Finite temperature field theory and phase transitions

74 Pages·0.68 MB·English

by  Mariano Quiros

IEM-FT-187/99
hep-ph/9901312
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1 Finite Temperature Field Theory and Phase

n Transitions
∗
a
J

4 Mariano Quiro´s
1 Instituto de Estructura de la Materia (CSIC), Serrano 123

E–28006Madrid, Spain
1
v
2
1 Abstract
3 Wereviewdifferentaspectsoffieldtheoryat zeroandfinitetemper-
1 ature, related to the theory of phase transitions. We discuss different
0
renormalization conditions for the effective potential at zero tempera-
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ture, emphasizing in particular the MS renormalization scheme. Finite
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temperature field theory is discussed in the real and imaginary time
/
h formalisms, showing their equivalence in simple examples. Bubble nu-
p cleation by thermal tunneling, and the subsequent development of the
-
phase transition is described in some detail. Some attention is also de-
p
e voted to the breakdown of the perturbative expansion and the infrared
h problem in the finite temperature field theory. Finally the application
: to baryogenesis at theelectroweak phase transition is done in theStan-
v
dardModelandintheMinimalSupersymmetricStandardModel. Inall
i
X caseswehavetranslatedtheconditionofnotwashingoutanypreviously
r generated baryon asymmetry by upperboundson theHiggs mass.
a
IEM-FT-187/99
January 1999
∗Based on lectures given at the Summer School in High Energy Physics and
Cosmology, ICTP, Trieste (Italy) 29 June–17 July 1998.
1 The effective potential at zero temperature
The effective potential for quantum field theories was originally introduced
by Euler, Heisenberg and Schwinger1, and applied to studies of spontaneous
symmetrybreakingbyGoldstone,Salam,WeinbergandJona-Lasinio2. Calcu-
lations of the effective potential were initially performed at one-loop by Cole-
manandE.Weinberg3 andathigher-loopbyJackiw4andIliopoulos,Itzykson
and Martin5. More recently the effective potential has been the subject of a
vivid investigation, especially related to its invariance under the renormaliza-
tion group. I will try to review,in this section, the mainideas and update the
latest developments on the effective potential.
1.1 Generating functionals
To fix the ideas, let us consider the theory described by a scalar field φ with a
lagrangiandensity φ(x) and an action
L{ }
S[φ]= d4x φ(x) (1)
L{ }
Z
Thegeneratingfunctional(vacuum-to-vacuumamplitude)isgivenbythepath-
integral representation,
Z[j]= 0 0 dφexp i(S[φ]+φj) (2)
out in j
h | i ≡ { }
Z
where we are using the notation
φj d4xφ(x)j(x) (3)
≡
Z
Using (2) one can obtainthe connectedgeneratingfunctional W[j] defined as,
Z[j] exp iW[j] (4)
≡ { }
and the effective action Γ[φ] as the Legendre transform of (4)
δW[j]
Γ[φ]=W[j] d4x j(x) (5)
− δj(x)
Z
where
δW[j]
φ(x)= (6)
δj(x)
1
In particular, from (5) and (6), the following equality can be easily proven,
δΓ[φ] δW[j] δj δj
= j φ = j (7)
δφ δj δφ − − δφ −
wherewehavemadeuseofthenotation(3). Eq.(7)impliesinparticularthat,
δΓ[φ]
=0 (8)
δφ
(cid:12)j=0
(cid:12)
(cid:12)
which defines de vacuum of the theory(cid:12)in the absence of external sources.
We cannow expandZ[j](W[j]) in a powerseriesofj, to obtainits repre-
sentation in terms of Green functions G (connected Green functions G c )
(n) (n)
as,
∞ in
Z[j]= d4x ...d4x j(x )...j(x )G (x ,...,x ) (9)
n! 1 n 1 n (n) 1 n
n=0 Z
X
and
∞ in
iW[j]= d4x ...d4x j(x )...j(x )G c (x ,...,x ) (10)
n! 1 n 1 n (n) 1 n
n=0 Z
X
Similarly the effective action can be expanded in powers of φ as
∞
1
Γ[φ]= d4x ...d4x φ(x )...φ(x )Γ(n)(x ,...,x ) (11)
1 n 1 n 1 n
n!
n=0 Z
X
where Γ(n) are the one-particle irreducible (1PI) Green functions.
We can Fourier transform Γ(n) and φ as,
n d4p
Γ(n)(x)= i exp ip x (2π)4δ(4)(p + +p )Γ(n)(p) (12)
(2π)4 { i i} 1 ··· n
Z i=1(cid:20) (cid:21)
Y
φ˜(p)= d4xe−ipxφ(x) (13)
Z
and obtain for (5) the expression,
∞ n d4p
Γ[φ]= i φ˜( p ) (2π)4δ(4)(p + +p )Γ(n)(p ,...,p ) (14)
(2π)4 − i 1 ··· n 1 n
n=0Z i=1(cid:20) (cid:21)
X Y
In a translationally invariant theory,
φ(x)=φ (15)
c
2
the field φ is constant. Removing an overall factor of space-time volume, we
define the effective potential V (φ ) as,
eff c
Γ[φ ]= d4xV (φ ) (16)
c eff c
−
Z
Using now the definition of Dirac δ-function,
d4x
δ(4)(p)= e−ipx (17)
(2π)4
Z
and (15) in (13) we obtain,
φ˜ (p)=(2π)4φ δ(4)(p). (18)
c c
Replacing (18) in (14) we can write the effective action for constant field con-
figurations as,
∞ ∞
1 1
Γ(φ )= φn(2π)4δ(4)(0)Γ(n)(p =0)= φnΓ(n)(p =0) d4x
c n! c i n! c i
n=0 n=0 Z
X X
(19)
and comparing it with (16) we obtain the final expression,
∞
1
V (φ )= φnΓ(n)(p =0) (20)
eff c − n! c i
n=0
X
which will be used for explicit calculations of the effective potential.
Let us finally mention that there is an alternative way of expanding the
effective action: it can also be expanded in powers of momentum, about the
point where all external momenta vanish. In configuration space that expan-
sion reads as:
1
Γ[φ]= d4x V (φ)+ (∂ φ(x))2Z(φ)+ (21)
eff µ
− 2 ···
Z (cid:20) (cid:21)
1.2 The one-loop effective potential
We are now ready to compute the effective potential. In particular the zero-
loop contribution is simply the classical (tree-level) potential. The one-loop
contribution is readily computed using the previous techniques and can be
written inclosedformfor any field theory containingspinless particles,spin-1
2
fermionsandgaugebosons. HerewewillfollowcloselythecalculationofRef.3.
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Scalar fields
We consider the simplest model of one self-interacting real scalar field, de-
scribed by the lagrangian
1
= ∂µφ∂ φ V (φ) (22)
µ 0
L 2 −
with a tree-level potential
1 λ
V = m2φ2+ φ4 (23)
0
2 4!
The one-loop correction to the tree-level potential should be computed
as the sum of all 1PI diagrams with a single loop and zero external momenta.
DiagrammaticallytheyaredisplayedinFig.1,whereeachvertexhas2external
legs.
+ + + :::
Figure1: 1PIdiagramscontributingtotheone-loopeffectivepotential of(22).
The n-thdiagramhas npropagators,n verticesand2n externallegs. The
n propagators will contribute a factor of in(p2 m2+iǫ)−n a. The external
−
lines contribute a factor of φ2n and each vertex a factor of iλ/2, where the
c −
factor 1/2 comes from the fact that interchanging the 2 external lines of the
vertex does not change the diagram. There is a global symmetry factor 1 ,
2n
where 1 comes fromthe symmetry of the diagramunder the discrete groupof
n
rotationsZ and 1 fromthesymmetryofthediagramunderreflection. Finally
n 2
there is an integration over the loop momentum and an extra global factor of
i from the definition of the generating functional.
Using the previous rules the one-loop effective potential can be computed
as,
V (φ )=V (φ )+V (φ ),
eff c 0 c 1 c
aWeareusingtheBjorkenandDrell’s6 notation andconventions.
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with
∞ d4p 1 λφ2/2 n
V (φ ) = i c
1 c (2π)42n p2 m2+iǫ
n=1Z (cid:20) − (cid:21)
X
i d4p λφ2/2
= log 1 c (24)
−2 (2π)4 − p2 m2+iǫ
Z (cid:20) − (cid:21)
After a Wick rotation
p0 =ip0, p =( ip0,~p ), p2 =(p0)2 p~ 2 = p2, (25)
E E − − − E
Eq. (24) can be cast as,
1 d4p λφ2/2
V (φ )= E log 1+ c (26)
1 c 2 (2π)4 p2 +m2
Z (cid:20) E (cid:21)
Finally, using the shifted mass
1 d2V (φ )
m2(φ )=m2+ λφ2 = 0 c (27)
c 2 c dφ2
c
and dropping the subindex E from the euclidean momenta, we can write the
final expression of the one-loop contribution to the effective potential as,
1 d4p
V (φ )= log p2+m2(φ ) (28)
1 c 2 (2π)4 c
Z
(cid:2) (cid:3)
where a field independent term has been neglected.
The result of Eq. (28) can be trivially generalizedto the case of N com-
s
plex scalar fields described by the lagrangian,
=∂µφa∂ φ† V (φa,φ†). (29)
L µ a− 0 a
The one-loopcontributionto the effective potentialinthe theory describedby
the lagrangian(29) is given by
1 d4p
V = Tr log p2+M2(φa,φ†) (30)
1 2 (2π)4 s b
Z h i
where
∂2V
(M2)a Va = (31)
s b ≡ b ∂φ†∂φb
a
and Tr M2 = 2 Va, where the factor of 2 comes from the fact that each
s a
complex field contains two degrees of freedom. Similarly Tr 1=2 N .
s
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Fermion fields
We consider now a theory with fermion fields described by the lagrangian,
=iψ γ ∂ψa ψ (M )aψb (32)
L a · − a f b
wherethe massmatrix(M )a(φi)is afunctionofthe scalarfieldslinear inφi:
f b c c
(M )a =Γaφi.
f b bi c
The diagramscontributingto the one-loopeffective potentialare depicted
in Fig. 2.
+ + + :::
Figure2: 1PIdiagramscontributingtotheone-loopeffectivepotential of(32).
Diagrams with an odd number of vertices are zero because of the γ-
matrices property: tr(γµ1 γµ2n+1) = 0. The diagram with 2n vertices has
···
2n fermionic propagators. The propagatorsyield a factor
Tr [i2n(γ p)2n(p2+iǫ)−2n]
s
·
where Tr refers to spinor indices. The vertices contribute as
s
Tr[ i2nM (φ )2n]
f c
−
whereTr runsoverthedifferentfermionicfields. Thereisalsoacombinatorial
factor 1 (fromthecyclicandanticyclicsymmetryofdiagrams)andanoverall
2n
1 coming from the fermions loop. One finally obtains the total factor
−
1 Tr(M2n)
f
Tr 1.
−2n p2n · s
The factor Tr 1 just counts the number ofdegreesoffreedomofthe fermions.
s
It is equal to 4 if Dirac fermions are used, and 2 if Weyl fermions (and σ-
matrices) are present. So we will write,
Tr 1=2λ (33)
s
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where λ = 1 (λ = 2) for Weyl (Dirac) fermions. On the other hand we have
grouped terms pairwise in the matrix product and used,
p˜2 =p2
where p˜stands either for p γ or p σ, depending on the kind of fermions we
· ·
are using.
Collecting everything together we can write the one-loop contribution to
the effective potential from fermion fields as,
∞ d4p 1 M2 n i d4p M2
f f
V (φ )= 2λiTr =2λ Tr log 1
1 c − (2π)42n" p2 # 2 (2π)4 " − p2 #
n=1Z Z
X
(34)
As in the case of the scalar theory, after making a Wick rotation to the
Euclideanmomentaspace,andneglectinganirrelevantfieldindependentterm,
we can cast (34) as
1 d4p
V = 2λ Tr log p2+M2(φ ) (35)
1 − 2 (2π)4 f c
Z
(cid:2) (cid:3)
Gauge bosons
Consider now a theory described by the lagrangian,
1 1
= Tr(F Fµν)+ Tr(D φ )†Dµφa+ (36)
µν µ a
L −4 2 ···
In the Landau gauge, which does not require ghost-compensating terms, the
free gauge-boson propagatoris
i
Πµ = ∆µ (37)
ν −p2+iǫ ν
with
pµp
∆µ =gµ ν (38)
ν ν − p2
satisfying the property p ∆µ =0 and ∆n =∆, n=1,2,....
µ ν
The only vertex which contributes to one-loop is
1
= (M )2 AαAµβ + (39)
L 2 gb αβ µ ···
where
(M )2 (φ )=g g Tr Ti φ †Tℓ φj (40)
gb αβ c α β αℓ i βj
h(cid:0) (cid:1) i
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+ + + :::
Figure3: 1PIdiagramscontributingtotheone-loopeffectivepotential of(36).
In this way the diagrams contributing to the one-loop effective potential are
depicted in Fig. 3.
A few comments about Eq. (40): (i) g is the gauge coupling constant
α
associated to the gauge field Aα; if the gauge group is simple, e.g. SU(5),
µ
SO(10),E ,...,thenallgaugecouplingsareequal;otherwisethereisadistinct
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gauge coupling per group factor. (ii) T are the generators of the Lie algebra
α
of the gauge group in the representation of the φ-fields and the trace in (40)
is over indices of that representation.
Taking into account the combinatorial factors, the graph with n propaga-
tors and n vertices yields a total factor
1 Tr((M )2)n
gb
Tr(∆)
2n p2n
where
Tr(∆)=3 (41)
whichisthenumberofdegreesoffreedomofamassivegaugeboson. Collecting
together all factors, and making the Wick rotation to the euclidean momenta
space, we can cast the effective potential from gauge bosons as,
1 d4p
V =Tr(∆) Tr log p2+(M )2(φ ) (42)
1 2 (2π)4 gb c
Z
(cid:2) (cid:3)
1.3 The higher-loop effective potential
CalculatingtheeffectivepotentialbysumminginfiniteseriesofFeynmangraphs
at zero externalmomentum is anextremely onerous task beyond the one-loop
approximation. However, as has been shown in Ref.4, this task is trivial for
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the case of one-loop, and affordable for the case of higher-loop. Here we will
just summarize the result of Ref.4 b.
Wewillstartconsideringthe theorydescribedbyarealscalarfield,witha
lagrangian given in (22-23), and an action as in (1). We will define another
lagrangianLˆby the following procedure:
L
δS[φ ]
d4xˆ φ ;φ(x) S[φ +φ] S[φ ] φ c (43)
c c c
L{ }≡ − − δφ
Z c
where we have used in the last term the notation (3). In (43), φ is an x-
c
independent shifting field. The second term in (43) makes the vacuum energy
equal to zero, and the third term is there to cancel the tadpole part of the
shifted action.
If we denote by φ ;x y the propagator of the shifted theory,
c
D{ − }
δ2S[φ]
i −1 φ ;x y = (44)
c
D { − } δφ(x)δφ(y)
(cid:12)φ=φc
(cid:12)
(cid:12)
and (cid:12)
i −1 φ ;p
c
D { }
its Fourier transform, the effective potential is found to be given by4:
i d4p
V (φ ) = V (φ ) logdeti −1 φ ;p
eff c 0 c − 2 (2π)4 D { c }
Z
+ i exp i d4xˆ φ ;φ(x) (45)
I c
L { }
(cid:28) (cid:20) Z (cid:21)(cid:29)
The first term in (45) is just the classical tree-level potential. The sec-
ond term is the one-loop potential, where the determinant operates on any
possible internal indices defining the propagator. The third term summarizes
thefollowingoperation: Computeall1PIvacuumdiagrams,withconventional
Feynman rules, using the propagator of the shifted theory φ ;p and the
c
interactionprovidedbytheinteractionlagrangian ˆ φ ;φ(xD) {,and}deletethe
I c
L { }
overallfactorofspace-timevolume d4xfromthe effective action(16). Itcan
be shown that the last term in (45) starts at two-loop. Every term in (45)
R
resums an infinite number of Feynman diagrams of the unshifted theory.
In the simple example of the lagrangian(22-23) it can be easily seen that
the shifted potential is given by
1 λ λ
Vˆ φ ;φ = m2(φ )φ2+ φ φ3+ φ4 (46)
c c c
{ } 2 3! 4!
bTheinterestedreadercanfindin4 allcalculational details.
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