202 The very early universe
Supersymmetry The symmetries we have considered so far relate bosons to bosons
and fermions to fermions. However, there may be a beautiful symmetry, known as
supersymmetry, which relates bosons and fermions. If supersymmetry is a true
symmetry of nature, thenevery bosonshould haveat leastone fermionic super-
partner, with which it is paired in thesupermultiplet. Every fermionshould also be
partnered with at least one boson. Under a supersymmetry transformation, bosons
and fermions in the same supermultiplet are “mixed” with each other. It is clear
that the supersymmetry generator which converts a boson to a fermion should be a
spinorQ,and in the simplest case it is achiralspinor of spin 1/ 2 .Because bosons
and fermions transform differently under the Poincar ́e group, supersymmetry trans-
formations, unlike gauge transformations, cannot be completely decoupled from
spacetime transformations. In fact, the algebra of the supersymmetry generatorsQ
closes only when the generators of the space and time translations are included.
Hence, if we try to make supersymmetry local, we are forced to deal with curved
spacetime.Local supersymmetry, calledsupergravity, thus offers a possible way to
unify gravity with the other forces.
Unfortunately, as in the case of Grand Unification, there are too many potential
supersymmetric extensions of the Standard Model. First, the supersymmetry can be
global or local. Second, we could include more than one boson–fermion pair in the
same supermultiplet, an idea known asextendedsupersymmetry. In principle, all
particles could be the members of asinglemultiplet. Extended supersymmetry is
characterized by the number of supersymmetry generatorsQ^1 ,Q^2 ,...,QNwhich
determine the particle content of the supermultiplets. For instance, forN=8 the
supermultiplet contains bothleft- and right-handedparticles of spins 0, 1 / 2 , 1 , 3 / 2
and 2; thusN= 8 supergravitywould be an ideal candidate for unification. Unfor-
tunately (or fortunately, depending on one’s attitude) Nature does not act upon our
wishes and in the absence of experimental data we must consider a diverse range
of theoretical possibilities.
All supersymmetric theories have features in common and we concentrate on
those which are relevant for cosmological applications. As we have mentioned, in
these theories bosons and fermions are paired. Disappointingly, all combine known
fermions with unknown bosons and vice versa. Hence, supersymmetric theories
predictthat the number of particles should beat leasttwice as big as the number of
experimentally discovered particles. To understand why the supersymmetric part-
ners of the known particles have not yet been discovered, we are forced to assume
that supersymmetry is broken above the scale currently reached by accelerators.
It is only when supersymmetry is broken that supersymmetric partners can have
different masses; otherwise they are obliged to have the same mass.
In the minimal supersymmetric extension of the Standard Model, usually called
MSSM, every quark and lepton has a supersymmetric scalar partner, called asquark