4.6 Beyond the Standard Model 201measurements of the proton lifetime and the discovery of neutrino masses, rules
out the minimalSU( 5 )model as a realistic theory. Nevertheless, we will take this
model a little further in order to explain some important features common to more
realistic theories.
TheSU( 5 )group has 5^2 − 1 =24 generators and correspondingly there are
24 gauge bosons. Eight of them should be identified with the gluons responsible
for color transitions within theSU( 3 )subgroup. Three bosons correspond to
theSU( 2 )subgroup and together with oneU( 1 )boson are responsible for the
electroweak interactions. The remaining 12 bosons form two charged colored
triplets,
Xi±^4 /^3 , Yi±^1 /^3 ,wherei=r,b,gis the color index and the upper indices correspond to the electric
charge. TheX,Ybosons form a “bridge” between theSU( 2 )andSU( 3 )subgroups
ofSU( 5 ).After symmetry breaking (for instance, via the Higgs mechanism), they
acquire masses of order∼ 1015 –10^17 GeV and the transitions betweenSU( 2 )and
SU( 3 )subgroups are strongly suppressed. At high energies, however, theX,Y
bosons can “convert” quarks to leptons and vice versa very efficiently.
TheX boson can decay either into a pair of quarks (X→qq)or into an
antiquark–antilepton pair
(
X→q ̄ ̄l)
.The baryon numbers of the final products
areB= 2 /3 andB=− 1 /3 respectively; hencebaryon number is not conserved
in theSU( 5 )model. On the other hand, the difference between baryon and lepton
number is equal to 2/3 in both cases andB−Lis not violated. HenceB−Lcannot
be generated and any baryon asymmetry will be washed out in subsequent topo-
logical transitions. Thus the baryon asymmetry problem cannot be solved within
theSU( 5 )model.
The larger symmetry groups look more attractive for several different reasons.
First of all, they have a richer fermion content than the minimalSU( 5 )model.
In particular, one can incorporate a right-handed neutrino, which is needed to ex-
plain the neutrino masses. The other attractive feature of these theories is that
they not only violate baryon number, but also do not conserveB−L.This opens
the door to an understanding of the origin of the baryon asymmetry of the uni-
verse. Finally, changing the fermion content influences the running of coupling
constants, and thus one can hope that they will yet meet at one point. There is ex-
tensive literature dealing with large gauge groups, for example,SO( 10 ),SO( 14 ),
SO( 22 ),...,E 6 ,E 7 ,E 8 ,.... Thecorresponding theories contain many particles
not yet discoveredand hence many candidates for dark matter. However, in the
absence of solid data, we see no point in going into further detail of unified theories
here.