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Thermodynamics of the high temperature
Quark-Gluon Plasma
J.-P. BLAIZOT
Service de Physique Théorique, CEA/DSM/SPhT,
Unité de recherche associée au CNRS
91191 Gif-sur-Yvette Cedex, France
Abstract.
These lectures are a brief introduction to some of the weak coupling techniques used to decribe
the physics of the quark-gluon plasma at high temperature.
INTRODUCTION
Because of the asymptotic freedom of QCD, one expects matter at high temperature
and/or high density to become simple: a weakly interacting quark-gluon plasma. The
most compelling theoretical evidences for the existence of such a quark-gluon plasma
are coming from lattice gauge calculations (for recent reviews see e.g. Refs. [1, 2]).
These are at present the unique tools allowing a detailed study of the transition region
where various interesting phenomena are taking place, such as colour deconfinement
or chiral symmetry restoration. In these lectures, we shall not consider this transition
region, but focus rather on the high temperature phase, exploiting the fact that at sufficiently high temperature the effective gauge coupling constant should be small enough
to allow for weak coupling calculations.
The physical picture which emerges from these weak coupling calculations is a
simple one, and in many respect the quark-gluon plasma is very much like an ordinary
electromagnetic plasma in the ultrarelativistic regime (with however specific effects
related to the nonabelian gauge symmetry [3, 4, 5]). To zeroth order in an expansion
in powers of the coupling g, the quark gluon plasma is a gas of noninteracting quarks
and gluons, with typical momenta k ∼ T . This is the ideal quark-gluon plasma. The
interactions appear to alter only slightly this simple physical picture: they turn plasma
particles into massive quasiparticles, and generate collective modes at small momenta.
However, while the relevant degrees of freedom have been identified long ago, and
effective theories to describe their dynamics are known[3, 6, 7, 8, 9, 10, 11, 12, 13],
the calculation of the thermodynamics of the quark-gluon plasma using weak coupling
techniques has remained unsuccessful until recently.
Strict expansions in power of the coupling constant have been pushed up to order
g5 , both in QCD and in scalar field theories. In both cases, one observes a rather poor
apparent convergence, the successive contributions oscillating wildly unless the coupling
is very small and the deviation from ideal-gas behaviour correspondingly minute. As we
shall see, the difficulty may be related to the fact that what determines the accuracy
of a weak coupling calculation in thermal field theories is not only the strength of the
coupling, but also the magnitude of the thermal fluctuations. These vary according to the
relevant momentum scales, so that the accuracy of the weak coupling expansion depends
on which momentum scale contributes dominantly to the quantity under consideration.
In these lectures, after a brief introduction of QCD and its phase diagram, I recall some
techniques of quantum field theory at finite temperature needed to treat the interactions.
I discuss the difficulties of strict perturbation theory when applied to the calculation of
the termodynamical functions. However, weak coupling techniques are not limited to
perturbation theory. To shed light on the difficulty met in perturbative calculations, I
then proceed to an analysis of the various important scales and degrees of freedom of
the quark-gluon plasma and focus on the effective theory for the collective modes which
develop at the particular momentum scale gT , where g is the gauge coupling and T the
temperature. A powerful technique to construct the effective theory is based on kinetic
equations which govern the dynamics of the hard degrees of freedom [14]. Some of the
collective phenomena that are described by this effective theory are briefly mentioned.
Then I turn to the calculation of the entropy and show how the information coded in the
effective theory can be exploited in (approximately) self-consistent calculations. Finally,
I present some selected results.
Now, a word of warning concerning these notes: they contain parts which are a
succinct editing of the oral presentation. Other parts use published material. In order
to keep the discussion elementary, explicit treatments are presented only in the case of
the scalar field. For QCD only selected results are given at the end. Useful complements
to these lectures, for the parts dealing with general aspects of the problems discussed
here, can be found in refs. [14, 15, 16, 5, 17, 18].
QCD, SYMMETRIES AND PHASE DIAGRAM
This section is a short summary of basics of QCD.
The QCD lagrangian is
1
L = − TrFµν F µν + ∑ ψ̄ f (iD
/ − m f )ψ f ,
2
f
(1)
Dµ = ∂µ − igAµ
(2)
where
Fµν = i[Dµ , Dν ],
or, making more explicit the color structure of the gauge field Aµ and the associated field
strength tensor Fµν ,
Aµ = Aaµ t a
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + g f abc Abµ Acν .
(3)
∑ f is a sum over the N f quark flavors.
The QCD Lagrangian has several symmetries relevant to the discussion of the phase
diagram:
FIGURE 1. The thermodynamical functions in SU(3) gauge theory. From [20].
• Local color (gauge) symmetry
• Global SU(N f ) × SU(N f ) chiral symmetry (when m f = 0)
• Conformal symmetry (scale invariance, when m f = 0)
Chiral symmetry is broken in the vacuum, but restored at finite baryon density and
finite temperature.
Some consequences of these symmetries, such as the conservation of local currents,
may not survive at the quantum level because of anomalies.
An essential property of QCD, which determines its thermodynamics at asymptotically high temperatures, is that of asymptotic freedom: the coupling constant g (or
α = g2 /4π) becomes weaker and weaker as the temperature increases, roughly as
α(T ) ∼
1
.
ln(T /ΛQCD )
(4)
A simple model where one can verify explicitly that the temperature is indeed the
relevant scale which determines the running of α in the calculation of thermodynamical
functions is the Gross Neveu model which is asymptotically free in one dimension [19].
A natural implication of asymptotic freedom is that matter becomes simple at high
T : At very large temperature quarks and gluons form a weakly interacting quark-gluon
plasma. That the thermodynamics functions go over to those of free quarks and gluons at
high temperature is well established by lattice calculations. An illustration is provided in
fig. 1. At very high baryonic density, the baryon chemical potential plays the same role as
the temperature in eq. (4) and determines the running of the coupling: one then expects
quark-matter to turn into a weakly interacting fermi gas at high density. However, this
is not so because of the familiar BCS instability: quark-matter becomes instable against
the formation of Cooper pair, the attractive interaction being the one-gluon exchange in
the 3̄ channel (for a review see [21] ).
In these lectures, we shall focus on the thermodynamical properties at zero baryon
density. We shall argue that weak coupling techniques can lead to accurate calculations
of thermodynamical functions down to temperatures T >
∼ 3Tc . We shall see that these
calculations suggest that the dominant effect of the interactions is well accounted for in
T
Quark-Gluon Plasma
Hadronic matter
Colour
superconductor
Nuclei
µB
FIGURE 2. A schematic picture of the QCD phase diagram
a quasiparticle picture. However, as one approaches the phase transition, new degrees of
freedom may become important.
Hadronic matter and quark-gluon plasma are separated by a phase transition or a
rapid cross-over, and the thermodynamical functions in the vicinity of this transition can
only be calculated by lattice techniques. These techniques provide robust predictions at
µB = 0, and recent progress have been made for finite (but small) µ.
An illustration of the expected phase-diagram is given in fig. 2
QUANTUM FIELDS AT FINITE TEMPERATURE
We briefly recall here some essential formalism needed to deal with quantum field theory
at finite temperature [22, 23, 24, 25, 26, 27].
We do this by studying the thermodynamics of a scalar field with Lagrangian
L
=
1
m2
∂µ φ ∂ µ φ − φ 2 −V (φ ).
2
2
(5)
All thermodynamic observables can be deduced from the partition function:
Z = tr e−β H ,
(6)
where β ≡ 1/T , with T the temperature. Here are some examples. Energy density and
pressure (V is the volume):
ε =−
1 ∂
ln Z
V ∂β
P=
1 ∂
T
ln Z = ln Z .
β ∂V
V
(7)
Conserved charges, and associated susceptibilities:
∂ ln Z
hNi i = T
∂ µi
∂ ni
∂ 2P
χi j =
=
.
∂ µ j ∂ µi ∂ µ j
(8)
In order to calculate the partition function, one may observe that e−β H is like an
evolution operator in imaginary time:
t → −iβ
e−iHt → e−β H ,
(9)
and use approximation schemes developed for the evolution operator.
For instance one may use the perturbative expansion. We assume that one can split
the hamiltonian into H = H0 + H1 with H1 H0 , and define the following “interaction
representation” of the perturbation H1 :
H1 (τ) = eτH0 H1 e−τH0 .
(10)
Using standard techniques (see e.g. [25]), one obtains
Zβ
−β H
−β H0
e
=e
T exp −
dτH1 (τ) ,
(11)
from which one gets the following expression for the partition function Z :
Zβ
Z = Z0 hT exp −
dτH1 (τ) i0 .
(12)
0
0
The symbol T denotes time ordering, Z0 = tr e−β H0 and, for any operator O,
!
e−β H0
hOi0 ≡ Tr
O .
Z0
(13)
In field theory, it is often more convenient to use the formalism of path integrals
rather than the operator formalism. Let us recall that for one particle in one dimension
the matrix element of the evolution operator can be written as
−iHt
hq2 |e
|q1 i =
Z q(t)=q2
q(0)=q1
D (q (t))
ei
R t2 1
2
t ( 2 mq̇ −V (q))dt
1
.
(14)
Changing t → −iτ, and taking the trace, one obtains the following formula for the
partition function:
Z β
Z
1 2
−β H
Z = tr e
=
D(q) exp −
mq̇ +V (q)
.
(15)
2
q(β )=q(0)
0
This expression immediately generalizes to the case of a scalar field, with Lagrangian
given by eq. (5) which we rewrite in the form, isolating the time derivative:
1
1
m2
L = (∂t φ )2 − (∇φ )2 − φ 2 −V (φ ).
2
2
2
(16)
The corresponding partition function reads:
Zβ Z
Z
1
1
m2 2
3
2
2
Z = D(φ ) exp −
dτ d r
(∂τ φ ) + (∇φ ) + φ +V (φ )
,
2
2
2
0
(17)
where the integral is over periodic fields: φ (0) = φ (β ).
Remarks. i) At high temperature, β → 0, the time dependence of the fields play no role.
The partition function becomes that of a classical field theory in three dimensions:
Z
Z
1
m2 2
3
2
Z = D(φ ) exp −β d r
(∇φ ) + φ +V (φ )
.
2
2
ii) Note the Euclidean metric in (17). Since the integrand is the exponential of a negative
definite quantity, it is well suited to numerical evaluations, using for instance the lattice
technique.
An important feature of the path integral representation of the partition function is
the boundary conditions to be imposed on the fields over which one integrates. For the
scalar case considered here, the field has to be periodic in imaginary time, with a period
β . Similar conditions hold for the fermion fields, which are antiperiodic in imaginary
time, with the same period β .
The propagator of the free scalar field ∆(τ) = hTφ (τ1 )φ (τ2 )i, where τ ≡ τ1 − τ2
satisfies the differential equation
2
−∂τ1 − ∇21 + m2 ∆(τ1~r1 ; τ2~r2 ) = δ (τ1 − τ2 )δ (~r1 − ~r2 ),
(18)
and obeys periodic boundary conditions. It admits the Fourier representation
∆(τ) =
1
e−iων τ ∆(iων ),
β∑
ν
(19)
where ων = 2πν/β , with ν an integer (ων is called a Matsubara frequency), and
∆(iων ) =
1
εk2 − ων2
,
(20)
√
where εk = k2 + m2 is the energy of a mode with momentum k. The sum over Matsubara frequencies in eq. (19) can be done explicitly and yields:
o
1 n
∆(τ) =
(1 + Nk )e−εk |τ| + Nk eεk |τ| ,
(21)
2εk
with Nk = 1/(eβ εk − 1).
1.04
1.02
¼
1
0.98
0.96
0.94
0.92
0.2
0.4
0.6
0.8
FIGURE 3. Weak-coupling expansion through orders g2 , g3 , g4 , and g5 for the pressure normalized to
that of an ideal gas as a function of g(2πT ) in φ 4 theory.
PERTURBATION THEORY AND ITS DIFFICULTIES
The weak–coupling expansion of the pressure in the case V (φ ) = λ φ 4 /4!, m = 0, has
been computed to order λ 5/2 (or g5 ), and reads[28, 29, 30]:
"
15 g 2 15 g 3 135
µ̄
g 4
log
+ 0.4046
P = P0 1 −
+
+
8 π
2 π
16
2πT
π
#
µ̄
4
g
g 5
405
log
− log − 0.9908
+ O(g6 log g) ,
(22)
−
8
2πT 3
π
π
where P0 = (π 2 /90)T 4 is the pressure of an ideal gas of free massless bosons, and
g2 (µ̄) ≡ λ (µ̄)/24 is the MS coupling constant at the renormalization scale µ̄. This
formula calls for several remarks.
First we note that, to order λ 5/2 , P is formally independent of µ̄ (in the sense that
dP/d µ̄ involves terms of order λ 3 at least). This is easily verified using the renormalization group equation satisfied by the renormalized coupling λ (µ̄):
µ̄
dλ
λ2
=3
+O λ3 .
2
d µ̄
16π
(23)
But the approximate expression (22) for P does depend numerically on µ̄. In order to
avoid large logarithms ∼ ln(µ̄/2πT ) in the expansion (22) one may choose µ̄ ' 2πT.
With this choice λ (µ̄) becomes effectively a function of the temperature. In the case
of QCD where the β -function is negative, this leads to the expectation that the effective
coupling becomes small at large temperature.
In Fig. 3, we show the successive perturbative approximations to P/P0 as a function
of g(2πT ). Each partial sum is shown as a band obtained by varying µ̄ from πT to
4πT , which gives an indication of the theoretical uncertainty in the calculation. Thus, in
drawing this plot, g(µ̄) has been calculated from g(2πT ) by using the renormalization
group equation (23). The lack of convergence of the weak-coupling expansion is evident
in Fig. 3. We can infer from Fig. 3 that the expansion in powers of g ceases to make
sense as soon as g(2πT ) & 0.4.
Another remark concerns the presence of terms non–analytic in the coupling constant
λ , i.e., terms of order g3 ∼ λ 3/2 , g5 ∼ λ 5/2 , and also terms involving ln g, in the
perturbative expansion in eq. (22). As we shall see shortly, these terms result from
infinite resummations of subsets of diagrams of perturbation theory.
At this stage, it is instructive to compare with the corresponding expansion of the
screening “Debye” mass mD , defined from the the static propagator by
p2 + m2D + Π(0, p) = 0
at p2 = −m2D ,
(24)
where Π(p0 , p) is the self–energy. For the massless case (m = 0), the weak coupling
expansion of m2D is known to order g4 , and reads[30]:
n
o
g 9
4 g
µ̄
g 2
2
2 2
mD = g T 1 − 3 − ln
− ln − 2.415
+ O(g3 ) .
(25)
π 2
2πT 3 π
π
Note that, with our notation g2 ≡ λ /24 for the coupling constant, the leading–order
contribution in eq. (25) is simply
m̂D = gT ,
(26)
which happens to be the same relation as in pure-glue QCD to leading order. As in the
case of the pressure, non–analytic terms (g3 , ln 1/g) occur in the expression of m2D .
In Fig. 4, we show the screening mass mD normalized to the leading order result m̂D
as a function of g(2πT ), for each of the three successive approximations to m2D . As
in Fig. 3, the bands correspond to varying µ̄ from πT to 4πT . The poor convergence
is again evident, with a pattern similar to that in Fig. 3: a large difference between
the order-g2 and order-g3 approximations, and a larger sensitivity to the value of the
renormalization scale for the order g4 .
In order to get some insight on the peculiar behaviour of the expansions (22) and (25),
we shall briefly recall how the various terms are obtained, up to order g3 .
We start with the zeroth order calculation. We have P0 = −Ω0 /V , with
Ω0
1
=
T
V
2 ∑
n
1
=
2
Z
Z
d3 k
ln(ωn2 + k2 + m2 )
(2π)3
d3k
εk + T
(2π)3
Z
d3k
log(1 − e−εk /T ),
(2π)3
(27)
where ωn = 2nπT is a bosonic Matsubara frequency and εk2 = k2 + m2 . The first term
in the second line of eq. (27) is the sum of the ‘zero-point’ energies of free massive
bosons. This is ultraviolet divergent, but independent of the temperature; it can therefore
be absorbed in the redefinition of the vacuum energy, and discarded. The second term is
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
FIGURE 4. Weak-coupling expansion to orders g2 , g3 , and g4 for the screening mass normalized to the
temperature as a function of g(2πT ). From [31].
temperature–dependent and finite; this is recognized as minus the pressure of the ideal
gas of scalar particles. For m T this can be expanded as:
P0 (m) = −T
Z
d3k
π 2 T 4 m2 T 2 m3 T
−β εk
−
+
+...,
ln(1
−
e
)
=
(2π)3
90
24
12π
(28)
where the neglected terms start at order m4 ln(m/T ).
FIGURE 5. Lowest order correction to the self-energy in scalar φ 4 theory.
Consider now the contributions of order g2 to the self–energy and the thermodynamic
potential, denoted as Π2 and Ω2 , respectively. Π2 is given by the “tadpole” diagram of
Fig. 5. This gives:
Π2
λ
=
T
2 ∑
n
λ
=
2
Z
Z
d3 k
1
λ
≡ I(m)
3
2
2
2
(2π) ωn + k + m
2
d 3 k 1 + 2n(εk ) λ
≡ [I0 (m) + IT (m)] ,
(2π)3
2εk
2
(29)
where n(ω) is the Bose-Einstein thermal distribution:
n(ω) =
1
.
−1
eβ ω
(30)
The first term in the second line of eq. (29) is ultraviolet divergent, but independent of
the temperature, so it can be absorbed into the definition of the (zero–temperature) mass.
FIGURE 6. The lowest order correction to the thermodynamic potential in φ 4 theory.
For instance, is we choose the mass parameter m in the Lagrangian to be the physical
mass at T = 0, we must add a counterterm 12 δ m2 φ 2 in the Lagrangian, with δ m2 chosen
so that:
λ
I0 (m) + δ m2 = 0.
2
(31)
Then, the lowest–order self–energy correction due to the thermal fluctuations reads:
λ
m2
λ T 2 mT
m2
Π2 = IT (m) ≈
ln
−
−
+ 2γ − 1
,
(32)
2
2 12 4π 16π 2
(4πT )2
where the expansion in the r.h.s. holds when m T . In particular, when m = 0, we
recover the result Π2 = m̂2D , cf. eqs. (25)–(26). This quantity is generally dubbed a “hard
thermal loop” (HTL), because the dominant momenta in the one–loop integral (32) are
hard (k ∼ T ).
The contribution Ω2 is given by the 2–loop diagram in fig. 6, and reads:
λ
1
λ 2
Ω2
= hφ 4 i0 + δ m2 hφ 2 i0 =
IT (m) − I02 (m) ,
V
4!
2
8
(33)
where we have also included the contribution of the mass counterterm and used eq. (31)
to verify that the temperature–dependent ultraviolet divergences cancel out, as they
should. The second term in the last line in eq. (33) is divergent, but contributes only
to the vacuum energy density, so that it can be discarded. For m = 0, eq. (33) yields:
P2 = −
λ
15 g 2
T4 = −
P0 ,
1152
8 π
(34)
in agreement with eq. (22).
At the next order in the loop expansion, the self–energy receives contributions from
2–loop diagrams such as the one displayed in Fig. 7. This particular contribution is
infrared divergent in the massless limit m = 0. Indeed, the upper loop (together with
the corresponding mass counterterm, not shown explicitly in Fig. 7) is recognized as
the “hard thermal loop” computed previously. Thus, this particular 2–loop contribution
reads:
Z
λ
d3 k
m̂2D
(a)
Π2L = − T ∑
.
(35)
2 n
(2π)3 (ωn2 + k2 )2
The three-dimensional integral over k in eq. (35) has a linear infrared divergence, coming
from the term with ωn = 0. (For the other terms, the Matsubara frequency ωn 6= 0 acts
effectively as an infrared cutoff.) A non–zero mass m would cut off this divergence, and
give a contribution ∼ λ m̂2D (T /m) ∼ g4 T 3 /m. As we shall see shortly, when m gT , a
physical cut-off is actually generated dynamically by the thermal fluctuations in the form
(a)
of a “thermal mass” m̂D = gT . Then Π2L ∼ g4 (T 3 /m̂D ) ∼ g3 T 2 . This is how nonanalytic
terms arise in eqs. (22) and (25): potential infrared divergences are cut-off by ‘thermal
masses’ which, through their dependence on g, modify the original expansion in powers
of the coupling constant.
To see this more explicitely, we carry out again the calculation of the diagram in
Fig. 5, but this time with the dressed propagator
D̂(iωn , k) ≡
1
ωn2 + k2 + m̂2D
.
(36)
This corresponds to resumming the diagrams of Fig. 7 (each of the inserted tadpoles in
Fig. 7 is accompanied by a mass counterterm which removes its ultraviolet divergence,
cf. eq. (31)).
..
FIGURE 7. Diagrams that are resummed when computing the tadpole in Fig. 5 with the dressed
propagator in eq. (36).
Since m̂D = gT , this resummation involves arbitrarily high powers of g. However, the
final result can be read from eq. (32) in which we set m = m̂D = gT :
Πtadpole = g2 T 2 −
3 3 2
g T + O(g4 ln g) .
π
(37)
This formula exhibits the next–to–leading order contribution to the thermal mass, of
order g3 , consistent with eq. (25). This order–g3 contribution comes from the term linear
in m in the r.h.s. of eq. (32). It thus is related to the non–analyticity of the thermal integral
IT (m) as a function of m2 and to the infrared divergences which occur when one attempts
to evaluate this integral after expanding the integrand in powers of m2 .
The infrared divergences are easily identified in the sum of the diagrams in Fig. 7 with
at least two loops:
Z
λ
d3 k
1
1
Πring =
T
− 2
2 ∑
(2π)3 ωn2 + k2 + m̂2D
ωn + k2
n
l
Z
d3 k
1
−m̂2D
2
= 12g T ∑
(38)
∑ ωn2 + k2 ,
(2π)3 ωn2 + k2 l≥1
ωn
where the subtracted term is the massless tadpole. The term with l = 1 corresponds to the
2–loop diagram that we have considered separately in eq. (35). Each of the terms with
ωn2 > 0 in eq. (38) starts to contribute at order g2 m̂2D ∼ g4 . But the terms with ωn = 0 are
individually infrared divergent for any l, and the divergences cancel only in thwe sum of
all the terms. If, in eq. (38), we separate the contribution the static mode, i.e., the mode
with zero Matsubara frequency, we obtain w
Z
1
d3 k
1
3
(3)
2
Πring = 12g T
− 2 = − g3 T 2 ,
(39)
2
3
2
(2π)
k
π
k + m̂D
where we recognize the order g3 contribution in eq. (37).
It is instructive to look at this contribution from another perspective. As we have
argued before, the dominant contribution to IT (m) in eq. (32) comes from momenta
k ∼ T . The subleading contribution is coming from soft momenta k T . To isolate it,
we subtract the hard modes contribution, which amounts to subtract IT (0) from IT (m) in
eq. (32), and write:
Z
d3k
1
1
m̂D T
IT (m) − IT (0) ' T
− 2 =−
.
(40)
2
3
2
(2π) k + m̂D k
4π
After multiplication by λ /2 = 12g2 , one recognizes the g3 term of eq. (37). In writing
the approximate equality above, we have used the fact that the dominant contribution in
the integral (40) comes from momenta k T , so that we can use
n(εk ) ≈
T
εk
for
εk T .
(41)
Note that the above resummation of the thermal mass, focusing on the static mode, hides an
ultraviolet problem that we have ignored so far. Namely, in order to obtain the finite result
in eq. (37), we had to cancel an ultraviolet divergence, cf. eq. (31), which now involves
the thermal mass, and is therefore temperature dependent. The T –dependent divergent piece
is of order g4 however (since proportional to g2 m̂2D ), and thus does not formally affect the
order–g3 calculation. But it signals that our calculation is not complete. The divergence
is identified in eq.(38) in the term with l = 1, i.e., in the 2–loop diagram in Fig. 7, after
performing the Matsubara sum. The overall divergence of the 2–loop diagram arises as the
product of a vacuum divergence in the 4–point function subgraph (the lower loop) times the
finite–temperature piece of the upper loop. Thus, the divergence is removed by the vacuum
renormalization of the 4–point function, i.e., by adding a 1-loop counterterm to the vertex in
the tadpole.
FIGURE 8. The “ring” diagrams responsible for the order–g3 effect in the pressure.
The contribution of order g3 to the pressure, also known as the “plasmon effect”,
comes again from resumming diagrams which are infrared divergent in the naïve diagrammatic expansion. These are the “ring” diagrams in Fig. 8, in which the central loop
is static (ωn = 0), while each outer loop represents an insertion of m̂2D . This series can
be easily summed up, with an infrared finite result[26]:
P3
l
(−1)l−1 m̂2D
∑ l
k2
l≥2
Z
m̂3D T
m̂2D
m̂2D
T
d3 k
log
1
+
−
=
= −
,
2 (2π)3
k2
k2
12π
T
= −
2
Z
d3 k
(2π)3
(42)
which is the order–g3 contribution in eq. (22).
FLUCTUATIONS, HIERARCHY OF SCALES
At this point it is useful to view the problem form a broader perspective. Strict expansions in power of the coupling constant have been pushed up to order g5 , both in QCD
and in scalar field theories. In both cases, one observes a rather poor apparent convergence, the successive contributions oscillating wildly unless the coupling is very small
and the deviation from ideal-gas behaviour correspondingly minute. As we shall see, the
difficulty may be related to the fact that what determines the accuracy of a weak coupling calculation in thermal field theories is not only the strength of the coupling, but
also the magnitude of the thermal fluctuations. These vary according to the relevant momentum scales, so that the accuracy of the weak coupling expansion depends on which
momentum scale contributes dominantly to the quantity under consideration.
Scalar field
To get orientation into these effects, let us then consider a massless scalar field
theory with g2 φ 4 interactions. In the noninteracting case the magnitude of the thermal
fluctuations of the field is given by the simple formula (with εk = k):
2
hφ i =
Z
d3 k Nk
,
(2π)3 εk
(43)
where Nk = 1/(eβ εk − 1) is the Bose-Einstein distribution function. We shall use this
formula as an approximation for the magnitude of the fluctuations also in the interacting
case. In order to get a criterion for the expected validity of perturbation theory, we shall
compare the expectation value of the “kinetic energy” h(∂ φ )2 i with the “interaction
energy” which we approximate as g2 hφ 2 i2 .
For the plasma particles εk = k ∼ T and hφ 2 iT ∼ T 2 . Then indeed, when g is small, the
kinetic energy of a typical particle, k2 ∼ T 2 , is large compared to its interaction energy ∼
g2 hφ 2 iT : The short wavelength, or hard, fluctuations produce a small perturbation on the
motion of a plasma particle. However, this is not so for an excitation at the momentum
scale k ∼ gT : then the kinetic energy k2 ∼ (gT )2 is comparable to the contribution to
the interaction energy coming from the coupling to the hard modes, g2 hφ 2 i2T . Thus, the
properties of an excitation with momentum gT are expected to be nonperturbatively
renormalized by the hard thermal fluctuations. And indeed, the scale gT is that at
which collective phenomena develop. The emergence of the Debye screening mass
mD ∼ gT in a plasma is one of the simplest examples of such phenomena. More
generally, this renormalization of the soft mode dynamics by hard mode contributions
invites a description in terms of an effective theory for the soft modes, the parameters of
this effective theory being determined by the hard modes. The building blocks of such an
effective theory have been dubbed hard thermal loops (HTL). In the scalar field theory,
there is only one such HTL, which is a mass. But in QCD, there is an infinite number of
them[3, 4].
Let us now consider the fluctuations at the soft scale gT T . These fluctuations can
be accurately described by classical fields. In fact the associated occupation numbers Nk
are large, and accordingly one can replace Nk by T /εk in eq. (43). Introducing an upper
cut-off gT in the momentum integral, one then gets:
2
hφ igT ∼
Z gT
d3 k
T
∼ gT 2 .
2
k
(44)
Thus g2 hφ 2 igT ∼ g3 T 2 is still of higher order than the kinetic term g2 T 2 . In that sense
the soft modes with k ∼ gT are still perturbative. Note however that they generate
contributions to physical observables which are not analytic in g2 , as shown by the
example of the order g3 contribution to the energy density, which goes schematically
as follows:
ε
(3)
∼
Z gT
0
d3 k εk
T
∼ T (gT )3 ∼ g3 T 4 .
εk
(45)
In fact, in contrast to the perturbation theory for the hard modes, the parameter which
controls the perturbative expansion for the soft modes is g instead of g2 so that is it less
accurate in general. Much of the difficulties with the strict perturbative expansion can be
attributed to the poor accuracy of perturbation theory in the soft sector.
QCD
The previous arguments concerning the fluctuations can be extended to QCD, where
one should compare the kinetic energy (∂ A)2 with the magnitude of the nonabelian
interactions g2 A4 (with A the gauge potential). For the plasma particles εk = k ∼ T
and hA2 iT ∼ T 2 . The associated electric (or magnetic) field fluctuations are hE 2 iT ∼
h(∂ A)2 iT ∼ k2 hA2 iT ∼ T 4 and are a dominant contribution to the plasma energy density.
As already mentioned, these short wavelength, or hard, gauge field fluctuations produce
a small perturbation on the motion of a plasma particle. This is not so for an excitation
at the momentum scale k ∼ gT which is nonperturbatively renormalized by the hard
thermal fluctuations. And again perturbation theory for the soft modes is governed by g,
not g2 .
However a new feature emerges in QCD: at the ultrasoft momentum scale, k ∼ g2 T ,
the unscreened magnetic fluctuations play a dominant role. At that scale, it becomes
necessary to distinguish the electric and the magnetic sectors (which provide comparable contributions at the scale gT ). The electric fluctuations are damped by the Debye
screening mass (εk2 = k2 + m2D ≈ m2D when k ∼ g2 T ) and their contribution is negligible, of order g4 T 2 . However, because of the absence of static screening in the magnetic
sector, we have there εk ∼ k and
hA2 ig2 T ∼ T
Z g2 T
0
d3 k
1
∼ g2 T 2 ,
k2
(46)
so that (∂ A)2 ∼ g6 T 4 ∼ g2 (A2 )2 : the fluctuations are no longer perturbative. This is the
origin of the well-known breakdown of perturbation theory at order g6 .
Whereas one may argue that these non-perturbative magnetic fluctuations can contribute to the deviation of the pressure from the ideal gas result, as observed on the
lattice, this contribution, being of order g6 , is presumably numerically rather small sufficiently above the transition temperature. At any rate, it is clear that the magnetic fluctuations have a priori no responsibility for the lack of convergence of the weak coupling
expansion at lower orders (i.e., up to order g5 ). This is further confirmed by the fact that
a similar lack of convergence is seen also in other theories, like the scalar field theory.
RESUMMATIONS AND EFFECTIVE THEORIES
One of the simplest way to account nonperturbatively for the dominant effect of interactions in the soft sector, which in the case of the scalar field is to generate a mass for the
excitations, is the so-called “screened perturbation theory”[32, 33, 34]. This consists in
a reorganization of the perturbative expansion, based on the following rewriting of the
Lagrangian:
1
1
0
L = L0 − m̂2D φ 2 + m̂2D φ 2 + Lint = L00 + Lint
,
(47)
2
2
with L00 = L0 − (1/2)m̂2D φ 2 . A perturbative expansion in terms of screened propagators
(that is keeping the screening mass m̂D as a parameter, i.e. not as a perturbative correction
to be expanded out) has been shown to be quite stable with good convergence properties. However, m̂D depends on the temperature, which makes the propagators explicitly
temperature dependent, and the ultraviolet divergences which occur in high order calculations become temperature dependent. Thus, at any finite loop order, the ultraviolet
renormalization gets artificially modified.
In the case of gauge theory, the effect of the interactions is more complicated than just
generating a mass. But we know how to determine the dominant corrections to the selfenergies. When the momenta are soft, these are given by the hard thermal loops[3, 4].
By adding these corrections to the tree level Lagrangian, and subtracting them from
the interaction part, one generates the so-called hard thermal loop perturbation theory
(HTLPT)[35, 36]:
0
L = L0 + LHT L − LHT L + Lint = L00 + Lint
.
(48)
The resulting perturbative expansion is made complicated however by the nonlocal
nature of the hard thermal loop action, and by the necessity of introducing temperature
dependent counter terms. Also, in such a scheme, one is led to use the hard thermal loop
approximation in kinematical regimes where it is no longer a justifiable approximation.
Some of these drawbacks may actually be absorbed by the flexibility in the choice of the
mass parameter m̂D in (47). Obviously, the results of SPT would be independent of this mass
if they were obtained in an all order calculation. But for a finite–order calculation, the choice
matters. One may try to optimize perturbation theory by a principle of minimal sensitivity
resulting in a self–consistent gap equation[32, 34]. This has been shown to reduce greatly
(though not completely) the dependence on the subtraction procedure[37] (see also below,
Fig. 12). Indeed, in Ref. [34] the calculation of the pressure of scalar φ 4 theory has been
extended to 2–loop order for the screening mass and 3–loop order for the pressure (meaning
a perturbative accuracy of order g5 in both cases), with results showing an excellent apparent
convergence: i.e., the difference between the 2–loop and 3–loop results for the pressure
remains rather small for all g . 1.
Other resummations will be presented later. In the remaining part of this section we
present an alternative to resummations, which exploits the hierarchy of scales discussed
above, and the strategy of effective theories.
Classical field approximation and dimensional reduction
In the high temperature limit, β → 0, the imaginary-time dependence of the fields
becomes unimportant and can be ignored in a first approximation. The partition function
(17) reduces to:
Z
Z
3
Z ≈ N D(φ ) exp −β d x H (x) ,
(49)
where φ ≡ φ (x) is now a three-dimensional field, and
H =
1
m2 2
(∇φ )2 +
φ +V (φ ) .
2
2
(50)
We shall refer to this limit as the classical field approximation.
Ignoring the time dependence of the fields is equivalent to retaining only the zero
Matsubara frequency in their Fourier decomposition. Then the Fourier transform of the
free propagator is simply:
G0 (k) =
T
.
εk2
(51)
This may be obtained directly from (19) keeping only the term with ων = 0, or from
eq. (21) by ignoring the time dependence and using the approximation
N(εk ) =
1
T
≈ .
− 1 εk
eβ εk
(52)
Both approximations make sense only for εk T , implying N(εk ) 1. In this limit, the
energy per mode is ∝ εk N(εk ) ≈ T , as expected from the classical equipartition theorem.
The classical field approximation may be viewed as the leading term in a systematic
expansion. To see that, let us expand the field variables in the path integral (17) in terms
of their Fourier components:
φ (τ) =
1
e−iων τ φ (iων ),
β∑
ν
(53)
where the ων ’s are the Matsubara frequencies. The path integral (17) can then be written
as:
Z = N1
Z
D(φ0 ) exp {−Seff [φ0 ]} ,
where φ0 ≡ φ (ων = 0) depends only on spatial coordinates, and
Zβ Z
Z
3
exp {−Seff [φ0 ]} = N2 D(φν6=0 ) exp −
dτ d x LE (x) .
(54)
(55)
0
The quantity Seff [φ0 ] may be called the effective action for the “zero mode” φ0 . Aside
from the direct classical field contribution (i.e., the restriction of eq. (5) to the static mode
φ0 ), this effective action receives also contributions which, diagrammatically, correspond
to connected diagrams whose external lines are associated to φ0 , and the internal lines are
the propagators of the non-static modes φν6=0 . Thus, a priori, Seff [φ0 ] contains operators
of arbitrarily high order in φ0 , which are also non-local. In practice, however, one wishes
to expand Seff [φ0 ] in terms of local operators, i.e., operators with the schematic structure
am, n ∇m φ0n with coefficients am, n to be computed in perturbation theory.
It is then useful to introduce an intermediate scale Λ (Λ T ) which separates hard
(k & Λ) and soft (k . Λ) momenta. All the non-static modes, as well as the static ones
with k & Λ are hard (since K 2 ≡ ωn2 +k2 & Λ2 for these modes), while the static (ωn = 0)
modes with k . Λ are soft. Thus, strictly speaking, in the construction of the effective
theory along the lines indicated above, one has to integrate out also the static modes
with k & Λ. The benefits of this separation of scales are that (a) the resulting effective
action for the soft fields can be made local (since the initially non-local amplitudes
can be expanded out in powers of p/K, where p Λ is a typical external momentum,
and K & Λ is a hard momentum on an internal line), and (b) the effective theory is
now used exclusively at soft momenta. This strategy, which consists of integrating
out the non-static modes in perturbation theory in order to obtain an effective threedimensional theory for the soft static modes, is generally referred to as “dimensional
reduction”[38, 39, 40, 41, 42, 43, 44].
As an illustration let us consider a massless scalar theory with quartic interactions;
that is, eq. (5) with m = 0. The effective action for the soft fields takes the form:
Z
1
1 2
3
4
2
2
2
Seff [φ0 ] = β F (Λ) + d x
(56)
(∇φ0 ) + M (Λ)φ0 + g3 (Λ)φ0 + · · · ,
2
2
where F (Λ) is the contribution of the hard modes to the free-energy, and the dots
stand for higher–order, local operators. We have changed the normalization of the
√
field (φ0 → T φ0 ) so as to absorb the factor β in front of the effective action. The
parameters of the effective Lagrangian in eq. (56), like M 2 (Λ), g23 (Λ), etc., are computed
in perturbation theory (they involve diagrams in which all the internal lines are hard), and
depend on the separation scale Λ, the original coupling g, and the temperature T (note
that the effective theory depends on the temperature only through these parameters). To
lowest order in g, g23 ≈ g2 T , and M ∼ gT , as we shall see shortly.
Note that the scale Λ acts as an explicit ultraviolet (UV) cutoff for the loop integrals in
the effective theory. Since it is arbitrary, the dependence on Λ of soft loop contributions
must cancel against the dependence on Λ of the parameters in the effective action. Let
us verify this cancellation explicitly in the case of the thermal mass M of the scalar field,
and to lowest order in perturbation theory. To this order, the mass parameter M 2 (Λ) in
the effective action is obtained by integrating over hard momenta within the one-loop
diagram in fig. 5 (cf. eq. (29)). This gives
d3 k (1 − δn0 ) + θ (k − Λ)δn0
(2π)3
ωn2 + k2
n
Z
1
T
d3 k
n(k)
2
= 12g
+
− θ (Λ − k) 2 .
(2π)3
k
2k
k
M (Λ) = 12g T ∑
2
2
Z
(57)
The first two terms within the last integral are the same as in eq. (29) with m = 0. The
first of them, involving the thermal distribution, gives the HTL contribution m̂2D = g2 T 2 ,
while the second one, involving 1/2k, is removed by the vacuum renormalization. The
third term, involving the θ -function, is easily evaluated. One finally gets:
6 Λ
6g2
2 2
2
2
.
(58)
M (Λ) = m̂D − 2 ΛT ≡ g T 1 − 2
π
π T
The Λ-dependent term above is subleading, by a factor Λ/T 1.
The one-loop correction to the thermal mass within the effective theory is given by
the same tadpole diagram, fig. 5, evaluated with the massive propagator 1/(k2 +M 2 (Λ)),
coupling constant g23 ≈ g2 T , and ultraviolet cut-off Λ. We obtain
2
d3 k
1
Θ(Λ − k) 2
3
(2π)
k + M 2 (Λ)
6g2
πM
M
6g2
3g2
T
Λ
1
−
arctan
'
T
Λ
−
T m̂D ,
=
π2
2Λ
Λ
π2
π
2
Z
δ M (Λ) = 12g T
(59)
where the terms neglected in the last step are of higher order in m̂D /Λ or Λ/T .
As anticipated, the Λ-dependent terms cancel in the sum M 2 ≡ M 2 (Λ) + δ M 2 (Λ),
which then provides the physical thermal mass within the present accuracy:
M 2 = M 2 (Λ) + δ M 2 (Λ) = g2 T 2 −
which agrees again with eq. (37).
3 3
g T,
π
(60)
Dimensional reduction puts emphasis on the static field φ0 : the contribution of the
hard modes, the modes which dominate the thermodynamical functions, are hidden in
the coefficients of the effective lagrangian. While this may be an efficient method to
treat the ultrasoft sector of QCD, in the soft sector it may be physically more transparent
to keep the soft collective excitations as explicit degrees of freedom. The properties of
these collective modes are reviewed in the next section. Then, in the following section,
we shall present a formalism which allows us to take into account these collective
phenomena in a consistent fashion.
COLLECTIVE PHENOMENA IN THE QUARK-GLUON PLASMA
We discuss briefly some collective phenomena that are expected to occur in Abelian
and non Abelian ultrarelativistic plasmas (Debye screening and Landau damping), and
which are relevant for the thermodynamics. More details can be found in [14] and
references therein.
Collective modes
The gluon propagator ∗Gµν in Coulomb’s gauge has the following non-trivial components, corresponding to longitudinal (or electric) and transverse (or magnetic) degrees
of freedom:
G00 (ω, p) ≡ ∗∆L (ω, p),
Gi j (ω, p) ≡ (δi j − p̂i p̂ j )∗∆T (ω, p),
∗
∗
(61)
where:
∆L (ω, p) =
∗
−1
p2 + ΠL (ω, p)
,
∆T (ω, p) =
∗
−1
ω 2 − p2 − ΠT (ω, p)
,
(62)
and the electric (ΠL ) and magnetic (ΠT ) polarization functions are defined as:
ΠL (ω, p) ≡ −Π00 (ω, p) ,
ΠT (ω, p) ≡
1 ij
(δ − p̂i p̂ j )Πi j (ω, p) .
2
(63)
Explicit expressions for these functions can be found in [14].
The dispersion relations for the modes are obtained from the poles of the propagators,
that is,
p2 + ΠL (ωL , p) = 0,
ωT2 = p2 + ΠT (ωT , p),
(64)
for longitudinal and transverse excitations, respectively. The solutions to these equations, ωL (p) and ωT (p), are displayed in fig. 9.b. Both dispersion relations are timelike (ωL,T (p) > p), and show a gap at zero momentum (the same for transverse and
longitudinal modes since, when p → 0, we recover isotropy). With increasing momentum, the transverse
branch becomes that of a relativistic particle with an effective mass
√
m∞ ≡ mD / 2 (commonly referred to as the “asymptotic mass”).
(a)
(b)
FIGURE 9. Dispersion relation for soft excitations in the linear regime: (a) soft fermions; (b) soft
gluons (or linear plasma waves), with the upper (lower) branch corresponding to transverse (longitudinal)
polarization.
The dispersion relations of soft fermionic excitations exhibit also collective feature
(see [5] and references therein) with a characteristic splitting at low momenta (see
fig. 9.a).
Debye screening
The screening of a static chromoelectric field by the plasma constituents is the natural
non-Abelian generalization of the Debye screening in classical plasma physics [45]. In
coordinate space, screening reduces the range of the gauge interactions. In momentum
space, it contributes to regulate the infrared behaviour of the various n-point functions.
Screening properties can be inferred from an analysis of the effective photon (or
gluon) propagators (62) in the static limit ω → 0. We have:
ΠL (0, p) = m2D ,
ΠT (0, p) = 0,
(65)
and therefore:
∆L (0, p) =
∗
−1
p2 + m2D
,
∆T (0, p) =
∗
1
,
p2
(66)
which clearly shows that the Debye mass mD acts as an infrared cut-off ∼ gT in the
electric sector, while there is no such cut-off in the static magnetic sector.
Landau damping
For time-dependent fields, there exists a different screening mechanism associated to
the energy transfer between the fields and the plasma constituents. In Abelian plasmas,
this mechanism is known as Landau damping [45]. The average energy loss is related to
the imaginary part of the retarded polarization tensor:
µν
Im ΠR (ω, p)
=
− πm2D ω
Z
dΩ µ ν
v v δ (ω − v · p) .
4π
(67)
Since in ultrarelativistic plasmas v is a unit vector, only space-like (|ω| < p) fields are
damped in this way.
Consider the effective photon (or gluon) propagator (62), and focus on the magnetic
propagator. For small but non-vanishing frequencies the corresponding polarization
function ΠT (ω, p) is dominated by its imaginary part:
ΠT (ω p) = −i
π 2 ω
m
+ O(ω 2 /p2 ) ,
4 D p
(68)
and therefore
∆T (ω p) '
∗
1
p2 − i (πω/4p) m2D
.
(69)
Thus Im ΠT (p) acts as a frequency-dependent IR cutoff at momenta p ∼ (ωm2D )1/3 . This
cut-off plays an essential role in the calculation of the transport coefficients [46].
SELF–CONSISTENT APPROXIMATIONS
Φ-derivable approximations[47] provide a more natural approach to propagator renormalization than screened perturbation theory, although the general goal is similar. It
allows us to exploit fully the spectral information that one has about the plasma excitations. In particular the hard thermal loops enter as essential ingredients,[48, 49, 50]
but in contrast to HTLPT, they are used only for soft external momenta where they
are accurate. One of the main advantages of such approximations is that they lead to
remarkable simplifications in the calculation of the entropy [51]. This allows one, in
particular, to bypass some of the difficulties of hard thermal loop perturbation theory.
Besides, the entropy makes transparent the underlying physical picture of quasiparticles, which (in simpler models) has been shown to provide rather accurate fits to lattice
data[52, 53, 54, 55]. The quasiparticle picture assumes that the dominant effect of the
interactions can be incorporated in the spectral properties of suitably defined quasiparticles with small residual interactions; this is precisely what comes out of the entropy
calculation.
1/12
+1/8
+1/48
+...
FIGURE 10. Skeleton diagrams in scalar theory with φ 3 and φ 4 interactions
Skeleton expansion for thermodynamical potential
A systematic way to take into account collective phenomena in the calculation of
the thermodynamics is to use the representation[56, 47, 57, 58] of the thermodynamic
potential Ω = −PV in terms of the full propagator D:
β Ω[D] = − log Z =
1
1
Tr log D−1 − Tr ΠD + Φ[D] ,
2
2
(70)
where Tr denotes the trace in configuration space, β = 1/T , and Φ[D] is the sum of
the two–particle irreducible (2PI) “skeleton” diagrams with no external legs: The full
propagator D is expressed in terms of the proper self–energy Π by Dyson’s equation (D0
is the free propagator):
D=
1
D−1
0 +Π
,
(71)
and Π itself, which is the sum of the one–particle irreducible (1PI) diagrams with two
external lines, is obtained from Φ[D] by:
δ Φ[D]/δ D =
1
Π.
2
(72)
By using this relation, one can check that Ω[D] is stationary under variations of D (at
fixed D0 ) around the physical propagator:
δ Ω[D]/δ D = 0
for
−1
D = (D−1
0 + Π) .
(73)
We shall usually refer to eq. (71) in which Π is given in terms of D by eq. (72) as a ‘gap
equation’.
An explicit expression for Ω is obtained by performing the summations over the
Matsubara frequencies in eq. (70), using standard contour integration techniques. One
obtains:
Ω/V =
Z
d4k
n(ω) ℑ log(−ω 2 + k2 + Π) − ℑΠD + T Φ[D]/V
4
(2π)
(74)
where n(ω) = 1/(eβ ω − 1), and the imaginary parts are defined with retarded prescription. For instance:
ℑD(ω, k) ≡ ℑD(ω + iε, k) =
ρ(ω, k)
,
2
(75)
where ρ(ω, k) is the spectral function, which enters the following representation of the
analytic propagator D(ω, k), valid for frequencies ω off the real axis:
D(ω, k) =
Z ∞
dk0 ρ(k0 , k)
−∞
2π k0 − ω
.
(76)
The stationarity condition (73) can be used as a variational principle to deduce non–
perturbative approximations for the physical propagator. A convenient approximation
scheme of this type is the self–consistent (or “Φ–derivable”) approximation[47] obtained
by selecting a class of skeletons in Φ[D] and calculating Π from eq. (72). We shall see
also that the stationarity property (73) brings simplifications in the calculation of the
entropy.
A simple model
As a simple illustration of the general formalism of the previous subsection, we
consider here the 2–loop approximation to Φ, given by the middle diagram in fig. 10.1
The corresponding skeleton for Π is the dressed tadpole (see Fig. 5), and is local. Thus,
the self–consistent self–energy is simply a mass, and we shall write Π ≡ m2 . (The
vacuum mass is set to zero.) The gap equation (72) reads then:
m2 =
λ
I(m).
2
(77)
It is possible to renormalize eq. (77) without introducing a thermal counterterm. This
is done through coupling constant renormalization. To proceed, we first rewrite eq. (77)
1
The self-consistent solution to 3-loop order has been worked out using mass expansions in Ref. [59, 60]
with satisfactory numerical results. However, the negative conclusion therein concerning renormalizability
of Φ-derivable approximations beyond two-loop order has since been refuted by Refs. [61, 62, 63, 64].
as:
m2
µ m = −λ0
32π 2
ε
2
Z
2
µ̄ 2
d 3 k n(εk )
+ log 2 + 1 + λ0
+ O(ε),
ε
m
(2π)3 2εk
(78)
and relate λ0 to the renormalized coupling λ by:
µε
1
1
=
+
.
λ
λ0 16π 2 ε
Then, the equation takes the following, manifestly finite, form (for ε → 0):
Z
λ
λ m2
m2
d3 k n(εk )
2
m =
+
log 2 − 1 .
2 (2π)3 εk
32π 2
µ̄
(79)
(80)
One should stress a subtle point in the procedure: expressing the bare coupling λ0 in
terms of the renormalized one λ according to eq. (79) is not sufficient to render finite
the r.h.s. of eq. (78) for arbitrary values of m2 , but only for these values which satisfies
the finite equation (80).
Note that the 1–loop correction to the 4–point function that is considered here is
iterated only in one channel among the three possible ones. Accordingly, the β –function
for the renormalized coupling constant (79):
dλ
λ2
=
,
d log µ̄
16π 2
(81)
(this follows by noticing that λ0 is independent of µ̄ in eq. (79)) is only one third of
the lowest-order perturbative β -function for this scalar field theory (see eq. (23)). This
is no actual fault since the running of the coupling affects the thermodynamic potential
only at order λ 2 which is beyond the perturbative accuracy of the 2-loop Φ-derivable
approximation. (In order to see the correct one-loop β -function, the approximation for
Φ would have to be pushed to 3-loop order.) By using eq. (81), one can check that the
solution m2 of eq. (80) is independent of µ.
By using the previous formulae, one can easily compute the pressure[65]:
P = −T
Z
d3 k
m2
−β εk
log(1
−
e
)
+
(2π)3
2
Z
d 3 k n(εk )
m4
+
,
(2π)3 2εk
128π 2
(82)
which differs from the pressure P0 (m) of an ideal gas of massive particles, eq. (28), by
the last two terms in the r.h.s.
By construction, the 2–loop Φ–derivable approximation is perturbatively correct to
order g3 , and, indeed, it can be easily verified that the weak coupling expansions of
eqs. (80) and (82) include the correct perturbative effects of order g2 and g3 (together
with incomplete contributions of higher orders). But the complete, self–consistent results, as obtained by numerical evaluation in eqs. (80) and (82), show a much better
behaviour than the respective perturbative expansions to order g3 , in the sense of being monotonic with g (unlike the perturbative approximants of order g3 ), and showing
1
0.8
0.6
0.4
0.2
0.5
1
1.5
2
2.5
3
FIGURE 11. Numerical solution of the mass gap equation (80) with µ̄ = 2πT in comparison with the
perturbative result to order g2 and g3 for m (long and medium dashed lines, resp.). The short-dashed line
just below the full line corresponds to the quadratic gap equation (86) following from the requirement of
only soft self-consistency.
¼
1.2
1.1
1
0.9
0.8
0.5
1
1.5
2
2.5
3
FIGURE 12. The 2-loop Φ-derivable pressure (82) in comparison with the perturbative results to order
g2 and g3 (long and medium dashed lines, resp.). The short-dashed line just above the full line corresponds
to the pressure evaluated perturbatively for hard modes and self-consistently only for the soft modes, eqs.
(83) and (87) with (86).
saturation at very large g & 2 (in contrast to the order–g2 estimates). This behaviour is
illustrated in Figs. 11 and 12 for the self-consistent mass (80) and pressure (82), respectively.
We have previously argued that the expected accuracy of perturbation theory is a
priori different in the soft and the hard sectors. Within the present solvable model,
it is possible to verify this argument explicitly. To this aim, we shall reformulate the
self-consistent calculation of the thermal mass and the pressure in an approximate way,
exploiting the distinction between the hard and soft sectors. That is, we shall build on
the strategy of effective theories to obtain perturbatively, through a calculation in the
hard sector, the coefficients of an effective Lagrangian for the soft sector. We shall then
look for an “exact" solution in the soft sector, which amounts here to use the two-loop
self-consistent approximation. As we shall see, this strategy, which mimics that used in
calculations based on dimensional reduction, turns out to be remarkably accurate. It also
provides a clearer physical justification for approximations that have been motivated
previously by other considerations.
Thus, we shall approximate the fully self–consistent 2–loop result for the pressure,
eq. (82), by P = P(2) + Ps , where P(2) is the perturbative result to order g2 , which is due
to the hard modes alone:
"
#
π2 4
15 g 2
P(2) =
T 1−
.
(83)
90
8 π
and Ps is obtained via a 2–loop self–consistent calculation within the effective three–
dimensional theory with Lagrangian (56). To the order of interest, one can set M 2 ≈ m̂2D
and g2E ≈ g2 T in eq. (56), and neglect all the other interaction vertices.
The corresponding free–energy functional reads then:
Z β Ωs [Πs ]
1
Πs
2
2
=
ln(k + m̂D + Πs ) − 2
V
2 k
k + m̂2D + Πs
Z
2
λ
1
+
,
(84)
2
8
k k2 + m̂D + Πs
R
where k denotes the momentum integral in 3 − d dimensions. We use here dimensional
regularization, with the scale µ chosen as the separation scale Λ between soft and hard
momenta. This has the advantage that all the power–like divergences in integrals like
those in eq. (84) are now set to zero, so, conversely, the parameters of the effective
Lagrangian are independent of Λ to the accuracy of interest[30].
In eq. (84), Πs is the scalar self–energy within the 3 − d effective theory, and is
determined from a gap equation which follows by demanding Ωs [Πs ] to be stationary
with respect to Πs :
Z
λ
1
Πs = T
,
(85)
2
2
2
k k + m̂D + Πs
or, equivalently, with m2 ≡ m̂2D + Πs ,
3
m2s = m̂2D − g2 T ms .
(86)
π
This is a second-order algebraic equation for ms , whose r.h.s. is recognized as the
expansion of the integral in the general gap equation in four dimensions, eq. (77), up
to terms linear in m/T . Note that, to this accuracy, there is no ultraviolet divergence at
all, so the coupling constant λ (or g2 ) in the equations above is the renormalized one.
The free energy (84) can be easily evaluated in terms of the solution ms to the above
gap equation. One obtains (Ps = −Ωs /V ):
T m3s T ms 2
3 g 2 2 2
Ps =
−
(ms − m̂2D ) −
T ms .
(87)
12π
8π
16 π
Note that this contribution starts at order g3 , but includes terms of all higher orders,
via ms . The sum P = P(2) + Ps of eqs. (83) and (87) should be compared to the fully
self–consistent result in eq. (82). Clearly, these two approximations coincide up to order
g3 , but deviate in higher orders, with the deviations restricted, however, to contributions
due to the hard modes. In Figs. 11 and 12, we also compare the numerical solutions to
the full gap equation (80) and its “soft” approximation (86), and also the corresponding
predictions for the pressure. We see that these predictions are extremely close to each
other, for both m and P, which confirms that the higher order effects associated with the
hard modes are small indeed.
The entropy
We have mentioned already that the stationarity property (73) allows one to simplify
the calculation of the entropy:
S = −∂ (Ω/V )/∂ T
(88)
or, more generally, of any first partial derivative of the pressure. Indeed, the thermodynamic potential (74) depends on the temperature through the statistical factors n(ω) and
the propagator, or equivalently its spectral function ρ. Because of eq. (73) the temperature derivative of the spectral density cancels out and one obtains[66, 51]:
S
with
d 4 k ∂ n(ω)
ℑ log D−1 (ω, k)
(2π)4 ∂ T
Z
d 4 k ∂ n(ω)
ℑΠ(ω, k)ℜD(ω, k) + S 0
+
4
(2π) ∂ T
= −
Z
Z
∂ (T Φ) d 4 k ∂ n(ω)
S ≡−
ℜΠℑD.
+
∂T D
(2π)4 ∂ T
0
(89)
(90)
Loosely speaking, the first two terms in eq. (89), which have a one–loop structure,
represent the entropy of ‘independent quasiparticles’, while S 0 accounts for a residual
interaction among these quasiparticles [51]. For 2–loop2 skeletons we have the following
additional simplification[66, 51, 50]:
S 0 = 0,
(91)
a property which holds independently of the self-consistency condition (73).
The formula (89), together with S 0 = 0, makes the entropy a privileged quantity
to study the thermodynamics of ultrarelativistic plasmas. Let us discuss some of the
attractive features.
2 For the scalar field theory with φ 4 self–interactions, the property (91) holds also for the 3–loop skeleton
in eq. (??).
First, we note that, for a finite function Π(ω, k), the integrals in eq. (89) with S 0 = 0
are manifestly ultraviolet finite, since ∂ n/∂ T vanishes exponentially for both ω → ±∞.
This reduces the problem of the UV renormalization of the 2–loop Φ–derivable approximation for the entropy to that of the renormalization of the gap equation for
the 1–loop self-energy. In general, this is still a difficult task (because this is a non–
perturbative approximation), but as we have seen in the previous subsection, progress
can be made at least in specific cases. In particular, it has been recently demonstrated,
in Refs. [61, 62, 63, 64], that for scalar field theories at least, the self–consistent approximations are renormalizable to all orders, i.e., they can be made finite by adding
temperature-independent (vacuum) counterterms to the Lagrangian, which are themselves determined in a self–consistent way.
Furthermore, as already mentioned, the entropy has a more direct quasiparticle interpretation than the pressure. To see this more explicitly, use the following identity:
ℑΠ
−1
ℑ log D (ω, k) = arctan
− πε(ω)θ (−ℜD−1 ),
(92)
−1
ℜD
to rewrite S as S = S pole + Sdamp , with
d 4 k ∂ n(ω)
πε(ω)θ (−ℜD−1 (ω, k))
4
(2π) ∂ T
Z
o
d3k n
(1
+
n
)
log(1
+
n
)
−
n
log
n
=
k
k
k
k ,
(2π)3
S pole =
Z
(93)
where nk ≡ n(εk ), with εk solution of ℜD−1 (ω = εk , k) = 0. This is the on-shell energy
of the dressed single–particle excitations in the system, or ‘quasiparticles’. The quantity
S pole is the entropy of a system of ‘non-interacting’ quasiparticles, while the quantity
Sdamp =
Z
d 4 k ∂ n(ω)
ℑΠ
ImΠℜD − arctan
,
(2π)4 ∂ T
ℜD−1
(94)
which vanishes when ℑΠ vanishes, is a contribution coming from the continuum part of
the quasiparticle spectral weights. We see that for the entropy, unlike for the pressure, the
main effects of the interactions come via the change in the spectrum of the excitations,
and therefore can be taken into account by simply dressing the propagator.
The above formulae, which are generic (i.e., they hold for any field theory, including
QCD), become even simpler when the 2–loop self–consistent approximation is applied
to the scalar theory with φ 4 self–interactions. Then, the self–energy reduces to a local
mass term, as we have seen in the previous subsection, and therefore S = S pole , with
S pole given by eq. (93) with εk2 = k2 + m2 and m determined by eq. (80). That is, in
this approximation, the entropy of the interacting scalar gas is formally identical to the
entropy of an ideal gas of massive bosons, with mass m. As emphasized after eq. (82),
such a simple interpretation does not hold for the pressure.
Approximately self–consistent calculations of the entropy
In this subsection, the calculation of the self–consistent 2–loop entropy will be considered from a new perspective, which will suggest a different strategy for implementing the self–consistency in an approximate way. This strategy will be especially useful
in view of subsequent applications to QCD, where a fully self–consistent calculation
seems prohibitively difficult, and not even desirable [50].
It is first instructive to see how the perturbative effects of order g2 and g3 get built
within the self–consistent 2–loop entropy, as given by (cf. eq. (93)):
d 4 k ∂ n(ω)
ℑ log(k2 − ω 2 + m2 )
(2π)4 ∂ T
2π 2 T 3 m2 T
m3
=
−
+
+ O(m4 /T ).
45
12
12π
S (m) = −
Z
(95)
The first two terms in the expansion in the second line come from integrating over hard
momenta (k ∼ T ) within the loop integral in the first line, while the third term, non–
analytic in m2 , comes from the soft loop (k ∼ m).
The expansion in eq. (95) must be considered simultaneously with the perturbative
expansion of the solution m to the gap equation (80). To order g3 , this is the same as the
standard perturbative expansion, eq. (37): m2 ' g2 T 2 (1 − 3g/π).
If one replaces m in eq. (95) by the LO thermal mass m̂D = gT , then the resulting
approximation for S reproduces the perturbative effect of order g2 , but underestimates
the correction of order g3 by a factor of 4. But to order g3 , the thermal mass itself must be
corrected, as discussed above. By replacing m2 → m̂2D + δ m2D in the second order term
of eq. (95), the correction δ m2D to the thermal mass generates the 3/4 of the plasmon
effect missing. To conclude, the order–g3 contribution to the entropy comes up in a
subtle way: 1/4 of it comes directly from the soft loop in eq. (95), with m replaced by
the LO screening mass m̂D , while the other 3/4 comes from the hard loop, via the NLO
correction to the dispersion relation of the hard particles.
Of course, within the present context, where the self–energy is independent of the
external momentum, there is no distinction between the screening mass and the quasiparticle mass: both are solution to the same gap equation (80). But in QCD, these are
distinct quantities already at lowest order, and this difference plays an important role
in the construction of approximations. Thus, although this may look at first somewhat
artificial here, we shall introduce different notations for the two masses. We shall keep
the notation mD for the screening (Debye) mass, and denote the hard quasiparticle mass
as m∞ : the p
latter characterizes the excitation spectrum at large momenta, typically of the
form εk = k2 + m2∞ . Thus the mass m2 in the second order term of eq. (95) should be
read as m2∞ , while m3 in the cubic term is m3D . That is, we rewrite eq. (95) as
S (m) '
m3
2π 2 T 3 m2∞ T
−
+ D.
45
12
12π
(96)
In perturbation theory, mD and m∞ coincide with each other up to order g3 , but differ
Ë Ë¼
1.2
1.1
1
0.9
0.8
0.5
1
1.5
2
2.5
3
FIGURE 13. Comparison of the fully self-consistent 2-loop entropy with the perturbative results to
order g2 and g3 (long and medium dashed lines, resp.). The short-dashed line between the order g2
result and the full line corresponds to the “HTL approximation” where the 2-loop entropy is evaluated
with just the HTL mass; the short-dashed line almost on top of the full line corresponds to the “NLA
approximation” using (97).
in order g4 and higher.3 Within the 2–loop Φ–derivable approximation, they coincide
if computed exactly, as already mentioned. But if the self–consistency is imposed only
in the soft sector, the corresponding approximations may a priori look different. We
have seen earlier that self-consistency in the soft sector is enough to obtain an excellent
approximation for the pressure, and we want to verify here the status of such an approximation for the entropy. In fact we shall examine the accuracy of two non–perturbative
approximations to the entropy, that have been implemented in the case of QCD. In both
approximations, the integral in the first line of eq. (95) is evaluated exactly (i.e., numerically), but for values of m which are only approximate solutions to the gap equation.
i ) In the ‘HTL approximation’ we set m = m̂D . The ensuing approximation for S
includes the correct LO effect of the interactions, of order g2 , but only 1/4 of the plasmon
effect. In Fig. 13, the resulting approximation is compared to the exact (i.e., fully self–
consistent) result, and also to the weak coupling expansion.
ii ) In the ‘next–to–leading approximation’ (NLA), the mass m∞ of the hard particles
is determined with NLO accuracy, so as to restore the correct plasmon effect in S . That
is,
m2∞ −→ m2∞ − (3/π)g2 T mD ,
(97)
where mD is the self-consistent mass, given by the solution to the gap equation (86).
Of course, the mass m∞ thus obtained coincides with the Debye mass mD , but this
‘degeneracy’ is specific to the present scalar model, as we have already emphasized.
3
Note that only the corrections to mD can be calculated by means of dimensional reduction. Dynamical
masses in general require the resummation of the nonstatic hard thermal loops (for simple examples see
Ref. [67]).
1.05
1
0.95
0.9
ÈÅË
0.85
0.8
1.5
2
2.5
3
3.5
4
5
4.5
FIGURE 14. The two-loop DRSPT result for pure-glue QCD (gray band) in comparison with the twoloop HTLPT result of Ref. [68] (dashed lines) and the “NLA” result of Ref. [50] (dash-dotted lines), all
with µ̄ varied around 2πT by a factor of 2.
The quality of the NLA based on eq. (97) can be appreciated from its comparison, in
Fig. 13, with the “exact” (fully self-consistent) result, and also with the HTL approximation alluded to before. Although the HTL approximation is seen to deviate from the
fully self-consistent result at large coupling, it remains smaller than the ideal-gas result despite the presence of a quarter of the plasmon effect, which, when treated purely
perturbatively, would result in a nonmonotonic behaviour leading to an excess over the
ideal-gas result at large coupling.
SELECTED RESULTS FOR QCD
An illustration of the results obtained with the techniques presented in these lectures,
when properly extended to QCD, is given by Fig. 14, taken from Ref. [69]. The main
message that I wish to extract from this figure is that weak coupling calculations,
contrary to what a naive examination of perturbative results may lead us to conclude,
can indeed account accurately for the deviations of the pressure from the free gas values
down to temperatures T ∼ 3Tc . The physically most transparent method is that based
on the entropy calculation (the NLA approximation in Fig. 14): it uses the full spectral
information on the quasiparticles and the collective modes in the system. This method
can be extended easily to finite chemical potential.
ACKNOWLEDGMENTS
Most of the material presented in these lectures is drawn from work done in collaboration
with E. Iancu and A. Rebhan.
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