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PRESENT STATUS OF THE ATTEMPTS AT A REALISTIC GUT IN EXTENDED SUPERGRAVITY
THEORIES
M. Günaydin
To cite this version:
M. Günaydin. PRESENT STATUS OF THE ATTEMPTS AT A REALISTIC GUT IN EXTENDED SUPERGRAVITY THEORIES. Journal de Physique Colloques, 1982, 43 (C3), pp.C3-328-C3-330.
�10.1051/jphyscol:1982366�. �jpa-00221921�
JOURNAL DE PHYSIQUE
CoZZoque C3, suppZ6ment au n o 12, Tome 43, de'cembre 1982 page C3-328
PRESENT STATUS OF THE ATTEMPTS AT A R E A L I S T I C GUT I N EXTENDED SUPERGRAVITY THEORIES
CERN, Geneva, SwitzerZmd
The l a r g e s t extended supergravity theory (ESGT)
( N =8 ) as written down by Cremmer and J u l i a has an on-shell E7(7)global x SU(8)local invariance
[1-3 1 . In the "gauged" version of the theory a s given by De Wit and Nicolai t h e on-shell E7(7)
globalinvariance i s l o s t and one has an S0(8)loca1 x SU(8)1 oc invariance on the Lagrangian level [4 I . Below, we shall r e s t r i c t ourselves t o the "ungauged"
version of the
N = 8ESGTC 5 1 .
AshasobservedbyGell-Elann
[6 1 , the fundamental f i e l d s of the
N =8 ESGT do not have .a rich enough s t r u c t u r e t o accommodate the basic f i e l d s of a r e a l i s t i c gauge theory of strong, weak and electromagnetic interactions and thus t o make contact with elementary p a r t i c l e phenomenology some of the f i e l d s entering such a gauge theory night have t o be made composite C7 1 . The f i r s t important development i n t h i s direction came from Cremmer and J u l i a who suggested t h a t the c l a s s i c a l l y non-propagat- ing composite gauge f i e l d s of SU(8)10ca1 nay become dynamical on the quantum level D l . Their suggestion was motivated by the analogy with two-dimensional
C P ~models whose study a t l a r g e
Nl i m i t shows t h a t the composite gauge f i e l d develops a pole a t
p2 = 0i n i t s propagator and hence becomes dynamical on the quantum level [81 . The same phenomenon has been shown t o occur in three-dimensional supersymmetric non- l i n e a r sigma models [91 . The suggestion of Cremmer and J u l i a motivated the study of SU(8) a s a possible
GUTgroup 1101 .
The next important attempt t o make contact with p a r t i c l e physics came from E l l i s , Gaillard, Maiani and Zumino (EGMZ) who suggested t h a t in the
N = 8ESGT, in addition t o massless gauge f i e l d s , other massless bound s t a t e s (fermionic and bosonic) may form whose low energy e f f e c t i v e theory i s a
GUTbased on SU(5) with three
generations of quarks and leptons
[11 - 12 1 . All the "elementary" p a r t i c l e s entering t h i s GUT (quarks, leptons, gauge f i e l d s , Higgs f i e l d s ) a r e composites of the basic f i e l d s of the N
= 8ESGT (referred t o as haplons o r preons) with a binding scale of the order of
14planck.In the candidate "super GUT" proposed by E l l i s , Gaillard, and Zumino 112 I-(EGZ) t h e "elementary" p a r t i c l e s were chosen from the "gauge super multiplet" of bound s t a t e s which containsthe 63 gauge f i e l d s of SU(8),o, C131 .
However, theUgauge super rdul t i p l e t " contains many unwanted he1 i c i t y s t a t e s a s we1 1.
In f a c t , t o give SU(5) or SU(3)C x SU(2)W x U(1) invariant superheavy masses (of the order of
Flpor MGU) to t h e unwanted h e l i c i t y s t a t e s and be l e f t with three massless chiral families of quarks and leptons i s a major problem of the
EGZscheme
112,14-171and of the subsequent attempts t o extend i t 118-191 . In t h e i r recent important work related t o t h i s problem A l t a r e l l i , Cabibbo and Maiani have shown t h a t i t i s possible t o give SU(3)C x U(1) invariant masses t o a l l the members of a f i n i t e number of massless N
=8 supermultiplets without, a t the same time, the mass matrix being SU(8) invariant
[201 . However doing so e n t a i l s problems w i t h exotic SU(5) representations f o r s
=1/2 bound s t a t e s and/or low mass
( z Flw)high h e l i c i t y bound s t a t e s which have not been observed. The problem of unwanted h e l i c i t y s t a t e s can be solved t r i v i a l l y i f one assumes i n f i n i t e supermultiplets of bound s t a t e s 1141. In f a c t , there a r e many theoretical arguments f o r the possible existence of an i n f i n i t e s e t of bound s t a t e s i n ESGTs [14-17, 21,221 . One argument i s the
)ti P r e s e n t a d d r e s s : L a b o r a t o i r e d e P h y s i q u e Thgorique d e 1 ' ~ c o l e Normale Supdrieure, 24, r u e Lhomond, 75231 P a r i s Cedex 05, F r a n c e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982366
M. Giinaydin C3-3 2 9
r e g g e i z a t i o n o f s c a t t e r i n g amplitudes i n ESGT s [23]
.
Another argument i s based on t h e f a c t t h a t i n two 124,251 and three-dimensional 1 9 1 g e n e r a l i z e d n o n - l i n e a r sigma models t h e bound s t a t e s f a l l i n t o l i n e a r r e p r e s e n t a t i o n s o f t h e g l o b a l i n v a r i a n c e group. I f t h e same phenomenon occurs i n four-dimensional t h e o r i e s then one expects i n f i n i t e towers o f bound s t a t e s i n ESGT s f o rN
= 4-
8 s i n c e t h e i r g l o b a l i n v a r i a n - ce groups a r e non-compact [17, 261.
The r e l e v a n t u n i t a r y r e p r e s e n t a t i o n s must be those t h a t can be constructed u s i n g t h e f i e l d operators o f these t h e o r i e s . The o s c i l l a t o r - l i k e u n i t a r y r e p r e s e n t a t i o n s o f t h e non-compact groups o f ESGT s (N = 4-
8 ) have been constructed using boson a n n i h i l a t i o n and c r e a t i o n operators, t r a n s f o r m i n g l i k e t h e i r v e c t o r f i e l d s 127,281.
One can a l s o c o n s t r u c t t h e u n i t a r y r e p r e s e n t a t i o n s o f these groups over t h e s c a l a r f i e l d s i n t h e r e s p e c t i v e t h e o r i e s [22, 291.
Remarkably enough, i n b o t h cases one o b t a i n s t h e same c l a s s o f u n i t a r y r e p r e s e n t a t i o n s [ 29, 30 ].
I n t h e unitar.y qauqe which i s m a n i f e s t l y f r e e o f clhosts, t h e qenerators o f t h e non-compact group a c t i o n p r e s e r v i n g t h i s gauge do n o t commute w i t h supersymmetry generators [ 14, 22 ]
.
The l a r g e r algebra i n t o which t h e y c l o s e i s an i n f i n i t e - dimensional superalgebra i n v o l v i n g generalized supersymmetry generators and general- i z e d momenta 122 ]. The r e s u l t i n g i n f i n i t e superalgebras have been w r i t t e n down f o r N = 4 andN
= 8 ESGT s i n t h e r i g i d l i m i t o f s p a t i a l i n f i n i t y where a l l f i e l d s vanish a s y m p t o t i c a l l y except f o r t h e s c a l a r f i e l d s and t h e v i e r b e i n [ 2 2 ].
I findeed t h e bound s t a t e s o f a given h e l i c i t y f a l l i n t o u n i t a r y r e p r e s e n t a t i o n s o f t h e non-compact i n v a r i a n c e group then t h e bosonic and f e r m i o n i c bound s t a t e s t o g e t h e r must correspond t o a u n i t a r y r e a l i z a t i o n of t h i s i n f i n i t e superalgebra. The u n i t a r y r e a l i z a t i o n s o f t h e i n f i n i t e superalgebra generated by non-compact SU(1,l) and supersymmetry generators i n t h e
N
= 4 case have been i n v e s t i g a t e d on t h e " c l a s s i c a 1 "l e v e l 122 1
.
One f i n d s t h a t t h e a c t i o n o f a generalized supersymmetry generator maps a s t a t e belonging t o a u n i t a r y r e p r e s e n t a t i o n w i t h h e l i c i t y A i n t o a s t a t e belonging t o a u n i t a r y r e p r e s e n t a t i o n w i t h h e l i c i t y A+
1/2. I f on t h e quantum l e v e l t h e i n t e g r a t e d charges o f these generalized supersymmetry t r a n s f o r m a t i o n s a r e a l l u n i t a r i l y i n e q u i v a l e n t then one w i l l have an i n f i n i t e s e t o f generalized supercharges i m p l y i n g bound s t a t e s o f a r b i t r a r i l y l a r g e h e l i c i t y 1 221. T h i s i s c o n s i s t e n t w i t h t h e r e g g e i z a t i o n o f s c a t t e r i n g amplitudes i n ESGT s and t h e f a c t t h a t these i n f i n i t e superalgebras correspond t o g e n e r a l i z a t i o n s o f Kac-Moody algebras o r s t r i n g algebras.I f t h e bound s t a t e s o f extended s u p e r g r a v i t y t h e o r i e s do indeed correspond t o u n i t a r y r e a l i z a t i o n s o f t h e i n f i n i t e superalgebras generated by t h e i r non-compact symmetry and supersymmetry generators, then a l l t h e ESGT s f o r N b 5 can i n p r i n c i p l e accommodate a r e a l i s t i c GUT based on SU(5) as an e f f e c t i v e "low energy" theory.
However, t h e N = 8 ESGT may t u r n o u t t o be t h e o n l y p h y s i c a l l y v i a b l e t h e o r y when one faces t h e c r u c i a l q u e s t i o n o f how these l a r g e symmetries break i n such a way as t o be l e f t w i t h t h e observed SU(3)C x SU(2)k, x U(1) symmetry (and p o s s i b l y a simple supersymmetry) and t h e r i g h t spectrum of "elementary p a r t i c l e s " a t low energies.
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