Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 89
for each degree of freedom to which the Higgs boson is coupled, the zero temperature
one–loop correction to the effective potential is (see Anderson & Hall,
1992)
¯V1(φ) =± 1
64π2
m 4 (φ) ln m2 (φ)
m2 (φ0) −
− 3
2 m4 (φ)+2m 2 (φ)m 2 (φ0) − 1
2 m4
(φ0)
(160)
where ± is for bosons (fermions) and m(φ) is the mass of the particle in the
presence of the background field φ. In addition to these quantum corrections,
we must also include the interaction between the Higgs field and the hot EW
plasma. Taking the Higgs boson sufficiently light, the effective potential for the
standard model can be reliably written as (e.g. Anderson & Hall, 1992)
V (φ, T )=D(T 2 − T 2 0 )φ2 − ET φ 3 + λT
4 φ4 . (161)
All the parameters in equation (161) depend on the particle content of the
theory (e.g. Mégevand, 2000). Parameter D contains contributions from all the
particles that acquire their masses through the Higgs mechanism and is given
by (Anderson & Hall, 1992)
D = 1
8φ2 2
2mW + m
0
2 Z +2m2
t , (162)
while the coefficient ot the term linear in temperature E, which has only boson
contributions, is given by (Anderson & Hall, 1992)
E = 1
4πφ3 3
2mW + m
0
3
Z . (163)
In the SMPP we have D ∼ 10−1 and E ∼ 10−2 while in the MSSM, due to the
larger particle zoo (see e.g. Table 11), D and E can be more than an order of
magnitude larger than in the SMPP (e.g. Mégevand, 2000).
The temperature–dependent φ4 coupling can be written as (e.g. Gynther,
2006)
3
λT = λ −
16π2φ4
2m
0
4 W ln m2W cBT 2 + m4Z ln m2Z cBT 2 − 4m4t ln m2t cF T 2
(164)
where the masses are evaluated at 〈φ〉 = φ0 and we have cB 5.41 and
cF 2.64 (Anderson & Hall, 1992). Although the parameter λT is temperature–
dependent, it is almost constant in the range of temperatures in which the phase
transition can take place. However, this parameter is very sensitive to the Higgs
mass (e.g. Mégevand, 2000).