Analytical methods: correlators and dispersion relations

3.1. Holographic correlation functions 49

we have β = ±ω/(4πT) and only one of the two signs produces a regular solution. In Minkowski signature this is completely different since there we computeβ =±iω/(4πT)and both signs can produce regular solutions, thus leaving an ambiguity which needs to be fixed by another requirement. Now the main achievement in [30] was to single out such a requirement

U=0

V=0

R

P L

F

Figure 3.2: The Penrose diagram for AdS containing a black hole as shown in [30].

which is in general applicable to anyn-point correlator. This requirement involves applying boundary conditions at the boundaries of different quadrants of the Penrose diagram shown in figure 3.2 and forming a superposition of those. The diagram shows the causal structure of our asymptotically AdS space (which contains a black hole) in Kruskal coordinates. In our earlier attempt to fix boundary conditions we only considered the R-quadrant and its boundaries. The prescription of [30] takes into account that the full space-time contains four quadrants.

Nevertheless, in what follows it will be sufficient to use a simplified boundary condition, the incoming wave boundary condition which allows us to restrict ourselves to the R-quadrant, to use the original Minkowski coordinates and it finally enables us to calculate two-point functions as discussed in the following section 3.1.2. It is argued in [30] that the general but also more complicated prescription involving Kruskal coordinates in the case of two-point correlators reduces to the (simple) prescription that we are about to use.

50 Chapter 3. Holographic methods at finite temperature

a method has been developed to find the correlators analytically for a field theory at finite temperature and without quarks. The main idea of this approach is to use the ratio of four-momentum and temperature~k/(2πT) := (w,0,0,q)² as an expansion parameter. Then the fields are expanded in a perturbation series in orders of w andq² and exact solutions to the equations of motion can be obtained up to the desired order inwandq². This kind of expan-sion is known from statistical mechanics and goes by the name of hydrodynamic expanexpan-sion.

Note that we only consider the diffusive modes with this choice. In order to find for example the sound modes and their damping we would have to consider an expansion inw, qn[104].

From the solutions expanded inwandq²we will obtain the correlators of the operatorO. The poles of these correlators can be read off directly from the analytical expressions giving the dispersion relations w(q). Note, that we work in the geometry described in [28] where the fluctuations are chosen along thex3-direction such that~x = (x0 =t, 0, 0, x3 =z). Further-more we choose the gauge in which A4 ≡ 0and we assume that the remaining space-time directions have already been compactified such that we have to consider a five-dimensional theory only.

The correlator recipe Let us review the three-step recipe to obtain two-point correlation functions motivated and developed in [27]. We calculate the retarded two-point correlatorG^R of the operatorOin Minkowski space. The operatorOis dual to a field which we denote byφ, whereφcan be a scalarΦ, vectorAµor tensor fieldTµν merely changing the index structure.

Step number one is to find the part of the action which is quadratic in the fieldφdual toO S⁽²⁾ =

Z

dud⁴xB(u)(∂uφ)²+. . . , (3.7) where the factorBdepends onuand the momenta only, collecting metric components and all other factors in front of the derivatives(∂uφ)². Now the second step is to solve the equation of motion for the fieldφ. We rewrite the space-time equation of motion in Fourier space such that all derivatives except∂uφ=:φ^′ can be expressed in terms of four-momenta~k

0 =φ^′′+a(~k, u)φ^′+b(~k, u)φ . (3.8) This second order differential equation in special cases can be solved analytically in the hydro-dynamic limit of smallw,q² ≪1.³ The general solution can be split into the field’s boundary valueφ^bdy(~k)and the bulk functionF(u, ~k)

φ(u, ~k) =φ^bdy(~k)F(u, ~k). (3.9) To clearly illustrate this step, we will consider details of this general procedure in the specific example below. In step three we finally assemble the solutionF(u, ~k)obtained in step two and the coefficientB(u)from step one to obtain the retarded correlator in Fourier space

G^R(~k) =−2B(u)F(u,−~k)∂uF(u, ~k)

u→0. (3.10)

2This choice for the four-momentum is adapted to the symmetries of the problem we will consider in this section.

3If the coefficientsa, bare sufficiently complicated (they might be given only numerically) we have to reside to numerical methods, two of which are explained in [34, 33] and [59] reviewed in section 3.1.3 of this work.

3.1. Holographic correlation functions 51

An example To illustrate the three steps in more detail we consider the example ofN = 4 supersymmetric Yang-Mills theory with anR-charge currentJ^µdual to the vector fieldAµin five-dimensional supergravity. The part of the action quadratic in the gauge fieldAis given by

S⁽²⁾ =− N² 16π²

Z

dud⁴xp

−g(u)FµνF^µν. (3.11) In order to place our field theory at finite temperature, we will work in the dual AdS black hole background

ds² = (πT R)²

u [−f(u)dt²+ dx²] + R²

4u²f(u)du²+R²dΩ52, (3.12) with the radial AdS-coordinateu ∈ [0,1], the horizon atu = 1, spatial infinity atu = 0and the functionf(u) = 1−u². This metric is obtained from the standard AdS black hole metric with radial coordinater by the transformationu = (r0/r)². The temperatureT =r0/(πR²) is a function of the AdS-radiusRand the black hole horizonr₀.

Applying step one of our recipe to the quadratic super-Maxwell action (3.11), we find the coefficient

B(u) =− N² 16π²

p−g(u)g^uug^νν′, (3.13)

(hiding the index structure on the left hand side).

Hydrodynamic expansion and equation of motion Now in step two of the recipe we take a closer look on the method for solving the equation of motion for our field Aµ. Us-ing (3.11) in the Euler-Lagrange equation, we get the equation of motion

0 = ∂_ν[p

−g(u)g^µ̺g^νσ(∂_̺A_σ −∂_σA_̺)]. (3.14) We make use of the Fourier transformation

Ai(u, ~x) =

Z d⁴k

(2π)⁴e^{−iωt+ik·x}Ai(u, ~k). (3.15) Rewritten in Fourier space we may split the equation of motion (3.14) into five separate equa-tions labeled by the free indexµ= 0,1,2,3,4

A^′′_t − 1

uf(q²At+wqAz) = 0, (3.16) A^′′_x,y+f^′

f A^′_x,y+ 1 uf(w²

f −q²)At= 0, (3.17)

A^′′_z +f^′

f A^′_z+ 1

uf²(w²Az+wqAt) = 0, (3.18) wA^′_t+qf A^′_z = 0. (3.19)

52 Chapter 3. Holographic methods at finite temperature

Note thatA_tandA_zneed to satisfy the coupled set of three equations (3.16), (3.18) and (3.19) while the transversalAx,ydecouple and merely have to satisfy the stand-alone equation (3.17) separately. However, we can decouple the system forAt, Az rewriting (3.16) as

Az = uf

wqA^′′_t − q

wAt, (3.20)

and use it to substituteAz in (3.19) yielding a single second order equation forA^′_t A^′′′_t +(uf)^′

uf A^′′_t + w²−q²f

uf² A^′_t= 0. (3.21)

Note that the appearance of the third derivativeA^′′′_t is a generic feature of this particular ex-ample and has nothing to do with the general method. Since this equation does not depend on any of the other field components we will solve it separately and impose conditions for the other components afterwards. Note that (3.21) has singular coefficients at the horizonu = 1 (and at the boundary as well). We have to invoke the indicial procedure in order to split the singular behavior(1−u)^β from the regular partF(u)of the solution

A^′_t= (1−u)^βF(u). (3.22)

The indicial exponentβ characterizing the singular behavior is determined by settingA^′_t → (1−u)^β, expanding the singular coefficients of (3.21) around the horizonu= 1keeping only the leading order term and evaluating (3.21) with these restrictions. The result is a quadratic equation forβ giving

β =±iw

2 . (3.23)

By the variable change to a radialξ = −ln(1−u)with0 < ξ <∞we see that the positive sign inβ describes an outgoing wave at the horizonA^′_t(ξ)∝ e^−iwξ/2 while the negative sign gives an incoming waveA^′_t(ξ) ∝ e^iwξ/2. We select the latter solution to be the physical one since no radiation should come out of the black hole. This is often referred to as the incoming wave boundary condition.

Now we are ready to write down the hydrodynamic expansion in momentum-temperature ratiosw,q² ≪1for the regular partF(u)of the solution

F(u) = F0+wF1+q²G1 (3.24)

+w²F₂+q⁴G₂+wq²H₁+. . . . (3.25) We will refer to the first line (3.24) as the leading order or first order hydrodynamics terms, while we coin the second line (3.25) second order hydrodynamics terms. Substituting the leading order hydrodynamic expansion (3.24) into the equation of motion (3.21) withA^′_t = (u−1)^−iw/2F(u)and comparing coefficients in the ordersO(1),O(w)andO(q²)yields three equations for the three hydrodynamic functionsF0, F1, G1

F₀^′′+(uf)^′

uf F₀^′ = 0, (3.26)

F₁^′′+(uf)^′

uf F₁^′ + i 2[ 1

(u−1)² − (uf)^′

uf(u−1)]F0 = 0, (3.27) G^′′₁ +(uf)^′

uf G^′₁− 1

ufF0 = 0. (3.28)

3.1. Holographic correlation functions 53

Note that we can compute higher order corrections in this hydrodynamic perturbation ap-proach by inclusion of higher order terms, e.g. the second order terms (3.25). We would have to compare coefficients up to the desired order of accuracy and would end up with e.g.

three further equations added to (3.26), (3.27) and (3.28) for three additional hydrodynamic functionsF2, G2, H1 in the case of second order corrections.

The solutions to (3.26), (3.27) and (3.28) can be obtained analytically if we start out noting that we may setF0 =constant=C. Then we get⁴

F₁ =C₂ +iC

2 ln(u−1)−C₁lnu+C₁

2 ln{(u+ 1)(u−1)} (3.29) with two undetermined integration constantsC1, C2. These can be fixed by recalling that we have already chosen the constant order in F(u)independent fromu, w, q²to be given by C.

So we now have to impose the condition on our solution forF1 that it gives no corrections to this constant C, meaning lim

u→1F1 = 0. In this limit two of the terms in F1 become divergent and the constants have to be chosen such that these cancel each other. After application of this procedure toG1as well, we are left with

F1 = iC

2 ln 2u²

u+ 1, (3.30)

G1 =Cln1 +u

2u . (3.31)

Now we have a first order solution for the derivativeA^′_t. We can also fix the constantC in terms of boundary valuesA^bdy for the physical fields. This is important becauseC contains thew-pole structure of the solution as we will see shortly. First we recall thatlim

u→0At =A^bdy_t and lim

u→0A_z = A^bdy_z . Now substitute the solution for A^′_t into equation (3.20) and take the boundary limit of this expression. This yields

C = q²A^bdy_t +wqA^bdy_z

iw−q²+O(w²,q⁴,wq²). (3.32) The denominator of (3.32) contains the poles of the solution which are the poles of the retarded correlator as well.

Taking our third and final recipe-step we assemble the correlator for time components of theR-charge current

G^R_tt =−2 lim

u→0(− N² 16π²)√

−gg^uu δ²

δA^bdy_t δA^bdy_t (g^ttA^′_tAt+g^zzA^′_zAz), (3.33) where the double functional derivative encodes the step of selecting the terms in the action which are relevant (meaning quadratic in the field A^bdy_t ) in order to be more illustrative here.

4Note, that the complex logarithmlnz being a multivalued function has branch points atz = 0, ∞and in general a branch cut is defined to extend between these points on the negative axis. Here we define the complex logarithm on the first Riemann sheet, such that e.g.ln(−1) = +iπ. All the equations here should be read with this in mind.

54 Chapter 3. Holographic methods at finite temperature

We finally get

G^R_tt = N²T 16π²

q²

iω−Dq² , (3.34)

with the constantD = 1/(2πT)which is identified with the diffusion coefficient. This inter-pretation is best understood by noting that the diffusion equation

∂tJ^t =D∇²J^t (3.35)

can be Fourier transformed to

−iωJ^t=D(iq)²J^t. (3.36) This suggests that the retarded correlator we found has the correct pole structure to be the Greens function for a diffusion problem, in our case this is the diffusion ofR-charges.

Dispersion relations The dispersion relation for theR-charge currentJ^µto first hydrody-namic order is given by

0 =iw−q² +O(w²,q⁴,wq²). (3.37) Computing the second order hydrodynamics corrections as described above, we obtain the dispersion relation

0 =iw−q²+ ln 2(w² 2 + i

2wq²−q⁴) +O(w³,q⁶,w²q⁴). (3.38) Since this equation is quadratic in wone at first suspects that two solutions exist, but if we solve (3.38) and then (recallingw,q² ≪1) expand both solutions inw, we get

w=−iq²−iln 2q⁴+O(q⁶), (3.39) w_discard =− 2i

ln 2 +iln 2q⁴+O(q⁶). (3.40) Only the first (3.39) of these two solutions is compatible with our initial assumption thatw∼ q² ≪1since the second solution (3.40) has a constant leading order with an absolute value of order one.

Dispersion relations and correlators of other operatorsO (e.g. the energy-momentum ten-sorT^µν) dual to other fieldsφare obtained in the same way.

Im Dokument Holographic quark gluon plasma with flavor (Seite 55-60)