Volume 118B, number 4, 5, 6 PHYSICS LETTERS 9 December 1982

TOWARDS A REALISTIC SUPERSYMMETRIC SU 8 MODEL*

J. LEON, M. QUIROS lnstituto de Estruetura de la Materia, CSIC, Serrano 119, Madrid 6, Spain

and

M. Ram6n MEDRANO Departamento de Ffsica Te6rica, Facultad de F{sicas, Universidad Complutense, Madrid 3, Spain

Received 24 May 1982

Flavour superunification is accomplished via a SU8 supersymmetric model, free of anomalies, asymptotically free, and with three families of light fermions, through a SU s -+ SU s -+ SUa x SU2 x U1 breaking. Possible patterns of supersym- metry breaking are described.

A few years ago, Georgi [1] initiated a program on flavour unification in non-supersymmetric theories. This program was developed by several authors [2] later on. Recently, a SU s GUT supersymmetric model has been worked out by Dimopoulos and Georgi [3]. Therefore, it seems natural to study a supersymmetric model based on a gauge group SU n (n > 5).

On the other hand, we know [4] that N = 8 ex- tended supergravity contains an effective SU 8 gauge theory at energies higher than Mp. Furthermore, the possibility of a breaking to a supersymmetric N = 1 theory was speculated recently [5]. Then, a study on SU 8 unification appears as specially attractive.

It is convenient to place fermions in a reducible representation R of SU 8 which has the following properties: (a )R is in the form of E m [8, m] in order to avoid the existence of exotic fermions under SU~, (b) R is complex but leads to a real representation under SU~ X U~ m, (c) R is free of anomalies at the SU 8 level for the theory to be renormalizable, (d) the theory has to be asymptotically free. R should contain at least 3 families of light fermions.

A study of the SU n representations that satisfy the above conditions was carried out by King [6] for non-

* Partly supported by Comisi6n Asesora de Investigaci6n Cien- tifica y Tdcnica, under contract 3209.

supersymmetric gauge theories. But supersymmetry conditions reduce the solutions very drastically. In fact, we find, in SUB, as the only possible candidates able to contain a Higgs in the adjoint, the following representations:

R 1 = p ( 4 . 8 + 2 8 ) (p = 1 , 2 , 3 ) , R 2 = 5 6 + 2 8 + 8 ,

R 3 = 5 - 8 + 56, R 4 = 9 " 8 + 28+ 56. (1)

We should emphasize here that, for a symmetry breaking (sb) via SU5, only R 1 wi thp = 3 and R 4 con- tain 3 families.

Obviously, we do not know a priori which super- multiplets correspond to ordinary fermions or to Higgses. Then, we leave this separation to be deter- mined by the different patterns of symmetry breaking.

The conditions for having a supersymmetric vacuum are k~ = 0 and V¢ = 0 [7], where

ka = ~R ~T°'¢' V¢ = O V(cb)/Odp, (2)

and V being the gauge invariant superpotential, limit the possible nonzero vev's. We start studying the con- dition ka = 0 which is the most restrictive one. A sys- tematic study of this question has been performed by Buccella et al. [8]. For the representations we are con- sidering (1) we write explicitly:

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k a(dp[c1 ""Cn] ) = --n ~1 "On Tqci ~)C'l C17'

~c i...cn T e l ko~(dP[cl ...on 1) = n _ o~ c i ~c 1...c n '

k a c - . ( % ) -- (3)

The k s = 0 condition can be carried through in two ways: (i) a unique vev, kc~ = 0 comes simply from TrTc~ = 0, (ii) two or more vev's that conspire to can- cel out k~.

Cases (i) and (ii), together with the symmetry break- ing pattern to which they lead, are summarised in table 1.

A few comments on table 1 are now in order. The vev's which contain a trilinear invariant with non- vanishing gradient are protected by the condition VO = 0 and they are indicated with a (P). SP6 and SP4 solutions are also excluded as they do not contain SU 3 X S U 2 X U l .So lu t i onSU m XSUn × U l ( m + n = 8) could posses a horizontal symmetry through the sb SU 8 to SU 5 X SU3H × U1, which was studied al- ready in the context of non-supersymmetric gauge theories [9]. This latest sb pattern together with solu- tion SP8 are both compatible with supersymmetry. At present, they are under our current study.

Finally, the symmetry breaking pattern SU 8 to SUs, which follows the scheme proposed by Georgi [ 1 ] for flavour unification in non supersymmetric theories, will be the one outlined in this letter. Then R 4 (1) is the only representation which is compatible with this scheme.

At a certain energy scale A1, let us suppose that the following vev's are produced:

Table 1 Symmetry breaking patterns in supersymmetric SU 8. 0 [al' "ak] stands for the k-fold antisymmetric tensor, (ja is the funda- mental representation, q5 a its conjugate, while q)~ stands for the adjoint representation. (P) indicates the existence of a trilinear invariant with non-vanishing gradient Vq5 :~ 0.

Representation Symmetry breaking pattern

¢[a,b,c,d] SU 4 X SU4 (P)

~ SUm× SU n x U 1 ~[ab] Spa

~ ( A = 1,2),~ [c,d] Sp6 ~ ( A = 1,2,3,4),~ [c'd] Sp4

OAa(A = 1 ,2 ,3) ,~ [d'e'f] SUs

OAa(A = 1,2),~ [c'd] SU6 (P)

(m +n = 8)

PHYSICS LETTERS 9 December 1982

~[1,2,31 = Ul, 4)Aa = lalSAa (tl = 1, 2, 3). (4)

These vev's will cause a breaking of SU 8 to SU 5. The representations of SU 8 are written, according to the chain SU 8 D SU 5 X SU 3 D SU5, as:

8 = (1 ,3 )+ (5, 1) = 31 + 5 ,

28 = ( 1 , 3 ) + (10, 1)+ (5, 3) = 31 + 1 0 + 3 5 ,

56 = (1, 1) + (1--0, 1)+ (5, 3) + (10, 3)

=1 + 1 0 + 3 5 + 3 1 0 ,

63 = (1, 1) + (1, 8) + (3, 3) + (5, 3) + (24, 1)

= 9" 1 + 35 + 3 3 + 24. (5)

The chiral supermultiplets, displayed in (5), can acquire a mass as a result of replacing the vev's in the kc~ and Ve terms.

We consider first the k s terms, emphasizing the fact that only those SU 8 supermultiplets which con- tain non zero vev's are able to acquire a mass. Namely, the supermultiplets (1, 1) and (5, 3), contained in 56, obtain a mass through the expression:

kc~(56)= - 6 1 J l ( T 2 a q f 23 + Tgac)a31+ T 2 a ~ a l 2 ) , (6)

where a = 1, 2, 3 is the singlet SU 5 × SU 3 and a = 4, .... 8 corresponds to (5, 5). Notice that (i-0, 1) and also (10, 3) remain both massless. The supermultiplets 8A, for A = 4, ..., 8 continue to be massless. On the contrary, the 8A for A = 1, 2, 3 will get a mass through the term:

3

The most general superpotential, compatible with the SU 8 symmetry, can be written as:

a b a b c V(O) = eAB~Aa(~abd~Bb + Xl(Pb(Pa + X2C~bd~cd~ a , (8)

where eAB (.4, B = 1, ..., 9) is an arbitrary antisym- metric matrix and X i are arbitrary real parameters. Supersymmetry condition Ve = 0 imposes CAB = 0 (A, B = 1, 2, 3) as a constraint on (8).

Analogously to the case of the K~ terms, the vev's in V 0 give masses to the supermultiplets coupled to them. As ( ~ ) -- 0, for energies of the order A 1, the X i terms coming from (8) are irrelevant from the point of view of mass generation. Nevertheless, we will show afterwards that the (q)~)'s will generate a supersym-

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metric breaking of SU 5 to SU 3 X SU 2 X U 1. Let us consider now the first term of the super-

potential:

goa b = eABOAaOBb ~ I.tleABOBb~Aa, (~Aa>

A = 1 , 2 , 3 , B > 3 , (9)

then, all the 88 with zero vev's can be made massive at will. We choose an CAB such that the last three re- main massless. In a similar way, from the expression:

UOA a = eAeOabOBb <~Aa > llleAB ~ab6Bb'

A > 3 , b = 1 ,2 ,3 (10)

we have that the (1, 3) or 28, for a = 1, 2, 3, acquires a mass of order/ l 2, and also the (5, 3) of 28 does it fora = 4, ... 8. The states (10, 1) of 28 remain massless.

Summarizing, at energies lower than the energy scale A 1 we have an effective SU 5 supersymmetric theory with the following content in massless super- multiplets: 3(10 + 3) + 10 + 1--0 + 9(1) + 24 + 3(5 + 5) + 9(1), where 24 + 3(5 + 3) + 9(1) come from the 63, 10 from the 28, and 10 from the 56 of SU 8.

The next step consists in the SU s breaking at an energy scale A 2 ~ A 1. This symmetry breaking could happen via a non-zero vacuum expectation value for the adjoint representation, following Witten's proposal [ 10] which has been already applied for a supersym- metric SU 5 model by Dimopoulos and Georgi [3] and Sakai [1 I] .

Contrary to a non supersymmetric SU5, at energies lower than A1, the SU s superpotential is given by:

= Xl~bdPa + X2ObdPe Oa, (11)

then the usual breaking, in the adjoint o f SU 5, is not protected by (11). We will choose the vev that breaks SU 5 to SU 3 × SU 2 X U 1. Therefore, at energies lower than A 2 we have the standard SU 3 X SU 2 X U 1 effec- tive theory with a gauge potential given only by the residual terms kc~k c~. Finally, the chiral supermultiplet

content is: 3 {(1, 2 ) -1 + (5, 1)2/3 + (1.1)2 + (5, 1)_4/3 + (3, 2)1/3} + {(1, 1)_ 2 + (3, 1)4/3 + (3, 2)_1/3 + (1, 1)2 + (3, 1)_4/3 + (3, 2)1/3) + 9(1, 1)0.

Still, two fundamental problems do remain: (a) supersymmetry breaking at an energy scale A 3 >~ 1

e n l TeV, (b) the breaking of SU2L X U 1 to U 1 at an energy scale A 4 ~ 300 GeV.

In our scheme, we do not dispose of supermulti-

plets 5 + 3 to break SU 2 X U 1 in the manner of Dimopoulos and Georgi [3]. Furthermore, the states 10 + 10 should disappear from the spectrum of low energy.

On the other hand Dimopoulos and Raby [ 12] have proposed a scheme with supercolor which, breaking supersymmetry dynamically through the building of fer- mion condensates, breaks at the same time SU2L X U 1 to U] m .

Now, once we have arrived at this point, we would like to describe briefly and without details two pos- sible mechanisms able to break supersymmetry and SU 2 X U 1 invariance.

(i) Dynamical breaking of supersymmetry: The appearance of fermion condensates able to

break supersymmetry can induce, at the same time, a breaking of the weak interactions in a suitable way, The simplest mechanism consist in imposing a non zero vev of the form

<~t~> = <(1, 2)1> = <10 X 10> ~ 0 ,

( ~ ) = ((1, 2)_1> = (10 X 10) 4: 0, (12)

which presents the problem of having the same scale for both breakings (A 3 = A4).

For a more realistic hierachy breaking we could force a breaking of the supersymmetry through

<66> = <10 X 1-6> = (1, 1)0> = A 3 , (13)

then, at energies below A 3 we would have a SU 3 X SU 2 X U 1 non supersymmetric theory. For a SU 2 X U 1 breaking we need a pair of 5 + 3 additional Higgses. A brief inspection of the ZAB matrix (8) shows we can leave such states massless with the proposed breakings. Hence, at a scale A 4 < A 3 we would have

(5) = ((2, 1)1)--~ A4, (5> = ((2, 1)_1) ~-- A 4 (14)

that would give us the usual low energy spectrum. We do not have a candidate to play the role of the

supercolor group, i.e. there is no place for it in our scheme. On the other hand, following de Wit and Nicolai [13], who found that the effective gauge theory contained in N = 8 supergravity is SU 8 X S08, we could believe in SO 8 as the group playing the supercolor role.

(ii) Explicit soft breaking: Recently, Girardello and Grisaru [ 14] showed how to break explicitely global supersymmetry without introducing quadratic diver-

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gences in the theory, through mass terms bilinear in the scalar field components of the chiral supermulti- plets and in the fermion field components of the vec- tor supermultiplets. In this way, it is possible then to give a mass to matter bosons and gauge fermions. Fur- thermore, Dimopoulos and Georgi have conjectured [3] that Girardello and Grisaru's mechanism could be applied to the masses of the fermions contained in chiral supermultiplets. In this case ttiggs fermions could be made also massive. Therefore, a scheme would emerge that could provide a realistic particle spectrum at energy scales lower than the breaking of supersymmetry.

We realize that the main problem of our model (and of other supersymmetry models) is the breaking of supersymmetry. The two proposed mechanisms although plausible, are difficult to carry through at a

practical level. Therefore, more detailed calculations in both directions are in urgent need and results will be published elsewhere.

Refcrences

[1] H. Georgi, Nucl. Phys. B156 (1979) 126.

[2] P.H. Frampton, Phys. Lett. 88B (1979) 299; 89B (1980) 352; P.H. Frampton and S. Nandi, Phys. Rev. Lett. 43 (1979) 1460.

[3] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150.

[4] E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141. [5] J. Ellis, Grand unification in extended supergravity, in:

Second Europhysics Study Conf. on Unification of the fundamental interactions (Erice, Sicily, 1981); TH 3206 CERN.

[6] R.C. King, Nucl. Phys. B185 (1981) 133. [7] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev.

D20 (1979) 403. [8] F. Buccella, J.P. Derendinger, C.A. Savoy and S. Ferrara,

Symmetry breaking in supersymmetric GUTs, in: Second Europhysics Study Conf. on the Unification of the Fun- damental Interactions (Erice, Sicily, 1981); TH 3212 CERN; Phys. Lett. l15B (1982) 375.

[9] J. Chakrabarti, M. Popovic and R.N. Mohapatra, Phys. Rev. D21 (1980) 3212; J. Chakrabarti, Phys. Rev. 1923 (1981) 2719.

[10] E. Witten, Nucl. Phys. B185 (1981) 513. [ 11] N. Sakai, Z. Phys. C, Particles and Fields 11 (1981) 153. [121 S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981)

353. [13] B. deWit and H. Nicolai, Nucl. Phys. B188 (1981) 98;

Phys. Lett. 108B (1982) 285. [ 14] L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982)

65.

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