SUPERGRAVITY, INTERMEDIATE SUPERSYMMETRY BREAKING AND PARTICLE PHYSICS

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SUPERGRAVITY, INTERMEDIATE

SUPERSYMMETRY BREAKING AND PARTICLE PHYSICS

S. Ferrara

To cite this version:

S. Ferrara. SUPERGRAVITY, INTERMEDIATE SUPERSYMMETRY BREAKING AND PARTICLE PHYSICS. Journal de Physique Colloques, 1982, 43 (C3), pp.C3-293-C3-296.

�10.1051/jphyscol:1982358�. �jpa-00221913�

JOURNAL DE PHYSIQUE

ColZoque C3, supple'ment au n o 1 2 , Tome 43, de'cembre 1982 page C3-293

SUPERGRAVITY, INTERMEDIATE SUPERSYMMETRY BREAKING AND PARTICLE PHYSICS

S. Ferrara

Division T', CERiV, I211 Geneva 23, Switzerland

Since the discovery of supergravity1)'2) it has been realized that one of the most important issues to solve in order to make contact with the real world is to understand the relation between the gravitino mass and the other fundamental scales of Nature.

In fact, when supersymmetry is local, the gravitino mass is the order parameter of spontaneously broken supersymmetry in much the same way as the gauge vector boson masses are related to the,symmetry breaking scales of spontaneously broken Yang- Mills theories.

The crucial difference is that for gauge vector boson masses one has a dimen- sionless coupling constant

while in supergravity the gauge coupling is dimensionful

,

^k = Mp -1

,

and one gets

where ch> is the v.e.v. of some (auxiliary) non-propagating dimension two field which, in terms of the elementary scalar fields, has a model-dependent form.

From Eq. ( 2 ) it is fairly obvious that if <h> % p2 where 1-1 is a low-energy scale, then m ,2 is extremely small and, as pointed out by ~ a ~ e t 3 ) , the helicity +1/2 componenzs of the gravitino essentially behave as a Goldstino of a spontaneously broken globally supersymmetry theory.

It was later realized4) that in globally supersymmetric models with spontaneously broken supersymmetry it is difficult to obtain a physically acceptable renormalizable theory with the correct pattern of gauge symmetry breaking unless the primordial supersymmetry breaking occurs at an intermediate scale

with

~6 4 Tev

In this situation the gravitino mass becomes of order 1-1 and supergravity may play an important role in particle physics at low energy5).

I would like to report on some recent work on the general coupling of n = 1 supergravitytosupersymmetric matter Yang-Mills systems which in particular indicates the possible relevance of supergravity to particle physics if the gravitino mass is of the order of the weak interaction scale. The supergravity Lagran ian which in- corporates arbitrary matter and Yang-Mills interactions is obtained67 by introducing vector multiplets in the adjoint representation of a generic Yang-Mills group G and chiral multiplets in arbitrary complex representations of G. The only restriction on the chiral multiplets is to form an anomaly-free set of representations of G.

The final form of the Lagrangian is given by Cremmer et al. in Ref. 6) and all results which follow remain valid with the inclusion of U ( l ) factors in G with or

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982358

C3-294 JOURNAL DE PHYSIQUE

without a F a y e t - I l i o p o u l o s term. The f i n a l Lagrangian depends on two a r b i t r a r y f u n c t i o n s Q ( z , z + + ) , f a g ( z ) which have t o f u l f i l t h e f o l l o w i n g p r o p e r t i e s : ~ ( z , z * ) i s r e a l and G i n v a r i a n t , i . e . ,

T Y ~ a r e t h e g e n e r a t o r s of t h e gauge group G. fol6(z) i s a n a n a l y t i c f u n c t i o n o f t h e s c a l a r components z i o f t h e c h i r a l m u l t i p l e t s , which t r a n s f o r m s a s t h e symmetric pro- d u c t o f t h e a d j o i n t r e p r e s e n t a t i o n o f G.

it h a s been p o i n t e d o u t by Crenuner e t al.') t h a t a n o n - t r i v i a l c h o i c e o f f a B ( z ) f f 6aB may i n t r o d u c e d i r e c t gaugino mass terms a s w e l l a s CP v i o l a t i n g terms:

Gaugino mass:

o ( ' - CP v i o l a t i n g i n t e r a c t i o n :

i ₄

^{I m}

e. ^{F F}

p r r

These terms may be r e l e v a n t f o r phenomenological a p p l i c a t i o n s .

We d e f i n e a s a "minimal" c o u p l i n g t h e s i t u a t i o n which corresponds t o

I n t h i s c a s e t h e i n t e r a c t i o n o n l y depends on a s u p e r p o t e n t i a l f u n c t i o n g ( z ) s i n c e

z

= r

I Z ; ~ ~ c

e ~ ~ r ~ c . ) ~ ~

^{( 6 )}

F o r t h e s e i n t e r a c t i o n s , model-independent mass r e l a t i o n s have been d e r i v e d ⁶

where N is t h e number o f c h i r a l m u l t i p l e t s , go t h e gauge coupling, and D" t h e a u x i l i a r y component o f t h e v e c t o r m u l t i p l e t . I n most c a s e s D~ = 0 a t t h e minimum o f t h e po- t e n t i a l and one g e t s

T h i s mass formula proves6) t h a t when m3/2 = O ( p ) , then s u p e r g r a v i t y e f f e c t s a r e n o t n e g l i g i b l e f o r p a r t i c l e physics. I n t e r e s t i n g l y enough, i t a u t o m a t i c a l l y p r o v i d e s a p h y s i c a l l y a c c e p t a b l e mass s p l i t t i n g between quarks, l e p t o n s and t h e i r s c a l a r s u p e r p a r t n e r s without any need f o r e x t e n d i n g t h e gauge group by a t l e a s t a n e x t r a

g ( 1 ) f a c t o r a s p r e v i o u s l y proposed3). It should be s t r e s s e d t h a t f o r f i e l d s which g e t a vacuum e x p e c t a t i o n v a l u e , such a s t h e normal Higgs d o u b l e t s o f weak i n t e r a c t i o n s , t h e mass formula by no means i m p l i e s t h a t Higgs d o u b l e t s have p o s i t i v e q u a d r a t i c masses. I n f a c t , i n g e n e r a l t h e o p p o s i t e is t r u e s i n c e n e g a t i v e mass terms a r i s e from t h e s u p e r p o t e n t i a l c o n t r i b u t i o n t o t h e mass matrix. The f i n a l s u p e r g r a v i t y s c a l a r p o t e n t i a l i s :

with

A general formalism has been proposed7) to study the low-energy limit of the super- gravity Lagrangian and in particular of the potential given by Eq. ( 9 ) . ^{It is} useful to separate the matter fields into a G-singlet z and fields yi transforming under G.

The superpotential g(z,yi) can be taken to be of the form 7

(10) so one can use the results of Cremmer et a1.*) to cancel the cosmological constant with

One can expand the Lagrangian around the v.e.v. of z and then neglect non-renormaliz- able interactions which are damped by inverse powers of Mp. The final result is:

with

where fi(yi) is an analytic polynomial at most cubic in the fields yi:

In the limit k + 0 (~~-;m) with m3/2 fixed, the z-field completely decouples and it corresponds to two massive neutral free fields. ~ecentl~g), it has been shown that a different parametrization of g(z,y)

leads to a different choice of 6 such that the final low-energy potential can be written as

+ ^ma $*:Iz

V = = ^l

^'a

A possible advantage of (17) is that the study of the potential merely reduces to the solutions of the (quadratic) equations

Ei

^: 0 Yi. The potential danger is that many degenerate solutions may exist whose degeneracy is only removed by radiative corrections. In all cases, the soft breaking does not introduce new quadratic di- vergences since the mass formula is field independent. It is now easy to provide models10 ) for QED

,

electroweak interactions and GUTS where, for instance, the weak interaction scale is induced by the gravitino mass and the GUT mass has to be put in the original superpotential g(yi). Irrespective of any particular model, the forma- lism developed in Ref. 7) seems to be a powerful tool for approximating in the low- energy regime sponta~eously broken locally supersymmetry in terms of explicitly softly broken globally supersymmetric theories.

JOURNAL DE PHYSIQUE

REFERENCES

1) FREEDMAN D.Z., VAN NIEUWENHUIZEN P. and FERRARA S., Phys. Rev.

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3) FAYET P., in "Unification of the Fundamental Particle Interactions!', Eds.

S. Ferrara, J. Ellis and P. van Nieuwenhuizen (Plenum Press, N .Y.

,

1980), P. 587.

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C13

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7) BARBIERI R., FERRARA S. and SAVOY C.A., CERN preprint 3365 (1982), to appear in Phys. Lett. B.

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29B

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10) See Ref. 7) and also: ARNOWITT R., CHAMSEDDINE A.H. and NATH P., Northeastern preprints 2559 and 2565 (19821;

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