Some Topics in Grand Unified Models and the Cosmological Baryon Asymmetry Citation Reiss, David Benjamin (1981) Some Topics in Grand Unified Models and the Cosmological Baryon Asymmetry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7629-m351. https://resolver.caltech.edu/CaltechTHESIS:03132018-141623091 Abstract In part I of this thesis some of the parameters relevant to the production of a cosmological baryon number asymmetry are considered in the context of grand unified models. General expressions for the average baryon number generated in the free decays of bosons are derived . The CP violation necessary for the generation of a baryon excess is discussed for a variety of SU (5) models. The kinematics of baryon number production in an illustrative SO (10) model is discussed in detail. In part II a viable SO (10) model is constructed which reproduces the phenomenological fermion mass and mixing angle values. A detailed discussion of the beta function for this model is presented. This analysis includes the effects of scalars. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Frautschi, Steven C. Thesis Committee: Unknown, Unknown Defense Date: 19 May 1981 Record Number: CaltechTHESIS:03132018-141623091 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03132018-141623091 DOI: 10.7907/7629-m351 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10774 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 20 Mar 2018 15:54 Last Modified: 26 Feb 2025 19:09 Thesis Files Preview PDF - Final Version See Usage Policy. 53MB Repository Staff Only: item control page

SOME TOPICS IN GRAND UNIF1ED MODELS AND THE COSMOLOGICAL BARYON ASYMMETRY Thesis by David Benjamin Reiss In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1.981 (Submitt~Cl May 19, 1981) -ii- Acknowledgements My sincere thanks go to _my advisor Pierre Ramond for his support and encouragement over these last few years; I have been lucky to have worked with and learned from him. My thanks also go to my other collaborators, J. A. Harvey, E.W. Kolb and especially S. Wolfram. The love of many people has made it possible and worthwhile for me to do the work required to produce this thesis. Without their support and warmth it would have been a very empty road indeed . In a pseudorandom order they are: Helen, Andy, Peter~ Joshua, Carrol, another Andy, David, Emily, Shirley and the rest of the Weixel clan, Blatts, Margaret, Liz, Roger, Charlene, Tom, Barbara, Tony, Jonathan, Jesse, David Edward, Edward David, Cynthia, Duno, Mephisto, Teddy, the man in white spats ... Certainly there are more and, to anyone I have left out, my only excuse is that it is 2:30 in the morning. There are two people, however, that I have not forgotten; this thesis is dedicated to them because I love them: Gertrude and Julius --who else? -iii- Abstract In part I of this thesis some of the parameters relevant to the production of a cosmological baryon number asymmetry are considered in the context of grand unified models . General expressions for the average baryon number generated in the free decays of bosons are deriv~d . The CP violation necessary for the generation of a baryon excess is discussed for a variety of SU(5) models. The kinematics of baryon number production in an illustrative SO( 10) model is discussed in detail. In part II a viable SO( 10) model is constructed which reproduces the phenomenological fermion mass and mixing angle values. A detailed discussion of the beta function for this model is presented. This analysis includes the effects of scalars . -iv- Contents 1) Introduction Part. I 2) Cosmological Baryon Number Generation in Grand Unified Models 3) B and B-L Violation in Models With SU(5) Singlet Fermions 4) Baryon Number Generation in Free Decays 5) SU(5) Models 6) CP Violation in the Minimal SU(5) Model 7) CP Violation in Alternative SU(5) Models 8) SO( 10) Models 9) Analysis of an illustrative 80(10) Model Part. n 10) Mass Matrices in SU(5) 11) A Viable 80(10) :Model 12) The Beta Function in this 80(10) Model Appendices A) Notation for Fermion Fields B) The CP Operation in Grand Unified Models C) Momentum Space Weights: Double-cut Diagrams -v- D) lmO E) Symmetries of the Illustrative SO(lO) !v!oO.el References -1- 1) Introduction In the beginning, the idea of grand unification ( 1] was introduced as an economizing gesture to reduce the number of possible Yang-Mills [2 ] couplings . An immediate consequence of this was (under some circumstances) to reduce the plethora of parameters that appear in the Yukawa couplings of a model. The Higgs self-couplings, however. do not fare so well under this treatment for two reasons: the number of unconstrained parameters generally increases and the vial containing the noxious problem of hierarchies [3,8] is uncorked. There is also another problem endemic to grand unification that is shared by both fermions and scalars (especially when one considers models larger than the minimal SU(5) scheme). This is the proliferation of degrees of freedom. One is forced to consider the possibility of some (presently) unobserved fermiens and many unobserved scalars . In the end, the program of grand unification, although its original aim was to father simplicity, has given rise to a rather large amount of complexity. Nonetheless, in spite of or, perhaps, because of this complexity, grand unified models possess a number of interesting features . One of the first things grand unification forced one to consider was the possibility of the decay of the proton [ 4,5]. Vector induced proton decay did not exist in a theory based on the product of a flavor group with SU(3) of color: G1,®SU(3}c. However, scalar induced proton decay certainly could have been put into a G1 ,®S U(3}c theory ad hoc through the inclusion of scalar representations with appropriate quantum numbers. The context of grand unification is a natural one in which to CODBider vector-induced proton decay. -2- A similar situation exists for other esoteric processes. Those arising from the presence of the extra fermions (either charged or neutral) could certainly have been considered in a G1 ,®SU(3)c theory and recently such ideas have been examined, prompted by their appearance in grand unified models [6]. Grand unification beyond SU(5) requires one to consider massive neutral fermions and the associated neutrino oscillation and lepton number violation phenomena [7]. The presence of a large number of degrees of freedom in larger grand unified models necessitates the examination of their effect on the renorrnalization of the parameters in the model (notably the gauge couplings and, hence, the Weinberg angle) [8,9, 10]. In grand unified models the global symmetry structure can generally be very rich, allowing one to experiment with a large number of natural- ness conditions in an attempt to reproduce phenomenological mass an~ mi.xi.ng angle (and CP violation parameter) values. The possible presence of of zeroth order mass relations and of the soft breaking of symmetries each allows one to consider exactly calculable quantities, permitting, perhaps, the construction of a model in which the electron family's parameters are strictly perturbative . To explore the ideas mentioned above, it is by no means necessary to introduce grand unification, but it does act as a natural matrix in which to consider them together. Just as one was able to consider the SU(2)LeU(l)y model as "a framework for organizing huge quantities of u:perhnenlal data," [ 11] so too grand unified models can be considered as a framework for considering a large number of theoretical possibilities. -3- One may wonder about the possible significance of grand unifie d models on a level somewhat deeper than the pragmatic and the organizational. Unification without gravity is only partial unification. So too is unification without a criterion for choosing which of many possible models is the most correct one. Grand unified models may be criticized on both of these accounts. The following operational philosophy is certainly a reasonable one to adopt. Grand unified models are worth exploring both from the point of view of being a laboratory for theoretical ideas and from the hope that one such model will turn out to be a limit of a fundamental theory yet to be discovered . A few such theoretical ideas are considered in this thesis . In the first part we discuss some aspects of the cosmological baryon asymmetry in the context of grand unified models. There are two perspectives that one may take in considering baryon number violating processes in the very early universe. First is the grand unified modeler's perspective . From this point of view one notes that there are only two laboratories in which the effects of the "intermediate vector baseballs• [53]" are manifest. One is caverns in salt mines where, it is hoped by some, the decay of the pro- ton may be observed. The other is at the superhigh temperatures that were possibly present in the very early universe : T~ 10 15 Ge V. At such temperatures the rates for baryon number violating interactions are competitive with those that conserve baryon number. The second perspective is that of the cosmologist who poses the so-called initial condition question: which observational cosmological facts must be taken as initial conditiaos (isotropy? homogeneity? thermal equilibrium? ... ) and 4l'!1umk JDU Sid. -4- which ones may o:r must be derived? In this context grand unified models act as a self-consistent setting in which to discuss how the cosmological baryon number asymme Lry does not need to be imposed as an initial condition on the evolution of the univerie . In the body of part I a number of topics relevant to t.he calculation of the baryon asyrnmetry are discussed . To set the stage for this we review, in chapter 2, the method of calculating the magnitude of the cosmological asymmetry in an arbitrary grand unified model•. In the second part of this thesis we discuss the construction of a grand unified model based on the simple Lie group SO( 10) which acts as a natural generalization of SU(5) . In this model we are able to reproduce the phenomenological fermion mass and mixing angle values. It is a general feature of models based on gauge groups larger than SU(5) that there may be more than one level of symmetry breaking; SO(lO) has this feature. The various predicitons of a grand unified model depend upon the complexity of the symmetry breaking . Notable in this regard is the effect of multiple symmetry breaking scales .on the running of the gauge couplings; hence we present an analysis of the · beta function in this model. In summary then, this thesis asks again the age old question, "What is the eat's laSt name?" This question cannot be answered as is pointed out in the poem which follows, nonetheless it is great fun to try. Eliot has put it (54]: -The text of chapter 2 i8 -.entially that of a paper by J. A. Harvey, E. W. Kclb, D. B. Re-. .adS. 1faifrmD r-eeMilly submitted to PhJSical Review Letters. -5- The naming of cats is a difficult matter, It isn't just one of your holiday games; You may think at first that I'm as mad as a hatter When I tell you, a cat must have THREE DIFFERENT NAllES . First of all, there's the name that the family use daily, Such as Peter, Agustus, Alonzo or James, Such as Victor or Jonathan, George or Bill BaileyAll of them sensible everyday names . There are fancier names if you think they sound sweeter, Some for the gentlemen, some for the dames: Such as Plato, Ademetus, Electra, DemeterBut all of them sensible everyday names. But I tell you, a cat needs a name that's particular, A name that's peculiar, and more dignified, Else how can he keep up his tail perpendicular, Or spread out his whiskers, or cherish his pride? Of names of this kind, I can give you a quorum, Such as Munkustrap, Quaxo, or Coricopat, Such as Bombalurina, or else JellyorumNames that never belong to more than one cat. But above and beyond there's still one name left over, And that is the name that you never will guess; The name that no human research can discover But THE CAT HliiSElF KNOWS. and will never confess . When you notice a cat in profound meditation, The reason, I tell you, is always the same: His mind is engaged in a rapt contemplation Of the thought, of the thought, of the thought of his name : His ineffable effable Effaninefiable Deep and inscrutable singular Name. -6- PART I -7- 2) Cosmological Baryon Generation in Grand Unified Models Cosmology is potentially an important source of information on the be.ryon number (B) violating interactions expected in most grand unified 1auge models. Any net B imposed as an initial condition on the universe lilhould have been rapidly destroyed by any B-violating interactions. To account for the observed baryon number density to photon number density ratio, nBin., ~ 10-9 , a net ba.ryon number must subsequently have been generated. This requires, in addition to B violation, the violation of C and CP (and hence T) invariance, along with departures from thermal equilibrium [ 12, 13]. This chapter oullinE:s the complete calculation of nBI n., generation in specific grand unified models in the context of the standard hot big bang model of the early universe. The method we present allows for the exact treatment of an arbitrary number of superheavy bosons as well as the presence of nonthermalizing modes [14]. We summarize results for several realistic SU(5) models. Many details and extensions are discussed in ref. [ 15]. We denote heavy bosons generically by x and light ferrnions by a, b, .. .. The number density ~ of a particle i and that of its antiparticle n 1 are parametrized by i+=(~+n;)ln., and i_=(~ -1l.r)ln.,. The time development of these quantities is described by a set of coupled Boltzma~11 transport equations. For the heavy bosons these are [ 13, 15] X+ =- 2;<r(x ... ab )> (x+ -x!q) (2.1a) a,b x- =-I;<r(x~ab )> (x-- (a_+ b_) x!q) (2.1 b) a,b where dots denote time derivatives and the expansion of the universe is accounted for through division by n., in the definitions of i± .. The first -8- terms on the right side of eqns (2 .1a) and (2.1 b) correspond to free decays of x and x with partial rates <r(x-.ab )> averaged over the decaying x. energy spectrum. The second terms account for back reactions in which the x. decay products interact to produce X· The equilibrium number density x!q is obtained by integrating the exp[ -Ex/ T] equilibrium Maxwell-Boltzmann phase space density. In equilibrium, X+=x~q and X+=O; the expansion of the universe produces deviations from equilibrium at temperatures T - mx . The densities of fermion species develop according to i- = L: <r(x-.ab )> {N1 )ab Hx+- x!q) R(x--ab) + 2x-- (a_+ b_) x!q~ a.~>.x + L: a,b,c ,ct.x ~ [(N1 )ab -(N1 )cctJ ~a_+b_-c_-d_J <lv la'x(ab _.cd)> , . (2 .2) where (N1 )a denotes the number of .particles of type f in the state ab. R(x-.ab) denotes the difference in branching ratios between the CP conju- gate decays x~a b and x_.a b divided by the full rate for X decay; it vanishes if CP is conserved . The first part of the first term on the right side of eqn (2 .2) therefore represents the production of an asymmetry in fermion number densities as a result of CP-violating decays of a symmetrical x. x mixture. The second part causes asymmetries ~ x_ ~ between x and xto be transferred to the ferrnions when the x ( x) decays. The third part gives a correction to the rate for inverse decays resulting from the deviation of the fermion number ciensities from their equilibrium value. The second term in eqn (2.2) represents the production and destruction of fermions by two-to-two scattering processes. rl x is the cross-section for this scattering Inediated by x exchange, but with the term correspondhTig to a real intermediate x removed (since this is already account~d for by x -9- decay and inverse decay processes). The number of independent particle densities to be treated in eqns (2.1) and (2.2) may be reduced by using unbroken symmetries (gauge and global). For non-Abelian groups, any asymmetries are shared symmetrically among members of each irreducible representation. If only a subset of the interac;tions that may potentially contribute to eqn (2. 2) is included, there may be additional symmetries leading to further conserved combinations of fermion number densities (e.g., TI conservation in the absence of Riggs-fermion couplings for the models discussed below). Let Ji_ (i=1. · · · .N1 ) be the independent fermion asymmetries and x~ (cx=1, . . . , Ny) the independent supermassive boson asymmetries. convenient to form a set It is Q which consists of independent quantum number densities B .L. etc .. . related to f=~fi_, x~~ by a unitary transformation, Q=H F, F=H- 1 Q. The thermalization of a quantum number Q through reactions of a particular boson x is given from eqn (2.2) by "'= ~ x:q Md Q;, where X Jl.d=l: llQ;,(x ... !Jc f')<r(x_.!~c /l)>(H~c-/+Hij ) and llQ(x_./Jc !') represents the 1 lc.l change in the value of Q, through the reaction x_./t f'. Boltzmann's H theorem requires that the eigenvalues of J.IX are all real and nonpositive. Any zero eigenvalues reveal additional s:ynunetries; the corresponding eigenvector of number densities is then conserved in x reactions (e.g., TI in vector boson exchanges in SU(5)). · We consider two grand unified models based on SU(5). In each case a family of fermions transforms as a reducible representation ( 5 Eel O)i, labeled by the family index i. The following Higgs representations are taken to couple to fermions: in model I (minimal SU(5)), ~ single 5 of -10- Higgs, H 5 ; in model IT, He and an additional 5 of Higgs, H 5•. The Yukawa couplings in these models have schematic the form It is shown in chapter 4 that a CP-violating nonzero R(x-+ab) enters through an imaginary part of the product of the couplings in diagrams in which one boson is exchanged between the ab produced in the x decay. The sum over a and b in eqn (2.2) runs over all types and families of fermiens; thus, for fixed fermion types. R(x-+ab) is proportional to a family space trace of Yukawa coupling matrices. In model I the first diagram exhibiting CP violation involves only Higgs bosons and is of eighth order in the Yukawa couplings [ 16,1 7,15]. This is discussed in chapter 6. It is proportional to the imaginary part of the family space trace, 17'[ UutUD2 [JiD 2 ], suggesting the rough estimate R ..... as (mFI mw) 6 e/ (128~a 3 ) = 4x10-9 (m.p/mr) 6 e, with !el~l. where mF is the mass of the heaviest fermion. (Stability of lhe effective potential requires that m.F~ v'Smw [ 18] and hence R~lo-e £, making the production of an adequate baryon asymmetry implausible in this model.) In model II (discussed in chapter ?), both H'5 and H 5 • have only the single B-violating component •, (3, 1, -l /3); since 5 is a complex representation one may form complex linear combinations so that the (3, 1, -1 /3) in both Sand 5' is separately a mass eigenstate. This suffices tc show that no CP violation may occur for gauge boson decay with Higgs scalar exchange (or vice versa). CP violation may occur at O(cx(mp/ T.'1. 1r) 2 ) through 5 decay with 5' exchange (and vice versa) [19]. • In this notation the first entry mthe SU{3) multiplicity. the second is the SU(2)L multiplicity and the last the value of the weak hypercharge Y normalized so that the charge operator is given by QcTst.-Y. -11- SU(3)®SU(2)L®U(l)y symmetry allows the 15 independent fermion fields in a family of an SU(5) model to be reduced to the set UL, (UC)L, (Dc)L, ELand (Ec)L (the subscript L denotes the left-handed hel- icity state and c denotes charge conjugation). The model contains a (3, 2, 5/6) of B-violating vector bosons X (with number densities parametrized by X_ and X+) · We consider the case where there are 71.s (= 1 or 2) scalars , S I> S 2 , . .. , Sn5 , transforming as (3, 1, -1 /3) (with number densities parametrized by S.;,- and Si+) . These models possess a locally conserved weak hypercharge whose initial value we assume to be zero. The models exhibit two iurther zero eigenmodes. The first is B -L which hds zer o eigenvalue (is conserved) in all boson interactions . A second zero eigenmode, TI=-3(Dc)L--2EL-, is present if scalar-fermion interactions are removed [ 14]. TI (termed rrfiveness") corresponds to the net number density of the fermion species appearing in the 5 representation. A density IIo generated through Higgs decays would be distributed as B = -TI / 10, 0 v_= -n0 I 5 through TI-conserving X interactions. no may be destroyed through exchanges of light Higgs bosons . A convenient choice of independent combinations · of fermion densities is no/ n 7 = B = 2Dr_-(UC)z_-(Dc)r_, TI and v_=EL- · For model I, according to the estimate for R(S ... ab) given above, an adequate baryon number asymmetry will be generated only if very heavy fermions exist (mF-miY )•. Fig. 2. 1a shows the baryon asymmetry (taking mx=5x10 14 GeV and a:=l/40) as a function of m 5 1mx for mFim r,. =l and mp/ m.r=3 obtained by numerically integrating the Boltzmann transport equations (2.1) and (2.2). When ms/mx>>l, X exchanges thermalize the B • Similar conclusions have recently been reached in ref. [20]. -12- 1011 t6'~o3L...---,o.L-2~--,.o-, . . --,.L.o-=-o--,~6-:-' _..j ms;mx Fig . 2.la: Baryon number density as a function of the Higgs boson ( S) mass generated in the minimal SU(5) model in which the heaviest fermion has mass mF. Results are for a= 1/40. mx=5x10 14GeV. The CP violation parameter F: is unknown but less than 1. -13- produced in S decay to the value -rv 10; meanwhile, II is reduced by light Higgs interactions. The final B attained is determined by the reduction in Il that occurs before X exchanges cease to be important and B becomes fixed. For m.5 /mx<1 the X is not effective in destroying the baryon number built up through S decay. The enhancement in the final value of B around ms/mx=l is a result of the transition between these two regions. The dotted curve shows the final baryon number if all .X interactions are artificially set to zero. Fig. 2.1 b shows the temperature development of the quantum number asymmetries B. II and v _ for the case m;-/ mw =3, m.:,·/ mx= 1C with the solid (dashed) curves indicating the effect of including (excluding) the destruction of II and v_ by the interactions of the light Higgs doublet. For model ll the final baryon number density as a function of ms 11 mx is shown in fig . 2.2 for different choices of ms(mx. Note that, when m 1=m 2 • we have (assuming (P51 )totaL =(r52)totcat in the Born approximation) R(S 1-+a.b)=-R(S 2 ... ab) and hence noB is generated. For msi>mx the addi- tional decay mode S;. -.x +9' (where 9' is a light Higgs boson) decreases the effective CP violation, R(S;. -.ab ), in S;. decay. For m 52 >mx and m 51 >mx. the final B is negative and determined by vector thermalization of the positive II produced in S 2 decay. For m 52 >mx but m 51 < 0.1 mx, the final baryon number is positive and determined mainly by inverse decays into S 1 • The dominant term governing the time evolution of B for T';<:,ms 1 is iJ IX s~~ <f sl>( i4v_-l2B+7Il) with similar equations for i;_ and n_. Since ll>O, il>v_ and II>B. this term tends to drive B positive. In general there are three linear combinations of B, v_ and II which decrease as exponentials until cut off at temperatures below m.51 . The final value of B thus -14- 165--~-----.-----r----1 m5 ;mx = 10 Fig. 2.lb. Evolution of independent quantum number densities as a function of temperature in the minimal SU(5) model. B denotes the net baryon number, v_ the asymmetry between v and 11 densities and TI the total asymmetry between ferrn.ions in the 5 and 5 representations of SU(5); 1TieV = 1o24 eV. In these. graphs the parameter t has been scaled out. The dashed curves are results obtained by neglecting light Higgs boson exchange processes. -15- 8/ IE" I -165~2----------~--------~----------~--------~ 10 10 1 10° 10 1 162 ms 1/mx Fig . 2.2: Baryon number density for an SU(5) model with two baryon number violating Higgs bosons (S 1 , S 2 ) as a function of the S 1 mass for different choices of the 8 2 mass . The results are for a=l/40 and rnx=5x10 14 GeV. The CP violation parameter~ is unknown but less than 1. -16- depends sensitively on the initial values of TI, v_ and B. For this reason, it is inadequate to assume that B is produced and damped in successive independent stages as in simple models which treat only one quantu_rn number [13,21 ]. For both ms 2 <mx and ms 1<mx inverse decays into 8 1 are no longer able to change the sign of the negative B produced through S 2 decays and hence the final B produced is negative . The possibility of changes in the sign of B associated with detailed features of the boson spectrum indicates that no generic relation may be found between the definition of "matter" as given for the J(O -XO system and that determined from the cosmological baryon nun1ber asymmetry . -17- 3) B and B-L Violation in Models With SU(5) Singlet Fermions At temperatures at which baryon number production is thought to occur, SU(3)®SU(2)L®U( l )y symmetry will be unbroken. With this assumption we may analyze the possible baryon number violating vector and scalar bosons which may occur in a renormalizable theory. This analysis has been done for fermions with the quantum numbers of the 6ffi 10 representation of SU(5) [ 19]. For this case the baryon violating vec- tor bosons come in two varieties, X and X ' , with SU(3)®S U( 2)L® U(l)y transformation properties (3. 2, 5/6) and (3. 2, -1 /6) respectively. The possible baryon violating scalars are s- (3, 1, 1/ 3), S 1-(3, 1. 4/ 3) and S 2 -(3 . 3, 1/ 3) . Fermi statistics require that S 1 and S 2 couple to fermions antisymmetrically in family space in order to violate baryon number (hence, they cannot give a tree-level contribution to the proton decay rate). With the conventional assignments of baryon number (B) and lepton number (L), it is found that all of these baryon number violating bosons preserve B -L. With the SO( 10) model in mind we extend this analysis to include an S U( 5) singlet fermion NL-(1. 1. 0) . We assume here that the l'h has a Majorana mass; consequently it may not carry any quantum numbers . The fermion fields considered in our analysis are listed in table 3.1. We assume that this pattern of fermions is repeated for the heavier families . Lorentz invariance requires that renormalizable vector couplings have tile form i'Ja~b V.u while renormalizable scalar couplings have the form i'[a 2 ~b S with VJ.~. and S vector and scalar fields respectively. By taking the relevant products of fermion fields appearing in Table 3.1 we obtain the possible SU(3)®SU(2)L®U(l)y transformation properties of the vector and scalar fields along with their values of B and B-L . These values of B and -18- Field l=~] q=~] I I i i I S [! ( 3)~S C (2) L ® U ( 1) :r (1, 2 , 1/2) i i I I I I I (3, 2, -1 /6) ec (1, 1, -1) uc (3, 1, 2/3) d.c (3, 1, -1/3) N (1, 1, 0) Table 3.1 Fermion fields and their associated SU(3)®SU(2)L®U(l)y representation content. -19- B-L are listed in tables 3.2 and 3.3 respectively. Bosons which may violate B or B-L are indicated by a dash. We find no new B-violating vector or scalar bosons . However, the X ' vectors and S scalars are now capable of violating B-L due to their interactions with NL . The additional B-L violating vectors transform as (3, 1. -2/3) and (1, 1, 1) and are gauge fields for the SU(4) and SU(2)R subgroups of SO(lO) respectively. The additional B -L violating scalars transform as ( 1, 2, 1 /2) (the ordinary Higgs doublet of SU(2)L®U(l)y), (3, 2, -l/6) and (1, 1, 1). These scalars are found in the following SO( 10) representations which may couple to fermiens : (1,2,1/ 2) c 10, 120, 126 (3.1) (3,2,-1/ 6) c 126 ( 1, 1, 1) c 120' 126 If the effective symmetry is SU(4)®SU(2)L®SU (2)R or SU(4}®SU(2)L®U(l)R, then a Majorana mass for the NL is forbidden by the SU(2)R or U(1)R symmetry and the NL must be treated similarly to the other ferrnions. In particular, the gauged B-L symmetry present in S0(10) will be unbroken and we must assign a value B-L = 1 to the 1VL. If the effective symmetry is SU(4}®SU(2)L®SU(2)R. then, as will be discussed in chapter 8, the presence of an unbroken charge conjugation .yrnmetry forbids the production of a baryon asymmetry [22]. For the tables vkich follow, in (i, j, k), i denotes the SU(3) representation, j denotes the SU(2) representation and k denotes the U(1)y charge Y. Y is given here corresponding to a definition for the electric charge Q = T3 -Y , where T3 is the diagonal generator of SU(2) normalized to 7r[TJ]=1/2 in the 5 representation of SU(5) . -20- B-I H (8 , 3, 0) 0 (8, 1, 0) 0 (8, 1, -1) 0 0 (6, 2, -5/6) 2/3 2/3 (6, 2, 1 /6) 2/3 2/3 (3, 3, -2/3) 1/3 4/3 (3, 2, -1 /6) - - (3, 2, 5/6) - -2/3 (3, 1, -2/3) 1/3 - (3, 1, -5/3) 1/3 4/3 (3, 1, 1/3) 1/3 1/3 (1, 3, 0) 0 0 (1, 2, -3/2) 0 2 (1, 2, -1/2) 0 1 (1, 1, 0) 0 0 (1. 1, 1) 0 - 0 i 0 I I Table 3.2 Vectors that may couple to the ferrnions of table 3.1. Their SU(3)®SU(2)L®U(l)y representation content is given along with the asso- ciated values of Band B-L. Vectors that may have more than one value for these quantities are indicated by a dash. -21- B-L B (B, 2 . 1 /2) ! I 0 I 0 I (6, 3, -1/3) I I I 2/3 2/3 (6, 1, -1/3) 2/3 (6, 1' -4/3) 2/3 2/3 (6, 1, 2/3) 2/3 2/3 (3, 3, 1 /3) - -2/3 (3, 2, -7 /6) 1/3 4/3 (3, 2, -1 /6) 1/3 - - - (3, 1, 4/3) - -2/3 (3, 1, -2/3) 1/3 1/3 (1, 3, 1) 0 -2 (1, 2, 1/2) 0 - (1, 1, 1) 0 - ( 1. 1, -2) 0 2 (1, 1, 0) 0 0 I (3, 1, 1 /3) 2/3 I Table 3.3 Scalars that may couple to the ferm.ions of table 3.1. Their SU(3}®SU(2)L®V(l)y representation content is given along with the asso- ciated values of B and B -L . Scalars that may have more than one value for these quantities are indicated by a dash. -22- 4) Baryon Number Generation in Free Decays In this chapter we describe the calculation of the average baryon number produced in the free decays of an equal mixture of particles x and their antiparticles (CP conjugates) X· This asymmetry is parametrized by the quantity (4.1) where T'(x~ f) denotes the partial width for decay of x to the final state f, rx is the total x decny width and B1 is the baryon number of the state f (so that B1 =-B1 ). In treating the statistical mechanics of baryon number production it is convenient to choose a basis so that the x are mass eigenstates . For (4.1) to be nonzero, CP must be violated in the decays of x and X· As discussed below (and proved in general in the first reference of [ 19] and in [ 13]), this requires interference between the Born amplitude for the decay and a one-loop correction with an absorptive part. In addition, the couplings of the particles participating in the decay must be relatively complex. We consider first the simplest nontrivial case: two massive bosons, X and Y, coupled to four fermion species i 1 , i 2 , i 5 and i., through the vertices of fig. 4.1 and their CP conjugates•. In the Born approximation, and the corresponding CP-conjugate processes. We denote the coupling -These vertices may be represented schematically by the interaction Lagrangian L ,.._iP'i 1+ilXi3+i lYi3+i~Yi4 +h. c . where all Lorentz .t.ructure.luls been suppressed. -23- Fig . 4.1: Boson-fermion-fermion vertices. The jagged lines indicate generic bosons. -24- in, for example, the vertex in fig . 4.1a by <i 2 1Xli 1> so that the CP- conjugate coupling becomes <i 2 IX li 1> •=<i 1 lxt li 2 >. The quantity X here may be considered as a matrix of couplings in the space of possible fermion states i; . Note that the set of vertices in fig . 4.1 is invariant under the combined transformations x .. y and i 1 .. i 4 . This invariance will be used below to obtain results for Y (Y) decays from those for X (X) decays . The couplings <ii IXIi.t> do not include the Lorentz structure whi~h determines, for example, which helicity states of the fermions ii may contribute . Born approximations to the X and Y decay rates may be obtained directly from the vertices of fig . 4. 1. For example (4.2) Here /} 2 accounts for the kinematic structure of the process X -+i 2[ 1; it gives the complete result if all couplings are set to one. Expressions for Ix for the cases where X is a scalar and a vector are given in appendix C. From eqn (4.2) it is evident that r(x -+i2il)Born = r(.X -+i2il)Bo17l , and hence Rx vanishes in this approximation. To obtain a nonzero result for Rx , one must include corrections arising from interference of the one-loop contributions shown in fig. 4.2 with the Born amplitudes of fig . 4.1. Consider, for example, the interference of the diagrams of fig. 4.1a and fig . 4.2b. The resulting term in the squared amplitude is shown as fig. 4.3a. There the dotted line is a "unitarity cut"; each cut line represents a physical on-mass-shell particle. The amplitude for the diagram fig . 4.3a is then given by -25- Fig. 4.2: One particle exchange corrections to the diagrams of fig. 4.1. The jagged lines indicate generic bosons. -26- Fig. 4.3: Lowest order (non-Born) contributions to the decay rate of the generic boson X. The dashed line is a unitarily cut. -27- (4.3) while the complex conjugate diagram, fig. 4.3b, has the complex conjugate amplitude where the kinematic factor !}?54 accounts for integration over the final state phase space of i 2 and [ 1 and over the momenta of the internal i 4 and l 3 . Introducing notations for the quadratic and quartic combinations of the couplings of the Born terms and of fig . 4.3 -)' - ,-)' )i -= ;:.jk ::.jk I <~. IX I'Lj. > 12 -- <'L.t. IX lI 'Lj. ><'Lj. IXt I 7-,t. > I 012s4 =<is I rt li1><ilj.xt ji2><i2 I·Yli4><i4 !X Iis> (4.5) one may write the one-loop approximation to the X -.i 2i 1 decay rate which is obtained by adding the results (4.2), (4.3) and (4.4) as (4 .6) The kinematic factors, lx., of the Born approximation are always real. However, the kinematic factors lxy for loop diagrams may have an imaginary part whenever the internal fermion lines have sufficiently small masses that they may propagate on their mass shells in the intermediate state. In the one-loop diagrams of fig. 4.3, this occurs when the threshold conditions (4.?) -28- and (4 .8) are satisfied. With light intermediate ferrnions therefore, IXY always exhiResults for ImJii1Z in a variety of cases are given in bits an imaginary part. appendix B. We now consider the CP-conjugate decay X-+ i 2i 1 . To obtain the CPconjugate amplitude all couplings must be complex conjugated . The kinematic factors , however, are unaffected by the CP-conjugation (this is n1anifest in the fact that reversal of the direction of ferrnion lines in a closed loop does not affect the associated Dirac trace). Thus, to one-loop order, the complete result for r(.x -+[2i 1) becomes (4.9) The diagrams for the decays X-+ i.i[3 and X -+ [ 4i 3 are sho1vn as figs. 4.3c and 4.3d. The loop diagrams differ from those for the decays X ~ i;::t1 and X .... i 2i 1 only in that the unitary cut is taken through the i 3 and i 4 rather than the i 1 and i 2 lines. In analogy with eqns (4.8) and (4.9) we obtain · -;-) = 1xs.-~34 + 1XY 34120 (/3412) ....n r(x -+t•ts 1234 • + XY Tui234 • (4.10) - -+'Z..t'l.s = J31r3412Q fJ3412Q ) , r(x X ~34 + i n · 1234 + \ XY 1234 · (4.11 ) and 0 • ) Using the results of eqns (4.7) through (4.11) together with eqn (4.1) we can compute the average baryon number produced in the free decays of an eqt1 al number of X's and X's. The one-loop contribution to this -29- (4.12) The analogous result for the 34 final state is (4. 13) The kinematic factors lm[J}P 4 ] and lm[Jjf12 ] are obtained from diagrams involving two unitarily cuts (as in fig. 4 .4): one through the i 1 and i 2 lines and the other through the i 3 and i 4 lines. The resulting quantities are invariant under the combined interchanges i 1 .. i 3 and i 2 ~ i 4 and consequently are equal: Im[Jjp-4] = Im[Ilf12 ]. (4. 14) mediate fermions have zero mass, then the Jjp-4 are completely independent of their upper indices; corrections from small fermion masses are of order• (m1 /mx) 2 . ~rrectiom~ of order m 1 /mx vanish due to the helicity structure of the relevar. . t diaerams. -30- Fig. 4.4: The double cut diagram that represents the contribution of the generic boson X to the baryon asymmetry in its free decays. -31- Upon adding the contributions (4.12) and (4.13) we obtain the final result: 4 ]Irn [0 Rx = ...L1mr -B·1.1)] . 1234 ] [B·~·. -B·1.3 -(B· rx :.. J.l2.3 XY ~ 1.2 (4. 15) The conditions for the kinematic factor Im[Ijf34 ] to be nonvanishing were given in eqns (4.5) and (4.6). A further condition for Rx to be nonvanishing is that both X and Y interactions must violate baryon number . This js seen as follows . If X coupling~ were B-conserving, the two possible final states in .~Y dP.cay would have the same baryon number, so that r:J . -r:J . -V\1 ~2 =B· -B· "• "s ( 4.16) and Rx would vanish . Similarly, il Y couplings were B-conserving, (4.17) and Rx would again vanish. Thus, both the X and Y couplings must be Bviolating to obtain a nonvanishing Rx. Furthermore, even if X and Y are baryon violating, graphs which do not exhibit this quality do not contribute to Rx . Thus, although it is not necessary that. it. i 2 • i 3 and i 4 all be distinct (it is, of course, necessary that at least one be different from the others), they must be such that Bi._.-B;, 5 -(Bi 2 -Bi 4 ) is nonzero if Rx is to be so. This is as implied by the general theorem given in the first reference of [19] and in [13], that there is no contribution to Rx from graphs of lowest order in baryon number vi~latin.g and arbitrary order in baryon number conserving interactions. The asymmetry Ry produced in Y and Y decays may be obtah1ed from (4.15) by the transformation x~Y and is~i 4 , yielding -32- (4.18) and so (4.19) It follows that the average baryon number produced in the free decay of an equal number of X, X, Y and Y is Even if the Rx and Ry are nonvanishing on their own, for the total to be nonzero, the terms in the brace must not cancel. Tills requires that the particles X and Y be distinct either in mass or in the Lorentz structure of their couplings (e.g., one vector and one scalar) and that rx#ry. The brace typically vanishes ii X and Yare in the same irreducible representation of an unbroken synrmetry group. If more than the minimal set of four fermion species is present, the result (4 .20) must be summed over all possible contributing ~ii ~ . It must also be summed over all possible (X, Y) pairs. Whenever the fermions have equal masses on the scale of mx, the corresponding kinematic factors may be factored out of the summation as follows from the comment immediately following eqn (4 .14). Eqn (4.20) is also valid, with slight modification, when the intermediate particles in fig. 4.4 are bosons rather than fermions, as illustrated in fig . 4.5. If the intermediate bosons Z 1 and Z2 themselves have B-violatLTJ.g decays, their weight B1 in eqn (4.1) is the average baryon number -33- Fig. 4.5: A digram like that shown in fig. 4.4 but involving a three-boson vertex. -34- produced in their decays. It gives the contribution to Rx . (4.21) Here !lz251z 2 depends on the masses of i 1 • i 2 and i 3 as well as on the masses of X, Zt and Z2. Since Zt or Z2 may be baryon number violating, Bz 1 and Bz 2 are defined to be the mean baryon numbers produced in the free decays of Z 1 and Z2 , respectively. We can now make the following argument: if X is baryon number conserving then we have Bi. 2 -Bi. 1 = Bz 2 - B21 a.rJd this contribution to Rx vanishes. Further, if both Z 1 and Z 2 are baryon number conserving, then Bz2 = B;. 1 - B.;. 3 and Bz 1 =Bi2 - Bi 3 and the diagram again vanishes. Because it corresponds to a double cut diagram, the expression for Im(Iiz~z 2 ) has the threshold conditions (4.22) The expression for 0 is (analogous to eqn (4.5)) (4.23) The individual baryon asymmetry parameters Rx for X decays enter the complete Boltzmann transport equations discussed in chapter 2 and in [ 13]. These parameters determine the final baryon asymmetry by themselves onJy if back reactions (inverse decays) and 2-+2 scatterings are ignored [ 13]. The total contribution to the baryon asymmetry from decays of two species of bosons, X and Y, thus is not generally a simple sum of their corresponding parameters Rx and Ry: if X and Y have different masses, the importance of back reactions and 2 ... 2 scatterings may be different in the two cases. -35- The discussion above concerns the one-loop contributions to baryon asymmetry. In the generic case , an asymmetry occurs at this order if it is to occur at any order. However, in some simple models (such as the minimal SU(5) model considered in chapter 6) the one-loop contribution vanishes, but ).here are higher loop contributions which are finite . In such cases the detailed analysis given above must be suitably generalized by summing over all possible unitarily cuts through the multiloop diagram . We now discuss the value of the CP-violating coupling parameter Im[O] defined in eqn (4. 12) (a general discussion of its structure is presented in appendix D). We assume here that the ii are all fermions V{ith masses much smaller than mx and my . In a grand unified model based upon a gauge group G a family of fermions will transform either as a reducible or an irreducible representation. These models are conveniently cast in terms of left-handed fermion fields . The two simplest examples to keep in mind are the SU(5) model where a family of fermions transforms as the reducible representation SEB10, and an 80(10) model where a family can transform irreducibly as a 16. In writing down eqns (4.15) and (4.18) we have assumed that Ll-Ie fermion mass eigenstates are states of definite baryon number . This is guaranteed by the unbroken SU(3) symmetry if no exotic assignments of baryon number are made to the weak ei.genstates (i.e ., all SU(3) singlet fermions have B=O and all SU(3) triplet ferrnions have B=O); no Majorana mass terms for quarks may appear in the Lagrangian. We further assume (though this does not affect our discussion very much) that all ferrnions are SU(3) triplets or singlets. Thus , fermions may mix within and between families so long as the mixings respect the quant urn numbers of -36- the unbroken local and globaJ symmetries of the model. In the boson sector of the bodel there may be mbdngs (when allowed by the other quantum nutnbers of the model) among the baryon number conservi:1g bosons and mixings among the baryon number violating bosons, but no 1nixlng may occur beJ:..ween these twc classes of bosons . As discussed in appendix B, the coupling of gauge vector bosons to massless fermions may always be taken as reaL Hence if both X and Y are gauge vector bosons, the CP violation parameter Im[O] will always vanish in this case: contributions from processes which only involve vectors come from the fermion mass matrix. We now consider the case in which X is a gauge vector boson V and Y is a Higgs sealar boson S, as illustrated in fig. 4.6. (Interchange of the identifications of X and Y is irrelevant for this discussion since this merely complex conjugates 0; however, this interchange does effect the Born rate.) The diagonal nature of the gauge couplings requires that the fermions i 1 and i 2 lie in the same irreducible representa.tion f 1 of the gauge group (and similarly i 3 and i 4 lie in the same irreducible representation f 2 ). Scalar bosons contributing to fig. 4.6 must lie in irreducible representations Bt such that (4.24) The exchanged mass eigenstate scalar boson S is in general a linear combination of components which have the same transformation properties under some subgroup of the gauge group (e .g ., for SU(5) the relevant subgroup is SU(3)8SU(2)L®U(1)y): (4.25) -37- 13 I II I I I I +s I I I Fig. 4.6: Vector decay with scalar exchange . -38- Note that this linear combination may include both mixing betweEn the irreducible representations Sa and within a given irreducible representation (an example of both such m.ixings is discussed in the context of an illustrative 80(10) model in chapter 9) . We shall assume for now that at most two components are present; the generalization to an arbitrary number is immediate. In this case , (4.26) x(al <i2 l S 1!i4>+a2<i2 l S2li4> )]] where we have dropped the real factor corresponding to the gauge boson couplings, and the trace represents a sum over all fermion representations (usually "families"). <i 2 1S 4 ji 4 > and <i 1 l S 4 ji 3 > Since· i 1 ,i 2 cf 1 and i 3 ,i. 4 cf2 , the couplings are related by a real Clebsch-Gordan coefficient: (4.27) Hence lm[O]=Im[ Tr[(a 1 •<is l SI !i 1>b:2 •<isl st li1>( C1a1<it l S 1lis>+C2a2<i 1!S2 !i3>) ]] (4.28) Thus:, i! C 1=C2 , then this contribution vanishes. This is inevitable if all relevant Higgs bosons lie in replications of the same irreducible representation of the gauge group, and if this representation contains -39- only one B-violating component. Examples of cases in which C1 "# C2 are the SU(5) model with a 5s and a 45H (case Bin chapter 7) and an S0 (1 0) model with a 10H and a 120H or a 126s. In these models, CP violation may occur at the one-loop level from scalar boson exchange in vector boson decay. Notice that since in the absence of spontaneous symmetry breakdown only one of the o..i is nonzero, the result (4.28) yields no CP violation in this case. The case of vector boson exchange in scalar boson decay (illustrated in fig . 4. ?) is exactly analogous to the case of scalar exchange in vector decay discussed above . When flg . 4. 7 contributes, il is often important by virtue of the large value of the ve c tor couplings relative to the scalar ones . We now consider CP violation arising from scalar boson (s') exchange in scalar (S) boson decay, as illustrated in fig. 4.8. If only one B-violating Higgs boson is present, then the decaying and exchanged boson must be identical, and the results discussed above show that :fig. 4.8 Cfui giv·e no CP violation. This is the case for the minimal SU(5) model. (Hovlever, as described in chapter 6, CP violation may occur in higher-order diagrams .) i We consider for now the case in which all fermions are effectively massless. Then, in analogy with (4.24), the contributing scalar bosons must appear in representations sa such that (4.29) -40- -->s -~-­ s Fig. 4. 7: Scalar decay with vector exchange. -41- II '3 I I _....;...rs As' I I --~- s I I '2 I F1g. 4.8 : ScaJnr decay with scalar exchange . -42- H all the left-handed fermions lie in the same complex irreducible representation, f (or sequence of such identical representations), then f1 =12 =13 = f 4 and these constraints become (4.30) For low-dimensionality representations, this requires sa and s~ to be real representations. Hence in S0(10) models where all fermions lie in the 16 representation, only 10n or 120n may contribute to fig. 4.8; the 126n which appears in 161 ®161 is complex. (For high-dimensional fermion representations., some complex Higgs representations may satisfy (4.30) : an example is the 126n occurring in the symmetric product 1441 ®1441 of S0(10).) After spontaneous syrnrnetry breakdown, mixing between scalar bosons may occur, and the constraints (4.29) are no longer applicable. Thus, in both SU(5) models with several Higgs representations coupling to fermions, and in SO(lO) models, fig. 4.8 can yield CP violation. The discussion above has assumed that all relevant fermion species are effectively massless. With gauge groups such as SO(lO) or E(6), it is common for fermions with SU(2) singlet and thus potentially large mass terms to exist. The effect of such fermions in intermediate states of figs. 4.6 .through 4.8 and in vector decay through vector exchange diagrams is always suppressed by O(m}lm]). If only a single massive fermion exists, then it can introduce no CP-violati.ng effects into vector decay through .ector exchange; a single massive fermion is, however, sufficient to generate CP violation in figs. 4.6 through 4.8 even when ( 4.28) vanishes. It is certainly worth noting that, though the analysis of this chapter has focused on baryon number, the expressions that we have derived are by no means restricted to that quantum number. The expressions are -43- valid to describe the generation of any quantum number in the free decays of X, X , Y or Y . Thus, for example, to describe lepton number generation we need to replace the B;, 's by the relevant lepton number assignments. Furthermore, although ·our analysis focused on the diagrams of fig. 4.4 as the first nonvanishing contributions to Rx, it may be that those diagrams give a vanishing contribution to Rx for a particular model (an example is the minimal SU(5) model discussed in chapter 6). In that event the discussion given here goes through with very little change: the quantity /0 then arises from the lowest order contributing diagrams. -44- 5) SU(5) Models SU(5) is the simplest group (and the only one of rank 4) which con- tains the group which so successfully describes the existing low energy phenomenology• [1], SU(3)®SU(2)L®U(1). The vector bosons transform according to the adjoint representation, 24. The symmetry breaking SU(5)-..SU(3)®SU(2)®U(1) is typically effected by means of a 24H of Higgs scalars which is postulated to attain a superlarge vacuum expectation value so that the phenomenological constraints on the decay rate of the proton [23] may be maintained. A family of fermions, which consists of 15 left-handed fields, falls into the reducible representation "5ffi l O. Such a family has the {generic) particle content (5.la) (5.1 b) where the superscript c stands for charge conjugation and the vector sign indicates transformation as an SU(3) triplet. Thus VL transforms as 8 3. Scalar fields whic.:h couple to fermions must transform according to representations that appear in the decompositions of~. 5®10 or 10®10: (5.2a) 5®10 =5+45 1~10 = {5+50)s + (45)A . •Sametimes refered to as the "Holy Tr~Jty." (5.2b) (5.2c) -45- With the assignments. (5.1), of these representations the only ones which have a neutral (zero electric charge) component (and hence can have a vacuum expectation value and contribute to the fermion mass matrix) are 5, 15 and 45. The inclusion of a 15 of Higgs would allow the left-handed neutrino to have a Majorana mass (and would thereby violate the B-L symmetry usually present in the broken SU(5) theory with only 5's or 45's of Higgs [24]) . It appears difficult to make such a MajorruJ.a mass naturally small in an SU(5) model with the fermions (5 .1); the 15 is usually excluded on these grounds . The simplest viable set of Higgs is a single 5H (in addition, of e:ourse, to the 24n ) . A model with a single 45H, though theoretically sound, has phenomenological problems because it gives the mass relation 7'ntJ I mT = 1/3 at the unification scale and this is very difficult to reconcile with experiment. Alternatives to this so-called minimal SU(5) model that effect the charged fermion mass matrix while leaving the neutral mass matrix unaltered are to have more than one 5H or to have a 5H and a 4 5H or to have some arbitrary number of each. A discussion of the CP violation necessary for cosmological baryon number production in such models is discussed in chapter 7. Another possibility is to add a 50n of Higgs (this, of course, has no effect on the fermion mass matrix at tree level); this case is relevant for the "primordial" S0 (1 0) model discussed in chapter 9. The reducibility of the fermion representation implies that, even with a single Higgs representation (5y or 45y), there are two independent Yukawa coupling matrices. One couples to the product 5i®10; (where i and i index fermion families) and the other couples to the product 10i®10;. The former yields (after the Higgs have obtained their vacuum -46expectation values) the D and _,:; rnc.ss rr1aLrif'e:' nnd the 1aitcr Lh2 fi rn ass matrix. The SU(5) representations introduced above may be decomposed according to the embedding SU(5)::>SU(3)®SU(2)L®U(l)y as 5 = (3, 1. 1/ 3) + (1, 2, -1/ 2) (5 .3) 10 = (3, 2, -1/6) + (3, 1, 2/3) + (1. 1. -1) 1 (5.4) 15 = (6, ] ' 2/ 3) + (3, 2 , -1/ 6) + (1, 3, -1) (5.5) 24 = (8, 1, 0) + (3, 2, -5/6) + (3, 2, 5/6) + (1. 1, 0) + (1. 3, 0) (5.6) 45 = (1, 2. -1/ 2) + (8, 2, -1/ 2) + (6, 1, 1/ 3) + (3, 1, -4/ 3) + (3, 1, 1/3) +(3, 3, 1/3) + (3, 2, 7/6) (5.7) 50= (6, 3, -1/3) + (B, 2, 1/2) + (3, 2. -7/6) + (6, 1, +4/3) + (3 1, -1/ 3) + ( 1, 1 -2) 1 1 e (5.8) -47- 6) CP Violation in the Minimal SU(5) Model In this chapter we discuss the result that, for the minimal S U(5) model, the first possibly nonzero contribution to Im(O) (in the notation of chapter 8) occurs at eighth order in the couplings [ 16]. In minimal SU(5) one puts each family of fermions in the reducible representation 5tffi10t , where the index i is a family index. As we discussed above, the Higgs multiplets in the minimal model are taken to be a 5n and a 24n ; the 24H cannot couple directly to ferrnions. The coupling of the fermions to the 5H may be written schematically as (6.1) where summation over repeated indices is implied and where hD and h r; are the Yukawa coupling matrices in family space. In eqn (6 . 1) there should also appear group coupling coefficients to make the respective terms to transform as SU(5) singlets . Such coefficients may be taken to be real as discussed in appendix B. Furthermore, there is only a single representation of Higgs that couples to fermions and that representation contains only one baryon number violating scalar (up to SU(3) degeneracy); thus, there is no mixing among the relevant baryon number violating scalars. Consequently, any group coupling coefficients that appear in 0 can be factored out of Im(O); thus, for the present discussion, they cause us no consternation and we will therefore suppress them. This will also be the case for an extended SU(5) model with two 5H representa- tions; however, for an extended SU(5) model with a 5H and a 45H this will not be valid since the group coupling coefficients for coupling to the 5H differ from those for coupling to the 45H. The details of these cases are discussed in chapter 7. Thus, all matrices that we consider in this -48- chapter are matrices in family space and all traces are over family indices. Off-diagonal elements in these matrices represent transitions between different families. The coupling of the gauge vector bosons to the fermion fields is diagonal in family space. We write those couplings schematically as (6 .2) In the broken theory · the errors introduced by neglecting the mixing of fermions induced by the off-diagonal elements in hD and hr; are of D(mjl JJ 2 ), where m 1 is a typical fermion mass and M is the mass of an internal boson line (see appendix C for explicit calculations in the twoloop case). In· the unbroken theory (i.e., at high temperatures) the fermiens propagate as massless particles. Either way we are justified in neglecting fermion mixing. The types of vertices that may appear in a diagram fall into two classes: the couplings of fermions to bosons and the couplings of bosons to one another. For the minimal SU(5) model, the latter class has all real (non-CF violating) couplings . Thus, all factors arising from such vertices may be factored out of Irn(O) . The reasons for this are as follows. In the minimal SU(5) model there is no spontaneous CP violation (the 24n is a real representation and hence its vacuum expectation value is real; any phase appearing in the vacuum expectation value of the 5H can be rotate u away in exactly the same fashion as in the SU(2)L~ U ( 1 )y electroweak theory with one Higgs doublet.) Furthermore, there can be no intrinsic CP violation in the Higgs potential (which CP violation would be necessary for the three- or four-boson couplings to be CP non-invariant) because it is Hermitian. For example, -49- (6 .3) which is CP invariant. In treating CP violation in the minimal SU (5) model to lowest order, we therefore consider diagrams with no three or fourboson vertic-es . This leaves us with diagrams consisting of one or more closed fermion loops and with only fermion-fermion-boson vertices . It therefore suffices to determine what the lowest order is for w:bJch the family-space trace corresponding to a single fermion loop bubble diagram with no multiple boson vertices has an imaginary part. The fermion-boson coupling vertices in the minimal SU(5) model are shown in fig. 6.1. The lowestorder corrections to B-violating decays in this model are given in fig. 6 .2. Each diagram is proportional to a trace in family space over the products of coupling matrices occurring around the closed fermion loop. The trace for fig . 6.2(a1) and 6.2(a2) is trivial, hence there can be no CP violation from vector boson exchanges in vector boson decays. For fig . 6.2(a1), the relevant trace is Tr [(hD)t(hD)] , which is real. Similarly, fig. 6.2(b2) involves Tr [ (h ~ )t(h ~ )] which is again real. Figs. 6.2c yield the same traces and are thus also CP-conserving . Finally, fig . 6.2d gives Tr[(hD)t(hD)(h~)t(h~)] , which is manifestly real. Thus, none of the diagrams in fig. 6.2 can give rise to CP violation. A systematic investigation of possible three-loop diagr~TTis reveuls that none can have CP violation. For example , fig . 6.3 yields the trace (6.4) The first diagrams for which the corresponding traces are not necessarily -50- g ./2 lOt ~ I ~f I + 'ho I~H I Fig. 6.1: Boson-fermion-fermion vertices in the minimal SU(5) model. -51- (a2) (a I) (b2) ( b I) (c2) ( c I) ·-(d) Fig. 6.2: Lowest order (non-Born) diagrams for the decay of bosons in the minimal SU(5) model. Unitarity cuts are not exhibited. These diagrams do not give CP violation. -52- Fig . 6.3: An example of a sixth order diagram for the decay of a scalar boson in the minimal SU(5) model. Unitarily cuts are not exhibited. This diagram does not give CP violation. -53- (PAGE 53 IS BLANK, DUE TO ERROR IN PAGINATION) -54- real appear in the next order [ 16] . . The relevant diagrams are shown in fig. 6.4. and are proportional to the trace' (6.5) At this point it is worth noting that there is an additional freedom in this model which allows us to take either hD or hu to be real and diagonal. In earlier work [16] it was believed that this freedom was necessary to show that all of the diagrams of less than eighth order do not have a CPviolating imaginary trace; however. as we have shown above, it is not necessary to use this freedom. We review the discussion oi this synr metry here because it is interesting to see how, by using the available freedom (under the assumption of massless fermions). the CP violation may be isolated. With the neglect of fermion mixing we can perform a unitary redefinition of fields in family space 5; = Yv5; } o 1 1 = uj~c fb~c (6.6) (6.7) Upon applying these redefinitions and suppressing explicit family indices, the couplings ( 6.1) become (6.8) where (~n) = llt·(hn )· (htr) UJ . = ur. (hu ) · •In this expreaion ~ ill taken to be real and diqonal (this may always be done, as Ia diacu.ssed in what follows). For such an h.IJ all of the diagrams in fig. 6.4 have equal weight. (6.9) -55- (b) (c) Fig. 6.4: Eighth order diagrams that may give · CP violation in the Minimal SU(5) model. Unitarity cuts are not exhibited. -56- The 5's of ferrn.ions and the 1D's of fermions can be independently redefined as in eqn (6.6) because, in this limit, the naming of families is an arbitrary convention. Thus, eqn (6.6) amounts to a new choice of family convention. The specific form of the Lagrangian is altered but its predictions are not. Since it is the symmetric product of two 10's that couples to 5H , h ~; must itself be a symmetric matrix. Clearly hv must be symmetric as / well. It is clear from eqn (6.6) that, for suitable V and U, hD may be rendered real and diagonal. Given such a V and U, hv may not, in general, be cast.. into such a form. entries. It nlay therefore contain CP-violating COlllplex (W~ may also proceed differently by choosing U so as to render hv real and diagonal [25], then hD cannot be made real and diagonal in general.) The gauge couplings (6.2) may be written after the transformation (6.6) as and are therefore left unaltered. The arguments given above also app]y in an SU(5) model with a single 45n of Higgs (as well as the 24n ) . For the minimal SU(5) model the high order of the diagrams gen- erally renders possible CP-violating differences between the X and X partial widths very small and prevents the generation of an adequate baryon o.synu'Uetry. A rough estimate fo1· the rnagnitude of the paraJ.neter R of eqn (4. 1) arising from these diagrams is ( 6.11) -57- where ~ is a CP violation parameter (~=sino where o is a CP violating phase angle ) !1: 1~1. The Yukawa couplings are dominated by the heaviest fermion F. The momentum integration factor lm[I] is given very roughly simply by the volume of available phase space for each loop integration: lm[f]"' (1/ 8~ 3 . Thus one may estimate (6. 12) This is completely inadequate unless vP-ry heavy ferrnions exist in a iamily transformaing as 6EB1 0. With the usual symmetry breaking mechanism, mF ~ -v'3 m,, so that R ~ x 10-a~ . In principle, one may make unitariiy cuts through the diagrams of fig. 6.4 to obtain either two-body or threebody final states. However, the fact that the exchanged bosons have ihe same mass as the decaying bosons renders all but two-body final states energetically forbidden. Above we have considered only decays to fermion final states . CP violation can enter in the minimal model only through intrinsic complex mixings between fermion families: fermion intermediate states are therefore necessary for CP violation. Decays such as S .... X cy or X .... S;; to boson final states (where rp is an SU(2)L doublet scalar) therefore exhibit CP violation only through internal fermion loops and at very high order in perturbation theory (always at an order higher than if one has only fermion final states). -58- 7) CP Violation in Alternative SU(5) Mod~ls The minimal SU(5) model is economical in its choice of Higgs representations, but that choice is by no means necessary. From the point of view of generating an acceptable baryon number asymmetry, / there are two simple modifications of the SU(5) Higgs structure that may be :made: adding a further 5s (case A) or the addition of a 45R (c as e B) . Of cour~e. more complicated Higgs structures may be chosen. We first discuss case A, in which two 5n's, denoted 5n 1 and 5s2 , appear . The coupling of these Higgs to fermions may then be written in the form (7.1) The 5n 1 and 5n2 here may be chosen to be mass eigenstates. This is possible because any linear combination of 5n's is also a 5n (complex linear combinations are permitted since 5n is a complex representation) and furthermore there is only a single baryon number violating scalar in each 5n up to S U(3) degeneracy . The diagrams for corrections to gauge boson decay through Higgs . exchange are as in fig . (6.2), except that either of the two 5n's may be exchanged. In each case, to lowest order, the CP-violating part is proportional to Im[Tr(hahat)] =0. Similarly, no CP violation is generated by gauge boson exchange corrections to the decays of 5a 1 or 5n2 Higgs bosons. Exchanges of 5na in 5na decay may be treated just as in the minimal model which contains only a single 5n discussed in chapter 6: CP violation in such cases was shown to vanish until eighth order . However, exchange of 5e 1 in 5n2 decay (or vice versa) may lead to -59- CP violation at fourth order. The coupling factor associated vvith this / diagram is 0 =7r[(hD )(hD )t(ht.r )(hr,r )t]. 1 2 1 2 One may apply the unitary transformations (6.3) to render real and diagonal either hD 1 or hD 2 , but not, in general, both. Hence here, in general, Im[O] 7l 0, so that CP violation may occur in 582 decays through 5H 1 exchange (or vice versa) at the one-loop (O(a:)) order. This is shown in fig. 7.1. We now discuss case B defined above, involving a 5H and a 45H. Their coupling to fermions may be written in the form (7.2) In this case, Higgs with definite SU(5) transformation properties will not, in general, contain baryon number violating scalars that are mass eigen- states. A (3, 1, 1 /3) (B-violating) component exists in both 5H and 45n; the mass eigenstates will be linear combinations of these components. The presence of a 5H·45H·24H-24H term in the Higgs potential enforces such a mixing between the (3, 1, 1 /3) in the 5H and that in the 45H . This term cannot be removed by the imposition of a discrete symmetry without affecting the Yukawa terms. A cubic term such as 5H·45H ·24n can be excluded by the symmetry 24H .... -24H . We denote the (3, 1, 1 /3) mass eigenstates (assumed mixtures of that in the 5H and that in the 45H) by S 1 and S 2 . The couplings of fermions to these mass eigenstates are linear combinations of the hJJ, HD and hr1 • HrJ app l-: aring in (7.2). However, in general, the linear combinations will be different for different fermions within a single family by virtue of the different Clebsch-Gordan coefficients in coupling fermions to the (3, 1, 1 /3) in 5H and to the (3, 1. 1/ 3) in the 45H. H we call the (3, 1, 1 /3) in the 5H, Sg (where a is the -60- __ .__ ~HI Fig. 7.1: Scalar decay with scalar exchange in an extended SU(5) model with two 5's of Higgs. This diagram may give CP vioation. -61SU(3) index) and call the (3, 1, 1 /3) in the 459 , Sa., then we may ·write their couplings to fermions as follows. The coupling of Sa. coming from the 5·10·59 term is (7.3) The coupling of Sa. coming from the 5·10·459 term is Aside from having differing y·ukawa couplings, the preceding two expressions also differ in their Clebsch-Gordan coefficients (notably the crucis.l minus sign in (7.4) as compared to (7.3)) . Eqn (4.28) then shows that we may now have a contribution to the baryon asymmetry due to the CP violation in vector exchange in scalar decay (and vice versa). Similarly, gauge boson exchanges in Higgs boson decays may also yield CP violation. The structure of CP violation for Higgs boson exchanges in Higgs boson decay is analogous to the model A discussed above. It is worth noting that there are two more baryon number violating scalars in Model B: a (3, 1, -4/ 3) and a (3, 3, 1 /3), both contained in the 45n. We call them S 3 and S 4 respectively. Diagrams involving only an S 3 or those involving only an S 4 show no possible CP violation until eighth order in the couplings if there is no intrinsic CP violation in the Higgs potential. Furthermore, in the decays of gauge vector bosons there is no CP violation through the exchange of a single S 3 or S 4 as follows from eqn (4.28) since S 5 and S 4 are necessarily mass ei.genstates. Also, since each couples to fermions identically (within a factor of a real Clebsch-Gordan coefficient), it follows that a diagram that only involves Ss and S4 "\\ill show no CP violation until eighth order, although a diagram for, sn.y, S 1 -62- decay through S 3 exchange may have CP violation at fourth order. -63- 8) Introduction to SO( 10) Grand unified models based on SU(5) are the most economical as vtell as being the simplest for actual calculations. However, the assignment of a (left-handed) family to the reducible 5~10 representation has a number of ugly featw·es. So1ne of the particles belong to differe11t irreducible representations than their anti partie les and, although the anomalies cancel between the 5 and 10 representations of fermions [ 1] , this cancellation appears rather artificial · from the standpoint of SU(5). In addition, many SU(5) models contain a global quantum number corresponding to baryon number minus lepton number, B-L [24]. These features may be removed by embedding the SU(5) theory in an S0(10) model with the fermions assigned to the lowest dimensional spinor representation [26]. The defining representation of S0( 10), 10, is real and has the following SU(5) decomposition: 10 =5 + 5. (8.1) The lowest dimensional spinor representations of S0(10) are• 16 and its conjugate, 16. All representations of S0(10) can be built out of products of 16 and 16 among themselves. The SU(5) decomposition of 16 is 16 =1+ 5 + 10. (8 .2) We see that a single family of fermions can be accommodated in the 16 of S0(10) . The anomaly cancellation that occurs in SU(5) between the 5 and the 10 of fermions has a natural explanation by choosing the 16 as the •It is here that the (often confused) distinction between 0(10) and 80(10) occurs. The lowe3t dimensional spinor representation of 0(10) is 32 and is self-conjugate. Under 0(10)::>80(10), 32 = 16 +ill. 32 is irreducible in 0(10), but reducible in 80(10) . -64fermion representation in 80(10). The 16 is anomaly free as are all representations of SO(n) for n >6. This is equivalent to the statement that the symmetric product of the adjoint with itself does not contain the adjoint and hence the d. coefficients vanish. SO( 10) is of rank 5 and contains SU(5) as a subgroup with the maxi- mal embeddL11.g SO(lO) ::>· SU(5) ® U(l}. (8.3) The SU(5) model may be considered as being embedded in an SO( l O) model in a sense sirnilar lo the way the SU(3}®SU{2)L®[/p)y rn.odel is embedded in SU(5). This point of view is expressed by the following discussion. In the simplest SU(5) models (with only 5's and 45's of Higgs) there is a global U(l) symmetry in addition to the gauged SU(5) symmetry. When SU(5) undergoes spontaneous symmetry breakdo·wn, this global U( 1) is broken, as is the combination of generators through which the Z 0 boson couples. A linear combination of these generators, however, survives as a global symmetry• and corresponds to B-L [2?]. If we demand that there be no ul'lgauged continuous symmetries••, then ·we •We will refer to this process by which a global symmetry survives spontaneous symmetry breaking as the 't Hoeft mechanism [28]. This mechanism can be used for both discrete as well as continuous symmetries. In the latter case, however, it provides one with a method for avoiding Goldstone basons. In the case of discrete symmetries it allows for what may possibly be a very simple symmetry to transmute into a much richer symmetry. If one takes a. somewhat proletarian altitude towards model building (and, for example, postulating various messy discrete symmetries to force particular results from a given model), this fa.ct gives one hope that embedding such a model in a larger model may give rise to a S"impler natural symmetry structure. An exacple of tr..is is given in the context of an il1ust.rative S0(10) mode: in chapter 13 !tlld appendix E. •f!rus is, of course. a bias which is not necessitated at present by any well defined theoretical principles just as long as any global symmetries that are spontaneously broken are rendered harmless by the 't Hoeft mecbanism. Other alternatives are to explicitly break the symmetry with terms of dimension 2 or 3 or terms of dimension 4 in the Lagrangian. This cannot always be done as is evidenced by the SU(5) example where one bas to use the 't Hoeft mechanism of necessity. It was this feelmg that continuous global symmetries are in some sense incompatible with the ideas of locel quantum field theory that originally led Yang and Mills to -65- must consider S['(5)®U( l) as the gauge group . SO( 10) is then the smallest simple group containing SU(5)®U(l) . The gauge vector boson correspo11ding to the U(1) factor can then mediate B-L violating processes . :r rorn eqn (3.2) we see that the price to pay for this SO( 10) unification is an SU(5) singlet fermion. Since the electric charge operator is entirely contained in SU(5), this fermion is neutral. We denote this extra field by NL. It provides a charge conjugate partner for the left-handed neutrino and thus allows a /1/ r = 1/ 2 Dirac mass term for the neutrino. The potential disaster of neutrino masses of the order of the fll w= 1/ 2 breaking mdy be avoided if the NL acquires a very large, f).fw=O, Majorana mass, MN [29, 27]. The neutral lepton mass matrix will then have the form • (8.4) with 171.q a matrix with entries of the order of the observed quark masses and MN the Majorana mass matrix for the NL. F.or MN>>mq the eigenvalues of this matrix are given approximately by the eigenvalues of MN, which are the masses of the NL, and the eigenvalues of the matrix m{M.N 1TTLq, • which are the light neutrino masses. As a result of this mechanism, SO( 1 0) models naturally predict the existence of neutrino masses and hence neutrino oscillations. The NL can be given a large Major ana mass either directly through a 126 of Higgs that obtains a _large vacuum the concept of the eauae field [2]. ~e zero entry in this matrix is not necessary. All that is required is a condition such as bein& < O(n; ). Such a Majorana mass for "'L would have A/, =1 and must be relatively small so as not to disturb m r/ (mzCOIIIJ ,) 01 1. A Majorana mass for 11L can be accommodated in SU(5) by including a 15 of Higgs which obtains a vacuum expectation value or by means of a 10 of Hi&p. In the latter case, the left-handed neutrino's Majonm.a I'D.-s arises • a ce.leuleble correction to the zeroth order mass relation which requires it to vanish [ 31]. -66- expectation value aloJ14 its SU(5) singlet direction or through radiati\ e 7 corrections [30]. The vector bosons transform as the 45 dimensional (adjoint) representation of which 80(10) (s 0 ( 10) :J s u (4 )®S u (2) L ®S u (2) R) has the chiral decomposition I 45= (6,2,2) + (15.1,1) + (1,3,1) + (1,1,3) (8.5) The last three representations correspond to the gauge bosons of SU(4), SU(2)L and SU(2)R respectively. The (6, 2, 2) contains the usual leptoquark-diquarks (X, Y) of SU(5) transforming as (3, 2, 5/6) under SU(3)®SU(2)L®U(1)y, their antiparticles, an additional doublet of leptoquark-diquarks, (X', Y'), transforming as (3, 2, -1/6) and their antiparticles. The gauge bosons of SU(4) contain the gluons of SU(3) and an additional color triplet field transforming as (3, 1, 2/3), which we denote by V, its antiparticle and a color singlet field. The gauge bosons of SU(2)R transform as (1,1,-1), (1,1,0), and (1,1,1) which we denote by w;, w.R. and WR respectively. The scalar fields which couple directly to fermions must transform according to representations which appear in the Clebsch-Gordan decomposition of 16®16: (16®16) =(10 + 126)5 +(120)A , (8.6) where S and A denote the symmetric and the antisymmetric parts. The SU(5) decompositions of some SO(lO) representations are given in table 8.1; their SU(3)®SU(2)L®U(1}y decompositions can be obtained easily from this table and eqns (5.3) through (5.8). The 10 and 120 are real representations; the 126 is complex. One may also include other Higgs -67- 10=5+5 16=1+5+10 45=1+10+10+24 54=15+15+24 120=5+5+10+10+45+45 126=1+5+10+15+45+50 Table 8.1 SU(5) decompositions of some S0(10) representations. -68- multiplets which do not have direct couplings to fermions. Typically an adjoint, 45, ' or a 54, is chosen. The minimal set of Higgs necessary to break SO(lO) down to SU(3)®U(1)Ell and give masses to all fermions is lOB, 16H and 45H. With only these Higgs one finds the tree level mass rela- tions mv =rnu ='l'na = ~ (8 .7) at the unification scale, with the same relations holding for the heavier farrti.lies . Thus we see that for the minimal set of Higgs the mass of the everyday neutrino is predicted to be far too large. The generation of a net baryon number from symmetrical initial conditions requires the presence of both C and CP violation [ 12, 13]. In SU(2)L ® V(1)y weak interaction models and SU(5) grand unified models no C operator may be defined since there is no left-handed antLn.eutrino to act as the charge conjugate partner of the left-handed neutrino . In some larger modeB, such as S0(10) or E(6), each fermion has a potential charge conjugate partner or is an eigenstate of C hence a C operation may be defined which is a symmetry of the unbroken theory [32]. The production of a C-odd quantum number (such as B or L) in these models therefore depends on the interplay between the sources of C violation and the processes which violate the quantum number under consideration. The lack of B production in a C-symmetric theory may be seen by considering the decays of B-violating bosons x and their antiparticles x as well as the decays of their charge conjugate partners >f and )C. The B produced by the decays of an equal mixture of x and x into the specific final state i 1i 2 and the charge conjugate decays of >f and X: into the state iYi~ is proportional to the quantity (see eqn (4.12)) -69R )(12 + (R)(12 )c = lm/lmO (B·' l -B·'1 ) + Irnlc: Imnc (B.\~ -B.\~ ) (8 .8) where 1 represents t.he integral over the intermediate momenta and final state phase space for the decay and 0 is a product of the relevant couplings . The lowest order contributions to I and 0 were discussed in chapter 4. 1c and OC are the corresponding quantities for the charge conjugate reaction. In a C-symmetric theory, I = 1c and 0 = oc, while, since B is C-odd, Bi 2 = -Bii and Bi 1 =-Bii causing Ri_ 2 + (Ri_ 2 )' to vanish. We now restrict our attention to SO(lO} grand unified models. The presence of a charge conjugate partner for the neutrino, NL, allows the definition of a C operation for all fermion fields appearing in the theory. In terms of the SU(4) ® SU(2)L ® SU(2)R subgroup of SO(lO), C interchanges the two SU(2)'s, as well as conjugating them, and also conjugates I the representations of SU(4) [22]. It may be shown that all C violation in the fermion mass matrix must lie in the part of the 126 representation of SO(lO) which gives a Majorana mass to ~"~L· This C-violating mass term allows for the production of a nonzero B since lml is no longer equal to Imlc . Expanding I and /c in powers of MN I Mx gives (8.9) where Mx is the mass of the decaying boson. If all asymmetries can be expressed in terms of C odd quantum numbers then, (8.9) constrains the possible values of MNI Mx if we demand that the theory be able to produce the observed baryon asymmetry. However, in the general case, asymmetries which have no definite behavior under C must be considered. Large asymmetries in such quantum numbers may be produced even if the theory is in a C-conserving -70- phase (e.g., SO(lO) broken t.o SU(4)®SU(2)L®SU(2)R) [33]. These asyrnmetries may later be converted into a baryon asymmetry by B-violating reactions which occur in a C-violating phase of the theory. These reactions will be able to produce a sufficient baryon asymmetry only if there exist B-violating bosons with masses less than the transition temperature between the C-conser~...ng and C-violating phases of the theory. For S0(10) ~ SU(4)®SU(2)L®SU(2)R the SU(4)®SU(2)L®SU(2)R symmetry must not persist to temperatures below -10 12 GeV if an adequate B is to be produced. A detailed discussion of some of these ideas is presented in the context of an illustro.t.ive SO(lO) model in the following chapter. -71- 9) Analysis of an illustrative SO(lO) Model We saw in chapter 8 that NL can obtain a large Majorana mass if we have the fermions couple to a .i.26n of Higgs which obtains a large vacuum expectation value along its SU(5) singlet direction. Vacuum expectation values along other components of the 126n break SU(2)®U( ~ ) (see table 8.1 and eqns (5 .3) through (5.8)) and hence must be small relative to the singlet vacuum expectation value. If SU(3)®SU(2)r®U(l)y is unbroken* the vacuum expectation value of 126n is purely along its SU(5) singlet, <126H>"'1. To effect a complete symmetry breakdown of SO( l O) to S U( 3)®SU(2}®U(l) we need another Higgs representation** . Either a 54n or a 45n is usually chosen so as to conform with typical SU(5) models since both contain a 24 of SU(5). The size of the 54n (or 45n) vacuum * expectation value along the SU(5) 24 direction is 0(10 15 GeV) so as to conform with the bound imposed by the nonobservation of proton decay [23]. Depending on the relative sizes of <54n> (or <45n>) and <126n> one will have different symmetry breaking patterns for 80(10) . To consider the production of a cosmological baryon number asymmetry in an SO(lO) model with <126>"'1 we will consider the simplified case where there is only one family of fermions. We will discuss the contributions to the asymmetry due to the free decays of baryon number violating bosons. The free decays of the NL do not contribute appreciably to the baryon asymmetry because the baryon number violating decays of the NL are generally into a three-body fermion final state, whereas the main contribution to the Born rate is through the two-body final state -rbis is generally the case at the high temperatures present in the standard model of the early universe. "With <126>-1 alone, SO(lO) bree.b down cmly to SU(5) . -72NL -+Vr'P where rp is an SU(2)L doublet scalar field [34]. The contribution of the decay of the .NL is thereby greatly supressed. We assume that there is no intrinsic CP violation in this model; however, it has been shown [35,36] that there can be a source of high temperature spontaneous CP violation ~ith a calculable phase. The Higgs potential contains the following quartic terms involving only the 126n: (9.1) where A1 and~ are real. If we write <126n>=pe'' then (9.1) gives (9.2) If we choose A1, ~>0 then• the potential has its minimum when rp=±Tr/ 4 or ±31T'/ 4. These two cases are not independent in the following since the quantity that always enters into the calculations is <126>2 . Such a CPviolating phase enters the theory at a scale 0( I <1269 >1) and is the only CP violation present between that scale and the scale at which the next level of symmetry breaking occurs. Thus, if the vacuum expectation value of the 54n is greater thaD that of the 1269 (this is the most interesting case since the SU(4)®SU(2)L®SU(2)R chain -+ of symmetry breaking is SO(lO) -+ SU(3)®SU(2)L®U(1)R ), then this CP violation will be absent in the SU(4)®SU(2)L&SU(2)R phase; any decays that occur in this phase will conserve baryon number by virtue of the absence of CP violation. Even if there had been CP violation in this phase {for example .-ro insure the stability of the potential we must also have ~<A1 . The sign of , must be ao as to generate the correct sign of the baryon number esymmetry con8lstent with our conventions fer particles and antiparticles. Also, (126·126) 2 may couple to form a singlet in four ways [33,36]; however. only the term construc.t ed from havin& 128·1'2! transform as a sqlet has a pf term in its expansion (where , 1 til the SU(t5) singlet in the t26a) and thus the remaining quartic invariants do not Contribute to eqn {9.2). -73- due to the presence of intrinsic CP violation in the Lagrangian), a baryon asymmetry could not be generated directly by the decays of baryon number violating bosons because of the presence of an unbroken charge conjugation operation as discussed in chapter 8. The Yukawa tern1 that we currently have in this model, (16·16) · 126H, has a global Ux(l) symmetry with a charge X for which (by convention) X= 1 for the 16 a.nd X=Z for the 126n (X=-2 for 126H ). (This global Ux(l) is broken explicitly by the (126H) 4 term in the Higgs potential as well as by other terms to be discussed below.) When 126H gets its vacuum expectalion value; Ux(l) is broken as is the local U(l)R appearing in SU(5)®U(1)R (we call the charge ~orresponding to the local U(l)R. R). If the vacuum expectation value of 126H is along any one direction in its SU(5) decomposition {as is true in our case), then the 't Hoeft mechanism is operable, yielding a global U(l)z after symmetry breaking with a corresponding charge (for <126H>-1), Z=(x-f1 . (9.3) The analysis of this symmetry ls discussed in appendix E. This symmetry allows us to classify the possible scalar mass terms in the model, since they all must be Z (as well as SU(3)®SU(2)L®U(1)y) invariant. To this end we list all of the values of Z for 54H and the 126H. All components of the 54H have Z=O. For 126H the 1, 5, 10, 15, 45 and 50 have Z values respectively, 8 4 16 12 8 0. -g-: 5' 5' 5and 5 . (9.4) -74In this model CP violation may occur through the scalar mass matrix and hence, in the scalar's mass eigenstate basis, through the scalar rotation matrix; in particular this occurs when complex entries occur in the mass matrix of the baryon number violating scalars that couple directly to fermions . In addition to the terms (9.1) there are other quartic terms appearing in the Higgs potential•: (9.5) If we assume that the Higgs potential consists only of quartic terms•• we can immediately state, within factors of quartic coupling constants and Clebsch-Gordan coefficients, what the scalar mass terms are. We write the fields appearing in the 126n as rp 11 rpJ, rp 10 , rpf:> 9'4:> and rpJo; and we write the fields appearing in 54n as 1JI 1r,, +m and Tj/ 24 . Note that since 54n is self-conjugate, "lfj=Th5. From (126n) 4 we can only get (9.6) and from (126n) 4 we get the Hermitian conjugate of this, •At this point the term (1268 )2(54H) 2 could appear, but is excluded on the basis of the discrete residue of the U(l)x symmetry. Later we shall need to include these termB when we break the U(l)x symmetry further by the inclusion of a real 10 of lfi&gs. Furthermore, as mentioned above, there are, for example, four ways to eouple 126, 126, it! and 1E to form a singlet. MTbis is the case if the symmetry breaking is generated by the Coleman-Weinberg mecbaltism [37] which one would wish to occur so as to exclude dimensionful parameters from the bare Lagrangian. Dimensional transmutation may then ocear. Cubic terms can always be excluded from the Hiegs potential by means of a .-crete aymmetry {in fact the discrete residue of the X symmetry will suffice to llo Uda). The pr.en.ce of possible quadratic terma does not affect the following .aalJais yery much aince they yield only a diagonal contribution to the baryon aamber violat.iq acalar bose~ mass matrix. -75- (9.7) From (126H ·12GH )2 , if a 126H and a 126H get a vacuum expectation value we get (9.8) each multipt,ied by I<9' 1>1 2 . If 126H and 126H get their vacuum expectation values then we get (9.9) and if 126H and 126H are given their vacuum expectation values the term (9. 10) arises. From (126H·126H)·(54H} 2 , if both 54H's get their vacuum expectation values, then we have the same terms as in eqn (9.8) but multiplied by <~ 24 > 2 . Other possible terms such as 9'e~ are excluded, at this point, by the Z symmetry; we shall have occasion to include them later. Finally, if 126n and 126n get their vacuum expectation values ,we get the terms (9. 11) Since all the terms above which involve baryon number violating scalars have real coefficients, it is clear that one cannot have a contribution to the baryon asymmetry in the present form of this model because CP is not violated., There is a simple extension of the model, however, which yields nontrivial results. H we introduce a Hi&P reprnentation transforming as a -76- real 10, 10H, then it can couple to the fermions through (9.12) (16 ·16)·10H . (The 10 of SO(lO) has the SU(5) decomposition 10=5+5, where, for a single scalar representation, the 5 is conjugate to the 5; thus, ii we write ~ 5 for the 5 in the 109 and ~:; for the 5, we have ~~=~J.) The X symmetry is now explicitly broken down to the discrete symmetry generated by the transformations (9.13) All quartic terms that are SO( 10) singlets and which can made from 126n , 54n and lOn respect this discrete symmetry. These terms are listed in Table 9. 1. Before the X symmetry is broken down to the discrete symmetry (9.13) we have the Z symmetry for which Z=5 5 8 for the a< 12 I ...... and Z=5 for the 5 (9.14) After the X symmetry is broken down to the discrete symmetry (9.14) the Z symmetry breaks down as well in the following fashion. H, under U(l)x, a given field (say~) transforms as (9.15) then. after the 126H gets a vacuum expectation value along its SU(5) sin&let direction, the Ulgrangian continues to be globally invariant under the action of U( l)z: -77- (9.16) When the U(1)x symmetry is explicitly broken down to the discrete symmetry (9.13), then , upon comparing (9.13) with (9.15) , we see that Utis is equivalent to the constraint a=n rr/2 (with n an arbitrary integer) in (9. 15) and, hence, in (9 . 16) . With this restriction a field with Z= 12/ 5 and one with Z=-B/5 have identical transformations under U(1)z since e\(12/5)nn/2=e -i(6/5)n1f/2 I (9.17) as must certainly be the case since 's=~~- Thus, the U(1)z symmetry allows• the (3, 1, 1/3) in the 5 to mix with those in the 5, 45 and 50 that come from the 126H. If SU(5) is unbroken, then certainly the (3, 1, 1 /3) in the 50, for example, cannot mix with those in the 45, 5 or 5 since the mass terms from which these mixings arise must themselves be SU(5) invariant. The breaking of SU(5) comes, in lhis model, from the 54n obtaining a vacuum expectation value. Thus, any te_rm of the form f'J~ 50 or ~~~ 50 must be proportional•• to <'f/24 >. The term ,!,~ could therefore come from (126·126)·542 , (126 ·1 26) ·54·126 or (126·126)·54·126; however, one cannot form an 80(10) singlet with the latter two possibilities, nor, for that matter, are they permitted by the Z symmetry. The term 'J~!a could come from / (10'126) ·542 , (10'126) ·54·126 or {10'126)·54·126; however, none of these can •It also turns out that the 15 and the 10 in the 126 now haw the same transformation properties as one enat.her under the remaining Z .,mmetry; however, neither the 10 or the 15 contain any baryon number Yiolatinc scalars as can be seen from eqns (5.3) tbroueh (5.8) and table 3.3. "'l'he full mess term must. at the SV{5) leftl, be an inwuilmt before the 24C54 obtains its YaCuum expectation Ylllue; thus, any eandidate mas term must be checked to show that ane can farm a .tn&Iet from the reMimt product of S U(5) repr...entatiom. -78com~ine to form singlP-ts of S0 ( 10); thus, the term ~J~ 50 cannot occur . Mixing between ~'6 and ~will occur through the term 126126·542 . A sirnJlar analysis shows that ~ 45 mixes with the rp 5 and the ~ 50 • but not with the ~ 5 • and the mixing occurs again through the term 1262 ·542 . MiXIng between the 5 in the 10n and the 5 in the 126n is permitted by th~ Z symmetry and does indeed occur through the lerrn 10n ·126n ·(126n) 2 +h.c . (9.18) Further quartic terms consistent with the symmetries of the previous terms are (10n) 4 , (10n) 2 ·(126n) 2 I (10a) 2 ·(126n · 126n) (10n) 2 ·(54n) 2 (9.19) I and tileir Hermitian conjugates. The quartic terms appearing in the Higgs potential are summarized in table 9.1 [33,36]. The entries in the baryon number violating scalar boson mass matrix that may come from t.hese terms are summarized • in table 9.2. The suppressed coefficients of these mass terms are typically the product of a quartic coupling constant, a combinatoric factor and a Clebsch-Gordan coefficient. In light of our ignorance of these factors '(especially the quartic coupling constants) we will take all the coefficients to have the common value A. We choose <, > 1 2 / I <~ 2 1 > l =+i (the case <, > 1 2 / l <~ 1 > 12 =-i can be obtained easily from the following). Defining e=<l!24>2 / I <~ 1 > 1 2 we get the following scalar mass matrix in the(,_,, WJ.o. rp 5 , ; 5 ) basis: e 1+~ e 0 £ 0 0 l+e 1-i £ l+i l+t ..-e anly write t.ha8e ia;uhlq the (3. 1. 113) in the p61 p5 p46 and f1D since all oth_.. lead ka (at maat) d btq ~ with no CP violation. ._)tkl. (9.20) -79- 104 126· 126 I I I 1264 102·542 126 ·1 26 · 542 I I 1262 ·542 le>2·126· 126 102 ·1262 l I 10·126·(126·1 26) Table 9.1 - All quartic terms in a Higgs potential involving a 54, a 126 and a real . - 10. The number of possible terms of each type is not indicated . -80- 9'~1it'~ l <~1> 1 I I 2 ~~s;·5o ! <rp 1> i 2 Y'451'~ : <9' 1> 1 2 10:z·54~ ~rs'V'J<~24><.! 126· 126·54 2 lit'o~~<~24> I J ! 2 IP!t51it'~<~24> 2 ~45~l5<1Jt2-1> 2 r,t'r;~<1P24> 2 1262·542 ,~,J lit'451it'~ lo2·126·126 1 0·126· ( 126·126) ,o~J I <1it'1> 1 2 ,o'it'J I <~1> 12 "'J <1',1'1> 2 'Polit' Table 9.2 Terms in the Higgs potential that give rise to entries in the mass matrix of (3, 1, 1/3) baryon number violating scalar bosons along with the associated mass terms. -81If we write the eigenvalues of this mPttrix as JJ-2 and \\Tite ( 9. 21) then the characteristic equation is (9.22) We can write down one solution of this exactly, x=t, giving (9.23) The remaining solutions can be given approximately for large t as (9.24) (9.25) (9.26) Thus, for large E, J.£2 and J..Ls decouple from equilibrium at a temperature at which the CP violation that we are considering has not yet turned on, and thus a baryon asymmetry cannot be generated through their decays. To 1et a nomero baryon number asymmetry we need to have a scalar that decouples from equilibrium at a temperature that is less than the SU(4)8SU(2)L&S'U(2)R .... SU(3)®SU(2)L®U(1)R transition temperature•; for 4We ~ here the question of the supercooling and possible associated entropy pcend.iall that may occur when a phase transition occurs via tha GolemanWeill.,. m...,.nimn [3B]. This may be avoided by haVing small negative quadratic t8TID8 ill tile Hi&P potential; BS mentioned abOYe, such terms would not subpaetieJV .&et aur .aalysis. -82A< 1 this may occur. Thus, the decays of the lighter baryon number violating scalars may gjve rise to a baryon number asymmetry if they have CP violation in their decays. The eigenvectors associated with the above eigenvalues are respec- 1 ~ 1 v'2 (9.27a) 0 0 1 7 v'3 36"'3 1 1 7 -./3 - 36'\1'3 _1_+ ..J3 5 (9 .27b) 36..J3 e 2 ein/4[ ~£ J 1 V2e 1 ~t (9.27c) 1 - "2£ ein/4[-1 + _U 4t2J and (9.27d) -83- Of the twn light scalars, 1-LI and~ (,u4 <J,.t 1), only the eigenve ctor of the tighter one is associ&tcd with a comp]ex eigenvector . Thu~. its de(;a.y, through the exchange of a vector, may exhibit CP violation. The other light scalar, ~ 1 , may not violate CP in such a decay, although it may do so through the exchange of another scalar. Generally, the decay of a scalar thra\lgh the exchange of a vector produces a baryon asymmetry greater than that produced through the exchange of another scalar by a factor g 2 1 ye where g is a typical gauge coupling constant and Y is a typical Yukawa coupling constant. Thus, unless the Yukawa couplings are very large, an estimation of the baryon asymmetry produced through the free decays will be dominated by the vector exchange diagrams . For illustrative purposes we only ~nsider these diagrams . This model allows for scalar exchange in vector decay since the double-cut diagram for scalar exchange in scalar decay is related to that of vector exchange in scalar decay sim.ply by complex conjugation; however, the relevant vectors have masses that are obtained in the S0(10)-+SU(4)®SU(2)L®SU(2)R symmetry breaking and decouple from equlibrium at a temperature generally greater than that at which our liigh temperature CP violation has turned an (cert.ainly this is true if t>>1 as is the case that we are investigating) . Also, the Born rates for tile two processes are generally different, making the latter process larN;er than the former by the factor g 21 Jl2. If we write Ute eigenvectors, eqns (9.27), as (~ ~j ; j=l. 2. 3, 4 (9.28) -84(8 .29) and this may be 111verted tc give f!l5=a;Xi rp~={Ji .Xi rp5=7; •x; ,5=6; •x; (9.30) where we have taken a.i to be real by convention (this convention has already been imposed in the expressions given above for the eigenvec- tors) . If we write the SO( 10) Yukawa couplings as A(16 ·16)·126+B(16·16)·10+h.c. (9.31) (BOte that A and B may be taken to be real since we are considering the case of no intrinsic CP violation), we find the neutral fermion mass matrix to be (9.32) '1lW; is rendered real by working with the field N'L related to NL by (9.33) tilling tJlis we ftnd the effective SU(5) Yukawa couplings to be (9.34) • ..te ht!re that the Clebsch-Gordan coeMcients for the coupling of the C& 1. VS) iD a 50 to 10r ·101 are different from those for the coupling of -85- the (3 , 1. 1/ 3) in a 5 to the same quantity . The former case has the coupling (9 .35) whereas ·the latter case has the coupling (9.36) The couplings of the Xi to 51 · 101 do not appear in the diagrams of a X.i decay through X' or Y' exchange and therefore are not relevant to our calculation. Thus, in writing down the relevant Yukawa terms for these processes, we will ignore these couplings. In terms of the Xi's the Yukawa couplings therefore read (referring back to our eigenvectors, eqns (9.27), we write a J·=aJ· f'J] R.=bJ· ""'I J·=c J· and o J- =e'n.t-'r~ ....,. where a·J b · c· and d·J are all I I I ] l ] real) : (9.37) The couplings of the X' and Y' vector bosons (the baryon number violating vector bosons that are in S0(10) but not in SU(5)) are [33] (9.38) and (9.39) Note that the sum af eqna (9.36) and (9.39) is invariant under SU(2)L. -86- As discussed 1n chaper 8, in the hmiL of va11i~h.ing "h maE~ the be ~'yon asyrru11ety generated in free decays will vanish in this S0(10) model since, in this limit, there is an unbroken charge conjugation operator. Vve need therefore consider only those diagrams which involve an NL . There are two such diagrams as shown in fig . 9.1. Note that, since the X ' and the Y" constitute an SU(2)L doublet, the diagram of fig . 9.1a can be obtained from that of fig. 9 .1 b by an SU(2)L rotation; therefore, the contribution of the two diagrams are identical (this is also verifiable by explicit computation) . The weiiht of each diagram is ~=3g 2 (.Ac; +e """'"14Bd.; )(Aft-a; +c; J+e -in/ 4 Bd;)- (9.40) and its imaginary part is Im0=-3g 2 ~a.; d.; (9.41) For the scalar, Ji.4, we he.ve (9.42) To compute the aYerage baryon number generated in the decays of the light scalar we must divide t.he above results by the full (Born) rate for its decay as noted in eqn ( 4.1 ). The total rate for the free decay of the XJ is proportional to (here, u =mN I m;) -87- x.J --~-- x.J -~--- (a) x.J -~-- x.J --~--- (b) Fig. 9.1: Scalar decay with vector exchange diagrams in the "primordial" 80(10) model that can give rise to a baryon asymmetry (for j =4) . Only diagrams that involve NL may contribute. -88- (9.43) where u < 1 and, on the left hand side , the term proportional to ~ 1-u 2 ~ is that due to the Born graph involving NL in the final state (see eqn C-15) . For u >1 this term is absent and the Born rate is proportional to (9.44) Both A ·and B are presumably small (at best, B-~lmw and A-gmNim 41 where 7n.t is a typical mass of a vector boson that becomes massive when the transition SU(4)®SU(2)L®SU(2) 9 .... SU(3}®SU(2)L®U(l)R occurs) ; therefore, for large e, t.he Born rate is proportional to (9.45a) for u<1 and to (9.45b) for u>l. It follows that the ratio of ImO to the Born rate is _r1 B 12 a A 11 u<l (9.46a) a>l . (9.46b) -89, In computing the everagP baryon number produced in the free decays of these scalars we must multiply this resalt by t.he baryon number factor -1 and by the difference, lm[Isv(v, u)]-Im[Isv(v, 0)], where the momentum space weight, Im[Isv(v , u)] is given in eqn C-13. This gives for u <1 (where v =mx·l ms), and (9.47b) . 2 for u>l. We also know that Mx-g <1/124 >; thus v 2 -~. H t is large enough so that v is also large, then we can make a Taylor expansion of ~B in powers of 1/ t. Keeping only the lowest order term we have 1 1 B ~ 2\2 AB~---Gf-R[l-(1--u I] 2 6.1r t A ~-~ 2] 12 3 (9.48a) for u <1 (where, in t.he last equality, we liave used the fact that u =AI -JX), IIIld (9.48b) for ">1. We can make a rough estimate of the maximum MJ one can expect in tbis IDOdel as follows. Stability of t.he effective potential requhw u.at. if lU1 then (A. B)~ and, if ~ 2 • then (A, B)~'A [18]. -90- Although A and B are bounded above in the~e two cases respectively by g and >.., they are not bounded below. Thus, for fixed A, we can vary u from 0 to ,._,.g/ v'X in the first case, and from 0 to v'X in the second case. In the :first case since, for fixed A, M3 is monotonically increasing as a function of u in the region O<u <1 and is monotonically decreasing for u>l and, since the maximum permitted value of u is g 1 ....IX> 1, it is clear that M3 is maximized by choosing u-1. The maximum choice for A under these assumptions is A-g 2 giving a maximum liB of 6B-3a/ 2~ 2 . In the second case, for fixed A, the maximum value of u. is v'X; thus, for X<l, the maximum of l:lB is obtained at u-..JX., while for A>1 the maximum is obtained at u -1. However, for A> 1 no baryon asymmetry will be generated since then the mass of the J-L4 will be greater than the temperature at which the SU(4)®sU(2)L®SU(2)R .... SU(3)®SU(2)L®U(1)R phase transition occurs and so there will be no CP violation in its decays (this is independent of the statement that perturbation theory may not now be valid in the scalar sector and the arguments that we are using here will then probably not be v~d). Thus we want A.<l (we are of course assuming that g 2 <1). If we saturate this bound we find that MJ <31 Bn-~ 2 . If g is very small there can be a substantial difference between the results of these two cases. In practice, however, we have g 2 / 41r-l/ 40 and hence g 2 -11 3. The value of AB is an upper bound for the value of the baryon number to photon number ratio that can 'be produced in the context of a grand unified modeL In fact, if we ignore the dynamics contained in the Boltzmann transport equations (2.1) and (2.2), then the value of nBin 7 is related to MJ by a statistical factor that is generally O(Nx/ N) where Nx is the number of bosons participating in the free decay process and N is the total number of particles with mass less than that of the relevent -91- decaying bosons [ 12, 19]. In a grand unified model this factor has the potential of being quite small. It follows that (for nBin 7 -10-9) we can have a maximum £-104. This in turn gives a minimum value <126n>-1 11 if we assume the minimum value <54n>-10 15 which is required to keep the proton slJ..fficiently stable . Thus we have seeq., in a rather detailed example, how the breaking of a charge conjugation symmetry may lead to the generation of' a baryon excess. The magnitude of this excess depends upon the scale at which the breaking of the charge conjugation symmetry occurs. In the present example the magnitude of this scale is proportional to the mass of the NL. In the limit mN-+0, 6B vanishes. -92- PART II -93- 10) llass llatriees in SU(5) The phenomenological mass relations [39] ... m., 1 fnci ---10 rn. mj6 (10. 1) ( 10.2) must be reproduced in any viable grand unified model. Since both sides of (10.1) are renormalized in the same way, it is a. relation valid at all scales and, in particular, at the grand unification scale ; ( 10.2) is a relation valid at the grand unification scale. A somewhat weaker set of relations (weaker in the sense that use of the renormalization group may not be a valid procedure) may be obtained from current algebra as ( 10. 3) at the grand unification scale . It is the first of the relations in ( 10.3) that one is most insecure about; nonetheless, these relations are consistent with (10.1) and (10.2) . In an SU(5) model it is rather easy to institute the relations m;/ ms ~s and m..rff'nt,~l in a natural way by having the 11- family obtain its masses solely through coupling to a 45 of Higgs and having the -r family get its masses through a 5 of Higgs. However, to incorporate the relation ( j 0.1) in a natural way is a trickier business. Nonetheless, it has almost been done with the following choice of Yukawa terms [39] (the Higgs representations are three 5's~ 5g, 5'n. 5"y and a 45y) : +fD(10t·102)+E(10s · 10s)~·5'n+F( 10 2 · 1~) · 5 ''n . (10 .4) -94- The naturalness of these terms is maintained by several U(l) 's which must be broken softly (i.e ., by terms of dimension ~ 3) in the Higgs potential. The latter fact then allows for there to be calculable corrections to the Yukawa terms ( 10.4) and hence the possible inclusion of terms that do not have the required form. The predicted mass relations may there by be altered . The mass matrices obtained from (10 .4) are schematically M-vs= ~ A'C OJ 0 ( 10.5) 0 B 0 A' OJ (10.6) Mwpton = A -3C 0 0 0 B 1 (10 .7) where, for brevity, we have absorbed vacuum expectation values into the definitions of the couplings and we have omitted CP-violating phases. These mass matrices yield the desired mass relations under the unnatural assumption A ~A' (10.8) and the "fermion-mass-hierarchy" assumption B>>C>>A. ( 10. 9) One may also obtain a prediction for the magnitude of the Cabibbo angle : -95- (10.10) The details of such calculations are discussed below in the context of S0(10). -96- 11) Mass Jlatriees in A Viable 50(10) Model To construct a set of SO( 10) Yukawa terms that behave like those in ( 10.4) when restricted to the SU(5) level we proceed as follows [35,36 j . To reproduqe the (~ · 10 2)·45n term in (10 .4) we must use a 12~ coupling to (1~· 16~s. The terms [A~ · 101+A '5t · 102+B~ · 103 }5n+[D10 1 · 1~+E10s·l0sl5 'n can be easily obtained by coupling 1 6 1 · 1~ and 1 63 · 1~ to the same complex = 10n ( 10 1 +i 1~) of Higgs. The relevant SU(5) decompositions are (11. 1) 126n=l+B+10+15+45+50 ( 11. 2) 16=1+5+10 ( 11. 3) and (Note that we have emphasized that we have given the decomposition for a complex lOn in (11.1) by having 5~5 . ) To assure the coupling ( 1 0 2 ·10~) · 5" as in (2.4) without a (52· 103 ) ·5n or a (5~r 102)·5n term, we couple 162·165 to a 126s: ( 11.4) The Yukawa terms of this model are, thus far, ( 11. 5) If we now assume that, for a range of parameters in the Higgs potential. the 1262 has its vacuum expectation value purely along its SU(5) 45 direction and that the 1263 has its vacuum expectation value along its SU(5) 5 direction, -97- <1262>-~5H2J <126s>-5Hs ' ( 11.6) then the predictions of (10.4) for charged fermions are reproduced. This would not have been possible without choosing 126s different from 12~. When the Yukawa terms (11.5) are expanded with the decompositions ( 11.1) ,_( 11.2) and ( 11.3) we obtain the following terms which contribute to the neutral fermion mass matrix when the Higgs fields get vacuum expectation values: ( 11. 7) If we view the CP conjugate of the SU(5) singlet fermion as a right handed partner for the neutrino, then we see that the terms ( 11. 7) give Dirac masses to the neutrinos of the same order of magnitude as those of the charge 2/3 quarks. H, however, in addition to the Dirac mass, m, mentioned above, the neutral singlet fermion has a Major ana mass, M, then the neutrino mass matrix has the following form in the (vL ,NL) basis* [29.27]: [! :;;]. ( 11.8) If M >>m, then the eigenvalues of this matrix are approximately M and m(m/ M). Thus, one eigenvalue is naturally very large relative to the charge 2/3 quark mass and the other is very small. It is the latter eigenstate which we identtly with the garden variety neutrino. Such an eigenstate would be primarily vL with a small amplitude (proportional to m 2/ M) for helicity flip into VR ( =(NL )CP). •Also see chapter 8. -98- The mechanism described above is effected in this 80(10) model by ~ coupling the .relevant combinations of fermion multiplets (in the case of eqn (11.5). the combination e 16 1 ·1~+/ 16s·163 ) to a 126 1 which obtains a superlarge Yaeuum. e%pectation value along its SU(5) singlet direction . Further, to preserve the predictions of eqn (11.5) along with the assumption ( 11.6), we assume that the 1261 has vacuum expectation values only along its SU(5) singlet and 5 directions• : ( 11. 9) The complete Yukawa couplings are now (11.10) These couplings are natural, the naturalness being maintained by two global U(l) symmetries which will be explicitly broken down to discrete symmetries in the Higgs potential. We call the charges associated with these U(l) symmetries X and Y. Their values for the representations present in this model are summarized in the following table: 161 1~ 16s 10 1261 1262 126s X -3/2 1/2 -1/2 1 -1 1 0 y 1 -1 0 0 0 -2 -1 •u <581 >-o. at~ maa relation fallows that i8 phenomenologically unacceptable . This .acuum expectatian value i8 al8o required fer naturalness reasons {33,36]. -99- When the Higgs acquire vacuum expectation values, these phase symmetries will be spontaneously broken. Since they are not gauged, massless Goldstone-N ambu bosons will result. To avoid their presence we have to break X and Y in such a way as to preserve the naturalness of the Yukawa couplings. Remarkably enough this can be done through the following property of the 126 representation: the fourfold fully symmetrized product ( 12ot")s contains one SO( 10) singlet. Thus, we require that the Higgs potential contain terms like (11.11) The first term breaks X to a discrete symmetry mod 4; the second one brea.l<s Y to another discrete symmetry mod 8. These two discrete symmetries suffice to maintain the naturalness of the Yukawa couplings while avoiding the problem of massless bosons . The remaining Higgs selfcouplings are selected so as to honor the remaining discrete symmetries. The terrns, at the SU(5) level, in eqn (11.10) which are relevant for the computation of the fermion f!lass matrices are: for charged fermions and (11.13) for neutral fermions. We will discuss here only the charged fermion mass matrices. (The neutral fermion mass matrices are discussed in [33] and [36] .) The first -100- thing worth notmg is that the unnatural relation (10.8) is natural in this S0(10) scheme. This is obviated by comparison of (10.4) with (11.12). If we now write (11.14) and define ar=R, br=T, dq=Q, cs=S, then the charged fermion mass matricies are: charged -1/3 quarks: M -11s- Rei-6 0 i-6 SeiX 0 ~ Tei-6 0 ro (11.15) charged 2/ 3 quarks: M=r.' 213 6 Pe"~ 0 0 QeiJ.' QeiJ.' Vei( (11.16) charged leptons: M:=r~·~ Re'~ -3SeiX 0 T~'~l ( 11.1 ?) -101- By a suitable redefinition of the fermion fields (discussed below) the phases can be removed from these matrices . (The phases certainly reappear, for example, in the charge current sector of the theory.) For the mass matrices with the phases removed we write respectively M-113 , M2/ 3 and JJ, . The eigenvalues of both Ji-v 3 and ii, are easy to compute since they are both block diagonal. The eigenvalues of M213 are easily computed in the limit V>>Q>>P, while assuming one eigenvalue to be 0( V) and the remaining two to be <<V. In this limit and in the limit T>>S>>R for M_113 and 11, we may identify the fermion masses as (11.18) (11.19) and (11.20) The mass relations (10.1), (10.2) and (10.3) follow from (11.18), (11.19) and ( 11.20). These expressions can be used to solve for the parameters Ulat appear in iJ -u:s. 11219 and .M, in terms of the fermion masses. -102- M-us. M215 and M, are real and symmetric and are diagonalized by orthogonal matrices which we denote respectively by U -vs. U215 and Ut: (11.21) (11.22) (11.23) U _113 is easily computed to be U-1/S~ ~ -[~ 1 0 0 0 1 ( 11.24) Similarly for Ut : ( 11.25) U219 is somewhat messier but nonetheless straightforward: -103- ( 11.26) 1 If we write the column vector, in family (weak eigenstate) space, of charge -1/3 left-handed quarks as L_ 113 and that for the right-handed components as R _113 , the mass term in the Lagrangian is before any phase redefinitions of the fields have been made . We can redefine each of the fermion fields by an arbitrary phase (11.28) where h-vs and R-vs are diagonal matrices of the form e L....l/3= ia 1 0 0 0 0 e iot2 0 0 e (11.29) ia3 and (11.30) We want rwt ""' ,... =L-vaM-vsR-t;s. (11.31) -104- or . - M-v3-~113M-u3R-V3 · ~ ')\ ( 11 ~ -~ · -) Note that if this is true then it follows that (11.33) although one might wish to choose Lz and Rz different from L- 113 and R- 113 for reasons of convenience. Eqn ( 11.32) results in the following: ( 11.34) Similarly, if we write e "71 0 ~s= 0 e' 0 0 72 0 0 (11.35) e""s and Rets= 0 (11.36) 0 then from ~ . Ma~s=IetsMvsRets (11.37) c$+911r2=0 6+912r1=0 #J.+9'2/s=O #J.+V's/e=O ("+9'3/s=O ( 11.38) we get -105- Of the phasPs a 11 o: 2 , o: 3 • {3 1 , {1 2 and (3 3 c.nly five are independent ire sc rd:' e:-; effecting changes on M-us i3 concerned (in particular the transforrr:.::.tjon a.1 = a.2= as= (3 1=(3 2 =(33 =0 is sterile) . Thus, we choose a. 1 =0; and similarly far charge 2/3 quarks we choose ')' 1 =0. Thus, from eqns (11.34) we are left with one degree of freedom and from eqns ( 11.38) we are left -v\i+..h no degrees of freedom. The left-handed charged current coupling io the usual (SU(2)L) Wl boson is (11.39) or, from eqn (11.27) and its analogue for charge 2/3 quarks, j; =L~/S a#-J..l/3~1/sL-t/3 . ( 11.40) We can write L- 1/ 3 and 'l2/ 3 in terms of the mass eigenstates: (11.41) Thus, in the mass eigenstate basis, we have ( 11.42) The matrix that appears in this expression to be acting on the lefthanded charge -1/3 mass eigenstates, -106- ( 11.43) is explicitly (11. #) where ( 11.45) and The quantity a is completely determined by eqns ( 11.34) and eqns ( 11.33) to be a=x-19--2p+o+( . (11.46) The remaining unconstrained quantity in eqns (11.34) (i.e., {33) cru.1 be used to set b=O. (11.47) -107- Thus the matrix (11.44) has a CP-violating phase. Eqn (11.44) is not in standard Kobayashi-Maskawa form [40]; however, it can be rendered so, if necessary, by a phase redefinition of the left-handed mass eigenstate fields. -108- 12) The Beta Function in This SO(lO) llodel In general, the symmetry breaking in a grand unified mociel based upon a gauge group G will proceed in several steps before arriving at the low energy model G1 t®SU(3) where G1t is the electroweak gauge group aud is at least as large as SU(2) 1 ®U( 1 ). After the i th step there will be a remaining unbroken gauge group ~ which is a valid symmetry up to a scale 0(2M) where M is the mass of a gauge vector boson which is in ~- 1 and not in ~ (note that we are writing the sequence of symmetry breaking as G~Gr-..G2 -t · · · .... ~ .... · · · ~[G1 t®SU(3)] ). In the SU(5) model there is only one possible pattern of symmetry breaking compatible SU(3)®sU(2) 1 ®U(l)y. with low energy phenomenology: SU(5) ~ However, as soon as one considers larger groups, the possible patterns of symmetry breaking compatible with the world as we know it become more numerous. Such models offer some hope of partially filling the "desert" region between 300 GeV and 10 1:> GeV which is present ih the simplest grand unified models based on SU(5). In a generic SO( 10) model there are a number of possible symmetry breaking patterns as illustrated in :fig. 12.1 [38,41]. Of these we consider Ll-tose with the intermediate scale gauge group SU(4)®SU(2) 1 ®SU(2)R to be of particular interest because of the presence of SU(4) as a generalized (Pati-Salam) color group with lepton number as the fourth color [ 42]. In this chapter we discuss the running of the various Yang-Mills couplings for the symmetry breaking scheme of Ule model described in the preceding chapter. We compute the renor- malization of the m.-val'"" mass relations down from the unification -109- so (10) . ~ l --------SU(5) e U( 1) SU(4) s SU(2)L® U( I) SU(4)8 SU(2)LS SU(2)R SU(3)8 SU(2)L e U(l)y Fig. 12. 1: Possible paths of symmetry breaking for the group 80(10). By suitable choice of Higgs representations any given sequence of symmetry breaking may occur that is consistent with the ftow of the lines in this figure . -110- scale. We consider the effects of running the coupling~ nn the value of sin 2~ It' and on the lifetime of the proton. Furthermore, since this model is only temporarily free, we compute the position of the Landau '~singularity" . We call" the SU(4)®SU(2)L®SU(2)R, scale m 2; at which SO( 10) breaks dovv-n Lu the scale at which SU(4)®SU(2)L®SU(2)R breaks down to SU(3)®SU(2)L®V(l)y is called m 1 ; and, the scale at which SV(3)®SU(2)L®V(l)y breaks to SU(3)®V(l)EAI is called mo. mo is related to the mass of the W boson through mo ~2 Mw. m 1 and m 2 are similarly related to the masses of the vector bosons that become massive at those scales. With forethought we choose to normalize the generators of SO(lO) to 2 in the 16 representation: Tr[T(16) 2 ] = 2 . (12.1) This will give us the expression for the electric charge operator, (12.2) where TM,0) is the diagonal SU(2)L generator (the superscript indicates that it is normalized as embedded in SO(lO)) andY is the V(l)y generator as embedded in S0(10), so as to conform with the traditional SU(5) expression. This latter point is assured if we normalize the generators of any SU(n) subgroup of SO( 10) to 1 /2 in the n representation: (12.3) -111- By trbcking the three low energy couplings up to the scale m 2 at which S0( 10) first breaks, we will geL three expressions that depend on the parameters m 1 , m 2 and a 10 (m 2 ) (a 10 is the SO( 10) coupling squared divided by 4n). These can be determined by using as inputs the values of a.3 , sin219 w and agJJ at '1ng. The qualitative behavior of the couplings for a simple assumption about scalar thresholds is shown in fig. 12.2. We work with beta functions to lowest order in g and treat all mass thresholds in the theta function approximation. We write g 10 for the SO(lO) coupling (a 10 =gf0 / 4n); similarly we write 9• for SU(4), gu for SU(2)L, g 2R for SU(2)R. gy for U(l)y and e for U(l)EM · We use a similar notation for the quantities b that appear in the respective beta functions . In general a Yang-Mills coupling g runs according to (J.L<m) [43] ( 12.4) assuming that there are no mass thresholds between JJ. and m and that the coupling remains perturoative in that region. The only difficulty in running, say, what starts as the SU(3) coupling up to the unification scale is in determining the boundary conditions applicable as one goes from one reigon to an adjacent one. With the normalizations in eqns ( 12. 1) and ( 12.3) we have (12.5) -112- I SU(3) ® U( I )EM I I lsU(3l®S~U(~SUI4l®SU2\_®~ I : I I I I I I I I I SO(IO) 1 I I I I I I I Fig 12.2: The scaling behavior of the couplings in this 80(10) model. -113- and gu is continuous across the threshold at m 1 . In eqn (12.5), Y' is a mixing angle which specifies which linear combination of the SU(2)R diagonal generator TsR and the diagonal generator T us in SU( 4) becomes the U(l)y generator. Similarly, 77 specifies which linear combination of TSL and Y becomes Q, and f is a constant which normalizes the electron charge to -1. The conditions that Y=O for the SU(5) singlet fermion and that Q=O for ,the SU(2)L doublet neutrino give Sin9' = -Yl ; COS9' = sin77 =~ ~; ; coS?'} =~ ; ( 12.6) f=VJ. The expression for, say, gy 2 is obtained from the following argument (which can easily be generalized to cases more complicated than the linear combination of U(l)'s that we review here [9]). Say that we have vector fields A and B coupling respectively with generators X and Y and coupling constants g and g' : g AX+g'BY. Then, if symmetry breaking leaves the combination aX +bY unbroken (where a 2 +b 2 =1), we have gAX+g'BY=gC(aX+bY)+gD(cX+dY) where D is the vector which couples through the broken generator eX+ dY (this generator is not necessarily orthogonal to aX +bY), 11 is the coupling constant for the vector field C and g is that for the vector D. This gives -114gA=gaC+gcD and g B ='gb c +gd..D I The orthonormality of A and B then gives and from which follows It is from this that the last two boundary conditions in eqn ( 12.5) were obtained. (If the normalization conditions ( 12.1) or ( 12.3) are changed the conditions in eqn (12.5) change accordingly. For example, if we choose Tr[T(16) 2 ]=1/2 then we getg 4 (m 2) 2 =g 10 (m 2) 2 /4 .) Using eqns (12.4), (12.5) and the definition of the Weinberg angle [8], ( 12. 7) where ce=5/ 3 (again, the specific form of this equation depends upon the normalization conditions), we get the following three relations: -115- (12.B) and (1 + c'!-W)-r+ [b2L ( 1- W)+c2by ]y (12. 10) where ( 12. 11) and b 2L is the SU(2)L value for b in ( 12.4) between m, and m 1 and b '2L is the value between m 1 and m 2 . In calculating the various b 's we use the expression obtained from the one-loop beta function [ 43], (12.12) where r =rank of the group, and the second index of the representation r, / 2 (r) is given by /e(r) =7r[T(r)2] . r (12.13) -116- (Tables of / 2 for the simple compact Lie gr·oups are given in [44].) The summations in eqn (12.12) are over all relevant lefl-handed fermion representations f and real scalar representations s; complex scalar representations are counted twice since each complex scalar field contains two degrees of freedom; this comment also applies to doubled real or pseudoreal representations. Adj stands for the (vector) adjoint representation. To consider the scalar and fermion thresholds it is convenient to consider the SU(4)®SU(2)L®SU(2)R decomposition of the 10, 16, 54 and 126: 10=(61 11 1)+( 1 ,212) (12. 14) 16=(41 21 1)+(41 1. 2) ( 12.14a) 54=(11 1.1)+(20, 11 1)+(6,2,2)+(1.313) (12.15) 126=(61 1,1)+( 10,31 1)+( 101 113)+( 151212) (12.16) In order to decide at which of the scales fno, m 1 or m 2 a given Higgs obtains a mass, we adopt the following ansatz: a scalar multiplet gets the largest possible mass consistent with its vacuum expectation value and the symmetry present at that scale. Thus, a multiplet which gets no vacuum ~xpectation value (such as the (6, 1, 1) in the complex 10) will have a mass of O(m. 2 ). This assignment assures that all baryon number violating scalars will have a mass of O(m 2) . Certainly other assignments may be made since the only necessary constraints are that the SU{3}®SU(2)L®U(l)y theory be consistent with phenomenology at accessi- ble energies and that the proton not decay too fast. Our choice is the one with the fewest low mass scalars. -117- As m.sntion.ed above, in the complex 10 of Higgs the (6, 1: 1) ho.s a mass ~ m 2 ; the (1,2,2) will have a mass ~ mo. Since, by assumption, <54>-m 2 , we put all of the masses of the (real) 54 ~ m 2 . The representa- tions 12~ and 1263 have vacuum expectation values 0(1'no) (in the 1262 case it is along the SU(5) 5 direction and for the 125:3 it is along the 45 direction). In each case the vacuum expectation value is contained in the (15, 2, 2). The SU(3) decomposition of the 15 of SU(4), 15=1c+3c+3c+Bc, then shows that we will have a mass for the (lc, 2, 2) of O(mo) and masses for the remaining parti~les in (15, 2, 2) O(m 1). The other particles in 1262 and 1263 may all be given masses O(m 2) . For the 126 1 we have the addition of a vacuum expectation value along the SU(5) singlet direction D(m 1). The SU(5) singlet is contained in the (10, 1, 3) which will therefore have a mass"' m 1 . These results are summarized in table 12. 1. With this information we can now compute by, b2R, b2.L. b 3 and b 4 (and b 10). With the normalizations ( 12.1) and ( 12. 3) we get, for F families of ferrnions, 20 16n-2b =~. 16n2b r=~+~· 3 5' 2L 3 3' (12.17) For F=3 these give -118- scale scalars I (1. 2, 2) l 2 I 6 i I i multiplicity I mo (lc, 2, 2) fc (15, 2, 2)~ m.l m.2 (1, 2, 2) 2 (10, 1, 3) 1 (15, 2, 2) 6 10 2 54 1 126 3 Table 12.1 Scalars with masses less than or equal to the respective scales m 0 , m 1 and m 2 . The superscript c indicates that the SU(3) representation is being specified. In the remaining cases (except for SU(4}®SU(2)L®SU(2)R transformation properties are specified. case of m 2 , the SO( 10) representation is given. m 2) the For the -11916n2by=24/ 5; 16n2bu=-B/ 3 ; (12.18) Thus, in this scenario, the only couplings which diminish with increasing r;f are the SV(3) and SU(2)L couplings in the region from '1no to m 1 . The predictions for the ratio of the charged lepton to that of the charged -1/3 quark are valid down to the scale mb where SV(4) is broken. To determine m_ 115 / mt at m 0 we must renormalize through [4,45,46] (12.19) where F is the number of families. From eqns (12.5) and (12.7) we find that ( 12.20) If we write m_v 3 (fno)/m,(f71.o)=R(F) m_ 113 (m 1)/m,(m 1), then, using eqns (12 .4), (12 .6) and the two preceding equations, we find R(F)= [ 1-Brrb 3 a:3 ('1no )(y -z) p21 (33-4F) x[l-BnbyagJI(m, H ~ w-; )- ~ 1+ Hy-z)] 3/4F . ( 12.21) Given a:E.V(7'71o) and a:5 (7no ), this expression depends on F through by and b 3 and through the exponents. Neither y nor z depends upon F. Aside from the explicit dependence upon W, R depends upon W implicitly through 'II and z. In this expr ession for R we are neglecting scalar thresholds. - 120- To estimate t he prot on lifetime in this sch eme W f! use the r esult fr om. BEGN [ 4] for the S U(5 ) rnodel, r(p 4lepton +rmy )=1'rp for three families . For rough comparison we would wish to replace O..gum by_o.. 10 (m 2 ) an d replace Mx by m 2 . Thus the ratio of the BEGN estimate to the present one is (12.23) Using t he SU(5) result Mx~4x 1 0 14 Gev and o..gwn~l/ 40 , the requirement that the proton not decay too fast implies that ( 12.23) be greater than one : -r1 (S0(10)) _ {m2) 4[cx 10 (m 2)]-lO/? Tp (SU(5)) 2x1061 ( 12.24) >1 · It is unclear whether the relevant parameter for a perturbation expansion is ex or cxTr ( T 2) (where T is the generator in the representation relevant to a vertex in the diagram being considered) or any of a large number of other possibilities. Thus , to ask where, for a non- asymptotically free theory, the expansion parameter becomes nonperturbative is ambiguous. However. a measure of where perturbation theory fails that is independent of these questions is the position of the 80-Called Landau singularity. This is the value of m in eqn (12.4) for which J(m.)-e vanishes. In the present model it is given by -121- (12.25) It is a matter of personal opinion whether one wants mL to be greater than, less than or of the arder of the Planck mass, mp; ''aesthetic" arguments can be made for each point of view. The resolution of this question must certainly wait for a theory that includes the effects of gravitation at the quantum level. What seems clear nonetheless is that asymptotic freedom is by no means a necessary condition to impose on a grand unified model. The results of these calculations are given in figs. 12.3 through 12.6. (For these calculations, in computing mw we have taken into account its variation with sin2 (~r) given by ( 12.26) This has a negligible effect on mL, but does affect rP (SO{l 0})1 Tp (S U{5}) somewhat. In all calculation~ we have used aE.V(2mr )~1/ 128.) In figs. 12.3 we have ploted m 2 and mL versus sin2~ w· What is notable about these graphs is that the presence of the Landau "singularity" gives an upper bound for the Weinberg angle in each case; this upper bound is determined by the condition m. 2=m.L. However, it is necessary to note that it is not clear that the Landau "singularity" is a true physical singularity as opposed to, for example, be~ just a relic of perturbation theory. Thus, it is not clear whether this upper bound is physically significant or not. In figs. 12.4 we have plotted m 2 and m. 1 versus sin21J,. What should be noted from these graphs is that with our symmetry breaking scheme we have a lower bol.md for sma,,: the value for which m 1 =m. 2. The inclusion -122- of scalars in the calculation generally rfl1ses thi~ value (in thP. ab:::;ence of scalars the value is -0 .20 [40]). Independent of this is the fact that the value of sin~w becomes larger as the difference between the scales (at which the symmetry breakings occur) grows . In figs. 12.5 we show lhe values of R(3) and R(4) as a function of sin~, . These values are somewhat larger than those in the absence of scalars [48 J . In fig . 12.6 is plotted the values of 'Tp(S0(10))/-rp(SU(5)). It is clear that considering a model that is more complicated than the minimal SU(5) scheme can increase the proton's lifetime by many orders of mag- nitude. It is worth noting in this context that the actual value of the proton lifetime is very sensitive to the values of the various input parameters . Thus, a calculation that uses only the lowest order beta function cannot be used to obtain precise results since higher order corrections can effect the results significantly. Furthermore, uncertainties in the location of scalar thresholds can also effect the results• ( 47]. The results of this chapter show that one must be quite careful in making catagorical statements about the specific numerical predictions of grand unified models••. ~ is independent of the possibility that an 8Ill!latz different from the one that we have chosen will effect the lowest order beta function s~nificantly by having many more low JD8SS scalars. •ecalculatioru~ similar to t~e presented here have recently been given in [ 49]. -123- lo~' m~ IOU> lo 11 to' 11 to'~ lo'.. Fii. 12.3a: m~ and m.L as a function of sine,, for as(m..)=0.12. -124- c(.,(Wl,\ : .13 102. 1 GeV 10'2.0 m~ \o '" to'' to'., Fig. 12.3b: m, and mL as a function of sin21Jr for as(fno )=0.13. -125I 1~ .0 m~ GeV IO~ IOICI lo'' to'~ mAt \o'" fi&. 12.3c: m.a. and m.L as a function of sine,, for as(m.o )=0.14. -126- toa.' m~ Gev 10~ 0 1o'' \o'" Fig. 12.3d: m~ and mL as a function of sin~r for as(f'11o)=0.15. -12710' 1 1o"' GeV to'" lo'~ to 11 \010 Fig. 12.4a: m. 1 and m. 2 as a function of sin2, , . for as(77l.o )=0.12. -128- to'l GeV to'._ m~ oiJ.,W...): .13 \o's Fig . 12.4b: m 1 and m. 2 as a function of sine, r for as(7'Tl.o )=0 .13. -129- GeV Fig. 12.4c: m. 1 and m. 2 as a function of sin2~, for as(?n.o)==0 .1 4 . -130- to fl m4 to'" GeV ol,(tm.)= .15 lo'f to'' to'~ Ion. to" Fig. 12.4d: m. 1 and m. 2 as a function of sin~, for as(ffl.o )=0.15. -131- ~): .12 Fig. 12.5a: R(3) and R(4) as a function of sin21)w for as(mo)=0 .1 2. -132- Fig. l2.5b: R(3) and R(4) fli a function of sin~r for as(fno )=0.13. -133- Rt~) R(l) Fig. 12.5c: R(3} and R(4) as a function of sin2 ~r for as(mo )=0.14. -134- ~.o L-----1---'---.....__ _.....__ • :l,'fo s' t\). ew Fig. 12.5d: R(3) and R(4) as a function of sin2~, for a.s(f'no )=0.15. -135- ~tl sotto)) 1010 ~,tSUlS)) lO~ \0~ «-lrm.)= .15 to" \ 1o" 1o' lo.. lo\ to" '\ ctst-wrt.)• .12 lo ,~~~----~--~--._----~------~--------.2. .. 5 '"~ ew Fig. 12.6: 'Tp(S0(10))/;p(SU(5)) as a function of sin2 ~, for the various values of as('m.o ). -136- Appendix A) Notation for Fermion Fields We describe spin-1/2 fermions by two-component fields of definite chirality: left-handed fields are denoted by 'f/IL and right-handed fields by YlR · For massless fermions. chirality and · helicity are equivalent and the two chirality slates are independent. Only one of the states need therefore be present in a model. For the two-component fields, tf denotes the left-handed antiparticle of l'R• while 'f/1!? denotes the right-handed antiparticle of tL· For fields in which both helicity states are present, parity (P) serves to interchange L and R components, while charge conjugation (C) interchanges particles with antiparticles, according to : where a 2 is a Pauli matrix. These transformations are summarized in fig . A.l. Note the important feature that while the separate operations of C and P interchange L and R components, the combined CP transformation does not modify the helicity state . Hence while the definition of individual C and P transformation properties generally requires the presence of both L and R states, CP transformation properties may always be defined for massless particles with only a single helicity state . The two-component fermion fields may be collected into a fourcomponent Dirac spinor describing a fennion of arbitrary helicity: V=~~l · -137- p ~L~4------------~• ~R c c C ~y,c '¥L~c-------p------~~ YR Fig. A. l: Illustration of the action of the operators C, P and CP on the two helicity components of a four-component spinor or, equivalently, two independent two-component Weyl spinors. -138lt i:::: convenient to t..a.'ke the Dir~c gamme1 matrices which act. on t.hi s ~ pi- nor in the Weyl re:..1resentation: 1" =[~ ~] ' 0 "l= 1ai -ci'] 0 , where ai (i=1, 2, 3) are the usual Pauli matrices . (This representation differs from the more usual Dirac representation simply by the interchange, -y0 ~r.) The kinetic energy term in the fermion Lagrangian is given by Fermion fields for which both helicity states are present may give a Dirac mass term: If only one helicity is present, say 1/IL, no Dirac mass term may be con- structed, but a Majorana mass term is still possible: Here the charge-conjugate four-component spinor ¥ is given by -139- For a fermion field with only a single helicity state, it is sometimes convenient to define a four-component Major ana spinor, in terms of which the Majorana mass term becomes ';. ~M~.M. Note that fields with Majorana mass terms may not carry any U( l )Q charges since the mass term is not invariant under the global gauge transformation 1/IL .... e'""Q'f/IL. -140Appendix B) The CP Operation in Grand Unified Models The generation of a baryon excess from an initially symmetrical state requires CP violating interactions . In this appendix we discuss some of the properties of the CP operator in the context of grand unified models constructed from a standard Yang-Mills action •. We consider first a complex scalar field Y'(x ,t ). It is necessary to distinguish the field operator ~ from the "fields" Y' obtained as the ex"})ectation values of this operator in particular states. It is the q-number field operator which appears in the canonical quantization procedure; the cnumber field appears in the path integral formalism; we generally work with the latter. The actions of parity (P), charge conjugation (C) and time reversal (T) on a complex scalar field are given, up to arbitrary phases. by: P: Y'(x ,t) -. ~( -x ,t) C: ~(x ,t) -. ~ "(x ,t) ~(x .t) -. ~t(x ,t) T: Y'(x ,t) -+ ~ "(x ,-t) ~(x,t)-+ ~(x , -t) The transformations P and C are represented by unitary operators, which act on ~ just as on rp . T is an antiunjtary operator, which reverses the order of factors in products of field operators . It thus interchanges the bra and ket states in an expectation value and complex conjugates the ~This presumes that we are talking only about compact Lie groups. pact Lie groups the discussion presented here must be extended. For noncom- -141- field rp. The combined operation of CPT on rp(.z ,t) yields rp( -.z. -t) and is equivalent, as usual, to a generalized Lorentz transformation (total 4inversion) . The P, C and T transformations above are modified for particles with spin. Their action on spin 1/2 fermions is outlined in appendix A. Note that separate P and C transformations interchange chirality-states, while the combined CP or T transformations do not. Thus, massless particles with only one chirality or helicity state have definite behavior under CP, but the action of C or P may not be defined . For spin 1 fields, P and T transformations reverse respectively the space and time compon~nts of the polarization vector; they may therefore be considered to "raise" or '1ower" the Lorentz vector index on a vector potential A~. If a simple scalar charge is associated with the :field rp above, then the C operation serves to reverse this charge . When a field carries a non- Abelian charge the action of the C transformation on this charge is more complicated, and again may not be defined. We shall consider a field tran.sforming under a unitary representation r of some simple compact Lie group G that acts as the gauge group for a grand unified model. The analysis presented below can be easily extended to take into account any additional global symmetries of the model. The group G will, for now, be assumed unbroken. A coltiffi.J."'"l vector of fields n, transforms under the action of a finite element of G accordii"lg to 'I'J .... etPcP 'I'J (B.l) where the CJfJ are the group parameters of the transformation and the ra. are the generators of G in the representation r . If the CJa. are chosen to -142- be real, then the 1"' are Hermitian. "Charges" or "quantum numbers" are usually associated with the elements of the Cartan subalgebra or center of G. The elements of this algebra will be denoted H,, where l runs from 1 to the rank of the group . This algebra consists of the maximal set of commuting generators of G and thus generates the maximal Abelian subgroup of G (it therefore may be written as a product of U(l) factors) . The Ht may be rendered simultaneously diagonal by a unitary transformation on the representation spac e {which does not affect the Herrniticity of the 'r'). We denote the vector 7'} of fields in this basis by 7'JD, and the Ht by HP. In this basis, the CP transformation is defined by (B.2) where A is a matrix acting on Lorentz indices, and no sum is taken over j . (For the case of scalar fields A= 1, for fermion fields A=±a2 depending on whether 7'JD is left- or right-handed respectively (see appendix A) and for vector fields A=g JW or g~v; i.e., A lowers or raises an index on a vector field.) The phase aj is, for the moment, arbitrary, but we will show later that it may be taken to vanish. With this definition, the transformation (B .3) becomes for the CP conjugate fields : (B.4) Hence the CP transformation reverses signs of all the charges associated with H'. If a set of fields transforms according to a representation r, the CP conjuaate fields transform according to the conjugate representation :;;, -143- If the representation r is irreducible, then the phase factors e 'i.a; appearing in eqn (B.2) must all be equal, so that aj =a for all j, since one may perform an arbitrary group transformation on ,f before applying CP. The common phase, a, may then be removed by an overall phase redefinition of all fields . We shall usually assume below that r is irreducible; reducible representations may be treated by considering separate]y each of their irreducible components. We have defined CP transformations above in terms of the "diagonal" basis TJD; below we shall consider other bases 1'J obtained by unitary transformations: TJ=Urf. (B.4) The action of CP on fields in this basis is given from (B.2) and (B.4) by (B.5) since CP(ut11)=utCP(TJ). The action of CP on the ·vector A(TJD) • transforming as the conjugate representation f" is (B.6) · where, if A=a2 (A=g~) then A'=-a2 (A'=g~W), so A'A=l. (B.7) The phase fJ must be such that the kinetic term for '1D be CP invariant. The analogue of eqn (B. 5) is -144- (B .B) The form of CP transformations for fields in a representation r depends on the relationship of r to the conjugate representation r . Representations with three basic characteristics may be distinguished [50]: Complex r and r are completely inequivalent (e.g . fundamental representation of SU(n) for n>2); for complex representations the singlet does not appear in the decomposition of r®r . Real r is equivalent to r and there is a basis in which the representation matrices are purely real (in which basis r is equal tor) (e.g . all adjoint representations of compact simple Lie groups); for real representations the singlet appears in the decomposition of the symmetric part of r®r . Pseudoreal r is equivalent to r, but there is no basis in which r is equal to r (e.g . fundamental representations of ~ym­ plectic groups) ; for pseudoreal representations the singlet appears in the decomposition of the antisymmetric part of r®r . All pseudoreal represe n tations have even dimension. For our purposes the basis of greatest interest for a complex representation is that in which the Cartan subalgebra is diagonal; the CP transformation properties of a set of fields transforming according to r are simplest in this basis. For any representation, under the action of the group, 7'J transforms as -145- (B .9) where g is an element of the (unitary) representation r in the TJ basis . It follows that (B.lO) If r is a complex representation and if TJ are scalar fields then aJJ-TJt~TJ is a gaugin~ group (and Lorentz) invariant (we have not yet gauged G; replaces all derivatives by gauge covariant derivatives). If the 17 are (left-handed) fermi fields then i(TJtal-lBJ4r]-(aJ4r]'t)al-lTJ) is a group (and Lorentz) invariant. In both cases the requirement of CP invariance gives fJ=O. Thus (CP) 2 = 1 when acting on a complex representation. Real and pseudoreal representations both have the property that, for an arbitrary basis there is a unitary matrix V such that all of the representation matrices satisfy g •=Vgvt. If we apply this relation twice we find g =V#Vgvtv'l' . Thus the (unitary) matrix V#V commutes with the representation and thus, by Shur's lemma, must be proportional to the identity: V#V=al. Since V is unj.tary it follows that v" =aV. Applying this relation tV\Iice then gives n 2 = 1; thus, the matrix Vis either symmetric or antisymmetric. These two alternatives correspond to the real and the pseudoreal cases respectively. We now discuss these cases in turn. If r is a real representation, then one may choose a basis TJR in which all the representation matrices g are real (orthogonal). The action of the · group ca.IlLot mix the real with the i.rnaginary parts of TJR. Th:...:s, for scalar fields we may choose 1'/R to be real; this gives us the minimal set of fields needed to construct the representation. For fermi fields we are obligated to take an appropriate linear combination of fields transforming as r to assure definite transformation properties under the Lorentz -146- group; i.e., the representation must be "doubled". (There exist an equivalence class of representation TJR connected by orthogonal similarity transformations.) This basis is obtained by the unitary transformation U such that (B. ll) is real; hence (B . 12) so lhat the matrix which effects the equivalence between -r and r is ::<ymmetric (this is related to the fact that the singlet is in the symmetric part of r®r). The TJR is distinct from the basis TJD in which the Car tan subalgebra is diagonal, but may be obtained from it by a unitary transformation UR . In the TJR basis, the action of the CP operator is obtained from eqn (B .5) as (B.13) (For scalars the complex conjugation on the right-hand side of this equation is superfluous.) It is clear that we may choose UR so that each of its elements is either purely real or purely imaginary. It follows that UR u_k is a real (orthogonal) matrix and that (UR UJ} 2 = 1. By an appropriate orthogonal matrix we may diagonalize URU.l; its eigenvalues are l's and -l's . Consequently we may work with a basis in which each 17f is an eigenstate of CP (it is a me1nber of the equivalence class mentioned above) . If, choosing TJR to be real, we act on it a second time ~ith CP vre get (CP) 2 (TJR)=(URU1} 2M'TJR=,f. Thus (CP) 2 =1 when acting on a real represen- tation. -147- A. pseudoreal representation, r, is one for which r is equivalent tor, but there is no basis in which the representation matrices are real. In any basis there is a unitary matrix C such that g •=Cgct (B.14) and Cis antisyrnrnetric (this is connected with the fact that, for a pseudoreal representation, the singlet is in the antisyrnmetric part of r®r) : CT=-C . (B.15) Although there is no basis in which the representation matrices are real there is a basis (with representation matrices gZ) with the property (B.l6) where Z is the antisymrnetric orthogonal matrix [50] -1 0 Z= 0 0 0 0 0 0 0 -1 1 0 .. ·] = diag(i'T2, i'T2· ... ) . (B .17) Note that since Z is related to C by a unitary similarity transformation, and since ~=-1, it follows that c2=-1. If TJ transforms as a pseudoreal representation, then a conjugation matrix C may be defined so that the constraint CTJ=7'J ~ may be imposed as long as this constraint does not violate any other syrnmetr~es of the theory (in particular, Lorentz invariance; this constraint cannot be imposed on the fermi fields of definite chirality with which we work). This amounts to the statement that the minimal number of degrees of freedam for a set of scalar fields transform1ng as a pseudoreal representB.ticn -148- may be chosen to be equal to the dimension of the representation, as is certainly true for a real representation, but definitely not true for a complex representation. The obvious choice for this conjugation matrix is C=C. Thus, if 1'J is a set of scalar fields with this constraint imposed, the usual kinetic term vanishes identically; B~TJt~=B~TJTcT~TJ=O since C is antisyrnmetric [32,51,52]. A normal kinetic term can be formed for scalar fields transforming as a pseudoreal representation only if the representation is "doubled" by taking an additional set of scalars x satisfying x•=Cx and forming the combination p=f']+ix; then the kinetic term for p becomes B;.J>'t~p=2iB;.JX 7 C~r;, which is nonvanishing . We can act \-vith CP twice on the fields p as follows : (CP) 2p= CP(TJ •+ix•)= CCP(r;+ix)= c!-p= -p , since £:2=-1 [32]. We can build a fermi field (of definite chirality) as follows . If 1/1 and ~ satisfy t•=Ct/1 and ( •=C~ and transform, under CP as 'ljl .... a27f; •=Ca 21/l and ~ .... -a2 ~ •=-Ca2{, then the combination +=1{1+i~ transforms as a left-handed field. Under CP +. . a 2 +•= Ca2 (t-it) and, under the action of CP once again, (CP) 2 : +. . c2a2 ("+i()= -+, which is the same result as for the scalar case [32,51 ]. Given two vectors TJ and x transforming as the pseudoreal representation r, there are two ways to form invariants under G (for simplicity v;e write in the Z basis): (B.18) and (B . l9) Whether one of these expressions is useful depends upon whether it can be made to have useful transformation properties under the Lorentz -149- group . If ~z are left-handed fermi fields (and therefore we are taking a "doubled" pseudoreal representation) and >f =iu~~z, then it is eqn (B.19) that is the Lorentz singlet; if x_Z =TJz, then eqn (B.1i3) is the Lorentz singlet [32,51]. We now consider the action of CP on gauge vector fields A_! . Under an infinitesimal gauge transformation parametrized by r.>a, the gauge potentials behave according to (B .20) where the C:c are the structure constants for G defined in terrns of the infinitesimal generators (the Lie algebra) ra by (B.21) To discuss CP we need only consider global gauge transformations, for which the last terrn in (B .20) is absent. The gauge vectors are real if the generators ra are chosen to be Hermitian (and hence if the group parameters are chosen to be real). By an appropriate choice of basis on the group manifold one can choose C:c =if abc where f abc is completely antisymmetric and real. This is the choice that is usually made. By virtue of the fact that the structure constants satisfy the Jacobi relation one rnay choose them to be the elements of the generator matrices in the adjoint representation. In this case the choice C:c =if abc renders the adjoint representation matrices real. It is not possible to choose t he generators in the adjoint representation to be the structure constants, to have Hermitian generators and to have the Cartan subalgebra diagonal all at the same time. If a basis is chosen so that the ra are all Hermitian, then the fields A" may all be -150- taken as real . This basis makes the reality of the adjoint representation manifest, and, as discussed above, gives (B .22) where t;~=±1 (depending on the value of a). Using the above results we shall now discuss the conditions that the matrices of the group generators and the Cle bsch-Gordan coefficients must satisfy so that the action of CP on the fields transforms a given term in the Lagrangian into its Hermitian conjugate (when all couplings are set to unity). If 1J'i. is a column vector of (left-handed) fermions transforming as some representation r, then in its gauge invariant kinetic term we have the term (B.23) If we work in the basis of r in which the Cartan subalgebra is diagonal then, upon acting with CP on the fields, we get (note that a2 a~a2 =(af.I.)T) (E .24) where we have used the anticommutativity of the fermi fields and the thrice-repeated index a is summed over as are all other repeated indices . For eqn (B.24) to be the Hermitian conjugate of eqn (B.23) we must have ~=ta Tfi (no sum on a). (B.25) Thus we must have tCI=+l for the elements of the diagonal Cartan sub~ebra. The rem.aining generators (in the basis in which the Cartan subalgebra is diagonal) are purely real or purely imaginary accordingly as -151~=±1. These conditions also suffice to render the gauge invariant scalar kinetic term CP invariant. We may now digress for a moment to consider the gauge vector boson mass matrix and its eigenstates at tree level. This matrix is real and therefore cannot itself violate CP. Consequently its eigenstates, if chosen to be Hermitian fields, must also be eigenstates of CP . Generally, those eigenstates will not be states with definite U(l) quantum numbers (unless those quantum numbers all vanish). In that case there will be at least a twofold degeneracy in the mass matrix where the mass eigenstates have opposite CP eigenvalues. If we call such a pair of eigenstates A; and A; (where ±indicates the CP eigenvalue) then the combinations (A; ±A;)I -v'2 are states of definite U( 1) quantum numbers and (B.26) If T± is the linear combination of generators that A;! couples through, then (A; ±iA; )I~ couples through ( r+ ±iT-)/ v'2, which is purely real . To discuss t.he Yukawa terms we consider (left-handed) fermi fields 1/li and Xm transforming as irreducible representations r'¥1 and rx respectively. We also consider scalar fields rpa transforming as some irreduc ible representation r • appearing in the Clebsch-Gordan decomposition of "•®rx. A group invariant Yukawa term is then of the form (B.27) where Rtma are group-coupling coefficients to couple r 1 . rx and r, to make a singlet. We assume that the bases for the representations r 1 and rx are chosen so that the Cartan subalge bra is diagonal. If we also choose the basis for r, so that the Cartan subalge bra is diagonal, then the action of -152- CP changes (B.22) into ·,.,,t t tRima · 'Z. .,.., U2Xm~a (B.28) For this to be the Hermitian conjugate of (B. 27), Rma. must be real. If r, is a real representation in the real basis (we call the group-coupling coefficients Rllna in this basis) then CP('iO~)=ta9'a (no sum on a.) where ta=±l as discussed above. The action of CP then changes (B.27) int.o (B.29) We must then have (no sum on a) . (B.30) Thus, Rlin.a is purely real or purely imaginary accordingly as ta=±l. The group-coupling coefficients for the coupling of scalars to scalars can be treated along the same lines as above. -153- Appendii C) Momentum Space Weights: Dau.hle-cut Diagrams In this appendix we evaluate the momentum space weight of the "double-cut" diagram of fig . 4.4. We present the results normalized by the Born diagram with massless ferm.ions . We take the fermions i 1 , i 2 , is and i 4 in fig. 4.4 to have masses m 1, m 2 , m 5 and m 4 respectively. X and Yare bosons with masses mx and m y. The coupbngs to be used depend upon whether X ( Y) is a scalar or a vector boson. We have, in the center of mass system of the X I (C.l) (C .2) where 2(m.f +ml) 1/ 2 (C.3) m} and c d e 1 and g are various products of gamma matrices and chiral I I I projection operators which depend upon the specific diagram being evaluated. -154- Some further kinematical facts are : (C.4) (C.5) q q: -I k I ItHx } k · =k. 2 2 k =mi; q =mi (C.6) · As an example, take the case of vector exchange in vector decay with all of the fermion masses set equal to zero. We obtain 1 Im[/w]= J d:::-~1-~ 32n1'r [ c()',uP(q-p)~P] -I (k -q )2 -my -1 (C .7) where P=(l±)'0 )/2 and v=my/mx. The results for the remaining three cases are (C .B) Im[lsv(v)]=~ 1 n 6 1T' 1~2 1 +v 2 (C.9) 4 and (C.10) -155- Plots of these functions are given in fig . C. l. For the case m 2 =m.:!=m. 4 =0 we define u=m 1/mx . We then get the following exact results and Taylor expansions about u=O: 1 2(1+v 2 )ln,~+2v 2 +3 =~ 6 161Tl ~ r r v2 1] 2 -l2+(1--v2)lnl~ u l+v J u4 +--~ 2(1+v2) u6 - ( ~) 3,1 +v- (C.ll) (C . l2) -156- Im[lsv(v, u)]= t:::-< 1-u2) lnlr~2 8 1f ll+v 2 --u 2 1 u 6 (1 +3v 2) 3( 1 +v 2 ) 3 - u4(1+2v2) 2(1 +v 2 ) 2 .I (C .13) and (C.14) We note here the expression for the Born rate (when all couplings are set to nnity) of the vector and scalar decay diagrams (normalized by the rate with massless ferrnions) : 1-[!!l-! 12 _f ~2 ~l~ (C.15) -157- lo7L___~,o~2~--------,~o~,----------,o~o~--------,~o~-,--~ m 5 /m 5 • Fig . C.la: Momentum space weight for scalar decay with scalar exchange for massless fermions . -158- mx =0.5 IIev -Im lvs 1m lsv ~~~oL2L---------,~o-,---------~~o-o---------,~o--~---------~~o2 ms/mx Fig. C.1 b: Momentum space weights for scalar decay with vector exchange (SV) and for vector decay with scalar exchange ( VS)for massless fer- miens. The mass of the vector is taken to be 0.5 neV. -:L59- Appendix D) lm(O) In this appendix we discuss some general features of the quantity 0 appearing in the expression for the average baryon number generated in the free decays of a boson and its antiparticle, eqn (4.15) . We specialize to the case in which each fermion family transforn1s as a single irreduc.:ible representation, r, of the simple gauge group G. We "YtTite an element of a fermion multiplet as 'f/lf (all fermion fields are taken to be left-handed two component spinors; we suppress any helicity labels), where the latin index specifies the family and the greek index is the group index. The couplings of the gauge vector bosons to the fermions may now be VvTitten as (D.l) where T(1.~ are the (Hermitian) generators of G in the fermion representation r. With more than one family the fermion fields can couple directly to any set of scalar fields transforming as one of the irreducible representations in the decomposition (D.2) (if all fermions appear in a single irreducible representation, they can only couple to scalar representatins appearing in the symmetric part of (D.2)). In general there may be several scalar multiplets which transform according to a given irreducible representation (whether or not it appears in (D.2)); that is, the scalar sector may have its own family structure. We write the l th scalar family which transforms according to r" as (rp")l. The Yukawa couplings can then be written as -160- where all repeated indil'es are . summPd over (even those repeated thrice) . The (h" )i;t are the Yukawa coupling constants and the (R" )o.fh' are the Clebsch-Gordan coefficients coupling r11 to r®r to make a singlet. Tne (Ra. )a~J·; may be taken as real when all fermion and scalar representations are taken in the basis in which their Cartan sublage bra is diagonal (as discussed in appendix B) ; also, for a given r 11 , (Ra.)aftr is either symmetric or antisymmetric under the interchange of a and {3 . ~a. is a unitary matrix defined to vanish when rb is a complex representation. If ra is a real representation, then (fla. )?P is symmetric, whereas if r 11 is pseudoreal, (fl")?P is antisymmetric. Furthermore, if ra. is a real representation and a basis is chosen in which the representation matrices are real, then (fl11 )?P:o?P . The second term is an independent term in the Lagrangian only if r is a doubled real or pseudoreal representation. The various fields appearing in (D.l) and (D.3) may be rewritten in terms of mass eigenstate fields by the unitary transformations (D.4a) (rpCI >i=( VU)lprtJp +( ~)lprtJp (D.4b) (D.4c) In eqn (D .4b) if, for a given P, the field rpp is a Hermitian field, then (D.5) Furthermore, if the representation ra. is a (undoubled) real representation and the basis of the representation space is chosen so that the fields (rp11 )l are Hermitian (real), it follows that (D.6) -161- (this condition was used in writing eqn (D .5c); the vector mass malrix i:~ discussed in appendix B). These transformaUc:1s satisfy the unitary relations (D. ?a) (D. ?c) In eqn (D. ?b) the indices a and b refer only to representations that appear in the decomposition of r®r, i.e., representations that can couple to the fermion-fermion operator. There may be other scalar representations in the model which do not couple directly to fermions (indeed, in general, with fermions transforming according to the fundamental representation of the gauge group, there must be such scalar multiplets except in the case of E(6) • [27]). Thus, to invert the expression (D.?b), one must allow the indices a and b to run over a set of values which includes these multiplets as well. However, the inverses of the expressions (D.4) are not needed in the following. In terms of the mass eigenstate fields eqns (D.l) and (D.3) become (D.B) •In an E(6) model a fermion family (or, in some incarnations of the model, more than one fermion family) can be put into the fundamental 27 dimensional representation. To break E(6) down to SU(3)&S'U(2)L®U(l)y and then to ,SU(3)eU(l).uwe need only use scalar multiplets that appear in the decomposition of 21S27: 2?W27={2'7+35t')s+351A . E(8) is the only compact simple Lie group with this property. -162- a.nd Ua"' aU/• (f'a )aaty_ (R" )Ah~ [ (h a )iJl: ( VC )lp + (ha )iik [ ( }ia )fp) •(6a )7P) ~ p +[(h• k.c (il-)lp+(h•~ [(JIG )fp] •(6a )7P)9'J>~+h. c . =U.•. Uf,. ("-')<Pt"-H~)39'p+(f~)ij9'.PHRa)4h+h.c . , (D.9) where (D.lO) and (D. ll) (there is no sum on a in the two preceding equations). We have the folk:Jwing symmetry as a consequence of the Pauli principle : (D .12a) and (D . 12h) We can now consider the quantity (D .13) which appears in eqn (4.15). in the limit where the fermion m asses are much smaller than boson masses. In this case the imaginary part of the loop integrals. lm[/n] are (as discussed in chapter 4) independent of their indicies i 1 , i 2 • i 8 and i" and we may therefore sum over the relevmt indices (the fermion mass eigen~tate indices) in the express ~ on for A In -163, the 1 e~ u1ti~ expression ihc t:ontributions of fermion mixing a ro p due to the condition (D .?a) and the absence of mixing at tb:s order betweP.n fermions of different baryon number . This last fact allows us to implement eqn (D.?a) in spite of the presence of the factor For the case of vector exchange in vector decay (fig . D.l ) we find (D . l4) where N1 is the number of fermion families and T~=W7PT~ (D.l5) is real as discussed in appendix B. There is no summation on P or S in (D . l4) because the bosons are specific mass eigenstates (neither the decaying nor the exchanged boson species is summed over). As a consequence of the reality of rP and T5 it follows that Im(Aw)=O . (D . l6) The expressions for Asv, Avs and Ass are not, in general, real•. For Asv there are two types of diagram that may contribute as sho·wn in fig . D.2. Diagrams in which the directions of the fermion arrows are reversed are equivalent to one of the diagrams already listed. Fig. D.2a gives the contribution ~ey are unreal. -164- Fig. D. l: Vector decay witt_, vector exchange . Arrows on the ferrrJon lines indicate the flow of the left-handed helic1ty. -165- p p -~-- -~-- (a) p p -~-- -~-- (b) Fig. D.2: Scalar decay with vector exchange . Arrows on the fermion lines indicate the flow of the left-handed helicity. -16691 ~r:. (rJ)~ (Ra. )~[ (r.t )~ (!?" )"'4] -~ B a- B ~-(B v-B ~) ~ I (D.l7) and fi.g. D.2b gives (D.lB) H the exchanged scaucr ' boson cannes a nonzero conserved quantuxn number (such as electric charge), then one of the two preceding expression~ will vanish for each set of a:, {3, J1 and v. For ~ there are two corresponding types of diagrams as shown in fig. D.3. These diagrams differ by reversal of the direction of the fermion arrows. If we reverse the direction of the exchanged vector boson we do not get a new type of diagram. Note that fig. D.3a and fig. D.3b correspond to processes with distinct fmal states in contrast to the situation for Asv. Fig. D.3a gives the contribution , (D.l9) and fig . D.3b gives g2(f'J)~ (R 0 ) 4~[ (r~)~ (R*' )~....S] ..,~~ TC~ [B a+ B ~-(B ~+ B v)] . (D.20) If S or P carry a conserved quantuin number, then one of the two preceding expressions vanishes for each choice of a:, {3, J.L and v. Apart from the baryon number factors, eqns (D.19) and (D.20) are the complex conjugates of eqns (0.17) and (D.18) respectively. For A.ss we again have two types of diagrams as shown in fig. D.4. The diagrams corespond to processes with distinct final states. Reversal of the direction of the exchanged scalar boson yields no new type of diagram. Fig. D.4a gives the contribution -167- (a) (b) Fig. D.3: Vector decay with scalar exchange . Arrows on the fermion lines indicate the fiow of the left-handed helicity. -168- p p -~-- -~-- (a) p p -~-- -~-- (b) Fig . D.4: Scalar decay with scalar exchange . Arrows on the fermion lines indicate the flow of the left-handed helicity. -169- (D .21) while fig . D.4b gives [ (f$)~ (Ra )v~(rJ,)i~ (Rb )a.S6] ~(r~ )]le (Rc )~J#J(r~)~ (Rd. )ave (D.22) x[ -~Ba+B~+B~+Bvn . Again, if S or P carry a conserved quantum number, then one ~f the preceding expressions will always vanish for each choice of a:, {3 , J.L and 1.1. -170~pendix E) Symmetries of lhe Illustrative SO( 10) Model In the model of chapter 9 we have a single family of fermions , _51 , the obligatory adjoint of vectors, 45v and, for th e sake of the present discussion, we consider the following set of scalar representations: a 126H, a cornplex 10n (=10 1 -t-il~ where 101 and 102 are real rt:presentations) and a 54n . Later we may replace the complex 10 by a real 10 and there by spe- cialize to the discrete symmetry, 9.14, of the model of chapter 9. We choose the Yukawa terms in this model to be (E . l) Ignoring for the moment the terms in the Higgs potential, this model possesses a global U(1)x symmetry. If we call the generator of U(1)x, X, then we have X=1 for 161 , X=2 for 126n and X=2 for 10H; all other values for X are zero. These values corespond to the transformations (E.2) U(1)x may be broken down to a discrete symmetry by various terms in the Higgs potential: as long as this discrete symmetry is large enough the Yukawa terms (E.l) are natural. However, for the particular case where the discrete syrn.m.etry is that generated by the transformations (E.3) the Yukawa terms (E.1) must either be augmented by the term (E.4) -171- OJ the thcury may be modified so that the lDH is J·ea.l. The discrete sym- metry (E .3) may be considered as a special case of (E .2) for a=±Tt/ 2. We wilJ therefore treat the theory as if it had the full U( 1) symmetry, special- izing later to the case cx=±1T/ 2 (and thus adding in the parameter C if the 10H is con"lplcx). The symmetry (E .3) is the Sy'rrunetry of the most gen- eral Yukawa coupling in this model. We will see below that although this symmetry is very simple it leads to a richer symmetry after SO(lO) has been broken via the SU(5) singlet vacuum expectation value of the 126H · The global symmetry U( l)x is spontaneously broken when 126H obtains a vacuum expectation value, as is the local group U( l)R appearing in S0(10)~SU(5)®U(l)n. U(1)R distinguishes among the terms in the SU(5) decomposition of a given 80(10) representation. We call the gen- erator of U(l)n. R. At temperatures sufficiently large so that SU(3}®SU(2)L®U(1)y is unbroken, 126H will have a vacuum expectation value only along its SU(5) singlet direction. At such a temperature a linear combination of X and R that vanishes along that direction will still generate a global U(1)z symmetry even though U(l)R and U(l)x are separately spontaneously broken •. To determine the relevant linear combination we must compute the values of R for the components of . 126H (in particular the SU(5) singlet component) . We call the values of R for the components in the decomposition 126=1+5+10+15+45+50 respectively a., b, c, d, e and f . Similarly for 16=1+5+10 we use fl, 1 and 6; and for 10=5+5 (where 10 is real) we use p and -p ( where only in the last case have we implemented the •At this temperature therefore nc Goldstone baaons would appear even if U(l)x was respected by the Hi&P potential. However, at lower temperatures this new global U(l)z will cenent.lly be broken in a way that produces a Goldstone boson. Thus, we must break U(l)..r tknrn to a discrete symmetry in the Higgs potential. -172- tracelessness of X) . Two conditions on (:3, -y and o can be obtained by considering t h e antisymmetric product of two 16's (E.5) In the SU(5) decomposition of the (real) 120, 120=5+n+10+10+45+45, (E.6) the 5 and the 5 are conjugates of one another: similarly for the 10 and 10 and for the 45 and 45. Thus the values of R for each of these pairs are equal and opposite. The 5 in (E. 6) arises in the combination (E.7) appearing in (E.5). Similarly the 5 comes from (E.B) Thus it follows that -y+6=-(f3+r) . (E .9) Similarly considering the 45 and 45 gives us 1+6=-26 . (E.lO) -r=-36= -3; {J=56=5 . (E.ll) Thus (We choose to normalize so that 6=1.) If we now consider the symmetric product of two 16's, -173- (E 12) and demC!Ild U1at the trace o! R be zero, ~· z lind lhat a=lO, b=2. c=6, ct=-6, e=-2,/=2 andp=2. (E.13) The linear combination of X and R that we seek, Z, is then Z =X-2!]_ a (E.14) The values of Z for the components of 161 , lOy, 54n and 126n are summarized in table E.l. 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