-13) and (13-lU) we find 8j = 9C «a 13° and e3 ~ 8°. The t quark is presumably heavier than the b quark, otherwise it would have revealed itself in some of the experiments which have established the '.xistence of the T-particle. Thus'no direct infor mation is so far available on the elements of the third row in (13-10) and (13-11). Information on the angle 62 may be obtained by, for example, comparing b •*• c I 5 and b * u ft v, 1 • e, u, T, 39 vhich hav* respectively the coupling* aj • B2 • •"A *!*%• "tara ia alao aoaie theoretical information, on »2, coming froa the Aa • 2 weak proeeaaea responsible for the aaaa difference between K£ and Kg, which ia expected to be dua to X° (13-15) w dkV Y (k4n )lf mjPf L .i VbV^VVD r ht (ka-MJpW-mjKk* -*) jk where u and v stand for the appropriate Dirac spinors; n denotes the mass. Note that the momenta of the quarks (inside the K° and R°) are small and thus have been neglected compared to the momen tum flowing in the loop; furthermore the m's in the numerator drop out. Fjk is the product of the four coupling constants as the vertices in (13-15) F. = a(u,,s) • a(d,u.) a(vL,s) aU.uJ , Jk j J Tt Tt (13-16) ui • u, u2 = c, u3 • t. The a's are the elements of the unitary matrix U discussed in sec tion 12, viz. Uj2 = a(uj,s), a(d,Uj) = a*(uj,d) = V\:, etc. For the four and six quark models these quantities were given in (13-2) and (13-10). Mote that the leading term in (13-15), which is ob tained by putting m. = m. • 0 vanishes because a(u.,s) • a(a,u.) = (U+U)i J J 2 For the four quark model, estimation of AM, gave an impressive pre diction for m;,. Kow that we have more than four quarks, the old prediction should not get spoiled. Estimation of (13-15) for the six quark model gives 62 < 27° (see Ellis et al., Nucl Fnys. B109. 213(76)) were the phenomenology of the model is discussed). Ik. GRAND UNIFICATION So far, in these lectures, we have discussed the conventional theory for weak and electromagnetic interaction. Nowadays„ strong interactions are believed to be described by quantum chromodynamics (QCD), a nonabelien gauge theory based on the group SU(3)j viz. three colours and eight mediators (gluons). This theory is not broken and the gluons are ^assless. In this section, we discuss to the Oeorgi - Olubov model (Fnys. Har. letten .3J, «38(197*)) vhere the conventional 811(2) x U(l) «odel and chromodynaaica are (grand) unified. Orand unification aeaas that Y, W*. 7P, and the eight gluons ahould be the mediator* (gauge bosons) of one and the «aae theory, vhere the latter ia baaed on a simple or semisimple gauge group*, i.e., it contain* only a single coupling conatant. The 8U(5) model by Georgi, and Glashow ia the moat economical acheme for unification of SU(2) x U(l) and (SU(3)) colour. The SU(5) model does not explain the existence of the families 2 and 3 (see 12-1), so vhat we have to do is to write the theory for a hypothetical family where j denotes the family number: Q(v') • 0, Q(t') - -1, Q(u') » 2/3, Q(d') * -1/3; the indices 1,2,3 denote the colour. The primes on the fields emphasize that the quantities in (lb-l) do not necessarily represent the physical particles; to get the physical fields, we have to diagonal!ze the mass matrix (see sect-» tion 12). Once we have constructed the theory for the family j, we just take a sum over all families, whereby all families are democratically represented. In the following, we simplify the no tation by suppressing the primes, etc. and replacing (l4-l) by v, d e, U] 2 3. ] :• 3> we shall discuss mixings later on. The S0(5) theory may be constructed (analogously to theories discussed before) by storting with the kinetic term •I. (?L * £ fL + \ * £ Hf > • f " v> e' U' d (ll-2> f How should we put f in SU(5) multiplets? The left-handed ve and e may be put into a fundamental 5-plet of SU(5) + c (1I4-3) together with the flavour a (to be determined) in three colours. Here (fic = c Ji, where c denotes the charge conjugation matrix. The reason for choice {lb-3) is clear, viz., (ep, -Vp) ~ (vL, eL) are intimately connected and form a doublet under SU(2). In (li-3) these leptons are "unified" with a triplet of hadrons aj>2,3- Requiring invariance under 4>p •* exp(i 5(x)«T)i>R (where the T/2 are 5x5 matrices representing the 2*4 generators of SU(5)) gives 2k cnuco booono VJ, j = 1,.., 2li. Thus, we get a term i ^ T • V h • (Ik-It) *-'v(£- K The Matrices T say be choaen analogously as is dona in SU{2) and SU(3). ror example j - 1....6 JkJ • Gell-Kann matrices (8U(3)) •i 0 0 0 Tlo 0 0 .(llt-5) Ve continue by putting 1 and ±i in the same pattern. It remains nis « ^L T2K ,JL (llt-5a) -3 \ The properties of T's are TrtT"1] • 0, T3 • TJ so that the group elements exp(i 5(x) • T) are unitary and have determinant equal to unity. Furthermore ITJ.T15) « 2icj1™T,,l defines the SU(5) Lie alge bra vith cOkro • structure constants. Note that TrtTOT*] >= 2Sji[. Since our theory is supposed to include all interactions (except gravity), one of the gauge bosons, i.e., one linear combination of the V's in (llt-U) should be the photon field. The generator coup ling to photon is the charge operator, Q = Zc; constants. Thus Tr Q • 0 (llt-6) For the multiplet (l'i-3) Tr Q = 3Qa + Qv + Qe+ 0, (H.-7) Q. =• where, ve have used the fact that aj, a2 and a3, except for colour, are identical objects. Relation (lU—7) implies that a should he identified with the d quark U2 Clfc-B) We see that (lU-8) includes a doublet and three singlets under the weak iaospin group. The nature of mediators in 8U(5) model is easily established from (ll)-8), where any member can transform into any other by sending off a mediator. Thus we have ^R*-»eJ|, i.e., V£,**e£ via W* mediators; ds^+djj via gluons, etc. However, there are also lepton-quark transitions via new exotic mediators denoted w Y(±V3) v(±-j) , - , o a by x. , Y. » , j • i,a,3 d. b/3) + \^ (l/3) J>ww W y >~x< + v ft L (1) (2) (3) K ^^w~ 0,v,Z° ^^vwv. Y.Z° ^^"^: d. J d) (5) (6) Thus neither the lepton number nor the baryon number is conserved. The twelve exotics X and Y are sometimes referred to as lepto- quarks, although they are neither leptons nor quarks. They raust be heavy becaur.e otherwise the stable matter (protons and neutrons) would evaporate into leptons, as we shall discuss below. We have altogether 15 helicity states to account for if the neutrino is massless (v^, e^, eR, ui^» u:j, di^, dip). For a mass ive neutrino there is also a Vg, i.e., altogether 16 states. In (llt-8), 5 of these states are accounted for, leaving us with 10 (11) unaccounted ones if the neutrino is massless (massive). The remaining ten states can be easily placed into a 10 dimensional representation of SU(5), as we discuss shortly. For massive neu trinos we would need to introduce an additional singlet Vp. This is ugly and, therefore, we conclude that SU(5) has at least two *3 beautiful features, (i) the fractionally charted quark» are natural and (ii) the neutrino is aassless. Fro» the five objects b;, j • 1,.., 5 which transform as 5- plet of SU(5) ve can form a ten-plet by taking the antisymmetric product b 1 5 (H.-9) 'jk-i b2 *5 Evidently bjk » 0 if j • k and bjk « -bj,:. For j,k « 1,2,3 we have the antisymmetric product of two colour triplets. From SU(3) rela tion 3x3*6 + 3, where the six is symmetric and 3 antisymmetric, we see that bjk, j,k » 1,2,3 represent a colour antitriplet. Furthermore these three objects are singlets under weak - em SU(2) and have Q * - -^ . Thus b :k may be identified with Ejjm, Tw. Similarly bj^tbjs), j • 1,2,3 represents a colour triplet which has charge ^- + 1 « -| l^r- + 0 « ^ 1 and the third component of weak iso- spin i(-J). These can be identified with UJ^ and d;L. The only remaining state is bi|5 which is singlet under colour and weak iso- spin and has Q • +1, i.e., ej. Thus «1 -u2 "I "I u2 d2 u (llt-10) 0 3 &3 = bj k 0 e+ -u3 -«3 0 Here u means ^ijj = cyu, i.e., the charge conjugated spinor. The kinetic energy term, (11.-2) can be expresses in terms of (11.-8) and (llt-10) o I J 3x J ki 3x \l} 3 TrU„ Y Y 3x *R * *. 3^ (H-ii) 3_ i Tr< Y {*R [- 3x Here 2gQ is the grand unified coupling constant, T(f ) are the 2k •atrices representing generators, for the 5 (10) dimensional repre sentation. Furthermore 2» •R T«V •„ - a\ TV V'lj. Tr(i T-V a) . a-1 (lk-12) 2k 0 TrtfL^v,J-b.k£(T v°>.kj.,k, bjlk, Although the matrices representing generators look a hit frightening, for the left-handed 10-plet, they are quite simple be cause the bjk decomposes into objects with simple transformation properties under colour and veak isospin. This ensures that, al though some fermions are put into 5-plet and others into 10-plet, there is universality of strong, weak and electromagnetic inter actions. For example the right-handed u's form a colour triplet - SU(2) singlet and are thus treated exactly as the right-handed d's, except for veak hypercharge; ("jLidj^) form doublets under weak isospin and are treated as (v,e), etc. Let us look a bit into the 5-plet sector. There we have U diagonal generators T3, TB, Tis and T2i,, see (l4-5). T3 and Te couple to gluons G3 and GB. T*5 is proportional to the (right- handed electric charge operator, Q^ = -Vr T*s, i.e., V*5 is the 5- e. If we are going to achieve unifi 2 1 2 cation. V,* better be°the Zv and (^j = r T "*, where r is a constant Tr[TJl*) = 2 6;k, whereby TrlQjQJJ] = 0. This relation is normally not satisfied in the standard model [Q^ ~ (isj, - Q sin28y), eq. (8-3)] as is seen by substituting IJJ, and Q for the objects in (llt-8) and rememberinG that (e+, -v) + (v, e"^. We require z So sin eu = * where the sum goes over the members in (lt*-8), W Tr(Q2) .9 2 • l/'i 3 ... i.e., urn'1!),. = = -5- . Thus the unification hypothesis w 1 • 3 • f a fixes the value of the angle tu to be «bout 38 , at the unification energy, which is going to be. a very large energy. (At our energies 0„ is expected to be «nailer.] Kov ye consider the left-handed aultiplet, (lk-10). The gen erators f , in (lU-11) are easily obtained by taking infinitesiaal transformations (see lb-9) b. + [(1 + i B • T)b], Djk + [(1 • i 5 • T)b]j[(l + i a • T)b]fc - j ** k Comparison with (lfc-12) yields jk.j'k- ii' '*' •»• H' ' (lk_13) Tr(iL i'. V •,,)- 2 bjk T^, • V bjk, - 2 bjk T... V b.,k . Again V'5 (Vj*) are respectively the photon (Z°) fields. The reader can easily write down the various pieces of the interaction, e.g. the coupling to the Z° of the right-handed electron is derived from 15 g0 *h yv r *h z" - {v/| e {DS„ T»S bs„+ b„5 T'§ bM5}} This agrees with (8-3), where we find sineH cosflH (-*™\K *eR ^ *eR = • Vf *eB * *eR , 2 Here we have used sin 8tf « 3/8. 1$. MIXING ANGLES IN THE GEORGI-GLASHOtf MODEL In the previous section, we determined the structure of the couplinGS between the elementary fermions and the mediators. How ever, all our results are valid for the unphysical fermions (primes were suppressed) and massless intermediate bosons. Ve may now repeat the arcuments in section 12.and rewrite the theory for the lu,L ld E physical fermions, viz. uf, = A ' uL , df_ = A > 'dL , etc., where as before, 1>6 d s b •*l The six aatrices A(r,L , A"* , f » e.u.d (the neutrino is assuaed to be Baseless) are all unitary- Furthermore, these BB- tricea are independent of colour. We can new easily convince our selves that in the interaction tera r{j yMT. V •„ • ; Y" • *'. *J the charged currents (couplings to W~, Xj, and Ym 3 , m • 1,2, 3) the A's shov up, that is, there've find all kinds of Cabibbo angles. More explicitly, the charged currents from ?_(T * V)ip„ are of the form e'cy'£w~ (15-2) , ve get the charged currents of the form Prom *L(T • V)*L V U d r (15-3) \°i L L <»l •? In (15-2) and (15-3) ve have left out the ^-matrices, colour indi ces, etc. We may nov express the charged currents in terms of the physical states. For.example (U U + (d L, (u d) j£ dj! W = UL[A ' ] A ' dL W = ^ U ' dL W, (15-U) „. (e L) \l' er' W = v' A ' et W = vT e, tf (15-5) Jj it ii U h if Here • P''0]^ . \ eLW ' {'ei.e L * V "L * 'TLT L * • -} ' (15"6) The remaining interactions in (15-2) and (15-3) nay be treated analogously. Kote that there are additional angles and phases in couplings to X and Y which are, in principle, completely different from the generalized Cabibbo angles and Kabayashi-Maskawa phases which are introduced to parameterize U*u,d'. For example Again the y-matrices, etc, have bee» suppressed. The specific form of these new mixing matrices will depend on how one chooses to break the symmetry and generate the fermion masses. The mediators X and Y are expected to be very very heavy (1015-1016 GeV). Thus it is going to be very tough to study the mixing angles and phases (such as CP-violation) in the X and Y charged currents. The matrices A, as discussed in Bection 13, originate from diagonalization of the fermion mass matrix. Ve believe that the masses have to do with the spontaneous symmetry breaking a la Higgs. Below we describe, briefly, how the Higgs mechanism is done for the SU(5) model. 16. THE HIGGSES OF THE SU(5) MODEL SU(5) has 2'i mediators (XV , Yr1/3\ j = 1,2,3, VT, Z, y, 0., i = 1,..,0). Clearly X and Y should be very heavy so as to suppress the violation of bayron number and lepton numbers, W- and '/P should turn out to be an å. la Weinberg, (eq. 11-8), in order to be in accordance with experiments; the gluons and the photon should remain massless. Therefore the symmetry breaking should occur at two levels, viz. (i) SU(5) -*• SU(3) x SU(2) x U(l) colour standard model massless: G. W~, Z, y (l6-l) 1.8 ii) SU(2) x U(l) * (U(l)) . standard chare* At the first stage one or sore Higgles (which we hava not introduced yet) should acquire a stupendous vacuus expectation value whereby X and Y get tremendous sasses. Compared to these monsters V and Z are light as feather, i.e., their masses are due to a new class of Higgses which have acquired a much smaller vacuum expecta tion value (of the order of 100 GeV). After the second stage ve have nine massless mediators (gluons and photon) and all symmetries except for colour and charge are broken. The first stage of SSB is achieved by introducing 2» Higgses (in the adjoint representation, 5 x 5 • 1 + all) "P5, j*l,2 24.. The charge and colour quantum numbers of these spin zero objects are just the same as for the gauge bosons (see after eq. (lU-8)). Thus ve have 6 gluonic Higgses, «ft. = v , j = 1,..,8 colour octet, electrically neutral. There are 12 Higgses vith charges ±U/3, ±1/3. These form 2 colour triplets and two antitriplets J Similarly «fy are tvo colour singlet Higgses vith charges ±1, viz. ipj, j=22,23. Finally 2» . . » 2 L 3* 3x" *L 3x 3xS ( } j=i ii j.k (l6_2) = + Tr ^ » lr 9x 3x ' vhere 21. (16-3) The T are as1 given by (lU-5) and (lU-5a). How ve let • a Tls + B T2* vhere a and t are as yet arbitrary constant!. Of course, to do so we «ust add a potential to (l6-2) and check that SSB caa occur. V* leave that task as an exercise to our reader or refer her to the . literature quoted at the end of this article. As in section 10,a£> in (l6-2) is not invariant under the local SU(S); ve nist replace *x" ax" *°Lmios M r Here f are 24 matrices (2k by 2k). They represent the generators in the 2k dimensional (adjoint) representation. It is easy to con struct these matrices by taking infinitesimal transformations and remembering that Vhere a stands for a 5 (same quantum numbers as in (1I1-8) and T*1 are given in (lU-5), viz. I r> a + 5(1 - i 3 • T)TJ(1 + i 5 • T)a + (16-6) air1 - i[5 • i, I1] a . We have to compare eq. (16-6) with J k 1 k Thus we find r / r j k\ rjk Tjk ~ Tr^tT , T ] T 1 ~ i c , (16-8) where c J are the SU(5) structure constants. This result is fam iliar from SU(3). [We apologize for being sloppy with factors of 2; T/2 are matrices representing the generators.} Thus the vector boson mass matrix is obtained from f~f5+c[T. V,*] , (16-9) Tr(tT • V,<*>]{T . V, Here e is a constant and the vacuus expectation value <•> ia given in (16-b). We rewrite <•> in the font <*, m \ A" «/ (l6_u) where (16-12) 2«i 3 3 Here a and b ore arbitrary constants; a is the Pauli «atrix and In is the unit n by n matrix. Hote that Tr(«*>) • 0. From eqs. (l6-10), (16-11) and the definitions of r, ve see that the vacuum is invariant under the colour SU(3), viz. the unit matrix E com mutes with the colour generators T*, [r>, <•>] . 0 , j - 15, 2« , «Y - «J - 0. However for b * 0, in (16-12) the vacuum is noninvariant under the weak isospin and W* acquire masses. We avoid this by putting b * 0, without any good justification. Then the vacuum is only noninvari ant under 12 operations in the group (generators corresponding to X's and Y*s). The relevant generators are of the form (see (lU-5)) 'W) Eqs. (16-10) - (16-12) yield M| » M2. ~ g| a2 (16-H) where a is supposed to be tremendously large. Let_us summarize: we introduced Zh Iliggses and 12 gauge bosons (Xj, X:, Y: and YJ) have acquired masses. Therefore half of the Higgses can be gauged away. In order to see who remains and who goes away, ye go back to relation (9-13). Higgses corresponding to generator which break the invariance of vacuum must turn into Goldstone bosons, who get eaten up. The remaining Higgses are those corresponding to gene rators which leave the vacuum invariant. From (9-13) and (16-8) ve find Tj^V ~ Tr([Tr. «Jo-jT*) . (16-15) r [T , Thus, the Higgles in the theory, up to nov, are the te»\ / 0 \ B2 0 (16-17) 2 J 2 (a ) .fc v v* ~ g <$> r" T* Here v is a constant and the tilde stands for transposition. Clearly these nev Higgses leave the photon and the gluons mas-less, <6> j=l,...,8, and 15. Furthermore the mass of the X is not altered, T^<8>« 0, however the Y. are affected. We do not vorry about this point because |v| « |a|. The <6> breaks the invariance under the veak isospin whereby W and Z become massive. In fact (S^.S5) are analogs of (p+, ifP) of Wein berg and ve obtain his relation, eg.. (11-8), via. «W K{0,*) o ?(!)=K|v| K <"e> (T2*) z »6*i I = cos 8„ . (16-18) Again the angle is find «t the unification energy. Ms hare intro duced 2* + 10 - 3» Higgsea. Fifteen gaua* BOMB* (all except the gluons and the photon) hare baccate naasive. Therefore there are 3e - 15 • 19 Higgses in the theory. The reader can aasily convince herself that these consist of a nautral colour octet, three neutral colour singlets, two colour singlets with ±1 unit of charge, a col our triplet with charges -1/3 and the antiperticlee of the latter. Although the Higgs sector of the 8U(5) night seen outrageous, we oust take it seriously because we lack an alternatire for the gener ation of nasaes of the rector bosons. the feraion Basses are also believed to be due to the SSB aech- anisn. If the reader is familiar with the Young diagraas, she can easily see that *R~5 -D , iR~5* *L ~1 0 *fl ' *L ~10* " i b&~19\ e ~ 5 -.. , e ~ 5 » - Furthermore the charge conjugated spinors transfora as (*H)°~5* , (•L)°~10* (16-20) Hote also that (•},) » (* )JJ , etc. The Higgs-feraion invariant couplings connect left and right belicities. Fron (l6-19) and (16-20) ve find that C *L *R 8, h.c. , (*L) *L 6, h.c. (16-21) with appropriate Clebsch-Gordan coefficients are the origin of masses. We have, in a short-hand notation, for example 5 5 (d, d, + d2 d2 + d3 d3 + e e) where ve have left out the uninteresting factors, and the primes denoting that the d and the e are not necessarily physical particles. Similarly __ * _ 1 17. THE FHOTOH/KEUTROH DECAY I» SU<5) Let us recall that SU(5) has several nice features: (i) It unifies the nongravitational forces. (ii) The quantization of charge and fractional quark charges are naturally accounted for. (iii) It has no room for a right-handed v, unless the right- handed neutrino is, somewhat superficially, introduced as a singlet. So, we understand why the neutrino is massless. Furthermore the leptons and quarks are unified, as they appear in the same multiplets. The reader might not like the 10 dimen sional multiplets and the fact that the number of quark-lepton fam ilies is not explained. If she objects, ve encourage her to pro duce a better theory. What are the predictions of the model, in additional to the fact that there are 2k gauge bosons 19 Higgses, etc. as discussed before? First of all the model, by including the standard SU(2) x U(l)'description of the weak and electromagnetic forces and SU(3) colour for strong interactions, reproduces the standard predictions of QCD, etc. At our energies the weak, electromagnetic and strong inter action coupling constants are vastly different. The unification is expected to take place for energies well beyond all masses in the theory. The heaviest objects being the x and y gauge bosons, we have to go to energies Q2 >> M2; M2 in order to have unifica tions. As we move down to lower values of Q.2, the strong, weak and electromacnetic couplings seperate and follow different tra jectories. The renormalization of the coupling constant is due to higher order corrections (loops involving gauge bosons) and in-' volves factors such as log(H2 /Q2). One may ask how big should Mx(=My) be in order to reproduce the presently measured values of the coupling constants at our energies. The answer is (D.A. Boss, Jfucl. FJiys. BlltO. 1(1978)) • M = H ~ 5 x 1015 GeV '. Thus the unification happens for energies much larger than IO1* OeV, presumably not far trom the Planck value'of 101* OeV, wh.ere the gravitational interaction! have already taken over! Ve cannot hope to Mature the trajectory of coupling constant» up to such energies, however fro» g « e/sin6tf , g* • e/cosOy the «normalization of g and g' imply that value of 9y is also 2 2 changing as function of Q . The quantity sin »H which vaa 3/8 at the unification energy falls off slowly as ve go down in Q2. At 2 our energies sin eH ~ 0.2 tor three faailies of quarks and leptons (A. Buras et al. Nucl. Kiys. B135. 66(1978)). The most spectacular predictions of the SU(5) «odel (as well as several other grand unified models) is that the protons and neutrons can evaporate into "instable" natter (leptons and pions). The reason for proton decay is the existence of the bosons X and Y which violate the conservation of the baryon and lepton numbers. The proton decay has been treated by A. Buras, loc. cit and by the author and F.J. Yndurfiin (to appear in Nucl. Fhys. B). Here we follow the latter reference. The dominant decay mechanism is that two quarks inside the nucleon annihilate, by exchanging either a X or Y boson. The third quark acts as spectator, viz. + - (a) u + u -» e + d P * e ir , etc. + -*• Jl + S P * u* K° + o + - + M u + d •+ e + u F •» • T ~ IO33 yrs , T ~ 2 x IO33 yrs. The errors are large due to present uncertainties in the input 55 parameter». The lifetime! could be shorter or longer, experi mentally, the present beit limit on the proton lifetime, 30 TV 2 2 x IO yra, ia based on looking for Mions (r. Heines'and Hit. Crouch, Fhys. Rev. Letters £2, «93(19T*))> However, if the Georgi-Glashow model is correct, the branching ratio to muons is small, because the dominant channel (b) cannot produce anions. Other measurements give limits which are mora than an order of magnitude shorter than the quoted best limit. Truly, the proton lifetime is much longer than the lifetime of our theories and our universe (if we believe the standard 101* yra). Measuring it is a great challenge. ACKNOWLEDGEMENTS I vish to thank Anne Grete Frodesen for reading these notes and Janny Asphaug for her efficient typing of the manuscript. REFERENCES An incomplete list of references on gauge theories and phenom enology includes 1. E.S. Abers and B.W. Lee, Phys. Reports 9C, 1 (1973). 2. M. Bohm and H. Joos, DESY Report 78/87 (In German); this ar ticle includes an abundant li.st of references. 3. H. Harari, Phys. Reports ]i2C,' 235 (1978). 14. J. IliopouloB, CERN Yellow Report 77-18, pp. 36-78. 5. J.C. Taylor, Gauge theories of weak interactions, Cambridge University Press (1976). The SU(5) model and its consequences may be found in, for ex ample, 6. II. Georgi and S.L. Glashov, Phys. Rev. Lett. 32, U38 (1971*). 7. H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. Lett, 33, &51 (I97"i). 8. A. Buras, J. Ellis, M. Gaillard and D. Nanapoulos, Nuclear Phys. B135, 66 (l?78) and references therein. 9. C. Jarlskog, F.J. Yndurain, to be published in Nuclear Phys. 10. D.A. Ross, Nucl. Phys. BlUO, 1 (1978).