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GROUP THEORY FOR UNIFIED

MODEL BUILDING

R. SLANSKY

Theoretical Division, LosAlamosScientific Laboratory, University ofCalifornia,Los Alamos,

New Mexico 87545, U.S.A.

50

I93~

(IA.C

NORTH.HOLLANDPUBLISHING COMPANY - AMSTERDAM


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PHYSICS REPORTS(Review Section ofPhysics Letters) 79 , No. 1 (1981) 1 1 2 8 .North-Holland PublishingCompany

GROUPTHEORY FO R UNIFIED MODEL BUILDING

R. SLANSKY

Theoret icalDivision, Lo s A lamosScientific Laboratory*. University ofCalifornia, Lo sAlamos, N ew Mexico8 7 5 4 5 , U.S.A.

Received 1 0 April 1 9 8 1

Contents:

1 . Introduction 3 7. E~and subgroups 46

2 . The standard model 5 8. Larger groups 60

3. Unification 1 4 9 . Symmetry breaking 6 4

4. Dynkin diagrams 2 3 Tables 7 7

5. Representations 30 References 1 2 5

6 . Subgroups 3 7

Abst ract :The results gatheredhereon simple Liealgebras have been selected with attention to theneedsofunified model builderswho study YangM ills

theoriesbased on simple, local-symmetry groups that contain a s a subgroup the SU~x U ! x SU~symmetry of the standard theoryof electromag-netic, weak, and strong interactions.The major topics include, after a brief reviewof thestandard model and its unification into asimple group, the

use ofDynkin diagramsto analyze the structureof the groupgenerators and tokeeptrack of theweights(quantum numbers)of therepresentationvectors; an analysis of the subgroup structure ofsimple groups, including explicit coordinatizations of the projections in weight space; lists of

representations, tensor products a nd branchingrules fo r a number ofsimple groups; and other details about groups and theirrepresentations that

are oftenhelpful for surveying unifiedmodels,including vector-couplingcoefficient calculations.Tabulationsofrepresentations, tensor products, an d

branching rules for E6, SON, SU6, F4, SO9, SU5, SO8, SO7, SU4, E7, E8, 5U8, SO14, S O 1 8 ,SO22, and for completeness, SU 3 are included. (These

tables mayhave other applications.) Group-theoretical techniques for analyzing symmetrybreaking are described in detail and many examples are

reviewed, including explicit parameterizations ofmassmatrices.

Work supported b y theU.S. Department ofEnergy.

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PHYSICS REPORTS (Review SectionofPhysicsLetters)7 9, No. 1(1981)1 1 2 8 .

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R. Slansky, Group theoryforuni f ied mod el building 3

1 . Introduction

Thepurposeof thisreview is threefold: to present a pedagogical introduction to the use of Dynkindiagrams, and especially their application to unified model building; to summarize the representations,quantum numberstructure, andtensor products ofa number of simple Liealgebras that have attractedattention; and todescribe severalproblemsin unified model building. Table 1 and theContents providea fairly detailed summary of the topics covered. The choice of groups was suggested primarily byresearch into YangMills theories [1, 2] based on simple, local-symmetry groups that are supposed to

unify quantum chromodynamics (QCD) with quantum flavor dynamics (QFD). QCD is the candidatetheory of the strong interactions; it hypothesizes that the strong interactions are due to the interactionsofeight vectorgluons with theeight symmetry currents ofan SU3 locally-symmetric Yang Mills theory[3]. This local symmetry is denoted SU~,where c means color and distinguishes this use of SU 3from others. The gluonsdo not carry thecharges of the flavor interactions. QFD is also aYang Mills

theory with local symmetry G~containing the SU~x U~of the electromagnetic andweak interactions[4]; flavor bosons do not carry color charge. In addition to those, unified models hypothesize theexistence ofadditional interactions. For example, themodel based on alocalSU5 symmetry, which w as

thefirstexample based on asimple group, hasadditional bosons that can mediate protondecay [5].Section 2 contains a brief review of the standard model ofelectromagnetic, weak, and stronginteractions, based on the group SU~x U~x SU~.It is presented as background material for readersfrom outside particlephysics. Some kinematical featuresof thefermion mass matrix arereviewed, and

the main result, summ arized in table2 , i s thenecessary part of theparticle spectrum to be incorporatedin unified models.

The proposal of unification is examined somewhat critically in section 3 . It is assumed that thestandard modelcan be embedded in asimple group G, as reviewed in ref. [6]. A qualitative reviewofsimple Lie algebras (to be elaborated in later sections) is given. Section 3 also has an introductorydescription of specific models based on SU 5 [5], SO~[7], and E 6 [8], and some notes on grouptheoretical problems to be solved when analyzing such models. Sections 2 and 3 contain elementary

material.

Much of theanalysis of the SU 5 model is easily carried out using traditional tensor techniques, butfor a group as complicated as E6, those techniques often become quite cumbersome. Thus, thereis amotivation forwantinga simpler and more transparent notation. It i s thecontention ofthisreview thatthe use of Dynkin diagrams [9] is just such a simplification; although it is hardly needed for the SU 5work, it does make detailed discussions of largergroups like E 6 or SO22 quite easy. The Dynkin labelsof the representation vectors, used in conjunction with the Dynkin diagram, take the whole groupstructure into account. In contrast, sets of tensorlabels are easily made symmetric, antisymmetric, ortraceless, but further algebraic structure is oftentimes expressed rather awkwardly. For example, thecomponent-by-component analysis of the quantum number content of a representation is trivial usingDynkins techniques; there even exist simple computer programs that do the whole job [10] for any

representationofany simple group, although the results presented herewere derived mostlyby hand.

Consequently the use ofDynkin labels for the states simplifies the details of many group-theoreticalcalculations. Finally, the group theoretical structure o f the symmetry breaking takes on a transparentgeometrical character, especially i n thosecases where the concept of thebreaking directionin weightspace is applicable.

Sections 4, 5 and 6 are devoted to summarizing the group-theoretical results that are needed forunderstanding Dynkins approach to representation theory. Readers familiar with unified models may


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4 R . Slansky, Group theory for uni f ied model building

wish to skip directly to section 4. The root system of asimple Liealgebradescribes the effect of theraising and lowering operatorsof thegroups algebra on the eigenvalues (or quantum numbers) of thediagonal generators, and provides a geometrical interpretation of the commutation relations. TheDynkin diagram is aconvenient mnemonic for a special set ofroots,called simple roots, that carry al lthisinformation. Iti s thendescribed how to compute theeigenvalues of the diagonalizable generators,whichform the Cartan subalgebra.

Ourinformal treatment of theunderlying grouptheory is directed toward applications [11].Thus, theusefulnessof many theorems is emphasized, but no proofs are reviewed. Hopefully, thisapproach willsupplement themany rigorous and more detailed treatmentsof simple Liealgebras already available intextbook form for many years [12]. The reader whodesires proofin addition to an intuitive pictureshould look therefor derivations.

Section 5 describes how Dynkin diagrams can be applied to the analysis of the finite-dimensional,unitary, irreducible representations (irreps) of simple Lie algebras. Each representation vector in theHubert space is labeled by the eigenvalues of the diagonal generators. It is demonstrated how to

calculate easily and quickly the eigenvalues in any irrep of any simple algebra. As examples that arenontriviali n othernotations, acomplete analysis ofthe 27 and7 8 ofE6 is given, including acalculation

of theelectric charge, weak charge, and colorcharge of each component. (This calculation is begun insection 5 andcompleted in section 6.)To complete the analysis of standard-model physics, as embedded in a unified model, we study

further the subgroup structure of the unifying group. Dynkins analysis of subgroups is outlined insection 6; although for many purposes it is quite adequate to have a list ofmaximalsubgroups, a few ofthegeneral results give important insights into model building. Explicit matrices that project the rootsystem ofa group onto theroots and weights ofits subgroupsare derived. The basis independentresultsusedto establish theuniqueness (up toan equivalencetransformation)of theseprojections arereviewedhere andin ref. [6]; many examples areprovided. We also discuss reflections ofthe generators that canbe used forcharge conjugation Cand forCP [13]; these are associated with symmetric subgroups.

Sections 7 and8 havediscussions of the tablesofirreps, tensorproducts, andbranchingrules. Section7 is a detailed account ofE 6 and its subgroups. The purpose of the text is to offer comments on the

content, conventions, andapplications of thetables. Then a consistent set ofprojection matrices from

E 6 through al l the physical subgroup chains to U~mXSU~is derived. These matrices are helpful for

calculations where explicitly labeled field operators are used. Finally amethod for calculating vector-coupling coefficients is outlined andsome examples are worked out.

Therei s some interest in using largergroups; a sketch ofthe groupsE7 , E8 , SU8, and the use of the

complex spinor representations ofthe SO4,,+6 groups [14] is given in section 8 .Section 9 covers several topics in the application of group-theoretical techniques to symmetry,

breaking. For the case where the breaking is done by a single irreducible representation, Michel hasconjectured a classification ofsolutionstoa ll possible (realistic) breaking mechanisms [15].His solutionsrely on the notion of a maximal stability group or maximal little group. The breaking in unifiedmodels is done by a reducible representation; a proposal forsolving this more complicatedsymmetry-

breaking problem (without explicitly minimizing comp licated Higgs potentials) is described [16]. Eachcandidate little groupm ay be a minimum for a range of parameters of the Higgs potential, includingradiative corrections; theminimization problem is reduced to a one-dimensional problem where afinitenumberofcandidate answers aresubstituted into the (effective) potential andcompared. We then lookat some specific problems in YangMills theories with the field operators labeled according to aconvenient basis; in this context the calculation of the vector-boson and fermion mass matrices are


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described in a number of examples [17]. There are some interesting examples where the concept ofsymmetry-breaking direction in weight space greatly simplifies mass-matrix calculations. In con-clusion there are some comments on the possible role of the charge conjugation reflection of theunifying group i n symmetry breaking.

There are a numberofimportanttopicsin unified model building that arenot covered in this review:

one that has received much attention is the use of renormalization group techniques to calculate theproton lifetime (and the mass of the bosons that mediate the decay) in terms of the experimentallymeasured strong and electromagnetic couplings [1820].There have been many other papers on the

phenomenologyofsmall effects (rare or forbidden decays, neutrino masses, and other effects notexpectedfrom the standardtheory ofelectromagnetic, weak, and strong interactions) predicted atsomelevel in many unified models; these too are not discussed. We havealso ignored the developments incomputing thespin ~ fermion mass ratios at low Q2 in terms of thesymmetry ratios.

Many ofthe results containedherewere derivedin collaborationswith M. Gell-Mann, J. Patera, P.Ramond and G. Shaw, in our investigations of unified models and Lie algebras. It is a pleasure toacknowledge their contributions andhelpful co nversationswith J. Ginocchio. The drafts of thisreviewwere cheerfully and excellently typed by Marian Martinez. R. Roskies and H. Ruegg provided manyhelpful comments on the manuscript.

2 . The standard model

The purpose of thissection is to review briefly the standard model [3,4] of electromagnetic, weak,andstrong interactions based on the local symmetry SU X U~X SU~,with focus on some elementaryfeaturesalso basic to unified models [21]. This section is intended to provide for those outside particlephysics someexplanation of the languageused in later sections: the relationships ofthe vector bosons,the adjoint representation, and interactions [2]; the structure ofthe representation of all left-handed,spin ~fermions, including particles and antiparticles and the construction ofthe kineticenergy and massterms in the Lagrangian; and the symmetry breaking of SU~x U~down to the U~mof quantumelectrodynamics (QED). The group-theoretical language relied on so heavily in unified model building

is not meant to hide thephysics, as it may appear at first glance, but is intended to communicate veryefficiently much of the physical content of these theories; our object here is to set up the physicallanguage so that the translation to group-theoretical language is explicit. The particle spectrum toappear in the Lagrangian of the theory is listed in table 2; the way that spectrum appears in theLagrangian can be restated in terms of representation theory.

A YangMills theory based on a local symmetry G is a field theory with the symmetry currentscoupled minimally to vector-boson fields in a form analogous to QED, where the coupling of thephoton field A,~(x)to the electromagneticcurrent jm(x) has the form, ejm(x)A~(x);eis the electriccharge of a particle contributing to the current. The space integrals of the time components of thecurrentsdefine formally thecharges or generatorsof the Liegroup, which, in the case of QED, is a U

1

or phase symmetry. These generators are the elements of the Liealgebra of0. Thus each generator ofG is associatedwith a vectorboson that is coupleddirectly to the symmetrycurrent; it is in this fashionthat Yang Mills theoriesaccount for theinteractions of Nature.

For thestudies ofYang Mills theories described here, the propertiesof the Liealgebra of the LiegroupG are all that are needed; theglobal or topological properties of 0 are not used at this level of

model building, as they are in instanton physics. Thus we may follow the traditional but incorrect


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usage, where the term group is often used when discussing infinitesimal transformations, which arecompletely described by the Liealgebraof0, and where thesame symbol 0is oftenused to denote thegroup and its Lie algebra. Perhaps a short, technical description of the problem will help the moredemanding reader; others should skip to the next paragraph. The identification of the groupwith thealgebra is ambiguous because usually there are several extensions of a given Liealgebra to the finite

transformations of the Lie group; the choice of extension depends on the choice of discrete groupelements factored out of the center of the covering group. For example, because electric charge isquantizedin ~units, Qern generates aU 1 andnot its covering group, obtained if the spectrumofchargeswere to cover all real numbers. Similarly, the Lie algebras of U~

mx SU~and U3 are the same, but

strictly speaking,those symbols refer to differentextensions ofthe Lie algebra to finite transformations[22].In order to refer to the factors separately, we call the unbroken part of thetheory U~

mx SU~when, in fact, the extension from thealgebra to finite transformations should becalled U

3, because oftheconnection between trialityofcolor andelectriccharge. In thisreviewwe need the properties of theLie algebra for most discussions, so all names really applyto the algebra; similarly heretheterm grouptheoretical almost always means Lie-algebra theoretical. Viewed in this way, ournotation is not assloppy as it first appears.

A boson field B(x) has thetransformation properties ofagauge field andis coupled directly to theath symmetry current J~(x);J~(x)depends on B~(x),so thetheoryi s nonlinear. The generatorsofthegroup, and consequently the currents from which the generators are constructed, transform as theadjoint representation. In order for the coupling of the currents to the vector bosons to be invariantunder G, thebosons must also transformas the adjoint irrep, since groupsinglets (or invariants) occuronlyin the products ofan irrepwith its complex conjugate, and adjoint irreps are always selfconjugate.The vector-boson fields transforming as the adjoint irrep are a necessary part ofa YangMills theory.

A YangMillstheory mayalso have otherparticlesi n the Lagrangian. For example, the leptonsandquarks, whichare spin~fermions, are usually assumedto be fundamental fields in theLagrangian. (Theymay also be tightly-bound composites that behave like fundamental fields in an effective Lagrangian.)Each particle field must beassigned to an irrep of 0, so that when the field is put in the Lagrangian, the

invariance under G is kept manifest. Thus an important step i n understanding the structure of thesetheories is to know the irreps and the action of the generators of 0 on them. This is the same asknowing the contributions of those particles to the currents and how those currents interactwith thevector boson fields,which explains whythe group-theoretical language is so powerful.

QCD is an unbroken SU3-symmetric, YangMills theory. (The meaning of broken is discussedlater.) The vectorbosons (calledgluons) mediating thestrong interactionsare gaugeparticles coupled totheeight symmetry currents o fSU~,so the gluons transform under SU~transformations as the adjointirrepor octet 8 ofSU~.(In thisreview irreps aredesignated by theirdimensions, with conjugate irreps

marked by an over bar, e.g., 3C, and other inequivalent irreps of the same dimension with primes orsimilarmarkings. Most practitioners find these conventions convenient, even though there are somejus tifiable objectionsto them. Other labelingsarestudied here too.) Thegluons carry no flavor charges,whichmeans that they are singlets under anytransformation in the flavor group; stated more formally,

thegluons transform as (1,W ) under G~x SU~.Several features of QCD due to its quantum mechanical structure should be noted; they are not

important for an elementaryunderstanding of therole of QCDin model building, but they will helpinforming aphysical picture of it. Isolated hadronic systems are composites of quanta carrying colorchargesand are assumedto be colorsinglets.An individual color charge cannot beisolated in spaceandtimefromothercolorcharges; that is, color i sbelieved to beconfined inside hadrons. (In spiteof much


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effort on theproblem, the confinementconjecture has been difficult toprove in QCD; it is not crucial tounifiedmodel building asdescribed here.)As the spatial resolution ofthe probe ofhadronic structure isshortened, the effective coupling of thegluons to the color currentsdecreases, so that the constituentsof hadrons appear to interact more weakly. This behavior is called asymptotic freedom, and isjus tified by perturbation theory calculations [23]. As a result, it appears possible to view hadrons ascomposites of strongly interacting elementary quanta with fractional electric charges [24] that gain anelementary identity, not by isolating them, but by probing hadrons at short distances, as is done indeep-inelastic lepton scattering [25] and high-energy electron-positron annihilation [26]. Thus quarksare elementary, not nucleons,pions, kaons, and other observed hadrons. The quarks are coupled to thegluons through their contributions to the color currents, so theymust transformas anontrivial irrep ofSU~:the quarks are assigned to the 3C ,the antiquarks to 3C~The representation theory ofSU 3 is used toillustrate moregeneral resultsseveral times later on in thispaper.

In YangMills models a central problem, both physically and mathematically, is relating particlestates to representation vectors; each particle degree of freedomi s in one-to-one correspondencewithone vectorof a representation. Thus one tricolored, four-component Diracquark of agiven momentumis describedby 3 x 4 = 12 Hubert space vectors: red,green, or blue (or whatever your favoritenames for

the three colors) times the label, left-handed quark, right-handed quark, left-handed antiquark, orright-handed antiquark, so the vectors have the labels, color, handedness, particle or antiparticle). (Inthe limit ofzero mass, left-handed means the spin projection is antiparallel to themomentum vector,andright-handed means the spin projectionis parallel to themomentum; for a massive particle at rest achiral eigenstate is a 50 50 mixture of spin projections.) The fermion field operator t 4 r that annihilates(i.e., removes) a particle from the state (,~s~particle)= vacuum)) carries the same set of labels as thestate, but with signs of appropriate quantum numbers reversed. The reason for using chirality(handedness)ratherthan somespin componentof thefermion is that the chiralityprojections ~(1y~commute with thegauge andproper Lorentz transformations; theleft-handed fermionstransforming as

fL do not mix with the right-handed fermions in fR under a gauge transformation, so the set of al lfermion states cannot belong to an irrep. However, fL (and co nsequently fR ) by itself can be an irrep.

Let usexamine in general theconstruction of thefermion kineticenergyand thefermion mass in any

YangMills Lagrangian [2]. Our object is to show that the kinetic energy couples fL and fR, that fR

transforms as fL(the conjugateof fL), and to discuss some group-theoreticalaspects of theconstruction;we thenshow that the mass operatorcouples fLto fL (and fR to fR),and it transforms as representationsin thesymmetric part of fL XfL The kinetic energymust begauge invariant, but it is notnecessary for

(fLx fL + fRx fR)5to contain a singlet, since all fermion mass can arise from symmetry breaking.

The usual covariantkinetic energy has theform

~iy(88. igBTa)ili, (2.1)

where i/i is acolumn vectorof real (Majorana) anticommuting fields, ~ = fr~Yo,Ta is an antisymmetricmatrix representation of thegroup, so that the current contribution 1 /7 M Tale is nonzero and transforms

as the adjoint irrep, and B(x) are the vectorboson fields, also transforming as the adjoint irrep. (Theadvantages of beginning with Majorana spinors and constructing Dirac spinors later will becomeapparent.) Since thechirality projections ~(1y4,commutewith y0y5~,the kinetic energycan also bewritten * 1 .. ~7 DilL+ t l R iy DilR, where D,. = igB~Ta and each term by itself is gauge in-variant.

The field operator il(and any other operatorin the theory)carries a definite change of quantum


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numbers;when .f r acts on a state,it changes the quantum numbersofthat stateby amount A (calledtheweight of il) and a change in angular momentum for a field with spin, whether it creates orannihilates particles.The Hermitian conjugate ilt must change the quantum n~imbersofa state by A ,and also make the oppositechange in spin for i l . Looking at the 114(70y . t9)~i~..piece of the kineticenergy, we note that the c-number pieces in parentheses cannot change quantum numbers, and thecombination ~ il~cannot either without destroying the global gauge invariance of the kineticenergy.Thus, if ilL~~fL( means transformsas), then ilLfL, since a Hermitian conjugated fieldoperator ilt acts on the labels ofthe dual vector ~ in thesame way that ilacts on thelabels of theket . .). Moreover, when1 /4 .acts on the ket ),i t changes theeigenvalues ofall thediagonalizable

generators by an amount thati s the negative of thechange due to i/iL, including chirality, so i / i L ~fR;these important results are summ arized by

fRfL. (2.2)

The kinetic energycan bewritten group theoretically as fR(Op)fL + fL(Op)fR.The reflection that takes fLto tR includes changes of the signs of the internal quantum numbers

(often done by a charge conjugation C)and the handedness (often done by parity F). Individually C

and P do not have to exist in a theory, but CP must, since it is necessary to have a reflection thatexchanges fLand fRin such a way that fL X fRcontains a group invariant and an adjoint, as needed forthegauge-invariant kinetic energy (2.1). As discussed in section 6, it happens in many unified modelswhere Cexists that Creverses the signs of only some of the quantum numbers, with P reversing the

remaining ones.This is because Cmust reflect fL onto itself; if fLi s not selfconjugate, then Ccannotreflect the signs ofall the quantum numbers [13].

W e can now relate fL to the column of Nfour-component spinors i/i in the case where fL isirreducible;note that if thedimensionof fLis N, then i /i has 2N independentcomponents, so theremustbe 2Nconstraints. The simplest example (N= 1) is a 4 component Majorana spinor, where f L is anontrivialone-dimensional irrep of a U

1, and fR has the opposite charge. A Majorana spinor hastwo

independent components, so there are two constraints relating i/i and i/,~:it is c i = Cy0i/i~,which is

called the Majorana condition. The 4-by-4 matrix Cis defined so C , C -1 5 and Cy5-y,. areantisymmetric,and Cy,. and Co~,,aresymmetric. Thus, ~lfrTCy Di/, = ~d~ey. Dt/i as in (2.1) is Lorentz andU 1 invariant,andis a suitable kinetic energy for a Majorana spinor. It is nonzero because Cy . D is antisymmetricandfermion fields anticommute.

The only case where group-theoretical complications might occur is for the irrep fL to be selfconjugate, because thematrix partofthe relation between 1 / i andilt must beable to reverse the signsof

all weights within the irrepfL ; thus, it is the matrix part of the unitary operator, ~ (~~)il(~P~=C(CP)I/i, where the unitarymatrix Cacts on thespin degrees offreedom and the unitary matrix (CP)acts on the weights in fL (Obviously, Cand (CP) commute.) The simplest generalization of theMajorana condition is c u t = ilT(CP)Cyo, and the kinetic energy (which is Hermitian) is~ii/i(CP)C-y Di/i ~ii/ry Di/i, where i/i = i/.

TC(CP) =1frt70 thelast equality followsfrom the Majorana

condition. Clearly, (CP) i ssymmetric.Thisconstruction is completely adequate for realirreps: iffLis real, then (fL X fL)5 contains thesingletand (fL X fL)acontains the adjoint, so (CP),which is thematrix coupling fLto fL to form the singlet,must be symmetric. Moreover, if fL is not self conjugate, then (CP) i s outside the group and can bedefined to be symmetric. However, there is also another kind of self conjugate irrep in some simplegroups: iffLi s pseudoreal, then (fLXfL)a contains the singlet and (fL X fL)5 contains the adjoint, so the


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(CP) matrix is antisymmetric. In this caseR IJT(C P)Y . Di/i vanishesidentically, so theconstruction above

must be revised. In the pseudoreal case, the kinetic energy must be written ~iilT(CP)Cy5y. Dcli

D~,whichis nonvanishing for theantisymmetric part of (CP).(This modification is not availableforspinlessparticles.) The Majorana conditionshould now bewritten with a y~, = ilT(CP)C757o, sothat i/i = I/,T(CP)CY = l/ityo is again theusual Dirac conjugate. It should be emphasized that within a

unified theory ofthis type, QCD andQED can still betransformed into theirusualform [27]. However,it is not even possible to writedown a kinetic energy term for spinless bosons if they transform as asingle, pseudoreal irrep, at least not without making some drastic revisions in thetheory [28].

The fermion mass term has the general form

+iFys)i/i, (2.3)

where S and P are Hermitian matrices in the group space and need not have gauge-group singletsin aspontaneously broken theory. Since this term can be rewrittenas ./i~(S iP)t/iL + IIIL(S+ iP)ilR, itfollowsfrom (2.2) that the first termtransforms as fL x fL(since ~ fL)and the second as its Hermitianconjugate, fR X fR. Moreover, since yo and YoYsbehave antisymmetrically between Majorana fields and

fermion fields anticommute, the mass operator is in thesymmetric part, (fL X f~+ fR X fR)s.The mass operatorofasinglequark is abilinearform on thetwelve quark states. Onemay concludethat the mass matrix of a single quark is 12-by-12, but of course it is never written in that form sincesymmetry requires most of the components to be zero and the remainder to have equal magnitude; the144 parameters are reduced to oneby rotational invariance, color conservation,andthephase freedomin defining the field operators. The analysis of asingle c1uark mass matrix is an important prototype forillustrating some ofthe physics ofthe choiceof thespin ~fermion representation in unified models.First

of all, since the local symmetry transformations commute with 1y~,the mass operator i/it/i =I/iL*R +1/iRcIL breaksup into two six-by-six pieces. Group theoretically, for a single quark,

fL=3c+3c; (2.4)

themost general quark mass matrix has the form,

qR qR qL qL

q~ / 0 0 M 11 M 12q~I 0 0 M~M~

~ Mt1 Mt1 0 0 (2.5)qp \M~M~ 0 0

where therows andcolumns of (2.5) are labeledby states in such aw ay that the matrix is manifestlyHermitian and the M 1 1 are elements of a symmetric 6-by-6 matrix M; the M,, are 3-by-3 matrices in

colorspace. (Theformulation here,which may appear clumsyfor asingle quark,i s set up to be triviallygeneralized to an arbitrary spin ~-fermionmass matrix;some comments about the general case will bemade. The labeling of the rows with a vertical column vectoron the left andthe columns by arow ontop is somewhat unorthodox, butmakes itpossible to fit some of theexamples below on a page.)Th eupper right-hand corner of (2.5) is themass matrix associatedwith i/fRilL; the lower left-hand matrix isthe Hermitian conjugate Mt, since ~(*RR/iL)t = ~LilR. The notation in(2.5) is thereforeredundant; we


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quark called b (for bottom in some parts of the world) is a constituent of a new set of particles,including theupsilon Y[33].

One noteworthy feature is the large number ofquarks; this proliferation is offundamentalconcern,andi t is one motivation for thisreview. There are atleast 30 3C and 3 components of 1L required byphenomenology. For example, ifall 30 of these are elementary and belong to one irrep of a unifyinggroup, thenthat irrep must be quite large.This, among otherreasons, helps motivate the development

of powerful group-theoretical techniques.The leptons are similar to quarks in that they arebelieved to be fundamentalspin ~fermions. They

do not interact strongly, which completes thedefinition ofa lepton; they are assigned to singlets 1 C of

SU~.Leptons and quarks both carry the charges of the weak interaction, which brings us to anintroductory discussionof QFD.

One of the most significant developments of the 1970s w as theconfirmation that a certainSU2x U~

YangMills fieldtheory [4] providesan excellent description ofthe com bined electromagnetic andweakinteractions. The vector bosons mediating thecharged- and neutral-current weak interactions havenotyet been observed, as their masses areexpected to be oforder 80 to 90GeV, beyond the range of themachines of the 1970s. Since SU~x U~has four generators, there are four vector bosons that are

coupled to four currents. The vacuum is invariant under a subgroup of the SU~x U~,namely the U 1generated by the electric charge Qern The electric charge is a linear combination of the two diagonal

generatorsofSU~x U~0ern17+ywI2 (2.6)

where I~is the neutral component of the SU (where w stands for weak, and distinguishes thisapplication of SU2 in physics from others) and Y

5 is the generator of the U~.The orthogonalcombination ofI~and ~W is a weak interactioncharge, and the current associatedwith it is coupled tothe Zbosonthat mediates the neutral current w eak interactions. That vectorboson is expected tohaveamass of 90 0eV. (Althoughthe Lagrangian of the Yang Mills theory possesses that U

1 symmetry, thevacuum does not, so the Zboson has amass.) Thecharge raising andcharge lowering currentsin the

SU~are coupled to the W and W~vector bosons,which mediate thecharge-current weak interactions.They are expected to have masses of about 80 GeV, which indicates that the vacuum carries al l theweakcharges.

The phenomenon of vector bosons becoming massive in YangMills theoriesi s called spontaneous

breaking of local symmetry [34].When we talk about broken local symmetries, we do not mean that asymmetry breaking term is added to the Lagrangian. A vectorboson mass termby itselfi s not invariantunder the local symmetry transformations. However, a vector boson mass term plus some additionalinteractions can be, and that is what happens in a broken symmetry. (In this paper, broken means

spontaneously broken.) The Lagrangian is still invariant under the symmetry transformations, and thecurrent are conserved, or in the language of quantum field theory, the Ward identities (except foranomalies) hold, which is crucial for the perturbative renormalizability of the theory. However, thecharge of a broken symmetry does not annihilate the vacuum, which corresponds to the chargebeing

spread through thevacuum. The mathematical description ofelectromagnetic radiation in a plasma iscompletely analogous to that of a vector boson in a broken vacuum [34]. The main point of thisdiscussioni sto emphasizethat even forvery badly broken symmetries, the Lagrangian is stillinvariant,and so it is still important to study the invariants that can be formed from the fields in the theory.However, certain numbers in the theory, like fermion mass matrix elements, will often havenontrivialtransformation properties. A complete study of thesymmetry properties is still needed.


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Although the weak vector bosons of the standard SU~x U~theory have not been observed, theform ofthe currents ofthe low-mass elementaryparticles has been wellestablished experimentally [35].Aside from the photon and the gluons, the only low-mass particles that have been accessible in thelaboratory are the quarks and leptons. As we discussed earlier, the left-handed quarks, antiquarks,leptons and antileptons must be assigned to representations of thegauge group and the right-handed

fermions to the conjugate representations. SU2 has singlets, doublets, triplets, and so on, and theassignment that works is to place the left-handed fermions in SU~doublets and the left-handedantifermions in SU~singlets. This unsymmetrical leftright assignment is rather startling, but isconfirmed by a hugequantity ofexperimental data; the maximal parity violation in the charged currentweak interactions already provided a strong hint over20 years ago [4]that this i s asensible assignmentphenomenologically.The choice ofgroups discussed in this paper is strongly influenced by the desireto

maintain these unsymmetrical assignments in a natural fashion. Specifically, only when fL is not selfconjugatei s it not required for every left-handed doublet to be matched with aright-handed doublet.Theories where fLis self conjugate arecalled vectorlike [36];otherwise, the theory is called flavorchiral[61.

We now discuss the specific assignment of the quarks to representations of SU~x U~.The

left-handed u and d quarks form a doublet, where the YW values are determined by (2.6) and theassignmentthat u hascharge~and d hascharge ~.(Theprime indicatesstates coupled to thecurrents;they are linear combinations of the mass eigenstates.) Thus, Y~= Y~= ~.(All members of an SU~multiplet must have the same value of Y~W,or else the ~VV generator will not commute with the SU~generators, andthe group will not factor.)The left-handed and d are SU~singlets, and havedifferentvalues of ~W in accordancewith (2.6). The CPTtransformationrelating theleft-handed fermions to theright-handed fermions justinverts thisarrangement: uandd are right-handed singlets and and dforma right-handed doublet.Sincethisi sa basic featureof all fermion representationsi n localfield theories,we only need to discuss theleft-handed fermion representation fL

This patternof quarkassignments repeats itself at least once with the c and s quarks, andis oftenpresumedto repeatitselfagain withthecharge ~bquark and aconjecturedcharge ~t(top)quark. The bquark has charged-currentweakdecays [37],whichis consistent with anonzero weakisospinassignment;however, a zeroweak isospin assignment is not yet ruled out.

Next we briefly indicate the relationships among theseassignments to a representation of thegaugegroup, the currents, and themasses. For simplicityw e considerfirstthe charge ~ quark mass, including

justthe d and s quarks, and ignoringany mixing with the b quark. Themass matrix (2.5) connects(theleft-handed) d to the d and s , and this combination hasdefinite, nontrivial SU~x U~transformationproperties, with Ii~I~=~and ~Yi =1, so that ~Qern =0. Thus, the existence of the quark massindicates already that SU~x U~is broken down toU~.This also means that there can be mixingamong all the fermions with the same color and electric charge. Thus it is important to distinguishbetween the states connected by the currents andthose connected by themass matrix. If we call themass eigenstates dand s , thenthe charge ~quarkscoupled to the u and c by the charged currentsoftheweak interactions arecalled

d =cos o~d+sin O~s, s = sin 6~d+ cos O~s (2.7)

where the canonical transformation that connects themass eigenstates to the states coupled by thecurrents is parameterized by &. The transformation with three quarks (e.g., d, s andb ) depends onfour parameters, includinga CP-violating phase [30].[Recallthediscussion below (2.5).] We summarize


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this discussion in table2 , again emphasizing that those designations summarize an incredible amount ofphenomenology in apowerful way.

Next,w e reviewthe leptonsector. Theleft-handed charged leptons e (me = 0.5 1100 X iO~GeV), the~z (m~.= 0.10566GeV) andmost likely the r (mr= 1.81 GeV [38])are all I~= ~members of weakisodoublets. To ahigh degree of accuracy, the mass eigenstates are the statesin the currents, with the

charge raisingcurrent transforminge 1 1 to ~eL, ~ ~ito ~ and(most likely) r~to V~L.If the neutrinos aremassless, but are only coupled to the vector bosons as in (2.1), then their current eigenstates can bedefinedby convention, and the charged leptons cannot mix through theneutrinos. If the neutrinos havesmallmasses, then(2.3) need not be diagonal i n the same basis as (2.1), andtherecan be mixing; therehas been considerable effort to search for muon-number violating processes [39] and neutrino oscil-lations. The left-handed positron, ~z+ andr~are each assigned to be weak singlets. The analogy withthe quarks wouldbe complete ifthereexisted left-handed antineutnnos.There is no argument againstthem since they would be neutral, JW = 0, and, consequently, YW = 0 particles, and would have noelectromagnetic, weak, or strong interactions. Thus they would be hard to see in present dayexperiments, although they mighthave left their markon the featuresof theuniverse, as described bythe big-bang cosmology. A conservative attitude is that there are no left-handed antineutrinos, since

they havenot beenobserved. For most unified models, thisattitude must be relaxed.The modelof the symmetry breaking of the standard SU~xU~theory down to the U ?

tm symmetry ofQED is doneby giving certain scalarfieldscarrying thecharges of theweak interactions, but no electric

or color charge, a nonzero vacuum expectation value [40]. This problem can be studied in the Higgsmodel [41] of symmetry breaking. Theboson andfermion masses are then proportional to the vacuumexpectedvalues of the scalarfields, as are theinteractions inducedb y thesymmetry breaking, atleast inthe classical limit. There is controversy whether the scalar fields doing the breaking of SU~XU j arefundamental fields in the Lagrangian of Nature, or are effective scalars formed, for example, asfermionfermion bound states. In the considerations in this paper, we often treat the symmetrybreaking in the Higgs model, but withhold judgement on the physical reality or origin of the scalarfields. A study of the group theoretical structure of symmetry breaking is useful, no matter what thebreaking mechanism.

In the standard model, SU~x U~is broken by the neutral member of an I= ~, Y~=1 scalardoublet b(x). CPTinvariancerequires a conjugatedoublet ~t(x) with Y~= 1 , so thereare four spin0degrees of freedom. Thus,up to threevector bosons can acquire masses,using thecorrespondingscalarfields as the longitudinal degrees of freedom. The fourth scalar is physical and defines the symmetrybreaking direction. The covariant kinetic energy of the scalars has the form (D~4i/)tD,4~,withD,4 = ig(rJ2)B~(x)4(x) i(gI2)B,~~(x)4~(x),whereg/2is thecoupling ofU~andB,~(x)is thevector boson coupled to the U~current, and B~(x)(a = 1 ,2 , 3) are the bosons coupled to the SU

currentswith gauge couplingg; the weak isospinor representation matrices are r0/2, where ~a are thePauli matrices. In the unitary gauge the constant part of 4, of which only the neutral component isnonzero, provides(classically) masses to the three weak bosons, while leaving the photon massless. Atthis point, weleave thecalculation, sincethe details areno w considered straightforward [21].However,the discussions of symmetry breakingin section 9 do rely on a thorough knowledge of thisw ell-knowncalculation.

In summary we can say that the standard model hasprovided an attractiveframeworkfor organizinghuge quantities of experimental data. Nevertheless, from a theoretical viewpoint it is somewhatawkward, asdiscussed at the beginning of the next section; it does not appear to be theultimate theoryof theinteractions in Nature.


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3. Unification

The purpose of this section is to give abrief, but critical introduction to the unification of thestandard model into a YangMills theory based on asimple group. The SU 5 example, which is aprototype,is thendescribed. Finally, the methodof embedding flavor and colorinto the simple group,

ascarried out in ref. [6], is described and atthesame time a briefintroduction togeneral features of thesimple algebras is given. (In this review, unified models based on simple groups are the only kindanalyzed; at the present stage ofdevelopment, thisrestriction is physically quite arbitrary, but many oftheresults discussed here are applicable to other kinds ofgauge models.)

The standard SU~x U~x SU~YangMills theory of the electromagnetic, weak, and stronginteractions has provided a detailed phenomenological framework in which to analyze and correlate

many experimental data. Although the constraintsof thismodel appear to be satisfied experimentally,thechoiceofsymmetry group, theassignment of scalarsand fermions to representations, and the valuesof many masses and coupling constants must be deduced from experimental data. Asidefrom beingderived from local symmetries, the threeinteractions are not related to each otheri n anyspecificway,and each gauge coupling for the factors SU~,U~and SU~is afree parameter. Moreover, the standard

model ignoresgravity and it gives no relationships am ong particlesofdifferentspins. Thus,i n spiteofitsenormous success, the standard modelappears to be partofa morecom plete theory; it leaves too muchunsaid. It is an obvious question to ask whether there are more complete theories that include theresultsofthe standard model and also interrelate the interactions and correlate the many assignmentsand parameters that are put into the standard model by hand. None of the unified models discussedhere is likely to be the ultimate theory either because they only partially solve some of the problemsabove, but one can hope that studying them will lead to a step i n the right direction.These models aresometimes called Grand Unified TheorieS, but not here.

Early attempts to find such theorieswere made by Pati andSalam [42], who argued that a theorywith integer charged quarks [43] (which is not QCD)can beembedded into a larger theory, includingnew interactions that violate baryon number. Shortly after, Georgi andGlashow [5]pointed out that the

standardmodel (including QCD)can be embedded into the simple Lie groupSU5. This means that theelectromagnetic, weak andstrong interactions are al l contained in alarger set of interrelated inter-actions.Sucha theory mustinclude the color andflavor interactionsplus new interactionsthat mix colorandflavor quantum numbers. The Pati Salam and GeorgiGlashowmodels havedifferent mechanisms(in detail) for protondecay, but in these models and others like them, proton decaydoes result from

additional interactions not in the standardmodel. Such theorieshave helped tomotivate moresensitiveexperiments on the proton lifetime [44] and neutrinomasses [45].

Ifthere were no spontaneous symmetry breaking, al l the vectorbosonswould be massless anda ll thevector boson-current coupling constantswould be equal or related by group-theoretically determinedconstants. The symmetry breaking thendistinguishes between thedifferent interactions: theleptoquarkbosons, which couple quarks to leptons andcan mediate proton decay, acquire very largemasses; theweak interaction bosons acquire much smaller masses; and the photon and gluons remain massless. The

separation of the underlying interaction into electromagnetic, weak, strong, and other components isdue to the specific pattern of spontaneous symmetry breaking. It is important to realize that thishypothesis ofunification is very speculative, at least until some experimental evidence,such as protondecay, is found to support it .

Unification by a simple group means thereis only one gauge coupling constant, but it also impliesthat the ratio of thestrong and electromagnetic coupling constantsis ~ in the limit that spontaneous


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symmetry breaking can be ignored [18] (in the SU 5 model).Experimentally,however, the ratio of thestrong coupling a5 and finestructureconstant aei svery different. At Q2 = 1 0 GeV

2, the ratio ofa5 to ae

is about 50 (with large theoretical andexperimental uncertainties). As long as a,, and a~aresmall andthe number of fermions and spinless bosons having color is not too large, this ratio decreases as a

logarithmof Q2 , andapproaches ~around Q2 of order (1014GeV)2 in the SU

5 model [19,20] . The new

interactions implied by unification aredramatic, but are so weak that very sensitive experiments areneeded to detect them. U nification does have a chance ofbeing viable.

Before looking at the general structure ofunified models, it may be helpful to describe the SU 5model proposed by Georgi andGlashow [5].The simple group SU 5 has24 generators, and therefore theYangMillstheory has24 vector bosons coupled to 2 4 different currents. SU 5 contains SU2 x U 1 xSU 3as a subgroup, and those 1 2 currents are identified with those of the standard model of theelectromagnetic, weak and strong interactions. The other 12 vector bosons mediate new interactions,but they are very weak because the bosons are expected to be very massive: they_include a 3~withelectric charge ~ in aweak isodoubletwith a charge ~boson, togetherwith their 3C antiparticles.

The assignment of the fermions in the SU 5 model is fairly complicated. The u, d , electron, and itsneutrino are assigned to one family ofparticles; the c, s, muon, and its neutrino are assigned to a

second family; and the b, conjectured top, tau, andits neutrino are usually assigned to a third family.Eachfamily itself is assigned to areducible representation.

We next go through the steps of searching for an SU 5 representation to which a family can beassigned.Theunifying group S U5 contains several subgroups,but among the SU 5 generators theremustbe an SUx U~x SU~to be identified with the standard model. In fact, as we shall discuss in greatdetail in section 6, SU2 x Ui x SU3 is amaximal subgroup ofSU5. We may carry out the embedding bybreaking up the fundamental five-dimensional, unitary,irreducible representation (irrep) of SU 5 intofundamental representations of SU2 x U 1 XSU 3 as follows,

5 =(2, 1)(1) +(1, 3)(2/3); (3.1)

in the entry (x,y), x is an irrep of SU2 and y i s an irrep of SU3, and the irreps are denoted by theirdimensions.The second parenthesis contains thevalue o f theU 1 generator when acting on the statesinthe (x, y). It is normalized so that i t can be identified with Y

5, if the SU2is identified with SU~and the

SU 3 is identified with SU~.The relative value of the YSV eigenvalues is determined from therequirement that all generatorsofSU5 must betraceless. The contentsof an irrepof a group i n termsofthe irreps of a subgroup, like (3.1), is called a branching rule. Knowledge of branching rules iscrucialfor studying the content of models, so this review contains many examples. (The SU 5 branching rulesare contained in table 30.) There is no other way to fit nontrivial SU2 x U 1 xSU3 irreps into a fivedimensional irrep aside from conjugating the 3 , so the choice (3.1) is unique once the embedding isestablished.(Since the5, 2 and 3 arefaithfulirreps of SU5, SU 2 and SU3, respectively, (3.1) already goesa long way toward establishing the embedding. What must be done is to show that the generators of

SU 5 contain the generatorsofSU2 , U1 andSU3, which can beproven by looking at thebranching rule

for the adjoint. See below.)The next question is whether anyof the known particles can be assigned to the 5, given the above

embedding ofSU~x U~x SU~in 5U5. From(3.1) it is seen that the 5 contains a lepton weak doubletwith electric charges +1 and 0, and a charge ~ quark weak singlet. Such a multiplet may beappropriate for the right-handed (e

4~,1e) doublet and the right-handed down quark, or equivalently,

fL-~fR maycontain a 5 with the (v,,,e~doublet and thed L singlet. If this assignment is made, then w e


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must find anotherrepresentation of SU 5 for the left-handed e~, , u and d.Consequently, thereis usefor alist of irreps ofSU 5 (or any other group that i sa candidate forunification), such as the onein table28, and the branching rules to see ifthere exists a suitable candidate. It is clear that the 1 0 is such a

candidate.This possibility can be derivedfrom (3.1) and the tensor product,

(5X5)a=10, (3.2)

which can be found in table 29 , where sub ameans that the irrep is found in theantisymmetric part ofthe product. From(3.1) and(3.2) it follows that

10 = (5x 5)a = [(2, 1C)(1) + (1, 3~)(2I3)]~

= [(2,1C )x ( 2 , 1 ~ ) ] ~ ( 2 )+[(2, 1C )x (1 3C)](1/3) +[(1, 3C )X (1, 3C)]a(_4/3)

= (1, 1c)(2) + (2, 3C)(1/3) + (1, jC)(_4/3) (3.3)

where the ~ values add, since the Y5 is a U

1 generator.Thus it is immediately foundfrom (3.3) and(2.6) that the (1, 1C ) is acharge 1 , singlet lepton, to which the positron may be assigned; the (2, 3C) is aquark doublet suitable foru andd ; and(1, 3 C ) hascharge ~ and is suitable for the singlet.

In summary, the electron family with e~,u ,,, u, U , d andd may be assigned to a 5 + 10 ofSU5. Thefirst stepofmodel buildingconsists of embedding QED, SU~andQCD in the unifying group, and thensearching for a representation fL that reproduces the standard model. With three families, fL=5 +10+ 5+1 0 + 5+ 10 , which is not selfconjugate, thetheory is flavor chiral. We are certain to recoverthe results reviewed in section 2 , as long as the additional interactions implied by unification do notmodify the SU~X U~X SU~interactions too much.

Letus make afew comments concerning the progress made by going from the standard model tothe SU 5 model. (1) The three independent gauge couplings of the standard model are reduced to a

single coupling, and given the experimental value of the QCD and QED coupling, SU 5 gives areasonable account of the relativeamount of the vector and axial-vector weak neutral currents, which

depends on the free parameter sin ~ relating the two independent couplings in the standard model[1820].(2) It describes the charged current weak interactions correctly. (3) If it is assumed that thesymmetry breakingi s lacking certain terms, the neutrinoi sexactly massless due to aconservation law inthe theory. In more complicated models the neutrino often acquires asmallbut finite mass; dependingon future experimental results, the prediction of massless neutrinos may (not) continue to be anadvantage [45].(4) It qualitatively predicts the mass of the b quark.

This impressive list of successes should be compared with the ambitions of unification: (1) Thenumberoffermion families is not predictedbythetheory nor are most ofthemasses. (2 )The SU 5modeland others like it require ratios of boson masses to be of order 1012. This requirement is difficult tosatisfy with al l symmetry breaking done with scalars appearing in the Lagrangian because quantumcorrections tend to obliterate large mass ratios unless very special values of thecouplings arechosen

[46]. This may be a devastating criticism of the usual form of the SU 5 model. (3) Gravity has not yetbeenunifiedwith the other interactions, although massesoforder iO~ofthe Planckmass are involved.Ofcourse, therestill remain the questions, why SU5? why families? and wh y thereduciblechoice 5 + 1 0for each family? (Of course, not everyone considers the above to be requirements of unification, inwhichcase, they are not shortcomings.)

Let usdwell for a moment on anotherissue that SU 5 and related models do not face: theydo not


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relate particles of different spin. It would seem that a truly unified model ought to relate the gaugebosons to the other fundamentalfields in the theory. Such relations do exist i n theoriesin which thesymmetry structure has been extended beyond that of Lie algebras to include fermionic generators

satisfying anticommutation relationswith oneanotherand commutation relations with themembers ofthe Lie algebra. Such algebraic systems are called graded Lie algebras or supersymmetries, and

model Lagrangians withglobal and local supersymmetries have been proposed [47].Supergravity is perhaps themost interesting example proposed so far, especially wheni t is based onextendedsupersymmetry, where the supersymmetry charges carry an extraindex that ranges from 1 toN. The N = 8 version Lagrangian contains explicitly the following particle spectrum: a single spin 2graviton is accompanied by an SO 8 octet of spin ~ particles, a set of spin 1 particles belonging to the28-dimensionaladjoint representationof SO8, aset ofMajorana spin ~particles belonging to the 56, andscalar and pseudoscalar multiplets in two 35-dimensional representations of SO8. The SO 8 represen-tation theory is summarizedin tables36 and3 7. If this particle spectrum is identified with what we call

elementaryparticles, then theshortcomings ofsupergravity are severe: SO8is too smallto include colortimes a sufficient flavor group and the 56 of spin ~ fermions cannot include all threecharged leptons.Moreover, thereis aserious problem if the S O 8 is gauged: measuredvalues of thegauge coupling imply

acosmologicalconstant that is too largeby a factor 1 0 6 0 . So it is possible that this interpretation of theN = 8 supergravity Lagrangian is misleading, and al l the particles normally considered elementary,such as the e, j.t, ~, etc., arecomposites on distance scales oforderof thePlanckmass [48],and the S O 8is notgauged. The elementarySO8fields in theLagrangian may be tied up in bound states (except thegraviton), even at mass scales of order 1 0 1 5 GeV, where the effective Lagrangian could be locally S U5

symmetric. More will be said about aneffectivetheory derivedfrom N = 8 supergravity in section 8;its fate is not yet settled.

One may wish to speculate about afuture unified theory of all interactions and all elementaryparticles that would resembleS O 8 supergravity but involvesacrificing some principle now heldsacred,so that the notion of extended supergravity could be generalized. In such a hypothetical theory, aninternalsymmetry group 0larger thanSO 8 would be gauged by spin 1 bosons, and both the spin ~andspin ~ fermions would be assigned to representations of G. It is then very natural to suppose that thespin ~ fermionswould belong to some basic representation of G andwould include only color singlets,triplets and antitriplets. The spin ~particles would thenpresumably beassignedto a morecomplicatedrepresentation. These speculations are a major motivation for thisreview, as theywere for ref. [6].

Let usdiscuss briefly two other rathersimple models. The simple groupSO10 contains SU 5 X U~as amaximalsubgroup. (Thelabel r on theU 1 merely distinguishes it from otherU1s.)It is possible to putthe 5 and 10 of an SU5 family together into an irrep of ~ The 16-dimensional spinor irrep of SO10has abranching rule intoSU5 x U~irreps,

16 = 1(5) + ~(3)+10( 1), (3.4)

as can be found in table 43. There is a (possibly) important feature of (3.4). If afamily of left-handed

fermions is assigned to the 16, then both the antiparticles andparticles ofafamily are assigned to thesame irrep in fL. In fact, i n SO~~there exists a group operation that exchanges the left-handed particlesand antiparticles: the quarks are reflected with their antiquarks, the electron with positron, and theleft-handed neutrino is exchanged with the SU5 singlet in (3.4) [13].This chargeconjugation operation

Cmay have great significance in (for example)coming to asystematic understanding of the symmetrybreaking (and fermion masses). For example, in the SO~~theory, the mass matrixelement connecting


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the SU 5 singlet1Lto the neutral lepton t~Lin the ~ is even under C, but the Coperationtransforms the

diagonal SU5 singlet mass into the diagonal M i = 1 neutrino mass. The

1 L mass will be small if theJW= 0 mass violates Cmaximally (i.e., it is odd under C), so that the VL ~L element iszero and the

~L weak singlet term is huge. This is called a Majorana mass, since a iL ~L mass term violateslepton number. The Coperation in unified models is discussed in sections 6 and9 .

The thirdmodel, which we onlymention fornow, is basedon E

6. The fermions are assignedagain i nfamilies to the fundamental27-dimensional irrep. In fact, it is onegoal of this review to be able to treatthe E 6 representation theory smoothly and without undo complication, and so we defer until later

discussion of thisexample.Ineach of thesemodels, we needalso to discuss the symmetry breaking and other details, which will

be examimed later. Without further example or motivation, we now begin our plunge into thegroup-theoretical details that should easethose discussions.

There are,essentially, two methods forstudying the embedding ofcolor andflavor in asimple group

G. Themost directmethod is to pick out the generators (or linear combinationsofgenerators) ofG thatmay beidentified as the color andflavor generators.This method can be complicatedin practice (untiltheshort cuts are learned),but it provides so much knowledge about the structure ofG that i t is often

obvious how to proceed to a study of symmetry breaking, masses, and so on. It is thismethod that isdeveloped in some detail in sections 4 7 , inclusive. The other method, advocated in ref. [6], requireslisting thepossible flavor and color structure of the fundamental irrep of G and then checking that theconjectured embedding gives thecorrect behavior ofthe groupgenerators. Embedding through an irrepis completely general and is easily implemented. Both methods provide insights into unified models.The remainder of this section contains an informal restatement of the arguments and results ofembedding through the fundamental irrep, and at the same time, provides an elementaryintroduction

to thesimple Liealgebras. A moreformal andcompletediscussionof this approach can be foundin ref.[6],in section II and theAppendix there.

According to the Cartan classification, whichhas been proven completein numerous ways,therearefourinfinite series ofsimple Liealgebras: thealgebraA ,, generates the groupSU,,+1, which is the groupof transformations that leaves invariant the scalar products ofvectors in an (n + 1)-dimensional complexvector space; B. generates SO2,,+1, the group of transformations leaving invariant scalar products ofvectorsin a real (2n + 1)-dimensional vector space; C, , generates Sp2,,, the groupof transformations thatleaves invariant a skew-symmetric quadratic form in a real 2n-dimensional vector space; and D,,generates SO2,,,which is analogoustoB,,,but has a differentspinorand rootstructure. (Aswarnedearlierwe often use thegroupname even whenthe algebraic properties are all thatis needed.) Inaddition therearefive exceptionalalgebras, denotedbythe symbols,02, F4 ,E 6, E7 and E 8, where the subscripts denote therank.The exceptional groups leave invariant certainforms with octonions. The Jacobi identity of Liealgebras requires that the commutation relations[A,B] = Ccan be realizedby associativematrices,whichmeans that (AB)C=A(BC). Since matrices of octonions can be associative only in special cases, theexceptionalalgebrasare restrictedin number. Inbuilding Yang Mills theories, the representationtheoryplaysthe central role; although thegeometrical propertiesof octonions are not requiredfor analyzing

theoriesbasedonexceptional groups,they can be helpful forsomecalculations andmay also suggest awaytotranscend the confinesofYang Mills theories [49].

The embeddingthrough the fundamentalrepresentation is done as follows. If thereis an embeddingof the form,

GJG~xSU~, (3.5)


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then the fundamental representation ofG has abranching rule,

(3.6)

where x~is an irrep of G~and y~is an irrep of SU~.Moreover, the adjoint irrep, which is the

representation of dimension equal to the order of the group with matrix elements made from thestructure constants, is always in the tensor product f x f. (This statement is true of any nontrivialirrep.)

This embedding procedurewas worked out in ref. [6] only for the cases where the irreps of SU~in(3.6) for some irrep are restricted to the set ~C, 3C and 3~.If, for example, the spin ~fermions wererestricted to leptons, quarks andantiquarks, thentherewouldbe at least onesuch irrep. The discussioncan be madecomplete, due to the following theorem: Ifany irrep ofG has abranchingrule(3.6) with y~restricted to theset ~C, 3C and3, then the fundamental irrep must also satisfy thesame restriction. (Seethe Appendix of ref. [6] for a formal proof.) Once the color content of the fundamental irrep isconstructed, the color content of other irreps is found from their branching rules, which are easilyderived from tensor products. Although there is no direct evidence that spin ~ fermions other thanquarks and leptons exist, it may be somewhat artificial to require no other color states from, say,10 0eVto the unification mass. In fact, if theknownquarks and leptons arefit into one irrep, then it isquite usual to expect other color states. (For exceptions, see the SO 4,,+6 models in section 8.)Nevertheless, the procedureof embedding thisway is quite adequate, because embeddings where thefundamental irrep hashigher color states seem quite awkward, and little appears to begained by thenew embeddings. A possible exception may be found in models based on E8 [50],where thefundamental irrep i s the adjoint, which must have an8~.

We nowexamine the simple groups, discussing their fundamental irreps, theembedding ofcolor and

flavor, thecalculation ofthe generators, anda numberofspecial featuresof the groups and their fieldtheories.The features mentionedherecan all be derived with thetechniques discussed in section 4 tothe end.

The unitary groups, SU,,. The fundamental irrep of SU,, (or A,,_1) is the n, which is n dimensional.The most general branching rule (3.6) with the 1C , 3c 3Ccolorrestriction is, obviously, of the form,

n =(n1,1c)+ (n3,3~)+(ni, ~), (3.7)

where the a, are irreps ofthe flavor group, and n = n 1 +3n 3+ 3n~.The identificationof theflavor grouprequires astudy ofthe groupgenerators. The flavorgroup ofanSU,,theory isnonsemisimple because ofthe

U 1 factor(s). The adjoint of SU,, is computed from the n from the tensor product,

nxfl=Adj+1, (3.8)

where the adjointAdj has dimension n2 1 . Iti s quitetrivial mathematically to workout (3.8) from the

generalform of(3.7),but themost interestingcasesphysically appear to be those with n~=0 andn

3 = 1,so that n 1 = n 3 . Then the adjoint has the branchingrule of the form,

Adj= [(n 3, 1C ) +(1, 3C)] x ~ r) +(1, ~)1 (1, 1C )

= ~ (3.9)


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The product n 3 x n 3 should contain flavorgeneratorsonly, justas the (1, W) generatorsofQCD areflavor singlets. Thus n 3 must be the fundamental irrep of SU,,_3, or else that product would havecomponentsthat do not transformas flavor-group generators and G~wouldnot be as large aspossible. The

extrasinglet in n 3X n 3implies a U 1factor, so (leaving U1 values implicit)

SU,, i SU,,_3 XU 1 XSU~, n= (n 3, 1c)+(1 3C)(3.10)

Adj(SU,,) =(Adj(SU,,_3), ic) +(1,1 C ) + (1,8~)+(n . . . 3 3c) +(n3 3C)

The color content of theother irreps of SU,, can bederived fromtensor products ofthe nwith itself;this procedure generates al l irreps of SU,,. The basic irreps (basic has a special significance whenthe Dynkin labeling of irreps i s discussed) are obtained from (nx n X~ Xn),. In particular (ii)a is anirrepof dimension (~)(abinomial coefficient). It is not difficult to multiply out (n), with n given in(3.10) to find that the color statesin each ofthese irreps are restricted to the set F, 3C and 3C~Any otherirrep must havepieces in (n)5, and must havehigher color terms. For example, (a x n)5 is an irrepofdimension n(n +1 ) 12 , containing apiece [(1,3C ) x (1, 3c)]~= (1,6c).

The irreps of SU3 fall into threecategories, distinguished by their triality. The quantum numbers inone trialityclassnevercoincide withthoseof another trialityclass.In theEightfoldW ay 5U3,theelectriccharge is a representation vector label. In thetriality zeroirreps, such as the8, 10 , 27, etc., the electriccharge eigenvalue is an integer; triality oneirrep~,such as the 3, 6, 15 , etc., havecharge eigenvalues plus an integer; and triality two irreps (3, 6, 15 , etc.) have charge eigenvalues + ~plus integer. Theconcept oftriality forSU3 generalizes to that of n-ality forSU,,, where it again means that the quantumnumbers of the representation vectors in one class never coincide with those in another class.As a generalization (but trivial for SU,,), it is possible to distinguish irreps of any group in ananalogous way; it is convenient then to speak of congruency classes [9, 51]. The number ofcongruencyclasses is mentioned in thissection, and i t is more fully exploited in section 5. Irreps of SU,, fall intoncongruency classes, distinguished by n-ality.

The GeorgiGlashow SU5 model is an example of (3.10), with n = 5, 5= (54),, and 10= (5 x 5),. The

grouptheoryi softenimplementedby writing the 10 as anantisymmetric tensorA~withi ,j = 1,. . . , 5. Thisprocedure isperfectlyadequate for SU5, but i tshouldbe noted that the 10is abasic irrep,whichmeansthatthe antisymmetricpair, ij ,can in a sensebereplacedby asinglelabel. This kindofsimplificationof the grouptheory will be emphasized in the next few sections. The distinction among basic, simple,fundamental, and composite irreps ismadei n section 5.

Other examples of(3.7),onewith n~= 0 and n3>1 , and finally, n~>0 and n3> 1 are workedout inref. [6], andall threeof theseembeddings ofQCD in SU,, are listed in table3 , cases 1 , 2 and3 .

The tensor product of the SU,, adjoint irrep with itself contains an adjoint symmetrically; theexistence of a completely symmetric d~1~symbol that couples three identical adjoints (that is, threevector bosons) into an invariant occurs only for the SU,, (n >2) groups. (The coupling through thestructure constants is completely antisymmetric. Also note that SO6 ~SU 4 has a d symbol.) The

existence of the d1~symbol can be aproblem, because fermions loopscan contribute to the triangleanomaly and destroy the theorys renormalizability. If fL is self conjugate, the uncontrollable piecescancel out automatically; in that case the theory is called vectorlike [36]. If fLi scomplex, then_it mustbe reducible andofa special form for the cancelation to takeplace. The SU 5 modelwith fL = 5 +1 0 isan example where theanomaly is cancelled. However, it is noteworthy that in SU5, thereis no complexirreptowhich fLmay be assigned without parity violation of thestrong or electromagnetic interactions


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or the existence of massless charged particles. More generally, in unified models it is not difficult toavoidthe triangle anomaly problem. In theories where theCoperator exists as a group operation,thereis never any difficulty. In any event the SO 4,,+6 and E6 groups, although they have complex irreps,do

not have a d,~symbol, so there is no anomaly problem; the adjoint occurs only in the antisymmetricpart ofadjoint times adjoint, as is required by the commutation relations. We place most emphasis on

flavor chiral models, which are defined by the requirement that fL is complex and the triangleanomalyproblem i s absent [36].

The orthogonalgroups. Cartans classification indicates two kinds oforthogonalalgebras B,,(SO2,,+1)andD,,(SO2,,). In both cases (SOm,m even or odd) thedefining or vector irrep mis m dimensional andthe adjoint irrep is constructed irreducibly from (m xm)a. However, it is noteworthy that not a ll irrepsof SOm can be obtainedfrom productsofm with itself, but thereare spinor irreps, just as fundamentalas the m, from which all irreps can be formed. The reason that m cannot contain a spinor is essentially

thesame reason that a 3 cannot be constructedfrom any numberof8 sin SU3the congruenceof irrepsof SOm prohibit it [9, 51]. The irreps of B, , fall into two congruence classes; the irreps of D,, fall intofour classes. For S 0 2 ,, + ,, spinor times spinor has ordinary irreps only, but spinor timesordinary hasspinor-like irreps only. For D,,, two classes are spinor like. This classification is helpful when deriving

tensor productsand is described later i n detail.Themost notable difference in the spinor irreps of B, , andD ,, is that B,, has onlyonesimple spinor,

of dimension 2~and always self conjugate, but D,, has two inequivalent simple spinors, each ofdimension 2.When n is odd, the spinors arecomplex and conjugate to each other, and when n iseven, they are selfconjugate andinequivalent.Thus 5010, SO14, SO18, . . . a l l havecomplexspinor irrepsthat can beused for fL to makea flavor chiral theory.

In carryingout theembedding of SU~in SO,,,, the mplaysthe fundamental role. If the spinors satisfythe colorrestriction, thenso does m, but theconverse is not necessarily true. Thus, our procedure is toembed through m and then construct thebranching rule for the spinors.

The mis a selfconjugate irrep,which impliesthat the branchingrule must alsobe selfconjugatewithjustas many 3~as 3c soi t has the form,

m =(n1,1c)+ (n3, 3c)+ (113, 3 c ) m = n 1 +6n3, ( 3 . 1 1 )

wheren 1 in (3.11)must be selfconjugate. Calculating the generatorsfrom (mxm)a, wefind easilythatthe flavor groupmust beSO,,, xSU,,, x U 1. Ifn3>1 , then the simple spinor hashighercolor states.Thecase that has attractedthe most interest is n3 = 1 , in which case,

SOm JSOm_6~


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22 R. S!ansky, Grouptheoryforunifiedm ode l building

to be a deep connection between S 02,,+1 andSp2,, [52],i n a d d i t i o n t o t h e f a c t t h a t t h e y h a v e t h e s a m enumber ofgenerators. The symplectic groups haveplayed almost no role in model building, but wemention afew properties. The fundamental irrep is the 2n, and all other irreps are found in (2n),k = 2, 3 The p r o d u c t ( 2 n x 2 n ) ~gives irreducibly the adjoint, and the singlet is found i n (2n x2n),.The embeddingwiththe colorrestriction is found i n table 3, case 6. There are twocongruence classes,

which coincide with real and pseudoreal.The exceptional groups. The e x c e p t i o n a l g r o u p s , e s p e c i a l l y t h e E series, have received considerable

attention from model builders. Perhaps one of the main motivations i s that if oneof the exceptionalgroupswere part of acomplete theory, then theremight be a chance of going beyond the Yang Mills

construction and explaining why it is thecorrect choice of gauge group. At present, such hope isspeculation, although the phenomenology of the E 6models being s t u d i e d a p p e ar s q u i t e ad e q ua t e .

G2 is not large enough to contain QCD and any piece of the flavor group, since it is only rank 2 .M o r e o ve r , i t i s n o t l i k e l y to show up as a relevant subgroup because the f u n d a m e n t al i r r e p 7 h a s t h eb r an c h i n g r u l e i n t o S U3 irreps, 1+3+3; t h e a d j o i n t 1 4 b r an c h e s t o 8+3+3. Thus i f G 2 3 SU~,s e t s o f

~C, 3C and 3 a r e l i k e l y t o a p p e a r w i t h e q u a l f l a v o r quantum numbers. Since 02 has selfconjugate irrepso n l y , i t i s n o t l i k e l y t o m a k e a good flavor group either.Nevertheless, it is oftentimes helpful to refer to

the propertiesofG2when studying general propertiesofother algebrasand their representations. Thereis only onecongruence class for 02 irreps.F4has rank 4 and52 generators;its irreps are all selfconjugate,so al l F4theories arevectorlike.The

embedding of colorcan be done through themaximal subgroupSU3 X SU~,with the fundamental irrep26 branching to (8, 1 C ) +(3 3C) +(3 3C); then no other irreps o f F4satisfy the color restriction. For other

(inequivalent) embeddings of SU~in F4, there are no irreps of F4 with color states restricted to 1~,3Cand 3C F 4 is investigated in some detail later on because it is a subgroup of E 6. There is only onecongruence class forF 4 irreps.

E 6 has rank 6 and 78 generators, and holds a prominent position in this review. It is the only

exceptional group with nonseif conjugate irreps, so it is the only exceptional group for which aflavor-chiral theory is possible. Moreover, it is a generalization of SO~~(SO10may be classifiedas E5),w h i c h i s i t s e l f a g e n e r a l i z a t i o n o f SU5 ( S U 5 may b e c l a s s i f i e d a s E 4 ) , s o t h e c h a i n o f s u b g r o u p s

E 63 SO~x U~3 SU 5x U~x U~contains many of the interesting flavor-chiral theories. E 6 irreps havetriality.

Theonly maximal subgroupdecomposition of E 6containing QCDas an explicit factoris E63 SU 3 XSU 3x SU~, and the_ fundamental 27-dimensional irrep has the branching rule, 27 =

(3,3, 1C ) +(3, 1 , 3C) +(1,3, 3C) Of course, depending on the symmetry breaking hierarchy from E 6 toU ?

t m x SU~,i t may b e t h a t o t h e r maximal s u b g r o u p s , s u c h a s F4 , S O 1 0 xU 1, or SU 2 x SU6, could play a

moresignificant role. Nevertheless, in each case the same generatorsof E6 can be chosen to generateSU~,and the same diagonal generator can be identified with Q~

mit is in thissense that these othersubgroups do not give a new embedding ofcolor. In each case, the 2 7 has three3, three 3C , a nd ni ne 1 ,with thesame distribution ofelectric charges. We haveignored the embeddingswhere the 27 has onecolorsinglet and a coloroctet, since obtaining a sensible looking lepton sector is awkward. The analysisofE

6 and itssubgroups is carried out in great detail in sections6 9 .E 7 has rank 7 and 13 3 generators. All of its irreps are selfconjugate and either real or pseudoreal.

The 56 is the only irrep that can have color restricted to ~C, 3C a nd 3C, a nd t h e e m b e d d i n gE7 3 SU6 x SU~exhibits the embedding through the branching rule, 56 = (20,ic) + (6,3)+ (6, 3). Al-though models based on E7 have beensuggested, they are not currently popularbecause those theories

a r e v e c t o r l i k e , s o an e x pl anat i on o f t h e w e a k n e u t r a l c u r r e n t i s t a n g l e d u p w i t h a d e t a i l e d u nd e r s t a n d i n gof the symmetry breaking.


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The f u n d a m e n t al i r r ep o f E8 , w h i c h h a s r a n k 8 , i s i t s a d j o i n t , s o f o r a n y e m be d d in g o f SU~t h e r e

must be color states beyond ~ 3C and 3C It is the only group for which the adjoint cannot beconstructedfrom some simpler irrep. The SU3x E 6 d e c o m p o s i t i o n o f E 8 of the248-dimensionaladjointi s ( 8 , 1) +(1,78) +( 3 , 2 7 ) + ( 3 , 2 7 ) , s o t h e 2 4 8 c an b e a r r an g e d t o h a v e o ne 8 , 7 8 1 ~ ,2 7 3 and 2 7 3cw h i c h i s al m o s t w i t h o u t h i g h e r c o l o r s t a t e s , a nd i t h a s t h r e e f a m i l i e s o f 2 7 . A g a i n , a l l E 8 t h e o r i e s a r e

vectorlike,and the irreps of E 8 are a l l i n a s i n g l e c o n g r u e n c e c l a s s . E 7 and E 8will both receivesomea t t e n t i o n i n s e c t i o n 8 .

4. Dynkin diagrams

T h e r e e x i s t s m u c h l i t e r a t u r e d e s c r i b i n g t h e t h e o ry a nd a p p l i c a t i o n o f g r o u p t h e o r y , a nd e s p e c i a l l y L i eg r o u p t h e o r y , t o p r o b l e m s i n p h y s i c s . O f t e n f a i r l y s m a l l L i e g r o u p s su c h a s SU 2 , S U3 , SO4o r SO5 a r e o f

i n t e r e s t ; f o r t h e s e , o ne may g e t t h e i mp r e s s i o n t h a t d e r i v i n g t h e co m m u t a t io n r e l a t i o n s , representations

a nd t he i r c o n t e n t , t h e s u b g r o u p structure, tensor products, vector coupling coefficients, recouplingc o e f f i c i e n t s , and s o o n i s s t r a i g h t f o r w a r d , b u t t e d i o u s . How ever, u n i f i e d m o d e l s a r e b a s e d o n m u c h

l a r g e r g r o u p s ( r a n k f o u r o r m o r e ) , s o t h e r e a p p e a r s t o b e c a u s e f o r a n x i e t y ove r t h e a l g e b r a i cc o mp l e x i t y t h a t m u s t b e fa c e d i n d e r i v i n g t h o s e r e s u l t s . Our p u r p o s e i n t h e n e x t t h r e e sectionsi s to setu p t h e a n a l y s i s o f s i mple g r o u p s i n su c h a way t h a t s o m e o f t h e c h o r e s e n c o u n t e r e d i n u n i f i e d m o d e l

b u i l d i n g are n o t a s t e d i o u s a s m i g h t b e e x p e c t e d .Simplegroups, their representations, and subgroup structure have been studiedby many mathema-

ticians and physicists, but perhaps the most convenient approach for dissecting YangMills theoriesbased on largesimple groups is the one introduced by Dynkin in the earlyfifties [9]. Of course, it iswidely understood that fieldtheory is an especiallyconvenientformalism for describingsymmetries, andputting internal quantum number labels on field operators in a YangMills theory is conceptuallysimple. The reason why Dynkins labeling is so useful is that the action of a generator, or a tensoroperator,on a state is designateda little moreconveniently forb ig groups than,say, by tensorlabels; itis easier to do thebookkeeping. For example, an important stepin exploring a theory is identifying the

color andflavor quantum numbersofa field that transforms as a componentofsome representation ofthe Yang Mills group 0, andfindingout how it is transformed through its interactions with the vectorbosons in the theory. This problem is reduced to integer arithmetic, and constant reference to thecommutation relations is not needed except through the Dynkin diagram. Our account is brief,informal, anddescriptive, with emphasis on the results needed to derive the many tables. Rather thanproving theorems, we useexamples forguidance. The mathematics can be found elsewhere [9, 11 , 12].

The maximum number of simultaneously diagonalizable generators of a simple Lie algebra G iscalled its rank 1 ; the total number of linearly independent generators is called its dimension. A simplegroup has no invariant subgroups, exceptthe whole group and theidentity; analogously, asimple Liealgebra contains no proper ideals. A semi-simple algebra can be written as a direct sum of simplealgebras.Excepting the study of subalgebras,we discuss simple algebras only; U 1 is notsimple.

In the standard CartanWeylanalysis, the generators arewritten in abasis where they can bedividedinto two sets. The Cartan subalgebra, wh

slansky: group theory

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