fRSOOlM^ LAPP-IH-09 November 1979
NEW QUARKS AND LEPTONS *5
Mary K. Gaillard LAPP, Annecy-le-Vieux, France
Luciano Maiani Univ. of Rone, Italy
CONTENTS
1. Introduction: a glimpse of quark and lcpton history
2. Elementary properties of quarks and leptons
3. Weak interactions of b and t quarks
4. Grand unified theories
5. Conclusions and outlook
' Lectures given at the Cargeae Institute, July 1979.
I. A. P. r. (HtviN ur nmfwE Boiri. rosrAtr iv> ?ion ANNTCY IE.VICL'X crorx rnirnoNl TABLE OF CONTENTS P»ge
1. Introduction: a glimpse of quark and lepton history 1 2. Elementary properties of quarks and leptons 1
2.1 What do He really know ? 1 2.2 Renormalizable gauge theories 11 2.3 The fermion mass matrix and the structure of the veak currents 18
3. Weak interactions of fa and t quarks 21
3.1 The veak mixing angles 21 3.2 Zounds on the veak mixing angles 23 3.3 Weak decays of b and t 29 3.4 Multilepton configurations in b and t decays 34
4. Grand unified theories
4.1 The unification of veak, electromagnetic and strong interactions 37 4.2 The minimal model: SD(5) 42 4.3 Feraion masses and constraints on the number of generations 45 4.4 Proton decay 49 4.5 Sounds on the t-quark mass 54
5. Conclusions and outlook t. IHTKOWJCTIQW; A CL IMPS E OT QUAME àM XMTtQK HISTOit
nowadays M era eeeustaeved to think of partiel* intereeeioas 1B »m of alaeaotary eouplinfs of quarks and leptona. In Table 1 w* list major step* which bave contributed to our presa&t picture, totriaa ia MUM bracket* refer to theoretical speculations; round brack*tad entries ara obuntd pturaoawaa which daavnatrat* tha ejnaaical reality of quark*, coaPleatttiting thair aliabraic reality which emerged fro* aa analysis of Che haaron spactrun. la this court* we will ba mainly concerned with tha weak couplings of quark* and Upton*, The prevailing view among theorist* it that quarks and leptoos hare a doublet structure with rcsptct to walk interaction*: e ).(';•). (ï),(;u:uîi..r that thnaa doable» ara charge daganarata for laptona (0,-1) and for quarki («2/3, -1/3), that thair cbargad current couplings ara of uatvaraal straagth and *a»a a T-A atruccura, and that thara ara aa many lapton doublata aa (color-criplat) quark doublota. The aubacript c in Iq. glbliotraphy C,S. vu and S.A. Hoakovski, "lata Da car" (Intancianca Inoliahara, X.T., 1966). "Waak lotaractiora", tin, ad. H.K. Gaillard and M. Nikolic IN2P3, Paria, 197?), M.K. Gaillard in "vaak and glactrpsagaatic Intaractiona at High Enargiaa, Carglaa IB!", ad. M. ISvy, 3-1 laadavaat, ». Spaiaar and «.. Gaataana (Planua traaa, 1.1-, 1976 ). 2. EiJuarnjtT ncrami or genu A» anow S.t What do va «aalty fcnovt Laptona. To tha autant that laptona aaasaa ara uniajportast, a V-A coupling iapliea that only laft-handal laptona and righc-handad anti-Uptons intaract vaakly. this raatricta al low* kinaaatical configuration»In a »«7 vhich allova aiapl* ca«ts of tha thaory. CaMidar .M^HMMVW •f Tabu 1: Quark and Upton History. 71 , iro 1M2-M M»-.t 1?H, If»-?' lMti4_ JSISL. t?7?-7* 1224z2t_ »7? Bacqvaral ffauU.Fa: taints and H* » I • t) Co***: rj v iataraetioa tutbarfortii aoclava aiaaaaargïj «<*> J" CkaaVickf i «troa Cocvarsl at al: M waafcly Utaractiaf Gall-Maaa, atraaax Caaiate» ate. /aw \ partidas 10(3). fniiki: i Fanllab _1 carramta . Sdnrinftr- Daaby at. al. HiabijiHt M* at: V„jf v v.. iittractioc "ïara, ate m. sue, laato*-a,uai DfST., TaraifOuxmBoxmSt ab , -WW a Cham aum j SUCf KST, CEBK farmlab: __jnsrJLv.sV Atéraction Kobayaatii- Barb •t al: OUI Kaikava: •(bo.)? CP T(bE) violation: 6 flavor* for KXaapïa g-deeay 2* — Z *e- \, (2.1) For a collinsar configuration, there is no orbital angular aesenttai «loci tha decay axis; angular awwsntuai conservation therefore reatrieta th« allowed configuration* vhish dapaad on th* Upton halicities, aa illustrâtad in Tabla 2. For a |ij| - 0 nuclear transition, only tha parallel (t~»v) configuration ia allowad; this Mans that tha electron and anti-neutrino helicitiea are opposite which in turn means that Upton helieity ia conservad in tha decay «ad the interaction ia necessarily vector and/or axial vector. Heaauraawot of a negative electron bclicity pin* it do*» to V-A. For m fAJJ - 1 transition, tha anti parallel config uration is alao allowed, and if the decay meats are eoliinaar with ûî (for «xaatpl* in the decay of polarized cobalt) th« configuration is necessarily antiparalUl» with the Uft-hand- ed electron aaitted in the direction opposite to A$. Table 2 Allowed collintar decay configurations for a univers*! V-A inter action: (avail arrow* denote spin direction). For decays into positive Uptons, all helicitie* are reversed. S«c»yj Aîlomà CoaCituzstiooi - M - 0 ": «» ** "•»'" z * z • • • v «-4J 4J - 1 V«f'\* «-""'* -3+v~ lT * •" + v, • Ï, ,'t >V' ^ *» r-~vr*Zv M V , " " 1% T" * i" • \ * vT " "'VT The structure of the vdecey awtrix element ia partially inferred frosi the electron apactnae which will be hard (aa observed) if the configuration of Tabu 2 ia allowed. Thia ewana chat the neutritu* and eaci-navtriooahav* apposite heUcit£aa, «nd tha fact that tha electron is eaitted preferentially opposite to tha spin of poUriaad suons laçlit* a left- handed electron. The neutrino haiicities cannot b* datanaiuad without the «ttasptios that the four f amfon interaction factorisa» into aauwicaod en electronic vertex, e.g. via «-«change aa in Fig. la. Then the observation* described above ia*>W that the coupling ia V-A st both vevticee. •- /r o) b) Fig. 2 Factorization hypothesis for leptonie decays of leptons. ODC reaction vhaxc neutrino halicitiaa caa be «ensured is in th* 2-body leptonlc dacay of a pgeudoscalar swson as fr -* *.- -*-3>» / -' (2.2) Thaaa presses ar« forbidden ia th» limit of vanishing laptoa sue because «oacntisg con servation raquitaa tha |j| - 1 parallel configuration, wheraa* the initial atata has |j| - 0. Tha dacay «nplitudas ara therefor* proportional to tha charged lepton suss, and ita polar isation will be in th* "wrong" direction in otd« to balanc» th* halitity of the (presuaed mastless) neutrino. Existing aeasureaeats are crude but indeed indicate that the (anti-) neutrino is (right) left-handed. With the V-A hypotheeis established for («\>t> end (pv ) couplings, the factorisation hypothesis. Fig. lb, can be used to extract the Cry_> coupling fro» the final atate lepton spectra* is the decay. (2.3) heiicitics» and therefore « V-A (rwt) coupling. Just as the abaence of the dacay If -* €*X B.4) and of th* infraction (2.5) where tha incident neutrinos originated fro» pion decay (2.2), estsblishsd that the nonobservation1) of decsys lib* •C -» Jit, 3jt,//iee. (2.6) •nd of tt» Miction*) V* U -* <• +X (2.7) for neutrino* originating fro* * end £ dece^e, iaply that »* * *,£ An alternative poaaibility would bt J4 «r Va, with laptone foraine ^^ iaotriplet* like (e~v T } instead of th* conventional doublet* (1.1). Than tha decay rate for alactrooic T-dacay (Fia;, 2) would be twice that for the ani onic Bod*, and in Addition the electron «pectrua would be «oft becauac identical neutrino* in the final atata forbid the configuration of Tabla 2. t_ v. Fig. 2 Zxaaple of an unconventional lepton coupling hypothesia. The decay datet as veil ai the abaence of reaction (2.7) i also iaply Jft +\,Vf. (2.9) Litht quart». Th* Miiurtd fi-d*c*y aaplitudai, including the relative sign, for Ferai tranaitioni AJ - 0: V or S, and Caaww-Teller transition* ÛJ - 1: A or T, together with the observed lapton dacay configuration* which alUinate the halicity violating aaplituda* S and T, tell u* that the weak (np) coupling i« approxiaately V-A. Whet ve are ifit*r*at«d In axa «uart coupling*. If the (ud) coupling i* V-A, then the (1»V * (3',,", * 1 (2.10) whereas the «assured values are The nonrclativistic quark aodal is not expected to be very reliable. A better probe of quark couplings is deep inelastic electron-nucleon scattering» which can be described as a sua of incoherent neutrino-quark scatterings. Fig. 3: dx d where f ._(x)/x is tha probility of finding the quark q. in the nucléon with noawntuai fraction x. 7T-* rig. 3 Partem aodel for daap inelastic v - M scattering. Tha V ± A natura pf the quark coupling ia reflected in tha dependence of the cross section on tha energy transferred to the hadrons: <2.J3«) whert Gf it the fatal coupling constant and y it tha «nergy fraction transferred to tha hadrons and ia siaely related to tha neutrino-quark c.au scattering angle: £v *• The y-dapindanca In Eq. (2.13) can b* understood fro* tha fact that backward scattering] e»s9em • -f t y * i - 7 - is allowed by momentum conservation if the incident neutrino and quark have the same helicity, giving zero net angular momentum component along the beam axis» and ia forbidden for opposite incident hclicities. The data show a nearly flat y-distribution for V-induced reactions and roughly a (l-y>2 distribution for v-induced reactions, indicating that tha couplings of the predominant "valence quark" component of tha nucléon ara V-A. The deviations from exact T-A predictions ar« attributed to a smell component of anti-quarks, with V+A couplings, in the nucléon quaxk-antiquark "sea". The affects of strong interactions discussed in tha course of Ellis and Sachrajd* do not change this simple interpretation of the data. Strange quarks. Tha most precise information on strange quark couplings comes from semi-laptonic hyperon decay. Strange quarks couple to light quarks through Cabibbo mixing as indicated in Eq. (1.1) where (neglecting heavier quark mixing) Jt«toi8td * s-vie.s «.1» so the charged currant for the (ud ) coupling is: 1 « eos&c û1(1-*s)d + si»9c û^.(t-ïr) s . (2.16) In the flavor 50(3) symmetry limit the vector currents OYV 're conserved, which means that their matrix «laments between baryons at zero-momentum transfer are completely determined in terms of SU(3) transformation properties. The axial vector currents qY Y.q* transform like an octet under SU(3), and their matrix elements between octet baryon states are determined in terms of two reduced matrix elements, conventionally labeled F and D. The measurement of three baryon decay rates (including neutron decay) are sufficient to fix the three unknowns 8c, F and D assuming (2.16), with tha overall sign of F and D fixed by the sign oE gA/gy in 8-decsy. Then all hyperon decay matrix elements ara fixed by the theory and the data is in good agreement with (2.16). In particular, a V+A (ua) coupling ia ruled out. Ve note in passing that in the nonrelativistic quark model tha parameters f and D ara alao calculable. The prediction (2.ICQ for the absolute magnitude F + 3> = (1A> is not in very good agreement with the measured value (2.11), but the prediction for the ratio _ is not far from the experimental value 0.658 ± 0.007. The difference in the degree of suc cess for these two predictions is believed to be connected to the relationship between "current" quarks which are the field operators defining the currents like (2.16) and the "constituent" quarks which era tha affective constituents of tha nucléon. Charmed quarks. The best information on the weak couplings of charmed quarks again comes from deep inelastic scattering, specifically from neutrino induced dilepcon avants which are attributed to the charm produc tion and decay mechanim of Fig. 4. Neglecting heavier quarks the charned quark couples to the combination st=eos8e s - s;*ô£d (217) orthogonal to (2.15)• ?he elementary mech- anisms which can contribute to dilepton production arc listed in Table 3, where the dominant machanls*» are starred. Be cause charm productioa from valence quarks is suppressed by a Cabibbo factor, strange Pig. 4 Parton model for quarks froa the nucléon sea are expected v-induccd dilepton production to play an enhanced role. Elementary processes contributing to neutrino induced dilepton production. Process ÎWAFVV v + d*u" + c • in\iàM - •—Ht* +... J V + s-fp" + ç co. 6cfg(x) » v + c •*• jT + â(d) 2 co«*ec£c(x)(i - y) c * £*•... V + d •*• U + C I-»».- »... J 5 + » * v* * ë cof 6cfj(x> « •—»t »... 2 V * c •*• \i + s(d) co»*ecfc(x)U - y) ! c » C *... (•in ec) The qq sea can be thought of as arising from gluon emission by a valence quark and subse quent materialization of the gluon into a qq pair, (Fig. 5). hadrons » Avalenc e quark Fig. 5 QCD mechanism for generation of qq sea. Since energy is shared at each vertex, tie expect on average KS < *4firert ^ XV*.»««tC . (2,18) The same mechanisn can generate ss and cc pairs, but we do not expect heavy quark material ization to be significant until energies such beyond the corresponding flavor threshold are reached. So for an isoscelar target and with Che CERN or Ferailab neutrino beams we expect h - -f, « k ' -fs * ^ -- {a * (cos-oioX^-i ). (2. 19) Neglecting v-c scattering in table 3, we see that the conventional V-A theory predicts a flat y distribution in dilepton production. In addition, v-induced events should have com parable sea and valence quark contributions, whereas ^-induced events cone only from the sea component. Thus the analysis of inclusive dilepcon production tests both the space-time and the flavor structure of the char» changing current. The observation of flat y distributions: — ""t-9"' (2.20) for both v- and v- induced events confirms the (V-A) nature of both (cd) and (cs) couplings, while the softer «-distributions: p —«y**/ V-*/t A * / (2.21) confirm the enhanced strange quark contribution. Information from charmed quark decays also confirms the dominance of the c -* s transition. b (bottom or beautiful) -qu'jks. At present there is scant information. Three narrow states have been observed:3) which are believed to be 1~ s-wave b"b bound states, analogous to the narrow 1" charmonium states. «J/JCS.O , 4'13.U The lowest "naked charm" states are the 0~* (eu), (• d) states with maas -*1 (2.22) Eq. (2.22) would suggest that the lowest 0" naked bottom state has a mass 7»B-=>! \W£ +30I>M * S The observation of the narrow T" restricts W-8 > « 6«" C2.24, to prevent the rapid decay T" •+• BB but Eq. (2.23) (as well as more sophistocated arguments based on potential models) implies that the b mass should not be much higher than (2.24) and there is indeed evidence at CERN1*' for a B with a mass of 5.3 GeV. hadrons a) hadrons gluons "\\ ^\ b) Fig. 6 Mechanisms for (a) pp * T + X, T •*• u+u~ and (b) e*e~ +T + hadrons. Information on the charge of the b can be inferred from the observed T production in both pp and e a collisions which are in a sense inverse processes, as illustrated in Fig. 6, and which both involve the y-T vertex, which in the quark aodel, Fig. 7, is proportional to the quark charge and the wave function at the origin: Uwf «cQwIV11 The factor (V (0)|2 is not known, b-.it naive extrapolations5' Eton charmonium as well a» •ore aophietocated acudiea of potential models" demanding consistency for observed T and T' ratci give consistency with the data for IQwl = y» «•*> and not for (qj - 2/3. What about the weak interactions of the b-quark? The naked bottom states of lowest mass are expected to be the charged pieudoacalar stataa B~(bû) end B (bu) because the empirical inequalities — . lead u* to believe that a < a.. and therefore Bu_(bû) < *_0(bd). In this case the B* will be atabla unless it is coupled to the lighter quarks. The non-observation'' of a stable charged particle with atsss in the 5 GaV region says that <£S(.Q**0 £ 10"* «ft. if as axpectad. On the other hand, the non-observetion of a signal for B production in neu trino induced reactions says that b couplings to light quarks is suppressed relative co the uaual four fermion coupling strength* Since b production fro» a valence quark can occur only in anti-neutrino charged current reactions via the proctsa 7 * u -*JL* * b, (2,27) the data aeta a lower liait on a V+A •www/ ?? < I Fig. 7 Quark aodal for T - y coupling. 2.2 glHOKHALIZAlLE GAUGE THEORIES If all interactions are described by renorulizable gauge theories the part of the Lagrangian relevant to fermions (neglecting Kiggs acalar particles for the tine being) is: X^^ZIJ^+ÏM , J*.** (2.28) where the invariant derivative D contains the fermion couplings to gauge bosons as well as their kinetic energy. In the atandard nodal9' of weak, electromagnetic and strong inter actions we have t ~ * :» £•£ 4. ;« <.U(W- )+ i. :„'Y Y TV VVtiK*^* 3' ^ - .29) where G ia an octet of colored gluon fields, X the act of 3 it 3 SU(3) natrices acting on quark color indices, W a weak isotriplet of vector fields, T the set of 2 x 2 Paul! matrices, B a vactor field which ia invariant under 50(2), and SU(3) and Y the (diagonal) weak hyper- charge matrix related to electric charge and weak iaospin by Q = Y+Ij (I.*$). (2.30) When spontaneous breaking of the SU(2)L ® U(l) gauge symmetry is introduced, the components Wi^j of H are identified with the charged heavy intermediate bosons H±, and the neutral fields K, and B become coupled via a non-diagonal mass matrix whose eigenstates are the masslesa photon and the heavy neutral wefckly coupled bosun Z, aa we shall describe in more detail in Section 4.1. In tht original Cabibbo theory of charged currants there was one weakly coupled quark doublet with left-handed couplings 1 lb -(1) • +**+ •l^l (2.31) giving the conventional V-A charged currents. According to Eq. (2.29) we must also intro duce a neutral currant: ï>t £4( « £(aa-3.d«)u „.„. which contains a strangeness changing piece [cf. Eq, (2.15)]: (2.33] The coupling of the current (2.33} to the Z would induce (see Fig, 8) transitions K. •+ w and K° *-•• K* • with amplitudes characterized by the usual ferai coupling strength, whereas they arc observed to be highly suppressed. This unwanted component of the neutral current is removed by the mechanism10* of Clashow, Iliopouloa and Maiani (CIH) whereby a (thti. con jectured) new quark witl. the same weak quantum numbers as the up quark is used tc complete • second weak isodoublet of quarks: - . (2.34) +.-(U S, • fi% - Sr'/lBt <*, r r KL a) K° J ^wwwK' b) Fig. 8 Strangeness changing neutral current processes which suit be eliminated frost the theory. implying a contribution to the neutral currant «Ul which exactly cancels that of (2.33) provided that the new doublet i|>2 Is also left-handed, i.e. the associated charm changing couplings havs the usual V-A structure -- and that ! w sH'c U* KL ÎW li. •V w u- a) b) w s u,c d K° I u.c 1—n U- Li I c) u,c Fig. 9 Strangeness changing neutral current effacta induced in higher order. Even with the introduction of the doublet (2.34) strangeness changing neutral current transitions can arise through higher order effects aa illustrated in Fig. 9. Because of the orthogonality of a and d and the identical strength and V-A structure of their coupling u- and e- quark exchange in th* diagrams of Fig. 9 cancel exactly in the liait of mass degeneracy • - m . Tor the physical case a » a theac amplitude» are characterized by an effective Ferai coupling strength H &' (AS*O, Aq-o) - ^2^ Si«ec „.„, so that their measured transition rates allowed an a priori estimate1') of the chcrmed quark aasa. It turns out that the leading (* In •jj/'O contributions to the K. •+ u*u~ amplitude cancelIs between Figs, 9a and 9b, ao that estimates of the charmed quark aasa relied on the Kj^ - Kg suae difference vhich ia related to the transition K° ++ K° by [see Section 3.2]: 1*1.. aw inwj (2.37) 3ir wî L 'WV J As explained in Section 3.2» the result (2.37) ia obtained b» neglecting strong interactions which can be important in the low momentum region of loop integration in Fig. 9c, and by estimating the matrix element of the effective four-quark operator in a nonreletiviatic quark modei approximation. while these approximations ere rather drastic, the order of mag nitude ia expected to be correct and indeed the resulting inference that a =» 1.5 GeV was borne out. The simple requirements of a renormalizable theory and the absence of the pro cesses of Fig. S were sufficient to determine the weak interaction properties10»12' of charmed hadrons. There is another problem in construct ing a renormalizable theory of weak inter actions which appeara only in higher orders and which is related to the presence of axial vector couplings. Explicitely "anom alous" traingle diagrams of Fig. 10 with an internel fermion loop, three external vector fields and an odd number of axial couplings are divergent and the theory can be made fully renormalizable only by removing them through cancellations of the (mass independent) infinities among different fermions. This clearly con strains the fermion content of the theory. We examine explicitely the case of Fig. 10 Divergent traingle diagram SU(2)L • UCDj it ia sufficient to study the neutral current couplings; tit 0 J (2.38) The anomalous triangle diagrams which can appear arc ahovn in Fig. 11; the sum over fermion ipeciea £• equivalent to taking the trace of the coupling matrices acting on internal quaat- vm nusbera. For cxaaple the Zgg (g - gluon) coupling of Fig. Ha vanishes identically since A and T act in different quantuai nuaber apacet: — •>* Tr XJL, Jt?$ e^, - T * it-\ *j TV *, - O for aach color triplât veafukc isodoubletiaodoublet. Similarly the Z3 ccouplino g of Fig. lib vanishes for each weak isodoublet: Fig, 11 Anomalous triangle diagrams arising IT- the standard «odel. The potentially non-vanishing contributions are Fig. lie: TrCts^Q = TrQU (2-39) where the subscript L inplies that only the charges of the weakly coupled left-handed fetmion component» contribute, and Fig, lid which gives the same result. Since Y takes a conmon value Y. for members of the i weak isodoublet we have so; Tr «3 Q* * T* r3 CY- % )* - tte," Y • T.Y,, « V^+O -"G-Oi.- Therefore the condition for a renormalizable theory SU(2), «0(1) is: Tr QL ' 0. (2A0) Restricting fermions to the conventional doublet structure, esch lepton doublet contributes Qf + fy = - i , «.*» and each colored quark doublet contributes 3.| +3-1-4) •**. so we see that triangle anomalies are eliminated if there «re equal numbers of quark and lepton doublets. Thus in constructing a finite theory of weak interactions from the observed V-À charged currents one is led to the requirement of lepton-quark syonetries. finally, the Lagrangian describing fermions written in Eqs. (2.28-29) has two defects. It is gauge invariant only fir zero mass fermions: M = 0. This is because SUU). trans formations act only on left-handed quarks and leaves their right-handed components invari ant. Since mass terme couple left and right components, they cannot be invariant under these transformations. Secondly the theory described so far baa no manifest CP ' violation, whereas CV violation le kno^n to occur, albeit with a very small amplitude. Both these de fects have a solution in the introduction of scalar (Higgs) melons which are necessary in any cass to generate the spontaneous symmetry breaking by which some of the vector neions acquire masses. Rare we shall concentrate on the scalar-fermion couplings in the "standard" modal which includes a single complex isodoublet of scalar fields f We disregard here non-pcxturbativc effects in QCD which could induce CP violation in strong interactions. - 17 - leu as raquirad by renonalizability. They ara tha (auge couplings* vbarc if G* ia a vector field vith coupling matrix (gl)1: the scalar potential and the Yukawa couplings vhara also transfers* as • weak isodoublet, and a, b ara generation indicée. If there ere 2n quark flavors* than Th* fielda C can be chosen real and the T heraitienf then hemiticity requires g • g* in (2.45) aa wall as u2 • ul* and X • X* in (2.46). Therefore neither the gauge couplings nor th* Higg* potential caa contain CP violation* although CP violation could ba introduced in tha Higgs potential If «ore scalar fields vara included,11*' On the other hand tha Yukawa coupling aatrices C^p wd 0^ ara arbitrary conplax Mtrice* which contain, for axaapla, 2na phase oaraaatars for the couplings to quark*. Aa va shall see in detail in the Section 2.3» not all of these lead to observable CP violating affecta. Up to now va atill have no feraion suss tern. If y2 > 0 in Bq. (2.46) tha ainiaua of tha Biggs potential occurs not for |f| • 0» but for i«pr - ¥/\ ^5 m\ (2.50) In order to do ccntentionaLonal perturbation theory around th* loweslevait anenerga y state—by defini tion the vacuua—v* definfine a scalar field which vanishes at tha ainiauaainia : i (2.51) Ue can always define a baaia in SU(2), apace in which <4> " {%) • (2-52> This defines electric charge vhicb ii automatically conserved becauie the charged component of n 2 iOT bosons through the tern | 1>| •** f«*mioo» through the Yukawa coupling: (2.53) «he» The structure of (2.53) will be discussed in detail in tha following section. 2.3 The femion -nasi matrix and the structure, of tha week currents. Ue heve aeen in the previous section that maaaea can be given to fermions through the spontaneous breaking of the gauge symmetry. Ue shall now analyse in more detail the struc ture of the resultant mesa matrix» and that of the weak currents. For simplicity we con sider a theory where there is only one» isodoublet» Higga field. Quarks arc assigned to H left-handed doublet» (later we shall sat N - 3), all right-handed fielda being ainglcts: (Ul) ("*•) (u*) 2 5 \ d, JL K di JL ,•'•>• <*. / c . < - " Tht SV(Z) x U(l) invariant coupling» of th» Hifia field • defined in (2.4Ï) to tha quark fialda to.. (2.5*) ara o[ th. form [au Zq. «.«)] *J . « — V* *i i < " '" -*• «J.4. (2.5:» #Y"î... M«0 *ïu-^««j«P where <*<., C°. arc M x X coupling matrices (in general not harmitian) and 9 ie the charge- con juge te doublet defined in Zq. (2.46). If ve substitute into (2.55) tha vacuum expectation value «f *•» Eq. <2.52)..£L gives rise to a quark macs term of the formt [see Eq. (2.53)1 C« - fa.; M" uRi + d4i Al^. J^ •+*.«. ; <».*> whare tf^ . Jv/VÏ)!!^, ate. rf1 »nd rf1 ara in general not real nor diagonal matrices* so thacj^ti| la not diagonal In flavor and it contains ys terms. This simply means that the fields u,L, u,_ etc introduced above do not correspond to fialds with definite masa. To find tha latter fields, wa make separate» unitary transformations on left-handed and right- handed field» according to*); (!) •«#(?) •*(?)• \:\ *VL \:/K lu„/R (2.5J: and eimilerly for the doua qustea» with matrices V. and T-. Mum expressed in tents of the fields u,c,t, . . d,s,b, » . , tha BUS Lagrangien Eq. (2.56) raads: whara wa have denoted by e.g. u. th* tf-row vector of components a,, c-, t, , . , . One can usep at this point» « wall known matrix theorem which aays that: given any (in general non tiermitian) matrix H, one ca& always find a pair of unitary matrices Vt and 0R, such that ULMUa *«*•*, ciiftMOntxt IMtttrlX 25 4% (2.58) (the proof, of this theorem ia immediate, if one racalla that any matrix can be written as the product of • unitary aMtrix times as hermitien Matrix; «poo expressing the hermitian sMtrix aa the unitary transform of a real diegooal matrix, one obtains tha result of Eq. (2.58). Eq. (2.58) implies that we can determine the four matrices l)t, . . . VR such that, when expressed in terms of the "physical** fields u, c» tt . , 4, s, b, . . v the mass ta- graogian reduces to the real, Y, tree form: Xmti * "l*^"UR + ^«•'»?J dL + ^ = «* ^"WA * *>»J <**.«•»> Tha diagonal values of the reel and diagonal matricaa irf and Hf give tha masses of the various quark flavors. Tha transformations Zq. (2.57) are obviously to be performed in all terne of the Lagreagian, and it ia interesting; to see the result for the complete Yukawa coupling, Eq. (2.55). and for the couplings of the harmiena to the gauge fields. Yukawa couplings. If we «at: K représenta the neutral Higga boson field (i.e. the only remnant of the Higgs doublet, after the spontaneous symmetry breaking has taken place). Substituting (2.60) into (2.55), *> He are neglecting hare tha problems related to tha fact that chirel transformations sre effected by anomalies. The transformation we are performing could induce parity and CP violations in the strong interaction sector. Sea e.g. Isf. 15). va get the sass tern (2.56) and a further ter», which représenta the Yukawa interaction of U with the fermions. Since the coupling «trices CV-, G.., are proportional to the mass aatricss H?., and Mr., one sees i—sdietely that in the physical quark field bssis, the H- intcraction is parity and flavor conserving. This is a vary nice property of the one Higgs doublât nodal. Xt insures that no AS jf 0 transition is mediated by H, thus avoiding any conflict with th* observed smallnass of K, **• \t*\T and of the K0 - K.Q transaction [the way to enforce naturally this condition if one has mora Higga fields has bean first studied in Kef. 16) and, sure generally in taf. 17)]. Couplings to the «auia fielda. The coupling of the quark fielda to the vector basons of SU(2) « 11(1) (<**, Z and Y) is entirely dictated by the doublet structure, Eq. (2.54) and by tha electric charges of u and d quarks. The interaction Lagrengiaa has the for* (see Sec tion 4.1): ** - £(ty J" *u) +±% 2, tf - «A. $L where C is the rami constant, e is the electric charge and 0v is the Glasbow-tfeifibarg-Salsm angle. The week currents» in terms of the fielda (2,54) are: Upon substitution of Zq, (2.57) into Eq. (2.61), one sets that: i) the neutral current, J?, and tha electromagnetic current, •£'*', maintain the same form in terms of tha physical fields. In particular they remain diagonal in flavor, so that no l-t order AS •f 0 or AC ^ 0 transitions occur in weak, neutral current mediated processes{ ii) on the other bead, J is non diagonal in flavor: ^-t«.£,f4...;Lyyiu/s| U-4.VJ , U*U -1 The currant (2.62) ia the generalization to H doublets of the Cebibbo-GIM current* and the •atrix element» of the weak mixing Matrix 0 determine the atrength of flavor changing weak practice*, The matrix U defined ia Bq. (2.63) contains «till some erbitrarineaa. In fact we can change independently the phase of each quark field without changing the physica, and thia would aultiply ÏÎ by a corresponding diagonal phase matrix. Aa a consequence» the nuar bar of physically relevant, real parsemer* upon which Ef dépende is not Sz (we have not counted an overall phase change, which would leave U invariant). It ia interes ting to note that an orthogonal BiH matrix hat: NU-n s. real parameters in it» so that by redefining the phases of quark fields» we cannot» in gen eral, auks U real. Therefore we may expect naturally some CP violations in the weak current. In fact the masher of uneliainable complex phases in U la just; ^(pluses) . (fit- t)* - gt"-l> - (tf-Hj*'** »••*> Eq. (2.64} «hovi that U is always real for • £ 2; it contains exactly one CP violating phase for B » 3, aa first observed by Xefeayashi and Haskawa11', and even more for H > 3. All we aaid above can be repeated for the leptos doublets. If ve assume that all neu trino masses are exactly zero, however, the coupling matrix G.. [sec Eq. (2.53)] vanishes and the corraspoodiog transformations λL» S^ sra entirely arbitrary. It follows, In this cast, that we can adjust thaïe matrice* (i.e. we can define the physical neutrino fields) in such a way that the weak mixing aatrix for the Upton system is equal to the unit matrix. If all neutrinos have zero mass, no leptoa flavor violation can occur in weak interactions and e.g. the moon number is exactly conserved. It neutrino masses are not exactly zero, but atlll vary small (say in the eV region) we can still neglect C., in weak decaya and in scattering procasses, but the difference between the "currant neutrinos" (i.e. those coupled with U - 1 to the charged leptons in the weak current) and the "definite mass neutrinos" would give rise to the peculiar phenomenon of neutrino oscillations11', possibly including also some Cf violation3*'. 3. WEAK IHTKACTIOKS OF b AMD t QOAKKS 3.1 The weak mixing angles From now oa we specialise to the east K - 3. Assuming massless neutrinos, we focus on the only non trivial mixing matrix, the quark matrix defined in the previous section. Allowing for the phase arbitrariness discussed in the previous section, the aixing matrix Ï1 can be parameterized in terms of three angles and one CF violating phase. One - 22 - parameterization which ie frequently used is the Kobayasbi-Haekawa parameterization11': C, S,CS S,S| ;ff <31) V = | -5.C, c.C^-^s, ct<,%-e s^ 1 iS S ;5 • s*c» c, % c, - ô 4S3 where c., a, = coafi., sinfl. (i - 1, 2, 3). 6 f 0 gives ris* ta CP violation, and in the liait B2, 9j •*- 0, the matrix U couples d and s to u and c but it couple» b to t only. In this limit, 6l ia to be identified with the Cabibbo angle, 6 . It turns out, for the raaarns discussed below, that it is «ore convenient to use a different parameterization for U21\ neatly; (3.2) Again w« have three real angles (9,D»Y) and one phase (which we continue to call 6, even though it differs fro* the previous 6), and wt have used the notation eg, s„ = cos8, sind etc. It oust b* noted that there are still sign ambiguities in U. IE we let 6, Y. 8 and 6 vary in their full range (from 0 to 2n) on* doaa not always obtain phyaically inequivalent determinations of U. The range of the mixing angles can in fact be restricted to: 0 «8,/* ^ % The advantages of the parameterization Eq. (3.2) ara: i) U can be expanded to first order in the angles, around 9 - & - Y • 0, which ia the region of interest, as we shall s«e shortly; it) in any case th* Cabibbo angle coincidea with 0, since: of COUTH, Ctbifcb/M.11o tmlvtr..lit* +1 Ui4r iI*s broVt * n<:«} for *8^0 < 1: . »•«> iii) the angles Ô and y have a staple meaning in terms of the weak transitions of the b quark: I Utb • (J.3) As a consequence of Eq. (3.4) we shall see that 8 has to be very small, (see next section) $ £ ^8 < lO"1. Neglecting terms of order B2, 62 and B8, the matrix U takes the simplified tarai •i e ^/»* We see frost Eq, (3.6) that the so called Cabibbo suppressed charm coupling (c -*• d) may differ from the value given in the 4-flavbr theory: tty = tant). Similarly V^ t -%b, but |«bcl a* l«t,[. 3.2 Bounda on the weak nixing angles. a) **°uPd f _ ESSSTS ,t*nda>rd gh e^omenology* ), We have already indicated that the weak angle 6 oust be small, because of the observed validity of Cabibbo universality2*'. To be «ore precise, we recall that from the y-lîfetime and from the superallowed nuclear 6-decays one can extract two values of th«j Ferrai constant) which we indicate by G and Gp, respectively (R stands for renormalized). Using Eq. (3.2), one has the theoretical prediction: G-R where %iV and ^& are, respectively, the radiative corrections to G and G , which, in their finite part are expected to be different for Li and for 0-decay. Using the experimental value of sinS determined from « comparison of K«3 with f. decays (tin 8 - • 225) or from s comparison of nuclear versus hyperon decays (sin 6 • .241), one obtains: K - A*^ --p = lz.it o.z)7. •u "*. -f> = ini«3;y( (3.B, where the error comes mainly from the uncertainty in 6. The radiative corrections in the SU{2)$U(t) theory have been computed by Sirlin25', and it turns out that they agree in sign and in magnitude with the central value given in the right-hand tide of Bq. (2.8). We can therefore calculate that the effect of B2 cannoc be greater than the error in the right- Bounds to the mixing angles in the Kobayashi-Maskawa parameterization from standard phenomenology have been discussed in &ef. 5, 22, 23). - 24 - hand side of Eq. (3.8). This leads to the bound: /* ** as anticipated in the previous section. We now turn to the bounds on Y> The first bound obtained in the literature 5) arises from an analysis of the X, - Kg mass difference, or equivalently, of the KQ - E0 transition amplitude. To compute the Kj - KQ amplitude, one computes first the box diagran Fig. 12 given in Fig. 12. This diagram gives rise to Quark diagram for the effective, an effective Lagrangian for AS - 2 transi- AS • 2» Lagrangian. The wavy line «>». represents the intermediate boson. AS'3. &'€# where m ÏB Che charmed quark nass. X ia, in general, a complex quantity which is given by (we set m « 0): (3.11) The K- - K mass difference is determined by the real part of X. In what follows* we shall at first neglect Che CP violation» and consider X to be real. Note chat X - 1 in ".he liait where t does not couple tod and B ((3 • Y - 0), while X may become very large for large coupling: X Z 100 for Y " n/2, 6-0, and m % IS GeV. To obtain the K. - Kg mass difference» one has to make a further step» and compute the ma trix element ot ^L*f thi* i* usually done in the approximation where one of the s fields and one of the d fields in Eq, (3.10) couple directly to the a and d quarks in K » and the other fields in Eq. (3.10) create the tQ. Keeping into account the possible waya in which this can be done» and using the appropriate color factors» one obtains: (3.12) Finally» relating the matrix elements of quark bilineari to tiie rate K+ •*• u*\> in an obvious way: Inserting the experimental values of the various quantities. on* obtain» finally: a s a! -c *s«kLs# r*xfc (3.W) Eq. (3.14) tell» u* that I » 1 is compatible with the observed velue of a, - Og. Eq. (3.14) ceci provide a combined bound on Y *nd 8. if *** "y that» given the theoretical uncertainties in the computation, we may tolerate values of X up to but not larger than, say, 5. The sensitivity of the bounds on y and 0 to the assumed bound on X is shovn in Fig. 13» wh&re we have plotted the level curves of X at a function of the mixing angles. (He have assumed 5-0 for simplicity.) Requiring 1 £ X <, 5, approximately gives the bound; |s>nrj è .S1 It has been argued recently"' that a com putation of the matrix element of X.*« Eq, (3.10), in the bag model would reduce the right-hand aide of Eq. (3.12) by a factor of about 2.S. This means that X - 1 would not be compatible, so that one could obtain also a lower bound on y. Keeping a safety factor of 2, the authors of Ref. 22) then require: tîO" The corresponding bounds on y can be read off Fig. 13. The validity of the lower bound thus obtained can however be criti led on the basis of the many uncertainties which go into the calculation. In parti cular, in the evaluation of ^,« we have neglected all strong interactions. In QCD Fig. 13 Level curves in Che plane one My expect hard gluon corrections3') B - ainy for the real part of X, see to the oon-leptonic vertices in Fig. 12 •• text Eq. (3.14), fsoa K^ - Kg mass difference (continuous lines). Dash- well as softer gluon exchange between the id curvesî bounds on 8 end siny froa internal quark lines. Indeed, hecause the K. •* M+y". Ue have taken a. - IS CeV » 10 m. GIH sechantba provides a cutoff at the nets of the heavier quark, internal quark vir tual «omenta range from p « • up to a « a , i.a. they go into the region where strong inter action corrections are expected to be large. Furthermore, it is unclear why the bag model computation of the matrix element has to be preferred to the standard calculation. All con sidered, le seem* safer to rely only on the upper bound given in Eq. (3.15), An independent, more reliable, bound on y and 0 has been derived recently27' fro» the IL •* U*M" tût*. Also in thi» case the t contribution is potentially vtry dangerous. Using again a diagram of the type of Fig. 12 (with one quark line replaced by a lepton line: u •* v •* p) one finds: (3.16) é 0.2 r f«- w^ > »s6».v. The bound resulting from Eq. (3.16} is also reported in Fig. 13» the allowed region being inside the dotted lines, and it is very similar to uhat was gotten before. Before closing this part, we consider briefly the violation of CP in the K. - K- sys tem. When 6^0, the effective Lagrangian, Eq. (3.10), gives rise to an imaginary contri bution to the K mass matrix. Indeed, it has hten shown that this is, to a good approximation, the only sizeable CP violating effect which arises in this model1**22'. The CP violating nixing parameter,Ê, is given by the expression11»2*': ReX where X is defined in Eqa. (3.10) and (3.11). Xm(X) is explicitly proportional to sin 6. From the above equation, given tbe experi mental value: € as 10"*, one can obtain sin 5 as a function of 8, Y end the quark masses. Ve have reported in Fig. 1ft the level curves of tin 6 for m„ « 15 GeV • 10m . c c A closer inspection of ImX» Eq. (3.11), shows that ImX may vanish far special val ues of & and Y* *v«n for "i" 5 1* 0. For those values, an infinite value of [sin 6| would be required to obtain € f 0. Thus in the J3-y plane we may expect the presence of forbidden regions, where |sin ô| > 1 would be required to obtein the given 6 . However, since £ is small, these regions are quite narrow, and are essentially re stricted to the coordinate axes (y • 0 or 6 • 0) and to a narrow neighborhood of the dashed curve in Fig. 14. Otherwise, |sin5| ranges approximately from 10"' to 10~s. Tig. 14 Level curves for «inS from the observed value of the CP violating parameter in K decay, a - 15 CeV « l0m_. j) Kixing angles and cham decaya. The currant x current product relevant for charmed particle non Uptonic decays is: °SWiL where d and a arc the appropriate combination» of d and s field» implied by Eq, (3.6). Neglecting the (ûb)(Bc) term, and dropping for simplicity the Dirac matrix structure, one hat: (S---1) X/,oVu C U.*U»„tOJ)«« + Uu where for each group of terse we hare indicated the strangeness of tha finui atatct. In any caaa S • -1 atataa dominate, but tha pretence of further mixing angle* could affect the pat tern of tha ao called Cabibbo suppressed, US • 0, transitions. This fact has focused much theoretical attention**»11', especially in connection with the SPEAR determination of the branching ratios12): r rir-,T'r) x (U^ r\ * Cs.3 * *•*)% The branching ratios Eq. (3.19) deviate from the common value, ean20, which is expected in tha four quark modal and in tha exact SU(3) limit. The breaching ratios Eq. (3.19) can be analysed in terms of D-spin*' properties of the affective Lagrangian (3.18). He may rewrite tha S • -1 and 5 - 0 terms in Eq. (3.18) according to: ffll'' ' CMU0S)(8J)UÉJ +iCU»jUtj -UnUÎsHcaJKâe) -Ms)(7c)] Tha S --1 and the first 5-0 tana ara aura tf-spin one, while tha last earn is purt lf-spin. zero. At a consequence, desseins; by A and I the amplitudes for producing the final maton *' B-aain is tha Stt(2) subgroup of flavor SO(3) which mixes d and a quarks. pair in U-spin equal 1 or 0, respectively, and using the matrix Eq. (3.6) one finds (we aet 6-0); ACD'-»*'*») -csA 7 I (3 2^" Thus if Bt 0 Hi vt can split th* K*lT fro» the irV* rat*. Eq. (3.20) givea rise to the sua ruU"»M>: Correcting the rate* for the appropriât* phase «pace factors, and assuming real amplitudes, the left-hand aid* of Eq. (3.21) £•: while the right-hand aid* 1st 3 4(fto*»(i + ^£)* «O + ^êTC* *»). (3.23) A good air.aa.iit ia thua found for tha uppar aolution, (which corraaponda to oppoaita aign but ia not naadad with eartainty. The probla* with thia analyaia ia in tha larga U-ipin •aro aaaUcada naadad to raaroduc. tht obi.rvad diff.rmc. batvaan R(IT+IT_) and K(K+K~). Froa Zaa. (3.20) «ad (3.19) ooa finda: |A(D*-»ïï»r)+A(3>'-K+K-) 3 le+%/4,4 "7" (3*24) Al^rri-^^Pirl A «lVAeUt\*li)i\*\ (3.25) for 6 & IB. |aitvy| 4*5. Thus a valu* of |*|M 5|A| is raquirad. This may load on* to sus pect that there are other U - 0 contributions, besides those given in Eq. (3.18). The so called penguin diagram, Pig. 15 (s*« next pate) gives a U - 0 hamiltonian, and it has been proposed a* an explanation of the large D • 0 contribution. Tha situation is however un clear (i.e. inr*resting). c W u Fig. 15 Penguin dlsgran from the charm- changing non laptonic bcniltonian. Dashed lint: W, iravy line: gluoni. To conclude thie «action va may eay that the etandard 6-decey and K phe us a rather stringent bound on 0, and a such less restrictive on* on yi /i £ \b ~ s*icr I&» tin. M - or Theee considerations do not exclude vanishing valuee for one of the two angles (the vanish ing of both if excluded by th* bounda on the stability of b-aatcer). D-dccays could give soae inforaation on 0 and yt and indeed tba argument discussed previously would indicate that B*t«B has to ba a sixeable fraction of 8, if one has to explain in this way why T(0 •*- n*t~) 4 r(D •* K*K") with not too auch U « 0 «nhancsawnt. It is however clear that we need a aore direct phenoaenological input to dateraine 8 and Y> This «hould be provided by the study of b and t decays, which we shall consider in the next section• 3.3 Weak decays of b end t. The weak inclusive decays of hadrons containing lore quarks can be easily studied in the s&proxiaation where the doainant decay mechanism is the heavy quark semileptonic or non leptonic decay: ^-> JL" *U +^tt (3.27) fa-* If Ma + 3U <3-28> and similarly for t: t _»jl++Jj[ +fj <3.29) q (q.) denotes an "up" ("down") quark such at u or c (t quarks cannot appear in the final state since a. < a.) and q,^ is dû or any other allowed coabination of « down quark and an up aotiquark. The decay rates of processes (3*27) to (3.30) can be coaputcd in taras of quark aassaa and of the weak mixing angles* so that inforaation oa the latter quantities can ba obtainad fro» the sieasurement of the decay rataa of procttses like e.g.: + unch*n«té h*éro*s. îhia i« whet we shall do in the following sec-.ura. Before going into this, however, v* would like to discuss two pointa: the reasons in favor of quark decay dominance, and tha atrong interaction correctiona to the processes (3.27) to (3.30). Quark decay can ba thought to dominate in b and t decaya because m. and a arc auch larger than any typical atrong interaction aaas. In this regiae: i> the parton picture ia thought to be valid, so that tha decay rata of a b or t con taining hedron should coincide ttith tha rata coaputed from eleaentary weak pracaiaea involving pointlike quark*; ii) in competition with quark decay, we aay hava a weak interaction process of the b or t quark with a spectator light antiquark (or quark in tha case of baryons); this interaction involves a factor ^(0), the aaplitude for finding b and the spec tator at tha ease point; since we expect the radius of the hadron not to decrease in proportion to aT1 or a"1 (but rather to atay approximately constant), the rate for quark decay which involves tba 5th power of a. or a should doainete over the rat* for quark interaction, which involves |^(0)|x « m3. a'. In tha eaaa of ehara decay, tha dominance of quark decay, which predicts equal life times for all weakly decaying hadrona with tha saae flavor, is contradicted by recent data33) which indicates that tha lifetime of D° (where quark interaction is Cabibbo allowed) is shorter than that of B* (vhare it ia Cabibbo auppraesed). If this ia because a is not sufficiently large, ao that tha arguaent (ii) does not apply, or because of a fault of the parton picture, or because of a shrinking of tha hadronic site with tha heavy quark aaaa, «a canuat tell, at present. For tha tiaw being wa chooaa the first option, and assuae that quark decay doainance will work in b and t decays. Strong interactions affect process (3.Z7) and process (3.Z0) differently. QCD correc tions to the scaileptonic rate and to tha chargad lapton energy apectrua hava been computed3*). To lowest order, they are similar to the QED corrections to V decay, and ara of order OS (a. ) (or a (m )), a being the running QCD coupling constant. Thus, QCD corrections tend to ba smaller, tha larger tha heavy quark mesa ia. Lowest order QCD corrections to the non leptonic rata, on the other hand, turn out to ba of tha order of a tn(mA/m? ). In the leadlag log approximation, such corrections can be auaaed by renoraalization group techniques, and give an enhancement factor to tha non itptonic with raapact to tha sealleptonic rate"*"*: (X L enhancement) - 2f * + f * where m - m^ or »t, and tht inomalour dimensions d± hava baaa computed in tha one loop ap proximation. In tha absence of atrong Interactions: a((m) * (^(m^ •*• 0, and the enhance- •tant factor reduces to 3, coinciding with tha color Multiplicity of H.L. over 5.L. processes. The further corrections of order a^(») to the non leptonic rate involve a two loop calculation, and are unknown, at preaent. Thus, to be consistant, in the case of b decays we shall include tha enhancement factor (3.31) in N.L. rates, but we shsll evaluate S.L rates to zaroth ordar in a (m. ). For »t £ 15 Gev, the distinction between leading and non-taadïng QCD corrections is not very meaningful, and tha leading corrections are myvay saall. For this reason w* shall ornât QCD corrections to t decay. Vc turn now to discuss the partial decay rates of tha b quarks, processes (3.27) and (3.28). Kates arc convianently given in unite of the natural decay rate: and ara reported in Table * for tha various allowed channels*1'. First coluam decay rates of the b quark in units of To, sea text. It denotes either. « or U> v« the corresponding neutrino; • = einv, c = cosy ate. Third to fifth colum: probability for each channel to yield 0,1 or 2 proapt charged Uptons. p Decay chvu.1 r/r\ ? . *, 2 0.41 .» c| 0.00 0.80 0.20 CT-;T 0.0» .» c| 0.53 0.40 0.07 •I 0.00 1.00 0.00 U T~ V 0.39 .| 0.66 0.34 0.00 0.10 0.20 0.00 cH.c 0.25 ej Ç eg 0.64 0.32 0.04 u û d 1.00 0.00 0.00 u c » 1.4 .Jc? 0.(0 0.20 0.00 The numerical factors include; i> phase space correction* (ve use: BV - 5 CeVt m - 1.75 CeV, m • 1.80 GeV); ii) in the case of non leptonic decays» the enhancement factor: The semileptonic rates give the best way to detemine the relative size of 0 and y: As for the non-laptonic rates, we observe that the decay into one (C - +1 or C - -1) charmed particle plus C « 0 hadrons is insensitive to the value of 0/Y. What discriminates between 6 « Y *nd B » T » the rate into states with no charmed particles, with respect to the rate into states with a CC pair: pft,-» nocktrmej ftriitltiU P(b-»1cJ<*rmeA ftrtitle) : If the recent observation.*) of bottom particle through the decay: is confirmed, it would strongly indicate Y » B. Adding up the entries of the first column of Table 4, one obtains the total b rate as a function of 6 and Y* The bounds previously obtained for 0 and Y give then11*17': Thia bound is interesting, in that it predicts a b lifetime in a range still suitable to emulsion study. tfe consider finally the decay of » t quark. The rate for any given channel can ba written ast Va have denoted by XQ the weak coupling factor, including color multiplicity, while ^in phase space correction. Table 5 gives the expressions of X and the numerical valu* of g for tfae three cases: tit - 15, 20, 25 CeV. Table 5 Weak couplings and phase space factors for t decay, sat text* t denotes either e or Mi 4 denotes either à or e. In evalu ating X the rates for the latter mo cases have been added. The • quark suits has been neglccteà in phase space factors. «a • - 20 Decay channel *„ .,-15 t •t-« 0.45 0.63 0.74 *«.%„ CB *Y b T* v 0.38 0.58 0.71 T •K ,«*v + C 1.0 1.0 1.0 t •Y Y'I 9 T* V 0.89 0.M 0.96 T 'Y^Y'S b u q 3 b c q 3^c|(c»*.'.|) 0.3? 0.58 0.71 1.0 1.0 1.0 i>i "!<'Y*CY'!' q»i 3(c> • .» .|X.* • e» .|) 0.89 0.93 0.96 q S c 0.38 0.59 0.71 3 «Y CS(,Y * q b u 3 ,J(.; * c} .|> 0.45 0.63 0.74 b S c 0.08 0.30 0.49 3=Y.Jc5 b S u 3 Irrespectively of the velues of the mixing, eagle», the t life tie» is expected to be extremely shore, «bouc 10'17 see. for •., - 15 CaT. 3,4 Huit il tp too, confifturationa in b and t decay» ' According to the pictur* discussed in the previous section, the disintegration of heavy flavora into normal, strange or non strange hadtoaa and lcptonaf involve» many steps in cascade. For instance, t decay may occur through the sequence: where X, Y and Z are normal hadrona or lepton». X and Y «ay themselves contain charmed had- rons, which would give rise to additional cascade decays. Since at each step there ia a aizaable probability (20 - 30Z) of emitting a. charged lapton, the probability for finding one or acre lepton* among the products of the over-all decay turn* out to be rather large. Thia fact auggsat* a possible signature for identifying the production of new flavors, for example in «*e~ collisions, which could be used in conjunction vith other proposed signa tures, such as a high sphericity of the éventa1*'. Furthermore, a measurement of the details of the cascade, e.g.» the relative probabilities of events with a different number of charg ed leptona, may give valuable information on the values of. the weak angles. He consider b decay first. The last three colms of Table 3 give, for each decay channel, the probabil ity for the final state to contain xero, one or two prompt charged lap tone (either toru). The leptona can be produced either directly, or front the subséquent c and T decay, tfa have assumed! ?>(e-»ey,) *o.*.(> , B(«-*tf,y») * 0.S4 . From Table 3, on* can eaaily compute the probability for the emiaslon of fc prompt charged leptona in b decay, V(k - 0, 1, 2). Neglecting terme higher than second order in sing and linY* on* obtains! & = o.fgsi»*t/R By dépend', only upon the ratio s in*67s in2Y. For all values of this ratio, the probability for emitting one lepton ia rather high: B, « OMO iA't>) B., «• 0.33- ((h»f) *' w* follow hara, alaoae Utaralljr, tha analgia »l»ao ta «at. 36). Bj is auch •mailer, but not negligible, for email &: Bj, * "\*\o~ One can alto consider the probability for the emiiaion of n charted leptoas in the weak de cay of a pair of B$ hadrons, auch aa one produced in e*e" or hadron collision*. This pro bability, which we call (BÏ) , can be obtained directly from the IV given above. In Fig. 16 we have rsportsd (1Ï),, (BB), and the probability for two or more charged leptons, (BB).2, aa function! of sin26/sin2Y (tBB) is dominatad by the n • 2 configuration). Tig. 16 Probabilities for prompt lepton emission in BÏ weak decay 2 versus sit^B/sin ^. a » (»)0i b - CBB).;_c - (Bl)., = A» anticipated, lepcon emission occurs with high probability: (Bl)# is smaller than SOX. The probability for two or «or* charged leptona ia the most sensitive to the ratio ain'd/ain'Y* Va turn now to t-decay. Combining Table 4 with Tabla 3, one may obtain the probability for the emission of k prompt charged leptoms in t-decay, Tfe(k • 0, 1, . . . , 5). For a given t mass. T. depends now on g cod Y separately, this dependence arises from two different effects. The first is the dependence of b decays upon the ratio sin*f)/ein*Y« the second is due to the appear ance, for appreciable ain3Y» oi decay channels which do not involve b particlca, and which value of Y« To illustrate this, we have reported in Fig. 17a (aaa following page) the prob- 1 abilitisa T,, T, and T * 2, «a a function of sin Y,,for the case 0 • 0, »c - 1.5 CeV. Simi larly to the ll case, on* can compute tb* probabilities for multi-lepton configurations in the waak decays of « TÎ pair, which we denote by given by our bounds. In Fig. 17b (see following page) we give the values of Until now, w* have reatrictsd the analysis to total rates. Further information is con tained, in the lepton spectrum, ly restricting the lepton energies in such a way as to select th* first generation leptons (i.e. those arising from b •*• c or t -*• b decay) one may r study vary interesting phenomena, luch as B( - S( mixing and, possibly, detect a CT viola tion in the neutral B-mesoa or T-Boson system5»"). fit. 17* Probabilities for prompt lap too emission in t-decay versus «iny. a * T,, b - T.» c • T-, = T, + . . .+ *9. Fig. 17b Probabilities for prompt lapton «miiiioa in the-vesT* dieay of a TÎ pair» versus atnv. « » (rt)i,t d - (IT)>% * 10 5 [(TÎ) • .. 4. CHAMP UWIFLED THEORIES 4.1 The unification of weak, electromMgnttic and strong interactions. The theory uaed to deicribe "low-energy*' phenomena it a direct product of gauge groups SW3U® S0U)Lt(t8>UU) Wil) which axa characterized by three distinct coupling constant* which we write as §•"«• ' 3**2' h'l' {4.2) «here the parameter c reflects the fact that the coupling constant in a U(l) gauge theory it not uniquely defined because the currant normalization £• not constrained by consultation relations as {or son abaiias groupa. Utiat do ve Baas by a coupling constants the "bare" coupling constant is infinitely renoraaliced and is not an observable. Instead we define en "effective" coupling constant ï-Gi1) which is the value of, say» the vector-fernion-anti- feraion coupling strength at son* normalization, point p1 - U2 for the external noaanca. To order o. - g|/4n the diagram* which contribute to t^(V2) sre those of Fig. lfla-f. i. y- Y ,Nf • Y* Y a) W c) d) •T rig. 18 Lowest order contributions to coupling renorsMlixation Calculated pa««bativaly is terse of • fixed coupling constant they diverge logarithmically: where h i$ a eut-*ff in the loop mesjtntusj integration, m is the suis» of internai particle and x is « parameter of order 1 which arises fro» Fayomsn integrations. However the dif ference between couplings defined at two différant normalisation points is finite: (4.4) - 38 - The perturbation, expansion (4.4) converges as long as the logarithm does not become too large; taking u and u' infinitesimaliy different, and since g. - g. + 0(g.2)> we get a dif ferential equation ?J» ^(fifol** H 2>J*2ii/')/Z>iU/<. with solution 2 2 2 2 2 2 where b. = £(b.; m. < u ) and by definition m^ < u , u0 < m. . At diccussed in the course of Ellis and Sschrajda, Eq. (4.6) is equivalent to the summation of the leading terns in a.ln(|iz/iio2) in ordinary perturbation theory. He see from E^. (4.5) that the evolution of 2 2 2 the coupling constant for values of the external momenta in «one range uB &-p £ u is governed by loops with internal -JJUS m,2 < u2, )isz. It turns out that the logarithmically divergent parts of Fig. 18b-e cancel one another (after wave function renom» lizat ion) be cause of gauge invariance, «o the parameter b- is determined by Fig. 16f, g, h, which de pend explicitely on the number of vectors, fermions and scalers, retrtece'vely, with m1 < ps. For an SU(n) gauge theory with fermions and scalers in multiplets of dimension n one finds b, ' ( 11 n - (lt - Us A )/llV (4.7) where H, is the number of two-componane-fermion multiplets end N the number of scalar mul tiplets with ma < V2. Now let us examine the fermion contributions to the evolution of SU(2) and SU(3) coup lings arising from the loops of Fig. 18g which are proportional to: SOW- f#»t'%$ ' *$jf** «-a) where N- is the number of fermion doublets. In the conventional picture described in Sec tion 2, with universality between color triplet quarks and color singlet leptons we have: where H„ is che number of quark flavors and HL • H_ ie the number of lepton types. Also: SUH): ]Î0TTr£% = ^ff (..IO) where the number of fcrmion triplets NT is twice the number of quark flavors since both left end right-handed components axe coupled. Since SU(2) and SU(3) have the same dépend ance on the fermion concent of the theory, their relative evolution (neglecting Higga scalers - 39 - uhîcii «re relatively uniaportant, as iiut in the aisisal nodal) ia governed only by the 1 rank of the group. Since b, > b2» %t decreeeee relative to g2 with increasing V , and since at present energiea g > g, thay will eventually becosw equal* If the group (4.1) represents the ultimate theory, we «ill have g < g for \i -*• <*, A «or* attractive possibil ity"0' is that there ia a larger gauge group C 3 SOW® SOMOVa) (4>11) with additional, very «stive vector bosons X which do not contribute to tha «volution of couplings at present energies, but which assure thtir equality at sufficiently high energy: „ >HX. For this scenario to be viable*'there ssast be a point y2 where all the g_ n - 1,2,3, attain a coeawm value. Since the U(l> coupling gats so contribution froa diagrams like ISf, we will have ^3 ^ ^1 ^ kl (4.12) as long as the fersion contribution to b i» identical to that for b . The latter eoa- 1 3»! dition is in fact necessary for 11(1) to be iabedded in a sisple group containing SU(2) and SVO)—and if the presently observed feraion spectrum forets a coaplcte {possibly reducible) représentation—and serves to determine the normalisation of U(l). In this case the loops of Fig. 18g ara proportional to where (q - I ) - 1/6 for each left-handed quark, -1/2 for each left-handed lepton and is just the electric charge for each right-handed feraion. The U(l) coupling constant g is thus determined by ¥P* s h Now let us recall tha structure of the SU(2), • U(l) part of the theory. In the eya- aatry Unit tha fanion gang* coupling* ara given by X-e„. 'I9'1 +/'8Y< (*.15> where f and If ara shorthand for tha appropriate feraion currants. After syi^ietry breaking, tha neutral vector bosons which are aaaa aigenatates arc Z* and the photon A related to W and a byt *1 We assume a single step breaking of C •* S8<3) « S8{2) 4 U(l); sure complicated patterns are possible*1' in sufficiently large groupa C« - 40 - The structure at the observed currant» is obtained by substituting (A.16) in (4.15Î Sinca the photon couples onlly to «ltctric charge Q with coupling strength c we get the constraints so that the "weak" neutral currant coupling becomes #C " 5s5L ^9^S " -?S#V£„ 1 . <4.19) 2 Experiments on neutrino and electron induced neutral current* determine th* value of sin 6u; th* present world average is*2* sm% * o.l% to.ots . (4.20) Because of the result (4.9), a neccfaary condition far unification is that at present energies V * h' P • «.2» using the résulta (4.M, 18, 19) we have f'jjt ' /| Û * {% t*»0u H «J.?I < I , (4.22) Ia the symmetry limit for the "grand unified theory" However the symmetry is obviously badly broken, and the relations (4.7) can be realised only for \i such larger than the masses of «11 the gauge bosons associated with the group G. In order to aea whether grand unification is indeed possible, w* can uaa two of the measured low energy paras*tars, which we choose to be g and e, together with the evolution equations (4.6) to daterais* the mus H where unification occurs. Then the remaining fre* parameter, which we take to be ain*9 ia uniquely determined"' and its calculated value can be com pared with th* experimental result (4.20). v shall audio* this calculation in the "lead ing log" approximation of Eq. (4.7), neglecting threshold corrections in the ragions Vs a H,2 which contribute only over • small region of the int*cr*l. The fine structure constant il determined on the mua thall of the photon and electrons*): a = a(m *), For u2 < w^3 it» «volution it determined only by fermion loop», io wa hava to integrate Eq. (4.5) over the range • * < ji2 < uj1 tafciac a atap function for b; the result is*****) dCv\w) , 1.o7c( For u2 > «y2, the electromagnetic coupling ia alio «normalised by W* loopi in Fig. 18fs in thii domaine wa hava to un the SU(2) Cj U(l) evolution aquations. Cling Eq. (4.18) wa nay writei and we rewrite Eq. (4.6) ai Tha strong coupling constant is cbaracterixid by a aits parasiter A, which is Measured for example in deep inalaatic leptoo-nucleoa scattering as discussed by Ellis and Sechrajda: «V/}"» - gig*JL(*Vtf) ' b,A^%0. <*•«) If there is some value H for which o(, wa may combine Eqi. (4.27-29) to eliminate tha unknown a^: Evaluating (4,30) at p2 ••*•*, using Eq. (4.7)» recalling that tha farmion contributions to the 1>£ are identical and neglecting scalers we gat which determines tha unification point in terms of low energy parameters» The effective value of A in Eq. (4.29) depends on tha number of quirk flavors with Hy < yj tha parametari- •ation (4.29) neglects threshold effects when M_ • u and A must be adjusted 10 that a, re mains continuous aa a threshold is croisad. Four flavors are relevant for tha experimental ,ff value A - 500 MeV, so if "^ - 6 we get A * 250 MsV for u - m„ > m£ b„ giving unification at ft * ¥.4 * ID** CtV. 1 (4.31) *) In practice one should first extrapolate to space like extermal momenta] uz - -m *. In fact a measured on shell is the same for all fermions, but the logarithmic diverg ence at small space-like u will he scaled by the mass of the lightest fermion. We can ua« this valut in Eq. (4.28) to aolva for cu» taking i • 3, say: and for *in2fl taking 1 • 1,2: giving**' SiV*9„ * 0.30 ± 0.t>1 (A.33) vhich it not incompatible with (4.20). Tha calculation* daacribad abova bav* in fact batn corrected"**5**7' for higher order contribution* and for thraihold affacti. Than ara uncertainties in the valuta of A and N_ which determine tba input value of o,("y) and alao on tha effect of Higgs acalara. What ia the aeaning of the unification MIS H? It ii approximately tha (coaawn) mats acale of the remaining—ao far unspecified—gauge boaona of the theory which ara assumed to cootributa via Fis» !••' ** tn« couplings for u2 > H2 in auch a vey that bi • bj - bj and the couplinia remain equal. 4.2 Tha minimal modal: SU(S). In order to proceed further we need a specific modal for C in Eq. (4.11). Since the rank of the direct product group ia 4 wa necessarily ham R$ ^ T . (4.34) Caortt and Claahow observed*'' in 1974 that tha only croup oi rank 4 allowing the correct low-energy structure ia SO(5). Zn thia case there are 5a - 1 » 24 sauge bosonsi in addition to the octet of colored gluona of SU(2) e>D(l) there arc 12 super heavy bosons, •_ «H which transform according to a complex (3.2) representation of JU(3): f X, Xa Xâ \ her hi. fw '!• The résulta of the last section show that unification ia compatible with the assumption that each lS-compoaent set of fermions vith tha quantum numbers of (u. d, e, Vfl) form a com plete representation of the unifying group (although additional components» auch as a right- handed neutrino» vhich decoupU from the observad currents ara mot ruled out). Tha lowest dimensional irreducible representations of gtf(5) with their SU(3) a>SU(2) content ere: £-(3,t) +(1,D W * (ff^t, - (3A) + (3,1) + (1,1) l <4W is • (s»s\fJttric -ti/Ly+tk,» +H.3). - 43 - The representation 15 is unaccaptable as it contains a color aextat and a weak ieospin tri plet which axa not observed. However the representations 5 and 10 have the correct quant» nuabers to accomodate the observed spectruai: S = S jA (4.: (4.37b) 10' = X. where the superscript c denotes a chart* conjugate state» a ia a "generation" index and the prisas indicate possible Cabibbo mixing* relative to fermion Bass eigenststes. We note that each representation 5+10 contains exactly 15 states per generation, so there is no room for a right-handed neutrino. This aaao* that all neutrino» are identically massless: H* S 0 (4.38) (unless one introduces vR*s aa 50(5) singlets or Biggs representstions which could give rise to Kajorana messest these complications are required neither by theory or experiment). Tb* icalar content of the theory most be such a* to reproduce the "observed** symmetry breaking. The ficat atep SU(S) —* soc3>e • si;u)L «t/co «.a,, ,f which occurs «t a masa scale «g ?* I0 GeV conserves the rank of the group. This condi tion will be mat if the scalar field which acquires a non-vaniahing vacuum expectatif-\ value transforms according to the adjoint representation of the group, in this case 24. We can represent a 24-plet of scalers as a 5 x 5 tracaless matrix •. Then its vaeuu* expectation value can always *« written in diagonal form; 2 <:$> (••"J (4.41a) <&>• f-U- <4.*lb) In addition, the case (4.41a) la eteble ageinat radiative corractiooa to tba Biggs potsotial*'). and it in fact corresponds to tha patten (4.39) giving large messes ^y-0(^«V) to the vector fields (4.35) and leaving tha others aassleas. Thus we aae that the first stage of symmetry breaking can be naturally realised via a simple Higgs potentir" This picture becoswa much leas appealing neon introduction of the second stage which «cars at a ansa scale «^ 2** 10* CeT. The difficulty lies in preserving the empiri cal condition in tha presence of radiative correction* co the Biggs potential**'1*). Va shall not diacuis this problem here) we consider only the group structura of tha additional scalar fields which nuit be introduced to realise (4.42). In addition to breaking the electroweak gauge symmetry (4.42) we wish to give Basses to fermions. The desired fermion mass teres are Dirae masses of the form vhara C la tha Dirac charge coajugatioa aatrix. laterriaf to the uiiroaaoti («.37) v. ••a that the eerraet feraioil Maaaa will be geaeratae by Tufcave coveliagi of tha for. (4.45) <£, 3 H5J0U + HlOLC10L -H>.c. So additional liggs nul. tplets must he contained in: J* to » s®*s 7 tot IO * J© is9so J <**6) Of the irreducible representations conteinsd in (4.46)» 50 does not contain a part vhich transform* like (1,2) trader 50(2) «11(1), as required by the empirical fora of the electroweek breaking (4.42). There are some arguments'" which fevor the assumption that only one weak iaospin doublet * acquires a noa-vanishing vacuum expectation value; a) In this case electric charge is necessarily conserved; by an SO(2) transformation vhich leaves the Lagrangian invariant we can always write <$> *(%.) <*•"> which defines the electric charge direction as discussed in Section 2.2. If there is more than one doublet, the potential must be chosen so as to assure that their vacuum expectation values are parallel. b) As remarked in Section 2.3 a single isodoublet coupled to fermions assures") the absence of neutral stramgenasa changing effects like that of Fig. 19 because its couplings axe automatically diagonalized along with the mass mi true. The aimplest assumption is to introduce 2 single complex 5-plet of scalar fialds on the SB<5) level; (4.46) white H, and 4 are respectively an SU(3) tri- Mg. 19 Kscbeaisa allowing *Ut «* « ««^L doublet. If the miui.ua of K, •* u*u~ for a multi-liggs system. the potential for the coupled * - H system sat isfies (4.41a), It will also satisfy either or WO, Hature has chosen the case of tU(2) breaking, (4.49a). 4.3 Faralea massss ami constraints on the mmaer of «anaratioaa. Ihe introduction of a single mon-adjoint scalar rsprasentation, Eq. (4.48), rsstriets the fanion, mus metric» to torn» txtaat. K o, &,.. at* 50(5) indices and a, b... are generation indices, the Tnkawa coupling* taxa the form* where C and C are arbitrary complex matrices. The fermioft smss matrices are obtained by the lubatitutioat K •*• l v £" = $, C* = CimG,, G> ~0. (4.5l, In addition to aaailais aaatrînoa, va obtain idantical aaaa aatrîcai for chartad laptona aad for quarka of charga -1/3. Thia maaaa that thay beva tba aama aiganvaluaa ao that in tha aymaatry limit laptooa and quark* will ba aair-viia dasauarata: ^l " W/L (*.52) Wt - Wt. Juat aa tha lyaawtry ralatioaa among eoaplioaa, Eq. (4.23), ara ranormalisad by eym- matry braahiac affacta, ao will ba taa IUI ralatioaa. Tbaaa affacti can ba avaluatad using raooraalisation grog* tacbmiquaa aa bafora. To thia and wa dafima an affactiva "running" •lit by dafiaisg tha draiaad faraion propagator. Fig. 2D» aa 1 S -£2i_ . A.... rig. 20 CoatribatSoaa to aaaa renormaliiatioa. z Por axtaraat aomanta |p*| » «J T tha ralationa (4.52) will ba latiafiad, but for |p I < a£ Y tha X,Y axchanaa contributioma drop out ao that lapton and quark nauai ara «normal it ad diffaraatly. Tor a1 • -p* tha lowaat ordar eorractiona in lit* 19 ara logarithmically di- vtrgtat; for i - 1, 2 or 3i ..t **u> » MU') + otiif)bm Yn(M)JULf'/j*) •«•••• (4.5*) agala liviaf a diffaraatial aamatioas vtilcb o *• 1.WCHU4 i» tfc. Mlu4he lot1* ^.roxl.at£oa utiot tfc. utprciilon (4.6) far a(v). hltlil I • li ii n hn 4 Suaaîag chc contributions for i - 1, 2 and 3 va jet 1 Sb"a ™<£l - rrfS*i!£L\h : In the notation of Sections 2.2 and 2.3, we know that for each generation •d. Th* dependence on a, drops out of (4.56) because charged Uptons and quarks of charge -1/3 have cba aaan weak isospin properties, and in fact the s»st important effect ia gluon ex change contribution* to quark propagators. The result is rather sensitive to the nuaber of (eoeratieu; explicite!)» we get**»"* for A - 0.5 • Is I - S h- Up vhar. tha «..liar ratio, on th. right includ. fiait. iui> «ffacta; including higher ordtr tun in th. b*' incrsasss thesa owe.» sonawhat"'. Th. lapton MIIII ar. naasurcd on lh.ll but this askas littla diffaraaca aim:, thair variation with u it vsak; taking N£ • 6 in Eq. (4.57) nt fini for u - 10 GaVi 7»7t » S.t GtV inÀ * At Mt\f In ordtr to test the success of the prediction (4.57) we aust have a aessorable defini tion of a quark aaaa. A prescription proposed by Ceorgi and Foliteer"' ii to define the "constituent" quark MSS at the threshold for qq production; tlKit 2 fti.ljtr.?»?.) • («.5» Using thia «.finition v. fin."' "*!,, ' (S-i)âtV, W, * tQ.*-t.s)GnV (4.60) coaat. •4 ' «MM to «atarnlais as It noaH iaralrs «trapolstioa of !.. (4.54) to V « 1 CaV vhar. th. p.xtwbativ. txaatawnt br.aka «own. Tta. rasulta (4.60) aapaar vary satisfactory, but a maasar of .iff ic.lit.e hav. bM« raia^. - 48 - a) Tha definition (4.59) ia not sauge invariant. In fact the only point «here the •ass ia gauge invariant ' in perturbation theory is on the mais shell: • = • (u - • ). This point ia disliked in QCD becauae it is sensitive to the 9 9 q infrared region [as is. in fact (4.59) if extrapolated to p* > 0, the actual threshold]. b) The "•aasured" constituent s-quark aass in indeed about 500 K«V, but it is argued that such of that ia generated dynamically and that the aechanical quark tuts con tributes only about 150 KeV for v = 1 GeV. These arguments are not entirely com pelling3s ' and in any case tha extrapolation of (4.55) to u = 1 Ct.v is less reliable than tha prediction for ajj00"*. c) A «ore serious problem is the conflict of Eq. (4.56). which predicts lw*Aj,>Jû^ ~ Tlf. " ZOO «•«» with tha current alccbr. ri«oltsc' &) (4.62) ws/^ f;„« *° where tha "bare" •asses «assure the aawtmt of chiral syaaetry breaking in the La- grangiaa, and their ratio ia equal to that of the running suisses defined by Fig. 19 for sufficiently high y so that finite auus effects are uniaportsnt. On the other hind a and •. are ao tiny that any outside contribution to femion masses—possibly associated with gravitational interaction*, for ev-^ule3S'—could change their ratio appreciably with a negligible affect on heavier MSSCS. Ignoring these difficulties, tba success of the prediction for the b-qu.iik masn strongly •uggssta H, * •[ if w* incraaaa tba nuaber of flavors with • i m^ _ we increase m^ apprecia bly, ea can be seen from Eq. (4.So). Of course this conclusion is aodel dependant. If in stead of a S-plet of Higga we had taken a 45-plet, instead of (4.52) we would get while this appears Intuitively bicarré» if wa take H». • 10 wa get for v • 10 CcV •Wj * ifo Mt V v>j » /.3 fhf which «ay be as acceptable aa (4.51). Mowever there are other argiaeeata which tend to liait tht nuabar of generations. While the relative «volution of tha couplings is unaffected by their nuaber. their coaaon value at the unification point rises with Kp ae can be seen from tqs, (4.7) and (4,2ft). Thia affecta tba stability of tha proton, to be discussed below, and present sxperlaantal 1iaits in tha proton lifatiaa require*7' K. < 16. la addition* cosao- logical arguments bated on the abundance of helium in the universe limit5" the nuaber of aaiflc» ncutrinoa to 3 or *V and hence—in the present theory where each generation includes An alternative approach is to exploit the possibility of a aore coaplex Higgs sector in order to produce aess relations in accord with a priori theoretical prejudice. Including bocb 5-plets and 45-plcts of scalers and imposing appropriate discreet symmetries it is pos sible*1*5*) to arrange that in the symmetry liait: which fins inataad oE (4.(0) and (4,61)-. Rovsvar thii picture antaila tha drawbacks9*' ot a sulti-Hitta syata» diacuaaad in Section 4.2. 4.4 Frnton decay. Tha coupliofta of tht aupar heavy X «ad Y. boeona violate eepsret. lepton and baryon Dun bar conservation. Explicitai? tbair eooplinge to tenions la given byr (4.67) where i, j, It ara color indices and d'T » u-, s'.v,;... ) u.'T * (.u',c',t',...> a» eigsr.vsctor* of SU(5), Eq. (4.37), but not of •••», Kota that (4.67) eoniarvas a - L; this conservâtion law ia specific to SU(W and trill in general not ba valid for larger groupa. In order to determine the effective Lagrangian zor proton, decay ve have to xevrite (4.67) in terms of mass eigenstates which will introduce Cabibbo anglea into the currente (4.67). There anglea are determined by diagonal!zing the Yukawa coupling matricea of Eq. (4.50). A priori there ie a possible Cabibbo tranefoxmatlon among generation» for each of the 15 component» of the 5 + 10. tfc ahell show however that, in the simpleet model described above, and apart from additional phases, generation mixing it completely determined by the Cabibbo matrix discussed in Section 2.3 which rotates (d, s, bf,..) relative to (u, c, t,.,.).. First note that we are free to make any SD(5) invariant transformation on generation indices without affecting the gauge couplings which are generation diagonal: + T (n. -$*s ^ . + T G=V FU. Z*U FV, {.-ytf T,uff «.. where U and V act on generation indicée, so the Yukawa coupling (4.50) becomes: T (4 70> #Y » H I FT + HT FT + !,.«. - while the gauge coupling (4.68) remains invariant: ^l"je " \ff I +^ $T Given any C, we may chose D and V such that F «^Mi is real and diagonal. Then mil the Cabibbo angles are contained in the matrix F which ia related to the charge 2/3 quark mast matrix by a unitary transformation: ^F -V*^U. <4.72, Ve see from (4,50) that C is necessarily symmetric; then so is F, so T + T F»F =*• V {AHU » FF+ , T * ^ yt^^OXO* . F F (4j 74) l which implies that the unitary matrix S - W cornantes with the diagonal matrix Mu - 10 that S U diagonal, and V - U* up to a diagonal phase operator: V -Su* , S»w- <*'** Sa, - 51 - After symmetry breaking and the substitution H -*• (4.75) 10 that mu «igcaicat«f «re related to SU(5) ciaenitatcs by d'-d , fi'--*, u'L *U*ut , < -- u*$u'L . „.7M By definition U 5 Uia the Cabibbo matrix and it ia in fact the only obaarvable nixing matrix apart free, the phase transformation S which dota not appear in low energy phenomen ology. Substituting (4.76) in (4.67) w« gat = T + ^6-AL*» 4 *ï (."^RJC *^kS UUj K C ) (4.77) t iW(3.a4*t«iv&-lUJLi.K,turL) Since no new mixing angles appear in (4.77), it is not possible to "rotate" avay proton decay in the minimal SU(5) model. The effective Lsgrangian ia completely determined up to an overall phase ((inobservable in lowest order) and tales* the form***11) -W* vir*L -/;*%* i where is the affective farmi-coupUng constant. This coupling ia rationalized by radiative cor rection* of which the most important are gluon exchanges. They give in the "leading log" approximation*1 for the affective coupling at u • m with H_ - 6; W. 2 and Y exchange corrections "•*') re- normalize further by a factor of abeat 1.5. Once the structure and strength of the effective four fermi interaction relative to proton decay has been determined, one still has to evaluate decay matrix elements. In a seai-incluaive approach'*''1 inspired by the parton model, one assumes that two valence quarks in the proton annihilate giving q + I and that the final atate q combines with the specta tor quark to give a hadron system with total probability 1 (see Pig. 21). i _ (mesons I _ [mesons "!:>-<;' "I;—r-v a) b) Fig. 21 Schematic diagram* for proton decay: a) annihilation and b) exchange mechanisms The nonrelativistic quark model is used to determine the spin, isospin and color dependence of the initial state annihilation amplitude, and the matrix element is proportional to the amplitude J4> (0)( for two valence quark* to be at the same point. Its order of magnitude can be inferred from fits to non-leptonic hypcron decay data where weak baryon-baryon trans itions are proportional to With this picture one calculates*''") Since the dependence on BL, „ {m» • m^fl + OCm^/m-1)!} *• quartic, an accurate estimate of this mass is important. In Section 4.1 we calculated the unification mass M>-X.Y where the coupling g. become equal. Zn order to extract the vector boson mass, threshold affects*" must be treated correctly. The result depends sensitively on A and in the range of interest can be approximately represented by"' ™X,Y & Jr.3A"/o'y6eV (4.82) T«kitiJ A • (0.3 - 0.7) -Cp * 10 yetrs - 53 - where the uneezteinty in the exponent also reflects the uncertainties in the decey matrix element. This prediction is not far above the present experimental liait; assuming the Lagrengian (4.78) is relevantin determining relative leptonic branching ratios, analysis of the data gives"' Calculation of specific decey modes is even e»re problematic than calculation of the decay rate. In Teble 6 we «how two différent titivates***'*' of seed-inclusive branching ratios; the differences should be taken as « Masure of present calculations! uncertainties, Table 7 shows estimates'*' of specific chancels. An encouraging result is that the easily detectable channel n •* ir~e* stasia co heve a healthy branching ratio. Table 6 Estimates of Cabibbo allowed semi-inclusive proton decey branching ratios (X) in SU(5). Hod. Jarl.kot «ad Yodurnin61) K.ch«c.k«5> p » 1* + X tl as P * % * X 13 13 f - 11* • K • X S 1 1 1 n * •* + X ao 7a n •» V, + X 19 21 n •> v* • X • X 0 0 D - v„ • X • X 1 1 A further uncertainty is the role of the SU(3) triplet of scalar mesons which remein in the theory *s physical states. Keglectimg radiative corrections» the effective ferai coupling for protom decay induced by scalar exchange is The «ass of the ligg* triplet if not calculable, but a "natural"1**1'1) order of magnitude is - 54 - au i pu. „. A sufficiently low H, mess would not only louer the proton life-tine but could alao enhance final states involving <ë, u+, v) since the acalar coupling is propor tional to the "generation Baas". Not* however that non-Cabibbo suppressed final states in volving the second generation necessarily go via annihilation, Fig. 21a, rather than exchange, Fig. 21b, channels. Since the Ha has charge ±1/3, it can create a V* s pair but not a u+I pair, so only the former modes can be enhanced in this way if the Minimal model is correct. In fact the easily visible decay is doubly Cabibbo suppressed in this model because for annihilation the combination (v*s) necessarily arises from a uu initial state in the nucléon; i.e. the decay ia allowed, but the annihilation channel is suppressed by wave function effects as aeen in Table 6. Table 7 Estimates of Cabibbo allowed two-body proton decay branching ratios (X) in SU(5). Mod. r p •.,'.* 40 n * it'v 20 14 n -mve p •+• u.* 26 - + 66 n -* T . - + P*P*v, 34 n •+ p . t * uV 30-50 4.5 lot»., oa th. t-ouirlc —•• In 10(5), th. BUUI of th. lon-lyio» f.mion («Htitiou irii. «ntir.ly fro. th. •pon- UMOIU .rulcia* of 10(2) • 0(1), via th. Tokau coupling of tin f.raiou to oo. limit gift, •oltipl.t (. I of I0<5». 7k* Hittictlon. ineli.1 by 10(5) on th. TukMr. couplint. lead to the mat» relation* for the "down" quark* (d, s, b) and the "down" lepton* (e, U, *) which we have discussed in the previous notion. No restriction arises» however, for the Yukawa couplings of the "up" quarks (u, c, t) and therefore no prediction for the t-quark mass» a is obtained. SOM information on *c can still be obtained in SU(5), however» in the for» of bounda. This is whet we shall discusa in this section; «ore precisely we will obtain57': i) an absolute upper bound on • ; l ^ 100 &t\l (4.90) ii) a combined bound involving m and the mass of Higgs boson» ou. The origin of these bound? can be understood as follows. Recall that in SU(5) there are two mass scales: (i) the grand-unified mass» M_-, which defines the mass scale where the gauge couplings reach their SU(5) symmetric values» (ii) the vacuum expectation value of the Higgs iaodoublet field: < It turns out c the renormalization group equation! for the Yukawa couplings of the ferm ions to the Higgs field and for the self-coupling of the Higgs field, X» have an unstable behavior. Therefore when we integrate the equations over the very large energy range, m £ q £ H_, or» in a lagarithmic scale, in the range: "GO" 0 st-jkC&0*Lw we may obtain various unwanted effects. First, both the Yukawa coupling and the Higea self coupling, X, may become strong for t < L_, unless their initial values at t * 0 (or q - <*>) are suitably restricted. If this were to happen, two or more loop contributions to the rc- normaliistioo of the gauge couplings (involving e.g. the Yukawa interaction) would spoil 3 the nice one loop prediction for sin 6w. The restriction on the Yukawa coupling of the t-quark at t • 0, needed to avoid the above affect, gives directly the upper bound Eq. (4.90). An upper bound on the Biggs boson mass can also be obtained in a a?*nilar way. Secondly, if the Yukawa interaction is large compared to the gauge coupling*, the minlmun in the Bigg* potential at the value of f gives by Eq. (4.91) <+> • n could become unstable1*) because of one loop corrections*7'. This happens if the running Higgs self coupling, A, becomes nega tive (for some value of t). Indeee*, for X < 0, Che Higgs seIf'•interaction: J. - -iW)* (4.93) - 56 - become» attractive and configurations with $ » v//f become energetically mora favorable. In this situation» we say expect that the atabla vacuum (if it exists) corresponde to <$> » v//F, and presumably <$> •* M—. and va loose the needed hierarchy of the two syunetry breakings, Eq. (4.92). Again» one aay avoid this affect by a suitable restriction on X and on the Yukawa coupling at t • 0, which leads to the second bound, mentioned before. Va turn now to consider tha renormalization group equations for the Yukawa coupling of the t-quark, in order to obtain the result (4.90). For simplicity, w* neglect the Yukawa couplings of all tha lighter quarks and leptons. In the notations of Section 2» this amounts to set: V 0, •O -for i.j'^3. Futhsrmore, we define: so that: vnt « Vfed(t=o) The renormalization group aquation for h has been studied in Ref. 68), and reada: 1 •«•iif.-CW-fcir*)., WlW where a ia the running strong interaction gauge coupling, and we have neglected terms of ordar a and ay. The evolution of h(t) implied by Eq. (4.94) is easy to discuss. For a suf ficiently large initial value» ha(0), tha term proportional to h* in the right-hand aide of Eq. (4.94) dominâtes, and n(qx) develops a singularity ae a finite value of q2. Zf a were constant, this would happen for: corresponding to m, > 250 GaV. Of coursa, in tba vicinity of the singular point» Eq. (4.94) does not describe the correct behavior of h2, in that higher order corrections become sig nificant. The presence of a singularity in the solution of Eq. (4.94), for a certain value of T, is simply a signal that» at that energy aeale, the interaction becomes strong. This should not happen for T smaller than L-, and this condition leads to an upper bound for the initial value, h(0). He note that the stronger condition that h(0) remains small for T < Ly, leads essentially to tba same bound. This is due to the fact Ly is very large» and that the solution to Iq. (4.94) varias vary rapidly near tha singular point. We shall therefore adopt the weaker bornai discussed amove. Zf we allow atf to vary with t, according to what - 57 - «as found in Sac. 4.1, «a can deteraine the upptr bound for h(0) by solving numerically Eq. (4.94). Using ag{0) * 0.1, ona finds tb« rtsult atacad ia Eq. (4,94). Concerning this raiule we aake tha following rasuurks: i) an uppar bound to quark assses in tha SIT(2) 0 U(l) theory has been darivad ia Ref. 69), fro» tha raquiraasne that th* Yukawa coupling ba saell at tha energy seals v/vT. Tha resulting bound (a i 1 TeV) ia considerably largar than the out derived hare, which is no surprisa since va usad tha stronger assumption that the coupling reaains asmll in the vhole energy range, froa v/*T up to H—,; ii) an independent bound on the «ass splitting of a feraion doublet has bean darivad by Veltaan"), within the S0(2) #0(1) thaory. Applied to the casa of the t-quark, Veltaan result gives*': ™t < ato the result of Eq. (4.90) is compatible with this bound, and somewhat stronger; iii) one can repeat the analysis described above for tha case where there are mora than three feraioa generations; one obtains a aore coaplicatad inequality17,7" but in any case faraion Basses ao greater than about 250 GeV are implied; iv) finally» wa stress that th* above considerations hold in the case where fanion aassas arise entirely froa the Itf(2) afU(l) breaking: BO bounds can ba obtained in this way for fermions which possess a $0(2) Wa consider now the renoraaliaatioa group aquations for the Riggs self coupling, \t Eq. (4.93), Restricting «gain to the case where only the t-quark Tukawa coupling is retained, one finds tha following aq«*tionM': dt ° «.95) In the above equation we have used the couplings g and g* of tha weak S0(2) and 11(1), respect ively. (See Eqs. (4.15) and (4.M).) I*. (4.95) is easy to analyse in the case where h, g and g* are constant. We bava repeated in rig. 22 (see following page) B as a function of A for the two interesting cuss: (a) h large, i.e. t and i (b) h eaell, i.e. the cue where the inequality (4.9o) ia not obeyed. with respect to the result quoted by Veltaan, we have inserted, an appropriate factor of 3 (for the color Multiplicity) and we have assuaad a 5X error on the experimental deter mination of the ratio of the neutral to charged current Ferai constants. Pig. 22 The function g, Eq. (4.95). Th* arrows indicate» for the two cases* tbc direction in which X aorta when t increases. In caae (a), if X(0) is below tha positive root of B, X(t) goes toward zero, and «ay becoM negative. In case (b), 6 > 0 and X increases vith q*. We aay obtain in thia case only an upper bound97' on X(0), siaiilarly to the case of the Yukawa coupling. Caae (a) is «ore interesting Not only, if X(0) is too large, X(t) grows too large before H™ is reached (thus giving an uppar bound on MO)), but, in addition, if X(0) is ••aller than the positive root of &, X(t) can become negative. As discussed previously, we Bust require that this does not happen for t < L.. Thia condition givea ua a lowar bound on A(0), which depends explicitly upon h, g and g* (i.e. • , au and •_). Since the Higgs boson aes* ia related to X(0) according to; we amy obtain, in thia way, a coabined bound on »t, a_ and tha known vector boson nurses. In the «ore complicated case where h varias with t according to £q. (4.94) and, eiai- larly, g and g* vary with t as diacuased in Section 4,1, the lower bound still exista but it has to be obtained numerically. In Fig. 23 Che corresponding bound in the a - BL. plane is reptatad (continuous curve, (a)), the allowed region being on the left of curve (a). Pig. 23 founds oa the «ass of the Higgs boson, m., as e («action of the t-quark mass, a . We have uaed «in^ - 0.2. Ths dashed line and the full lines represent tha upper and lover bounds, respectively. The dotted line ia the prediction of tha theory where eyaaetry breaking arises ttorn radiative corrections only. The latter curve ends ia correspondence to the bound onat givea by Zq> (4.M). - 59 - Hot* that, according, to our previous considerations, the bound exists only for sufficiently large values of •, Eq. (4.96), i.«. » £ 80 Gt?. For completeness w* have reported, in Fig, 23, alto the upper bound on au obtained froai the condition that A(0) remains ««all up t9(*L (dashed curve) a* well as the lower bound on BL, diacusaad earlier by Iiinde72' and by Weinberg71' (continuoua curve (b), ace alao Kcf> 57)). These three curvea delimit the region of the m - au plane which ia allowed in the SI) (5) scheme (for the caae of three fcradon genarationa) and tbey swamarise all the discussion of this section. If these bounds were Co be exceeded, we would kmow that something has to happen in between vf/z and !£_,, contrary to what is asinmifl in the standard 511(5). It is interesting to noce that for large valuta of mt, the upper and lower bounds on m„ tend to coincide. Thia is easy to understand. As h(0) increases, the right-hand side of Eq. (4.95) varie* more rapidly with A, and the equation becomes more and more unstable. As m consequence, the range of initial values X(0) such that A{c) neither blows up nor be comes negative (for t < L.) narrow* down. Eventually, wham m» tends to its upper bound, au is tastntiilly determined to be7**: y)jH*iao C*Vm Finally» we note that the allowed region includes the values of au (as a function of m ) one would predict in the case where the Stf(2) •V(L) breaking arises from radiative corrections only*" (dotted cum in fig. 23). Thia situation has been recently suggested'" •» * plaus ible explanation of tb* large value of L_. 5. COWCLUSICHS A» OUTLOOK The present theory of elementary perciclei is based on very simple ideas: spin % ferm- iona interacting with gauge fields and with Higgs bosons. The gauge group ia SU(2) • u(l) •> 5U'^color* *°* *' d**cr*k#f °°tn **• *tromg and the electro-weak interactions. This is the picture discussed in these lacturas. Sections 1 to 3, paying special attention to the weak interactions of the b end t quarks. The verification of the theory ia not complete: quark confinement in QCD ia still an open problem, tf, 2 and the Higgs bosons have not been seen. Yet, the evidence in favor is strong» end the acquisition of these substantial results may be only a matter of time aad energy. If this is so, M we bellave, then: whet is MxtT It ia herd to believe that the $U<2) • tT(l) • CU(3) theory U the final theory of elementary partie Its, even in its simplest version, with three leptoa and quark weak «oublats and one Higgs boson. It simply leaves too many open questions: -why ere there quaiks (I.e. color triplets) and leptors (i.e. color singlets); -why is charge quantisedi -why is the mass apectrum (i.e. ligga couplings) ao complicated; -why are there three independent sets of gauge transformations; -why ao memy quark and lapton generation* (at ltaat three). Soma of these question cam be aeewsrad by grand unified thaoriaa, the most simple and de- - 60 - velopcd ona being the SU(5) scheme of Georgi and Claahow. Tba SU(5) acbaae haa achieved a number of important auecaaaaa: va have reviewed then in Section 4. It alao laada to one striking prediction, and it raiaaa a few new problem. The prediction (abarad alao by other grandmnifiad schemes, such as the rati-Salan one) is that proton is unstable, with a lifetine within a couple of order of magnitudes fron tha present experimental liait. It is hard to overemphasize tha importance of testing this pre diction. Its verification would be the first unequivocal sign of grand-unification. The problem* of SU(5) are well known. One is tba hierarchy problem, i.e. the vary large ratio of the grand unified mass, H^, to the V or Z mass. The other is the family problem, which i* tha same aa before: why are there at leaat three identical groups (families) of fermions. A further very peculiar aspect (eae Section 4) ia that nothing ia expected to happen between energies of order 102 CeV up to about 10l*CeV, where unification sets in. Such a large "desert" may look auspicious» although for no rational reasons*). There has beam some progress on tha hierarchy problem: Weinberg hea shown that a ratio Hgg/Bg aa large aa needed can be obtained under the simple requirement that tha curvature of the Biggs potential at • - 0 vamishee"'7**. why this should be so» however, ia still mys terious. Tha family problem haa lad many to believe that even SU(5) ia too email a theory, and it has to be iahsddad into a larger atructure. Orthogonal groupa ara being explored as pro mising candidates, notably 0(U) or 0(22)7**. Such large groupa imply tha existence of new» independent, gauge interactions, which could possibly be saturated (i.e. strong) in the TeV region* To *wca mew interactions there should be associated new types of very heavy quarks, aa in the technicolor scheme advanced by lusshind"*. Technicelorad quarks would fill the SU(S) desert, which ia «ertaialy good far our futur*. Bowever, such extensions are some what arbitrary mad, in on* raapect, deceiving: to explain three SU(5) families, we need to postulat* a much larger somber of alemaatary fermions, not to speak of the Higgs fields, which have now to be eowatnd with 5-figur* numbers. Actually, oaa may argue that grand unified gauge théorisa cannot explain tha number of farmion generation*. In fact, while gauge fields are exactly specified by the group, no thing, in * Tang-Mi 11a tb*ory, prafers a given typa or « given number of matter multiplets. If so—on* succeeds in claaaifying all fermions into one aingla multiplet, we can ask, with equal fore* aa before: why only em* family! What we really need, to solve the family pro blem, la a symmetry relating gang* fiald* to spin \ (and 0) fields: wa need auparaymnatry or, for a real unification, separgravity. This lead* us to consider ta* 0(1) supergravity, uniquely determined aa the largest acharna containing only on* graviton. 0(1) supergravity give* a solutUm to th* family problem. ** wall aa to the nature of the gauge group (the con tent of all particle types ia exactly apecifUd) but on* which la too email: it doe* not contain th* muoa and it doe* not contain th* w*. The* also this way out eaeme to be closed. This is th* situation a* va a** it, no», and w* could aa wall cloe* our discussion here. Before doing so, however, w* would lik* to Illustrât* some wild speculation* which might *) to quota seme precedents: it v*« atremgly argued that ADOK «ns amine to *— nothing but th* electromagnetic mee*rt following ta* p-tailsi alao, a «esart ocean van strongly believed t* jel* Spain to Japan. - 61 - indicate a direction for further progress. Uc nay auppoat that tha fitlds va usa now (quark*, leptona» th« proton, the gluons» «te) ara all composite fields, coottruetad out of «ore fundamental objecta (preons). Sup pose, further that the natural snarly acale of tha bound state» is the Plane* nasi, JL aiio1* CeV, and that for aoae raason (see below) a number of bound states have, in aoaa approximation, aero nass. These bound states (the fields we use now) would have very con- plicated, effective, interactions. However» if tha affective coupling constants are deter mined by A-, at the present, very low, energy only a vary specific claaa of interactions would survive; thM« determined by diaensionleaa couplings» i.e. the renoraalizable inter actions. In fact, * son reuormalisable interaction would have e coupling of order Ap^Cm > 0) end it would be entirely negligible (except for gravitation, which is always attractive and Mediated by a strictly «assisse particle). The effective interactions be tween aasBlesB bound states would be precisely what we ..." (gauge and Yukawa interactions) and one could justify empirical relations like: o - [in (jyiCeV)]*1 Indeed, it has been shown in Kef. 71), that the conaiatençy condition that the present theory is near an infrared stable point7*) gives a reasonable picture of the coupling constants of su 5U(2) 9 VlH 4 (3)oolor. Of course, U this picture, one should also be able to compute the ratio \J*f» but this has, at arasent, defied all attempts (similarly and perhaps relat ed to tha hierarchy problem"* im grand —Ifled theories). A possible realisation of the above ideaa can be baaed en the 0(1) swpergravity, by identifying: preoae with tha particles of the basic 0(«) awpermultiplet.**) It has bean shown by Julia and Creemer7*) that tha 0(8) supergrevity baa a hidden local syametry SOW* The local eynmetry ia realised in s non linear fashion, and chare ara not, in tha original thaory, the corresponding gauge fields. •y analogy with 2-dt—a lunal CfB model"), one nay suppose that tha SU(B) gauge fields arise dynamically, aa bound states of preens. Tha SU(t) gauge fields would be naturally maealass, aa far aa t(J(l) Is exact, and gu(t) is certainly large enough so that we may class ify bound «tares in terms of fU(5) familial. Most important, the lupersyametry of the whole schema implies the existence of similarly masslass bound states with spin 0» 1/2 and possibly 3/2 or higher. Tha composition of tha bound states associsted by suaorayamvtry to tha SP(6*> gauge fields should be, in fact, completely determinad. unfortunately it has not bean pos sible, till now, to pat these ideaa into « concrete schema. Tha first attempts failed to obtain a reasonable spectrum. The simplest version la plagued by tha presence of higher spin, strictly messleas, colored particles. The general scheme is attractive, however» and we think it deserves further study. *> Aa far aa a* know, this observation {Voue to G. rarlii. what follows la am Illustration of am inasueliehed work moan im collaboration with **) J. nils and 1. Zemino. toe also Kef. 7»). r - 62 - Ackaowladganants Va would lik« to thank all tb* participants at tba CargCc* School for tha «any undtr- th«-tr«« diacnaaiona which halpad shaping thaaa lactams, in tba preparation of Sactioa 3, em* of us (L.M.) bss greatly profita* from comrsatlou with M. Cabibbo and G. Altaralli. 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