Transport equations for quark-gluon plasma

The distribution of hard particles in the QCD plasma is described by a Hermitian density matrix in color space. For a SU(N_c) color group, the distribution of hard (anti-)quarks is a N_c×N_c matrix while for gluons it is a (N_c²−1)×(N_c²−1) matrix. The linearized kinetic equations for the QCD plasma look formally as Vlasovequations [75, 69]

V ·D_Xδn(k, X) = −gV_µF^µν(X)∂_νn(k), V ·D_Xδn(k, X) =¯ gV_µF^µν(X)∂_νn(k)¯ ,

V · D^XδN(k, X) = −gVµF^µν(X)∂νN(k). (2.19) In the above equations,δn,δ¯n andδN are the fluctuating parts of the density matrices for quarks, anti-quarks and gluons, respectively. We discuss here how the plasma, which is (on

2.3 Transport equations for quark-gluon plasma 15

average) colorless, homogeneous and stationary, responds to small color fluctuations. The distribution functions are assumed to be of the form

Q_ij(k, X) = n(k)δ_ij +δn_ij(k, X), Q¯_ij(k, X) = ¯n(k)δ_ij +δn¯_ij(k, X),

G_ab(k, X) = N(k)δ_ab+δN_ab(k, X), (2.20) where the functions describing the deviation from the colorless state are assumed to be much smaller than the respective colorless functions. Note that δn and δn¯ transform as a color vector in the fundamental representation (δn = δn^at^a, δn¯ = δn¯^at^a) and δN transforms as a color vector in the adjoint representation (δN = δN^aT^a). Here, t^a and T^a are the SU(N_c) group generators in the fundamental and adjoint representations respectively. The velocity of hard and massless partons is given byV_µ= (1,k)ˆ ≡(1,k/k). D_µand Dµare the covariant derivatives which act as

D_µ=∂_µ−ig[A_µ(X), . . .], Dµ=∂_µ−ig[Aµ(X), . . .], (2.21) withA_µ and Aµ being the mean-field or background four-potentials

A^µ(X) =A^µ_a(X)τa, A^µ(X) =A^µa(X)Ta. (2.22) F_µν andFµν are the mean-field stress tensors with a color index structure analogous to that of the four-potentials.

The presence of covariant derivative on the left-hand side of Eq. (2.19) is a new feature of the non-Abelian theory. These equations are covariant under local gauge transforma-tions [76, 77].

In the weak-coupling limit, we neglect terms of subleading order in g and the theory becomes effectively Abelian as D_X → ∂_X and F_µν → ∂_µA_ν −∂_νA_µ. The color channels decouple and Eq. (2.19) can be written for each color channel separately [69, 78]

V ·∂_Xδn^a(k, X) = −gV_µF^{µν a}(X)∂_νn(k), V ·∂_Xδn¯^a(k, X) = gV_µF^{µν a}(X)∂_νn(k)¯ ,

V ·∂_XδN^a(k, X) = −gV_µF^{µν a}(X)∂_νN(k), (2.23) and the induced current in the fundamental representation reads

j_ind^{µ a} =−g Z

k

V^µ{2N_cδN^a(k, X) +N_f[δn^a(k, X)−δn¯^a(k, X)]} . (2.24) As seen, the induced current occurs due to the deviation from the colorless state.

It is more convenient to solve these linearized equations in the momentum space. Taking the quark distribution as an example, we firstly perform theFourier transform

(−i ω+ip·v)δn^a(k, P) +gV_µF^{µν a}(P)∂_νn(k) = 0. (2.25)

16 2 Kinetic theory for hot QCD plasmas

It is now straightforward to obtain the fluctuating part for quark distribution δn^a(k, P) = −gV_µF^{µν a}(P)∂_νn(k)

We introduce a small positive number ǫ in order to get the retarded solution. Similarly, we can solve the linearized kinetic equations for anti-quarks and gluons and the induced current finally reads where we combined the gluon and quark distributions in

f(k) = 2N_cN(k) +N_f[n(k) + ¯n(k)] . (2.28) We also present the more general solution for the induced current which reads [70, 77]

j_ind^µ (X) =g² where the gauge parallel transporter defined in the fundamental representation is

U(X, Y) =Pexph

and the retardedGreen function satisfies

Kµ∂^µGK(X) =δ⁽⁴⁾(X), (2.31)

or

GK(X) =E_k⁻¹Θ(t)δ⁽³⁾(x−vt). (2.32) In Eq. (2.30), P denotes the ordering along the path from X to Y. The Θ function is the usual step-function. There is an analogous formula of the gauge transporter U(X, Y) in the adjoint representation. Consider the contribution at leading order in g, the Fourier transform of Eq. (2.29) gives the same result as Eq. (2.27). Note that in this approximation, the gauge transporters are approximated by unity. However, with the approximation, the induced current is no longer gauge covariant.

The higher order terms in Eq. (2.27) represent gluon vertex corrections. In fact, beyond the linear approximation, we can express the induced current as a formal series in powers of the gauge potentials [76, 77]

j_{ind a}^µ = Π^µν_abA^b_ν+1

2Γ^µνρ_abcA^b_νA^c_ρ+. . . . (2.33)

2.3 Transport equations for quark-gluon plasma 17

These amplitudes are the “hard-thermal-loops” [79, 80, 81] which define the effective theory for the soft gauge fields at the scalegT. The linear approximation in QCD holds as long as the mean-field four-potential is much smaller than the temperature [69]. Actually, for the soft scales we consider here, the mean-field four-potential is at the order of √gT and the linear approximation holds in the weak-coupling limit.

By functional differentiation

Π^µν_ab(P) = δj_{ind a}^µ (P)

δA^b_ν(P) , (2.34)

we obtain theretarded gluon self-energy Π^µν_ab(P) =g²δ_ab

The distribution function in the above equation is arbitrary at this point. Actually, the kinetic theory of quarks and gluons has been shown to be fully consistent with the QCD dynamics not only for equilibrium systems but also the systems which are far from equilib-rium. However, the space-time homogeneity must be invoked. The reliability of the kinetic theory methods is proved in Chapter 3 based on the QCD diagrammatic approach where we calculate the self-energy tensor explicitly for both equilibrium and non-equilibrium systems.

Now we are going to show that this self-energy tensor is symmetric Π^µν = Π^νµ and transverseP_µΠ^µν = 0. We demonstrate the transversality

P_µΠ^µ0=−g²p^j Z

k

∂f(k)

∂k^j . (2.36)

Since the energy density carried by partons is expected to be finite,f(k→ ∞) must vanish.

Consequently, the above integral vanishes as well. To prove thatP_µΠ^µi = 0, we first perform partial integration and assume thatf(k→ ∞) = 0, show the symmetry of the self-energy tensor.

As we discussed above, in linear approximation the current that is induced by the fluctu-ations can be expressed as

j_ind^µ (P) = Π^µν(P)Aν(P). (2.38) Insert Eq. (2.38) intoMaxwellequation

−iP_µF^µν(P) =j_ind^ν +j_ext^ν , (2.39) we obtain

P²g^µν−P^µP^ν + Π^µν(P)

A_µ(P) =−j_ext^ν (P), (2.40)

18 2 Kinetic theory for hot QCD plasmas

where j_ext^ν is an external current. Note that the mean-field stress tensor in Eq. (2.39) is effectively Abelian which is consistent with our approximation. The self-energy is gauge invariant in HL approximation therefore we can write Eq. (2.40) in terms of a physical electric field by specifying a certain gauge. In temporal axial gauge defined byA₀ = 0, we have

−P²δ^ij−pⁱp^j+ Π^ij(P)

E^j(P) = [D⁻¹(P)]^ijE^j(P) =iωj_extⁱ (P). (2.41) The response of the system to the external source is given by

Eⁱ(P) =iωD^ij(P)j_ext^j (P). (2.42) In the next section, we will study the dispersion relations for the collective modes. They can be obtained by finding the poles in the propagatorD^ij(P) defined in Eq. (2.41).