Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 88
Figure 35: Schematic plots of the EW (left) and QCD (right) phase diagrams
in terms of temperature and the relevant chemical potentials (µL and µB are,
respectively, the leptonic and the baryonic chemical potentials). The solid lines
correspond to the critical lines for a number of different Higgs/strange quark
masses and the dotted lines indicate the location of the endpoint of the first–
order phase transition line as the masses are varied. The arrows the order of
magnitude in which masses increase along the dotted lines (Gynther, 2006).
value as (e.g. Anderson & Hall, 1992)
U(φ) = λ0
4
2 2 2
φ − φ0 (155)
where φ0 is the expectation value of the Higgs field and λ0 is related to the
Higgs boson mass by (e.g. Anderson & Hall, 1992)
m 2 H =2λ0φ 2 0. (156)
The EW phase transition takes place when the expectation value of the Higgs
field passes from its high temperature value 〈φ〉 = 0 to its nonzero value in the
low temperature broken phase (e.g. Mégevand, 2000).
To reliably analyze the dynamics of this field, we need to include the interactions
of the Higgs field with virtual particles and with the heat bath (Anderson
& Hall, 1992). The one–loop, zero temperature potential, V (φ) can be written
as the sum of the classical potential and a one–loop correction (Anderson &
Hall, 1992)
V (φ) =U(φ)+ ¯ V1(φ). (157)
If we adopt the renormalization prescriptions (e.g. Anderson & Hall, 1992)
V ′′ (φ0) =m 2 H
(158)
V ′ (φ0) = 0 (159)