P H Y S I C A L R E V I E W D V O L U M E 7 , N U M B E R 6 1 5 M A R C H 1973

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Comments on Gauge Theories Without Anomalies*

Murat Giinaydin P h y s i c s Depar tmen t , Y a l e Un ivers i t y , N e w Haven, Connect icut 06520

(Received 27 October 1972; revised manuscript received 4 December 1972)

Anomaly-free gauge theories corresponding to "safe" simple Lie groups classified by Georgi and Glashow a r e further analyzed under the requirement that they contain no frac- tionally o r doubly charged particles. It i s found that orthogonal groups SOW) can satisfy this restriction when the leptons transform a s an irreducible "fundamental" representation of SOW). For the symplectic groups Sp(2N), this i s true if the lepton representation i s the adjoint representation or the representation pa defined below. If the leptons a r e to transform a s the lowest-dimensional representation, i t i s necessary to consider product groups of the form Sp(2N)B U(1). The exceptional group G2 i s ruled out.

Georgi and Glashow' have classified all the pos- sible gauge theories corresponding to "safe" simple Lie groups, which a re free from triangle anomalies and hence may be renormalizable. They call a representation of a Lie algebra safe if its representation matrices T , satisfy

T~({T,, T,}T,) = 0 , (1

and a Lie algebra is safe if (1) holds for all its nontrivial representations. They find that the fol- lowing simple Lie algebras a r e safe:

The remaining Lie algebras a r e in general not safe but can have safe representations.

In the standard formulation of gauge theories 3 8 4

one considers, the spinor lepton fields a s trans- forming like some nontrivial representation of the corresponding Lie group. The gauge fields that a re introduced through the usual Yang-Mills -

type formalism6 transform a s the adjoint repre- sentation of the Lie group.

It is possible to further restrict the possible gauge theories corresponding to safe simple Lie groups under the requirement that no doubly o r fractionally charged particles be allowed in the adjoint representation and the "lepton represen- tation" according to which the leptons transform. The physical reasons for this requirement a r e twofold. First , no fractionally charged particles have ever been observed. Secondly, there seems to be no direct experimental evidence for the existence of doubly charged currents in weak in- teractions .7 This requirement would also be valid for possible gauge theories which attempt to com- bine weak, electromagnetic, and strong interac- tions, and in which gauge particles a r e taken to be the intermediate vector bosons, the photon, and the vector mesons.

Analysis of the simple Lie groups that accom- modate only the integral charges (+I, -1, O)le( in the lepton and the adjoint representations simul- taneously is quite simple if we assume that elec- t r ic charge is an additively conserved quantum number and hence its generator can be taken a s an element of the Cartan subalgebra of the cor- responding Lie algebra. We will investigate only the cases when the leptons transform according to an irreducible "fundamental" representation of

7 - C O M M E N T S ON GAUGE T H E O R I E S W I T H O U T A N O M A L I E S

the simple Lie group.' Below, the (spinor) fields corresponding to the lepton representation will be denoted by {+,) and the gauge fields by {$J.

Case I. Let us first consider the case when the lepton representation is the lowest-dimensional nontrivial representation of the simple Lie group.

For the orthogonal group SOW), the lowest non- trivial representation has dimension N and the adjoint representation has dimension $ N (A' - 1). Hence, in this case, there will be N fields q i and $ N(N - I ) fields 4, . As in the quark model of hadrons, we can consider the particles associated with the fields +, a s some kind of bound states of the particles associated with fields +, obeying certain statistics in an abstract charge space. Denoting the charge-space states corresponding to fields I), and 4, by q, and b,, respectively, we have that in the case of orthogonal group SO(N) the states b, correspond to the antisymmetric tensor product of {q,) with i t ~ e l f , ~ i.e .,

Therefore, if there a r e two different charge states q, and q j with the same unit charge, then they can combine to give a doubly charged state. Hence under the above requirement we cannot allow two states with the same unit charge in the lowest- dimensional representation. We can only have one state q, with charge +le 1 , one q j with charge -/e / , and (N -2) states q, with zero charge. [The total charge must add up to zero since the representa- tion matrices of the Lie algebra of SO(N) a r e traceless.] This gives, in the adjoint representa- tion, (N - 2) states b, with charge + le 1, (N - 2) states b,withcharge - / e l , a n d g ( ~ - 2 ) ( ~ - 3 ) + 1 states b, with zero charge.

For the symplectic groups Sp(2N), the lowest representation is 2 N-dimensional and the adjoint representation has dimension N (2 N + 1). The ad- joint representation is obtained from the sym- metric product of the lowest-dimensional repre- sentation with itself. Now the 2 N-dimensional representation of the generators of Sp(2 N) can be transformed into convenient form1':

where Z is an N x N complex Hermitian matrix and Y is a symmetric complex matrix. Therefore Sp(2N) has U(N) as a subgroup, and the charge- space states that transform a s the 2 N-dimensional representation of Sp(2 N) can in general be written a s

and tj denote the quark and antiquark states as- sociated with group U(N). The charge states a s - sociated with the ad joint representation of Sp(2 N) can be given in the following 2N x 2N matrix form:

where qq a re the quark-antiquark states and (qq), a r e the diquark states (symmetrized). There a r e N2 qq states and $ N (N + 1) (qq), states. Since U(N) is a maximal compact subgroup of Sp(2N), one can take a s the Cartan subalgebra of Sp(2N) that of U(N). It is a well-known fact that one can assign integral charges (+ / e 1 , - le 1 , 0) to quark and quark-antiquark states simultaneously within the unitary group U(N) =SU(N)@ U(1). For example, for the generator of electric charge take

where 1, is the N x N identity matrix, and a=-(N-I)/N (or ~ = I / N ) .

Yet this assignment necessarily gives doubly charged (symmetric) diquark states. Hence, with- in gauge theories corresponding to the simple Lie groups Sp(2N), fractionally o r doubly charged states a r e unavoidable if we require the leptons to transform a s the lowest nontrivial representa- tion of Sp(2 N). Under this requirement, the prob- lem of double or fractional charges can be rem- edied if one adjoins the Abelian group U(1) to the group Sp(2N). Since the group U(1) is not safe, the anomaly due to it has to be removed by some other mechanism.

Under Sp(2N)8 U(1), there a r e several ways of choosing the charge generator that avoids the problem of doubly charged states. The most con- venient choice seems to be

Then the 2N-dimensional charge space decom- poses a s

where where q' and q2 transform a s the quark states of the subgroup SU(N) but have different charge as-

signments. Let

Then the charges of the states qq a r e

Q(q:)=O for i + l ,

Q(qi2) = O for i = I ,

and

Q(q:)=-lei for izl.

The adjoint representation of Sp(2N) 8 U(l) con- s is ts of an Sp(2N) scalar with zero charge and the following states given in matrix form:

where q,q, and q,qz a r e the same a s the qq states above. Diquark states (q,q,), now have different charges than (qq), states above, and they can only have charges +le 1 , -1e 1 , 0.

In this charge assignment, there a r e (N -1) qq states with charge +/e 1 , (N - 1) qij states with charge -1e 1, and [ (N - 1)' + 11 qq states with zero charge. In addition, we have one (q,q,), state with charge + / e l , 2(N -1) (q,q,), states with zero charge, and (N - (q,q,), states with charge -lei together with the oppositely charged @,q,), states.

Of the exceptional Lie groups, the group G, i s 14-dimensional and its lowest nontrivial repre- sentation i s 7-dimensional. G, i s a subgroup of S0(7), and hence naively one would expect it to accommodate only zero-charge or singly charged states in 7- and 14-dimensional representations simultaneously. Yet G, has the peculiar property that the requirement of having only the charges (+I, -1, 0)leJ leads inevitably to two states with charge +/e 1 , two states with charge - /e I , and three states with zero charge in the 7-dimensional representation. This in turn gives r i se to doubly charged states in the adjoint representation." The analysis of the charge-space structure of the other exceptional groups (F,, E,, E,, E,) is con- siderably more complicated and will be given else- where.

Case II. Let us now consider the case when leptons transform a s the adjoint representation. As was shown above, double and fractional charges can be avoided in the adjoint representation of orthogonal groups. For the orthogonal groups SO (2 N + 1) (N = integer), the charge assignment scheme given above is the only one to satisfy this

constraint. For the orthogonal groups SO(2N) there i s another charge assignment scheme that gives r i se to integral charges (+le I , -1e l ,O) in the adjoint representation. In this scheme one assigns charge ++ (e I to N charge states qi , and charge -$lei to the remaining N charge states q, in the lowest nontrivial representation, and by the arguments given above this leads to $ N (N - 1) states with charge +le 1 , i N(N - 1) states with charge -/e I , and N 2 states with zero charge in the adjoint representation.

Even though it i s not possible to avoid double and fractional charges in the lowest and the ad- joint representations of Sp(2 N) simultaneously, one can do this within the adjoint representation alone. To do this, simply assign charge 4 le 1 to M charge states q, , and charge -$ le / to the r e - maining (N -M) states q, . This will give, in the adjoint representation

[ ( N -M)'+ M2] states qq with zero charge, (N -M)M qij states with charge +le 1 , and (N - M)M qij states with charge - le 1 , plus

BM (M + 1) (qq), states with charge +le 1 , i (N - M ) (N - M + 1) (qq), states with charge -1e 1 , and M (N - M) (qq), states with zero charge. Hence one can avoid the problem of double and fractional charges in gauge theories correspond- ing to Sp(2N) in which the fermions transform like the ad joint representation.

In the case of exceptional group G,, going to the adjoint representation for the leptons does not help. The adjoint representation of G, decom- poses with respect to i ts SU(3) subgroup a s

and it is a well-known fact that the integral charge (+le 1 , -1e 1 , 0) requirement for the octet representation of SU(3) inevitably leads to frac - tional charges in the triplet representations.ll Thus, within G,, fractional or double charges a r e unavoidable.

Other Cases. As classified by Tits,* the or- thogonal groups W(2N +1) have irreducible "fundamental" representations pi with dimensions

and the groups SO(2N) have irreducible funda- mental representations pi with

Of these representations p, and p, a r e the lowest

7 - C O M M E N T S ON GAUGE T H E O R I E S W I T H O U T A N O M A L I E S 1927

and the adjoint representations, respectively. The remaining representations pi (i 2 3) corre- spond to the totally antisymmetric tensor product of i copies of the lowest-dimensional representa- tion. The charge assignment scheme given under case I that leads to integral charges (+le 1 , -le 1 , 0) in the adjoint representation of SO(N) gives rise, by the same arguments, to integral charges (+le / , - ( e 1 , 0) in the representations p, (i '- 3) a s well. Hence, it is possible to avoid double and fractional charges in gauge theories correspond- ing to orthogonal groups in which the leptons transform according to an irreducible fundamental representation pi (i 3 3 ) .

The symplectic groups Sp(2N) have, in addition to the lowest and the adjoint representations, i r - reducible fundamental representations pi of di- mensions

whose representation spaces correspond to the antisymmetric tensors tk l . . . , , of rank i in 2N di- mensions, satisfying the condition

where gklk2 i s an antisymmetric Sp(2 N)-invariant tensor.

The construction given under case I1 that avoids double and fractional charges in the adjoint rep- resentation will in general lead to fractional o r

double charges in the representations pi (i 3), However, for the representation p,, this problem does not ar ise . In terms of the charge states q i (i = 1, . . . , N) defined in case I1 for Sp(2 N), the charge states associated with the representation p, can be written in the 2N x 2N matrix form a s

where (q,), represent the traceless quark-anti- quark states of SU(N), and (qq), the antisymmet- r ic diquark states of U(N). All these states will have integral charges (+je 1 , - / e 1 , 0) if we use the charge assignment scheme of case 11.

Conclusion. Among the gauge theories corre- sponding to safe simple groups in which the lep- tons transform a s an irreducible "fundamental" representation of the Lie group,' the theories in- volving the orthogonal groups avoid the problem of double and fractional charges for a l l their fundamental representations. For symplectic groups, this constraint i s satisfied when the lep- ton representation is the adjoint representation o r the representation p, considered above. If the lepton representation is taken to be the lowest- dimensional representation, then it i s necessary to consider product groups of the form Sp(2 N ) @ U(1). The exceptional group G, never satisfies this constraint.

I would like to thank Professor Feza Giirsey for several enlightening discussions, and Dr. J. R. Primack for bringing Ref. 7 to my attention.

*Research supported by the U. S. Atomic Energy Commission under Contract No. AT(11-1) 3075.

'H. Georgi and S. L. Glashow, Phys. Rev. D 6, 429 (1972). 's. L. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell

and R. Jackiw, Nuovo Cimento 60, 47 (1969). 3 ~ e e , for example, S. Weinberg, Phys. Rev. Letters 3 1968 (1971).

4 ~ . Georgi and S. L. Glashow, Phys. Rev. Letters 2, 1494 (1972). ere the term "lepton field" i s used in a general

sense. It could refer to the known leptons o r to the a s yet unobserved heavy leptons o r even to the spinor quark fields (with integral charge) that a r e introduced in certain gauge theories (see, e.g., Ref. 4).

6 ~ . N. Yang and R. L. Mills, Phys. Rev. 96, 191

(1954); R. Utiyama, ibid. 101, 1597 (1956); M. Gell-Mann and S. L. Glashow, Ann. Phys. (N.Y.) 15, 437 (1961).

?see E. W. Beier, D. A. Buchholz, A. K. Mann, and S. H. Parker , Phys. Rev. Letters 2, 678 (1972), and the references contained therein, for the experimental status of doubly charged currents.

8 ~ h e s e representations a r e classified in J. Tits, in Lecture Notes in Mathematics (Springer, Berlin, 1967), Vol. 40.

' ~ o t e that the tensor product here r e f e r s exclusively to charge space and i s independent of the space-time properties of the states..

'Osee, e.g., S. Helgason, Differential Geometry and Symmetric Spaces (Academic, New York, 1962), p. 341. or a detailed account on G2 see M. Giinaydin and

F. Giirsey, Yale Report No. 3075-27 (unpublished).