素粒子論研究 ･ 電子版 Vol. 8 (2011) No. 3

Soryushiron Kenkyu

Quantum Fields at Finite Temperature“from tera to nano Kelvin”
Jean-Paul Blaizot∗
Institut de Physique Théorique, CEA-Saclay,
91191 Gif-sur-Yvette Cedex, France.

(2011 年 5 月 30 日受理)

These lectures introduce techniques that are used in the description of systemsof particles and ﬁelds at high temperature (or density). These methods have abroad range of physical applications. We shall discuss two speciﬁc applications: one
related to hot and dense matter composed of quarks and gluons, with temperaturesin the tera Kelvin range, the other related to Bose-Einstein condensation in ultracold gases, with temperatures in the nano Kelvin range. As we shall see, in both
systems, long wavelength collective phenomena lead to similar features, in spite of
the huge diﬀerence in orders of magnitude of the respective energy scales.

∗

jean-paul.blaizot@cea.fr

Soryushiron Kenkyu

2
I.

LECTURE 1

INTRODUCTION

Let me start with a few words of explanation about the title of these lectures, “Quantum
ﬁelds at ﬁnite temperature, from tera to nano Kelvin”. Tera is a Greek word which means
1012 , while nano means 10−9 (this latter word is presumably very familiar to you because of
the “nano technologies”). The Kelvin is the unit of temperature. In dealing with systems
with so vastly diﬀerent orders of magnitude it is useful to think in terms of the corresponding
energy scales. Energies are conveniently measured in electron-volt. To convert Kelvin into
electron-volt, recall that 300 K is about 1/40 eV or, if you wish, 1 eV is about 120 000 K.
So 1012 K translates approximately into 100 MeV. This is an energy scale typical of (highenergy) nuclear processes. At the opposite end, the nano Kelvin corresponds to a subatomic
energy scale. It is for instance realized in experiments with ultra cold atoms, where thephenomenon of Bose-Einstein condensation has been observed. The lectures will discusstheoretical techniques that are relevant to the calculation of the properties of matter in these
two extreme energy regimes. The beauty of theoretical physics is that the same techniques
are indeed capable to provide an adequate description of some important aspects of thesevastly diﬀerent systems.
That quantum ﬁeld theory appears as an essential tool in the description of hot and dense
matter, composed of quarks and gluons, is a priori natural: the dynamics of quarks andgluons is governed by a quantum ﬁeld theory, called Quantum Chromodynamics (QCD), of
which I shall say more in today’s lecture. What is perhaps more surprising is that ﬁeld theoryis also useful for understanding the behavior of cold atoms. Atoms are objects which we canstudy in isolation, and their dynamics obey non relativistic many-body quantum mechanics.
However, collections of atoms can undergo collective, long wavelength oscillations. By longwavelength, I mean a wavelength much larger than the typical distance between the atoms.
Such long wavelength oscillations necessarily involve collectively many atoms. And thesecollective excitations can be described by (classical) ﬁeld theory. These long wavelength
phenomena provide the connection between the two topics that I plan to discuss, hot anddense matter composed of quarks and gluons, and the Bose-Einstein condensation of weaklyinteracting cold atoms
A central theme of our discussion will be that, in both systems, the eﬀect of the interaction
can be large, although the strength of the interaction between the elementary constituentsis small. At ﬁrst sight, this looks like a paradox. But the clue to resolve this apparent puzzle
has just been mentioned: collective, long wavelength phenomena involve many degrees of
freedom, and the cooperation of these degrees of freedom compensates for the weakness ofSoryushiron Kenkyu

3
the coupling. Technically, this feature shows up in infrared divergences in the Feynman
diagrams of perturbation theory. Because of these divergences, perturbation theory in fact
breaks down, and other techniques have to be developed to perform calculations in the weakcoupling regime. As we shall see, in both systems that we shall consider, a simple eﬀective
theory will allow us to overcome the diﬃculties met in perturbation theory (albeit only
partially in the case of QCD).
After this brief and general introduction, and the explanation of the title, let me say a
few words about the plan of the lectures. There will be six lectures. The ﬁrst four lectures
will be mostly devoted to the physics of hot and dense matter, with in mind the quark gluon
plasma. That will give me the opportunity to introduce techniques of quantum ﬁeld theory
at ﬁnite temperature which can be used in many other contexts. In these lectures, I shall bemainly concerned with the calculations of thermodynamical quantities (like the pressure).
I shall use the scalar ﬁeld as a prototype of a quantum ﬁeld theory in order to illustrate
the main diﬃculties that one encounters in perturbative calculations at ﬁnite temperature.
Many of these diﬃculties are common to other ﬁeld theories, in particular QCD. In the lattercase further complications arise, that I shall brieﬂy indicate. I shall also introduce a simple
eﬀective ﬁeld theory that allows us to handle the infrared divergences of perturbation theory.
This eﬀective theory can be extended to the case of QCD, but this will not be covered in
the lectures. We shall rather ﬁnd a direct application of this eﬀective theory in the study ofBose-Einstein condensation, to which the last two lectures will be devoted. Bose Einsteincondensation is a phase transition which takes place in the ideal Bose gas, that is, without
interaction. The question that I want to address is how the interactions between the atoms
modify this remarkable phenomenon. More speciﬁcally, I shall be interested by the shift in
the transition temperature Tc caused by a very small repulsion between the atoms. Naively,
since the interaction between the atoms can be chosen as small as desired, you may thinkof using perturbation theory in order to calculate the shift in Tc . But we shall discover that
perturbation theory is meaningless, it is plagued by infrared divergences. And we shall seethat the eﬀective theory introduced in the ﬁrst part of these lectures can be used to obtain
an elegant solution to this problem.
Finite temperature aspects of many-body physics or quantum ﬁeld theory are presented
in a number of textbooks, for instance [1–7]. Complements to the present lectures canbe found in several lecture notes or review articles. Thus, the lectures 1-4 are based more
speciﬁcally on [8, 9, 11–13]. Lectures 5-6 are based on [14]. These papers should be consulted
for systematic references to the original literature. Further references will be given in the
text about speciﬁc results that will be used or referred to.

Soryushiron Kenkyu

4
A.

A brief introduction to QCD and its symmetries

Let me now begin the discussion of hot and dense matter. At suﬃciently high temperature and/or density, one expects nuclear matter –the matter that makes atomic nuclei–
to turn into a plasma of quarks and gluons, whose interactions are described by Quantum
Chromodynamics (QCD). I shall then remind you a few basic properties of QCD that areimportant to understand the bulk features of the quark-gluon plasma.

1.

Quantum Chromodynamics

As I just said, QCD is the theory that governs the dynamics of quarks and gluons. Quarks
are spin 1/2 fermions, that I shall represent by a ﬁeld Ψf (x). The index f refers to the socalled “ﬂavor”. There exists six such ﬂavors, denoted u, d, s, c, t, b, and the corresponding
quarks have diﬀerent masses: the masses of u, d are very small, of order 2-4 MeV, that of
the strange quark is in the hundred MeV range, that of the charm quark is of the order of a
GeV, the same for the bottom, 4-5 GeV, while the top quark is much heavier, ∼ 170 GeV.
We shall be mostly concerned with matter that results from the “melting” of neutrons andprotons, that is, matter made of up and down quarks. In addition to ﬂavor, quarks carryanother internal quantum number, color. There are Nc = 3 diﬀerent colors for quarks.
Gluons are somewhat similar to photons. They are the modes of a vector gauge ﬁeld
Aµ . As photons, they are massless bosons, with spin 1 (and two polarization states). In
contrast to photons, which are electrically neutral, gluons carry color charge (gluons exist
in Nc2 − 1 = 8 colors), and interact directly among themselves.
The interaction between quarks and gluons is coded in the QCD lagrangian which takes
the following form.
1a
L = − Faµν Fµν+ Ψf (iD/ − mf )Ψf ,4(1)

where mf is the mass of the quark with ﬂavor f , D/ ≡ γ µ Dµ , with Dµ the covariant derivativeDµ = ∂µ + igAµ , with Aµ the gauge ﬁeld. This is a non-Abelian gauge ﬁeld, i.e., a color
matrix, Aµ = Aaµ ta , where ta is a generator of the gauge group SU (3) (in the fundamental
representation). The ﬁeld strength tensor reads
Faµν = ∂ µ Aνa − ∂ ν Aµa + gfabc Aµb Aνc .

(2)

The ﬁrst two terms are just those you would get in Quantum Electrodynamics (QED). In
a non-Abelian gauge theory you have the additional term, quadratic term in A, which is
responsible for the interactions among the gluons (fabc are the structure constants of thegauge group).

Soryushiron Kenkyu

5

g

gP

g2

P

FIG. 1. Feynman diagrams describing the elementary QCD interactions.

Now the main thing that I want to do here is to remind you how the interactions amongquarks and gluons, and among gluons themselves, can be red oﬀ the QCD lagrangian. Let’slook ﬁrst at the quarks. Their interactions with the gluon ﬁeld is contained in the termΨ̄γ µ Aµ Ψ, and I shall represent this interaction by the ﬁrst diagram in Fig. 1. The strengthof the coupling is g. Let us now turn to the gluons. If one would ignore the last piece of
the ﬁeld strength tensor (2), i.e., set g = 0, then (Fµν )2 would be quadratic in the gauge
potential. And a lagrangian which is quadratic in the ﬁeld describes only normal modes, orfree particles. These modes are what we call the gluons. When g 6= 0, the gluons interact,
and the interaction vertices can be obtained by analyzing the (Faµν )2 , which, aside from thepart quadratic in the gauge potential, contains also the product of ∂ µ Aνa and gfabc Aµb Aνc ,which generates a three gluon vertex, proportional to g and to the derivative of the ﬁeld (sothat the strength of the three-gluon interaction is proportional to the momentum of one of
the gluons). We have also a four gluon vertex that is proportional to g 2 . These vertices are
displayed in Fig. 1.

2.

Symmetries of QCD

The symmetries of QCD play an important role in the analysis of the phase diagram of
hot and dense matter. I shall brieﬂy review here these symmetries.
There is of course the color symmetry, a local gauge symmetry. The QCD lagrangianis invariant under (local) color rotations of the quark ﬁeld, accompanied by an appropriate
transformation of the gauge potential:iAµ → hAµ h−1 − h∂µ h−1 ,g

ψ → hψ ,Dµ ψ → hDµ Ψ

(3)

with
a

a

h(x) = eiθ (x)t .

(4)

This gauge symmetry is actually the guiding principle behind the whole construction of the
QCD lagrangian (1).

Soryushiron Kenkyu

6
Color symmetry is responsible for color conﬁnement, the fact that color charges cannot be
isolated. The quarks in Nature combine to form color singlet states, the hadrons. However,when matter is at extremely high temperature conﬁnement “disappears”, during the socalled conﬁnement/deconﬁnement transition. In these lectures, we shall be interested in thethermodynamics of this deconﬁned phase, called the quark-gluon plasma.
When the mass of the quark is strictly zero, there is another symmetry, called chiralsymmetry. Let me digress here on an issue that will appear at various occasions. The massof the quark is something which is not determined by QCD, but by physics at a higher
energy scale, by the electro-weak physics of the standard model. So the mass of the quark issomething which is not “negotiable”. However as a theorist, I can play games and consider
a theory which is not quite QCD, but which diﬀers from it only by the fact that mf = 0. If
I allow myself to do that, I observe that the lagrangian possesses another symmetry, chiralsymmetry, which corresponds to the independent rotation of the left and right quarks. It isa global symmetry.
What I did for the quark masses can be done for all the other parameters which arearound. I have told you that Nc = 3, and indeed in Nature there are three colors of quarks
and eight colors of gluons. That is an experimental fact. But it is sometimes instructive
to consider theories which look like QCD, but in which Nc can be varied. In particular
you can sometimes obtain interesting insights by considering the limit where Nc is inﬁnite.
The theory corresponding to Nc = ∞ has many features similar to those of the theory with
Nc = 3. But letting Nc going to inﬁnity allows you to do calculations which you cannot do
if Nc is ﬁnite. We can also play with the number of ﬂavors, etc.
Returning to chiral symmetry, we note that it is spontaneously broken in the vacuum.
This feature has many consequences, for low energy nuclear physics in particular: pions,the lightest hadrons, can be understood as (pesudo) Goldstone bosons. Chiral symmetryis also important in characterizing the bulk properties of hot and dense matter, as it isrestored by thermal ﬂuctuations at high temperature. Thus, the quark condensate hΨ̄Ψi,an order parameter for chiral symmetry, is non vanishing in the vacuum, but vanishes athigh temperature.
One last symmetry which I want to mention, again without going into too much detail, isscale invariance. The QCD lagrangian (1) is left invariant in the rescaling of the coordinatesx → λx, where λ is an arbitrary number, accompanied by a corresponding rescaling of theﬁelds (Aµ → λ−1 Aµ ). This symmetry (which holds for massless quarks) is easily veriﬁed bynoticing that the gauge ﬁeld Aµ has mass dimension 1, and that there are no dimensionful
parameters in the lagrangian. In particular the coupling constant g is dimensionless. This is
in fact a rather remarkable property because, as you know, the QCD lagrangian is supposed
to allow you to calculate the mass of the proton for instance. So how are you going to getSoryushiron Kenkyu

7
the mass of the proton in GeV, if there is nothing in the lagrangian that “knows” about
GeV ?
At this point, I need to remind you a few things about renormalization and the running
coupling constant. When we go beyond the classical level, and calculate physical processes,
we need to take into account the eﬀects of the short wavelength ﬂuctuations and that usuallyleads to divergent quantities. In order to control these divergences, you need to introduce
some cutoﬀ, and this is where an energy scale enters. In the case of QCD this energy scaleis called ΛQCD . It’s value is of the order of 250 MeV. The key point now is that the original
QCD coupling constant becomes a “running coupling constant”, i.e., it depends on the scaleof the processes that one considers, or more precisely on the ratio of that scale to ΛQCD ,according to the (one-loop) formulag 2 (µ) =8π 2,
b0 ln(µ/ΛQCD )b0 =11Nc 2− Nf .
3
3(5)

This formula has a remarkable consequence, called asymptotic freedom: when µ is muchbigger than ΛQCD , g(µ) goes to zero. In the high temperature quark-gluon plasma, thetypical energy scale is of the order of the temperature. Thus, when the temperature is large
compare to ΛQCD the coupling constant becomes small. This is essentially the argumentthat leads one to expect that the quark-gluon plasma is a weakly interacting system at hightemperature.
Now returning to the scale invariance, one notes that the the symmetry that exists at
the level of the classical lagrangian is broken at the quantum level. This manifests itself
in particular in a so-called “quantum anomaly”, that can be measured by the trace of theenergy momentum tensor. That trace, which should normally vanish for a system of masslessparticles (reﬂecting the scale invariance), is given (at one-loop) byT µµ =β(g)Tr(F µν Fµν ) .
g2(6)

The function β(g) is called the beta function. It describes the variation of the running
coupling with the scale µ. It is given byβ(g) = µ

b0 3
dg
=−
g .
dµ
16π 2

(7)

At ﬁnite temperature, (and after subtraction of the vacuum contribution), one can write thetrace of the energy momentum tensor as  − 3P , where  is the energy density and P thepressure . This quantity is non vanishing above the deconﬁnement transition, and (slowly)
goes to zero with increasing temperature, in agreement with asymptotic freedom.
I shall end here the discussion of the symmetries of QCD, and how these can help tocharacterize the bulk properties of dense matter. I shall return brieﬂy to the phase diagram

Soryushiron Kenkyu

8later today. At this point, the main message that I want to leave you with is that, becauseof asymptotic freedom, one expects matter at high temperature to be simple: a weaklyinteracting system of quarks and gluons. Since the interactions are weak it is natural totry and calculate their eﬀects using perturbation theory. We shall learn in these lecturesthat the situation is in fact more subtle. But anyway, before calculating the eﬀects of theinteractions, it is important to recall some well-known properties of the non interacting
system.

B.

Thermodynamics of relativistic particles

Let me then remind you about the thermodynamics of free relativistic particles. This
will also oﬀer us the opportunity of a short reminder of basics of statistical mechanics thatwill be useful later at various points in the lectures.

1.

Some reminders

As you know, the statistical description of quantum systems involve the so-called density
operator D, which, for systems in equilibrium at temperature T = 1/β, is of the form
1 −β(H−µQ)
e
,
(8)
Z
where H is the hamiltonian, Q is the conserved charge which can be the electric charge, theD=baryon number, the strangeness, etc, and µ is the associated chemical potential. The factthat Q is a conserved charge means that H commutes with Q, [H, Q] = 0. It follows thatthe the eigenstates of the hamiltonian can be classiﬁed according to the eigenstates of Q. In
other words one can write
H |ψn i = En |ψn i ,Q |ψn i = qn |ψn i .

We can also rewrite the density operator as∑D=|ψn i pn hψn | ,

(9)

(10)

n

where pn is the probability to ﬁnd the system in the particular eigenstate |ψn i of the hamil∑
tonian. We have n pn = 1.
The partition function
Z = Tr e−β(H−µQ)(11)

is the central object of most calculations, since most thermodynamical functions can beobtained from Z. In particular, the thermodynamic potential readsΩ = −T ln Z .

(12)Soryushiron Kenkyu

9(I am using natural units where kB = 1. In other words, I am measuring the temperaturein unit of energy or mass.) The thermodynamic potential is also
Ω = E − T S − µN ,

(13)

where
E = hHi =1 ( −β(H−µQ) )
Tr He
,
Z

(14)

and similarly N is the expectation value of Q, N = hQi, which is calculated in the sameway. S is the entropy.
S = −kB Tr D ln D = −kB∑

pn ln pn .

(15)

n

It is a positive quantity because pn is a positive number smaller than 1. You can verify thatE=−

∂ ln Z
,
∂β βµ

N =−

∂ ln Z.
∂(βµ) β(16)

The relation between the pressure and the thermodynamics potential is (V is the volume)
P =−

∂Ω
.
∂V T,µ(17)Ω is the a function of temperature, chemical potential and volume. It is proportional to thevolumeΩ(T, µ, V ) = V ω(µ, T ) ,(18)from which one deduces that Ω = −P V .

2.

Free particles

Now I would like to consider free particles. Let me write a typical free particle hamiltonian
in second quantization
H0 =∑εp a†p ap .

(19)

p

Here p represents the quantum numbers for one particle. It is typically the momentum,
but I am not going to separate the momentum, the spin, the color, etc, and just use p asa generic symbol for the set of all the quantum numbers that are needed to characterizeentirely the state of a single particle. I am assuming that the hamiltonian is diagonal in thisrepresentation. And εp is just the energy for a single particle in the state which is labeledby p. The operators a†p and ap are creation and annihilation operator and these satisfy

Soryushiron Kenkyu

10commutation or anticommutation relations, depending on whether the particles are bosons
or fermions. That is, for bosons we have
ap a†p − a†p ap = 1 ,(20)

ap a†p + a†p ap = 1 .

(21)

and for fermions

Now we want to calculate the partition function.
()∑†
Z0 = Tr e−βH0 = Tr e−β p εp ap ap .

(22)

I have to tell you a bit more about what is this trace here. We deal with free particles.
The partition function involves a sum over states that have arbitrary numbers of particles:
the state with zero particle (the vacuum), the states with one particle, with two particles, and so on. How do you characterize the state with n (identical) particles? You canlook at the diﬀerent states of one particle. Assume that the values of p are all discrete:
p0 , p1 , p2 , . . . , pk , . . . is the list of all the possible states that a single particle can occupy.
To characterize a state with a large number of identical particles, it is enough to tell what is
the number of particles which occupy each single particle state p. The operator n̂p ≡ a†p ap
is the operator which counts the number of particle in the state p. Now, I can easily rewrite
Z0 in terms of this operator. I get
Z0 =

∑
{np }

where

(

∏

)
e−βεp np

=

p

∏
p



∑

e−βεp np  ,

(23)

np

∑

{np } is the sum over all “conﬁgurations”, that is all sets of possible numbers np(eigenvalues of the operator n̂p ). I have to distinguish two cases, the fermions and thebosons. For fermions, you cannot put more than one particle in a given state, so np = 0, 1.
The sum is then very easy:
Z0 =

∏(

)
1 + e−βεp .

(24)

p

For bosons, np can be any integer, and we get
Z0 =

∏(
p

) ∏
1 + e−βεp + e−2βεp + · · · =1.
1 − e−βεpp(25)Knowing the partition function, you can calculate all thermodynamic quantities according
to the formulae recalled above. In particular let me call fp ≡ hn̂p i. This is usually referred
to as the occupation number. In equilibrium, there is not an exact number of particles inSoryushiron Kenkyu

11each individual state, but each single particle state is occupied with some probability, andthere are ﬂuctuations. The occupation number can be obtained from the formula
fp = −∂ ln Z1= βεp,∂(βεp )e ∓1(26)with − for bosons and + for fermions.

C.

The quark-hadron transition in the bag model.

I now return, as promised, to the phase diagram of hot and dense matter, with a shortdigression on a simple model that mainly exploits the formulae that we have just recalled.
Further details on this model may be found in [10].
The phase diagram of dense hadronic matter has the expected shape indicated in Fig. 2.
There is a low density, low temperature region, corresponding to the world of ordinary
hadrons, and a high density, high temperature region, where the dominant degrees of freedom
are quarks and gluons. The precise determination of the transition line requires elaborate
non perturbative techniques, such as those of lattice gauge theories. But one can get roughorders of magnitude for the transition temperature and density using a simple model dealing
mostly with non-interacting particles.

T
Quark-Gluon Plasma

Tc
HadronsµcµB

FIG. 2. The expected phase diagram of hot and dense hadronic matter in the plane (µB , T ), where
T is the temperature and µB the baryon chemical potential

Let us ﬁrst consider the transition in the case where µB = 0. At low temperature thisbaryon free matter is composed of the lightest mesons, i.e. mostly the pions. At suﬃciently
high temperature one should also take into account heavier mesons, but in the present
discussion this is an inessential complication. We shall even make a further approximation
by treating the pion as a massless particle. At very high temperature, we shall consider that
hadronic matter is composed only of quarks and antiquarks (in equal numbers), and gluons,
forming a quark-gluon plasma. In both the high temperature and the low temperature

Soryushiron Kenkyu

12phases, interactions are neglected (except for the bag constant to be introduced below).
The description of the transition will therefore be dominated by entropy considerations, i.e.
by counting the degrees of freedom.
The energy density ε and the pressure P of a gas of massless pions are given by:
ε=3·

π2 4
T ,
30

P =3·

π2 4
T ,
90

(27)

where the factors 3 account for the 3 types of pions (π + , π − , π 0 ).
The energy density and pressure of the quark-gluon plasma are given by similar formulae:
π2 4
T +B ,30
π2
P = 37 · T 4 − B ,90
ε = 37 ·(28)

where 37 = 2 × 8 + 78 × 2 × 2 × 2 × 3 is the eﬀective number of degrees of freedom of gluons(8 colors, 2 spin states) and quarks (3 colors, 2 spins, 2 ﬂavors, q and q̄). The quantity
B, which is added to the energy density, and subtracted from the pressure, summarizes
interaction eﬀects which are responsible for a change in the vacuum structure between the
low temperature and the high temperature phases. It was introduced ﬁrst in the “bagmodel” of hadron structure as a restoring force needed to equilibrate the pressure generated
by the kinetic energy of the quarks inside the bag. Roughly, the energy of the bag is
E(R) =

4π 3
C
R B+ ,
3
R(29)

where C/R is the kinetic energy of massless quarks. Minimizing with respect to R, one ﬁnds
that the energy at equilibrium is E (R0 ) = 4BV0 , where V0 = 4πR03 /3 is the equilibrium
volume. For a proton with E0 ≈ 1 GeV and R0 ≈ 0.7 fm, one ﬁnds E0 /V0 ' 0.7 GeV/fm3 ,which corresponds to a “bag constant” B ≈ 175 MeV/fm3 , or B 1/4 ≈ 192 MeV.
We can now compare the two phases as a function of the temperature. Fig. 3 shows how
P varies as a function of T 4 . One sees that there exists a transition temperature
(
)1/4
45
Tc =
B 1/4 ≈ 0.72 B 1/4 ,
17π 2

(30)

beyond which the quark-gluon plasma is thermodynamically favored (has largest pressure)compared to the pion gas. For B 1/4 ≈ 200 MeV, Tc ≈ 150 MeV.
The variation of the entropy density s = ∂P/∂T as a function of the temperature isdisplayed in Fig. 4. Note that the bag constant B does not enter explicitly the expression of
the entropy. However, B is involved in Fig. 4 indirectly, via the temperature Tc where thediscontinuity ∆s occurs. One veriﬁes easily that the jump in entropy density ∆s = ∆ε/Tc
is directly proportional to the change in the number of active degrees of freedom when T
crosses Tc .

Soryushiron Kenkyu

13

FIG. 3. The pressure of the massless pion gas compared to that of a quark-gluon plasma, showing
the transition temperature Tc .

In order to extend these considerations to the case where µB 6= 0, we note that thetransition is taking place when the total pressure approximately vanishes, that is when the
kinetic pressure of quarks and gluons approximately equilibrates the bag pressure. Taking
as a criterion for the phase transition the condition P = 0, one replaces the value (30) for Tc
by the value (90/37π 2 )1/4 B 1/4 ≈ 0.70B 1/4 , which is nearly identical to (30). We shall thenassume that for any value of µB and T , the phase transition occurs when P (µB , T ) = B,
where B is the bag constant and P (µB , T ) is the kinetic pressure of quarks and gluons:

P (µB , T ) =37 2 4 µ2B 2 µ2B
π T +(T + 2 ) .
90
9
9π

(31)

The transition line is then given by P (µc , Tc ) = B, and it has indeed the shape illustrated
in Fig. 2.
The model that we have just described reproduces some of the bulk features of theequation of state obtained through lattice gauge calculations. In particular, it exhibits thecharacteristic increase of the entropy density at the transition which corresponds to theemergence of a large number of new degrees of freedom associated with quarks and gluons.
One should be cautious, however, and not attempt to draw too detailed conclusions about
the nature of the phase transitions from such a simple model. In particular this model
predicts (by construction!) a discontinuous transition; but this prediction should not be
trusted [10].

Soryushiron Kenkyu

14squark-gluon
plasma

spl

∆s

pions

sh
0

T

3
c

T

3

FIG. 4. The entropy density. The jump ∆s at the transition is proportional to the increase in thenumber of active degrees of freedom
D.

Perturbative evaluation of the partition function

I now begin the formal part of the lectures, and start introducing the techniques that willallow us to treat the eﬀects of the interactions.
The direct evaluation of the partition function of an interacting system is rarely possible
as this requires a complete knowledge of the spectrum of the hamiltonian H. Various
approximation schemes have therefore been devised to calculate Z. We brieﬂy describe oneof them, the perturbative expansion. We assume that we can split the hamiltonian into
H = H0 + H1 with H1  H0 , and that the spectrum of H0 is known H0 |ψn0 i = En0 |ψn0 i.
For convenience, we assume that the states |ψn0 i are also eigenstates of the charge Q, whichallows us to treat Q on the same footing as H0 . Thus, in the following, we shall assume thatthe term −µQ is included in H0 .

1.

The imaginary time formalism

We deﬁne the following “evolution operator”:
U (τ ) = exp (−τ H) ≡ U0 (τ )UI (τ ) ,(32)

where U0 (τ ) ≡ exp(−H0 τ ). The operator U (τ ) is analogous to the familiar evolution operator of quantum mechanics, exp (−iHt). It diﬀers from it solely by the replacement of
the time t by −iτ . Because of this analogy, we refer to τ as the “imaginary time” (τ isreal!). It has no direct physical interpretation: its role is to properly keep track of theordering of operators in the perturbative expansion (indeed in a “classical” approximation
where the operators are allowed to commute the time dependence disappears). The operator
UI (τ ) = exp (τ H0 ) exp (−τ H) is called the interaction representation of U . We also deﬁneSoryushiron Kenkyu

15
the interaction representation of the perturbation H1 :H1 (τ ) = eτ H0 H1 e−τ H0 ,(33)

and similarly for other operators. It is easily veriﬁed that UI (τ ) satisﬁes the following
equationdUI (τ )= −H1 (τ )UI (τ ) ,(34)dτ
with initial condition UI (0) = 1. By solving this equation one establishes the following
important formula
−βH

e

=e

−βH0

{ ∫ β
}
T exp −dτ H1 (τ ) ,

(35)

0

where the symbol T implies an ordering of the operators on its right, from left to right indecreasing order of their time arguments. Using this formula one can rewrite Z in the form
{ ∫ β}
Z = Z0 hT exp −dτ H1 (τ ) i0 ,(36)0where, for any operator O,( −βH0 )e
hOi0 ≡ Tr
O .
Z0

(37)

The operator H1 is usually expressed in terms of creation and annihilation operators a anda† . Then the calculation of Z reduces to that of the expectation values of time ordered
products of such operators. Whenever H0 is a quadratic function of these operators, Wick’s
theorem applies, and all expectation values can be expressed in terms of single particle
propagators. A diagrammatic expansion can be worked out following standard techniques.
The partition function can be written as Z = Z0 exp(Γc ) where Γc represents the sum of allconnected diagrams.

2.

Free propagators

The study of the free propagators will give us the opportunity to add a few remarks on
the structure of perturbation theory at ﬁnite temperature. Let us consider a system withunperturbed hamiltonian:H0 =∑p a†p ap ,pwhich commutes with the particle number operator Q =

(38)
∑

†
p ap ap . We deﬁne time dependent

creation and annihilation operators in the interaction picture:a†p (τ ) ≡ eτ H0 a†p e−τ H0 = ep τ a†p
ap (τ ) ≡ eτ H0 ap e−τ H0 = e−p τ ap .

(39)

Soryushiron Kenkyu

16
The last equalities follow simply from the equation of motion da(†) (τ )/dτ = [H0 , a(†) (τ )],and the commutation relations:[H0 , a†p ] = p a†p[H0 , ap ] = −p ap ,(40)

which hold for bosons and fermions. The single particle propagator can then be obtainedby a direct calculation:
Gp (τ1 − τ2 ) = hTap (τ1 )a†p (τ2 )i0= e−p (τ1 −τ2 ) [θ(τ1 − τ2 )(1 ± np ) ± np θ(τ2 − τ1 )] ,(41)

where:
np ≡ ha†p ap i0 =1eβp ∓ 1,(42)

and the upper (lower) sign is for bosons (fermions). The fact that G is a function of τ1 − τ2alone may be viewed as a consequence of the fact that H0 is independent of (imaginary)
time. Note that since 0 < τ1 , τ2 < β, one has −β < τ1 − τ2 < β. One can verify on theexpression (41) that, in this interval, Gp (τ ) is a periodic (boson) or antiperiodic (fermion)
function of τ :Gp (τ − β) = ±Gp (τ )(0 ≤ τ ≤ β) .

(43)

To show this, note the following useful relation:eβp np = 1 ± np .

(44)

Thanks to its periodicity (43), the propagator can be represented by a Fourier series
Gp (τ ) =1 ∑ −iων τe
Gp (iων ) ,
β ν

(45)

where the ων ’s are called the Matsubara frequencies:ων = 2νπ/βbosons,ων = (2ν + 1)π/β

fermions.

(46)

The inverse transform is given by∫ β

1.
H0 − iων(47)−β <τ <β(48)dτ eiων τ G(τ ) =G(iων ) =0

Using the property
δ(τ ) =1 ∑ −iων τeβ νSoryushiron Kenkyu

17and Eqs. (45) and (47), it is easily seen that G(τ ) satisﬁes the diﬀerential equation(∂τ + H0 ) G(τ ) = δ(τ ) ,(49)

which may be also veriﬁed directly from Eq. (41). Alternatively, the single propagator
at ﬁnite temperature may be obtained as the solution of this equation (49) with periodic(bosons) or antiperiodic (fermions) boundary conditions.
Remark 1. The periodicity or antiperiodicity that we have uncovered on the explicit formof the unperturbed propagator is, in fact, a general property of the propagators of a manybody system in thermal equilibrium. It is a consequence of the commutation relations of
the creation and annihilation operators and the cyclic invariance of the trace. I leave it to
you as an exercise to establish this general property.
Remark 2. The statistical factor nk can be obtained from the relation nk = ±G(τ = 0− , k).
In the approximation in which the sum over Matsubara frequencies is limited to the termωn = 0, one gets from (45): nk ≈ Tk . Such an approximation corresponds to a “classical ﬁeld”
approximation valid when the occupation factors are large. This approximation, typicallyvalid for long wavelength (small k), should not be confused with the classical limit reachedwhen the thermal wavelength of the particles becomes small compared to their averageseparation distance. In this limit, the occupation of the single particle states becomes small,and the statistical factors can be approximated by their Boltzmann form:1eβ(p −µ) ± 1≈ e−β(p −µ)  1

e−βµ  1 .

(50)

Soryushiron Kenkyu

18
II.

LECTURE 2

I brieﬂy remind you where we stand. Last time I started to explain to you how to do
calculations in ﬁeld theory at ﬁnite temperature, and I told you that we were going to use
two languages. One is the operator formalism that in many cases oﬀers the most direct
physical interpretation. This is a convenient formalism if you know the Hamiltonian. But(relativistic) quantum ﬁeld theory is more often formulated in terms of a lagrangian rather
than a hamiltonian, and the most appropriate formalism is then that of path integrals. I
shall remind you how the calculations proceed in the two formalisms.
Then, what I intend to do today is to apply these formalisms to the calculation of thethermodynamics of scalar ﬁelds. I would like to show you, on the simple example of the scalarﬁeld, how we go on calculating various Feynman diagrams, and alert you on the diﬃculties
which emerge in such calculations. That will take us slowly towards the problems speciﬁc
to QCD.

A.
1.

Calculation of the partition function

Operator formalism (reminder from last lecture)

We established last time the basic formula for the calculation of the partition function,
Z = Tr(e−βH ). We showed that it can be written as an expectation value of a time-ordered
exponential:
Z = Tre

−βH

= Z0

{ ∫ β
}
T exp −dτ H1 (τ )
,
0

(51)

0

where β = 1/T is the inverse temperature, and the expectation value is taken with thedensity operator associated with the free hamiltonian H0 . That is, for any operator O, wehave
hOi0 =

1Tr[e−βH0 O] .
Z0

(52)

Obviously, if you choose O to be the identity, this formula tells you that 1 = h1i0 =
1tre−βH0 , which indicates that Z0 = tre−βH0 : Z0 is the partition function corresponding to
Z0

the hamiltonian H0 . I have written H = H0 + H1 , assuming that H1  H0 , in some sensethat I shall specify later on. I remind you that H1 (τ ) is the interaction representation of
H1 , i.e., H1 (τ ) = eτ H0 H1 e−τ H0 . The expression (51) suggests a way to calculate Z, as an
expansion in powers of H1 . This is perturbation theory in the operator formalism.
Now, what does it mean expanding in powers of H1 ? After expanding the exponential in
Eq. (51), you have to take the expectation value. If H0 , as most often, is a quadratic form

Soryushiron Kenkyu

19

H1

H1

H1

FIG. 5. Examples of Feynman diagrams in φ4 theory.

of creation and annihilation operators, for instance H0 =∑†p p ap ap , then the calculationthat you have to do can be expressed in terms of Feynman diagrams. The lowest order
Feynman diagrams for the φ4 theory that we shall discuss soon are displayed in Fig. 5, with
H1 entering the vertices. So the left diagram will be of ﬁrst order in H1 , and the right one
will be of second order in H1 . The lines which join two vertices at times τ1 and τ2 are freepropagators G0p (τ1 − τ2 ). Such a propagator, as we saw last time, has two pieces dependingon whether τ1 is bigger or smaller than τ2 :G0p (τ1 − τ2 ) = e−p (τ1 −τ2 ) [θ(τ1 − τ2 )(1 ± np ) ± θ(τ2 − τ1 )np ] ,(53)

where np is the statistical factor (occupation number) for bosons or fermions, and the plus
or minus sign corresponds to each of these two possible situations. Remember also that
G0p (τ ) with τ ≡ τ1 − τ2 , is a periodic function of τ . Thus, it can be expanded as a Fourier
seriesG0p (τ ) =1 ∑ −iωn τeGp (iωn ),β n

Gp (iωn ) =1.
p − iωn(54)

where ωn = 2nπ/β is a Matsubara frequency.

2.

Path integral formalism

Now, I would like to move to the path integral formalism. The path integral formalism isbased on the famous Feynman formula for probability amplitudes. Let me ﬁrst discuss thecase of ordinary quantum mechanics. Consider a particle in one dimension, which is locatedat point x1 a the initial time t = 0. Its motion is described by a hamiltonian H, typically of
the form H = p2 /2m + V (x). At time t, the state of the particle is given by
e−iHt |x1 .

(55)

The probability amplitude that at time t, the particle is located at point x2 is
x2 |e−iHt |x1 .

(56)

Soryushiron Kenkyu

20
Now what Feynman told us is that this can be written as an integral over paths x(t) suchthat x(0) = x1 and x(t) = x2 :
−iHt

x2 |e

|x1 =

∫ x(t)=x2∫t 1

0

D(x(t)) ei 0 ( 2 mẋ −V (x))dt ,2(57)x(0)=x1

where we recognize in the exponent the action integral. Note that the notation in Eq. (57)
is symbolic: most of the paths involved in the sum are not smooth paths. But I’m assumingthat most of you have some familiarity with this expression.
What I shall do now is to use this expression (57) in order to calculate the partition
function Z. I shall rely on the analogy that I already pointed out last time – the analogy
between the evolution operator e−iHt and the density operator e−βH of statistical mechanics.
So I’m going to substitute it → τ , with τ real. Then I shall calculate the matrix elements
x2 |e−Hτ |x1 according to the rule we used in order to write the expression (57), and ﬁnd apath integral from x1 to x2 , where now the path is a function of τ . Watch out how things
get modiﬁed in the action integral: it is replaced by τ , idt is replaced by dτ , and ∂t by i∂τ ,so that the kinetic energy acquires a minus sign. A minus sign exists also in front of thepotential energy, and I pull this overall minus sign out of the integrand. Then I get∫ x(τ )=x2
∫τ 1
2
0
−Hτ
x2 |e
|x1 =
D(x(τ )) e− 0 ( 2 mẋ +V (x))dτ .

(58)x(0)=x1

where now ẋ = dx/dτ . This is the formula that will be useful to calculate the partition
function. Pay attention to the diﬀerence with Eq. (57): the factor i in front of the action in
the real time path integral (57) has been replaced by a minus sign. And there is a plus signin front of the potential instead of minus sign.
Now I return to the partition function Z = Tre−βH , which can be calculated as
∫
−βH
Tre
= dx x|e−βH |x .

(59)

The matrix element is given by the path integral (58), with x(0) = x(β) = x, and we shallwrite, symbolically,
Tre

−βH

∫
=

−βH

dx x|e∫|x =D(x(τ )) e−SE ,(60)x(0)=x(β)

where I have introduced the notation SE for the “Euclidean action”
)
∫ β(
1
2
SE =
mẋ + V (x) dτ .
2
0

(61)

Eq. (60) is the path integral expression for the partition function. It involves a sum overpaths x(τ ) that are periodic in imaginary time: x(0) = x(τ ).

Soryushiron Kenkyu

21
The extension of this formula to ﬁeld theory is easy. The scalar ﬁeld theory is speciﬁedby the lagrangian L
1
1
L = (∂µ φ)2 − m2 φ2 − V (φ) ,
2
2

(62)

where I explicitly extracted the mass term from the potential V (φ). I will very often consider
a speciﬁc form for V (φ), the so-called “φ4 ﬁeld theory”, whereV (φ) =

λ 4
φ .
4!

(63)

The notation (∂µ φ)2 is a shorthand for
1
1
1(∂µ φ)2 = (∂t φ)2 − (∇φ)2 .
2
2
2
I now move to imaginary time, and change it into τ . Then L is changed to
[
]
1
1 2 2
1
2
2
L → − (∂τ φ) + (∇φ) + m φ + V (φ) .
2
2
2

(64)

(65)

You see that the change of dt → −idτ produces the same minus sign here as in Eq. (60),and I can write the partition function of the scalar ﬁeld as the following expression:∫Z=
Dφ(t, x) e−SE .

(66)

φ(β,x)=φ(0,x)

Let me remind you that φ is playing here the role of the coordinate and is a function of timet and the three-vector x. I am summing over all ﬁeld conﬁgurations which are periodic in
the imaginary time τ , i.e., φ(β, x) = φ(0, x). The Euclidean action SE is an integral over τfrom 0 to β,
∫ β
SE =

∫
dτ

0

]
1
1
1 2 2
2
2
d x (∂τ φ) + (∇φ) + m φ + V (φ) .
2
2
2
[

3

(67)

This is the basic formula which we are going to use in speciﬁc calculations.
Remarks
In the operator formalism we deduced the periodicity of the Green’s functions or the
propagators from an explicit calculation in Fock space, calculating the time-dependence ofthe operators and observing the periodicity. I left it as an exercise to you to prove that thisperiodicity is in fact quite general and exists for the full Green’s functions (i.e., not only
for the Green’s functions of the non interacting system). In the path integral approach,the periodicity emerges directly from the boundary condition in Eq. (66): The trace in thepartition function involves summing over ﬁeld conﬁgurations that are periodic in imaginary
time.
There are further remarks which I want to make at this stage.

Soryushiron Kenkyu

22• The ﬁrst remark concerns the Euclidean metric, and why the action is called Euclidean.
By Euclidean metric I mean that all the gradient terms come with positive signs.
This makes the action SE a positive deﬁnite quantity, provided of course that thepotential V (φ) is well-behaved. For example, if it is λφ4 , we require λ > 0. Then, one
can interpret this exponential e−SE as a Boltzmann weight. That is to say, one caninterpret e−SE as a probability distribution, which allows us, in particular, to calculate
the path integral using Monte-Carlo techniques: One discretizes the ﬁeld φ, puts iton the four-dimensional lattice and select the paths using “importance sampling”, i.e.,with a probability distribution proportional to e−SE . You would not be able to do thatin real time, because then, you would have to add wildly oscillating factors eiS , whichno computer knows how to do.
• The second remark concerns the periodicity condition in the expression (66). Let’s
assume for a minute that we forget about it. What do we get? We may rewrite theEuclidean action as
∫
SE =

[

]
1
1 2 2
2
d x (∇φ) + m φ + V (φ) ,
2
2
4

(68)

where ∇ represents the gradient in four dimensions. This may be interpreted as theenergy of a classical ﬁeld conﬁguration in four dimension, and the whole path integral
can be viewed as the partition function for a classical ﬁeld theory in four dimension(a sum over all classical ﬁeld conﬁguration weighted by the factor e−SE ). So, if one
ignores the temporal periodicity, one is left with a four dimensional statistical ﬁeld
theory.
Q: In this case, isn’t there a temperature factor in front of the action?
A: Indeed, strictly speaking, the Boltzmann factor is the exponential of 1/T times
the energy of the ﬁeld conﬁguration (here SE in Eq. (68)). However, this factor
1/T is just a multiplicative normalization. This is not the main point that I wantto emphasize here, which is that ﬁnite temperature eﬀects in a quantum ﬁeld
theory could be viewed as ﬁnite size eﬀects in a problem of classical statisticalmechanics in one dimension more. In fact, by abandoning the periodicity condition and letting the integration over imaginary time extend to inﬁnity, one islooking at the zero temperature limit, that is, one is doing quantum ﬁeld theory
for the vacuum.
• The third remark concerns again the integration over the imaginary time in SE . I
have argued that, if I let β → ∞, I can treat the time τ as an ordinary coordinate,

Soryushiron Kenkyu

23
and I end up with a four-dimensional classical ﬁeld theory. Now, I want to argue theother way. Let’s consider the limit where β → 0 or T → ∞, that is, the limit of verylarge temperature, large compared to all typical energy scales in the problem. For
instance, when you have a mass m, this limit applies when T  m. If β is very small,and unless extremely singular ﬁeld conﬁgurations play a role, I can ignore the timedependence of the ﬁeld φ. Then I can rewrite the Euclidean action as
]
[
∫
1 2 2
1
3
2
SE = β d x (∇φ) + m φ + V (φ) .
2
2

(69)

Because the ﬁelds are considered to be independent of τ , the τ -integration can bedone trivially and leads to the factor β in front. Now we get again a classical ﬁeld
theory, this time in three dimensions. The reason why physics is becoming classicalhere is because I am ignoring the time dependence. The role of the imaginary-timedependence is clear in the operator formalism. The time dependence comes because
the operators do not commute. You can look at the equation of motiondO= [O, H] .
dτ

(70)

If the hamiltonian commutes with the operator O, then O is time independent. Thatis, the imaginary-time dependence of an operator is related to the non-commutationof the operator in question with the hamiltonian. Thus, in the formula (51), the timeordered exponential is there to keep track of the time-ordering of quantum mechanicaloperators. Now, I would like to remind you of one thing. Recall the formula for the
propagator that I wrote at the beginning of the lecture
G0p (τ ) = e−p (τ1 −τ2 ) [θ(τ )(1 ± np ) ± θ(−τ )np ] ,(71)

where τ = τ1 − τ2 . Clearly, in the limit where τ → 0− :G0p (τ → 0−) = ±np .

(72)

Let me focus on bosons for whichnp =1ep /T − 1

.

(73)

When p /T is a small number, I can expand the exponential to obtain
np ∼

T
.
p

This is the occupation factor that you get in classical ﬁeld theory.

(74)

Soryushiron Kenkyu

24
Now remember also that G0p (τ ) can be expanded in a Fourier series
G0p (τ → 0− ) =1 ∑ iωn τ
1.
eβ n
p − iωn(75)

The approximate statistical factor in Eq. (74) is obtained by ignoring all Matsubara
frequencies except the one with n = 0.
These are features that we are going to meet repeatedly.
Q: You are assuming ~ = 1 here, but if you put ~’s back in this formula one of ~’s appears
in front of SE , and I guess ~ is something like temperature...
A: I am going to leave as an exercise to you to put back the ~ everywhere and understandhow the classical aspects enter as ~ → 0. But the classical physics which we are
discussing is actually subtle. One sometimes say, as a joke, that there are “two” ~’s:there is the ~ of Mr. Pauli and that of Mr. Heisenberg. These are of course the same~, that of Mr. Planck (and we set it equal to one !). What I want to emphasize hereis that there are two physics issues. We are going to deal with systems of particles
which are intrinsically quantum, like the black body radiation. There the ~ involved
is that of Pauli, that of the “Pauli principle”. And we are going to deal also with longwavelength excitations. These behave like classical ﬁeld oscillations whose wavelengthis large compared with the typical distance between the particles. That allows us to use
the small gradient expansion and that expansion is controlled by the ~ of Heisenberg(that of the “uncertainty principle”). These features are often mixed in a subtle way.
You can have long wavelength, classical-like, oscillations of a gas of particles which are
intrinsically quantum. So, taking the limit ~ → 0 is tricky here.

B.

Thermodynamics of the scalar ﬁeld

With all this preparation, we can now begin the discussion of the thermodynamics of thescalar ﬁeld. I shall do that using perturbation theory, and shall use a mixed formalism, justto train you to go from one language to the other. Let me view the scalar ﬁeld in terms of
the hamiltonian to start with. This is
(
)
∫
1 2 1
1 2 2 λ 432
H= dx
π + (∇φ) + m φ + φ = H0 + H1 ,2
2
2
4!
where π(x) is the canonical momentum conjugate to φ, and H1 ∼ 4!λ

∫

(76)

φ4 . When I said

earlier that H1 is small compared to H0 , it means in the present context that the couplingSoryushiron Kenkyu

25
constant λ is a small number. And I want to calculate the thermodynamic potential Ω as a
power series in λ,1Ω = − log Z = Ω0 + λΩ1 + λ2 Ω2 + · · · .
β(77)

The reason why I write the hamiltonian ﬁrst is that I don’t want to spend time onΩ0 , which is associated with the free hamiltonian H0 . I can write H0 in terms of thenormal modes of the ﬁeld, that is, as a collection of harmonic oscillators with frequencies√
ωk = k2 + m2 (corresponding to the wave numbers k):
)
∑ ( †
1
H0 =
ωk ak ak +
.
(78)
2
k
As you probably all know, from this expression one can easily calculate Ω0 as
1∑
1∑
ωk +
log(1 − e−βωk ), .
2 k
β k

(79)

that is, the sum over all the modes of the corresponding oscillator thermodynamic potentials.

1.

Short wavelength modes

This expression (79) reveals a problem which we are going to face repeatedly: ultraviolet
divergences. I want to spend a few minutes on this.
The sum over k for the zero point energy is a shorthand for∫∑
d3 k √ 2
ωk =k + m2 .
3(2π)
k
As it stands, this integral has no meaning, because

√

(80)

k2 + m2 ∼ |k| for very large k and if I

put a cutoﬀ Λ at the upper end of this integral the result will grow like Λ4 . So it is an inﬁnite
number when Λ → ∞. On the other hand, it does not depend on the temperature T , and canbe interpreted as a correction to the vacuum energy. Accordingly, we are going to redeﬁne
the zero of the energy and simply subtract it. This is a “poor man’s renormalization” andwe will do more sophisticated things soon. What I am doing here is simply redeﬁning thezero of the energy in such a way that the contribution of the zero point energies (the ﬁrst
term in Eq. (79)) drops out. The second term in Eq. (79) is ﬁnite because when k → ∞ thefactor e−βωk goes to zero.
Now I want to move on and proceed with the calculation of the ﬁrst-order correction, Ω1 .
It is given by the diagram in Fig. 6. Because this is the ﬁrst order of perturbation theory,
this is simply λ/4! times the expectation value of φ4 calculated in the non-interacting groundSoryushiron Kenkyu

26

FIG. 6. First order correction to the thermodynamic potential.

state. Since the non-interacting energy is quadratic in the ﬁelds, the average is a Gaussian
integral, and therefore hφ4 i0 = 3 hφ2 i0 hφ2 i0 . Now, hφ2 i0 is nothing but the propagator
evaluated at τ = 0 and x = 0. ThusβΩ1 /V =

=

λ 4
λ
λ
2
φ 0 = 3 φ2 0 = [∆(τ = 0, x = 0)]2 .
4!
4!
8

The boson propagator is, in a mixed representation,]
1 [∆k (τ ) =(1 + nk ) e−ωk |τ | + nk eωk |τ | .
2ωk

(81)

(82)

You can also verify that this is a periodic function of τ and it has the Fourier decomposition
like the propagators that I was discussing earlier. The Matsubara representation is∆k (iωn ) =1
ωn2 + ωk2.

(83)

The propagators I used before were those for the creation and annihilation operators, andthey had the singularity at τ = 0 because it matters whether you order a† on the left of aor a on the left of a† . For the propagator of the scalar ﬁeld, there is no singularity at τ = 0.
In particular for τ = 0 and x = 0,∆(τ = 0, x = 0) =∫d3 k 1 + 2nk.
(2π)3 2ωk

(84)

This is an important formula that we are going to use again and again. This is the expressionof the ﬂuctuation of the ﬁeld, hφ2 i0 in the absence of interactions.
The calculation can be completed now. The expression (81) is equal to
(
(
[
]2
)2 (
)2
)(
)
∑
∑
∑
∑
∑
1
λ
nk 
1 + 2nk
λ
1
nk
= 
+
+2
,
8 k
2ωk
8
2ωk
ωk
2ωk
ωk
k
k
k
k

(85)

where I rewrite the integrals as sums over k. Actually there are various problems in thisexpression.
Look at the ﬁrst term:()2 ( ∫
)2
∑ 1
1
d3 k
1
√
=
.
3
2
2
2ω
2
(2π)
k
k
+
m
k

(86)

Soryushiron Kenkyu

27

FIG. 7. First order correction to the mass.

This integral is divergent. But since it doesn’t depend on temperature, we can drop it, as
we did earlier: it represents an inﬁnite correction, of order O(λ), to the zero of the energy.
The second term is ﬁnite, because of the presence of the statistical factor.
The troublesome term is the third one. This is divergent and temperature dependent.
We know from general principles that this should not happen. What I want to show you isthat it will disappear. To that aim, it is useful to calculate the leading order correction to
the mass. This is given by the simple diagram in Fig. 7. Let me call that Σ. It is a number
which doesn’t depend on the momentum of the external lines and is equal toΣ==λ ∑ 1 + 2nk
λ∆(0, 0) =
.
2
2 k
2ωk

(87)

The propagator Gk , including the mass correction, is now given by the Dyson equation−1
G−1
k = ∆k + Σ ,(88)

where ∆k is given in Eq. (82).
Now, we observe that, in Σ, there is a contribution which is independent of temperature,
so which will exist in the limit of vanishing temperature. This contribution is divergent, but
I cannot handle this divergence the way I did for the energy – just subtract it – because thecorrection to the mass propagates in all the lines of the Feynman diagrams. So I need to doa more elaborate treatment. I introduce a term, called a “counter-term”,
1 2 2
(89)
δm φ ,
2and add this to the hamiltonian density. Then I require that this correction doesn’t change
the mass at zero temperature, that is,λ∑ 1+ δm2 = 0 .
2 k 2ωk(90)

This determines δm2 :δm2 = −λ∑ 1.
2 k 2ωk(91)

Soryushiron Kenkyu

28

FIG. 8. Three-loop contribution to the thermodynamic potential.

Of course, the term added to H contributes to the thermodynamic potential. It willin particular generate a contribution to Ω1 . Since δm2 is already of order λ, in order toobtain this contribution, one just needs to calculate the expectation value of φ2 in the
non-interacting system, which we know already (see Eq. (84)). Thus:()()∑ 1 + 2nk
1 2
λ∑ 1
1
δm ∆(0, 0) =
−
.
=
2
2
2 k 2ωk
2ω
k
k

(92)

Now if you compare what we have here with Eq. (85), you have λ/4 × (divergent sum) ×(ﬁnite sum) there, and now λ/4 with minus sign × (divergent sum) × (ﬁnite sum) here.
So they cancel, as anticipated. This simple example illustrates a general result: there
cannot be quantities which are (ultraviolet) divergent and which depend on temperature.
Such contributions are eliminated by a proper treatment of the subdivergences of Feynman
diagrams, of which I have given you a very elementary example. This property can be
veriﬁed to all orders. The ﬁnal result for Ω1 reads
λ
T 4.
(93)
1152
I would like to show you now the results of the three loop calculation, which involves inΩ1 =particular the diagram displayed in Fig. 8, in order to illustrate other important features ofﬁeld theoretical calculations at ﬁnite temperature. When we do such higher loop calculations, we need more sophisticated techniques for the renormalization procedure than what
I’ve done so far. In particular in gauge theory it is essential, and in the scalar case it isconvenient, to use dimensional regularization. The outcome is that, in higher orders, quantities like the coupling constant, or the mass, start to be dependent on the “renormalization
scale”. This dependence is what I would like to discuss now.
Let me just quote the result for m = 0 (Ω = −P V ):
{
[
](
)2 }
1 λ
1
π2T 4 1
31
λ
µ
−
+
3 log
+C
.
P =
+
9
10 8 16π 2 8
4πT
15
16π 2

(94)

You could have guessed (from last lecture) the ﬁrst term π 2 T 4 /90, which is just the pressureof non-interacting massless modes. C is a number which has an explicit expression in terms

Soryushiron Kenkyu

29

FIG. 9. One-loop correction to the coupling constant λ.

of dilogarithms and the Riemann zeta function, but this expression doesn’t matter here. In
this expression, µ is the renormalization scale which can be chosen at will; it can be 1 MeV,
can be 1 GeV, can be whatever you want, a priori.
Now, the pressure is a physical quantity. It cannot depend on what you choose for µ, andtherefore dP/dµ has to be zero:
dP
=0.
dµ

(95)

How is that possible since there is an explicit µ dependence which just comes from thecalculation of the diagrams? The point is that λ depends also on µ:µd
λ2λ(µ) = β(λ) = 3+ O(λ3 ) .
dµ
16π 2

(96)

(Note that the β-function is here positive, in contrast to QCD.) This dependence on µ
appears essentially because λ receives contribution from diagrams, like that in Fig. 9, thatare logarithmically divergent. The factor 3 comes from the three independent channels.
Now let us calculate dP/dµ. This contains two contributions, the µ dependence implicit
in λ, and the explicit one. We have
{
(
)2 }
dP
λ
π2T 4
1 1
dλ 3
µ
=
−
µ
+
+ O(λ3 ) .
22dµ98 16π dµ 8 16π(97)Here, in line with the weak-coupling expansion, I am ignoring the µ-dependence of the λ2
term, as this is of order λ3 . By using Eq. (96), one veriﬁes that the term within the braces iszero, In other words, if you do a calculation at the order of three loops, which is a calculation
at the order of λ2 , then the pressure P is independent of the renormalization scale µ at thisorder. That can also be veriﬁed in higher order calculation. It is a general result.
But there is more to be said. You see that µ enters in a logarithm. This is also a fairly
generic situation when doing high order calculations. Now, in the logarithm, µ is divided
by some scale, and the typical scale that appears is that in Eq. (94), namely µ appears
generically as µ/(2πT ). The logarithm can be large if µ is very big compared to 2πT , or ifSoryushiron Kenkyu

30µ is very small compared to 2πT . You don’t want that because you are doing an expansionin powers of λ, and in order for the successive terms to be as small as possible, you wantthese logs to be as small as possible. This is why it is natural to choose µ of the order of
2πT .
This is an important observation, which leads one to expect the thermodynamics of QCD
at high temperature to be very close to that of a free gas of quarks and gluons. The reason
is that, in QCD, there is a minus sign in the β-function, so that the coupling decreases as µincreases. Since the optimized µ is related to the temperature as we have just indicated, atvery large temperature µ is large and the coupling is small.
This is the summary of the arguments which deal with short wavelength ﬂuctuations –the modes with very large momenta. There are ultraviolet divergences that we can control
by the standard process of renormalization. Once this is properly done, we get ﬁnite results
at any ﬁnite temperature. I have shown to you one particular example where indeed terms,
that may appear in intermediate stages of a calculation, which are divergent and dependon the temperature, do cancel. And ﬁnally I’ve shown that in higher order, you have toworry about running coupling constant eﬀects. The running of the coupling is in particularessential to guarantee that the physical observables are independent of the renormalization
scale µ, at the order at which we calculate. I have also indicated that a natural scale may
be chosen, µ ∼ 2πT , in order to optimize the apparent convergence of perturbation theory.

2.

Long wavelength modes

I would like to go now into another regime, that of the long wavelength modes. I’m goingto address another technical issue which is related to infrared divergences. This is actually
where the major diﬃculties of perturbation theory at ﬁnite temperature lie, and this will
also occupy us next time.
To introduce the subject, let me return to my calculation of the correction to the mass,
Eq. (87). I am going to drop systematically the vacuum contribution. I shall work in thelimit m → 0 where I can do explicitly the relevant integral:
λ
2

∫

λ
d3 k n k=(2π)3 ωk
2

∫

d3 k
1
λ T2
1
=
.
(2π)3 ek/T − 1 k
2 12

(98)

Note that this integral would be quadratically divergent in the absence of the statisticalfactor. (This factor provides a cut-oﬀ at the scale k ∼ T , and dimensional analysis showsthat the integral is proportional to T 2 . ) Thus it is dominated by “hard” modes, with
k = O(T ). The modes which contribute to the integral are those plasma particles whichhave wavelength of the order 1/T . This mass correction is actually important, and I’m goingSoryushiron Kenkyu

31

FIG. 10. A second order correction to the mass.

to call it the Debye mass m2D ,
m2D =

λ T2
.
2 12

(99)

The inverse of mD plays the same role as the Debye screening length in an ordinary plasma.
This is an example of what we will meet later and what is called a hard thermal loop. Thisterminology has its origin precisely in the fact that the integral is dominated by hard modes.
This correction is also sometimes called “thermal mass”.
Let me now imagine calculating the second-order correction to the mass, given by the
diagram in Fig. 10. What do I get? There is a minus sign because there are two vertices.
Then I have a sum over the Matsubara frequency, and an integral over the momentum.
The insertion is m2D . We get (I am assuming again that the particle is massless at zerotemperature):
λ ∑
− T
2
n

∫

d3 k
m2D
.
(2π)3 (ωn2 + k 2 )2(100)

Now look at what happens in the particular term with n = 0 in this sum,∫d3 k m2D
λ
− T
.
(101)
2
(2π)3 (k 2 )2∫That is awful. Because k 2 dk/k 4 = dk/k 2 , this is infrared divergent as dk/k 2 . We have aproblem.
And this problem is going to be worse and worse as we try and calculate more diagrams.
Let’s indeed calculate the “Mickey Mouse” diagram in Fig. 11. This is proportional to theintegral∫d3 k (m2D )2∼(2π)3 (ωn2 + k 2 )3∫

dk.
(k 2 )2(102)

Soryushiron Kenkyu

32

FIG. 11. Mickey Mouse diagram for the mass correction.

FIG. 12. Ring diagrams for mass correction (left) and thermodynamic potential (right).

That’s indeed worse! The more mass insertions you add, the more serious the problem is.
But now something happens. You see that what we are doing here is something a little
bit stupid, once you understand the physics. I will have the opportunity to come back tothis later. What we have recognized is that, in the thermal bath, the particle acquires amass even if they are massless to start with. This is a non-perturbative eﬀect, although thecalculation of the mass itself can be done within perturbation theory. But the fact that theparticle acquires a mass is something that you should keep in mind when you do a higher
order calculation. When we do this calculation order-by-order, we extract the mass and wetreat it as a correction. We should not do that. If the particle has a mass, let’s take thisproperly into account! Let me do that by calculating the sum of an inﬁnite number of suchinsertions in Fig. 12, which I will call ring diagram. You can write the expression for that
because that is the same as Fig. 10 but with a slight modiﬁcation:
(
)
∫
λ ∑
d3 k
1
1
Σring = T− 2,
2
3
2 + k 2 + m2
2
(2π)
ω
ω
+
k
n
n
D
n

(103)

where the ﬁrst propagator, 1/(ωn2 +k2 +m2D ), is that of a massive particle with mass mD . You
can verify that if you expand this propagator in powers of m2D , then you generate back all thediagrams that we have considered before and that are infrared divergent. The second term inSoryushiron Kenkyu

33
Eq. (103) subtracts the contribution of the diagram in Fig. 7, which is calculated diﬀerently(there is no infrared divergence in this hard thermal loop contribution). The remaining
integrand is then dominated by soft momenta. Now, for the particular contribution withn = 0, I getλΣring → T
2

∫

d3 k
(2π)3

(

1
1
−
2
k + m2D k2

)

λ ∑
= T
2
n

∫

d3 k
−m2D
.
(2π)3 k2 (k2 + m2D )(104)

You see that the mass mD in the denominator provides an infrared cut-oﬀ that allows you to
calculate explicitly the integral. Simple dimensional analysis reveals that it is proportional√
to mD ∼ λT . An explicit calculation, using the explicit expression for mD given above,
yields
λT 2
Σring = −
8π

(

λ
24

)1/2
.

(105)

Likewise, you can calculate the ring contribution, Ωring , in Fig. 12 to the thermodynamic
potential and you get
( [
]
)
∫
1 ∑
d3 k
m2D
m2D
Ωring /V = T
log 1 + 2
2 − 2
3
2
(2π)
ω
ωn + k2
+
k
n
n
( [
]
)
∫
d3 k
1
m2D
m2D
log 1 + 2 − 2
→ T
2
(2π)3
k
k
(
)
3/2
λ
T4
.
=
12π 24(106)

This is an important result. It tells you that if you do an inﬁnite resummation, you get ridof the infrared divergences. Infrared divergences appear then an artifact of the fact that we
did not do the calculation in a proper way, i.e., we attempted an expansion that is not valid.
If we include, correctly, the thermal mass in the propagator, we get a ﬁnite result.
The second lesson is that you see something very funny appearing. I told you before that
we assume a power series in λ: Ω = Ω0 +λΩ1 +λ2 Ω2 +· · · . And this is indeed what you expect
if you calculate Ω by just calculating Feynman diagrams one after the other. But what you
get here is a non integer power, λ3/2 ! An unexpected term is sneaking in the series, which is
λ3/2 Ω3/2 . That means that the weak coupling expansion of the thermodynamic potential is
not what you naively expected. You get a contribution which is not analytic in the coupling
constant. And there is no way to get such a contribution by summing a ﬁnite number ofterms.
There is a lot of material in this lesson. We will come back on several points next time.
I will then try to explain what is going on in very simple terms.
Q: In this result for the thermal mass, λ should be positive, shouldn’t it?

Soryushiron Kenkyu

34abFIG. 13. The potential of the φ4 theory wihtout (m2 > 0) and with (m2 < 0) spontaneous
symmetry breaking.

A: The λ should be positive in any case, because the potential should be bounded frombelow for the system to be stable.
Q: Is the instability related to the appearance of the λ3/2 term?
A: No, but indeed, when λ is negative, you have to pay attention to the meaning of λ3/2 .
But this is not really related to this instability issue that I am referring to. If thesystem is truly unstable, worse things happen.
Q: If you have an attractive interaction then the mass term will be eaten up by theinteraction, which can give a vanishing mass. I guess that is what he is asking.
A: The potential of the φ4 theory is like this (Fig. 13, right). It could also be like this(left of Fig. 13), corresponding to m2 < 0. The eﬀect of thermal ﬂuctuations would
be typically to restore the symmetry, i.e., to transform the potential from its initial
shape (at T = 0) on the right hand side of Fig. 13 to the convex shape of the l.h.s ofFig. 13. That is, thermal ﬂuctuations contribute positively to the mass squared, andif m2 < 0 to start with (at T = 0), the thermal ﬂuctuations will eventually turn it
positive at suﬃciently high temperature.
Q: Is m2D proportional to λ?
A: Yes, m2D is proportional to λ (at least its leading contribution), and it is positive if λis positive, which we assume to be the case.
Q: It is unless you are on the top of the eﬀective potential.
A: Yes, but if you are here (a), you have to be careful about what you calculate. Usually,
one wants to expand from here (b), i.e., around the local minimum. But I shall not
discuss symmetry breaking in these lectures (except a little towards the end).

Soryushiron Kenkyu

35
Q: You mentioned that you need to put µ equal to T in perturbation theory. But I thinkin the end a physical quantity does not depend on µ...
A: Yes, that is a very good point. Indeed, a physical quantity does not depend on µ.
But remember that if you do the calculation in perturbation theory, the physicalobservables are independent of µ only up to terms that are of the same order ofmagnitude as those that are explicitly neglected. For example, in the case that I
treated, I considered terms up to, and including, order λ2 , and the µ dependence
will be of order λ3 . So at any ﬁnite order, there will be a residual µ-dependence.
What I have argued is that one can exploit this dependence in order to improve the(apparent) convergence of the perturbative expansion. Such a strategy is sometimes
referred to as the “principle of minimal sensitivity”, or “principle of fastest apparent
convergence”. At ﬁnite temperature, the coeﬃcients of the expansion in powers of
the coupling constant contain typically logarithms of the ratio µ/T . In order to avoid
these logarithms to become too large (and hence spoil the apparent convergence), it isjudicious to choose µ ∼ T (and detailed calculations suggest the more speciﬁc choiceµ ' 2πT ).

Soryushiron Kenkyu

36
III.

LECTURE 3

Summary of last lecture

Let me start by a brief summary of what we learned in the last lecture.
What we did last time was to consider a scalar ﬁeld with the hamiltonian density givenby1
1
1
H = π 2 + (∇φ)2 + m2 φ2 + V (φ) ,
2
2
2

(107)

where we explicitly wrote V (φ) = λφ4 /4!. Then we applied the general formalism to calculate
thermodynamics, essentially the partition function, Z = Tre−βH . Remember that this could
be written as an integral over ﬁeld conﬁgurations φ(x), each conﬁguration being weighted
by the factor e−SE [φ] , with SE the Euclidean action corresponding to the hamiltonian H.
The ﬁeld conﬁgurations included in the integral are periodic in imaginary time. Here thecalculation was organized as a power series, and we calculated the grand potential asΩ = −(1/β) log Z = Ω0 + λΩ1 + λ3/2 Ω3/2 + λ2 Ω2 + · · · .

(108)

In doing this calculation, we encountered two types of diﬃculties, namely divergences ofsome momentum integrals. Remember that loop integrals in Feynman diagrams involve thesum over the Matsubara frequencies ωn and the integral d3 k.
We met two types of divergences. Ultraviolet divergence – these are familiar in ﬁeld
theory and can be handled by the general procedure called renormalization. In doing so, we
introduced the notion of the running coupling constant, with the β-function which describes
how λ varies with the renormalization scale µ,µdλλ2= β(λ) = 3
+ ···
dµ
16π 2

(109)

Physical quantities, like the pressure, should be independent of µ, that is,
dΩ
=0.
dµ

(110)

Of course, this holds if Ω is calculated exactly. If you do an approximation on Ω, forexample if you calculate Ω up to order λn , Eq. (110) will hold up to corrections of order
λn+1 which are not included in the calculation. In that case some residual dependence on
the renormalization scale will subsist in Ω. This residual dependence can often be used as
an indication of the accuracy of the calculation.
Ultra-violet divergences are common in all ﬁeld-theoretical calculations and they havenothing to do with the temperature. The formalism to handle these is well-established.

Soryushiron Kenkyu

37
We encountered also another type of divergences, called infrared divergences. These are
intimately related to ﬁnite temperature eﬀects. At the end of the last lecture I showed you
that, in some particular cases, these infrared divergences are eliminated by performing an
inﬁnite resummation of Feynman diagrams, which corresponds to taking into account thegeneration of a “thermal mass”. We’ve seen also that this resummation is responsible forthe fractional power of λ that occurs in weak coupling expansion of the pressure, namely,
the term proportional to λ3/2 . Such a term is completely unexpected from the point of viewof perturbation theory because if you expand naively the partition function in powers of
λ, you will only generate terms with integer powers of λ, not λ3/2 . A term such as λ3/2
can only be obtained from resumming an inﬁnite number of Feynman diagrams. Finally,
we understood that there was one particular sector which is important, the sector with the
Matsubara frequency ωn = 0. Since ωn = 2πnT , as soon as n is equal to 1 or bigger than 1,
ωn = 2πnT provides an infrared cut-oﬀ in the propagator 1/(ωn2 + k2 ); therefore there is no
infrared divergence for these non vanishing Matsubara frequencies. This ends the summaryof the main issues that we discussed last time.

A.

QCD perturbation theory and its breakdown

My purpose today is to show you, from a very general perspective, why we have these
speciﬁc diﬃculties at ﬁnite temprature and give you hints about how you can get aroundthese diﬃculties by techniques that I shall explain in the next lecture.
Before I do that, let me summarize the results that have been obtained for the pressure
as a function of the coupling constant. To do so, let me change the notation slightly, andset g 2 = λ/24 (then λ3/2 becomes ∼ g 3 ). The pressure P of φ4 theory is known up to orderg 6 (in fact, up to order g 8 ln g [15])P = P0 + a2 g 2 + a3 g 3 + · · · + a6 g 6 + · · ·(111)

The result is plotted in Fig. 14, divided by the free gas pressure P0 = (π 2 /90)T 4 (this has
been calculated in the ﬁrst lecture), and as a function of the running coupling constantdeﬁned at the scale 2πT . As I argued last time, this is a natural scale to choose in thecalculation of the thermodynamical functions.
As the plot clearly indicates, perturbation theory does not appear to be very predictive
unless the coupling constant is very small, g . 0.3. This same feature is also met in more
complicated theories at ﬁnite temperature, such as QCD. In such theories, we know thatthe coupling constant decreases with the temperature, but unless the temperature is verylarge, so that the coupling is very small, the same pattern of bad apparent convergence isSoryushiron Kenkyu

38

1.04
1.02

g4

P=P0 1

g3

0.98

g5

0.96
0.94
0.92

g2
0.2

0.4

0.6

0.8

g( = 2T )FIG. 14. Weak-coupling expansion for the pressure normalized to that of an ideal gas as a functionof g(2πT ) in φ4 theory. Taken from the review by Blaizot, Iancu and Rebhan [9].

revealed. These calculations of high order contributions are technically demanding, but theresult is disappointing. One of the questions that we have to answer is why it is so bad.

1.

Breakdown of perturbation theory

In fact, before I turn to general considerations which will shed light on this question,
let me show you that in QCD, we can reach a point where perturbation theory completely
breaks down, whatever the strength of the coupling. Let me show you one particular example
of a class of Feynman diagrams where the problem in question manifests itself.
I shall consider the particular family of diagrams displayed in Fig. 15. The wavy lines
represent gluon propagators. With the experience that we gained last time, we can expect
infrared divergences to occur when ωn = 0. Since I am interested in these divergences, I am
going to assume that ωn = 0 in all the propagators. Let us then examine the contribution
to the pressure, or to the thermodynamics potential, of the `-loop diagram. For each loop,
there is a factor of T that normally accompanies the sum over the Matsubara frequencies. I
shall keep only the term n = 0, so there is no sum, but the factor T remains. And there isan integral d3 k/(2π)3 over the momentum (the factor (2π)3 is not relevant here and I shalldrop it). Now there are also vertices. It is not hard to count their number: If there is one
loop, there is no vertex; if there are two loops, there are two vertices; each time I add one
loop, I add two vertices. Therefore, the power of g is 2` − 2. Now, remember that in the ﬁrst
lecture I emphasized the fact that the three-gluon vertex carries a momentum. Therefore,

Soryushiron Kenkyu

39

1 2

l-1 l

FIG. 15. Gluon loop diagrams contributing to the QCD thermodynamic potential.

at each vertex is also attached a momentum k. Then there are propagators, of the form
1/(ωn2 + k 2 ). I set ωn = 0, but I shall add a term m2 which is a ﬁcticious gluon mass, sothat the propagator is 1/(k 2 + m2 ). The number of the propagators has to be counted. Thisis 3(` − 1). You can understand this number in the following way: Start with two loops(second diagram in Fig. 15), there are three propagators . Each time you add a loop, you
add two propagators. I can therefore write, very schematically, and for ` > 1)
( ∫)`g 2`−2 k 2`−2(`)
3
ΩQCD ' T d k.
(k2 + m2 )3(`−1)(112)

I am interested in what happens when all momenta are going to zero at the same rate. In
this situation, all individual momenta can be combined to form a big momentum, K, in aspace of 3` dimensions. Then, I separate the integral into an angular integral and a radialintegral, and focus on the radial integral. In other words, in the same way as d3 k = k 2 dkdΩ3∫∫
with solid angle dΩ3 , I’m going to write the integral as (d3 k)` → K (3`−1) dKdΩ3` , anddrop the angular integral because, from the angles, no trouble is going to come. Let me dothat. Then I get∫(`)ΩQCD ∼ T ` g 2(`−1)

dKK 3`−1

K 2`−2(K 2 + m2 )3(`−1)(113)

This integral is infrared divergent when m → 0. To see that more clearly, let me replace thedenominator K 2 + m2 by K 2 , and put m as a lower cutoﬀ in the integral:∫1(`)` 2(`−1)ΩQCD ∼ T g
dK `−3 .
K
m

(114)

If ` ≤ 3, this integral is perfectly well-behaved because the denominator has a vanishingor negative power of K. No problem. If ` = 4, we have a problem – we have an integral
dK/K. There is a logarithmic divergence and I have put a cutoﬀ m. In order to ﬁx thescale there has to be another factor: the natural scale here is T . Thus, if I was doing the
calculation at four loop, I would expect the ﬁnal result to be of the form(`=4)

ΩQCD ∼ T 4 g 6 log(m/T ) .

(115)

You see this is a divergent result: If I let m go to zero, which I should do because the gluons
don’t have any mass, then I would get an inﬁnity.

Soryushiron Kenkyu

40`≤3well-behaved integral`=4T 4 g 6 log(T /m)()`−4
g 6 T 4 g 2 T /m`>4

TABLE I. Order of magnitude of the `-loop contributions to the pressure. All the `-loop diagramswith ` > 4 are of the same order of magnitude when m = O(g 2 T ).

Let’s look at ` > 4. This is even worse because, for ` = 5 for instance, I get∫

dK/K 2 ,which is a power divergence ∼ 1/m. More generally, for ` > 4, we have
∫
(
)`−4
(`)
` 2(`−1).
ΩQCD ∼ T g
dK/K `−3 ∼ T ` g 2(`−1) (1/m)`−4 = g 6 T 4 g 2 T /m

(116)

m

This situation is not quite the same, but is very reminiscent of the situation that we met lasttime. I showed then that there were Feynman diagrams which where infrared divergent, thehigher the number of loops, the more severe the divergence. Now, in the case of the scalarﬁeld theory, you know that there is a cut oﬀ because, as I showed you last time, the modesof the scalar ﬁeld acquire a thermal mass of order of gT . Remember m2D ∼ λT 2 ∼ g 2 T 2(with λ ∼ g 2 ). You see, if the mass is of the order of gT , there would be no problem because
g 2 T /m ∼ g  1 in the weak coupling regime. In other words, the thermal mass cures thepotential infrared divergences, and makes the successive loop corrections proportional to g ` .
In the case of QCD, not all the modes acquire a mass of order gT . Only the electric modes
do (the thermal mass is then identical to the Debye mass related to electric screening). But
there are also modes in QCD which are magnetic in nature, and these modes may develop
a mass, so-called “magnetic mass”. However, this magnetic mass is expected to be of orderg 2 T . Then, because g 2 T /m = O(1), all the terms in the expansion (116) end up being of
the same order of magnitude! This is a situation where there is not much you can do witha standard weak coupling expansion: if the mass is of the order g 2 T , then all the terms in
the perturbative expansion are of the same order of the magnitude.
In the rest of this lecture, I would like to give you a simple understanding for why such
nasty things happen.
Q: Is there any diﬀerent kind of mass other than magnetic mass in QCD?
A: There are two types of modes in QCD, related to oscillations of the electric ﬁeld and themagnetic ﬁeld, respectively. The electric ﬁeld behaves like a scalar ﬁeld and it acquires
a mass which is related to the screening phenomenon about which I will say more in
the next lecture. That screening mass is of oder gT . So, there is no problem withthe electric modes. The diﬃculty in the QCD plasma comes from the long wavelength

Soryushiron Kenkyu

41
magnetic modes. Such modes also exist in electrodynamics (a static magnetic ﬁeld is
not screened). But in QCD the magnetic modes interact with themselves, and it is
this interaction which it is hard to calculate.

B.

Expansion parameter at ﬁnite temperature
1.

Harmonic oscillator at ﬁnite temperature

The above is the review of the diﬃculties. Now, I will try to give you an insight into
the physical origin of the diﬃculty and go through an elementary discussion of thermal
ﬂuctuations in quantum mechanics, going back to a very simple system, namely the harmonic
oscillator, which you all know for sure. Basically, what we are doing with ﬁeld theory is
playing with an inﬁnite collection of harmonic oscillators. You will see, this is a detour, but
a quite instructive one.
Let’s consider a harmonic oscillator in one dimension whose hamiltonian reads
1
1
H = p 2 + ω 2 x2
2
2

(117)

In any state at T = 0, the expectation value of p2 is equal to that of ω 2 x2 and given by
p2 = ω 2 x2 ∼ ~ω .

(118)

I shall drop the factor ~ in what follows (i.e., I shall use natural units where ~ = 1). Thus,
p2 ∼ ω,x2 ∼ 1/ω .

(119)Now, I need to do a little bit more elaborate things. Remember that, for the harmonic
oscillator, it is useful to introduce creation and annihilation operators such that[a, a† ] = 1.

(120)

In terms of a and a† , we can write1
x = √ (a + a† ) ,
2ω
1
1 2
2
x2 =
(a + a† )2 =(a + a† + aa† + a† a) ,
(121)
2ω
2ω
( )n
and the eigenstates of H are of the form |n0 i = a† 0 |0i. When we calculate the expectation value of x2 in a given eigenstate of the harmonic oscillator, e.g. hn| x2 |ni, we get
contributions only from the last two terms. Using the commutation relation, one easily gets
n0 |x2 |n0 =1(1 + 2n0 ).
2ω

(122)

Soryushiron Kenkyu

42
Here you must recognize something: the factor (1 + 2nk )/(2ωk ) is indeed that same as thatin the integral giving the ﬂuctuation of the scalar ﬁeld (see for instance Eq. (87)).
Now, at ﬁnite temperature, T 6= 0, we can apply the same results. On the average, states
will be occupied with a probability given by the Boltzmann factor, and the expectation valueof x2 at a given temparature T will be given by a formula analogous to Eq. (122), namely
x2 T ∼

1(1 + 2nT ) ,
2ω

(123)

where
nT =

1
eω/T − 1.

(124)

There is a limit which is interesting to us. It is the high temperature limit, T  ω.
In that case, ω/T is a small number, and we can expand the exponential in the statisticalfactor nT . In this regime, nT ∼ T /ω  1, and we can ignore the contribution of the zero
point ﬂuctuations (the “vacuum part”). What remains is
x2 T ∼

T
.
ω2

(125)

This result is familiar and reﬂects the equi-partition of the energy,
ω 2 x2 ∼ p2 ∼ T,
as expected from Boltzmann statistics.
Of course, the harmonic oscillator itself is ﬁne, but what I’m really interested in is to
study the eﬀect of the interactions. So let me add interactions by changing the hamiltonian
H → H0 + H1 . What’s a good choice for H1 ? To stay as close as possible to the φ4 scalarﬁeld theory, I take
λ 4
x .
(126)
4!
We may now calculate the energy levels as a function of λ which is supposed to be a small
H1 =parameter. So, the energy levels will be written in the form
1
En0 = (n0 + ) ω + a1 λ + a2 λ2 + · · · .
2

(127)

where the numbers ai depend on n0 . At ﬁnite temperature, the pressure, or the thermodynamical potential Ω, are functions of λ, and we are interested in the expansion of thesefunctions in powers of λ. Then, the question we want to ask is what controls the expansion. We want H1 to be small compared to H0 in some sense. Let this be measured by aparameter, γ:
γ∼

H1
.
H0

(128)

Soryushiron Kenkyu

43
At this moment this is not very well-deﬁned. Let me specify this parameter a little bit betterby taking some average values:γ∼

λ hx4 iλ hx2 i∼,ω 2 hx2 i
ω2

(129)

2

where I assume hx4 i ∼ hx2 i . This γ is my dimensionless expansion parameter. As you see,this depends on the strength of the coupling λ, and obviously λ should be small for the
expansion to make sense. But it also depends on hx2 i and ω 2 .
At T = 0, what is γ? At T = 0 we have calculated hx2 i ∼ 1/ω, so that
λ
λ 1
= 3 .
(130)
2
ω ω
ω
You can verify that the dimension of λ is the same as that of ω 3 , so that λ/ω 3 is dimensionless,γ∼

as it should. If you are doing a perturbative calculation of ground state properties at zerotemperature the calculation is valid provided that λ/ω 3 is small compared to one.
At T 6= 0, things are diﬀerent. In the high temperature regime, T  ω, we havehx2 iT ∼ T /ω 2 and
λ
λ T
λ T
x2 ∼ 2 2 = 3 .
(131)
2
ω
ω ω
ω ω
The factor λ/ω 3 is the familiar one, but there is another factor which can be big when theγ∼temperature is big. So, at ﬁnite temperature, it’s not enough to have λ/ω 3  1. If thetemperature is big enough, γ may become of order unity, even if λ/ω 3  1.
What happens physically at ﬁnite temperature is that the system expands, because stateswith large quantum numbers become occupied, leading to an increase of hx2 i. The situation
here is comparable to that of perturbation theory for the excited states at T = 0. There
the expansion parameter would be γ ∼ (λ/ω 3 )n0 , which may become of order unity if n0 islarge enough.
Remark. One way to improve perturbation theory is to deﬁne an eﬀective frequency as2
ω 2 → ωeﬀ
= ω 2 + #λ x2 ,(132)

that is, absorb part of the interaction into the “unperturbed hamiltonian”, i.e., replacex4 → x2 hx2 i, and adjust hx2 i self-consistently. This leads to an approximation which issimilar to the Hartree approximation of many-body physics.

2.

Field theory

Let’s move now to ﬁeld theory. The ﬁeld theory is almost identical except for one important feature, namely that a mode carries a given wavelength. We are thus led to distinguishbetween layers of ﬂuctuations at diﬀerent wavelengths.

Soryushiron Kenkyu

44
Q: I want to interrupt to make sure that this diﬃculty comes from some kind of divergenceat small ω.
A: In the harmonic oscillator there is no divergence because we have only one mode.
The diﬃculty comes from the fact that, if the temperature is big, i.e., T  ω, a large
number of quanta are excited, and this may make the expansion parameter of orderunity even if the zero temperature expansion parameter λ/ω 3 is small. If λ/ω 3 = 10−1
and if T /ω = 100, you are expanding in powers of 10’s — that is the problem.
Q: This factor comes from the singularity of the distribution function?
A: It comes indeed from expanding the distribution 1/(eω/T − 1). In the regime where thetemperature is huge compared to the distance between the energy levels, we need totake into account many levels, all of which contribute to produce a large value of hx2 i.
(A similar diﬃculty would occur if you were calculating corrections to the energy ofan excited state with large n0 .)Q: At high temperature the system becomes classical. Is that the reason why you can use
this mean-ﬁeld approximation to treat the higher-order terms?
A: Well, this Hartree approximation that you are referring to can be used also at zerotemperature. It turns out that this is a rather good approximation, even though it’smore useful at ﬁnite temperature. But I don’t think that this is related to the classicalapproximation. Of course, if you are in the classical regime, then hx2 i is big, and it isadvantageous to do this.
Q: I thought that the quantum ﬂuctuations become less and less important ...
A: Yes, that’s right. In the expectation value, hx2 i ∼ 1 + 2nT , the ﬁrst term, which isthe vacuum ﬂuctuation can be ignored if nT is large enough, which is the case at hightemperature where nT ∼ T /ω  1.
So let’s move now to the ﬁeld theory. We replace x2 by φ2 , and hφ2 i is now given by an
integral∫2

φ=d3 k 1 + 2nk∼(2π)3 2k∫

d3 k nk,(2π)3 k(133)

where nk = 1/(ek /T − 1). I am also going to assume that m = 0 so that k = k. In the last(approximate) equality, I have just ignored the vacuum ﬂuctuations. Compare this to theformula (122) that we had before for the harmonic oscillator. The main diﬀerence is that
we have now a sum all over the modes which are labeled by the momentum k.

Soryushiron Kenkyu

45
Now, I want to repeat the analysis of the expansion parameter; I want to compare “thekinetic energy” and the “the potential energy.” The kinetic energy is(∇φ)2 ,

(134)

λ 4
2
φ ∼ g 2 φ2 ,
4!

(135)

and the potential energy is2

where I assumed that hφ4 i ∼ hφ2 i , which is good enough for the qualitative discussion that
I want to present.
Now, comes an important remark. The integral in Eq. (133) is dominated by the large
momenta. One way to see that is that, in the absence of the statistical factor, it would bequadratically divergent. What the statistical factor does is to provide a cutoﬀ at k = O(T ).
Let us then consider the integral with an upper cutoﬀ κ:∫ κ 3
d k nk2
φ κ≡.
(2π)3 k

(136)

This integral, whatever κ is, is always dominated by the largest possible momenta, andtherefore what will contribute dominantly in this integral are the ﬂuctuations which have
momenta k of the order of κ. I’m going to call hφ2 iκ the “contribution of ﬂuctuations at
scale κ”. In the same spirit, I’m going to deﬁne the typical kinetic energy in (134), at thesame scale κ, as(∇φ)2 κ ∼ κ2 φ2 κ .

(137)

Now, I have all the tools to deﬁne the expansion parameter γ, as we did in the analysisof the harmonic oscillator. The big diﬀerence is that now this depends on κ. I call γκ the2

ratio the potential energy ∼ g 2 hφ2 iκ to the kinetic energy κ2 hφ2 iκ , that is,2

g 2 hφ2 i
g2γκ ∼ 2 2 κ ∼ 2 φ2 κ .
κ hφ iκκ

(138)

The rest of the discussion will be concerned with the analysis of the typical momentum
scales that appear in the ultra-relativistic plasmas in the weak coupling regime.

C.

Interplay of the various wavelengths
1.

Self-coupling of hard modes

There are “natural” scales in the ultra-relativistic plasma; one scale, which I call “hard”,
is κ of the order of T . This is the scale for the typical plasma particles which are the modes

Soryushiron Kenkyu

46
hard

soft

ultra-soft

hard-soft∗

κ

T

gT

g2T

gT -T

φ2

T2

gT 2

g2T 2

–

g2

g

1

1

κ

γκ

TABLE II. Expansion parameter for modes with typical scale κ. *see text for deﬁnitions.

of the ﬁeld that carry momenta of the order of the temperature T . Remember that thedensity of such particles goes like T 3 , and the average distance between them is thereforeof order 1/T . Note that 1/T is also the typical de Broglie wavelength, so that these hardparticles are always quantum (as are the photons in the blackbody radiation).
Then I want to calculate hφ2 iκ , with κ ∼ T . This is an integral that we have alreadydone. We get
γT ∼

g2 2
T ∼ g2 .
T2(139)

For these particular modes, there is therefore no problem: The expansion parameter is g 2 ,so that perturbation theory is OK, provided that g is small.. (In fact, if you refer to a
familiar quantum ﬁeld theory like QED, you know that the expansion parameter is g 2 /4π
rather than g 2 . )2.

Self-coupling of soft modes

In an ultrarelativistic plasma, there is another scale, which is called the soft scale, κ ∼ gT .
You will see why this particular scale emerges dynamically. We already have an example ofthat scale with the thermal mass, which is proportional to gT .
When κ = gT  T ,
2

φ

∫ κ

∼
κ=gT

d3 k n k∼(2π)3 k∫ κ
k 2 dkT 1∼ T × κ ∼ gT 2 ,
k k

(140)

so that
γκ ∼

g2
g2
2
φ
∼
gT 2 ∼ g .
κ=gT(gT )2(gT )2(141)

You see an interesting phenomenon here. You see that γκ is still a good expansion parameter,
because it proportional to the coupling constant, which is a small number. But you also see
that, while we had γT = g 2 before, now γgT = g. It means that the perturbation theory forthe self-interaction of the soft modes is still valid, but the expansion is not in powers of g 2
but in powers of g, and so is less precise. This new expansion parameter, g rather than g 2 ,Soryushiron Kenkyu

47

FIG. 16. Schematic representation of the expansion parameter γgT for the soft modes κ = gT andthe coupling γgT −T of the soft modes to the hard modes.

where g is the strength of the self-interaction between the soft modes, is the origin of theodd powers of g that we have encountered in the expansion of the pressure.

3.

Coupling between the soft and hard modes

But there is something more, which is very important. The parameter γgT ∼ g compares
the strength of the self interactions of the soft modes with their kinetic energy. Now, you
may ask another question which is as relevant as this one. You may ask what about thepossible coupling between the soft ﬂuctuations and the hard ﬂuctuations.
Kinetic energies of the soft modes are of order (gT )2 , while the interaction energy between
the soft and hard modes is ∼ g 2 hφ2 iT ∼ g 2 T 2 . So the motion of a soft mode, with wavelengthof order 1/(gT ), is non-perburbatively renormalized by its coupling to the hard degrees of
freedom. Here you cannot expand because the two eﬀects are comparable. So, this is a nonperturbative correction, which goes under the name of Hard Thermal Loops. The thermal
mass which I have introduced the last time, and also discussed today, is one particular
example of this correction.
So you see, at the soft scale the landscape complicates a bit. We know that we can treat
the soft modes perturbatively, although the perturbation theory is not as accurate as for the
hard modes. But the propagation of the soft modes itself is aﬀected by their coupling to thehard modes in a non-perturbative fashion. In order to treat this phenomenon, perturbation
theory is not enough. But this is well under control. The mass resummation that we havediscussed last time is a simple example of what needs to be done to handle this problem.
Q: I thought that you deﬁned the γκ as the ratio of the kinetic energy to the potential
energy of each mode at the scale κ, right?
A: Yes, this is what I’ve done before. But here I’m deﬁning another γ. I look at a modewith wavelength 1/gT , much bigger than the wavelength 1/T of the typical plasma

Soryushiron Kenkyu

48
particles. When such a mode propagates, it can interact with the hard modes throughloop corrections (see Fig. 16). The present expansion parameter is coming from thecomparison between the kinetic energy of the soft modes, which is (gT )2 , and thepotential energy coming from the hard loop, ∼ T 2 , times the coupling g 2 . I’m arguingthat they are of the same order of magnitude.

4.

Ultra-soft modes

Now let me move on to the another natural scale, and confront a real catastrophy. This is
the ultra-soft scale, κ ∼ g 2 T . The reason why this scale occurs is that it is at this particular
momentum that you have the complete matching between kinetic and potential energies.
At the scale κ ∼ g 2 T , we have indeed
φ2 g2 T ∼ T × κ = g 2 T 2 ,(142)(∇φ)2 g2 T ∼ (g 2 T )2 φ2 g2 T ∼ g 6 T 4 .

(143)

so thatThen you see that the kinetic energy and the potential energy are of the same order:
γg2 T ∼

g 2 hφ2 ig2 T
∼1.
κ2

(144)

No expansion is possible! In other words, the ultra-soft modes remain strongly coupled forarbitrary small coupling. Even if the coupling is 10−23 , you cannot expand in the powersof 10−23 because the kinetic energy and the potential energy are always of the same order.
Note that the contribution of the ultra-soft modes to the thermodynamic potential is2

g 2 φ2 g2 T ∼ g 2 (g 2 T 2 )2 = g 6 T 4 .

(145)This is the order g 6 T 4 at which QCD perturbation theory breaks down, as we have discussed
earlier.
You see that the diﬃculties with QCD perturbation theory at ﬁnite temperature can beunderstood from rather general considerations. The key point here is that ultra-relativistic
plasma should be viewed as multi-scale systems. These are diﬃcult to treat because, typically, approximations devised for one particular scale do not work for other scales.
Perhaps you have heard about “the strongly coupled quark-gluon plasma” and about
the AdS/CFT correspondence which allows us to calculate at inﬁnite coupling. From theperspective that I have just outlined, the problem with ﬁnite temperature ﬁled theory is not
so much connected with the absolute strength of the coupling. The picture provided by theSoryushiron Kenkyu

49
AdS/CFT correspondence is one in which all modes interact with inﬁnite strength. What
I have tried to argue is that this picture ignores the important fact that the eﬀects of theinteraction in the QCD plasma depend on the wavelength of the modes that one considers.
Q: This discussion reminds me very much of the critical phenomena. Near the criticalphenomena, we have long wavelength massless modes. They interacts very strongly
and there are many scales intermingled. Is there any insight to be gained from theexperience of the critical phenomena?
A: Well, in fact, in the last two lectures, we will be dealing with one particular criticalphenomenon, which is Bose-Einstein condensation. This very same problem will appear. The techniques to solve it will be borrowed from what I will shall discuss next
time. Here, as I said before, the ultra-soft scale does not occur for the ordinary scalarﬁeld, because a thermal mass is generated. But if you are in the vicinity of a second
order phase transition, you can adjust a parameter in such a way that the eﬀective
mass at some temperature vanishes. Then we are exactly in the same situation.
Q: I think the origin of the classical, long wavelength modes is very similar.
A: You will see that the structure of the infrared divergences in the analysis that I will
do quickly for the Bose-Einstein condensation is indeed very similar. That is why I
put the two topics together in these lectures. In the last lecture, I hope to have sometime to introduce you to some of the modern techniques of the renormalization groupto handle explicitly such multi-scale phenomena.
Q: Can the breakdown of perturbation theory at order g 6 be overcome by using otherweak coupling techniques? For example, the mass-screened perturbation theory, hard
thermal loop resummation, optimized perturbation theory, or others?
A: I have discussed here the origin of the diﬃculty. I have not discussed at all what weshould do in order to overcome it. We know that if you use perturbation theory in a
regime where soft momenta are integrated over you should not be surprised to meet
infrared divergences because the expansion that you are trying to use makes no sense.
That we know a priori without doing any calculation. So we know a priori that weshould do better.
Now what to do depends very much on what you want to calculate. I will brieﬂycomment next time on the application to the thermodynamics of QCD. For the thermodynamics, hard degrees of freedom are dominant. So you expect that perturbation
theory, plus some corrections, will work, and indeed it does. So, you can use screened

Soryushiron Kenkyu

50
perturbation theory, you can use the 2PI formalism, you can do hard thermal loop
resummation, all these techniques will essentially deal with this part of the problem.
If you want to address the situation described here, and if you want to calculate explicitly the contribution of the very long wavelength modes (the modes that I called
ultra-soft), then there is no other way, that I know of today, than to do lattice calculations.
But for the thermodynamics, observe that the long wavelength, small momentum,
modes have very small phase space, so their contribution to the energy density andthe pressure is presumably small. But if you are asking about correlations at long
distances in the plasma, or perhaps transport phenomena, you may need to worry
about these modes. Their analytical treatment remains an open issue.
Q: You said that the coupling between the soft and hard modes becomes in a sense
dangerous because the ratio becomes essentially one. If you consider the coupling of
the ultra-soft modes to the other modes, what happens?
A: This is an interesting question. I let you meditate about it. I shall just point out thatthe scale gT is uniquely determined by the requirement that κ2 hφ2 iκ ∼ g 2 hφ2 iκ hφ2 iT .
Next time I will show you the technique of the eﬀective ﬁeld theory. This will be directly
relevant to what I will do for the Bose-Einstein condensation. And I will also show you how
the hard thermal loop emerges in a dynamical context, i.e., from kinetic theory. This is aninteresting perspective, which may be also of special interest to those of you who are areworking on kinetic equations.

Soryushiron Kenkyu

51
IV.

LECTURE 4

Summary of lecture 3

One of the things that I have emphasized in the previous lecture is that the quark-gluon
plasma, or more generally an ultra-relativistic plasma, can be viewed as a multi-scale system:
there is only one scale to start with, namely the temperature, but at weak coupling, otherscales are generated dynamically. Such a plasma contains modes with various wavelengths,
in fact there is a continuum of wavelengths. And the important point is that the eﬀect of
the interaction between these modes depends very much on their wavelengths. In order tocharacterize this interaction I have introduced the quantity hφ2 iκ which I referred to as theﬂuctuations at scale κ:∫ κ
hφ iκ ≡
2d3 k nk,(2π)3 knk =1
ek/T − 1.

(146)(Remember that the dominant contribution to this integral is determined by momenta which
are of the order of the upper momentum cutoﬀ κ.) This allowed me to deﬁne an expansion
parameter which controls perturbation theory, that I called γκ :γκ =g 2 hφ2 iκ
g2T
∼
.
κ2
κ

(147)

The discussion of the last lecture can be summarized in the table below. We have considered three particular scales, T , gT , and g 2 T . I’m assuming here that g is a small number,g  1. (This is a condition which in fact does not need to be that strict because there arefactors 2π, for instance, that have been left out. )hφ2 iκ g 2 hφ2 iκ
κ=T

T

g T

2

g2

κ = gT

gT 2

g3T 2

g

2

4

1

2

κ=g T

g T

2

γκ

2

2

g T

2

The ﬂuctuations at scale T have interactions controlled by g 2 . For these, the perturbative
expansion behaves as at zero temperature (and is really an expansion in powers of g 2 /4π).
If we move down to the scale gT then we have again an expansion parameter which is small.
But it is of order g instead of order g 2 . That means that perturbation theory for theseparticular set of wavelengths will be less precise. The expansion will not be in powers ofg 2 but will contain odd powers of g. The contributions of order g 3 will be at the center ofour discussion today. And ﬁnally at the scale g 2 T the expansion parameter is of order one,which means that the very long wavelength modes will remain strongly coupled, however
small g may be.

Soryushiron Kenkyu

52
I also made another remark concerning the scale gT , which will introduce today’s discussion. The remark is that if I compare the kinetic energy of a mode with momentum κ ∼ gT ,with the contribution of its interaction with the ﬂuctuations at scale T , I ﬁnd that they areof the same order of magnitude, namely:κ2 ∼ g 2 T 2 ∼ g 2 hφ2 iT(148)

This indicates that even though the self-interactions of the modes with momentum gT
are of order g, the motion of these modes is strongly modiﬁed by their coupling with theﬂuctuations at the scale T . This feature is at the heart of what is known in the literature as
the “hard thermal loops”. And I will tell you a good deal about these hard thermal loops
today.
What I would like to do is to explain how we can handle this coupling between soft andhard degrees of freedom. This will be done by introducing an important construct which isthat of eﬀective ﬁeld theories. I shall do that both in Euclidean, or imaginary time formalism,
and also in real time, where the eﬀective theory takes the form of a kinetic theory. This
program would require a whole set of lectures in itself, which of course I cannot do in one
afternoon, so I shall have to skip some details, especially in the second part of the lecture.

A.

Eﬀective theory

Let me now discuss eﬀective theory. What we have to deal with is a situation wheredegrees of freedom with diﬀerent wavelengths are coupled together and interact diﬀerently
depending on their momenta. I’m going to approach the problem within the path integral
formalism. Let me remind you the formula we have for the partition function
∫
−βH
−βΩ
Z = Tr e=e=
Dφ e−SE ,(149)

φ(β,x)=φ(0,x)

where

∫ β
SE =

∫
dτ

d3 x L(φ, ∂φ).

(150)

0

What I am going to do is a more elaborate version of something that I have already introduced. Remember two lectures ago. We discussed the situation where β goes to 0. If β goesto 0, and if nothing singular happens then, we can ignore the time dependence of the ﬁeld.
Then SE simpliﬁes since the integration over time gives just a factor β:()∫
1
m2 2 λ 4
32
SE → β d x(∇φ) +
φ + φ .
2
2
4!

(151)

Since β is the inverse of the temperature, this corresponds to the high temperature limit.
In this limit, the quantum ﬁled theory reduces to a classical three dimensional ﬁeld theory.

Soryushiron Kenkyu

53
What I want to do now is essentially an elaboration of this remark. I will often refer tothis high temperature limit as the “classical ﬁeld approximation”. As I have also argued
earlier, this is the approximation that is obtained by ignoring the non vanishing Matsubara
frequencies when one expands the ﬁeld in Fourier space. That is, this is the approximation
where one keeps only the vanishing Matsubara frequency component of the ﬁeld.
So let me consider the Fourier expansion of the ﬁeld:
1 ∑ −iωn τφ(τ, x) =eφn (x)β n∑= T φ0 (x) + T
e−iωn τ φn (x).

(152)

n6=0

What is done in Eq. (151) is just keeping φ0 and ignoring all the rest. What I would like to
do now is to show you how we can take into account the rest. I do this in the framework of
a general φ4 scalar ﬁeld theory. (The procedure generalizes to more complicated theories, inparticular to QCD, but we shall not have time to discuss this in detail.) This approximation
will play a crucial role in the solution of the problem that I shall discuss in the next two
lectures, namely Bose-Einstein condensation.
How are we going to take into account the eﬀects of the components with non vanishingωn ? Note that I can make a change of variables, going from a path integral over ﬁeld
conﬁgurations in spatial coordinates × time, towards conﬁgurations in spacial coordinates× frequency. In other words I can write the measure of the path integral as a product∏
Dφ0 × n6=0 Dφn . (I am concerned here just with the time dependence, and I do not write
explicitly the dependence of the ﬁeld on the spatial coordinates.) Then I can rewrite thepath integral as follows
∫
Z=

Dφ0
∫

=

e

Dφn e−SE [φ0 ,φn ]n6=0

Dφ0 e−Seﬀ [φ0 ] ,where−Seﬀ [φ0 ]

∏

≡

∫ ∏

Dφn e−SE [φ0 ,φn ] .

(153)

(154)

n6=0

How can we calculate Seﬀ [φ0 ]? In fact, you know enough to be able to calculate Seﬀ fromwhat I have told you already. Look indeed at Eq. (149), and compare it with Eq. (153). You
see that exp (−Seﬀ ) is a partition function, for a system in which φ0 appears as a frozen,
given parameter. So we know how to calculate that. We know that the thermodynamic
potential Ω in Eq. (149) is given by the set of all connected Feynman diagrams. Similarly,
Seﬀ [φ0 ] ≡ set of all connected Feynman diagram with external lines “φ0 ”(155)Soryushiron Kenkyu

54

0
p

0

n
K

0

n

0

FIG. 17. One loop diagram with external lines φ0 , and internal lines φn .

We know the leading order contribution to Sef f [φ0 ]: this is the “tree-level” action, obtained by evaluating the original classical action with the ﬁeld φ(τ, x) = T φ0 (x), and it is√identical to Eq. (151). In fact it is convenient to rescale φ0 by a factor T , that is to deﬁne
φ0 as

√ ∫ β
dτ φ(τ, x)
φ0 (x) = T

(156)

0

With this deﬁnition, you see that a term such as βφ2 in Eq. (151) would lead a term T φ20
with the deﬁnition of Eq. (152). The new deﬁnition absorbs the factor T into φ20 , leaving
the simpler expression for the leading order eﬀective action:)(∫
m2 2 λT 4
1(0)
2
3(∇φ0 ) +φ +φ .
Sef f [φ0 ] = d x
2
2 0
4! 0

(157)

Corrections to the leading order are generated, for instance, from a loop expansion in thetheory for the hard modes, whose Euclidean action is given by
}
∑ {1
)
1( 2
2
2
SE [φ0 , φn ] =∇φn ∇φ−n +m + λT φ0 + ωn φn φ−n
2
2
n6=0
0
0
λT ∑
λT φ0 ∑
φn φm φk +φn φm φk φl ,+3! nmk4! nmklwhere the ﬁelds φn have been rescaled by the same factor

√

(158)

T as φ0 . For instance a one loop

contribution is easily obtained from the part of SE [φ0 , φn ] that is written in the ﬁrst lineof Eq. (158), and which is quadratic in φn . The one loop diagrams produce a correction to
the mass (to be discussed later), a correction to the four-point function (see the Fig. 17), a
contribution to the six point function, etc. There will be other corrections generated by theterms in the second line of Eq. (152), but these will be of higher order in λ. For instancethe ﬁrst correction to the mass generated by the term with three φn will be of order λ2 atleast.
At this point I need to specify more carefully the separation between what is meant by“soft” and “hard”. Naively we may attempt to call soft the mode with n = 0, and hard themodes with n 6= 0. This is what we have done so far, but this is not enough. If we just leave

Soryushiron Kenkyu

55
things at this stage, we are going to generate an eﬀective action which will contain arbitrary
powers of φ0 , but which will be mostly non local. For instance, consider the diagram in
Fig. 17. As it stands, it is a complicated function of the momenta carried by the variouslines labelled φ0 . We would like however to continue working with a local eﬀective action,that is, we would like to be able to consider this diagram as a correction to the coupling
constant. This can be achieved by introducing a separating scale Λ, chosen so that (here I
rename λ → g 2 )gT  Λ  2πT,

(159)

because I’m now assuming “soft” will involve some energy scale of order gT , while “hard”
will involve some energy scale of order 2πT . Then I redeﬁne the separation between softand hard in the following way. In the soft sector I consider the mode n = 0, but I also
assume that the momenta are limited by Λ. So, when I calculate an integral in the eﬀective
theory, I do all the momentum integrations up to the scale Λ. On the other hand, the hard
sector contains all the modes with non vanishing Matsubara frequencies, plus the sector
with the large momentum components (k ≥ Λ) of the mode n = 0. We have now a cleaner
separation. In all cases you see that, with K 2 = ωn2 + k 2 , the separation which I haveintroduced guarantees that the hard momenta satisfy K ≥ Λ and the soft momenta satisfy
K ≤ Λ.
hard

n 6= 0soft

n = 0, |k| . Λ+

n = 0 and |k| & ΛThus deﬁned, the eﬀective theory for soft external momenta is valid only when the momentum p carried by the ﬁeld φ0 is smaller than Λ. On the other hand, the loop integrals are
dominated by hard momenta.
Now, we have gained something with respect to the argument of locality that I alluded
to earlier. We know that the momenta which are inside the loop are all going to be large
compared to the momenta outside. The locality will result from the fact that one can expand
in p/K where K is a typical momentum inside the loop, and p an external momentum. The
expansion in powers of p/K means the expansion in the ﬁeld and its derivatives. Therefore
that means that the eﬀective action is going to be given by a series of monomials built fromthe ﬁeld φ0 and its derivatives:
L = aφ20 + b (∇φ0 )2 + cφ40 + dφ60 + e (φ0 ∇φ0 )2(160)

Note that this expansion preserves the symmetry of the original lagrangian under φ0 → −φ0 .
In principle we have an inﬁnite collection of terms, and we need a guiding principle totruncate this expansion, otherwise we cannot do calculations. We shall use here a weak

Soryushiron Kenkyu

56

FIG. 18. Tree level diagram contributing to g32 .

coupling approximation, where the successive terms in the eﬀective action can be calculatedusing perturbation theory.
Let me give you the form of the eﬀective action with more standard notation:()∫
1
1 2 2 g32 4 h 6
2
3
Seﬀ = fΛ + d x(∇φ0 ) + MΛ φ0 + φ0 + φ0 + · · · ,
2
2
4!
6!

(161)

where fΛ represents the contribution of the hard mode to the thermodynamical potential.
Note that the dimensions of the various terms are characteristic of a three dimensional ﬁeld
1

theory: φ0 have mass dimension 1/2, [φ0 ] = M 2 , g32 has a dimension of mass, [g32 ] = M , andh is dimensionless, [h] = M 0 .
The partition function of the system can be written as∫
Z = Dφ0 e−Seﬀ [φ0 ] .

(162)

If Sef f is calculated exactly, this is identical to Eq. (149). The strategy now is to obtain Z
from an approximate Seﬀ [φ0 ]. The coeﬃcients in Sef f can be calculated using perturbation
theory. Some are easy to get, because they exist already at tree level. This is the case for
instance of g32 (the corresponding tree level diagram is given by Fig. 18, whose value can be
red oﬀ Eq. (157):
g32 = g 2 T(163)

The next vertex, h, has no contribution at the tree level because there is no term like φ6 inthe original lagrangian. The term of order 6 in φ0 is induced, at leading order, by the one
loop diagram of Fig. 19. This diagram is proportional to g 6 , and is therefore subleading if
g is a small number.
Everything is ﬁne so far with the general strategy. We have a very systematic way of
calculating the contributions to the eﬀective action. One can do that in perturbation theory,
i.e., one can calculate the coeﬃcients of the eﬀective action from Feynman diagrams. Note

Soryushiron Kenkyu

57

0

0

0

0

0

0

FIG. 19. One loop diagram contributing to the φ60 vertex of the eﬀective theory.

that there is no contradiction here with the fact that perturbation theory cannot be used to
calculate within the eﬀective theory. I am using perturbation theory here in the sector where
it is safe, because in calculating the coeﬃcients of the eﬀective theory, I am only integratingover the hard modes. To calculate with the eﬀective theory, I have to do something moresophisticated.

B.

Calculation of the thermal mass M 2

What I shall do now is perform a simple calculation using the eﬀective theory, in order toillustrate how things work. I shall calculate the thermal mass. The calculation will proceed
in two steps. First, I shall calculate explicitly the one-loop correction to the coeﬃcient of φ20
in the eﬀective action (161). Then, I shall use the eﬀective theory to calculate the correction
due to the soft modes. One issue that I want to address is that of the dependence of theresults on the arbitrary scale Λ which comes in as soon as we consider loop corrections. I
shall verify that when the calculation of a physical observable is correctly performed, thisdependence disappears from the ﬁnal result.

1.

Contribution of the hard modes

The coeﬃcient of φ20 in the eﬀective action is given by all the Feynman diagrams whichhave two external φ0 lines. At leading order, there is a single diagram, often called the“tadpole” diagram, displayed in Fig. 20. This is easy to calculate. We have already done
so a number of times. But now, we have to pay attention to the fact that the loop integral
runs over hard momenta only. Thus, we have∫d3 k 1 − δn0 + δn0 θ(k − Λ)g2 ∑2
M (Λ) = T2(2π)3
ωn2 + k 2 ,
n

(164)

Soryushiron Kenkyu

58

FIG. 20. Tadpole diagram that contributes to the coeﬃcient of φ20 in the eﬀective theory

where I have taken into account that the hard degrees of freedom are all the modes with
n 6= 0 (hence the contribution proportional to 1 − δn0 ), as well as the contribution from themode n = 0 with momenta bigger than the dividing scale Λ (hence the term δn0 θ(k − Λ)).
Now, use the following relation between θ functions:θ(k − Λ) + θ(Λ − k) = 1,
1 − δn0 + δn0 (1 − θ(Λ − k)) = 1 − δn0 θ(Λ − k),and rewrite the integral in (164) as∫
g2 ∑d3 k 1 − δn0 θ(Λ − k)M (Λ) = T2(2π)3
ωn2 + k 2
n
∫
∫ Λ 3
dk 1
d3 k
1
g2 ∑
g2
.
= T
− T
3
2
3 k2
2
2
(2π)
ω
2
(2π)
+
k
n
n
2

(165)

The interpretation of the ﬁrst line of this equation is simple: the term δn0 θ(Λ − k) explicitly
removes from the loop integral the contribution of the soft momenta, which needs to be
calculated more accurately. This will be done by using the eﬀective theory. By performing
the sum over the Matsubara frequencies, one obtains
∫
∫ Λ 3
g2
dk 1
d3 k 1 + 2nk g 2
2
M (Λ) = T
− T
.
3
2
(2π)
2k
2
(2π)3 k 2

(166)

As we have done earlier, we shall drop the term which is ultraviolet divergent and independent of the temperature (this goes away with a standard ultraviolet renormalization of themass at zero temperature). The result can be then written as follows
(
)
g2T 2
Λ 6
2
M (Λ) =1−,Λ  T.
24
T π2

(167)

Since Λ is much smaller than T , the second term of (167) can be viewed as a correction, butwhich depends on Λ. As we shall see, this Λ dependence will cancel against an analogous
contribution from the eﬀective theory calculation.

Soryushiron Kenkyu

59

0

g32

0

0

FIG. 21. Tadpole diagram in the eﬀective theory. The loop involve the propagator of the ﬁeld φ02.

Contribution of the soft modes

What I have done so far is to calculate the coeﬃcient MΛ2 in the eﬀective action. What
I want to do next is to calculate the correction to the thermal mass in the eﬀective theory.
What is the correction to the mass? This is the correction that results from the self interaction of the ﬁeld φ0 . The correction is given by the same diagram as in Fig. 20, but redrawn
in Fig. 21 in order to emphasize the elements of the calculation. The diagram in Fig. 21 isa diagram in the eﬀective theory. So the vertex is g32 . And the loop integral involves softmodes only, that is, the momenta are limited to Λ. As for the propagator, it’s inverse canbe read oﬀ the eﬀective action itself in Eq. (161): k 2 + MΛ2 . The calculation then proceeds
as follows∫
1
g32 Λ d3 k
δM =
3
2
2
(2π) k + MΛ2
∫
Λ
g32
k2
= 2
dk 2
4π 0
k + MΛ2
[
]
g32
MΛ
−1 Λ
= 2Λ 1 −
tan
.
4π
Λ
MΛ
2

(168)

Remember that MΛ is of order gT (see Eq. (167)), so that Λ  MΛ ∼ gT . By expanding
the second term of (168) for large Λ/MΛ , and keeping only the leading order correction inMΛ , that is, MΛ2 ≈ MΛ2 = g 2 T 2 /24 (g32 = g 2 T ), one obtains
δM 2 '

g2T Λ
g2
−
M T.
4π 2
8π

(169)

It follows that the physical mass is given by
(
)
g2T 2
Λ 6
g2T Λ
g2
M → M (Λ) + δM =
1−
+
−
MT
24
T π2
4π 2
8π
g 2 T 2 g 2 T gT
√ .
=
−
24
8π 24
2

2

2

(170)

Soryushiron Kenkyu

60
You see that, as anticipated, the Λ dependent term coming from the parameter of theeﬀective theory, MΛ , and that coming from the cutoﬀ in the loop integral within the eﬀective
theory, precisely cancel. The correction, of order g 3 T 2 , agrees with the calculation done in
the second lecture (see Eq. (105)).
I have presented a very simple example of the construction of the eﬀective theory at thescale gT . In the particular context of QCD, such an eﬀective theory (also called dimensional
reduction), has been pushed to a high degree of accuracy. When I mentioned earlier in thelecture the calculation up to g 6 or g 6 log g for the QCD pressure, these were obtained byrelying on such techniques (see for instance [16] and references therein). Of course in QCD,
these techniques are more elaborate. In a gauge theory, you cannot simply put a cut oﬀ on
integrals, as we did. You have to use more sophisticated regulators. The technicalities are
more diﬃcult to master. But the basic concepts can be understood from the simple scalarﬁeld theory discussed in this lecture.

C.

Real time Hard Thermal Loops

In this second part of the lecture, I would like to give you another perspective on hard
thermal loops, using kinetic theory. As we shall see, kinetic theory emerges as the eﬀective
theory that allows us to eﬃciently handle the coupling between hard and soft degrees of freedom in ultrarelativistic plasmas. I shall not proceed through a systematic derivation, which
would require more lectures, but shall try to indicate the main steps in such a derivationand emphasize the main physical aspects.

1.

Real time propagators

As a preliminary, I would like to comment about connections between the imaginary time
and the real time formalisms, and in particular remind you of some relations based on theanalyticity of the propagators. In real time, we deﬁne]
1 [ −βHTr e
T (φ(t1 )φ(t2 )) ,Z
hT φ(t1 )φ(t2 )i = θ(t1 − t2 )hφ(t1 )φ(t2 )i + θ(t2 − t1 )hφ(t2 )φ(t1 )i ,G(t1 , t2 ) = hT φ(t1 )φ(t2 )i =

(171)
(172)

where I have omited the spatial coordinates (which play no role in the discussion) in order toalleviate the notation. We also deﬁne G> (t1 , t2 ) = hφ(t1 )φ(t2 )i and G< (t1 , t2 ) = hφ(t2 )φ(t1 )i.
Let us focus on G> (t1 , t2 ). By making explicit the time dependence, and expanding on a

Soryushiron Kenkyu

61

Im t

0

Re t

−β

FIG. 22. Analyticity domain of G> (t)complete set of eigenstates of the hamiltonian, we get)
1 (G> (t1 , t2 ) = Tr e−βH eiHt1 φ e−iHt1 eiHt2 φ e−iHt2Z
1 ∑ −βEn iEn t1eehn|φ|mie−iEm t1 eiEm t2 hm|φ|nie−iEn t2=
Z n,m
1 ∑ −βEn i(t1 −t2 )En −i(t1 −t2 )Em
2
e
=
e
e
hn|φ|mi ,Z n,mor, setting t1 − t2 = t,
G> (t) =1 ∑{ −βEn iEn t −iEm t2}ee ehn|φ|mi
.
Z n,m

(173)

(174)

In this expression, we can, as we have already done several times in another context, set
it = τ or t = −iτ . We then get
G> (−iτ ) =1 ∑{ −βEn τ En −τ Em
2}ee ehn|φ|mi
.
Z n,m

(175)

Let us see under which conditions this substitution is legitimate, or more generally, underwhich conditions the time can be given an imaginary part. In most cases, the convergenceof the sum will be controlled by the exponential factors. If τ > β, eτ En will win comparedto e−βEn and the sum will explodes. However if τ < β, e−βEn dominates and this will kill
the other terms as En gets large. Therefore the sum over states in Eq. (175) that allows
the calculation of G> (t = −iτ ) is ﬁnite if τ < β. One concludes that G> (t) is an analytic
function of t in the strip −iβ < Imt < 0, as indicated in Fig. 22.
After Fourier transform, the propagator enjoys also analyticity properties in the frequency
complex plane. Recall the form of the propagators in terms of Matsubara frequencies
1
1
=
.
(176)
2
2
2
2
ωn + k + m−(iωn ) + k 2 + m2

Soryushiron Kenkyu

62
Im ω
iω n

Re ω

FIG. 23. Analytic continuation in the complex ω Plane.

By changing iωn to ω, I transform this into the familiar propagator of a relativistic particle
1
1
→
.
(177)
2
2
2
2
−(iωn ) + k + m−ω + k 2 + m2
Once you have continued the propagator from discrete imaginary frequency to an arbitrarycomplex frequency ω (see Fig. 23), you make apparent the pole at a real frequency ωkcorresponding to the excitation energy of the system with plus or minus one particle, ωk =√± k 2 + m2 . Thus, the analytic continuation of the Matsubara propagator allows us to get
information on the excitation energies of the system.
However, it is sometimes diﬃcult to do the analytic continuation explicitly. For instanceyou may know the propagator only numerically, at all values, or only at a given subset ofvalues, of the Matsubara frequencies. Then, performing the analytic continuation in order
to extract the physical singularities may be an (almost) impossible task. Therefore, it may
be advantageous to be able to perform calculations directly in real time. This is what weshall do in this lecture.

2.

An example of hard thermal loop

After this reminder about analyticity property, I would like to show you one particular
Feynman diagram calculation of a hard thermal loop. This is actually how hard thermalloops were discovered, by explicitly calculating a series of Feynman diagrams, and making
the appropriate kinematical simpliﬁcations. I shall do a calculation of a one-loop self-energy
in a scalar theory with a φ3 interaction, in order to be able to compare with corresponding
results in electrodynamics that I shall consider next. The scalar φ3 ﬁeld theory is notcompletely stable, but this diﬃculty is not relevant for the present discussion.

Soryushiron Kenkyu

63

k
p

τ2

τ1

p

p+k

FIG. 24. Self-energy diagram in φ3 scalar ﬁeld theory.

The diagram that I want to calculate is displayed in Fig. 24. I am going to use the mixed
representation of the propagator, Dk (τ ), that I introduced in previous lectures. I shall callthe self-energy Π(τ, p), and I shall focus on the regime where p is a soft momentum while
the loop integral is dominated by hard momenta k ∼ T . I get ﬁrst∫
dD k2
Π(τ, p) = g
Dp+k (τ )Dk (−τ )(τ2 − τ1 ≡ τ ) ,
(2π)D
]
1[
(1 + nk )e−ωk |τ | + nk eωk |τ | ,Dk (τ ) =ωkso that, for τ > 0
Π(τ, p) = g

∫
2

(178)
(179)

dD k
1
1
(2π)D 2ωp+k 2ωk][][× (1 + nk+p )e−ωk+p τ + nk+p eωk+p τ (1 + nk )e−ωk τ + nk eωk τ . (180)Now, I take the Fourier transform by integrating over the imaginary time from 0 to β∫ βΠ(iωn , p) =dτ eiωn τ Π(τ, p)
0∫
1
1 { 1 + np+k + nk
1 + np+k + nk
dD k
2
+
=g
D
(2π) 2ωp+k 2ωk ωp+k + ωk − iωn ωp+k + ωk + iωnnk − np+k
np+k − nk }++. (181)
ωp+k − ωk + iωn ωk − ωp+k + iωn
At this point, I can perform the analytic continuation, iωn → ω. In doing so I may run into
trouble because the denominators may vanish. When the denominators vanish, a priori the
integral blows up, but this singularity is associated with well understood physics, that I am
going to discuss. Look at the ﬁrst term in the integrand1 + np+k + nk
1 + np+k + nk→.
ωp+k + ωk − iωnωp+k + ωk − (ω ± iη)(182)

Soryushiron Kenkyu

64

k

k

−
p

p
p+k

p+k

(1+n k )(1+n p+k )n k n p+k

p

p
−
p+k

k
n k (1+n p+k )p+kk
n p+k (1+n k )FIG. 25. The various processes contributing to Eq. (181).

When the denominator vanishes, it is a signal that there is a process which is allowed. Hereit is the process by which an excitation carrying momentum p (and energy ω) decays into
a set of two excitations carrying momentum k and momentum p + k. This translates into
an imaginary part, which is proportional to the rate of such decay. To get this imaginarypart, we add a little imaginary part to ω. In other words, in the continuation, we start from
Matsubara frequency and we continue up to the real axis but stop a little bit below or above(depending on which propagator we want to consider, retarded, advanced, etc). We just do
not touch the real axis (see Fig. 23).
Now let me turn to the numerators, and focus again on the ﬁrst process in Fig. 25. If
you view the imaginary part as part of a rate calculation (the denominator providing thedelta-function that expresses energy conservation), the numerator accounts for the statisticalfactors that accompany the direct process, by which the mode with momentum p decays
into two other modes, as well as the reverse process by which two excitations recombine
to form the initial excitation (the second term in the ﬁrst line of Fig. 25). The directprocess is enhanced by the factor (1 + nk )(1 + np+k ), while the reverse process will be
simply proportional to nk np+k , the probability that the modes k and k + p are occupied.

Soryushiron Kenkyu

65
Subtracting the two yields(1 + nk )(1 + np+k ) − nk np+k = 1 + nk + np+k ,(183)which is indeed the numerator in the ﬁrst term of Eq. (181). You may verify that all theother terms can be obtained from this simple reasoning.
Let us consider for instance the third term of Eq. (181). This represents a process bywhich the mode with momentum p can absorb a mode with momentum k and go into astate with momentum p + k (see Fig. 25). The reverse process is a mode with p + k going to
the modes with k and p. The statistical factors here are nk for the incoming line (np does
not count because it corresponds to the particle which I am looking at), and the induced
emission factor 1 + np+k on the outgoing line, giving a factor nk (1 + np+k ) for this process.
There is a factor np+k (1 + nk ) for the inverse process. You see again that the products
nk np+k cancel out, leaving a term linear in n:
nk (1 + np+k ) − np+k (1 + nk ) = nk − np+k .

(184)

Note that this process is a genuine ﬁnite temperature eﬀect, and it disappears at T = 0.
This is in contrast to the the one considered previously, which exists also in the vacuum(the 1 in the numerators of Eq. (181)), and which represents a decay process. The processes
that we consider now are scattering processes involving particles of the heat bath. As you
will see in a moment, these are the dominant contributions at high temperature.
Now comes the relations with the rest of the lectures. What we have done so far is anexact one loop calculation. Let us pursue a little bit the analysis of the diagram here. The
loop integral, for the same reason as I discussed already several times, is dominated by thelargest momenta, that is by momenta of the order of the temperature. Observe the ﬁrst two
terms in the self energy, which contain the vacuum contribution, and which represent decay
processes:
1 + np+k + nk
1 + np+k + nk+.
ωp+k + ωk − iωn ωp+k + ωk + iωn(185)

The denominators are of the order of k ∼ T (p ∼ gT ), which is big. In contrast, in the last
two terms,
nk − np+k
np+k − nk+,ωp+k − ωk + iωn ωk − ωp+k + iωn(186)

you have diﬀerence between two large energies ωp+k and ωk , andωp+k − ωk ≈ p ·∂ωk= p · vk ,
∂k

(187)

Soryushiron Kenkyu

66
where v k is the velocity, whose modulus is the speed of light (assuming massless particles).
Thus the denominator, if ω is of order gT , is a soft energy denominator. And indeed, athigh temperature, the dominant contribution is obtained form these last two terms∫
1 { nk − np+k
np+k − nk }dD k2+
ΠHTL (ω, p) ≈ g(2π)D (2ωk )2 ω + v · p−v · p + ω
∫
{
} ∂n
D
d k
1
1
1
k
≈ g2
v
·
p
−
,
(2π)D (2ωk )2
ω − v · p ω + v · p ∂ωk(188)

where I have used the relation
np+k − nk ≈ p ·∂nk∂
nk = p · v k
.
∂k
∂ωk

(189)

The energy denominators reﬂect the well-known phenomenon of Landau damping, that takes
place when the phase velocity of the soft mode, ω/p, equals the velocity of the hard particle,
v k , in the direction of the propagation of the soft mode.
What we will do now in the rest of the lecture is to recover similar expressions for QED,
starting from kinetic theory. My way of showing you the connection with the kinetic theory
is not a formal way. There is a formal route to deduce things but I shall only be able here
to give you hints of how things work and are tied together.

D.

Calculation of ΠQED (ω, v) using kinetic theory

What I shall do now is to do a similar calculation, but for electrodynamics, and using
kinetic theory. I shall show you how to get the self-energy, ΠQED (ω, p) by solving a simple
kinetic equation. Doing so, we shall in fact get immediately the hard thermal loop approximation for ΠQED . What is kinetic theory? It is a theory which describes the evolution of
distribution functions, fq (p, X), which are the phase space densities of particles at spacetime point X = (t, x), carrying momentum p, and electric charge qe (with q = ±1).

1.

Linearized Vlasov equation

The Vlasov equation is the simplest of kinetic equations. It describes the evolution ofparticles under the action of a force, and reads∂fq+ v · ∂x fq + F orce · ∂p fq = 0 .
∂t

(190)

The force, in the case of QED, isF orce = q(E + v ∧ B) ,(191)

Soryushiron Kenkyu

67
pwhere E is an electric ﬁeld and B is a magnetic ﬁeld and v = ∂.
∂pI assume that the system is initially in thermal equilibrium, with a distribution function0

f which is independent of X: ∂X f 0 = 0, and function only of the energy of the particle.
At time t = 0, the system is weakly perturbed away from its equilibrium state, and the
distribution function becomes
f 0 −→ f 0 + δf .

(192)

I’m going to assume that the perturbation is small so that I can treat δf as a small quantity
and linearize the Vlasov equation in order to determine δf :df 0
v · ∂X δfq (p, X) = −qv · E
,
dp

(193)

∂
∂
∂
=
+v·
µ
∂X
∂t
∂x

(194)

where a covariant notationv · ∂X = v µ

is used. To obtain Eq. (193), I have just replaced in Eq. (190) f by f0 + δf and took
advantage of the fact that f0 is an equilibrium distribution. In particular, I have used thefact that f0 is isotropic to eliminate the contribution from the magnetic ﬁeld.

2.

Induced current

Let me now introduce a new function W (X, v):
δfq (v, X) = −qW (X, v)

df 0
.
dp

(195)

If you compare (192) and this expression (195), you can write the following
fq (v, X) = fq0 (p ) − qW (X, v)

df 0
.
dp

(196)

You see this is just the beginning of Taylor expansion of the quantity fq0 (p − qW (X, v)):
fq (v, X) = fq0 (p − qW (X, v)) .

(197)

In other words, the distribution function in the presence of the linear perturbation is justthe thermal distribution function for an energy which is shifted by an amount qW which
depends on both the velocity of the particle and the coordinates and time.
The equation in terms of W reads simply
v · ∂X W (X, v) = v · E .

(198)

Soryushiron Kenkyu

68This is a ﬁrst order partial diﬀerential equation, which can be solved by the method ofcharacteristics. One gets
∫ 0
W (X, v) =−∞dt0 v · E(x − v(t − t0 ), t0 ) .

(199)

You can verify by a direct calculation that this satisﬁes the equation. The interpretation
is simple. The characteristic line is a straight line which is represented by x = v(t − t0 ).
Along the characteristic line the electric ﬁeld does the work v · E in time dt. The work of
the electric ﬁeld adds up to make the quantity W in Eq. (199).
µ
Once we know W (X, v), we can calculate the induced current jind(X), which is thecurrent generated by the perturbation (in equilibrium the current vanishes). This is given
by

∫
µ
jind
(X) = ed3 p µv [f+ (p, X) − f− (p, X)] ,
(2π)3

(200)

where the positive charges going in one direction contribute as the negative charges goingin the opposite direction. By using Eq. (199), we get∫∫
d3 p µ df 0 ∞µ2jind = −2evdτ v · E(X − vτ ) ,(2π)3 dp 0(201)

where τ = t − t0 (not the imaginary time !).
Now, there is a simple relation between the (retarded) polarization tensor and the induced
current:

∫
µ
jind
=

d4 y Πµν
Ret (x − y)Aν (y) .

(202)

Since
E = −∇A0 −

∂A
,
∂t

(203)

Πµν
Ret can be easily obtained from Eq. (201). It is in fact convenient to perform ﬁrst a Fouriertransform (F.T.), using
∫ ∞
F.T.
0

e−ητ f (X − vτ ) =if˜(Q),
v · Q + iη

where f˜(Q) is the Fourier transform of f (X). Then one gets
{
}
∫
dΩv
vµ vν
2
Ret
Πµν (ω, q) = mD −δµ0 δν0 + ω,
4π ω − v · q + iη
where

∫
2e2 ∞
df 0
2
mD = − 2
dp p2
.
π 0
dp

(204)

(205)

(206)

Soryushiron Kenkyu

69

A(x)

A (y)(a)A (y)j(x)(b)

FIG. 26. Diagrammatic representation of the (one-loop) polarization tensor (left) and induced
current (right).

The structure of this equation is very similar to that of Eq. (188). The energy denominators in particular are identical and reﬂect the Landau damping processes, while theintegration over the modulus of the hard momentum (which factorizes) involves the derivative of the equilibrium distribution function. The Debye mass (206) is the analog of thethermal mass of the scalar ﬁeld.
Let me summarize what I have done. I have obtained an approximate expression for the
QED polarization tensor which could have been obtained also by calculating the one-loop
diagram in Fig. 26 (a), and doing the appropriate kinematical approximation valid whenthe external momentum is soft. Instead of calculating approximately a diagram, I have useda kinetic equation to calculate the induce current, from which the polarization tensor was
obtained. In the kinetic theory, the hard particles are described by a distribution functionf (p, X) whose slow variations in space time describe the soft, collective, excitations. You
see how kinetic theory manages to separate the hard and soft degrees of freedom: thehard degrees of freedom are those which govern the momentum dependence of f (p, X). In
f (p, X), p is a hard momentum. The slow degrees of freedom are described by the slow
variations of the distribution function in space and time (i.e., the dependence of f (p, X) on
X). In a way, the kinetic theory based on the Vlasov equation can be viewed as an eﬀective
theory for describing real time phenomena, somewhat analogous to the (Euclidean) eﬀective
theory used earlier.
As I said before, it is possible to establish the correspondence between the kinetic theory
and the diagrammatic hard thermal loop calculation. Such an approach has been generalized to QCD, where it provides a microscopic eﬀective theory for the quark-gluon plasma
in the regime where the separation between the hard and soft degrees of freedom can bemeaningfully realized (see [13] for more details).

Soryushiron Kenkyu

70
V.

LECTURE 5
Introduction

Today we start exploring a new system, the dilute Bose gas. The speciﬁc phenomenon
that I want to discuss concerns the eﬀect of weak repulsive interactions on the critical temperature. As you know, in a Bose gas at high density or low temperature, there is a phasetransition, called the Bose-Einstein condensation. This phase transition occurs in the absence of any interaction among the atoms. The question I shall address is what happens tothis phase transition when the atoms repel each other very weakly. This problem is interesting in many respects. In the particular context of these lectures it is interesting because,
as you will see, the techniques employed to calculate the shift of the critical temperature arevery similar to the techniques that I explained to you in the previous lectures when we dealt
with general aspects of quantum ﬁelds at ﬁnite temperature. In particular the technique ofdimensional reduction, and of eﬀective ﬁeld theory will play an important role.
Let me be now more speciﬁc. The interaction between the atoms will be characterized
by a scattering length. I am going to assume that atoms interact only in the s-wave partial
wave, and I will denote the corresponding scattering length by a. This has the dimension ofa length. There is another parameter with the dimension of a length, namely n−1/3 , where nis the density of particles. I am assuming that the gas is dilute, which means that somehow
n is small. To be more precise, let me remark that together with a and n, I can form adimensionless parameter, n1/3 a. Then the diluteness condition reads n1/3 a  1.
Two issues arise in addressing the question that I was mentioning a minute ago. The
ﬁrst issue is whether the phase transition which is observed in the absence of any interaction
survives in the interacting system. I am simply going to assume that this is the case. The
second issue, assuming that the phase transition indeed takes place, is connected to theevaluation in the change of the critical temperature, and this is what I want to calculate,that is∆Tc(Tc − Tc0 )
=
,
Tc
Tc0

where Tc is the critical temperature in the presence of the interaction, while Tc0 is thecorresponding critical temperature (i.e., at the same density) in the absence of interaction.
What I will show you is that∆Tc
= c(an1/3 ) ,
Tc

(207)

where c is a dimensionless positive number.
This is a non trivial result. The shift in Tc is a quantity which is small if the scattering
length is small, which is the case when the interactions are weak. Now, given that ∆Tc → 0

Soryushiron Kenkyu

71
as a → 0, you could think naively that the change of the critical temperature can becalculated by perturbation theory. However perturbation theory is useless – well, I mean,
not completely– but strict perturbation theory is useless because if you start calculating the
Feynman diagrams order by order in an expansion in powers of a, you will meet infrared
divergences. That is a situation we have already met. It is an indication that we are doing
something wrong. What I shall do in the next couple of lectures is to show you the originof the diﬃculty, and present the techniques that can be used to overcome it, namely thetechniques based on eﬀective ﬁeld theory. The exact renormalization group sheds a more
complete light on this problem, but I shall not have time to discuss it.

A.

Bose-Einstein Condensation1.

Non-interacting uniform systems

Let me now remind you of a few basic facts about Bose-Einstein condensation. This is
textbook material, so I shall skip many details. I consider a collection of non-relativistic
atoms. These atoms are (spinless) bosons, and at ﬁnite temperature the average occupationof the single particle level of momentum p is given bynp =1e(p −µ)/T − 1

,

p =

p2
,
2m

(208)

where µ is the chemical potential.
Let us start by considering a dilute gas at high temperature. Then the chemical potential
is negative and large, so that the factor e(p −µ)/T is large, and I can approximate np ≈e−(p −µ)/T . In this regime, the gas is essentially classical, the eﬀects of quantum statistics(the −1 in the denominator) can be ignored. The density is easily obtained
∫
d3 p
n=
np ≈ eµ/T λ−3 ,
3
(2π)
where

√
λ≡

2π
mT

(209)

(210)

is the thermal wavelength, an important length scale in the problem. (As in most of theselectures, I am using the natural units with ~ = 1.) It is convenient to rewrite Eq. (209) asa formula for µ/T (valid when nλ3  1):|µ|= − ln(nλ3 ) .
(211)
TLet us now examine what happens when one decreases the temperature, keeping the density
ﬁxed. The formula above gives you the trend (when the gas is very dilute so that nλ3  1):

Soryushiron Kenkyu

72

FIG. 27. Phase Diagram of Ideal Bose Gas

the chemical potential decreases in absolute value. As the temperature continues to decrease,it eventually reaches the value µ = 0. When µ = 0, the number of particles is given by(quantum statistic can no longer be ignored then)∫1ζ(3/2)
d3 p
0
=
,
nc =
3

/T
p
(2π) e−1
λ3

ζ(3/2) = 2.612 .

(212)

At that point, something happens. The chemical potential can no longer increase: if itwould, the statistical factor would become singular (with in particular negative values forp < µ). What happens is that particles start to accumulate in the state with vanishing
momentum p = 0. Thus, the total number of particles is split into two contributions: one
contribution, n0 (T ), from particles in the state p = 0, another contribution from the particles
populating all other momentum states. That isζ(3/2).
(213)
λ3
Equation (212) can be viewed as the condensation condition relating the critical temperaturen = n0 (T ) +Tc0 to the density n0c . This leads to the phase diagram drawn in Fig. 27, where the phaseboundary is the curve that relates the critical density n0c to the critical temperature Tc0 :because λ ∼ T −1/2 , we get from Eq. (212) n0c ∝ (Tc0 )3/2 . Above this phase boundary, we
have the condensed phase and below we have the normal phase. It is not diﬃcult to verify
that the density of particles in the condensate is given by[( T )3/2 ]n0 (T ) = n 1 −(T < Tc0 ) ,
0
Tc

(214)

where n is the total density. The condensate density vanishes, as it should, at the criticalpoint. At zero temperature all particles are in the condensate.
At this point I want to empasize that the phase transition that we have just described, andwhich occurs in the absence of interaction, exhibits several unphysical features, which will beSoryushiron Kenkyu

73
cured by interactions, however small these may be. Consider for instance the compressibility(at ﬁxed temperature). This is given by
1 ∂V
1
1 dn
χ=−
= 2
= 2
V ∂P T
n dµ
nT

∫

d3 p
np (1 + np ) .
(2π)3

(215)

This integral is dominated by the low momentum region, where I can replace np in thevicinity of µ = 0 by
p→0,

np ∼

2mTc0
p2

(µ = 0),(216)

an approximation which I have already used in previous lectures. When p → 0, the integral
is infrared divergent. So the compressibility diverges at the transition, which reﬂects theexistence of anomalously large ﬂuctuations of the density. As we shall see, this pathologicalbehavior will disappear in the interacting system.

2.

A ﬁrst look at the eﬀect of interactions

As I already mentioned, I am working in a system where the interaction is dominated bys-wave scattering. I am going to assume that this interaction can be described by an eﬀective
two body potential, function only of the distance between the two atoms, and of the form
V (r 1 − r 2 ) = gδ (3) (r 1 − r 2 ). The relation between the coupling strength g and the scattering
length a can be obtained by solving the scattering problem and is g = 4πa/m. (In fact,
there are ultraviolet divergences in the scattering calculation, so an ultraviolet cutoﬀ needs
to be introduced. The coupling strength should be considered as a function of this cut-oﬀ sothat the relation to the scattering length remains valid for any choice of the cut-oﬀ. I shallnot discuss this in detail here since this play no role in our main discussion. I just want toalert you about this subtlety.) The hamiltonian density is composed of the kinetic energy
term and the interaction termH(r) = −1g∇ψ † (r) · ∇ψ(r) + ψ † (r)ψ † (r)ψ(r)ψ(r) ,
2m
2

g=

4πa
,
m

(217)

where ψ(r) and ψ † (r) are quantum ﬁelds which describe the atoms and obey the usual
commutation relation [ψ(r), ψ † (r 0 )] = δ (3) (r − r 0 ).
As I indicated at the beginning of the lecture, we expect the shift in the critical temperature to be proportional to the strength of the interaction, i.e., proportional to the scattering
length a. It is therefore natural to try and estimate it using perturbation theory at leading
order in a. Let us then calculate the correction to the single particle energy p2 /2m due tothe interaction. This is obtained from the simple Feynman diagram shown in Fig.28. Letme just give you the result: 2gn (it is proportional to g, and the loop integral is proportional

Soryushiron Kenkyu

74

FIG. 28. Self-energy diagram of leading order in ato the density, so the only hard work is the determination of the factor 2). In the presenceof the interaction, the single particle energy becomes p = 0p + 2gn, where 0p = p2 /2m. The
main feature of this correction is that it is a constant shift, by the quantity 2gn. What is theeﬀect of this shift on the transition temperature? Does it move the transition temperature
up or down when the interaction is repulsive, i.e., g > 0? That is, is the shift positive, negative, or zero? The right answer is zero! Let me explain to you why this is zero. Rememberthat the criterion for condensation is n(µ = 0, T ) = 2.612, where
∫
d3 p
1
(218)
n(µ, T ) =.
0 −µ)/T3((2π) e p−1This is for a = 0. When a 6= 0, the density is given by
∫
d3 p
1
n=
= n(µ − ∆µ, T ) ,∆µ = 2gn ,(219)0(2π)3 e(p −(µ−2gn))/T − 1
where µ − ∆µ = µ − 2gn can be viewed as a modiﬁed chemical potential since 2gn is aconstant. The condensation condition that we had for a = 0, n(0, T ) = 2.612, is unaﬀected;
it just occurs at a diﬀerent chemical potential (µ = 2gn), but the relation between the criticaltemperature and the critical density is independent of g. So the ﬁrst-order perturbation
theory or, if you wish, the mean-ﬁeld calculation, does not produce any shift in the criticaltemperature. This result, simple to establish, shows immediately that the linear relation
between the shift in Tc and a is not going to be obtained by a trivial procedure.
Q: If you have temperature dependence in the mean-ﬁeld, somehow...
A: Somehow? What do you mean by “somehow”?
Q: If you calculate only this diagram, of course you do not get a temperature dependence.
A: You can think of a more complicated calculation. But at this level, the message isfairly robust, although it has in fact been overlooked by quite a number of people.
Perhaps I should say that I am approaching the transition temperature from above
where the gas is classical, and there is no... Well let me not say that now. I was about
to say “symmetry breaking”, but I have not mentioned that concept. I shall do thatshortly.

Soryushiron Kenkyu

75
Although in leading order the transition temperature is not aﬀected by interaction, theinteraction has nevertheless a profound eﬀect on the properties of the system. Let me showthat in the case of the compressibility. I leave it as an exercise to you to recalculate n12 dn.
dµLet me just give you the result,∫1n (1 + np )
1 dn
T
T p p
∫
=
≈
.
2g
2
n dµ
2g
1 + T p np (1 + np )(220)

(You see why you get the denominator: when you take the derivative with respect to µ ofthe density
∫
n=

1
d3 p
,
0
(2π)3 e(p −(µ−2gn))/T − 1(221)

you have an explicit derivative, but you have also to take the derivative of the factor n inside
the integral, and that brings another factor dn/dµ (proportional to 2g/T ).) The momentum
integral in Eq. (220) are infrared divergent, but now the divergences actually cancel out since
the same divergent integral appears in the numerator and in the denominator. Of course, if
g is very small, the compressibility is very big but even for inﬁnitesimal g the compressibility
becomes ﬁnite. So there is a deep modiﬁcation of the properties of the system. What is being
done here, in terms of Feynman diagrams, is actually a resummation of a chain of bubble
diagrams which contribute to the screening of the long wavelength density ﬂuctuations.

3.

Symmetry breaking

There is another aspect which is qualitatively new in the presence of the interaction: we
can discuss the phase transition in terms of symmetry breaking. I am going to consider very
low, in fact zero, temperature and show you that Bose-Einstein condensation is what one
may call a quantum phase transition seen here as a change in the properties of the Bosegas as one tunes the chemical potential. To carry out the discussion in simple terms, let me
introduce a quantum state, which I call |N0 i,√|N0 i ∼ exp( N0 a†0 )|0i .

(222)

This is a coherent state, with a†0 the zero momentum component of the ﬁeld creation operator,[a0 , a†0 ] = 1. The expectation value hN0 |H − µa†0 a0 |N0 i contains no contribution from thekinetic energy because a†0 carries no momentum. The result isg
hN0 |H − µa†0 a0 |N0 i = −µN0 + N02 .
2

(223)

If g = 0 and µ < 0, the ground state in Fock space is N0 = 0: this is the vacuum statewith no particle. On the other hand if µ = 0 the system is completely degenerate: weSoryushiron Kenkyu

76

FIG. 29. Energy of the coherent state |N0 i as a function of N0 for µ < 0 and g = 0 (straight line)
and for µ > 0, in which case a minimum occurs for N0 = µ/g.

FIG. 30. Phase diagram of the weakly Interacting Bose Gas at zero-Temperature

have an arbitrary number of particles in the ground state because it cost no energy to add
one. This degeneracy is the source of the large density ﬂuctuations in the non interacting
system at the transition. If µ > 0, the system is completely unstable: you can decrease
its energy by an arbitrary amount by adding more and more particles. The presence of aninteraction controls this phenomenon and cures the instability. When µ > 0, the quadraticterm coming from the interaction generates a minimum at the value N0 = µ/g (see Fig. 29).
As a function of µ, the phase diagram looks like Fig. 30. This is a pattern which allows us toview this Bose-Einstein condensation using techniques which are familiar in quantum ﬁeld
theory, namely borrow all what we know about symmetry breaking and associated Goldstone
modes and related phenomena. The symmetry which is broken here is a U (1) symmetry:
Remember that the hamiltonian of the system does not depend on the (global) phase of thebosonic ﬁeld. However, the ﬁeld acquires an expectation value (that is explicitly taken into
account here in our description with a coherent state), the U (1) symmetry is broken.
Q: Excuse me, did you include −µN to hamiltonian?
A: Yes. Whenever I talk about the hamiltonian H, it is the true hamiltonian with kinetic
energy, potential energy and −µN . When I talk about degenerate states, I am referring
to eigenstates of H − µN .

Soryushiron Kenkyu

77
This is a very rapid digression, but it indicates to you that indeed even an inﬁnitesimal
interaction changes the properties of the system in a qualitative way (you can extend this
discussion to ﬁnite temperature). In the presence of interactions the ground state is indeed
diﬀerent and we can use languages which we are familiar with in ﬁeld theory to understandwhat is going on. This actually leads us to deep consequences; when there is a symmetry
breaking there are Goldstone modes associated with a rotation of the phase of the orderparameter and those Goldstone modes have a strong impact on the physics of Bose-Einsteincondensates. But I shall not discuss this too much here.
There is another digression that I want to make to illustrate the role of mean-ﬁeld eﬀects
on Tc : these are indeed very diﬀerent in uniform and ﬁnite sytems.

4.

Atoms in a trap

I would like to discuss brieﬂy what happens for the atoms in a trap. This for two reasons.
The ﬁrst is that the physics of cold atoms in traps is what has triggered the renewal of interest
in Bose-Einstein condensation and much of the works that I am discussing. As you know,
one is able now to cool atoms in a trap to a suﬃciently low temperature to observe BoseEinstein condensation, in a system where the interaction strength is very small. Till then,the prototype of systems in which one could observe a phase transition akin to Bose-Einsteincondensation was liquid helium. But in liquid helium the interaction between the atoms isvery strong and the fraction of the particles that are sitting in the condensate is never bigger
than about 10 %, even at very low temperatures. Nowdays, one is able to prepare small
condensates of few tens of thousands of atoms in a trap, tune their mutual interaction to
be as small as desired, and observe genuine Bose-Einstein condensation. However, and thisis the second motivation for this digression, the presence of the trap inhibits some of theeﬀects of the interactions that occur in uniform system. So the study of atoms in a trap
gives, so to speak by contrast, an interesting perspective on some aspects of the eﬀects of
the interactions, related to long wavelength phenomena that are characteristics of uniform
systems. These are these long wavelength phenomena that, in my view, make Bose-Einsteincondensation so interesting.
I am going to consider the following situation: the trapping potential is described by aharmonic oscillator, with a typical level spacing ~ω. I assume that the temperature T is
high, such that kB T  ~ω. Under this condition, I can use a semiclassical approximation todescribe the atoms. What does it amount to? The semiclassical approximation essentiallystates that one can consider the gas of the atoms locally as a piece of uniform matter whose
density is equal to the local density in the trap. In other words, one assumes that the energy

Soryushiron Kenkyu

78
of an atom can be written as(p, r) =mω 2 2
p2
+
r ,
2m
2

(224)

ignoring the fact that p2 and r 2 do not commute. The density is, as usual, obtained byintegrating the distribution function over the momentum of the particles∫
1d3 pn(r, µ, T ) =.
3((p,r)−µ)/T(2π) e−1This is now a function of r. The total number of particles is∫
N = d3 r n(r, µ) ,

(225)

(226)

and is kept ﬁxed. The number density at the center of the trap, where it is is the biggest,
is given by∫n(0, µ, T ) =

d3 p
1
.
3
((p,0)−µ)/T(2π) e
−1

(227)

When
n(0, µ = 0, T ) = 2.612 ,(228)

the atoms at the center of the trap will undergo Bose-Einstein condensation. That is,condensation occurs whenever the density at the center of the trap satisﬁes the same relation
with temperature as in a uniform system. The density at the center of the harmonic trap
can be calculated easily. You ﬁnd then, that in the absence of interactions,()1/3
kB Tc0
N
=
.
~ω
ζ(3)

(229)

So you see that if N is big enough (in typical experiments, N ∼ 105 ÷ 107 ), the condition
for the validity of the semiclassical approximation, kB T  ~ω is well fulﬁlled.
Now we can discuss easily the eﬀect of the interaction on the transition temperature of
the trapped gas. I can ask again my question on the shift in Tc : Will it be positive, negative
or zero? Let me tell you that this is not going to be the same result as before, namely,
this is not going to be zero. That leaves two possibilities: positive or negative. You shouldbe able to answer this question because that requires no calculation at all. What will theinteraction do if you put particles in the trap? Let us imagine putting particles in a trap,
and keep adding particles (at ﬁxed temperature) until the density at the center of the trap
is high enough for condensation to occur. When the particles in the middle of the trap justbegin to condense, let us switch on the interaction. What happens? Because the particles
repel each other, the gas will expand and the density in the middle of the trap will decrease,
destroying the condensation. How can one recover the condensation? By decreasing theSoryushiron Kenkyu

79

n
n c0
n c0
nc

(Tc0 ) 3 / 2
a>0

Tc0 TcT

FIG. 31. The critical lines nc (Tc ) for the non interacting (n0c ) and the interacting (a > 0 ) systems.

temperature: this will indeed decrease the kinetic energy of the atoms making them moresensitive to the eﬀects of the trapping potential (that pushes them towards the center of thetrap). This discussion shows that the eﬀect of the repulsion between the atoms is to shift
the transition temperature downwards. This eﬀect has been observed in experiments. It’s
magnitude has been estimated

where aho =

√
~/mω.

∆Tc
a 1/6
= −1.32
N ,
Tc
aho

(230)

This negative shift of Tc is purely a mean-ﬁeld eﬀect, and its physical origin is transparent,
as we have seen.

B.

Towards the calculation of ∆Tc in an uniform system

In a uniform system, the eﬀect of the interaction leads to a positive shift in the criticaltemperature, which is therefore opposite to the eﬀect that we just discussed for atoms ina trap. The physics responsible for a change in Tc in uniform systems has in fact nothing
to do with what happens in a trap where mean ﬁeld eﬀects dominate: as we have seen, in
uniform systems mean ﬁeld corrections do not produce any shift in Tc .
Let me ﬁrst of all indicate a useful relation, which is a purely geometrical relation, valid
in leading order in the interaction strength. It is a relation between the shift in the criticaldensity and that in the critical temperature. Figure 31 represents the phase diagram in thedensity-temperature plane in the cases where a = 0 and a > 0. I am assuming here thatin the presence of the interaction, there is a still a phase transition similar to Bose-EinsteinSoryushiron Kenkyu

80
condensation, and that if a is small the critical line is only slightly displaced from where itis when a = 0. The relation that I am referring to is
2 ∆nc
∆Tc
=−
,
Tc
3 nc

(231)

which just follows from the fact that the two curves are very close to each other, and thefact that I am looking at the leading order in a. The factor 2/3 has its origin in therelation n0c ∼ (Tc0 )3/2 , while the minus sign is obvious from the ﬁgure. Why is that relation
important? It is important because ∆nc is much easier to calculate than ∆Tc . This is so
because it is easier to calculate at a ﬁxed temperature than to calculate at a ﬁxed density(in the latter case, you need to adjust the chemical potential to keep the density ﬁxed as
you change a).
Let us go back to the interaction. The hamiltonian that we consider is of the form
{
(
}
)
∫
∆
g †
4πa
3
†
†
H = d r ψ (r) −g=, (232)
ψ(r) + ψ (r)ψ (r)ψ(r)ψ(r) ,2m
2m
which is the hamiltonian (217), and we have recalled the relation between the coupling
constant g and the scattering length a. When using this hamiltonian we assume that we canignore a lot of details of the atomic physics. The potential between the atoms is replaced
by a contact potential, which is of course meaningful only if the atoms are on the average
far from each other. Only then can we ignore the details of the atom-atom interaction, its
dependence on speciﬁc electronic levels, etc. To be more precise, this hamiltonian will be
used to describe modes of the Bosonic ﬁelds whose wavelengths 1/k are large compared to
the range r0 of the potential,
kr0  1 .

(233)

We have also mentioned another condition, that involves the scattering length a,na1/3  1 .

(234)

This is the statement that the distance between the atoms is large compared to a. Recall
ﬁnally that in the vicinity of the transition, nλ3 ∼ 1, where λ is the thermal wavelength.
The condition on the scattering length translates then into the relation a  λ.
In order to calculate ∆nc , it is useful to express the density in terms of the propagator(in the imaginary time formalism):n = lim− hT ψ(τ, r)ψ † (0, r)i = lim− G(τ, r) .
τ →0

τ →0

(235)

The Fourier transform of the propagator obeys the Dyson equation:
G−1 (iωn , p) = G−1
0 (iωn , p) + Σ(iωn , p) ,(236)

Soryushiron Kenkyu

81
where G0 is the free propagator, Σ the self-energy, and ωn = 2nπT is a Matsubara frequency.
Therefore, the expression for the density of particles can be written as∑ ∫ d3 pe−iωn Tn = lim− T,τ →0(2π)3 p − iωn + Σ(iωn , p)
n

(237)

0
where p = 0p − µ and G−1
0 (iωn , p) = p − µ − iωn . This is an exact relationship: It allows

me to calculate n, provided I know how to calculate the self-energy.
Now I need to say a few words about the condensation condition. I shall give you theresult and try to motivate it. I am going to assume that condensation takes place whenG−1 (ω = 0, p = 0) = 0 .

(238)Note ﬁrst that, in the non interacting case, the condition G−1
0 (0, 0) = 0 yields µ = 0, whichis indeed the condensation condition that we have already met. Turning to the interacting
system, one may recognize that G−1 (iωn = 0, p = 0) is just the second derivative of the“eﬀective potential” (the free energy expressed in terms of the expectation values of theﬁeld ψ and ψ † ). Assuming that the condensation is a second-order phase transition, thesecond derivative of this eﬀective potential vanishes at the transition. This is just thestatement that G−1 (0, 0) = 0.
By using the explicit expression G−1 (iωn , p) = 0p − µ − iωn + Σ(iωn , p), one sees that thecondition G−1 = 0 reduces toµ = Σ(0, 0) .

(239)

Now I can calculate the shift ∆nc in the critical density. I get
{
}
∫
∑
d3 p
1
1
−iωn T∆nc = lim− Te−. (240)τ →0(2π)3 0p + Σ(iω, p) − Σ(0, 0) − iωn 0p − iωnnThe ﬁrst term is the critical density of the interacting system, for which µ = Σ(0, 0), thesecond term is the critical density of the non interacting system at the same temperature,
for which µ = 0. Note that the only place where the interaction enters is the self-energy. So
we have to evaluate the self-energy. You see here that when the self-energy is frequency andmomentum independent, there is no correction, because if the self-energy does not dependon ω or p it is the same as for ω = p = 0, and then the two terms cancel and ∆nc = 0.
Q: This is precisely what happens in the mean-ﬁeld approximation?
A: Yes. As I said, the mean ﬁeld approximation leads to a self-energy which is independentof frequency and momentum. That produces no shift of Tc . To get a non-trivial eﬀect
you need frequency and momentum dependences in the self-energy.

Soryushiron Kenkyu

82

k,

p,

n'

p+q
n +
n''

g
n

k q,

n'

g

n''

FIG. 32. Second order diagram contributing tho the self-energy.

Let me start the calculation of the correction. We have already done the ﬁrst-order
calculation and shown this to be zero. Therefore, we have to go to the second-order. I shallcome back to it next time. The second-order diagram is shown in Fig. 32. The corresponding
contribution to the self-energy reads∑ ∫ d3 k ∫ d3 q
1
1
1
2 2
Σ(iωn , p) = −2g T,(2π)3(2π)3 k−q − i(ωn0 − ωn00 ) k − iωn0 p+q − i(ωn + ωn00 )
0 00
n ,n

(241)
where the factor T 2 comes from the double sum over the Matsubara frequencies. I am going
000

to focus on the contributions with n = n = 0. Remember that the infrared divergencesof the scalar ﬁeld theory are coming from this particular sector where all the Matsubara
frequencies vanish. I am going to look at this particular contribution, and follow the same
strategy as a few lectures ago when I discussed the divergences in high temperature QCD
and I exhibited special classes of Feynman diagrams which are infrared divergent. I used a
power-counting argument and replaced the multiple integrations over the momenta by an
integration over a big momentum vector in a large space and I ignored the angular integral.
Let me proceed in the same way quickly today (I shall come back to this result next time)
because I would like to show you that there is diﬃculty. Think therefore of the momentum
integrals over q and k as an integral of a big vector in 6 dimensions. Then the integral in
Eq. (241) behaves as∫∼

k 5 dk
=
k6

∫

dk
.
k

This is a logarithmically divergent integral.
Q: How about the external momentum p? It does not provide any cutoﬀ?

(242)

Soryushiron Kenkyu

83
A: Well, it does, but in a subtle way that we shall discuss more precisely next time. For
today, let me just observe that what we need to calculate is Σ(0, p) − Σ(0, 0), andthere is no p in Σ(0, 0).
What will come after is very much the same thing as what we met when we discussed
QCD at ﬁnite temperature. Namely, when I calculate the third-order, the fourth-order, etc.,
I ﬁnd increasingly divergent contributions. The pattern of divergences becomes worse andworse as the order increases. What I want to argue next time is that this situation is verymuch similar to what we have met already. You will see that this pattern of divergences isindeed that of 3-dimensional ﬁeld theory. Having recognized that, we will know what to do,namely, how to construct an eﬀective theory for the sector where the divergences occur. This
will allow us to establish the linear relation between ∆Tc and a, and to obtain an explicit
formula for the coeﬃcient c in the formula (207). I will then explain to you why it is hardto calculate explicitly this coeﬃcient.
Q: In the last integral, I think you used a large momentum approximation?
A: Not really. But, of course, the logarithmic divergent integral involves a ratio of momenta, and it is large when one of the momenta is large (or small) compared to theother. But I am really concerned here by the low momentum sector.
Q: This approximation is on the low momentum?
A: Yes. I am assuming that all the loop momenta are going to zero at the same rate.
Note that, for this second order case, I can calculate the integral explicitly and I shalldo that next time. I hope then that this question will be completely clariﬁed.

Soryushiron Kenkyu

84
VI.

LECTURE 6

Let me remind you that we want to establish the following formula for the shift in thecritical temperature of the Bose-Einstein condensation caused by weak repulsive interactions:∆Tc= c(an1/3 ),Tc

an1/3  1 .

(243)

We want to establish this formula to leading order in the strength of the interaction, measured by the scattering length a. The plot in Fig. 33 displays the critical line for the
3/2

non-interacting gas. It goes like nc ∼ Tc . Also drawn is the critical line for the interacting
system. We assume that it diﬀers very little from that of the non-interacting system whena is small, and the curve is drawn here for a positive scattering length.
I’m interested in the critical temperature Tc at point C on Fig. 33: Tc is the transitiion
temperature of the interacting system for some small positive value of a. Starting from thenon interacting system, a = 0, I can approach this point C either by moving the density
downward at constant temperature (going from A → C) or moving upward the temperature
at constant density (going from B → C). I argued last time, based solely on the geometricalpropeties of Fig. 33, that ∆Tc and ∆nc are related by the simple equation:
∆Tc
2 ∆nc
=−
.
Tc
3 nc

(244)

I also argued last time that it is much easier to calculate ∆nc , i.e., to work at ﬁxed temperature.
The last thing I want to remind you is the condensation condition, expressed as thevanishing of the inverse propagator for vanishing momentum and Matsubara frequency.
This translate into a condition on the self-energy:
G−1 (ω = 0, p = 0) = 0→

Σ(ω = 0, p = 0) = µ(245)

FIG. 33. The critical lines nc (Tc ) for the non interacting (n0c ) and the interacting (a > 0 ) systems.

Soryushiron Kenkyu

85
With this, we can write the formula for the shift ∆nc :∫∑d3 p−iωn τ
∆nc = lim− T
e
τ →0
(2π)3
n
}
{
1
1
.
× 0
−
p + Σ(iωn , p) − Σ(0, 0) − iωn 0p − iωn(246)

The ﬁrst term of the integral is the density of the interacting system at criticality. The
subtracted term is the density of the same system at the same temperature, but without
interaction (we used here the fact that at criticality the chemical potential in the noninteracting system is zero). From this formula you may observe that if the self-energy is aconstant, independent of frequency and momentum then Σ(iωn , p) − Σ(0, 0) vanishes, andso does ∆nc . This is what happens for instance in the mean ﬁeld (one-loop) calculation that
I did last time.
Now perhaps I should add some word of caution here. Sometimes you may be led toconsider a mean ﬁeld calculation where the dominant eﬀect of the interaction leads to a
momentum dependent potential, that is, Σ doesn’t depend on frequency but depends on
momentum (note that for this to occur, you need to go beyond the approximation thatconsists in replacing the atom-atom interaction by a contact potential). That dependence
on momentum can often be described by an eﬀective mass. Such an eﬀect would produce ashift, just because the eﬀective mass enters for instance the thermal wavelength. But thisis not the kind of eﬀect that I am interested in here.

A.

Power counting and infrared divergences

We also started discussing the interaction. I remind you that the typical hamiltonian weuse is given as
∫
H=

{
(
)
}
∇2
g †
†
†
d r ψ (r) −ψ(r) + ψ (r)ψ (r)ψ(r)ψ(r) ,
2m
2
3

g=

4πa
.
m

(247)

We argued that the leading order correction to the single particle energies, of order g,
produces no shift in Tc , and we started considering the calculation of the second order
diagram (see Fig. 34). Then I argued that one could expect infrared divergences when all
Matsubara frequencies are equal to zero. So, I’m going to calculate this diagram only in thisparticular case. Let me give you the complete expression, near criticality:
∫
∫
d3 k
d3 q
2 2
Σ(iωn = 0, p) − Σ(0, 0) = −2g T
(2π)3
(2π)3
(
)
1
1
1
× 0
−
,
(k−q − µ0 )(0k − µ0 ) 0p+q − µ0 0q − µ0(248)

Soryushiron Kenkyu

86

k
p+q

p
!n = 0kpqFIG. 34. Second order diagram contributing to the self-energy.

where µ0 is given asκ2cµ ≡ µ − 2gn = −
.
(249)
2m
I’m doing a calculation in second order perturbation theory. But I’m incorporating in0

the chemical potential the shift due to the ﬁrst order diagram. At the mean ﬁeld level,
the condensation takes place in exactly the same way as in the non-interacting system:µ0 is negative until one reaches condensation where it vanishes. The notation with κc isconvenient, as it allows me to express the energy denominators as follows:
0k − µ0 =1 2
(k + κ2c ) ,
2m

(250)

and κc will play the role of an infrared regulator in the integrals that I’m going to calculate.
Within the present perturbative setting, it measures the deviation from criticality.
Let me give you the value of the integral in Eq. (248). This can be calculated analyticallyfor ﬁnite κc :128π 2 ( a )2
Σ(0, p) − Σ(0, 0) =
2m
λ2

{

(
}
(
)2 )
p
3κc
p
1
tan−1
+ ln 1 +
−1 .
p
κc 2
3κc

(251)

This result allows me to do a couple of comments. The quantity Σ(p) − Σ(0) is plotted
schematically in Fig. 35. For small p, p  κc , it goes like p2 , while for large p, p  κc ,it goes like ln p. When p is very large compared to κc , Σ has a logarithmic behavior. And
when p is small compared to κc , Σ is a smooth function which has a regular momentum
expansion. Of course κc is a quantity that we would like to let go to 0, because we want toapproach the condensation. But when κc goes to 0 we get an infrared divergence. That’s
the infrared divergence that I mentioned at the end of the last lecture.
Now I will show you that this peculiar infrared behavior is actually not limited to thesecond order diagram. It will show up in higher orders. We are going to rediscover thepattern that we have already identiﬁed in the case of hot QCD, or in the case of the thermodynamics of the scalar ﬁeld that we discussed a few lectures ago. Let me repeat the analysis
that I did last time for the second order diagram (Fig. 34). Focusing on the region where

Soryushiron Kenkyu

87

FIG. 35. Σ(p) − Σ(0) given by Eq. (251) for small p and large p.

all the momenta are going to zero at the same rate, we regroup the two loop-momenta intoa single 6-dimensional vector K, and write
∫
2

g T

2

∫
a2 2 3K 5 dK
d6 K
3
(2m)
∼
T
m
(K 2 + κ2c )3
m2
K6
κc
∫
dK
1 ( a )2
∼
.
2
m λ
κc K

(252)

This is a two loop diagram, so there are two momentum integrations, hence the d6 K. Then
are three propagators of the form 1/(K 2 +κ2c )2 . And there are a number of coeﬃcients: there
is a factor g 2 coming from the two vertices, and a factor T 2 coming from the two summations
over the Matsubara frequencies ( I keep only the zero Matsubara frequencies, but the factor
T 2 that accompanies the double sum remains). The factor (2m)3 comes from the p2 /2m inthe three propagators. The last line is obtained by remembering that g ∼ a/m, and thatmT ∼ 1/λ2 . Also, to make it easier to see the divergence of the integral when κc → 0,
I have removed κc from the denominators, and put it at the lower end of the integration.
Then I ignored the angular integration, which would produce just a numerical factor. You
see now that in the limit where κc goes to 0 this integral is logarithmically divergent. Also,
it is proportional to (a/λ2 )2 .
Let us now generalize this calculation. I go from this 2-loop diagram to an l-loop diagram.
Let’s see what happens by adding a simple loop as shown in Fig. 36. When I add a loop I add
one vertex and two propagators. One can easily verify that this is generic. The calculationof the l-loop diagram is then going to take the following form:
1 ( a )2
m λ2

∫

d6 K
g l−2 T l−2(K 2 + κ2c )3∫(d3 K)l−2(2m)2(l−2) .
(K 2 + κ2c )2(l−2)(253)

In this calculation I have added l − 2 loops to the 2-loop digram of Fig. 34, in order to get
an l-loop diagram. By doing so, I have added l − 2 loops, l − 2 vertices, hence the factor
g l−2 T l−2 . Then I have an integration (d3 K)l−2 and 2(l − 2) propagators (K 2 + κ2c )−2(l−2) ,Soryushiron Kenkyu

88

FIG. 36. 2+1 loop diagram.

and a factor (2m)2(l−2) . You are still with me? I can rewrite these factors as( a )l−2( a )l−2 ( 1 )l−2
l−2
l−2
l−2 l−2m=.
(g) (T m) m ∼
m
λ2
λ2

(254)

In the integral, I remove κc in the denominator, and put it as a lower bound. After all this,I can rewrite the expression (253) as
1 ( a )lm λ2∫

K 3l−1 dK,
2 2l−1κc (K )

so that the l-loop contribution to Σ reads)l−2(∫
1 ( a )ldK
1 ( a )2 a 1(l)Σ (p = 0) ∼
∼
.
l−1
m λ2
m λ2
λ2 κc
κc K

(255)

(256)

As anticipated, we are indeed confronted to a situation which is very reminiscent to one
that we have already encountered. Namely, as κc → 0 all terms are infrared divergent. I
will show you later that there are good reasons to choose κc of order a/λ2 . But if you dothat, a/λ2 · 1/κc is of order 1. That means that all the diagrams in perturbation theory will
be of the same order of magnitude. Perturbation theory breaks down. It cannot be used
to calculate the shift of the critical temperature. But we know what to do. This is veryanalogous to what we have encountered earlier. And I have shown you in the case of thescalar ﬁeld that we can develop other tools to handle this problem. One of them is eﬀective
ﬁeld theory, to which I now turn.

B.

Eﬀective theory

Infrared divergences occur in Feynman diagrams only when all the Matsubara frequencies
ωn are set equal to 0. Let’s then focus on this particular sector and recall what we did forthe scalar ﬁeld. The eﬀective theory amounts to retain in the ﬁeld expansion∑ψ(τ, r) = T ψ0 (r) + T
e−iωn τ ψn (r),
n6=0

(257)

Soryushiron Kenkyu

89

n=0

n=0

6

0

0

0

6

0

n=0
6

(a)

0

0

0

0

0

0
0

0

(b)

(c)

FIG. 37. Some diagrams contributing to the parameters of the eﬀective theory

only the component ψ0 (r), which does not depend on the imaginary time. As you know
from previous lectures the eﬀective theory for ψ0 (r) will be an eﬀective theory for a ﬁeld in 3
dimensions. One dimension is lost because I have abandoned the imaginary time dependence.
If I just replace the ﬁeld ψ(τ, r) by ψ0 (r), I can perform trivially the integration of theimaginary time in the action, and get:∫∫ 3
Z ∼ Dψ0 e−β d r(H(r)−µN (r)) ,where(
H(r) − µN (r) = ψ0∗ (r)

)
)2
g(
∇2
−
− µ ψ0 (r) +|ψ0 (r)|2 .
2m
2

(258)

(259)

This is the leading order, where the eﬀective action is just β times the energy of the ﬁeld
conﬁguration. This is what I called earlier the classical ﬁeld approximation. We know how
to calculate the corrections to this simple approximation. Remember that I described to you
the systematic procedure to do that. What we have to do is to calculate Feynman diagrams
where the external lines corresponds to ψ0 (r) or ψ0∗ (r) and the internal lines corresponds
to the modes with non-vanishing Matsubara frequencies. Such diagrams are displayed inFig. 37. The diagram of Fig. 37a is proportional to a, and it is a correction to the chemical
potential µ. We shall not need to evaluate this correction because the chemical potential
is eventually adjusted so that the system is critical, or at the phase transition. But there
are other corrections. Let me ﬁrst look at the correction to the coupling constant, thediagram Fig. 37b. This correction is proportional to a2 . Since we are interested in leading
order calculation, it can be ignored. The correction in Fig. 37c represents a three-particle
interaction, and corresponds to a term not present in Eq. (259). This is of order a3 , and canagain be ignored. The lesson of this brief analysis, which we have done more extensively inthe case of the scalar ﬁeld theory, is that if one is interested in the leading order eﬀect of thecoupling constant, one can just use, as an eﬀective theory, the classical ﬁeld approximation,
whose energy density is given by Eq. (259).
However, we are a priori somewhat stuck here, because with this eﬀective theory perturbation theory cannot be used even though the coupling g can as small as one wants.

Soryushiron Kenkyu

90
Let me brieﬂy review that issue, and recall why a priori we should expect a problem with
perturbation theory. When we do perturbation theory, we are assuming in some way that,2

1
in Eq. (259), the kinetic energy 2m(∇ψ0 )2 is big compared to the potential energy g (|ψ0 |2 ) .

Naively we expect that if g is very small, the potential energy is correspondingly small. But
this is not always the case. Perturbation theory breaks down precisely at that particularscale where all the terms in the eﬀective action are of the same order of magnitude, that is,in particular, when (in average)()2
1(∇ψ0 )2 ∼ g |ψ0 |2 .
2m

(260)

Let me repeat here an analysis that we have done in a more general context. Rememberthat we can separate layers of ﬂuctuations corresponding to diﬀerent momenta or diﬀerent
wavelengths. Consider the particular ﬂuctuations of momentum κc , so that(∇ψ0 )2 ∼ κ2c |ψ0 |2 .

(261)

The density in the eﬀective theory here is h|ψ0 |2 i. This is also given by
∫
d3 p
1
2
n ∼ h|ψ0 | i ∼
.
3
(
−µ)/T
p
(2π) e−1(262)

Close to condensation, I can ignore the chemical potential (or a constant self-energy), andreplace (p − µ)/T by p2 /2mT . For long wavelength modes, p2 /2mT is small and I can
expand the exponential factor, and rewrite Eq. (262) as∫
d3 p 2mTn∼.
(2π)3 p2

(263)

(This approximation is in fact in line with the classical ﬁeld approximation.) This integral
diverges at large momentum. But it is supposed to represent only the contribution to thedensity of long wavelength modes, with p . κc . With κc as ultraviolet cut-oﬀ, this gives
2

2

2

κc
κc
n ∝ mT κc . Let’s then compare 2m|ψ0 |2 with g (|ψ0 |2 ) , or equivalently 2mand g|ψ0 |2 , withh|ψ0 |2 i ≈ mT κc . I can rewrite Eq. (260) as
kc2∼ gmT κc .
2m

(264)

Remembering that g ∼ a/m and mT ∼ 1/λ2 , one sees that the condition (260) implies that
κc ∼

a
.
λ2

(265)

This is the characteristic scale that we encountered before and which signals the breakdownof perturbation theory. We see here how this scale emerges from a simple analysis of theeﬀective action.

Soryushiron Kenkyu

91

FIG. 38. General behavior of U (p) as a function of p.

Now, if we cannot expand, what can we do? We can still make progress, because weknow that Eq. (246) is an exact formula for ∆nc , and we can deduce from it an equivalently“exact” formula for ∆nc in the framework of the eﬀective theory. This is very easy to do.
We just ignore all the terms which have n diﬀerent from 0 in Eq. (246). This leads to
{
}
∫
1
d3 p
1
∆nc =
−
.
(266)
(2π)3 0p + Σcl (p) − Σcl (0) 0p
where Σcl (p) ≡ Σ(iωn = 0, p). This formula is “exact” in the eﬀective theory, in the sensethat it does not involve any approximations, beyond those made to arrive at the eﬀective
theory.
I’m going now to introduce a new notation. Let me set
U (p) = 2m (Σcl (p) − Σcl (0)) .
Then, with a little algebra, I can rewrite the formula (266) as
∫ ∞
2
U (p)
∆nc = − 2
.
dp 2
πλ 0
p + U (p)

(267)

(268)

This is still an exact formula (within the eﬀective ﬁeld theory). And if I know the exact
expression for U (p), then I can conclude something. Now, I have to tell you a few morethings about the general behavior of U (p). Before, I have done the calculation with an
infrared regulator: remember that when the momentum was smaller than the regulator,
U (p) was analytic, ∼ p2 . But when one removes the regulator, the behavior is actuallylogarithmic (in leading order), and more generally a power law, p2 + U (p) ∼ p2−η whereη, a small number, is the anomalous dimension (a small number of order 0.1). This powerlaw behavior is characteristic of the scaling regime of a second order phase transition. The
main point at this stage is that the integral in Eq. (268), if the function U (p) behaves as I
indicated, is a ﬁnite integral.

Soryushiron Kenkyu

92
There is more that we can say, namely that U (p) > 0. I do not have a complete,analytical, proof to oﬀer for this property. One may just observe that it is veriﬁed in thesecond order calculation, Eq. (251). The critical ﬂuctuations correct the small momentum
behavior (p2 → p2−η ), but do not alter the fact that U (p) is a growing function of p at
small p. As for the large momentum behavior, it is correctly given by perturbation theory.
Detailed calculations conﬁrm that U (p) is an increasing function of p, with the general shapedisplayed in Fig. 38. Now, if U (p) is positive, so is the integral in Eq. (268). Therefore ∆nc
is negative, which implies that ∆Tc is positive. This answers the question concerning thesign of ∆Tc .
The last thing we have to do is to show that the shift of ∆nc is linear in a. This canbe done via a simple analysis. Let me ﬁrst remark that the eﬀective theory which I havewritten here in principle requires an ultra-violet cut-oﬀ. You have seen that already from thecalculation of the density in Eq. (263): if you want to calculate the density with a statisticalfactor of the form 2mT, you need an ultra-violet cut-oﬀ. This is in line with the philosophyp2of the eﬀective theory. Remember what we did in the scalar ﬁeld theory: we introduced anintermediate scale, i.e., a separation scale Λ, and we integrated the modes above this scale Λto get an eﬀective theory valid below that scale. Of course it may happen that the eﬀective
theory contains ultra-violet divergences, and these divergences are in principle compensatedby the cut-oﬀ dependence of the coeﬃcients of the eﬀective theory. Here you don’t have toworry about such issues because, as I argued, the density enters mostly the correction to thechemical potential. Observe also that in the integral of Eq. (266) we are taking the diﬀerenceof two contributions, and this diﬀerence is ultra-violet ﬁnite. But there are cases where weneed a cut-oﬀ Λ. What is the scale of this cut-oﬀ? To answer that question we need toask ourselves ﬁrst why we get ultra-violet divergences in the eﬀective theory. The answer
to that is contained in Eq. (262). The full statistical factor there, 1/(e(p −µ)/T − 1), kills all
the momenta which are bigger than the temperature. But we are using an approximation√
to the statistical factor that is valid only at small momenta, i.e., for p . mT , so that the√natural cutoﬀ for the eﬀective theory is of order mT ∼ 1/λ.
Now the eﬀective theory is a 3-dimensional theory, and is “super renormalizable”. What itmeans is that the only ultraviolet divergence is the one that I have discussed. It corresponds
to a correction to the chemical potential (or to the mass, in the language of ﬁeld theory). As
I have argued, such a correction to the chemical potential is innocuous because the chemical
potential is adjusted to be at criticality. Let us then go through some dimensional analysis.
As you’ve seen, there is a natural momentum scale in the problem which is a/λ2 . So let us
set p = x λa2 (with x a dimension-less variable). In principle, quantities calculated within the

Soryushiron Kenkyu

93
eﬀective theory depend also on an ultra-violet cut-oﬀ Λ ∼ 1/λ. Thus I can write( a )2 ( a 1 )aσ x, 2
U (p = x 2 , Λ) =
,
λ
λ2
λ Λ

(269)

where λa2 Λ1 ∼ a/λ if Λ ∼ 1/λ. In fact, since the theory is super renormalizable, I can let the
cut-oﬀ Λ go to inﬁnity, and get a ﬁnite result. But the fact that I can let Λ go to inﬁnity
does not guarantee that the results will not depend on Λ, if it is kept ﬁnite. Finite cutoﬀ
corrections will be truly negligible only when a/λ is suﬃciently small. In that case, I canreplace Eq. (269) by( a )2aU (p = x 2 , Λ) 'σ(x) ,λλ2(270)where σ(x) is a universal function of x (that is, independent of a).
Then I can rewrite ∆nc as
2 a∆nc ' − 2 2
πλ λ

∫ ∞
0

dx xσ(x),x x2 + σ(x)(271)

where I have extracted the logarithmic integration measure to emphasize that the variationsof the integrand are best visualized on a logarithmic scale (see Fig. 40 below). The integral
is just a number, so that ∆nc is proportional to a. Basically, the dependence in a justfollows from dimensional considerations, once we have made sure that possible ultraviolet
cutoﬀ eﬀects play no role (which, as we have argued, requires a/λ to be small enough).
At this point, it is useful to brieﬂy review the steps that led to this result. The ﬁrst stepis to recognize that perturbation theory doesn’t work. Naively one could think of expandingin powers of a, because a is very small quantity. But we have seen that this doesn’t work
because the calculation of Feynman diagrams is plagued with infrared divergences. We
have exploited the fact that these divergences occur when all the Matsubara frequencies
are vanishing. Then, by relying on what we did earlier, I explained to you how one candeal with all the Feynman diagrams at once, by constructing an eﬀective theory. In thisparticular case, the eﬀective theory is extremely simple because it just amounts to replace inthe original action the bosonic ﬁeld, which is a function of three space coordinates and theimaginary time, by a ﬁeld which is independent of time and depends only on three spatial
coordinates. This leads to a three dimensional eﬀective ﬁeld theory, which in principal allows
us to calculate ∆nc .
The eﬀective theory makes obvious the reasons for the breakdown of perturbation theory:
there exists a typical scale of ﬂuctuations at which kinetic energy and potential energy areof the same order of magnitude. Therefore, in order to calculate within this eﬀective theory,
we have to use tools other than perturbation theory, even when the coupling is small.

Soryushiron Kenkyu

94
But even without doing explicit calculations, we have been able to extract from theeﬀective theory the answers to the questions that we are addressing. Namely the fact that(a) ∆nc is negative or ∆Tc is positive and (b) ∆nc is linear in a.
Q: You take some kind of classical ﬁeld approximation somewhere?
A: Yes. The eﬀective theory that I have discussed is what I called the classical ﬁeld approximation. I have argued that Feynman diagrams are divergent when all Matsubara
frequencies are vanishing. Earlier in the lectures I have shown that we can construct
an eﬀective ﬁeld theory to handle systematically all these Feynman diagrams at once.
This eﬀective theory is in general an inﬁnite series in local operators, such as φ4 , φ6 ,φ8 , φ2 (∇φ)2 , etc. An important point is that the coeﬃcients can be calculated inperturbation theory. If you are interested in leading order, then the leading terms
in the expansion are enough, and this leads to the classical φ4 theory in 3 dimension(with O(2) symmetry). So, in the particular context of the present problem, things are
extremely simple because the eﬀective theory is just the leading order term. Namely
you just take the initial action, which is
(
)
∫ β ∫
∇2
3
∗
S=
dτ d r ψ (r, τ ) −ψ(r, τ ) + · · · ,
2m
0

(272)

and replace ψ(r, τ ) by ψ0 (r), to get∫
S=β(d r ψ0∗ (r)
3

∇2
−
2m

)
ψ0 (r) + · · · .

(273)

Q: Is that a classical ﬁeld approximation? The Heisenberg ~ goes away by this procedure.
A: Yes. If you wish. Another way to think about this approximation is as a high temperature (small β) approximation. In any case, this approximation allows me to do
calculations outside the framework of perturbation theory, but still in leading order ina. That is to say, all correction to the eﬀective theory are order of a2 or higher (these
corrections may for instance induce corrections of order a2 log a to ∆Tc ).
Q: In the ratio of ∆Tc /Tc ∝ n1/3 , both sides are dimensionless. But λ depends on T . If
you take the ratio, λ disappears or not?
A: If you go through the calculation you will see that the factor n1/3 can be read as
1/λ. And indeed λ depends on T . But the temperature is ﬁxed in the calculation of∆Tc /Tc . That is, λ is here λc which can be estimated by taking the critical temperaturein the absence of interaction. So this temperature dependence does not spoil the linear
relation between ∆Tc /Tc and a.

Soryushiron Kenkyu

95

FIG. 39. Self-energy in the large N expansion

Q: OK. Thank you.
Let me tell you a bit more before ending this lecture. We have established the relation()∆Tc= c an1/3 ,
Tc

(274)

in which c is a positive constant, given by an integral of the form
∫ ∞
σ(x)
.
c∝
dx 2
x + σ(x)0(275)

I remind you that σ(x) is essentially Σcl (p) − Σcl (0). So, provided you know how to calculate
the self-energy and its momentum dependence, then in principle we can calculate c. The
problem is to calculate σ(x).
To do so, we must use non perturbative techniques, of which there are not so many. The
ﬁrst technique that I want to discuss is the large N approximation. The idea is to replace theﬁeld ψ0 (r), which is a complex ﬁeld written as φ1 (r) + iφ2 (r) = (φ1 , φ2 ), by an N -component~
vector φ(r)= (φ1 , φ2 , · · · , φN ). Then we can write an action
{

∫
S=

d3 r

(

∇2
~
φ(r)
−
2m

)

g
~
φ(r)
+
2

((

~
φ(r)

) 2 )2},(276)

which is invariant under O(N ) transformations. The usefulness of this strategy is thatan analytic calculation is possible when N → ∞. For the calculation of the self-energy,
this amounts essentially to resum the chain of bubbles in Fig. 39 (this is actually a 1/N
correction; the leading order correction is the mean ﬁeld correction that leads to a momentum
independent self-energy). If you remember what I said in the last lecture, this chain of
bubbles is what produces screening of the long wavelength density ﬂuctuations. Because ofthis screening, the resulting calculation of the self-energy is infrared ﬁnite. The calculationof the self-energy in this order can be done analytically, and yields c = 2.3.
Another possible strategy is to do a “brute force” lattice calculation, as you do in QCD
for instance. This leads to the most accurate determination of c. The results obtained
almost simultaneously by two groups are c = 1.32 ± 0.02 [18] , and c = 1.29 ± 0.05 [19].

Soryushiron Kenkyu

96

FIG. 40. Schematic behavior of the integrand in Eq. (271).

After these results were obtained, the calculation of c has become a play ground fortesting various approximation in ﬁeld theory or many body physics (see for instance [14] forreferences).
Perhaps before closing I should tell you one word about why it is hard to calculate σ(x).
The reason can be understood from looking at the plot of the integrand in Eq. (271). As canbe seen on Fig. 40, the integrand is peaked around x ∼ 1, that is at the momentum p ∼ κc ∼a/λ2 , which delineates the frontier between two very distinct regimes: At momenta smaller
than κc , we have the critical regime, well described by the theory of critical phenomena
which allows in particular a precise determination of the anomalous dimension η. This iswell under control. The other regime, that of large momenta, is also very well under control
because in this regime perturbation theory can be applied (perturbation theory is accurate
at large momenta in a three dimensional scalar ﬁeld theory). But the quantity which weneed is sensitive to what happens at the border line between these two regimes. What weneed to get with accuracy is the precise point where this transition between the two regimes
occurs. And this is hard. That is why we need non-trivial non-perturbative techniques todo the explicit calculation (and why simple methods, like the large N expansion, are oﬀ
quantitatively by almost a factor two).
What was going to come after that was a discussion of this problem, from the perspectiveof the non-perturbative (or “exact”) renormalization group [14]. But that will be for anotheroccasion... as the time for these lectures is over.

Soryushiron Kenkyu

97
Acknowledgements. I thank the Physics Department of the University of Tokyo for
hospitality during the winter 2009 when these lectures where delivered. I am very gratefulto Tesuo Matsui, for his invitation, and for taking the initiative of recording the lectures.
Special thanks are due to the students who produced the transcript of the recorded lectures.
I also gratefully acknowledge the contributions of Professors Hirotsugu Fujii and Yusuke
Kato to the pre-editing of the transcribed text. Clearly, without all this help these notes
would not exist. I also gratefully acknowledge the hospitality of the Physics Departmentof Nanjing University, where I had a chance to give similar lectures in March 2011, and towork there on the ﬁnal version of these notes.

[1] L. Kadanov and G. Baym, Quantum Statistical Mechanics, (Benjamin/Cummings, London,1962).
[2] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in
Statistical Physics, (Dover, New-York, 1963).
[3] A. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems, (McGraw Hill, NewYork, 1971).
[4] J.P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, (MIT Press, Cambridge, 1986).
[5] J.I. Kapusta, Finite Temperature Field Theory, Cambridge Monographs in Mathematical
Physics, (Cambridge University Press, 1989).
[6] J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications,(Cambridge, UK: Univ. Press, 2006).
[7] M. Le Bellac, Thermal Field Theory, Cambridge Monographs in Mathematical Physics, (Cambridge University Press, 1996).
[8] J. P. Blaizot, “Quantum ﬁelds at ﬁnite temperature: A brief introduction,” Prepared forCargese Summer School on QCD Perspectives on Hot and Dense Matter, Cargese, France,
6-18 Aug 2001.
[9] J. P. Blaizot, E. Iancu and A. Rebhan, “Thermodynamics of the high-temperature quark
gluon plasma,” arXiv:hep-ph/0303185.
[10] J. P. Blaizot, “The quark gluon plasma and nuclear collisions at high-energy,” Les Houches
Summer School On Theoretical Physics, Session 66: Trends In Nuclear Physics, 100 Years
Later, 30 Jul - 30 Aug 1996, Les Houches, France.
[11] J. P. Blaizot, “QCD at ﬁnite temperature,” Prepared for Les Houches Summer School in
Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, Les
Houches, France, 28 Jul - 5 Sep 1997.

Soryushiron Kenkyu

98[12] J. P. Blaizot, “Theory of the quark-gluon plasma,” Lect. Notes Phys. 583 (2002) 117[arXiv:hep-ph/0107131].
[13] J. P. Blaizot and E. Iancu, “The quark-gluon plasma: Collective dynamics and hard thermalloops,” Phys. Rept. 359 (2002) 355 [arXiv:hep-ph/0101103].
[14] J. P. Blaizot, “Non Perturbative Renormalization Group and Bose-Einstein Condensation,”
arXiv:0801.0009 [cond-mat.stat-mech].
[15] J. O. Andersen, L. Kyllingstad, L. E. Leganger, JHEP 0908 (2009) 066 [arXiv:0903.4596[hep-ph]].
[16] K. Kajantie, M. Laine, K. Rummukainen, and Y. Schroder, Phys. Rev. D 67 (2003) 105008.
[17] G. Baym, J-P. Blaizot, M. Holzmann, F. Laloë and D. Vautherin, Phys. Rev. Lett. 83, 1703
(1999).
[18] P. Arnold and G. Moore, Phys. Rev. Lett. 87, 120401 (2001).
[19] V.A. Kashurnikov, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. 87, 160601 (2001).