162 Chapter 6. Transport processes at strong coupling
quarks with four-velocityvµ. In the rest frame of a heavy quark bound state,v = (1,0,0,0), the operatorsOE andOBare
OE =EA·EA, OB =BA·BA, (6.4)
whereEAandBAare the color electric and magnetic fields.
For the gluons the stress-energy tensor is given by Tµν = 14gµνGαβAGαβA
−GµαAGανA. (6.5)
Using this we may write
OE =T00− 14trG2, OB =T00+ 14trG2. (6.6) cE andcBare polarizabilities which may be determined from meson mass shifts,
δM =−hHIi=−(cB+cE)T00− 14(cB−cE)trG2. (6.7) If the constituents of the charmonium dipole are non-relativistic it is expected that the mag-netic polarizability cB is of second order in the four-velocity O(v2) relative to the electric polarizability cE. For heavy quarks we assume that cB can be neglected and set cB = 0.
Below we will generalize our results to the holographic context.
We expect the kinetics of the heavy meson dipole in the medium to be described by Langevin equations for long time scales compared to the medium correlations
dxi
dt = Mpi , (6.8)
dpi
dt =ξi(t)−ηDpi, (6.9)
hξi(t)ξj(t′)i =κδijδ(t−t′). (6.10) Here theξiare components of an arbitrary force acting on the heavy dipole,x, pare its position and momentum, respectively. In this context the coefficient κ is the second moment of the force applied to the dipole. The drag coefficientηDand the fluctuation coefficientκare related by the Einstein equation
ηD = κ
2MT , (6.11)
with the massM and temperatureT.
In the regime of times long compared to medium correlations but short compared to the time the system needs to equilibrate, we can neglect the drag coefficient in equation (6.9).
Then the fluctuation coefficientκis obtained from the correlation of microscopic forcesFi κ= 1
3 Z
dthFi(t)Fi(0)i. (6.12)
The thermodynamical forceF acting on the heavy dipole is determined by the gradientF =
−∇U of the potentialU identified as the interaction part of the Lagrangian U =Lint =
Z
d3xφ†vcE
E2
2 (x, t)φv(x, t). (6.13)
6.4. Charmonium diffusion 163
So the fluctuation coefficient is given by κ= − lim
ω→0
2T 3ω
c2E N4
Z d3q
(2π)3q2ImGRE2E2(ω,q), (6.14) where
GRE2E2(ω,k) =−i Z
d4xe−i~k·~xΘ(x0)hOE
2 (x, t)OE
2 (0)i. (6.15)
The three-momentum factor q2 in (6.14) comes from the derivative in the potential gradi-ent∇U and the term proportional toω2 vanishes in the zero-frequency limit.
In the case of QCD the integral in (6.14) evaluates to κ= c2E
N2 64π5
135 T9. (6.16)
The fluctuation coefficient κwhich we identified with the second moment of the force acting on the dipole gives the rate of momentum broadening. We also identify the coefficientscE, cB
as the electric and magnetic polarizabilities. These and analogous coefficients in the following are calledαwith an appropriate index (e.g.αF, αT).
Linear perturbations of N = 4 Super-Yang-Mills theory Our aim is to calculate the heavy meson diffusion coefficientκfrom gauge/gravity duality. This requires the calculation of the two-point correlators as well as of the polarizabilities in N = 4 Super-Yang-Mills theory.
To set the scene we transfer the results of the preceding section toN = 4SU(N) Super-Yang-Mills theory. We consider the effective Lagrangian
L =φ†viv·∂φv+ αT
N2φ†vTµνφvvµvν + αF
N2φ†vtrF2φv, (6.17) which is a linear perturbation ofN = 4Super-Yang-Mills theory by two composite operators.
The polarization coefficients αT, αF will be determined below from meson mass shifts in gauge/gravity duality.
The force on the dipole now becomes F(t) =−
Z
d3xφ†v∇h
αTN2Tµνuµuν + αF
N2trF2i
φv. (6.18)
Again there will be no cross-terms. In the gauge/gravity duality this is reflected in the fact that at tree level in supergravity, there is no contribution tohT00(x)trF2(y)i= δg00(x)δΦ(y)δ2 W = 0, withg00the metric component andΦthe dilaton.
We proceed by calculating the stress tensor andtrF2 correlators from graviton and dilaton propagation through the AdS-Schwarzschild black hole background. Moreover we determine the polarizabilityαT by considering the linear response of the meson mass to switching on the black hole. The polarizabilityαF is obtained by determining the linear response of the meson mass to a perturbation of the dilaton. As an example we choose the dilaton deformation of Liu and Tseytlin [134].
164 Chapter 6. Transport processes at strong coupling
AdS/CFT setup We consider two different gravity backgrounds, the thermal and the dila-ton one. Starting with the gravity dual of N = 4 theory at finite temperature given by the AdS-Schwarzschild black hole with Minkowski signature (see e.g. [28]). Asymptotically near the horizon the corresponding metric returns to AdS5 ×S5. The black hole background is needed in the subsequent both for calculating the necessary two-point correlators hT00T00i andhtrF2trF2i, as well as for obtaining the polarizability contribution for the linear response of the meson masses to the temperature.
We make use of the coordinates of [37] to write the AdS-Schwarzschild background in Minkowski signature as
ds2 =r R
2
−f2
f˜dt2+ ˜fdx2 R r
2
dr2+r2dΩ25
, (6.19)
with the metricdΩ25 of the unit5-sphere, where f(r) = 1− r4H
r4, f˜(r) = 1 + r4H
r4 , rH = T πR2
√2 , R4 = 4πgsNcα′2, λ = 4πNgs, gY M2 = 4πgs.
(6.20)
In this section we will work in a coordinate system with inverted radialAdS-coordinateu = R2/r2 used in e.g. [28]. In these coordinates, the deformed AdS5 part of the metric (6.19) reads
ds25 = (πT R)2
u −f(u)dx02+ dx2
+ R2
4u2f(u)du2, (6.21) withf(u) = 1−u2and the determinant square root√
−g5 = R104u(πT3 )8.
A further necessary ingredient is the polarizability contribution obtained from the linear response of the meson mass totrF2. The gravity dual of the operatortrF2 is the dilaton field.
Therefore, we consider a dual gravity background with a non-trivial dilaton flow. We choose the dilaton flow of Liu and Tseytlin [134] which corresponds to a configuration of D3 and D(-1) branes.
In order to fix notation, we write down the string frame metric of [134] in the form ds2string =eΦ/2ds2Einstein =eΦ
"
r R
2
d~x2 + R
r 2
dr2+r2dΩ25
#
. (6.22) The type IIB action in the Einstein frame for the dilatonΦ, the axionCand the self-dual gauge field strengthF5 =⋆F5reads
S = 1 2κ210
Z
d10x√
−g
R − 1
2(∂Φ)2− 1
2e2Φ(∂C)2− 1
4·5!(F5)2 +. . .
, (6.23) with the curvature scalarRand the ten-dimensional gravity constant
1
2κ210 = 1
(2π)7(α′)4g2s . (6.24)
6.4. Charmonium diffusion 165
Solving the equations of motion derived from (6.23), we obtain the dilaton solution eΦ=gs(1 + q
r4). (6.25)
Note that the parameter qwe are using here differs from that given in [134] in the following wayq= Rλ8qLiu&Tseytlin.
The dilaton is dual to the field theory operator trF2 appearing in the gauge theory ac-tionSgauge =R
d4xtrF2+. . .. So the expectation value or one-point function of this operator is given by
htrF2i= δS
δΦ = N2
2π2R8q . (6.26)
Correlators According to (6.14) and (6.18) the heavy meson diffusion coefficient is given by
κ=−lim
ω→0
2T 3ω
Z d3q
(2π)3q2αF
N2 2
ImGRF2F2(ω,q) +αT
N2 2
ImGRT T(ω,q)
, (6.27) where the bracket is the imaginary part of the force (6.18) correlator GRFF. We need to cal-culate the retarded momentum space correlatorGRT T of the energy momentum tensor compo-nentT00which is dual to the metric perturbationh00, and the 2-point correlatorGRF2F2 of the operator trF2dual to the dilatonΦ. On the gravity side both field correlators are computed in the black hole background (6.19) placing the dual gauge theory operator correlation functions at finite temperature.
For simplicity in this section we work in the conventions and coordinates of [28]. Espe-cially the radial coordinate is changed from r touwith the horizon atu = 1. These are the same coordinates we have used in section 3.1.2. We apply the method of [27] to find the two-point Minkowski space correlators from the classical supergravity action as described in section 3.1.2 .
The classical gravity action for the graviton and dilaton is obtained from (6.23) as S = 1
2κ25 Z
dud4x√
−g5
(R −2Λ)− 1
2(∂Φ)2+. . .
, (6.28)
where
1
κ25 = Ω5
κ210 = N2
4π2R3 . (6.29)
So comparing to (3.7) we get
BΦ =− 1 4κ25
√−g5guu. (6.30)
The equation of motion derived from (6.28) in momentum space reads Φ′′− 1 +u2
uf(u)Φ′+w2−q2f(u)
uf(u)2 Φ = 0, (6.31)
166 Chapter 6. Transport processes at strong coupling
with the dimensionless frequencyw=ω/2πT and spatial momentum componentq=q/2πT. The equation of motion (6.31) has to be solved numerically with incoming wave boundary condition at the black hole horizon. Computing the indices and expansion coefficients near the boundary and horizon as done in [33, 34], we obtain the asymptotic behavior as linear com-bination of two solutions. We get the correlators by applying the matching method described in section 3.1.3. Solving (6.31) and matching the asymptotic solutions, we obtain
ωlim→0
Z d3q (2π)3
q2
3ω[−2TImGRF2F2(ω,q)] =N2T9C1. (6.32) The corresponding result for the energy-momentum component correlator is obtained in an analogous way from the action and equations of motion already discussed in [29] . The final result is
ωlim→0
Z d3q (2π)3
q2
3ω[−2TImGRT T(ω,q)] = N2T9C2. (6.33) Note that the real numbersC1, C2here are numerical values which are currently being checked.
The final results will appear in [4].
Polarizabilities Looking at the meson diffusion formula (6.27) we realize that we have to determine the polarizabilities αT, αF. In analogy to the QCD calculation we consider the effective SYM Lagrangian (6.17) leading to the meson mass shift
δM =−αT
T00
−αF
trF2
. (6.34)
On the other hand the mesons are dual to the gravity field fluctuations describing the embed-ding of our D7-brane (cf. section 2.3) and their masses are determined by the dynamics of the gravity fluctuations. We have already reviewed how to compute meson masses from D7-brane embeddings in section 2.3. One of the major results there is the meson mass formula for scalar excitations (2.80) which depends on the angular excitation numberl as well as on the radial excitationn. From here on we will consider the case of the lowest angular excita-tionl = 0only. Picking up the QCD idea that the interaction with external color-fields shifts the meson mass linearly (cf. equation (6.7)) we write down an analogous relation for the gauge condensatehtrF2i
δM =−αFhtrF2i. (6.35)
The constant of proportionality αF is identified with the polarization. It can be calculated by determining the meson mass shiftδM at a given value of the gauge condensatehtrF2i ∝ q. Let us now determine the mass shift analytically. This requires the further assumption thatq¯= q/L4 is small. Next we derive the equation of motion for D7-brane fluctuations as shown in [135] and subsequently linearize that equation inq, which then gives
−∂ρρ3∂ρφ(ρ) = ¯M2 ρ3
(ρ+ 1)2φ(ρ) + ∆(ρ)φ(ρ), (6.36) where the operator∆(ρ)is given by
∆(ρ) =−4¯qTeaney
ρ4
(ρ2 + 1)3 ∂ρ. (6.37)
6.4. Charmonium diffusion 167
Setting the operator ∆ ≡ 0returns the case of vanishing gauge condensatehtrF2i ≡ 0. So the term ∆φdescribes the meson mass shift generated by the condensate on the level of the equation of motion. We consider the lightest of the mesons by choosing the lowest radial excitation number n = 0and the solution at vanishing condensate is φ0. Any deviationδφ0 from the solution φ0 of the case qTeaney = 0 may be written as a linear combination of the functionsφn, which are a basis of the function space of all solutions,
φ(ρ) =φ0(ρ) + X∞ n=0
anφn(ρ), an≪1, (6.38)
M¯2 = ¯M02 +δM¯02, δM¯02 ≪1. (6.39) Plug this Ansatz into the equation of motion derived in [135], make use of the radial fluctu-ation equfluctu-ation of motion at vanishingq (2.73) and keep terms up to linear order in the small parametersan,q¯andδM02to get
ρ3 (ρ2+ 1)2
X∞ n=0
anM¯n2φn(ρ) =δM¯02 ρ3
(ρ2+ 1)2 φ0(ρ) + ¯M02 ρ3 (ρ2+ 1)2
X∞ n=0
anφn(ρ) + ∆(ρ)φ0(ρ).
(6.40) We now multiply this equation byφ0(ρ), integrate overρ∈[0,∞[and make use of the fact that the φnare orthonormal and of the non-interacting lowest mode solutionφ0 = √
12/(ρ2+ 1) in order to rewrite
δM¯02 =− Z∞
0
dρ φ0(ρ) ∆(ρ)φ0(ρ)
= 4¯q Z∞
0
dρ ρ4
(ρ2+ 1)3 φ0(ρ)∂ρφ0(ρ)
=−8 5q.¯
(6.41)
FromδM¯02 = 2 ¯M0δM¯0 we therefore obtain δM0 = L
2R2 δM¯02
M¯0
=−
√2
5R2L3 q =−2π2
√2 5
R6 N2L3
trF2
, (6.42)
where we inserted the meson mass formula (2.80) and switched back to dimensionful quanti-ties. By comparison with (6.35) we may now identifyαF
αF = 2π2
√2 5
R6 N2L3 =
√2 20π
λ3/2
Mq3N2. (6.43)
The calculation of the polarizability αT is completely analogous. We are now looking for the proportionality constant of meson mass shifts with respect to deviations from zero temperature,
δM =−αT
T00
. (6.44)
168 Chapter 6. Transport processes at strong coupling
The vacuum expectation value
T00
= 1
2π2N2T4 (6.45)
is proportional to(temperature)4. We eventually obtain the polarizabilityαT as αT = 9√
2 160π
λ3/2
Mq3N2. (6.46)
These results (6.43) and (6.46) for small values ofq¯agree very well with the numerical calcu-lation we performed in parallel (not shown here, see [4] for details) relaxing the assumption thatq¯needs to be small.
Result Substituting our polarizations (6.43) and (6.46), as well as the correlators (6.33) and (6.32) into the Kubo equation for the heavy meson diffusion coefficient (6.27) yields
κ=
√2 20π
λ3/2 Mq3N2
!2
(C1N2T9) + 9√ 2 160π
λ3/2 Mq3N2
!2
(C2N2T9) =C3
λ3T9
Mq6N2 , (6.47) with numerical valuesC1, C2, C3 which are currently being checked. The final results will appear in [4].
This strong coupling result resembles the weak coupling result obtained from a perturbative calculation very closely
κ= ˜C3 λ3T9
a−60 N2 , (6.48)
where the inverse Bohr radiusa−01 replaces the quark mass Mq as the characteristic energy scale. In order to compare the weak coupling result (6.48) to the strong coupling result (6.47), we need to divide by the corresponding mass shifts(δM)2 such that the Bohr radius and the quark mass cancel from the results. The number C˜3 is still being checked. Nevertheless, our preliminary results indicate that the ratio κ/(δM)2 is about five times smaller at strong coupling compared to its value at weak coupling. It is reassuring that the viscosity to entropy quotient shows an analogous behavior being much smaller at strong coupling [26]. After the exact valuesC3,C˜3are confirmed we will draw a more precise conclusion [4].