Phase transitions in supersymmetric GUTs
51
For T > Tc. the system is in the SU(5) symmetric phase. for which tPc = O. and
all gauge bosons are massless. For T < Tc, tPc is non-zero, the system is in the
SU(3)c x SU(2)L x U(I)y symmetric phase, and only the electroweak gauge
bosons are massless. By the same sort of argument as in the previous section, Tc
should be of order 10 15 GeV.
However, if g~ » A I. A.2. then a discussion similar to that given for the
Higgs model in section 2.4 shows that a first-order phase transition takes place.
In the case of a GUT, this conclusion is not negated by Yukawa couplings of
fermions giving additional contributions to the coefficient B. because quarks and
leptons do not couple to the grand unified Higgses. This is an important difference
compared with the electroweak phase transition.
2.7 Phase transitions in supersymmetric GUTs
If elementary particle theories possess supersymmetry, then each (complex scalar)
spin-O particle is paired with one chirality of a spin-! particle in the same socalled 'chiral supermultiplet' and each spin-I vector particle is paired with a
spin-! particle of a single chirality in the same so-called 'vector supermultiplet'.
The quarks and leptons have supersymmetric partners referred to as 'squarks'
and 'sleptons·. the Higgs scalars have supersymmetric partners referred to as
'Higgsinos' and the gauge bosons are paired with 'gauginos·. In the absence
of supersymmetry breaking. particles in the same supermultiplet have the
same mass. Of course, since at the time of writing we have not observed
supersymmetric partners of the known particles ('sparticles·). there must be some
(spontaneous) supersymmetry breaking to produce substantial mass splittings
within supermultiplets.
The presence of these extra sparticles can be very important for the
discussion of phase transitions at temperatures large compared to the sparticle
masses. In addition. the supersymmetry transformations transforming particles of
different spin within a supermultiplet into each other strongly constrain the form
of the Lagrangian and the tree-level effective potential. with further implications
for phase transitions. These supersymmetry transformations may be local or
global depending respectively on whether the parameters of the transformation
do or do not depend on the point in spacetime. In this section the case of globally
supersymmetric GUTs will be discussed [7-12] and the locally supersymmetric
(supergravity) case will be discussed in the next section. (For a systematic
development of globally and locally supersymmetric theories see [14].)
In general. in a supersymmetric theory, the Lagrangian and the tree-level
effective potential are determined once the superpotential W is given. For
example. for a theory with a single scalar field tP together with its supersymmetric
partner. the superpotential for a renormalizable theory takes the form
w = !mtP 2 + !l.t/J3
(2.11S)
51
For T > Tc. the system is in the SU(5) symmetric phase. for which tPc = O. and
all gauge bosons are massless. For T < Tc, tPc is non-zero, the system is in the
SU(3)c x SU(2)L x U(I)y symmetric phase, and only the electroweak gauge
bosons are massless. By the same sort of argument as in the previous section, Tc
should be of order 10 15 GeV.
However, if g~ » A I. A.2. then a discussion similar to that given for the
Higgs model in section 2.4 shows that a first-order phase transition takes place.
In the case of a GUT, this conclusion is not negated by Yukawa couplings of
fermions giving additional contributions to the coefficient B. because quarks and
leptons do not couple to the grand unified Higgses. This is an important difference
compared with the electroweak phase transition.
2.7 Phase transitions in supersymmetric GUTs
If elementary particle theories possess supersymmetry, then each (complex scalar)
spin-O particle is paired with one chirality of a spin-! particle in the same socalled 'chiral supermultiplet' and each spin-I vector particle is paired with a
spin-! particle of a single chirality in the same so-called 'vector supermultiplet'.
The quarks and leptons have supersymmetric partners referred to as 'squarks'
and 'sleptons·. the Higgs scalars have supersymmetric partners referred to as
'Higgsinos' and the gauge bosons are paired with 'gauginos·. In the absence
of supersymmetry breaking. particles in the same supermultiplet have the
same mass. Of course, since at the time of writing we have not observed
supersymmetric partners of the known particles ('sparticles·). there must be some
(spontaneous) supersymmetry breaking to produce substantial mass splittings
within supermultiplets.
The presence of these extra sparticles can be very important for the
discussion of phase transitions at temperatures large compared to the sparticle
masses. In addition. the supersymmetry transformations transforming particles of
different spin within a supermultiplet into each other strongly constrain the form
of the Lagrangian and the tree-level effective potential. with further implications
for phase transitions. These supersymmetry transformations may be local or
global depending respectively on whether the parameters of the transformation
do or do not depend on the point in spacetime. In this section the case of globally
supersymmetric GUTs will be discussed [7-12] and the locally supersymmetric
(supergravity) case will be discussed in the next section. (For a systematic
development of globally and locally supersymmetric theories see [14].)
In general. in a supersymmetric theory, the Lagrangian and the tree-level
effective potential are determined once the superpotential W is given. For
example. for a theory with a single scalar field tP together with its supersymmetric
partner. the superpotential for a renormalizable theory takes the form
w = !mtP 2 + !l.t/J3
(2.11S)