Thermodynamics of the high temperature Quark-Gluon Plasma

Transcription

1 Thermodynamics of the high temperature Quark-Gluon Plasma J.-P. BLAIZOT Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS 99 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. [, 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, 0,, 2, 3], 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 g 5, 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

2 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 [4]. 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. [4, 5, 6, 5, 7, 8]. QCD, SYMMETRIES AND PHASE DIAGRAM This section is a short summary of basics of QCD. The QCD lagrangian is L = 2 TrF µνf µν + f ψ f (i /D m f )ψ f, () where D µ = µ iga µ F µν = i[d µ,d ν ], (2) or, making more explicit the color structure of the gauge field A µ and the associated field strength tensor F µν, A µ = A a µt a F a µν = µ A a ν ν A a µ + g f abc A b µa c ν. (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:

3 FIGURE. 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 = g 2 /4π) becomes weaker and weaker as the temperature increases, roughly as α(t ) ln(t /Λ QCD ). (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 [9]. 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.. 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 3 channel (for a review see [2] ). 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 > 3T c. We shall see that these calculations suggest that the dominant effect of the interactions is well accounted for in

4 T Quark-Gluon Plasma Hadronic matter Colour superconductor Nuclei µ B 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 µ B = 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 = 2 µφ µ φ m2 2 φ 2 V (φ). (5) All thermodynamic observables can be deduced from the partition function: Z = tr e βh, (6)

5 where β /T, with T the temperature. Here are some examples. Energy density and pressure (V is the volume): ε = V β lnz P = β Conserved charges, and associated susceptibilities: V lnz = T lnz. (7) V N i = T lnz µ i χ i j = n i = 2 P. (8) µ j µ i µ j In order to calculate the partition function, one may observe that e βh is like an evolution operator in imaginary time: t iβ e iht e βh, (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 = H 0 + H with H H 0, and define the following interaction representation of the perturbation H : H (τ) = e τh 0 H e τh 0. (0) Using standard techniques (see e.g. [25]), one obtains { β } e βh = e βh 0 Texp dτh (τ), () 0 from which one gets the following expression for the partition function Z : { β } Z = Z 0 Texp dτh (τ) 0. (2) 0 The symbol T denotes time ordering, Z 0 = tr e βh 0 and, for any operator O, ( ) e βh 0 O 0 Tr O. (3) Z 0 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 q 2 e iht q = q(t)=q2 q(0)=q D (q(t)) e i t2 t ( 2 m q2 V (q))dt. (4) Changing t iτ, and taking the trace, one obtains the following formula for the partition function: { β ( )} Z = tr e βh = D(q) exp q(β)=q(0) 0 2 m q2 +V (q). (5)

6 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: L = 2 ( tφ) 2 2 ( φ)2 m2 2 φ 2 V (φ). (6) The corresponding partition function reads: { β ( Z = D(φ)exp dτ d 3 r 0 2 ( τφ) 2 + )} 2 ( φ)2 + m2 2 φ 2 +V (φ), 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 = D(φ)exp β d 3 r 2 ( φ)2 + m2 2 φ 2 +V (φ). ii) Note the Euclidean metric in (7). 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) = Tφ(τ )φ(τ 2 ), where τ τ τ 2 satisfies the differential equation [ 2 τ 2 + m 2] (τ r ;τ 2 r 2 ) = δ(τ τ 2 )δ( r r 2 ), (8) and obeys periodic boundary conditions. It admits the Fourier representation (7) (τ) = β ν e iω ντ (iω ν ), (9) where ω ν = 2πν/β, with ν an integer (ω ν is called a Matsubara frequency), and (iω ν ) = εk 2, (20) ω2 ν where ε k = k 2 + m 2 is the energy of a mode with momentum k. The sum over Matsubara frequencies in eq. (9) can be done explicitly and yields: (τ) = { } ( + N k )e εk τ + N k e ε k τ, (2) 2ε k with N k = /(e βε k ).

7 P=P g 4 g 5 g g g(μ =2ßT ) FIGURE 3. Weak-coupling expansion through orders g 2, g 3, g 4, and g 5 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) = λφ 4 /4!, m = 0, has been computed to order λ 5/2 (or g 5 ), and reads[28, 29, 30]: [ P = P 0 5 ( g ) 2 5 ( g ) ( log µ ) ( g ) 4 8 π 2 π 6 2πT π 405 ( log µ 8 2πT 4 3 log g ) ] ( g ) 5 π O(g 6 logg), (22) π where P 0 = (π 2 /90)T 4 is the pressure of an ideal gas of free massless bosons, and g 2 ( µ) λ( µ)/24 is the MS coupling constant at the renormalization scale µ. This formula calls for several remarks. First we note that, to order λ 5/2, P is formally independent of µ (in the sense that dp/d µ involves terms of order λ 3 at least). This is easily verified using the renormalization group equation satisfied by the renormalized coupling λ( µ): µ dλ d µ = 3 λ 2 6π 2 + O ( λ 3). (23) But the approximate expression (22) for P does depend numerically on µ. In order to avoid large logarithms ln( µ/2πt ) in the expansion (22) one may choose µ 2πT. With this choice λ ( µ) 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/P 0 as a function of g(2πt ). Each partial sum is shown as a band obtained by varying µ from πt to 4πT, which gives an indication of the theoretical uncertainty in the calculation. Thus, in

8 drawing this plot, g( µ) 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 g 3 λ 3/2, g 5 λ 5/2, and also terms involving lng, 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 m D, defined from the the static propagator by p 2 + m 2 D + Π(0,p) = 0 at p 2 = m 2 D, (24) where Π(p 0,p) is the self energy. For the massless case (m = 0), the weak coupling expansion of m 2 D is known to order g4, and reads[30]: m 2 D = g 2 T 2{ 3 g π 9 2 [ ln µ 2πT 4 3 ln g π 2.45 ] ( g π ) 2 + O(g 3 )}. (25) Note that, with our notation g 2 λ/24 for the coupling constant, the leading order contribution in eq. (25) is simply ˆm D = 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 (g 3, ln/g) occur in the expression of m 2 D. In Fig. 4, we show the screening mass m D normalized to the leading order result ˆm D as a function of g(2πt ), for each of the three successive approximations to m 2 D. As in Fig. 3, the bands correspond to varying µ 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-g 2 and order-g 3 approximations, and a larger sensitivity to the value of the renormalization scale for the order g 4. 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 g 3. We start with the zeroth order calculation. We have P 0 = Ω 0 /V, with Ω 0 V = 2 T = 2 n d 3 k (2π) 3 ln(ω2 n + k 2 + m 2 ) d 3 k (2π) 3 ε k + T d 3 k (2π) 3 log( e ε k/t ), (27) where ω n = 2nπT is a bosonic Matsubara frequency and ε 2 k = k2 + m 2. 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

9 m=t 0.8 g g 4 g g(μ =2ßT ) FIGURE 4. Weak-coupling expansion to orders g 2, g 3, and g 4 for the screening mass normalized to the temperature as a function of g(2πt ). From [3]. 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: P 0 (m) = T d 3 k (2π) 3 ln( e βε k ) = π2 T 4 90 m2 T m3 T +..., (28) 2π where the neglected terms start at order m 4 ln(m/t ). FIGURE 5. Lowest order correction to the self-energy in scalar φ 4 theory. Consider now the contributions of order g 2 to the self energy and the thermodynamic potential, denoted as Π 2 and Ω 2, respectively. Π 2 is given by the tadpole diagram of Fig. 5. This gives: Π 2 = λ 2 T = λ 2 n d 3 k (2π) 3 ω 2 n + k 2 + m 2 λ 2 I(m) where n(ω) is the Bose-Einstein thermal distribution: d 3 k + 2n(ε k ) (2π) 3 λ 2ε k 2 [I 0(m) + I T (m)], (29) n(ω) = e βω. (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.

10 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 2 δm2 φ 2 in the Lagrangian, with δm 2 chosen so that: λ 2 I 0(m) + δm 2 = 0. (3) Then, the lowest order self energy correction due to the thermal fluctuations reads: Π 2 = λ 2 I T (m) λ { T mt ( 4π m2 m 2 )} 6π 2 ln (4πT ) 2 + 2γ, (32) where the expansion in the r.h.s. holds when m T. In particular, when m = 0, we recover the result Π 2 = ˆm 2 D, 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 Ω 2 is given by the 2 loop diagram in fig. 6, and reads: Ω 2 V = λ 4! φ δm2 φ 2 0 = λ ( I 2 8 T (m) I0(m) 2 ), (33) where we have also included the contribution of the mass counterterm and used eq. (3) 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: P 2 = λ 52 T 4 = 5 ( g ) 2 P0, (34) 8 π 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: 2L = λ 2 T Π (a) n d 3 k (2π) 3 ˆm 2 D (ω 2 n + k 2 ) 2. (35) The three-dimensional integral over k in eq. (35) has a linear infrared divergence, coming from the term with ω n = 0. (For the other terms, the Matsubara frequency ω n 0 acts effectively as an infrared cutoff.) A non zero mass m would cut off this divergence, and

11 give a contribution λ ˆm 2 D (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 of a thermal mass ˆm D = gt. Then Π (a) 2L g4 (T 3 / ˆm D ) g 3 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 ˆD(iω n,k) ωn 2 + k 2 + ˆm 2. (36) D 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. (3))... FIGURE 7. Diagrams that are resummed when computing the tadpole in Fig. 5 with the dressed propagator in eq. (36). Since ˆm D = gt, this resummation involves arbitrarily high powers of g. However, the final result can be read from eq. (32) in which we set m = ˆm D = gt : Π tadpole = g 2 T 2 3 π g3 T 2 + O(g 4 lng). (37) This formula exhibits the next to leading order contribution to the thermal mass, of order g 3, consistent with eq. (25). This order g 3 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 I T (m) as a function of m 2 and to the infrared divergences which occur when one attempts to evaluate this integral after expanding the integrand in powers of m 2. The infrared divergences are easily identified in the sum of the diagrams in Fig. 7 with at least two loops: Π ring = λ 2 T n = 2g 2 T ω n d 3 { } k (2π) 3 ωn 2 + k 2 + ˆm 2 D ωn 2 + k 2 d 3 ( k ˆm 2 ) l (2π) 3 ωn 2 + k D 2 l ωn 2 + k 2, (38) where the subtracted term is the massless tadpole. The term with l = corresponds to the 2 loop diagram that we have considered separately in eq. (35). Each of the terms with ω 2 n > 0 in eq. (38) starts to contribute at order g 2 ˆm 2 D g4. But the terms with ω n = 0 are individually infrared divergent for any l, and the divergences cancel only in thwe sum of

12 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 Π (3) ring = 2g2 T d 3 k { (2π) 3 k 2 + ˆm 2 D k 2 } = 3 π g3 T 2, (39) where we recognize the order g 3 contribution in eq. (37). It is instructive to look at this contribution from another perspective. As we have argued before, the dominant contribution to I T (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 I T (0) from I T (m) in eq. (32), and write: d 3 ( k I T (m) I T (0) T (2π) 3 k 2 + ˆm 2 ) D k 2 = ˆm DT 4π. (40) After multiplication by λ/2 = 2g 2, one recognizes the g 3 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(ε k ) T ε k for ε k T. (4) 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. (3), which now involves the thermal mass, and is therefore temperature dependent. The T dependent divergent piece is of order g 4 however (since proportional to g 2 ˆm 2 D ), and thus does not formally affect the order g 3 calculation. But it signals that our calculation is not complete. The divergence is identified in eq.(38) in the term with l =, 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 -loop counterterm to the vertex in the tadpole. FIGURE 8. The ring diagrams responsible for the order g 3 effect in the pressure. The contribution of order g 3 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

13 is static (ω n = 0), while each outer loop represents an insertion of ˆm 2 D. This series can be easily summed up, with an infrared finite result[26]: P 3 = T 2 = T 2 ) l d 3 k ( ) (2π) l ( ˆm 2 D 3 l 2 l k 2 d 3 [ ( ) ] k (2π) 3 log + ˆm2 D k 2 ˆm2 D k 2 = ˆm3 D T 2π, (42) which is the order g 3 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 g 5, 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 g 2 φ 4 interactions. In the noninteracting case the magnitude of the thermal fluctuations of the field is given by the simple formula (with ε k = k): φ 2 = d 3 k (2π) 3 N k ε k, (43) where N k = /(e βε k ) 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 ( φ) 2 with the interaction energy which we approximate as g 2 φ 2 2. For the plasma particles ε k = k T and φ 2 T T 2. Then indeed, when g is small, the kinetic energy of a typical particle, k 2 T 2, is large compared to its interaction energy g 2 φ 2 T : 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 k 2 (gt ) 2 is comparable to the contribution to the interaction energy coming from the coupling to the hard modes, g 2 φ 2 2 T. Thus, the

14 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 m D 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 N k are large, and accordingly one can replace N k by T /ε k in eq. (43). Introducing an upper cut-off gt in the momentum integral, one then gets: φ 2 gt gt d 3 k T k 2 gt 2. (44) Thus g 2 φ 2 gt g 3 T 2 is still of higher order than the kinetic term g 2 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 g 2, as shown by the example of the order g 3 contribution to the energy density, which goes schematically as follows: ε (3) gt 0 d 3 k ε k T ε k T (gt ) 3 g 3 T 4. (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 g 2 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 g 2 A 4 (with A the gauge potential). For the plasma particles ε k = k T and A 2 T T 2. The associated electric (or magnetic) field fluctuations are E 2 T ( A) 2 T k 2 A 2 T 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 g 2. However a new feature emerges in QCD: at the ultrasoft momentum scale, k g 2 T, the unscreened magnetic fluctuations play a dominant role. At that scale, it becomes

15 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 (ε 2 k = k2 + m 2 D m2 D when k g2 T ) and their contribution is negligible, of order g 4 T 2. However, because of the absence of static screening in the magnetic sector, we have there ε k k and g 2 A 2 g 2 T T T d 3 k k 2 g2 T 2, (46) 0 so that ( A) 2 g 6 T 4 g 2 (A 2 ) 2 : the fluctuations are no longer perturbative. This is the origin of the well-known breakdown of perturbation theory at order g 6. 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 g 6, 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 g 5 ). 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: L = L 0 2 ˆm2 Dφ ˆm2 Dφ 2 + L int = L 0 + L int, (47) with L 0 = L 0 (/2) ˆm 2 D φ 2. A perturbative expansion in terms of screened propagators (that is keeping the screening mass ˆm D 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, ˆm D 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]: L = L 0 + L HT L L HT L + L int = L 0 + L int. (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

16 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 ˆm D 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. 2). 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 g 5 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. 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 (7) reduces to: Z N { D(φ)exp β where φ φ(x) is now a three-dimensional field, and } d 3 xh (x), (49) H = 2 ( φ)2 + m2 2 φ 2 +V (φ). (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: G 0 (k) = T ε 2 k. (5) This may be obtained directly from (9) keeping only the term with ω ν = 0, or from eq. (2) by ignoring the time dependence and using the approximation N(ε k ) = e βε k T ε k. (52) Both approximations make sense only for ε k T, implying N(ε k ). In this limit, the energy per mode is ε k N(ε k ) T, as expected from the classical equipartition theorem.

17 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 (7) in terms of their Fourier components: φ(τ) = β ν e iω ντ φ(iω ν ), (53) where the ω ν s are the Matsubara frequencies. The path integral (7) can then be written as: Z = N D(φ 0 )exp{ S eff [φ 0 ]}, (54) where φ 0 φ(ω ν = 0) depends only on spatial coordinates, and exp{ S eff [φ 0 ]} = N 2 { β D(φ ν 0 )exp dτ 0 } d 3 xl E (x). (55) The quantity S eff [φ 0 ] may be called the effective action for the zero mode φ 0. Aside from the direct classical field contribution (i.e., the restriction of eq. (5) to the static mode φ 0 ), this effective action receives also contributions which, diagrammatically, correspond to connected diagrams whose external lines are associated to φ 0, and the internal lines are the propagators of the non-static modes φ ν 0. Thus, a priori, S eff [φ 0 ] contains operators of arbitrarily high order in φ 0, which are also non-local. In practice, however, one wishes to expand S eff [φ 0 ] in terms of local operators, i.e., operators with the schematic structure a m,n m φ0 n with coefficients a m,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 ωn 2 +k 2 Λ 2 for these modes), while the static (ω n = 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, 4, 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: S eff [φ 0 ] = βf (Λ) + { d 3 x 2 ( φ 0) 2 + } 2 M2 (Λ)φ0 2 + g 2 3(Λ)φ0 4 +, (56) 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

18 field (φ 0 T φ 0 ) 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), g 2 3 (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, g 2 3 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 M 2 (Λ) = 2g 2 d T 3 k ( δ n0 ) + θ(k Λ)δ n0 n (2π) 3 ωn 2 + k 2 = 2g 2 d 3 { k n(k) (2π) 3 + k 2k θ(λ k) T }. k 2 (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 ˆm 2 D = g2 T 2, while the second one, involving /2k, is removed by the vacuum renormalization. The third term, involving the θ-function, is easily evaluated. One finally gets: ( M 2 (Λ) = ˆm 2 D 6g2 π 2 ΛT g2 T 2 6 ) Λ π 2. (58) T The Λ-dependent term above is subleading, by a factor Λ/T. 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 /(k 2 +M 2 (Λ)), coupling constant g 2 3 g2 T, and ultraviolet cut-off Λ. We obtain δm 2 (Λ) = 2g 2 T = 6g2 π 2 T Λ d 3 k (2π) 3 Θ(Λ k) k 2 + M 2 (Λ) ( πm 2Λ arctan M Λ ) 6g2 π 2 T Λ 3g2 π T ˆm D, (59) where the terms neglected in the last step are of higher order in ˆm D /Λ or Λ/T. As anticipated, the Λ-dependent terms cancel in the sum M 2 M 2 (Λ) + δm 2 (Λ), which then provides the physical thermal mass within the present accuracy: which agrees again with eq. (37). M 2 = M 2 (Λ) + δm 2 (Λ) = g 2 T 2 3 π g3 T, (60)

19 Dimensional reduction puts emphasis on the static field φ 0 : 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 [4] and references therein. Collective modes The gluon propagator G µν in Coulomb s gauge has the following non-trivial components, corresponding to longitudinal (or electric) and transverse (or magnetic) degrees of freedom: G 00 (ω,p) L (ω, p), G i j (ω,p) (δ i j ˆp i ˆp j ) T (ω, p), (6) where: L (ω, p) = p 2 + Π L (ω, p), T (ω, p) = ω 2 p 2 Π T (ω, p), (62) and the electric (Π L ) and magnetic (Π T ) polarization functions are defined as: Π L (ω, p) Π 00 (ω, p), Π T (ω, p) 2 (δ i j ˆp i ˆp j )Π i j (ω,p). (63) Explicit expressions for these functions can be found in [4]. The dispersion relations for the modes are obtained from the poles of the propagators, that is, p 2 + Π L (ω L, p) = 0, ω 2 T = p 2 + Π T (ω T, p), (64) for longitudinal and transverse excitations, respectively. The solutions to these equations, ω L (p) and ω T (p), are displayed in fig. 9.b. Both dispersion relations are timelike (ω L,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 m D / 2 (commonly referred to as the asymptotic mass ).

20 (a) 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. (b) 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: Π L (0, p) = m 2 D, Π T (0, p) = 0, (65) and therefore: L (0, p) = p 2 + m 2, D T (0, p) = p 2, (66) which clearly shows that the Debye mass m D acts as an infrared cut-off gt in the electric sector, while there is no such cut-off in the static magnetic sector.

21 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: ImΠ µν dω R (ω,p) = πm2 D ω 4π vµ v ν δ(ω v p). (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 Π T (ω, p) is dominated by its imaginary part: and therefore Π T (ω p) = i π 4 m2 D ω p + O(ω2 /p 2 ), (68) T (ω p) p 2 i(πω/4p)m 2. (69) D Thus ImΠ T (p) acts as a frequency-dependent IR cutoff at momenta p (ωm 2 D )/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 [5]. 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.

22 /2 +/8 +/ FIGURE 0. 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: βω[d] = logz = 2 TrlogD TrΠD + Φ[D], (70) 2 where Tr denotes the trace in configuration space, β = /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 (D 0 is the free propagator): D = D 0 + Π, (7) and Π itself, which is the sum of the one particle irreducible (PI) diagrams with two external lines, is obtained from Φ[D] by: δφ[d]/δd = Π. (72) 2 By using this relation, one can check that Ω[D] is stationary under variations of D (at fixed D 0 ) around the physical propagator: δω[d]/δd = 0 for D = (D 0 + Π). (73)

23 We shall usually refer to eq. (7) 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 = d 4 k (2π) 4 n(ω)( Ilog( ω 2 + k 2 + Π) IΠD ) + T Φ[D]/V (74) where n(ω) = /(e βω ), 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) = dk 0 2π ρ(k 0,k) k 0 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. 0. 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 m 2. (The vacuum mass is set to zero.) The gap equation (72) reads then: m 2 = λ I(m). (77) 2 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) 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. [6, 62, 63, 64].

24 as: µ ε m 2 = λ 0 m 2 32π 2 ( ) 2 µ2 d 3 + log ε m 2 + k n(ε k ) + λ 0 (2π) 3 + O(ε), (78) 2ε k and relate λ 0 to the renormalized coupling λ by: λ = µε λ 0 + 6π 2 ε. (79) Then, the equation takes the following, manifestly finite, form (for ε 0): m 2 = λ d 3 ) k n(ε k ) 2 (2π) 3 + (log λm2 m2 ε k 32π 2 µ 2. (80) One should stress a subtle point in the procedure: expressing the bare coupling λ 0 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 m 2, but only for these values which satisfies the finite equation (80). Note that the 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 dlog µ = λ 2 6π 2, (8) (this follows by noticing that λ 0 is independent of µ 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. (8), one can check that the solution m 2 of eq. (80) is independent of µ. By using the previous formulae, one can easily compute the pressure[65]: P = T d 3 k (2π) 3 log( e βε k ) + m2 2 d 3 k (2π) 3 n(ε k ) 2ε k + m4 28π 2, (82) which differs from the pressure P 0 (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 g 3, and, indeed, it can be easily verified that the weak coupling expansions of eqs. (80) and (82) include the correct perturbative effects of order g 2 and g 3 (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 g 3, in the sense of being monotonic with g (unlike the perturbative approximants of order g 3 ), and showing

25 m=t g(μ =2ßT ) FIGURE. Numerical solution of the mass gap equation (80) with µ = 2πT in comparison with the perturbative result to order g 2 and g 3 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. P=P g(μ =2ßT ) FIGURE 2. The 2-loop Φ-derivable pressure (82) in comparison with the perturbative results to order g 2 and g 3 (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 g 2 estimates). This behaviour is illustrated in Figs. and 2 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