Exceptional ultraviolet finite Yang-Mills theories

Volume 143B, number 1, 2, 3

PHYSICS LETTERS

9 August 1984

EXCEPTIONAL ULTRAVIOLET FINITE YANG-MILLS THEORIES

J.-P. DERENDINGER, S. FERRARA and A. MASIERO CERN, Geneva, Switserland Received 27 April 1984

We show that a new class of ultraviolet finite N = 2 supersymmetric Yang-Mills theories exists in which some hypermultiplets belong to irreducible, pseudoreal representations of the gauge group G. Some of these theories contain fermions which are strictly massless before the spontaneous breaking of G. A classification of these theories and of those which admit at least three generations of quarks and leptons is given.

It is known that N = 2 supersymmetric Yang-Mills theories [1] can be made ultraviolet finite provided the one-loop beta function vanishes [2]. It has been shown that ultraviolet finiteness even persists if soft breaking terms of a particular form are introduced [3]. This opens up the possibility of constructing realistic models for unification of electroweak and strong interactions based on a softly broken global N = 2 supersymmetry [4]. A classification of ultraviolet finite N = 2 Yang-Mills theories has already been given [5] under the assumption that the matter spin1/2-spin-0 hypermultiplets belong to pair-conjugate representations R + R of the gauge group G [1-51. Under this circumstance, the condition of ultraviolet finiteness is simply given by [21

C2(G) = E n , T ( R , ) ,

(1)

i

~ , Q+~a, Q+Qj+f]~,

(2)

(4)

in which the Clifford vacuum has + 1/2 helicity, and is defined by

and a is a gauge group index. The four states given by eq. (4) and their antiparticles correspond in general to fields a

T(R), multiplied by the rank of the group, is known as the second-order Dynkin index. It can be obtained from a realization of the generators T A for the representation R with proper normalization through T( R )8 A~ = T r ( T A T S ) .

i = 1,2,

Qif~ ~ = O,

where C2(G ) is the quadratic Casimir of the adjoint representation of the gauge group G, and T ( R ) = C2 ( R ) dim R / d i m G,

The sum in eq. (1) goes over all hypermultiplet species, and n i is the multiplicity of each hypermultiplet. However, in deriving eq. (1), it has not been taken into account that hypermultiplets of N = 2 supersymmetry can belong to irreducible representations of G, provided the representation is pseudoreal. The reason why this can occur is that for pseudoreal representations of G, the hypermultiplet is P C T self-conjugate, and the doubling of states usually due to PCT-conjugation is not needed [6]. Consider, in fact, massless one-particle states given as follows:

(3)

~,A,,x

a~

,

(5)

in which + and X are Weyl spinors of opposite handedness and A T are two scalar fields. However, if the representation of the gauge group G is pseudoreal, the four states in eq. (4) are P C T self-conjugate and we can impose the conditions

0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

133

[71

(A~)* = Ci j ,t..abZ'l "~ ,,aj , xa~=¢~hcab(~/~) *

(6)

where Cab is the skew-symmetric metric of the pseudoreal representation of G. Therefore for pseudoreal representations Rp, the hypermultiplet contains only 4 dim Rp degrees of freedom instead of 8 dim R for non-pseudoreal RofG. Self-conjugate irreducible hypermultiplets appear in N = 8 supergravity if this theory is analysed in terms of N = 2 representations [8]. The fields of N = 8 supergravity transform in this case under SU(6). SU(2). The 70 scalar fields belong to (15 + 1--~, 1 ) + (20, 2), corresponding to 15 complex scalar fields belonging to 15 N = 2 vector multiplets, and 20 complex scalar fields belonging to a self-conjugate N = 2 irreducible hypermultiplet where eq. (6) reads (AI .... ~31)* = .0.

a[ . . . . . . 1

-[al." .a6]'"j

(7)

Ci,,...a61 is the invariant tensor of SU(6), and the threefold antisymmetric tensor 20 of SU(6) is pseudoreal. If we come back to the finiteness condition, it is clear that if a group G admits pseudoreal representations Rp,, then eq. (1) has to be modified according to

C:(G)= Z n , T ( R,) + ½]~np T( Rpj). i

(8)

j

(9)

with C real, skew-symmetric, and verifying C 2 = 134

--1. The simplest example is the 2 of SU(2) for which the generators are the Pauli matrices and C = io 2. All pseudoreal representations are evendimensional. The simple (compact) Lie groups which possess pseudoreal representations are [9,10] SU(4n + 2), Sp(2n), SO(n) with n = 3, 4, 5 modulo 8, E7. In order to satisfy eq. (8), we have first to enumerate all pseudoreal representations R p of these groups which r ( R p ) ~< 2C2 (G).

(10)

The result is given in table 1. The invariant T ( R p ) has been calculated using the standard normalization for the generators of the lowest dimensional representation: for SU(N), T ( N ) = 1/2; for SO(N), T ( N ) = 2; and for Sp(2N), T ( 2 N ) = 1. It is then simple to calculate T(R) for all other representations using either

T ( R 1 × R 2 ) = dim R 1 T ( R 2 ) + ( 1 ~ 2) T( Ra + R2) = T( R1) + T( Rz),

(11)

or the eigenvalues of a specific generator of the Cartan subalgebra. Tables of T ( R ) already exist in the literature [10,11]. The pseudoreal representations of SU(4n + 2), satisfying eq. (10), are only the (2n + 1)-fold antisymmetric tensors of SU(2) [ - S O ( 3 ) - Sp(2)] (2) and of SU(6) (20). For orthogonal groups, only spinorial representations can be pseudoreal. The invariant T(R) for the fundamental spinor is 2 n-2

Our task is to give a complete list of solutions to eq. (8) with some non-zero np. It is obvious that cases where all np, are even are particular solutions of eq. (1). New theories will be obtained when some n p, will be odd. Before discussing the solutions of Eq. (8) with non-zero np,, we have first to recall the precise definition of a pseudoreal representation. A representation of the Lie algebra of a group G is called pseudoreal (or symplectic) if the generators T A for this representation satisfy the property

CTA*C = T A,

9 August 1984

PHYSICS LETTERS

Volume 143B, number 1, 2, 3

for S O ( 2 n + l ) ,

2 n-3

for SO(2n).

Table 1 All pseudoreal representations with T(R p) ~<2C2(G). G SU(2) SU(6) SO(11) SO(12)

C2(G) 2 6 18 20

SO(13) Sp(2n) Sp(4) Sp(6) Sp(8) E7

22 2(n + 1) 6 8 10 18

Rp

T(Rp)

2 20: ~[Okl 32: spinor 32: spinor 32': spinor 64: spinor 2n: fundamental 16: (1,1) 14': (0,0,1) 48: (0,0,1,0) 56

1/2 3 8 8 8 16 1 12 5 14 6

(12)

Volume 143B, number 1, 2, 3

PHYSICS LETTERS

Then, since S O ( 3 ) - SU(2), S O ( 4 ) - SU(2) SU(2), and S O ( 5 ) - Sp(4) will be considered below, only SO(ll), SO(12) and SO(13) have pseudoreal spinors with T(Rp <~2C2(G ). The pseudoreal 2 N representations of Sp(2N) obviously always satisfy eq. (10). The other solutions are only the threefold antisymmetric traceless tensor of Sp(6) (14') and of Sp(8) (48) and the 16 of S p ( 4 ) - SO(5). This latter representation is an SO(5)-spinor obtained from 4 × 5 = 4 + 16.

(13)

For exceptional groups, only the lowest dimensional representation 56 of E7 fulfils condition (10). We must now construct all finite theories, solving eq. (8), with some pseudoreal matter multiplets. Table 2 shows all such solutions. In order to cancel the one-loop beta function, some of these theories contain, in addition to the pseudoreal part, hypermultiplets with fermions belonging to R + R of G (R can be real or complex). This is the case for all SU(6) models, except for the one containing 4(20). Since SU(6) contains SU(5), one can look for the embedding of quark and lepton generations 5 + 10, with the mirror fermions 5

9 August 1984

+ 10. The SU(6) models containing at least three generations are 3(20) + 3(6 + 6-) = 3(10 + 5 +1-0+ 5) + 6(1),

2(20) + (15 +15) + 2(6 + 6-) = 3 ( 1 0 + 5 + 1 0 + 5) + 4(1),

(14)

20 + 2(15 +15) + (6 + 6-)

= 3(10+ 5+10+ 5)+ 2(1). There is no place for additional Higgs multiplets. The pseudoreal spinors of orthogonal groups which appear in Table 2 decompose under the SO(10) subgroup according to S O ( l l ) : 32

= 16 +1--6,

SO(12): 32, 32' = 16 +1--6, SO(13): 64

(15)

= 2(16 +]-6). m

Each 16 + 16 contains one generation of quarks and leptons and the mirror states. Thus, finite theories based on SO(N) groups can accommodate up to five generations, or four generations if additional Higgs multiplets are required.

Table 2 Finite N = 2 theorieswith pseudorealmatter multiplets. G su(2) su(6)

S0(12)

SO(11) SO(13) Sp(2N) Sp(2N) Sp(4) Sp(6) Sp(8) E7

Pseudorealpart 8(2) n(20) 2(20) 20 20 20 n("32")

"32"' means: 32 or 32' n(32) 2(64) 64 4(N + 1)2N 8(2N) 16 4(4) 14' + 3(6) n(14')+(16- 3n)(6) 48 + 6(8) 6(56)

Real part + 3(4- n)(6+6) n _1, 2, 3,4 + (15+ ]-5)+ 2(6 + 6) + 2(15+~)+(6+6) + (15+~)+5(6+ 6) +(21 + 2-i)+ (6 + 6) + 2(5- n)(12+ 12) n=1,2,3,4,5 +(9-2n)(I1 + 11) n =1,2, 3,4 +3(13+ 13) + 7(13+ 13) N>:2 +([212N+[212N) N>~2 + 2(5+ 5) + (14 + 14) n=1,2,3

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PHYSICS LETTERS

M o d e l s b a s e d on symplectic groups c a n n o t be used for realistic m o d e l s except if very peculiar e m b e d d i n g s of S U ( 3 ) x S U ( 2 ) x U(1) into a very large g r o u p are accepted. T h e only e x c e p t i o n a l model, with gauge group E7, contains six generations of quarks a n d leptons, with the m i r r o r states. In fact all e m b e d d i n g s of SU(5) into E7 are equivalent a n d give 56 = 10 + 1---0+ 3(5 + 5-) + 6(1).

(16)

T h e p s e u d o r e a l r e p r e s e n t a t i o n s have an interesting p r o p e r t y : their q u a d r a t i c i n v a r i a n t is antisymmetric. This m e a n s that a s u p e r s y m m e t r i c mass t e r m is only allowed for two different m a t t e r multiplets. If a t h e o r y c o n t a i n s an o d d n u m b e r of s o m e p s e u d o r e a l r e p r e s e n t a t i o n Rp, then autom a t i c a l l y one m u l t i p l e t R p will r e m a i n massless as long as the gauge group r e m a i n s u n b r o k e n . Since soft terms b r e a k i n g s u p e r s y m m e t r y while preserving finiteness are n o t able to give masses to fermions in a m a t t e r multiplet, finite theories with p s e u d o r e a l r e p r e s e n t a t i o n s can give rise to massless fermions. S o m e interesting cases can b e f o u n d in table 2. Consider, for instance, SO(13) with m a t t e r m u l t i p l e t s t r a n s f o r m i n g a c c o r d i n g to 64 + 7(13 + 13). T h e 64 will r e m a i n massless, giving rise to two massless g e n e r a t i o n s of q u a r k s a n d l e p t o n s with m i r r o r fermions, as is clear from the e m b e d d i n g of SO(10) into SO(13) [eq. (15)]. T h e same result will h o l d for SO(12) with m a t t e r multiplets 3(32) + 32' + 2(12 + 12), the two massless generations b e i n g e m b e d d e d in 32 + 32'. It is however i m p o s s i b l e to find m o r e t h a n two massless generations (with m i r r o r fermions) w i t h o u t enforcing the absence of an allowed m a s s term. A n o t h e r interesting case is Sp(4) with m a t t e r multiplets 16. This finite m o d e l has no m a s s scale at all. T h e presence of p s e u d o r e a l m a t t e r m u l t i p l e t s a p p e a r s then as a g e n e r a l i z a t i o n to finite N = 2 Y a n g - M i l l s theories of the so-called " s u r v i v a l h y p o t h e s i s " of unified theories. It is w o r t h m e n t i o n i n g that p s e u d o r e a l representations Rp possess a s y m m e t r i c c o u p l i n g to the adjoint representation (gp × Rp)sy M = Adjoint + .... 136

(17)

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which allows one to construct the N = 2 c o u p l i n g to the gauge N = 2 multiplet. A l s o we notice that h y p e r m u l t i p l e t s b e l o n g i n g to p s e u d o r e a l represent a t i o n s of G r e m a i n irreducible when restricted to N = 1 s u p e r s y m m e t r y . In fact, b e i n g P C T self-conjugate, they have the s a m e n u m b e r of degrees of f r e e d o m as a N = 1 W e s s - Z u m i n o multiplet. T o summarize, in the present p a p e r we have shown that an a d d i t i o n a l class of finite N = 2 Y a n g - M i l l s theories exists in which s o m e hyperm u l t i p l e t s b e l o n g to P CT self-conjugate, irreducib l e r e p r e s e n t a t i o n s of the gauge group. These repr e s e n t a t i o n s allow n a t u r a l l y massless m a t t e r fields in N = 2 s u p e r s y m m e t r y . W e have given a complete classification of all finite N = 2 theories of this sort, i n c l u d i n g those which c o n t a i n particle states with the q u a n t u m n u m b e r s of three generations of q u a r k s a n d leptons. Finally, it should be stressed that our analysis is p u r e l y g r o u p - t h e o r e t i cal, a n d the c o n s t r u c t i o n of a n y realistic N = 2 S U S Y G U T m o d e l is still an o p e n p r o b l e m , even after the inclusion of s o f t - b r e a k i n g terms which preserve finiteness.

References [1] P. Fayet, Nucl. Phys. Bl13 (1976) 135; M. Sohnius, Nucl., Phys. B165 (1980) 483. [2] P. Howe, K. Stelle and P. West, Phys. Lett. 124B (1983) 55; P. West, Proc. of La Jolla Conference (January 1983), to be published by APS. [3] A. Parkes and P. West, Phys. Lett. 127B (1983) 353; P. West, Supersymmetry and finiteness, Lecture Shelter Island II (June 1983). [4] F. del Aguila, M. Dugan, B. Grinstein, L. Hall, G.G. Ross and P. West, Harvard preprint HUTP-84/A001 (1984). [5] I.G. Koh and S. Rajpoot, Phys. Lett. 135B (1984) 397; F.-X. Dong, P.-Y, Xue, T.-S. Tu and X.-J. Zhou, Phys. Lett. 140 B (1984) 333. [6] S. Ferrara and C.A. Savoy, Supergravity '81, eds S. Ferrara and J.G. Taylor (Cambridge UP. London, p. 47). [7] B. de Wit, P.G. Lauwers, R. Philippe, S.Q. Su and A. van Proeyen, Phys. Lett. 134B (1984) 37. [8] J.P. Derendinger, S. Ferrara, A. Masiero and A. van Proeyen, Phys. Lett. 140B (1984) 307. [9] R. Slansky, Phys. Rep. 79 (1981) 1. [10] W.G. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras. Lecture notes in Pure and Applied Mathematics, Vol. 69 (Marcel Dekker, New York, 1981). [11] M.T. Vaughn, Z. Phys. C2 (1979) 111.