**1.2. Phase transitions and finite temperature field theory**

Phase transitions are known to occur in the early universe. Examples
we mentioned are the quark to hadron (confinement) transition, which
QCD predicts at an energy around 1 GeV, and the electroweak phase
transition at about 250 GeV. Within grand unified theories (GUT), aiming to
describe the physics beyond the standard model, other phase
transitions are predicted to occur at energies of order 10^{15} GeV;
during these, the Higgs field tends to fall towards the minima of its
potential while the overall temperature of the universe decreases as a
consequence of the expansion.

A familiar theory to make a bit more quantitative the above
considerations is the
||^{4} theory,

(1) |

with *m*^{2}_{0} > 0.
The second and third terms on the right hand side yield the usual
`Mexican hat' potential for the complex scalar field. For energies
much larger than the critical temperature, *T*_{c}, the
fields are in
the so-called `false' vacuum: a highly symmetric state characterized
by a vacuum expectation value
<| | >
= 0. But when
energies decrease the symmetry is spontaneously broken: a new `true'
vacuum develops and the scalar field rolls down the potential and sits
onto one of the degenerate new minima. In this situation the vacuum
expectation value becomes
<| | >^{2}
= 6 *m*_{0}^{2} /
.

Research done in the 1970's in finite-temperature field theory [Weinberg, 1974; Dolan & Jackiw, 1974; Kirzhnits & Linde, 1974] has led to the result that the temperature-dependent effective potential can be written down as

(2) |

with *T*_{c}^{2} = 24
*m*_{0}^{2} /
,
*m*^{2}(*T*) = *m*_{0}^{2} (1 -
*T*^{2} / *T*_{c}^{2}), and
<| | >^{2}
= 6 *m*^{2}(*T*) /
. We
easily see that when *T* approaches *T*_{c} from below
the symmetry is restored, and again we have
<| | > = 0. In
condensed-matter jargon, the transition described above is
second-order
[Mermin, 1979].
^{(3)}

^{3} In a first-order phase transition the
order parameter
(*e.g.*, < | | >
in our case) is not continuous. It may
proceed by bubble nucleation
[Callan & Coleman, 1977;
Linde, 1983b]
or by spinoidal decomposition
[Langer, 1992].
Phase transitions can
also be continuous second-order processes. The `order' depends
sensitively on the ratio of the coupling constants appearing in the
Lagrangian.
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