PHYSICAL REVIEW 0 VOLUME 14, NUMBER 6 15 SEPTEMBER 1976

Group theory for Feynman diagrams in non-Abelian gauge theories*

Predrag Cvitanovic~Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

(Received 19 February 1976)

A simple and systematic method for the calculation of group-theoretic weights associated with Feynman

diagrams in non-Abelian gauge theories is presented. Both classical and exceptional groups are discussed.

I. INTRODUCTION

The increased interest in non-Abelian gaugetheories has in recent years led to the computa-tion of many higher-order Feynman diagrams. ' "Asymptotic form-factor and scattering amplitudecalculations are of special interest, because theysuggest that it might be possible to sum up dia-grams with arbitrary numbers of soft gluons justas one can sum up soft-photon processes in QED.In such a program the analysis of the momentumintegrals proceeds by the traditional techniquesdeveloped for QED calculations. The new aspect,characteristic of non-Abelian gauge theories, isthe emergence of a group-theoretic weight (orweight, "for short) associated with each Feynmandiagram. The dramatic cancellations among vari-ous diagrams occur through interplay of theirgroup-theoretic weights and their momentum-spaceintegrals. So the study of weights becomes ofinterest, as it might suggest cancellation patternsneeded for summations of diagrams.

In this paper we give a general method for com-puting group-theoretic weights, and give explicitrules for SU(n), SO(n), Sp(n), G„E„F„andE,symmetry groups. We restrict ourselves to themodels with quarks in the defining (lowest dimen-sional) representation, but the method can be ex-tended to higher representations. .4,s only globalsymmetry is assumed, we can compute weights notonly in symmetric gauge theories, but also in thosespontaneously broken gauge theories where all par-ticles within a multiplet have the same mass.

The evaluation procedure is very simple. Wethink of the weight itself as a Feynman integral(over a discrete lattice), and introduce Feynmandiagrammatic notation to replace the unwieldy al-gebraic expressions. Then we give two relations;the first eliminates all three-gluon vertices, andthe second eliminates all internal gluon lines. Theresult is a sum over a unique set of irreduciblegroup-theoretic tensors which form a natural basisfor all Lie algebras. All this is accomplishedwithout recourse to any explicit representation ofthe group generators and structure constants. Asa by-product, we learn how to count quickly thenumber of invariant couplings for arbitrary num-

bers of quarks and gluons, thus avoiding involvedreductions of outer products of representations byYoung tableaux.

In most calculations, one looks for propertieswhich arise solely from gauge invariance, andthere the explicit numerical values of weightsshould really not be necessary. While in somesuch calculations' ""it is appealing to expresssimple diagrams in terms of quadratic Casimiroperators (so that the form of the expression isindependent of the particular gauge group and theparticular representation), for higher-order dia-grams there is no simple way of relating weightsto generalized Casimir operators, '""and such anapproach becomes very cumbersome. Then theexplicit expressions for weights might be bothsuggestive and useful as checks for the cancella-tions among various diagrams. Another applica-tion of explicit weight expressions is 1/n expan-sions" for which the above evaluation method givessimple and direct estimates. "

Possibly, a novel aspect of this paper is itstreatment of exceptional groups. It is known"that exceptional groups arise from invariance ofnorms defined on octonion spaces, but the demon-stration is rather difficult (it involves Jordan al-gebras over octonionic matrices). We skirt thecomplexities of this underlying structure by givinga formulation of exceptional groups purely in termsof the geometrical properties of their definingrepresentations. Intuition so developed might beof use to quark-model builders. We give the fol-lowing example: Because SU(3) has a cubic invari-ant &'"q,q,q„ it is possible to build a three-quarkcolor singlet with desirable phenomenologicalproperties. " Are there any other groups thatcould accommodate three-quark color singlets'PIt turns out that the defining representations ofG„F„and E, are among groups with such invari-ants. A systematic discussion of such invariantsshall be given elsewhere. "

In the past, most weight calculations have in-volved SU(n) and, even more specifically, SU(3).This has led to the development of methods speci-fic to SU(n). 25 " For the sake of completeness andcomparison, we pursue this traditional line forawhileandfind ourselves at an impasse.

1536

GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1537

The organization of the paper is as follows. InSec. II, we state the evaluation rules. In Sec. III,we introduce diagrammatic notation and derivevarious relationships true for all Lie groups,while particular groups are defined in Sec. IV. Anexample of weight evaluation is given in Sec. V. InSec. VI, we discuss group-theoretic tensor basesand relations between basis tensors for specificrepresentations, while higher representations aretouched upon in Sec. VII. Full Feynman rules arestated in Appendix A. Appendix B is a long dis-cussion of an older method of weight evaluation,specific to SU(n). For readers interested only inmodels with classical symmetry groups, Figs.1-3 summarize all that is needed for weight com-putation.

F""= e"4"—8"A"+ «A'4"f/' g k)

D~» = 5»a" —iA,"(T,)„(2.1)

a, 5=1,2, . .. ,n, i,j=l, 2, .. . ,N

where the n complex quark fields q, transform asthe defining (lowest-dimensional cogredient) rep-resentation of a compact simple N-dimensionalLie group 8, and the N Hermitian gluon fields A,transform as its adjoint (regular) representation.In Yang-Mills theory the coupling constant e of theusual QED is generalized to quark-gluon couplingmatrices (T,)». They are generators of 9, close aLie algebra

II. RULES FOR THE EVALUATION OFGROUP-THEORETIC WEIGHTS

For our model we take a Yang-Mills theory withmassive quarks of n colors and N massless gluons,defined by the classical Lagrangian density

2 = —4F(""F)„„+q(i$ —m)q,

1iC)~» = Tr(T( TjT» ——T»T~T,.); (2.5)

man diagrams. There is no mixing between thespacetime and the gauge group 9, and the Feynmanamplitude associated with a diagram 6 factorizesinto 8'~M~, where 8'~ is the group-theoreticweight consisting of various (T, )» and C,», and

M~ arises from the integrals over internal mo-menta and is similar to QED Feynman amplitudes.Even though M~ will not concern us in this paper,we give the rules for its computation in AppendixA. We note that while in momentum space thereare four-gluon vertices, for S'~ there exist onlythree-gluon couplings, because the group-theo-retic factors in a four-gluon vertex have the form

«~»«a~The group-theoretic weight S'~ is a product of

the following factors (all repeated indices are sum-med over}:

(a) for each internal quark line, a factor 5»„a, 5

1 ) 2) 0 ~ ~ )n)(b) for each internal gluon or ghost line, a factor

5)), i)j=1,2). .. )N)(c) for each quark-quark-gluon vertex, a factor

(&)).',(d) for each three-gluon or ghost-ghost-gluon

vertex, a factor —i«,»,(e) for the four-gluon vertex, the factors

—(C, &C, »+ C,„,C&~) (multiplying g~g„,),—(C&~»C&„,+ C, &C», } (multiplying g»&g»),—(C,„gC» ~+C, »C, ~q) (multiplying g», gc„),where gluon group and Lorentz indices are pairedas (i, X), (j, p, ), (k, v), and (l, g) (see also Fig. 24).

The weight 8'~ for an arbitrary Feynman ampli-tude 6 is evaluated in two steps:

(1) Reexpress all three-gluon vertices —i C,» interms of the defining representation:

(a) If 9 is SU(n) or E,

f. ~g & ~jl ~Ci»i»t

TrT, =0)

(2.2)

(2 2)

(b) if 9 is SO(n), Sp(n), G„F„orE,

2i C,» = Tr(T, TzT»). —a

(2.6)

Tr(T)Tq) = a5 gg. (2.4)

For example, for SU(n), the conventional choice"is T, =&gX,. and a=&g'. In this paper a shall re-main a,rbitrary throughout. ~a can be thought of

as the overall coupling constant for a simple group

9, and powers of Wa count the number of verticesin a diagram. That the gluon self-couplings -iC;~~also scale as ~ais evident from (2.2).

The Lagrangian (2.1) generates the usual Feyn-

and can be chosen to satisfy a normalization condi-tion"

5~»5; — 5,'5~» for S-U(n),n'"—(Tg)»(T, )„'=( 2(5~5; —5"5»~) for SO(n),

» (5~5»+f"fM) for Sp(n),

(2.'I)

(2.8)

(2 9)

with n even, f"= f",f"f,»=5;. -

(2) Eliminate all internal gluon lines' ' ' (T,)~

~ ' ' (T,); by replacing them with gluonprojection operators:

PREDRAG CVITANOVIC 14

2 (3„'3f—3"5M) —f—'~,f"~ for G„(2.10)

for E3Q

21( 1)

(2.12)

for E(2.13)

(a) SU(n), E6

(We do not know how to evaluate E,.)The rules for the exceptional groups are supple-

mented by the identities of Sec. IV which define theassociated invariants. Graphically, the aboverules are summarized in Fig. 1. Section V givesan example of how the rules are used in a typicalcomputation.

III. LIE ALGEBRA IN DIAGRAMMATIC NOTATION

A group-theoretic weight 5'z can be visualizedas a Feynman diagram in which the internal linesrepresent sums over all colors of the associatedparticles, and vertices represent their couplings.There is never any need to label the lines andvertices; the equivalent points on the paper repre-sent the same index in all terms of a diagram-matic equation. " Besides automatically keepingtrack of indices, diagrams make it easier to rec-ognize the symmetries of more complicated ex-pressions.

In this section algebraic relations shall be trans-cribed into diagrammatic equations which applyto any semisimple Lie algebra with quarks in anyrepresentation. The diagrammatic Feynman rulesare given in Fig. 2. Figure 3 summarizes thebasic relations of a semisimple Lie algebra. Notethat Fig. 3(a} fixes the sign convention for -iC,.&~,

indices circle the vertex in anticlockwise direction.If the direction of the quark line were reversed,the right-hand side would change sign.

Figures 3(e) and 3(f) count the numbers of quarksand gluons, respectively: 5,'= n, 5,'. =N. The above

(b)

SO(n), Sp(n),

62, Fg, E7

SU (n)

20

for rea I representations:(a) a b= sa a,b=l, 2, '' n

a — b = Sab

(b) i j = 8 j i, j = I, 2, ~ ~ N

SO (n)I

(a) a -I b =(Ti)abfor real representations:

Sp (n) JE +02

02 i j

=-iCijk ~

E6(27) 6 it+-IS ~

I I I I )

4! + + 0 ~ ~

F~ (26)+ ~ ~ ~ ~

E7(56) = — 1~ ik +0 224 a

FIG. 1. Weight evaluation rules for the defining re-presentations of al.l simple groups except E8. (a) Elimi-nation of a three-gluon coupl. ing -i C;~&, (b) eliminationof an internal gtuon line. Further rules for exceptionalgroups are given in Sec. IV.

FIG. 2. (a) Quark propagator, (b) gluon or ghost pro-pagator, (c) quark-quark-gtuon vertex. The arrow de-notes the direction of multiplication of 7.'; matrices.Whenever omitted, it is assumed to be pointing to thel.eft for quarks going through the diagram, or counter-clockwise for closed quark loops, (d) three-gluon orghost-ghost-gluon vertex. Indices circle the vertexcounterclockwise, (e) symmetrization symbol. , (f) anti-symmetrization symbol, (a generalized Kronecker &).

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1539

c,Q=

FIG. 5. A diagrammatic computation of the quadraticCasimir operator for the fundamental representation.

(e)

/Xy r&definitions already enable us to perform somesimple calculations. For example, to calculatethe quadratic Casimir operator for the quark rep-resentation, Fig. 4(a}, we form a trace (join theexternal quark lines) and use Figs. 3(c), 3(e), and

3(f), as outlined in Fig. 5, to obtain

0 =N

NC~=a —.

n(3.1)

FIG. 8. (a) Lie commutator for the quark representa-tion, (b) tracelessness condition ("color conservation"),(c) normalization convention, (d) Jacobi identity (or Liecommutator) for the adjoint representation, (e) quarknumber, (f) gluon number.

In other words, if we know the g1uon projectionoperators [as those listed in (2.V) through (2.13)],we can comPute the dimension of the algebra bytracing the normalization relation (2.4):

(3.2)

(b) -=cA

(c) gy c2

21%

FIG. 4. (a) Quadratic Casimir operator for the de-fining representation, (b) quadratic Casimir operatorfor the adjoint representation. The remaining figuresare examples of the reduction of (c) a quark-quark-gluonvertex, (d) a three-gluon vertex, and (e) another quark-quark-gluon vertex.

Existence of the gluon Casimir operator C„[seeFig. 4(b)] is a necessary and sufficient conditionthat the algebra is semisimple. For compactgroups C„&0. If the group is simple, '""

Tr(T&T&) = /Tr(C&C&)

[where f is called the index of the representation,and the adjoint (or regular) representation of 9 isconstructed from matrices (C,)»= -iC,»], Figs.3(c) and 4(b) are compatible. For a semisimplegroup, this is generally not true. Joining gluonindices in commutators Figs. 3(a) and 3(d) leadsto relations in Figs. 4(c) and 4(d). Similarly, therelation Fig. 4(e) follows from the commutationrelation Fig. 3(a).

The antisymmetry of C,» leads to vanishing ofnonplanar diagrams of Fig. 6 as well as all dia-grams that contain these as subdiagrams. Thisfollows from the commutation relations of Fig. 3,but it is easily seen as a consequence of the skew-ness of C,», Fig. 2(d). For example, interchangeof vertices 1 2 in Fig. 6(a), and 1 2, 3 4,and 5 6 in Fig. 6(d) gives a factor (-1)' fromskewness of C,~~, while the diagrams are mappedinto themselves. The obscure diagram of Fig.6(d) is related to the Peterson graph' in graphtheory, while Fig. 6(a} is related to the nonplanarKuratowski graph. "'"

One quickly runs out of relations achievable byLie algebra manipulations. For example, at this

1540 PREDRAG CVITANOVIC 14

IV. WEIGHT EVALUATION

=0 Our objective is to express the group-theoreticweight of an arbitrary diagram as

(4.1)

(b)

where T' ' are some basis tensors which carrythe external particles' indices, and C are realcoefficients. If T' ' are independent, C can becomputed by solving a set of linear equations

W"=C f ", m, n=1, 2, . . . , P (4 2)

(c)

/ X

=0

(4) =0

FIG. 6. Some diagrams that vanish because of theskew symmetry of C;;&.

FIG. 7. A sixth-order quark-quark-gluon vertexgraph.

point we have no clue to the evaluation of the gluonCasimir operator C„of Fig. 4(b), let alone anymore complicated diagrams, such as the one ofFig. 7 (related to a quartic Casimir operator).For that, it is necessary to concentrate on specif-ic groups, as we shall do in the next section.However, it should be emphasized that for thestudy of general properties of gauge theories, thetechniques of this section are all that is needed.For vertex exponentiation this is evident frompublished calculations. " For quark-qua, rk andgluon-gluon scattering' ' the difficulty lies infinding a spontaneous -symmetry-breaking me-chanism which maintains the global symmetry(i.e. , equal gluon masses). When such a schemeis found (as for example in the Bardakci-Halpernmodel" ), one finds that nothing specific to thegroup enters into cancellations between relevantweights. 46

where t "=-T~'~' T "' ~=Tt™g T~' & is ob-tained from T' ' by a reversal of all quark lines,and the product is formed by a contraction of allpairs of corresponding indices. For example,

any gluon self-energy weight can be expressed interms of a single basis T"' = 6,&, W, &

= C,5, &(in

this case W'= W«and t"=X).W" and t "are weights of diagrams with no ex-

ternal legs, which we shall refer to as vacuumweights. From (4.2) it is clear that vacuum weightscarry all the information needed for weight evalua-tion. They also have a direct combinatoric signif-icance. We have already noted that single-loopvacuum weights count the number of ways in whicha loop can be colored [Figs. 3(e) and 3(f)]. Forarbitrary weights a hint is given by SO(3), wherethe weight of a gluon vacuum diagram is simplythe number of ways of coloring the lines of the dia-gram with the three colors meeting at each vertex. 4'

In general, a vacuum weight is a combinatoricnumber generated by some more complicated"graph coloring rule. "

How is this "coloring rule" built into vacuumweights? If we eliminate gluon self-couplings by(2.5), we note that the remaining couplings (T,);always appear in the combination (T,.)~~(T&)„'. It isthis combination that must implement the "coloringrule. " What is its significance? As (T, )~q~q, trans-forms as the adjoint representation (see Behrendset al. ,

52 Sec. V A for a demonstration), (T, )~(T, )fq~q,picks out the part of the quark-antiquark productthat transforms as a gluon. Repeated applicationsof (1/a)(T;);(T;)~ reduce to a single applicationthrough the normalization convention (2.4); hence,we shall refer to (1 a/)(T;) (~T&) ~as the gluon Pr oj ection oPexatcn . The problem of weight evaluationis solved once such projection operators are known.

A gluon projection operator is also a weight (dia-grammatically, a Born term in quark-quark scat-tering), and, according to (4.1), we can expressit in terms of quark-quark scattering basis ten-sors. To construct a complete set of these, weneed to know all invariants of the particular quarkrepresentation. There is no simple way to enu-

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . . 1541

merate the invariants of an arbitrary representa-tion; let us instead concentrate on models withguarks in the defining representation (the lowest-dimensional cogredient representation47"b). Allhigher representations can be constructed from thedefining representation; in particular, the adjointrepresentation emerges as the (T&)~q~q, term in theClebsch-Gordan series qq = A. S . . Further-more, in the defining representation a classicalgroup has a simple geometricalinterpretation[suchas length preservation for SO(n)]. The main thrustof this section will be to use such invariance prop-erties to characterize the exceptional groups aswell.

P(Gq) =P(q),

P(q)=g" 'q.q, " q~.

(4.3)

Infinitesimal parametrization G = 1+i&aT& gives usa differential statement of P(g) invariance,

sP(Gq)9&)

so that the generators (if a nontrivial group exists)must satisfy'

A. Invariants of the defining representation

Motivated by the existence of invariants such as5' q,q„ for SO(n), we study unitary transformationsG~ which preserve an arbitrary polynomial

of exceptional groups, the invariance conditionsare so constraining that they can be realized onlyin certain dimensions" (dimensional constraintsalready appear in classical groups; the symplecticinvariant can be realized only in even dimensions).

Our intention is merely to demonstrate that if weknow the invariants of the defining representation,we can construct the gluon projection operatorsand evaluate any weight. Hence, we shall simplystatetheprimitive invariants for each defining rep-resentation and show the conditions they must sat-isfy. Again, as we are computing vacuum weights,we shall find that no explicit realizations ofg' '

are needed, only some identities which implementthe "coloring rules. "

All simple Lie algebras are generated by a smallset of primitives which are either fully symmetric(d'b"") or fully antisymmetric (f'"'"'). All de-fining representations preserve 5b and the Levi-Civita tensor in n dimensions, E' '"~. Their fur-ther primitive invariant tensors are

SU(n): ~ ~ ~,

SO(n): 5, ,

Sp(n): f'b, n even

Gb 5abtfabc~

(g)s cb" f+ P )bgc f+ (T")f+b ~ c 0 (4 4)

Contracting this with (1/a)(T, ), we obtain an invati'ance condhtion for gluon projection operators.

Suppose g b' is an invariant tensor. Then

g b'g, ~„g"'g"+, and so forth automatically satis-fy (4.4) and give us no new constraints on T, . Letus therefore concentrate on primitive invarianttensors (primitives). They are defined by therequirement that any invariant tensor can be ex-pressed in terms of chains of their contractions(which, diagrammatically, can be disconnected orconnected, but cannot contain loops). We assumethat the number of primitives is finite [hence, thenumber of bases in (4.1) is also finite]. Any weightis expressible in terms of primitives; in particu-lar, the gluon projection operator will be of theform

{a)

(c)

dab"c =0 ~

ab- c

fab"c

ab- ~ c ab. ~ c

I I

) 1 ~ ~ (~ ~ & &

= 0

dab "c~ ~

)ab- c

= a

+ Cbg gbM+ ' ' ' . (4.5)

Substituting this into invariance conditions (4.4), weobtain conditions on C„and g b"'", which, as willbe shown, suffice to determine the gluon projectionoperator up to the overall normalization g. In case

FIG. 8. (a) Diagrammatic notation for fully symmetrictensors 4' '', d~q. ..~ and Mly antisymmetric tensorsf' '', f, t ..., , (b) invariance conditions for gluonprojection operators, (c) normalization convention forgluon projection operators, (d) normalization conventionfor cubic quark self-couplings.

1542 PREDRAG CVITANOVIC 14

d d'~ - o.5"abc

ycM ~6d

n for different groups need not be the same.

(4.6)

(4.7)

B. Special unitary groups SU(n)

The defining representation of SU(n) is a set ofall unitary (U~U = 1) and unimodular (detU = 1)[n x n] matrix transformations acting on an n-di-mensional complex vector space (n qua, rks). Theinfinitesimal transformations can be parametrizedby N=n' —1 traceless Hermitian matrices (T~);which close a Lie algebra (2.2). The invariantsare the Hermitian (sesquilinear") metric 5;(which imposes the unitarity condition; qq =—q'5g,is preserved) and the Levi-Civita tensor in yg di-mensions, &' '"f. The contragredient Levi-Civitatensor acts as an inverse to the cogredient one inthe sense that a direct product of the two can beexpressed as a generalized Kronecker 5 function[see also (6.4)]

E6

F&: &ao~d.ac~

E . y~~ d~~«

E,: 5,&, C,», unknown.

Before we proceed with the discussion of indivi-dual groups, let us make a few observations thatwill apply to all cases. Owing to the full (anti)symmetry of (f' "') d"'" tensors, the invarianceconditions can be stated very compactly (Fig. 8).f'~"' and d""' can be interpreted as quark self-couplings. Unlike quark-gluon couplings (T,)~,whose scale is fixed relative to C&» by (2.2), theyhave no a priori relation to gauge couplings, andto characterize their scale we introduce an arbi-trary normalization n. For cubic couplings we candefine n by

gpeoof gagee of~Pq e ~ ~ ff +Pq e ~ of4 ~ (4.8)

Gluon projection operator expansion (4.5) is ofthe form

1(T(—)t',(T))„'= A (5„'6;+b6f 6~),a (4.9)

C. Special orthogonal groups SO(n)

The defining representation of SO(n) is a set ofall orthogonal (R R = 1) and unimodular (detR = 1)[n x n] matrix transformations acting on an n-di-mensional complex vector space (n quarks). Thedefining invariant is a symmetric tensor d" =d"(and its inverse d„=d„) introduced diagrammati-cally in Fig. 10(a). The remainder of Fig. 10derives the gluon projection operator from the in-

(a)

sO (n)

abdab

which we give diagrammatically in Fig. 9(a). [+nypossible z'"'"' terms reduce to the above two by(4.8)]. Substituting this expression into E' '" in-variance condition Fig. 8(b), we obtain the equa-tion Fig. 9(b), which, when contracted with 6~ (inthe only way possible, the incoming line with anyoutgoing line) yields b= —1/n. We now see how aprojection operator" works; 6;6,' removes thesinglet from a quark-antiquark state, leaving N=n && n —1 gluons. Tracelessness of T, ensuresthat the gluon does not connect to the vacuum (i.e. ,that the group is semisimple). From the normal-ization convention Fig. 8(c) A = 1, and we can verify that the number of gluons is indeed N=n' —1 byevaluating (3.2).

(b) =A I +b +C

(a)t'

=A I'+b

(c) O=, , + b +C y ~ b=O, C=-~A

(b) 0&f 'll)'lF ~ ~ 1P

b = -)/n

FlG. 9. (a) The most general, form of the gluon pro-jection operator for SU(n), (b) the Levi-Civita tensorinvariance condition.

FIG. 10. (a) Diagrammatic notation for SO(n)-invari-ant tensor d, q, (b) the most general form of the gluonprojection operator for SO(n), (c) d, q invariance condi-tion, (d) gluon projection operator for SO(+).

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1543

(a)

Qb

dab

p even

defining representations. " Construction of thegluon projection operator (Fig. 11) proceeds as inthe SO(n) case.

E. Exceptional group G2 (Ref. 56)

(b)

(c)

FIG. 11. (a) Diagrammatic notation for Sp(n)-invari-ant tensor f,&, (b)f~q invariance condition, {c)gluonprojection operator for Sp(n).

variance of d" [Fig. 8(b)]. By diagonalizing d"and rescaling q' fields, we can always find a rep-resentation where d„=5,~. There is no distinctionbetween upper and lower indices (quark = antiquark,the representation is real), and in diagrams we canomit all d" tensors and all line arrows, and notethat because of (4.4).the generators are antisym-metric: (T,),~

= —(T, )~, They ar. e clearly trace-less, and it is easily verified that the Levi-Civitatensor &"" in n dimensions is preserved as well.

ln the conventional choice of SO(n) generators""4with only two nonzero elements +1, the normaliza-tion is fixed by p = 2g'.

D. Symplectic groups Sp(n)

The defining representation of G, (n= 7) pre-serves a symmetric invariant 5„[G,is a sub-group of SO(V)], and a fully antisymmetric cubicinvariant f'". lt is possible to show that G, is theonly nontrivial simple group that possesses suchinvariants, '4 and that f„,must satisfy the alterna-tivity relations'"" given in Fig. 12(b). By theserelations two out of three tensors f„,f,~, f„,f,M,and f~,f,», can always be eliminated in favor ofthe third and some combination of 5", s. As in the

SO(n) case, 6,~ invariance makes generators T,antisymmetric, and the gluon projection operator(4.5) has the form given in Fig. 13(a). From the

identity Fig. 13(b), we derive relation Fig. 13(c},which determines the gluon projection operatorthrough invariance of f,„[Fig.8(b)]. Actually,Fig. 13(c) (through a few more applications of thealternativity relations) leads to a very strongstatement" that any chain of three f,~, can be re-duced to a sum of terms linear in f,~, by the equa-tion of Fig. 13(d). This guarantees that eventhough the projection operator (2.10}replacesinternal gluon lines by internal quark lines, the

resulting weights can always be reduced to the

bases (4.1). The gluon number, evaluated by (3.2), isindeed N = 14. Further relations are given in Fig. 14.

An explicit realization of tensors f„,is given by

octonions. '"" In this framework 6, is the auto-

The invariant preserved by the defining repre-sentation of Sp(n) is a skew-symmetric metric4"'4f' = —f '(anditsinversef, ~= f~,). Aninv—erseexists only iff '' is nonsingular, det(f) e 0. The skew-symmetry off ' allows that only for even-dimensional

(a)Gp

=A I S+

(b)

(a)a b (c) =0~ b= —)

{b)(d) —6

(c)

FIG. 12. (a) Diagrammatic notation for the tensor f,~~for the exceptional group 62(7), (b) the "alternativity"relation which relates contractions of pairs of f,q, , (c)the invariance condition for f,q~ .

FIG. 13. (a) The most general. form of the gluon pro-jection operator for G2(7) pe~ invariance has alreadybeen imposed), (b) an identity between contractions ofthree f,&~ which arises from the skew-symmetry off,q~, and leads to (c) the invariance condition for thegluon projection operator, (d) identity that reduces anychain of contractions of more than two f, &~ .

1S44 PREDRAG CVITANOVIC 14

(a)

(b) I

2

(a)

I I

Vga f C =-)0

{e) —— ' = — -: -: +2 + —a6

FIG. 15. (a) The most general form of the gluon pro-jection operator for E6(27). (b) Springer's relation. To-gether with the invariance condition for the gluon pro-jection operator, it fixes the constants in (a).

36Q2

FIG. 14. Some derived relations between f,~, tensorsuseful in the computations of weights for G2.

morphism group of octonions, i.e. , it is a set ofall [7 x 7] real matrices G,~ such that the trans-formation

15(a)] . Evaluation of (3.2) yields the dimension ofthe algebra of E„N= 78.

Springer's relation arises from the characteris-tic equation for [3 x 3] Hermitian octonion matri-ces. The gluon projection operator (2.11) wasactually first constructed by Preudenthal" in avery different notation (as a derivation of a Jordanalgebra). His normalization convention is n= —,'.

preserves the octonionic multiplication rule

eaea = 5an+fabcsc' (4.10)

(o)

where f„,are given explicitly in Ref. 58; for ourpurposes, it is sufficient to note that octonionssatis fy the alternativity condition if

[xyz] —=(xy)z —x(yz),

[xyz] = [waxy] = [yes] = —[yxz],

(b)

tTII . A AI II I I &

"I I II {

vrhere x,y, z are arbitrary octonions. The alterna-tivity relation Fig. 12(b) follows from the multi-plication rule (4.10) and the alternativity condi-tion. " Equation (4.10) also fixes the normalization(4.7) g = —8. Then —n is simply the number ofdistinct colorings of diagram Fig. 8(d) allowed bythe octonion multiplication rule.

28I I

I II I

I I I

F. Exceptional group E6

The defining representation of E, (n= 27) pre-serves a fully symmetric cubic invariant d,~, (andits inverse d'~').""'""No condition relatingd„,d ' type tensors exists and the only nontrivialrelation" on d'~' tensors is a trilinear Springerrelation" [Fig. 15(b)] which arises from the re-quirement of d'~' invariance [Fig. 8(b)). This re-lation enables us to compute the gluon projectionoperator [whose general form is given by Fig.

lt W I &

FIG. 16. Diagrammatic notation for the tensor d, &,for the exceptional group F4(26), (b) "characteristic"relation which relates contractions of pairs of d, ~~, (c)expansion of this identity [which fol.lows from (b)] leadsto (d) a relation between contractions of three d, q, .Antisymmetrization in top legs and symmetrization inbottom legs yields (e) the Jordan identity which togetherwith the invariance condition for d, &~ fixes the gluon pro-jection operator for E4(26).

GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 154S

E7

(b)

=0

&a7

dabcd

a bed

(b)

=0

(c)

FIG. 17. Some derived relations between

daric

tensorsuseful in computations of F4 weights.

(d)

G. Exceptional group F4

The defining representation of F, (n= 26) pre-serves" ~'8' both d,~, and 5,~. To derive F4, it isnecessary to assume that a relation between bi-linear combinations d„,d„„exists. The only non-trivial relation" of such type is the characteristicrelation" of Fig. 16(b). The gluon projection op-erator is constructed the way it was constructedfor G, . The identity of Fig. 16(c) leads us to theJordan identity of Fig. 16(e), which together withthe d,~, invariance [Fig. 8(b}] fixes the projectionoperator up to an overall normalization. Thenormalization convention [Fig. 8(c)]then yields thegluon projection operator given in (2.12). There areN= 52 gluons. Further relations are given in Fig. 17.

An explicit realization of tensors d„, is givenby octonion matrices. In this framework" F, isthe isomorphism group of the exceptional simpleJordan algebra of traceless Hermitian [3x3] ma-trices x with octonion matrix elements. The non-associative multiplication rule for elements xcan be written as

(e)

FIG. 18. (a) Diagrammatic notation for the tensord, q, & for the exceptional group E, (56), (b) symplecticinvariant tensor f'~ relates + " and da&c&, (c) themost general form of the gluon projection operator (fa~

invariance has already been imposed), (d) Brown relationwhich relates contractions of pairs of da~cz, (e) reduc-tion of a one-loop diagram.

The characteristic equation for traceless [3 x 3]matrices

x' ——,Tr(x')x —3 Tr(x') 1=0

gives a relationship between contractions of pairsof d, „„drawn in Fig. 16(b). (Characteristic equa-tions are discussed in Sec. VI.) The Jordan iden-tity (xy)x' = x( yx') is automatically satisfied; it isjust the relation of Fig. 16(e). Normalization isfixed by (4.11), n= —, .

x=—x,e, g=1, 2, . . . , 26

Tre, =0, e, is a [3x 3] basis matrix,

~a~~a~g g~a 3

~ + agcec &

Trl = 3, 1 is a [3 x 3] unit matrix.

(4.11)

Transformations of F4 preserve the quadraticform Tr(x') [the length in 26-dimensional space,so that F, is a subgroup of SO(26)], as well as afully symmetric cubic form

Tr(xyz) = Tr(yxz) = Tr(yzx)=d, ~~,y,z, .

H. Exceptional group E7

The defining representation of E, (n= 56) pre-serves a skew-symmetric tensor f" [E, is a sub-group of Sp(56)] and a fully symmetric quarticinvariant, e'e4'" d'~~ [Fig. 18(a)]. f,~, f'~ raise andlower indices [Fig. 18(b)]. The gluon projectionoperator can have the general form of Fig. 18(c).The invariance of d,„~ gives the Brown relation"[Fig. 18(d)], which enables us to compute Fig.18(e), impose the normalization condition Fig.8(c), and derive (2.13}. The evaluation of the gluonnumber gives N= 133.

1546 PREDRAG CVITANOVIC

In the explicit realization of tensors d,b~ by oc-tonion matrices, "the conventional normalizationis @=3.

I. Exceptional group Es SU(0) n -) 2n2 n~ U—p2

The defining representation of E, (n=N= 248) isalso the adjoint representation, so our method ofreducing everything to the lowest-dimensionalrepresentation is of no help. Still, if the invari-ants of the defining representation of E, wereknown, we would be able to reduce higher-orderweight diagrams to a basic set just as for all othersimple groups. Known invariants are 5„andC„, and other invariants are certainly higher thanquartic. The Tits construction, ""which relatesSU(n) -E„SO(n)- F„and Sp(n) —E„suggests(extrapolating octonions -E,) that the E, invariantis a fully symmetric octet d„~,&~„. We do not knowwhether this is true and we hope we shall neverneed to know.

We should also point out that we have not provedthat our identities for F4, E„and E, suffice toevaluate any weight. We have only verified thisfor all vacuum weights up to 4 loops (F, and E,)and 3 loops (E,).

V. ILLUSTRATIVE EXAMPLES

Evaluation of any 8'~ is now almost trivial, es-pecially for classical groups. We just proceedapplying systematically the rules of Fig. 1, firsteliminating all three-gluon vertices, and then re-moving all internal gluon lines. Removal of eachgluon line reduces 8'~ into a sum of weights oflower order. Eventually we end up with a set ofirreducible tensor bases, each multiplied by somepolynomial in n (n is the number of quark colors).

As an example, we evaluate the SO(n) qua, dratic

SO(n)

SO(n) n(n-))2

(n-2)—2

Sp(n)

G2(7)

Fq (26) 52

Eg(27) 78

E,(56)

sQ7 L8

p2 U2

U2

p2 UYp2 U(2

p2

9p2 U24

+a

+

+a

FIG. 20. A tabulation of some simpl. e weight evalua-tions.

VI. RELATIONS BETWEEN BASIS TENSORS

The procedure outlined in Secs. I-V always leadsus to a unique set of tensors: (T~)~ and traces overT, matrices. In other words, we are expressingall 8'G. in terms of the defining representation.Let us illustrate this by writing all irreduciblebases T' ' for quark-quark scattering weights[see (4.1)]:

Casimir operator for the adjoint representation(gluons) in Fig. 19. We find that

C„=a(n —2).

Other such results are tabulated in Fig. 20. Qfcourse, dimensions and Casimir operators (orrepresentation indices) are all tabulated in theliterature4" 4' and our algorithm is unnecessaryfor their evaluation. However, we can now calcu-late the weight of any diagram. A typical examplewould be computation of all the weights that appearin the SU(n) quark-quark scattering calculation, 'or the order of the first nonleading term in 1/nexpansion for various groups. "

2 +

FIG. 19. A sample diagrammatic computation: quad-ratic Casimir operator for the adjoint representation ofSO(). (a) C~&~ are replaced by the defining representa-tion, (b) internal. gluons are replaced by gluon projectionoperators, and (c) the expression is expanded and evalu-ated.

SO(n): 3'5' O'O' V'3 (P = 3)

Sp(n): 5q5q, 5f5q, f' fM, (P = 3)

G2(7): &u&~ ~s &~ &"»a f'~.f"g,(P =4)

and so forth. These bases appear naturally in ourapproach, but they are by no means the only pos-sible choice. For example, we can replace the"color exchange" base 6„'5~ by the "color flip"base' ' (T,.);(T,)f using relations (2.7)-(2.13). Asanother example, we write down all irreducibletensor invariants for a process with t' externalgluons and no external quarks, the set of all dis-

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1547

.~ t ~ ~

~ b ~

O none (traceIessness)

(t ) su(z) + =0

2(c) SU(&) 6

I I

FIG. 22. (a) A characteristic equation for t4x4] mat-rices, (b) characteristic equation for SU(2) tthere areno d&,k coefficients; see Fig. 25(d)l, (c) Macfarlane et al .relation for- SU(3).

SO(n): P, = 1, P, = 6, etc. (6.2)

5 446 265

Further relations, dependent on the dimensional-ity of the defining representation, arise from thecharacteristic equations for [n x n] matrices (i.e. ,from the invariance of the Levi-Civita tensor).The characteristic polynomial" is defined as

P(~) =detl&

7 I8548 I4 833

FIG. 21. Tensor bases for processes with r =2, 3, . . .external gluons and no external quarks. These are alsothe complete and independent bases for SU{n) tensorsas long as n~~.

where

8(-+)"b —6+1+2"'abgbg ~ ~, Abb

~ bk a& ky-0

ap~bp ~fp

(6.3)

tinct traces overs T, matrices (Fig. 2].).P„, the number of all distinct tensors of rank r,

is the number of ways in which r T; matrices canbe grouped into traces over their products, withthe restriction that Tr(T, )= 0. P„can be calculatedin a number of arduous ways, such as by Youngtableaux, '""or by the method of Appendix B.However, it turns out that p„had already beencalculated in 1708,"'"and is known as a numberof derangements, or subfactorial

(6 1)

Not all tensor bases thus enumerated are neces-sarily independent, because they might be relatedthrough the invariants of the defining representa-tion. P„was calculated from a single condition,tracelessness. Thus, traces over T, form naturalbases for all simple Lie groups, SU(n) in particu-lar. For SO(n), Sp(n), G„F„and E„ the clock-wise and anticlockwise directions of loops in Fig.21 are related by 6„,f,b invariance, and the num-ber of independent bases is reduced:

5b6""'~= detpqe ~ e g

&eu' ' '

&yu

(6.4)

is the generalized Kronecker 6. Identity P(A) =0yields the characteristic equation for A:

n iqk0 Afg~ gggg2 o gk

bgb2" 'bk

Aby Ab2 ~ ' ' Abk

Now if we substitute A = a, T, , where T, are gen-erators of the group 9, for each n we obtain vari-ous relations between tensor invariants. As anexample, we work out the n= 4 case diagrammati-cally in Fig. 22(a). The indices are symmetrizedbecause the whole expression is multiplied by asymmetric factor a;a&aka, , summed over all i, j,k, and L. More familar relationships are workedout explicitly for SU(2) and SU(3) in Figs. 22(b) and22(c). The SU(3) relationship can be rewrittenin terms of d&+ tensors, the form of which hasbeen originally derived by Macfarlane et al."Higher SU(n) relationships have been worked outin Ref. 29. Such relations do not affect the cor-

1548 PREDRAG CVITANOVIC 14

rectness of our general procedure for weight eval-uation.

VII. HIGHER REPRESENTATIONS

tensor notation. To check this construction wecompute the dimension [Fig. 3(f)) and the index(3.3) and verify"'"'" that

In Sec. 1V we have constructed gluon projectionoperators from the invariants of the quark repre-sentation. This approach is by no means restrictedto the defining representation; in Appendix B weshall give an example of a calculation in terms ofthe invariants of the adjoint representation. Thatcalculation will exemplify the difficulties arisingin the study of higher representations; it is noteasy to find a complete set of invariants for anarbitrary representation, and even when those arefound, the evaluation ofweights can still be difficult.

However, we already have a simpler solution forone higher representation; we know how to com-pute weights of diagrams with all particles in theadjoint representation. We evaluate them by re-writing them in terms of the defining representa-tion. This suggests that we should attempt to ex-press the particular higher representation in termsof the defining representation; once that is accom-plished, the weights can be evaluated by the meth-ods of Sec. IV. In principle, we always know howto construct any representation from the definingone by the Young symmetrization procedure.

As an example we construct the antisymmetricsecond-rank tensor representation of SU(n)."The projection operator —,'(5;5~b —5t5b) picks out theantisymmetric part of a two-quark state q~~, andthe generator of SU(n) transformations is

(t, )~b = 2 [(T,);5,' —(T,);5~+ 5', (T, )", —5,'(T, )t],

—Tr(t, t;) =n —2.1

Further examples of projection operators for high-er representations are given by Behrends et gl."

We should also mention that there already existalgorithms for computing weights of arbitrary rep-resentations. For example, Agrawala and Belin-fante" have developed a computer program forevaluation of SU(n) invariants.

ACKNOWLEDGMENTS

I would like to thank the Aspen Center for Phys-ics, where a part of this work was done, for itshospitality, R. Pearson for help with the Youngtableaux calculations, and R. Abdelatif, T. Duncan,B. Durand, E. Eichten, P. G. O. Freund, T. Gold-man, F. Gursey, H. Harari, D. Milicic, P. Ra-mond, C. Sacharajda, G. Tiktopoulos, and B.Weisfeiler for stimulating discussions.

APPENDIX A: COMPLETE FEYNMAN RULES FOR Ng V~

With the definition of the group-theoretic weight8'~ given in Sec. II, the rules for M~ are easily

Factors for WG MG

I

a b a = b i(D~+m~) (quark)

where a, b, . . . =1,2, . . . ,n, and T,. are the gener-ators of the defining representation of SU(n) (Sec.IV B). This is a nice example of how compact thediagrammatic notation is" (Fig. 23) compared to

i ~wvx~rj v iI I j

—ig„, (g luon 3

(ghost 3

su(n)

i, )

2 I(j-- -4 kJ

(a)kiv

[( ~ &) q +(o2 4) q"" (oz oi)"9 ]

)kv pg x( yv

Vg )kg

n (n-1)2

k~u y g

I

I g g —g g

FIG. 23. (a) Diagrammatic notation for the antisym-metric second-rank tensor representation of SU(n), (b)computation of its dimension.

FIG. 24. Factors for the group-theoretic weights W~and Feynman momentum integrate ~ in the Feynmangauge.

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1549

constructed by consulting some standard refer-ence, such as Abers and Lee." In this appendixwe state the full rules for unrenormalized Feyn-man amplitudes in (unbroken) non-Abelian gaugetheories as an extension of the rules for construc-ting Feynman-parametric integrals given previ-ously. " Factors of rule 5, of Ref. 73, are now

replaced by the factors of Fig. 24. AdditionallyM~ gets a factor —1 for each quark or ghost loop.

0 I

I 0

2 I

APPENDIX B: EVALUATION OF SU(N) WEIGHTS

USING f- AND d-TENSOR BASES

In this appendix we extend the SU(3) method ofDittner" to SU(n). The generalized Gell-Mann[n & n] X matrices together with I, iI, and iX spanall complex matrices, "so we can write a multi-plication law for X matrices as

SU(n): A.;X~ = (g+ ib)5)g + (dg»+ if)»)X».

This relation, which has no obvious analogs forother simple groups, is the departure point formost of the earlier attempts at weight evalua-tion.""The tensors 5,&, d,», and f,.» are numer-ically invariant in the sense that they are the samefor all equivalent representations X, —u~&,u, u~u = i.They are real by definition. b = 0 because of theHermiticity of X, , while a is related to the arbi-trary normalization of Eq. (2.4), g = (qg' j4)g.

According to Sec. IV, we can evaluate any weightif we know how to evaluate vacuum weights. There&; matrices always appear in traces, TH(A, &,. A,„),and they can be eliminated by the repeated applica-tion of the X-multiplication rule [depicted in Fig.25(b)]. The problem of weight evaluation for SU(n)

4 l5

5 140

6 lel5FIG. 26. Construction of all simple d and f tensors

with r external gluons.

(a)

su(n}

=—4iIk

is then reduced to the problem of evaluation ofvacuum weights built solely from the adjoin. nt rep-resentation invariant tensors 5,&, f&», and g&»,Dittner solves this by setting up a chain of sets oflinear equations of type (4.2), which make it pos-sible (in principle) to compute weights with /+ Iloops once all vacuum weights with up to / loops

a +-r) 2 2

r ar-I ar

(e)

Id) ~ =~

FIG. 25. (a) Notation for the (ful. ly symmetric) numeri-cal tensor d;~&, (b) multiplication rule for SU() matricesT; =—

2 g &;, (c) decomposition of three external gluonquark-l. oop into real and imaginary parts, (d) d&&z as itsreal part.

4 2

5 5

6 I 4 I 680

FIG. 27. Catalan's trees.

1550 PREDRAG CVITANOVIC 14

are known. To achieve this, it is necessary toconstruct independent bases for processes withx=2, 3, . .. external gluon legs.

The simplest set of tensors for each x is easilyconstructed (see Fig. 26). To enumerate them,we start a systematic construction by drawing allCatalan'"" trees in Fig. 27, whose number isCatalan's number (the number of ways in which aproduct of n numbers can be evaluated)

(a)

su(n)

a 4aMn ~+

a 4an

(Jacobi identity)(2r —4)!

(r —1)!(r —2)! '

By (r —1)!permutations of all branches, and

factor 2 for each crotch (f or d tensor), we obtainthe number of all distinct connected tensors

n, =2" '(2r —5)!!, n, =o, n, =-l

where (2n —1)!!is the product of the first n-oddintegers, 7 1 t =-7 & 5 x 3 x 1. To relate Q.„to the

n„, the number of all distinct tensors (connectedand unconnected) we introduce generating functions

(b) Q / 4a/

4an

A(f)=- gr=O + ~

(B4)

The numbers of connected and disconnectedgraphs are related in the usual fashion,

A(t) eA(0) (B6)

By differentiation with respect to t, this can berestated as

(B7)

which enables us to calculate recursively Q.„ listedin Fig. 26.

However, tensors so constructed are redundant,and if we attempt to use them to expand an arbi-trary tensor with x external gluons, we would notbe able to calculate the expansion coefficients,because the determinant of the system of Q.„equa-tions vanishes for x& 3.

So our next task is to find all the relations be-tween „ tensors. These stem from the associ-ativity of T& matrices. For example, Tr(T, T&T~T, )can be evaluated in two ways, by pairing matriceseither as Tr(T, T&)(T„T,) or Tr(T&T~)(T, T), and thenusing Fig. 25(b). The two evaluations give therelationship of Fig. 28(a). There are (4 —1)!= 6distinct connected tensor bases (Fig. 21) withfour T,. each, giving us y~= 6 relationships. Wecast those in the form familiar from the litera-ture, '6 "three equations for the real parts [Fig.28(b)] and three for the imaginary parts [Fig.28(c) . Figure 28(c) states that d&z~ are invariant

FIG. 28. (a) Associativity of T~ matrices leads to re-lations between various d and f tensors. All relationsbetween (b) real and (c) imaginary parts of simple ten-sors with four external gluons.

y„=(r —1)!(a„,—1), r~2.For each x there are

p „=n„-y„= (r —1)! r ~ 2

(B9)

(B10)

independent connected tensors. The total numberof independent tensors P„ is given by

B(t) g ~fr„.0 r! (B11)

[see (4.4) and remember that (T&)»= —if&» for theadjoint representation of SU(n)]. The second andthird lines of Fig. 28(b) are two versions of theSU(n) generalization"'" of the SU(2) relationship

(B8)

Glancing back at the gluon projection operator forF, [Fig. 1(b)], we realize that this is the gluonprojection operator for models with quarks in thead joint representation of SU(n).

The number of associativity relations for arbi-trary r is again related to Catalan's number,which is nothing but the number of associativitypatterns

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1551

SU(n)

(a)

2 a2

=2a

+-an2

+ —2an2

(b) =2a + +n +

FIG. 29. Gluon "box" diagram evaluated in (a) twodifferent f and d bases and (b) T~ basis.

B(t)= Q "t"r=2 Y

= —f —ln(1 —t),e t

B(t)= e~"'=1 —t

(B12)

But B(t) is precisely the generating function forsubfaCtorill, so we have rederived the simplecounting of (6.1) in a complicated way.

Once a set of P„ independent tensors has beenconstructed, the tensor to be simplified is expan-ded in this basis. By contracting all its indiceswith each basis tensor, a set of P, linear equations

is obtained. Now it is necessary to solve theseequations —for the details, we refer the readerto Dittner's papers. " To illustrate the form of theresults, we give the reduction of a gluon "box" dia-gram in two (of many possible) choices of f, dbases [Fig. 29(a)]. For comparison with the meth-od of Sec. VI, we also evaluate the same diagramin T, bases, Fig. 29(b).

To summarize, for SU(n) the knowledge of theinvariants of the adjoint representation leads to afeasible method of weight evaluation. However,compared with the evaluation via the defining rep-resentation, it suffers from numerous drawbacks.It introduces a tensor d,» that does not appear inthe original interaction Lagrangian, and leads toarbitrariness in the choice of tensor bases (notethat the T, bases are unique). Finally, it involvessolving large sets of linear equations; already forx = 4 we found it convenient to do the algebra on acomputer. " By contrast, if we use the definingrepresentation, evaluation never requires solvingany equations (for classical groups, at least): Itis a systematic procedure of eliminating internalgluons one by one until only irreducible tensorsare left. If there are d, » couplings in the model,they are easily incorporated into our scheme byFig. 25(d).

*Work supported by the Energy Research and Develop-ment Agency.

~Present address: Institute for Advanced Study, Prince-ton, New Jersey 08540. After Oct. 1, 1976: Depart-ment of Theoretical Physics, Oxford University, 12Parks Bd., Oxford OX1-3PQ, England.

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~2These are not to be confused with Cartan's weight dia-grams.

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8J. Tits, Ned. Akad. . Wetensch. Proc. A69, 223- (1966)..H. Freudenthal Advan. Math. 1 145 (1964).B. D. Schafer, Introduction to Nonassociative Algebras(Academic, New York, 1966).

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matrix representation; King's College report, 1975

1552 PREDRAG CVITANOVIC

(1965).L. M. Kaplan and M. Resnikoff, J. Math. Phys. 8, 2194(1967).

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33A. McDonal. d and S. P. Rosen, J. Math. Phys. 14, 1006(1973).

34R. E. Cutkosky, Ann. Phys. (N.Y.) 23, 415 (1963).3~D. E. Neville, Phys. Rev. 132, 844 (1963).36For a semisimple algebra the symmetric bi1inear form

Tr(T& T&) is nonsingular and it can always be broughtto the convenient form (2.4). In the language of Cartan*sdiagrams, ~a sets the length scale for root vectors.Any representation can be used for normalization of theLie algebra. In mathematics this is usually done byfixing the value of the quadratic Casimir operator forthe adjoint representation.

37M. Gell-Mann, Caltech Report No. CTSL-20, 1961, re-produced in M. Gell-Mann and Y. Ne'eman, The Eight-fold Way (Benjamin, New York, 1964), p. 11.

~ Diagrammatic notation appears frequently in group-theoretic problems. Canning (Ref. 17) has used dia-grammatic equations for SU(n) which are identical toours, and similar notation has been developed by Pen-rose (Hefs. 39—40), J. Mandula (unpublished), YeungRef. 46), and Cahalan and Knight (Hef. 17). Diagram-matic methods for coupling coefficients for arbitraryrepresentations (related to Wigner's 3n-j coefficients)have a long tradition in atomic spectroscopy, nuclearshell theory, and many other areas (see Ref. 41).R. Penrose, in Combinatorial Mathematics and itsApplications, edited by D. J. A. Welsh (Academic, New

York, 1971), pp. 221-244.4 T. Murphy, Proc. Camb. Philos. Soc. 71, 211 (1972).

The reduction of diagrams with four external gluonsattempted in this paper is valid only for SO(3).

+See, for example, H. Biritz, Nuovo Cimento 25B, 449(1975); G. P. Canning, Phys. Rev. D 8, 1151 (1973);J. S. Briggs, Rev. Mod. Phys. 43, 189 (1971); H. p.Durr and F. Wagner, Nuovo Cimento 53A, 255 (1968);V. K. Agrawala and J. G. Belinfante, Ann. Phys. (N.Y.)49, 130 (1968).and references therein.

42E. M. Andreev, E. B.Vinberg, and A. G. Elashvili,Funct. Analysis and Appl. 1, 257 (1967); E. B. Dynkin,Trans. Amer. Math. Soc. (2) 6, 111 (1957).

43J. Patera and D. Sankoff, Tables of Branching Rulesfor Representations of Simple Lie Algebras (Univ. deMontreal, Montreal, Quebec, Canada, 1973).Group-theoretic weights have an amusing graph-the-oretic interpretation for SO(3). If we consider a planarvacuum diagram (no external lines) with normal. izationa =2, then+& is the number of ways of coloring thelines of the graph with three colors (see Ref. 39).This, in turn, is related to the chromatic polynomials,

Heawood's conjecture, and even the four-color problem.[See R. C. Read, J. Combinatorial Theory 4, 52 (1968)and O. Ore, The Four-Color Problem (Academic, NewYork, 1967)l.K. Bardakci and M. B. Halpern, Phys. Rev, D 6, 696(1972); H. T. Grisaru, H. J. Schnitzer, and H.-S.Tsao, ibid. 8, 4498 (1973).

4~In Ref. 7, this is not manifest because the weights arecomputed explicitly for SU(n) by a method discussedhere in Appendix B. However, Higgs particles contri-bute only as a correction to the three-gluon vertexwhich is proportional to C& [Fig. 4(b)j for any group,and the cancel. lations between remaining diagramsfollow from Lie algebra commutation relations a1one.[See also P. S. Yeung, Phys. Rev. D 13, 2306 (1976).]

VR. Gilmore, Lie Groups, Lie Algebras and Some ofTheir Applications (Wiley, New York, 1974).Sometimes the defining representation is referred toas the vector representation (see Refs. 13 and 49), theprincipal linear representation (see Ref. 50), or thefundamental n-tuplet. However, note that in Cartan'sterminology a group of rank r has r fundamental repre-sentations, and that sometimes higher representationsare called "vector" N, ef. 52, p. 25).

49T. P. Cheng, E. Kichten, and Ling-Fong Li, Phys. Rev.D 9 2259 (1974).

5 D. P. Zelobenko, Compact Iie Groups and Thei~ Re-presentations, Trans. of Math. Monographs 40 (Ameri-can Math. Society, Providence, Rhode Island, 1973).For this reason mathematicians refer to T; as "deri-vations. "R. E. Behrends, J. Dreitlein, C. Fronsdal, and W. Lee,Rev. Mod. Phys. 34, 1 (1962).

~~Such projection operators are sometimes called corn-pl.eteness relations. For SU(n) they were given by Mac-farlane et al. (Ref. 27), and for SO(n) by Cheng et al .(Ref. 49).

5 M. Hamermesh, Group Theory (Addison-Wesley, Read-ing, Mass. , 1962).This arises because Sp(n) is a complex representationof the quaternionic norm invariance group in n/2 dim-ensions. f is a representation of an imaginary unit ifor a quaternion written as C&+ iC2, C; complex. Skew-symmetry arises from i *=-i, and inverse fromi2=-1.See, for example, D. Finkelstein, J. M. Jauch, andD. Speiser, J. Math. Phys. 4, 136 (1963); M. Gourdin,Unitary Symmetries (North-Holland, Amsterdam, 1967).Besides Refs. 18-22, 42, and 43, some general. pro-perties of exceptional groups are given in M. L. Mehta,J. Math. Phys. 7, 1824 (1966); M. L. Mehta and P. K.Srivastava, ibid. 7, 1833 (1966); J. Tits, LectureNotes in Mathematics (Springer, New York, 1967),Vol. 40; R. Caries, Acad. Sci. Paris A276, 451 (1973);J. M. Ekins and J. F. Cornwell, Rep. Math. Phys. 7,167 (1975). 62 has been studied in Refs. 52, 58, andby G. Racah, Phys. Rev. 76, $352 (1949); R. E. Be-hrends and A. Sirlin, ibid. 121, 324 (1961); J. Patera,J. Math, Phys. 11, 3027 (1970); J. Patera and A. K.Bose, ibid. 11, 2231 (1970); D. T. Sviridov, Yu. F.Smirnov, and V. N. Tolstoy, Rep. Math. Phys. 7, 349(1975), and references therein; R. Casalbuoni,G. Domokos, and S. Kovesi-Domokos, Nuovo Cimento31A, 423 (1976), have an interesting model based onthree-quark coupling via f, q~ tensors. T. Yoshimurahas computed several weights for G2 using an explicit

14 GROUP THEORY FOR FEYNMAN DIAGRAMS IN NON-ABELIAN. . . 1553

(unpublished).~7K'. Cartan, Oeuvres Comp/etes (Gauthier-Villars,

Paris, 1952).~ M. Gunaydin and F. Gursey, J. Math. Phys. 14, 1651

(1ev3).~9The same relation has been obtained by R. E. Behrends

et al. (Sec. V D of Ref. 52) without octonions. Use ofoctonions greatly simplifies the derivation.A. Springer, Ned. Akad. Wetensch. Proc. A65, 259(1962).

@H. Freudenthal. , Ned. Akad. Wetensch. Proc. A57, 218(1954).

62F. Gursey, P. Ramond, and P. Sikivie, Phys. Lett.60B, 177 (1976).An explicit representation of F4 is given in J. Patera,J. Math. Phys. 12, 384 (1971).

84R. B. Brown, J. Reine Angew. Math. 236, 79 (1969);J. R. Faulkner, Trans. Amer. Math. Soc. 155, 397

. (1ev1).65F. Gursey and P. Sikivie, Phys. Rev. Lett. 36, 775

(1976); Report No. 68-540 (unpublished).We thank R. Pearson for carrying out a Young tableaucalculation to check our numbers. P„ is the number of

times the singlet appears in the decomposition of a pro-duct of r adjoint representations NxNx xN=P 1+ ''r

67An invaluable aid in identifying such combinatorialseries is N. J. A. Sloane, & Handbook of Integer Se-qlences (Academic, New York, 1973).

@Montmort, Essai d'Analyse sur les Jeux de Hasard(Paris, 1708); F. N. David and D. E. Barton, Com-binatorial Chance {Griffin, London, 1962); Louis Com-tet, Analyse Combinatoire (Presses Universitaires deFrance, Paris, 1970).

89P. I,ancaster, Theory of Matrices (Academic, New

York, 1969).Diagrammatic representations of Young symmetrizersare discussed by Penrose {Ref.39).V. K. Agrawala and J. G. Belinfante, BIT 11, 1 {1971).

~2E. S. Abers and B. W. Lee, Phys. Rep. 9C, 1 (1973).73P. Cvitanovib and T. Kinoshita, Phys. Rev. D 10, 3978

(19v4).~4Catalan, J. M. Pure Appl. 3, 508 {1838).

A. C, Hearn REDUcE 2 Stanford U'niversity ArtificialIntelligence Project Memo AIM-133, 1970 (unpublished).