energy-scale dependent, and one of them (g 1 ) is not asymptotically free, raising
the question of whether there is a short-distance problem with the definition of
this part of the theory.
One attempt to answer these questions is the idea of a “Grand Unified The-
ory (GUT)”, based on a large Lie group that includesU(1)×SU(2)×SU(3)
as subgroups (typical examples areSU(5) andSO(10)). The question of “why
this group?” remains, but in principle one now only has one coupling constant
instead of three. A major problem with this idea is that it requires introduction
of some new fields (a new set of Higgs fields), with dynamics designed to leave
a low-energyU(1)×SU(2)×SU(3) gauge symmetry. This introduces a new set
of problems in place of the original one of the three coupling constants.
48.4.2 Why these representations?
We saw in section 48.2 that the fundamental left and right-handed spin^12
fermionic fields carry specific representations under theU(1)×SU(2)×SU(3)
gauge group. We would like some sort of explanation for this particular pattern.
An additional question is whether there is a fundamental right-handed neutrino,
with the gauge groups acting trivially on it. Such a field would have quanta
that do not directly interact with the known non-gravitational forces.
The strongest argument for theSO(10) GUT scenario is that a distinguished
representation of this group, the 16 dimensional spinor representation, restricts
on theU(1)×SU(2)×SU(3)⊂SO(10) subgroup to precisely the representation
corresponding to a single generation of fundamental fermions (including the
right-handed neutrino as a trivial representation).
48.4.3 Why three generations?
The pattern of fundamental fermions occurs with a three-fold multiplicity, the
“generations”. Why three? In principle there could be other generations, but
these would have to have all their particles at masses too high to have been
observed, including their neutrinos. These would be quite different from the
known three generations, where the neutrino masses are light.
48.4.4 Why the Higgs field?
As described in section 48.3, the Higgs field is an elementary scalar field, trans-
forming as the standardC^2 representation of theU(1)×SU(2) part of the gauge
group. As a scalar field, it has quite different properties and presumably a dif-
ferent origin than that of fundamental fermion and gauge fields, but what this
might be remains a mystery. Besides the coupling to gauge fields, its dynamics
is determined by its potential function, which depends on two parameters. Why
do these have their measured values?