Phase transitions in field

theories at finite temperatures

Srivatsan

May 16, 2014

1 Motivation

In the usual textbook formulation of field theory,temperature is set to zero. In my presentation, I amgoing to focus on the formalism to be adopted whena temperature is turned on. This is especially im-portant when one wishes to study phase transitionsin gauge theories. One reason why one should be in-terested in this is the issue of baryon asymmetry. Itwas pointed out in [1] that departure from thermalequilibrium is required for a small baryon asymme-try to be produced in the early universe. In theStandard Model, the Baryon number violating tran-sitions (which are primarily the result of instantontunneling) are exponentially supressed at zero tem-perature. But in Electroweak Baryogenesis- they areexpected to occur frequently at temperatures com-parable to or greater than the Electroweak transi-tion temperature due to thermal activation. Thusit is important to determine this temperature. Ithad been observed in [3] that since the Coleman-Weinberg effective potential is gauge dependent, thecritical tempertaure derived from this is similarly de-pendent on gauge choice. I review a recent method[4] to calculate the critical temperature in a gaugeindependent way.

2 Matsubara formalism in thestatic limit for scalars

I now start by reviewing the method of construct-ing finite temperature partition functions in quan-

tum mechanics. This and the next section closelyfollow [5]. The starting point is the definition of thepartition function

Z = Tr[e−βH ]

β = 1/T(1)

We can evaluate the trace using a path integral inthe x-representation and the result is given by Eq.

[2] with SE = m2

(dxdτ

)2+ V (x(τ))

The second equality follows from Feynman’s pathintegral formulation of Quantum Mechanics. Theminus sign and the absence of the imaginary unitin the exponential can be understood from the fact

that instead of e−iHt as the evolution operator,

we have e−Hτ . Thus the usual formulae of quan-tum mechanics can be used with the identificationt → −iτ, L → −LE . We can now generalise thisargument to the case of a free scalar field

Z =

∫φ(0,x)=φ(β~,x)

∏x

Dφ(τ,x)exp

{−1

~SE

}where again

SE =

∫ β

0

dτ

∫x

1

2

(∂φ

∂τ

)2

+

d∑i=1

1

2

(∂φ

∂xi

)2

+ V (φ)

The recipe for constructing the partition function forscalar fields is thus that we restrict to fields which areperiodic in imaginary time. Thus, in the τ variable,instead of a Fourier integral we have to deal withFourier series.

3 Fermions

For fermions, we need to introduce Grassman val-ued fields in the path integral. Let a† and a bethe fermionic creation and annihilation operators re-spectively. They anticommute. Consequently, theHilbert space has only two basis states |0〉 and |1〉.If we introduce the Grassman numbers c and c∗ anddefine the states |c〉 = e−ca

† |0〉 then it is easy to seefor any operator A we have

∫dc∗dce−c

∗c〈−c|A|c〉 =

〈0|A|0〉+ 〈1|A|1〉 = Tr[A] The minus sign in the bra

1

Z =

∫dx〈x|e−βH |x〉 = C

∫x(β~)=x(0)

Dxe− 1~SE (2)

is because the Grassman numbers anticommute withthe integration measure. Using this relation for the

operator e−βH , we get

Z =

∫dc∗dce−c

∗c〈−c|e−βH |c〉 (3)

By the usual trick of insering complete states in be-tween, we get the following result for the fermionicpartition function which is given in Eq. [4]

For Dirac fields, we again use the prescrip-tion of going to imaginary time and changing thesign of the Lagrangian. Thus, the Euclideanversion of the Dirac Lagrangian reads LE =ψ[γ0∂τ − iγk∂k +m

]ψ

4 Ghosts and Gauge fields

Gauge fields require special care since not all statesin the Hilbert space in a gauge theory are physicaland consequently the statistical hypothesis cannotbe applied to all of them. In this section, we fol-low [6] and detail how the quantisation works in thiscase. We start with the quantum Hamiltonian in theAa0 = 0 gauge which is physical

H =

∫d3x

1

2

(g2 (Ea)

2+

1

g2(Ba)

2

)Inserting the complete set of states as usual, we

get for the amplitude 〈A′|e−βH |A′′〉∫DAexp−

(∫ β

0

dt

∫d3x

1

2g2Tr(A2 + B2

))

where A(β,x) = A′(x) and A(0,x) = A′′(x)Now, the trace of e−βH can be written as

a functional integral over periodic gauge fields.But we must only include physical states in thesum. The physical states satisfy D.E|ψphys〉 =

0. This can be shown to be equivalent to re-quire exp

(−i∫d3xtr [DΛ.E]

)|ψphys〉 = |ψphys〉 for

all Λ = ΛaTa with compact support. This showsthat one should require the physical states to beinvariant under gauge transformations whose pa-rameters vanish at infinity. Such states can be se-lected if one inserts the projection operator P =∫

Λ(∞)=0DΛexp

(−i∫d3xtr[DΛ.E]

)Thus the correct

partition function is given in Eq. [5] which exhibitsthe partition function as an integral over fields whichare periodic upto a gauge transformation.

The strictly periodic condition can be obtained ifwe redundantly insert the projector more than onceand we get the expression in Eq. [6]. Note that wehave renamed Λ as A0 in this equation.

The allowed gauge transformations are those thatvanish at infinity and are periodic in β. Moreover,the Faddeev popov determinant is now defined overperiodic functions only. Thus when we carry out theFaddeev popov quantisation, the ghosts are anticom-muting, periodic functions in β. This concludes oursurvey of the various fields at finite temperature.

5 Interactions

Interactions are treated the same way as in zero tem-perture field theory. The derivation of Feynmanrules and other combinatoric factors is exactly thesame. The only difference is that after Fourier de-composition, the timelike parts of the momenta aremultiplies of the period in imaginary time. Hencewe sum over the timelike parts of the momenta in-stead of integrating over them. Thus for example,the propagator for a real scalar field which at zerotemperature was 1

p2+m2 can be written at finite tem-perature as

G0(x− y) = T∑pn

eip·(x−y) eipnτ

p2n + E2

p

2

Z =

∫Dc∗(τ)Dc(τ)exp

{−1

~

∫ β~

0

dτ

[~c∗(τ)

dc(τ)

dτ+H(c∗(τ), c(τ))

]}(4)

Z = Tr(P exp−βH) =

∫Λ(∞)=0

DΛDA〈AU|e−βH |A〉

=

∫Λ(∞)=0

DΛ

∫A(β,x)=AU (0,x)

DAexp

(− 1

2g2

(∫ β

0

dt

∫d3xTr

(A2 + B2

))) (5)

Z = limN→∞Tr(Pe−βH)N =

∫Aµ(β,x)=Aµ(0,x)

DAµexp

(−1

4g2

∫ β

0

dt

∫d3xtr(FµνFµν)

)(6)

6 Applications

6.1 Symmetry breaking transforma-tions and the critical temperature

The critical temperature is defined as that temper-ature above which the scalar condensate vanishes.We now formalise this with the help of the Coleman-Weinberg effective potential V (φ2). At zero temper-ature, the theory is supposed to have a symmetrybreaking solution such that ∂V 0(φ2)/∂φa = 0, φa 6=0. At some finite temperature, if ∂V

β(φ2)∂φa

6= 0, φa 6= 0

then symmetry is restored. We assume that ∂V/∂φ2

is positive for large φ2. Thus persistence of symme-try requires ∂V/∂φ2 > 0 for φ 6= 0. Thus a necessarycondition for symmetry restoration is ∂V (φ2)/∂φ2 ≥0 at φ = 0. Decomposing the effective potential intoits zero and positive temperature parts, we get the

result ∂V βc

∂φ2 = −m2

2 where m is the renormalised

mass parameter and V β is the finite temperaturepart of the effective potential. We now evaluatethe effective potential in the φ4 theory. We useJackiw’s method [7] of shifting the fields by constant(spacetime independent) fields and then reading offthe one loop correction from the propagator (whichnow depends on the constant fields) and calculating

higher loops by evaluating vacuum diagrams usingthe shifted lagrangian (now, the coupling constantsdepend on the constant fields). The result for the la-grangian L = 1

2∂µφ∂µφ+ 1

2m2φ2 + λφ4/4! is simply

V β1 (φ20) = −1

2i

∫k

lnD−1 {φ0, k}

=1

2β

∑n

∫k

ln

(−4

π2n2

β2− E2

M

)with E2

M = k2 +m2 + 12λφ

20 and M2 = m2 + 1

2λφ2

The series can be summed and we find V β1 (φ20) =∫

k

[EM

2 + 1β ln(1− e−βEM )

]. Thus, the total effec-

tive potential to one loop order is given by

V (φ2) =1

64π2

[M4ln

M2

m2− 3

2(M2 − 2

3m2)2

]+

1

2π2β4

∫ ∞0

x2ln(1− e−(x2+β2M2))

where the temperature independent part has beencalculated in a problem set. The temperature depen-dent part gives the location of the critical tempera-ture. Note that naively setting the derivative withrespect to φ2 equal to zero at φ2 = 0 gives a resultfor βC that is complex. However we can obtain an

3

approximate value of the critical temperature if it islarge (for small βC). We expand the effective poten-

tial in the limit of small β and we get 1β2C

= −24m2

λ .

Note that since we have assumed symmetry breakingat zero temperature, m2 < 0

6.2 Effective potential in gauge theo-ries

This section follows [4]. Consider the Lagrangian

L =1

2DµΦiD

µΦi −1

4F aµνF

µνa − V (Φ)

We can again shift each scalar field Φi by a constantΦi = φi+ φiand look at the quadratic part to get theone loop effective action. This is a routine exerciseand we get in the Rξ gauge the quadratic part givenin Eq.[7] where M2

ij(φ) = ∂2V/∂φi∂φj the derivative

being evaluated at φ and m2A(φ)ab = g2(T aφ)i(T

bφ)iwhile m2

A(φ)ij = g2(T aφ)i(Taφ)j

We now get the one loop potential which is givenin Eq.[8]

This at finite temperature gives Eq.[9]. In thisequation, m2

i (φ, ξ),m2a(φ) are the eigenvalues of

M2ij(φ)+ξm2(φ)ij ,m

2A(φ)ab respectively. VCW is the

Coleman-Weinberg result at zero temperature andJ(z2) =

∫∞0dxx2ln(1− e−

√x2+z2)

We see explicitly from Eq.[9] that in the effectivepotential for gauge fields depends explicitly on thegauge fixing parameter ξ. Thus the critical temper-ature obtained from this potential is dependent ongauge choice which is unphysical. the reason for thisconclusion is that as noted in [3], the effective po-tential itself depends on gauge because the sourcedgenerating functional does not involve a conservedsource. The remedy for this and the correct methodto calculate the critical temperature is detailed inthe next subsection.

6.3 Gauge invariant calculation ofcritical temperature

The starting point is the flowing identity derived byNielsen [8] using BRST non-invariance of the gener-

ating functional. The identity reads

∂Veff∂ξ

= −Ci(φ, ξ)∂Veff∂φi

(10)

This implies that the value of the effective potentialis gauge independent when it is stationary. But thefield value that extremises it depends on the gaugechoice. Note that these identities are derived at zerotemperature. But since no non-trivial space-time orenergy-momentum integrations were involved in de-riving them, these identities are also valid at non-zero temperature.

Now we expand the effective potential in a loopexansion

Veff (φ, T ) = V0(φ) + ~V1(φ, T ) + ~2V2(φ, T ) + · · ·(11)

We want to solve

∂Veff∂φ

∣∣∣∣φmin

= 0 (12)

We expand φmin itself as a series in ~

φmin = φ0 + ~φ1(T, ξ) + ~2φ2(T, ξ) + · · ·

Here φ0 denotes the tree level minimum. We thenexpand Eq.[12] as

∂V0

∂φ

∣∣∣∣φ0+~φ1+~2φ2

+∂V1

∂φ

∣∣∣∣φ0+~φ1+~2φ2

+ · · · = 0

which is simply

∂V0

∂φ

∣∣∣∣φ0

+ ~

(∂V1

∂φ

∣∣∣∣φ0

+ φ1∂2V0

∂φ2

∣∣∣∣φ0

)+O

(~2)

= 0

The result of this pertubative expansion is then sub-stituted in Eq.[11] to get the following expansion

Veff (φmin(T ), T ) = V0(φ0) + ~V1(φ0, T )+

~2

[V2(φ0, T, ξ)−

1

2φ2

1(T, ξ)∂2V0

∂φ2

∣∣∣∣φ0

](13)

At each ~ order, the expression for Veff is indepen-dent of ξ in Eq.[13]. To find a critical temperature,

4

L2 =1

2φi[−∂2 −M2

ij(φ)− ξm2A(φ)ij

]φj +

1

2Aaµ

[(∂2gµν − (1− 1

ξ)∂µ∂ν)δab +m2

A(φ)abgµν]Abν+

η†a[−∂2δab − ξm2

A(φ)ab]ηb

(7)

Veff (φ) = Vtree(φ)− i

2

∫ddp

(2π)d[Tr ln(p2 −M2

ij(φ)− ξm2A(φ)ij)

]− i

2

∫ddp

(2π)d[(d− 1)Tr ln(p2 −m2

A(φ)ab) + Tr ln(p2 − ξm2A(φ))− 2Tr ln(p2 − ξm2

A(φ)ab)] (8)

Veff (φ, T ) = Vtree(φ) + VCW (φ) +T 4

2π

[ ∑scalar

JB(m2i (φ, ξ)/T

2) + 3∑

gauge,a

JB(m2a(φ)/T 2)−

∑gauge,a

JB(ξm2a(φ)/T 2)

](9)

we simply find each tree level minimum, substitute itin the expansion above, and find that temperature atwhich the two lowest different branches cross. Thislocation gives a critical temperature. Note that thiswhole process does not depend on ξ at all.

References

[1] A.D. Sakharov, JETP Lett. 91B, 24 (1967)

[2] M. Trodden, Rev. Mod. Phys. 71 1463 (1999)

[3] L. Dolan and R.Jackiw, Phys. Rev. D 9 3320(1974)

[4] H.H. Patel and M.J. Ramsey-Musolf, JHEP 7(2011)

[5] M. Laine, Basics of Thermal Field Theory. A setof online lecture notes from which parts of thisoutline have been adopted.

[6] D.J. Gross, R.D. Pikarski, L.G. Yaffe, Rev. Mod.Phys. 53, 1 (1981)

[7] R. Jackiw, Phys. Rev. D 9 1686 (1974)

[8] N.K. Nielsen, Nucl. Phys., B101, 173 (1975)

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