Phase transitions in the Higgs model
37
with aIL as in (2.13). The Fadeev-Popov ghosts ." cancel contributions to the free
energy from the two non-physical degrees of freedom of the gauge field AIL, as
discussed in section 2.2, and ~ is the gauge-fixing parameter. For spontaneous
symmetry-breaking to occur (without requiring radiative corrections to drive it),
m 2 must be negative.
To derive the finite-temperature effective potential using the methods of
section 2.3 it is necessary to shift the scalar field by its expectation value. We
write
tPc
(tP) = .,fi
(2.37)
where the factor of I/.,fi has no significance but has simply been introduced for
convenience, and tPc may be taken to be real because of gauge invariance. Then
real fields tPl and tP2 are introduced through
I
tP = .y'2(tPc + tPl + itP2).
(2.38)
The quadratic terms in the shifted Lagrangian density are
I (2 3A 2) 2 I (2 A 2) 2
Cquad = - 2" m + 4tPc tPl - 2" m + 4tPc tP2
e 2 2
I -
2 I - 2
+ "2tPcAILAIL + 2"(BlLtPl) + 2"(BIL tP2)
- ~ (ijILAIL)2 + aIL"'· aIL IJ
(2.39)
where we have adopted the Landau gauge ~ -+ 0 which removes an A lL a lL tP2
cross term. Then, in the notation of (2.28),
M~(tPc) = diag (m2 + 3: tP;, m 2 + ~tP;)
(2.4O)
and
~ 2
2 2
Mv(tPc) = e tPc'
(2.41)
The nature of the phase transition depends on the relative sizes of e 4 and A.
2.4.1 e 4 « 1.
The tree-level contribution Vo( tPc} to the effective potential may be read from
(2.33) by replacing tP by its expectation value ~tPc and is
m 2
-
2
).. 4
(2.42)
Vo(tPc) = TtPc + t6tPc'