Phase transitions in supersymmetric GUTs
53
Por the S U (3) x S U (2) x U (I) symmetric phase, the 12 gauge fields together
with their associated gauginos contribute 45 towards NB + ~ N F and, for the
S U (4) x U (I) symmetric phase, the 16 gauge fields together with their gauginos
contribute 60 towards NB + iN F. In each of these two cases, the matter field
content is the same as for the SU(5) symmetric phase and we find that
",2 (
7) -19
.
90 NB + gNF = -s-1r 2
SU(3) x SU(2) x U(l) symmetric phase
(2.127)
1r2 (
2
7) -61
-
NB+-NF = - 1 r
SU(4) x U(I) symmetric phase. (2.12S)
90
8
24
If a copy of 5 + 5 is included to provide the two electroweak Higgs doublets
(plus Higgsinos) needed to give masses to both up-like and down-like quarks in
a supersymmetric theory, then there is an additional contribution -5",2/12 to
(2.126) and _",2/6 to (2.127), in the latter case from the two SU(2)L doublets
that are all that survive from the 5 + 5 after spontaneous symmetry breaking of
SU(5) by the expectation values of the adjoint Higgs scalars. (The surviving
adjoint Higgs scalar states are too heavy to contribute to the temperaturedependent corrections to the effective potential for temperatures below the grand
unification scale.) In the SU(4) xU(I) phase, the completeS+Sbecomes massive
and fails to contribute to the temperature-dependent corrections. The values of
~(NB + tNF) in (2.126),(2.127) and (2.128) are then modified to ~1r2, ~",2
and ~1r2, respectively. Thus, the T4 term favours the SU(5) symmetric phase
over the SU(3) x SU(2) x U(l) and SU(4) x U(l) symmetric phases, which
remain on the same footing. This conclusion is only strengthened by the inclusion
of the adjoint Higgs supermutiplet which provides extra light states in the SU(5)
symmetric phase. If the theory contains more than one pair of Higgs multiplets
coming from 5 + 5, then the SU(5) symmetric phase continues to be favoured
over the other two phases but the SU(3) x SU(2) x U(l) symmetric phase is
favoured over the S U (4) x U ( 1) symmetric phase, which is the assumption we
shall make in what follows.
Clearly, the SU(5) symmetric phase, for which the scalar expectation value
is zero, minimizes the T2~~ term as well as the T4 term. Thus, the theory
appears to favour the SU(5) symmetric phase at all temperatures. However,
at temperatures below lOO GeV-I TeV the (non-perturbative) supersymmetry
breaking mechanism will lift the degeneracy of the three phases more than
the temperature-dependent terms and may favour the SU(3) x SU(2) x U(1)
symmetric phase. At higher temperatures the temperature-dependent terms
dominate. This suggests that the universe is in an SU (5) symmetric phase down
to temperatures of 100 Ge V-I Te V.
This conclusion is modified by the running of the gauge coupling constant
gS for SU(5) with temperature. This may result in gS becoming strong at
temperatures of order 10 9 _10 10 GeV. Then, confinement may result in fewer
53
Por the S U (3) x S U (2) x U (I) symmetric phase, the 12 gauge fields together
with their associated gauginos contribute 45 towards NB + ~ N F and, for the
S U (4) x U (I) symmetric phase, the 16 gauge fields together with their gauginos
contribute 60 towards NB + iN F. In each of these two cases, the matter field
content is the same as for the SU(5) symmetric phase and we find that
",2 (
7) -19
.
90 NB + gNF = -s-1r 2
SU(3) x SU(2) x U(l) symmetric phase
(2.127)
1r2 (
2
7) -61
-
NB+-NF = - 1 r
SU(4) x U(I) symmetric phase. (2.12S)
90
8
24
If a copy of 5 + 5 is included to provide the two electroweak Higgs doublets
(plus Higgsinos) needed to give masses to both up-like and down-like quarks in
a supersymmetric theory, then there is an additional contribution -5",2/12 to
(2.126) and _",2/6 to (2.127), in the latter case from the two SU(2)L doublets
that are all that survive from the 5 + 5 after spontaneous symmetry breaking of
SU(5) by the expectation values of the adjoint Higgs scalars. (The surviving
adjoint Higgs scalar states are too heavy to contribute to the temperaturedependent corrections to the effective potential for temperatures below the grand
unification scale.) In the SU(4) xU(I) phase, the completeS+Sbecomes massive
and fails to contribute to the temperature-dependent corrections. The values of
~(NB + tNF) in (2.126),(2.127) and (2.128) are then modified to ~1r2, ~",2
and ~1r2, respectively. Thus, the T4 term favours the SU(5) symmetric phase
over the SU(3) x SU(2) x U(l) and SU(4) x U(l) symmetric phases, which
remain on the same footing. This conclusion is only strengthened by the inclusion
of the adjoint Higgs supermutiplet which provides extra light states in the SU(5)
symmetric phase. If the theory contains more than one pair of Higgs multiplets
coming from 5 + 5, then the SU(5) symmetric phase continues to be favoured
over the other two phases but the SU(3) x SU(2) x U(l) symmetric phase is
favoured over the S U (4) x U ( 1) symmetric phase, which is the assumption we
shall make in what follows.
Clearly, the SU(5) symmetric phase, for which the scalar expectation value
is zero, minimizes the T2~~ term as well as the T4 term. Thus, the theory
appears to favour the SU(5) symmetric phase at all temperatures. However,
at temperatures below lOO GeV-I TeV the (non-perturbative) supersymmetry
breaking mechanism will lift the degeneracy of the three phases more than
the temperature-dependent terms and may favour the SU(3) x SU(2) x U(1)
symmetric phase. At higher temperatures the temperature-dependent terms
dominate. This suggests that the universe is in an SU (5) symmetric phase down
to temperatures of 100 Ge V-I Te V.
This conclusion is modified by the running of the gauge coupling constant
gS for SU(5) with temperature. This may result in gS becoming strong at
temperatures of order 10 9 _10 10 GeV. Then, confinement may result in fewer