36
Phase transitions in the early universe
+ ~: [ ~(Mj)i + 3 ~(M~)a + 2 ~(MF)~]
- ~[L(M1)i + 3 L(M~)a] + ...
121l'
i
a
1l'2T4 (
7)
= - 9() NB + gNF
T2
A2
A2
A2
+ 24 [tr Ms(t/Jc> + 3 tr Mv(t/Jc> + 2 tr MF(t/JC>]
A
A
T
2
3/2
2
3/2
- -[tr(MS(t/Jc}} + 3 tr(Mv(t/Jc}} ] +.... (2.32)
121l'
where Mj(t/Jc), M~(t/Jc) and Mi(t/Jc)are the scalar. vector and Dirac fennion mass
matrices of (2.28). (For fennions described by Weyl spinor fields there should be
no factor of 2 in front of the Mi term in (2.32).) The T4 term in (2.32) is just the
free energy density for an ideal ultra relativistic gas (in agreement with (2.19) with
NB and N F respectively the number ofbosonic and fennionic degrees of freedom,
in the sense described following (2.19). If some fields are heavy and some are
light on the scale of the temperature T, then NB and N F should be interpreted
as the degrees of freedom of light fields, and the traces over the mass matrices
should be evaluated only for light fields. since heavy fields do not contribute. as
discussed earlier.
2.4 Phase transitions in the Higgs model
Before studying phase transitions in electroweak theory and grand unified theory,
we warm up on the simpler case of the Higgs model. The Higgs model is the
theory of a complex scalar field coupled to a V (I) gauge field. which may be taken
to be the electromagnetic field, with the Vel) gauge symmetry spontaneously
broken. In other words. it is scalar electrodynamics with spontaneously broken
electromagnetic gauge symmetry. The finite-temperature Lagrangian density is
C = D",t/JD"'t/J* - m 2 t/J*t/J - ~(t/J*t/J>2 - ~F"'\lF"'''
4
4
- ;~ (8",A",>2 + 8",'1*8"''1
(2.33)
where
F",,, == 8",A" - a"A",
(2.34)
D",t/J == (a", + ieA",)t/J
(2.35)
and
D",t/J* == (a", - ieA",)t/J*
(2.36)
Phase transitions in the early universe
+ ~: [ ~(Mj)i + 3 ~(M~)a + 2 ~(MF)~]
- ~[L(M1)i + 3 L(M~)a] + ...
121l'
i
a
1l'2T4 (
7)
= - 9() NB + gNF
T2
A2
A2
A2
+ 24 [tr Ms(t/Jc> + 3 tr Mv(t/Jc> + 2 tr MF(t/JC>]
A
A
T
2
3/2
2
3/2
- -[tr(MS(t/Jc}} + 3 tr(Mv(t/Jc}} ] +.... (2.32)
121l'
where Mj(t/Jc), M~(t/Jc) and Mi(t/Jc)are the scalar. vector and Dirac fennion mass
matrices of (2.28). (For fennions described by Weyl spinor fields there should be
no factor of 2 in front of the Mi term in (2.32).) The T4 term in (2.32) is just the
free energy density for an ideal ultra relativistic gas (in agreement with (2.19) with
NB and N F respectively the number ofbosonic and fennionic degrees of freedom,
in the sense described following (2.19). If some fields are heavy and some are
light on the scale of the temperature T, then NB and N F should be interpreted
as the degrees of freedom of light fields, and the traces over the mass matrices
should be evaluated only for light fields. since heavy fields do not contribute. as
discussed earlier.
2.4 Phase transitions in the Higgs model
Before studying phase transitions in electroweak theory and grand unified theory,
we warm up on the simpler case of the Higgs model. The Higgs model is the
theory of a complex scalar field coupled to a V (I) gauge field. which may be taken
to be the electromagnetic field, with the Vel) gauge symmetry spontaneously
broken. In other words. it is scalar electrodynamics with spontaneously broken
electromagnetic gauge symmetry. The finite-temperature Lagrangian density is
C = D",t/JD"'t/J* - m 2 t/J*t/J - ~(t/J*t/J>2 - ~F"'\lF"'''
4
4
- ;~ (8",A",>2 + 8",'1*8"''1
(2.33)
where
F",,, == 8",A" - a"A",
(2.34)
D",t/J == (a", + ieA",)t/J
(2.35)
and
D",t/J* == (a", - ieA",)t/J*
(2.36)