Phase transitions in grand unified theories
49
re normalization group is deployed to derive their dependence on M. However,
for energy scales greater than MG. all gauge fields are on the same footing and
there is a single gauge coupling constant gG developing in accordance with the
renormalization group equation of the GUT.
The simplest example of a grand unified group that is large enough to contain
SU(3)c x SU(2)L x U(1)y of the standard model is SU(5) and we shall use this
example to illustrate phase transitions in GUTs. If the renormalization group
equations are used to run the low-energy values of the gauge coupling constants
to the scale M = MG at which the SU(5) relationships
gs(MG) = g(MG) = jig'(MG) = gG(MG)
(2.106)
hold, then upon inputting the values of the strong and electromagnetic coupling
constants at M = mz, the unification scale is found to be of order 10 15 GeV.
In addition, there is a prediction for sin 2 9w(mz) at M = mz of around 0.21,
which differs significantly from the observed value of around 0.23. Nevertheless,
we shall use the SU(5) GUT as a simple illustration of the way in which phase
transitions work in a GUT. In section 2.7, we shall consider supersymmetric GUTs
in which the prediction for sin 2 (mz) can be brought into line with experiment to
a high degree of accuracy.
In the SUeS) GUT, the grand unified phase transition is from the SU(5)
symmetric phase to the SU(3)c x SU(2)L x U(I)y symmetric phase and is
followed at a lower temperature by the electroweak phase transition described
in the previous section. We expect the critical temperature for the grand unified
phase transition to be of order 10 15 GeV (the energy at which the spontaneous
symmetry breaking occurs) and, at such high temperatures. the expectation values
of the electroweak Higgs scalars (of order 200 Ge V) are negligible. Thus, to
describe the grand unified phase transition we need only retain the Higgs scalars
responsible for breaking the SU (5) gauge group, whose expectation values are on
the 10 15 GeV scale.
The grand unified Higgs scalars belong to the 24-dimensional adjoint
representation of SUeS):
24
= L tPala
(2.107)
a=1
where la are the SU(5) generators in the fundamental five-dimensional
representation. Suppressing the gauge-fixing term and the Padeev-Popov ghost
term, the finite-temperature Lagrangian density (apart from a possible tr <1>3 term)
is
c = -m~ tr <1>2 - AI (tr <1>2)2 - A2 tr <1>4 + tr(Dp. <1»2 - t tr(Fp.v FP.V). (2.108)
In (2.108), the covariant derivative Dp. is given by
Dp.<1> = ap' <1> + igG[Ap., <1>]
(2.109)
49
re normalization group is deployed to derive their dependence on M. However,
for energy scales greater than MG. all gauge fields are on the same footing and
there is a single gauge coupling constant gG developing in accordance with the
renormalization group equation of the GUT.
The simplest example of a grand unified group that is large enough to contain
SU(3)c x SU(2)L x U(1)y of the standard model is SU(5) and we shall use this
example to illustrate phase transitions in GUTs. If the renormalization group
equations are used to run the low-energy values of the gauge coupling constants
to the scale M = MG at which the SU(5) relationships
gs(MG) = g(MG) = jig'(MG) = gG(MG)
(2.106)
hold, then upon inputting the values of the strong and electromagnetic coupling
constants at M = mz, the unification scale is found to be of order 10 15 GeV.
In addition, there is a prediction for sin 2 9w(mz) at M = mz of around 0.21,
which differs significantly from the observed value of around 0.23. Nevertheless,
we shall use the SU(5) GUT as a simple illustration of the way in which phase
transitions work in a GUT. In section 2.7, we shall consider supersymmetric GUTs
in which the prediction for sin 2 (mz) can be brought into line with experiment to
a high degree of accuracy.
In the SUeS) GUT, the grand unified phase transition is from the SU(5)
symmetric phase to the SU(3)c x SU(2)L x U(I)y symmetric phase and is
followed at a lower temperature by the electroweak phase transition described
in the previous section. We expect the critical temperature for the grand unified
phase transition to be of order 10 15 GeV (the energy at which the spontaneous
symmetry breaking occurs) and, at such high temperatures. the expectation values
of the electroweak Higgs scalars (of order 200 Ge V) are negligible. Thus, to
describe the grand unified phase transition we need only retain the Higgs scalars
responsible for breaking the SU (5) gauge group, whose expectation values are on
the 10 15 GeV scale.
The grand unified Higgs scalars belong to the 24-dimensional adjoint
representation of SUeS):
24
= L tPala
(2.107)
a=1
where la are the SU(5) generators in the fundamental five-dimensional
representation. Suppressing the gauge-fixing term and the Padeev-Popov ghost
term, the finite-temperature Lagrangian density (apart from a possible tr <1>3 term)
is
c = -m~ tr <1>2 - AI (tr <1>2)2 - A2 tr <1>4 + tr(Dp. <1»2 - t tr(Fp.v FP.V). (2.108)
In (2.108), the covariant derivative Dp. is given by
Dp.<1> = ap' <1> + igG[Ap., <1>]
(2.109)