Heavy Quark and Quarkonia Transport in AdS/CFT

Derek Teaney

SUNY Stonybrook

• This work was done in close collaboration with Jorge Casalderrey-Solana

– Jorge Casalderrey-Solana, DT; hep-th/0701123

– Jorge Casalderrey-Solana, DT; hep-ph/0605199

• Work on Quarkonia

– DT, Clint Young, Kevin Dusling, Johanna Erdmenger, Felix Rust, Matthias Gross

Observation:

x

y

bφ

There is a large momentum anisotropy:

v2 ≡

⟨p2

x − p2y

⟩⟨p2

x + p2y

⟩ ≈ 20%

Interpretation

• The medium responds as a fluid to differences in X and Y pressure gradients

Hydro models “work”

Estimate of transport times with Heavy Quarks

• Put a heavy quark in this medium

θδT

T mp~

mT~θδ

• The charm quark undergoes a random walk suffering many collisions

• The relaxation time of the heavy quark is:

τcharmR ∼ M

Tτ

lightR

If you think you know the relaxation time you should be able to compute the charm

spectrum.

Langevin description of heavy quark thermalization:

• Write down an equation of motion for the heavy quarks.

dx

dt=

p

Mdp

dt= − ηDp︸︷︷︸

Drag

+ ξ(t)︸︷︷︸Random Force

• The drag and the random force are related

〈ξi(t)ξj(t′)〉 =κ

3δij δ(t− t′) ηD =

κ

2MT

κ = Mean Squared Momentum Transfer per Time

• Einstein related the diffusion coefficient to the mean squared momentum transfer

D = 2T 2/κ

All parameters are related to the heavy quark diffusion coefficient or κ

Hydro + Brownian Heavy Quarks

x

y

The heavy quarks will either relax to the thermal spectrum and show the same v2

as all thermal particles or not depending on the Drag/Diffusion coefficients and pT .

Summary

1. Suppression and Elliptic Flow are in-

timately related.

2. From the suppression pattern, we

estimate that

D <∼12

2πT

With this diffusion coefficient, I can’t

produce enough elliptic flow.

Matching Langevin to a Microscopic Theory

• Heavy Quarks are Quasi Classical

λde Broglie ∼h√MT

� h

T

• Compare the Langevin process to the microscopic theory

Langevin

dp

dt= −ηDp+ ξ(t)

Microscopic Theory

dp

dt= qE(t,x)

• Match the Langevin to the Microscopic Theory

Langevin

κ =∫dt 〈ξ(t) ξ(0)〉

Microscopic Theory

κ = q2∫dt 〈E(t,x)E(0,x)〉

Diffusion Coefficient↔ Electric Field Correlator

QCD and Color: Heuristic Derivation

MTp=

T

• Use Wong equations for a classical color charge

Langevin

dp

dt= −ηDp+ξ(t)

Microscopic Theory

dp

dt= Tr [QE(t,x)]

∂tQ = −i[Q,A0(t,x)]︸ ︷︷ ︸Color Precession

• Then compare to the microscopic theory:

κ =CR

dA

∫dt⟨Ea(t,x)Uab(t)Eb(0,x)

⟩

Computing κ – Kinetic Theory vs. Correlators

• κ is the mean squared momentum transfer per unit time:

2

)q, ω(

p p’

=⇒ κ =∫p,q

q2 n(p)(1+n(p′))∣∣∣Mglue

∣∣∣2

• κ is the Electric Field Correlator:

q q

p

p’

=⇒ The Same Thing

Wilson Lines and the Diffusion Coefficient:

• Variations of Wilson Lines give electric field correlators

time1t 2t

space⟨1i2

δ2W [δx]δx(t1) δx(t2)

⟩∼ 〈T [E(t2)E(t1) ] 〉

• Put the Wilson line along the Schwinger-Keyldish contour.

κ =13

limω→0

∫dt e+iω t

⟨1i2

δ2W [δx]δx2(t) δx1(0)

⟩

The AdS/CFT correspondence and the Wilson Line – Maldacena, S.Rey

String

Event Horizon Test BraneN−1 D3 Branes

r=0 r0 r=Infinity

Temperature Mass

� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �

� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �

Measure the diffusion of the heavy boson with the dynamics of the string

Testing the string dynamics:

Boundary

xδ

tδ

time

1t

2t tδ

xδ

space

W [δx]

=⇒

Bulk

r=0 0r ∞r=

Scl[δx]

1〈WC [0]〉

〈WC [δy]〉 =1

eiScl[0]eiScl[δy]

Summary of Technical Steps

• Fluctuate the end point of a string at r =∞ at very small frequencies: u = 1/r2

SNG =R2

2π`2s

∫dt dr︸ ︷︷ ︸Area

[1− 1

2

(y2‖f− 4fu

(y′‖

)2)

︸ ︷︷ ︸Quadratic Fluctuations

]

• Solve for the waves along the string in the AdS5 × S5 metric

– Impose incoming boundary conditions for retarded propagator .

• Differentiating twice with respect to δy yields the retarded force force correlator

〈E(t)E(0)〉 ∼ δ2

δy(t)δy(0)〈W [δy]〉 ∼ δ2

δy(t)δy(0)eiSNG[δy]

κ is the imaginary part of the retarded correlator

Results:

• Mean Squared momentum Transfer

κ = π√λ︸ ︷︷ ︸

Large g2Nc

× T 3︸︷︷︸dimension

• Answer:

D =2T 2

κ

=2√λπT

QCD Guesses: Strong Coupling

• Strong Coupling: N = 4 SUSY. λ ≈ 5↔ 20

D =1√

g2Y MNc

42πT

−→ D ' 1.0↔ 2.02πT

• Weak coupling (Aleski Vuorinnen)

2 gluons + 6 scalars + 8 fermions 6= 2 gluons

DQCD

DSY M=

6

1 + Nf

2Nc

≈ 4

• Best guess for QCD from strong coupling

D ≈ 4.0↔ 8.02πT

Compare to weak coupling best guess D ≈ 6/(2πT )

Heavy Quark Diffusion is Parametrically Small

D =1√

g2Y MNc

42πT

η

e+ p=

14πT

c NYM2g

Transport

/(e+p)η

D

‘

Constraint On The Heavy Quark Mass

• To treat the heavy quark as a quasi-classical quasi-particle we need

τR �h

T

• Then we have

τR ∼M

TD D =

2√λπT

• This leads to a constraint on Mass/String Length

M � πT

2

√λ L� ro

• Substituting numbers we have

M � 1.7GeV

(T

0.250 GeV

) (αSY MN

1.5

)1/2

Generalize to Relativistic Heavy Quarks

Before After

LκTκ

• Average Energy Lossdp

dt= −η(p)p (1)

• Transverse Momentum Broadening of a heavy quark (analagous to q)

κT (v) = Mean squared transverse momentum transfer per unit time

κL(v) = Mean squared longitudinal momentum transfer per unit time

Moving Heavy Quarks (HKKKY and S. Gubser)

• Turn on an electric field to accelerate the quark

2 3 4 5u

!0.4

!0.8

!1.2

F

Figure 2: A plot of the function F (u) which determines the string profile, in units where uh = 1.

vv

Figure 3: Schematic drawing of the physical solution (left) in which energy flows toward the horizon,and the unphysical energy solution (right) in which energy flows away from the horizon.

This function is plotted in Figure 2. It vanishes as u → ∞ and diverges to −∞ as u → uh;its asymptotic behavior is

F (u) =

− u2

h

3u3+ O

(u6h

u7

), u → ∞ ;

− 14uh

ln( 2uh

u−uh

)+

π

8uh+ O

(u−uh

u2h

), u → uh .

(3.18)

The rate at which energy flows down the string is given by π1t . As seen in Eq. (3.12),

this energy flux is proportional to C. If C is positive, then energy flows down the stringtoward the horizon, and the string profile resembles a tail being dragged behind the movingquark, as illustrated on the left in Figure 3. If C is negative, then one has the time-reversedsituation: energy flows upward from the horizon and the tail of string leads the quark. Wepostulate (as in Ref. [49]) that the physical process we want to describe requires us to pickpurely outgoing boundary conditions at the horizon.13 We have to discard the time-reversedsolution in which energy flows from the black hole to the moving quark.

The resulting rates at which energy and momentum flow toward the horizon are

π1t

∣∣u=uh

= T0L2 u2

hv2

(1 − v2)2/d

d=4−→ π2

√λ T 2 v2

√1 − v2

, (3.19a)

and13Here and elsewhere, outgoing refers to moving out of the physical region and into the black hole.

– 15 –

• Find a quasi-steady state

• Calculate the momentum flow out of the string

dp

dt= − ηD︸︷︷︸

Constant!

p

– Drag is independent of momentum!

Fluctuating the String (J. Casalderrey, DT; S. Gubser)

or

orγ

Tπ

Tπγ

Results Relativistic Heavy Quarks

Before After

LκTκ

• Transverse Momentum Broadening of a heavy quark (analgous to q)

κT (v) =√λπ T 3︸ ︷︷ ︸

Non-relativistic result

×√γ

κL(v) =√λπ T 3︸ ︷︷ ︸

Non-relativistic result

× γ5/2 ⇐ (Gubser)

The strong dependence on the energy could have important phenomenological

consequences and theoretical implications.

First Look at Phenom Consequences: (M. Gyulassy and Horrowitz)

• Suggested looking at very high momenta quarks

dpdt ∝ p is very weird! 2

to the asymptotic behavior of the classical supergrav-ity (SUGRA) correlations. Unfortunately these corre-spondences are hard to interpret in terms of gauge the-ory energy loss mechanisms since infinitely coupled SYMdoes not support a familiar quasiparticle basis similar togluon and quark degrees of freedom in QCD. In contrast,the third model, while more easily interpretable, requiresa stronger form of the AdS/CFT correspondence. Allthree approaches remain under active debate (see, e.g.[21, 33, 34]).

We focus in this Letter on the third proposed AdS/CFT application that involves the most direct string the-oretic inspired gravity “realization” of heavy quark dy-namics [18, 19, 20]. A heavy quark in the fundamentalrepresentation is a bent Nambu-Goto string with one endattached to a probe brane and that trails back above thehorizon of a D3 black brane representing the uniformstrongly coupled SYM plasma heat bath. This geometrymaps the drag force problem into a modern string the-oretic version of the old 1696 Brachistochrone problemthat yields a remarkable, simple analytic solution for thestring shape and momentum loss per unit time.

AdS/CFT compared to pQCD Exploiting this AdS/CFT correspondence, the drag coefficient for a massivequark moving through a strongly-coupled SYM plasmain the λ = g2

SY MNc ! 1, Nc ! 1, MQ ! T ∗ limit isgiven in [19, 20, 21] as

dpT

dt= −µQpT = −π

√λ(T ∗)2

2MQpT , (1)

where T ∗ is the temperature of the SYM plasma as fixedby the Hawking temperature of the dual D3 black brane.Issues related to the relaxation of the strong assumptionsmade in deriving, and the momentum limitation of theapplicability of, Eq. (1) will be discussed later in the text.

Applying Eq. (1) to LHC requires an additional pro-posal that maps QCD temperatures and couplings tothe SYM world and its SUGRA dual. The “obvious”first prescription [35] is to take gSY M = gs constant,T ∗ = TQCD, and Nc = 3. However it was suggested in[35] that a more physical “alternative” might be to equateenergy densities, giving T ∗ = TQCD/31/4, and to fit thecoupling λ = g2

SY MNc ≈ 5.5 in order to reproduce thestatic quark-antiquark forces calculated via lattice QCD.

The string theoretic result for the diffusion coefficientused in the Langevin model is D/2πT ∗ = 4/

√λ [31].

This illustrates well the problem of connecting the T ∗and λ of SYM to “our” QCD world. Using the “obvious”prescription with αs = .3, Nc = 3, one finds D/2πT ∼1.2. However, D/2πT = 3 was claimed in [13, 31] to fitPHENIX data somewhat better. Note that D/2πT = 3requires an unnaturally small αs ∼ 0.05 that is very farfrom the assumed λ! 1 ’t Hooft limit.

We proceed by computing the nuclear modification fac-tors, neglecting initial state shadowing or saturation ef-

FIG. 1: (Color Online) RcAA(pT ) and Rb

AA(pT ) predicted forcentral Pb+Pb at LHC comparing AdS/CFT Eq. (1) andpQCD using the WHDG model [25] convolving elastic and in-elastic parton energy loss. Possible initial gluon rapidity den-sities at LHC are given by dNg/dy = 1750, from a PHOBOS[6, 38] extrapolation, or dNg/dy = 2900, from the KLN modelof the color glass condensate (CGC) [39]. The top two curvesfrom pQCD increase with pT while the bottom two curvesfrom AdS/CFT slowly decrease with pT . The AdS/CFT pa-rameters here were found using the “obvious” prescriptionwith αSY M = .05, τ0 = 1 fm/c, giving D/2πT = 3 (abbrevi-ated to D = 3 in the figure). Similar trends were seen for theother input parameter possibilities discussed in the text.

fects. In order to correctly deconvolute such effects fromthe final state effects that we compute below, it will benecessary to measure nuclear modification factors in p+Aas a function of (y, pT ) at LHC just as d+A was the crit-ical control experiment [1] at RHIC [2].

Final state suppression of high-pT jets due to a frac-tional energy loss ε, pf

T = (1 − ε)piT , can be com-

puted knowing the Q-flavor dependent spectral indicesnQ + 1 = − d

d log pTlog

(dσQ

dydpT

)from pQCD or directly

from p + p→ Q + X data. The nuclear modification fac-tor is then RQ

AA(pT ) =< (1− ε)nQ >, where the averageis over the distribution P (ε;MQ, pT , %) that depends ingeneral on the quark mass, pT , and the path length % ofthe jet through the sQGP. As in [25] we average over jetsproduced according to the binary distribution geometryand compute % through a participant transverse densitydistribution taking into account the nuclear diffuseness.Given dNg/dy of produced gluons, the temperature iscomputed assuming isentropic Bjorken 1D Hubble flow.As emphasized in [25], detailed geometric path lengthaveraging plays a crucial role in allowing consistency be-tween π0, η and heavy quark quenching in pQCD.

For AdS/CFT drag, Eq. (1) gives the average frac-tional energy loss as ε = 1 − exp(−µQ%). Energy loss isassumed to start at thermalization, τ0 ∼ 0.6− 1.0 fm/c,and stops when the confinement temperature, Tc ∼ 160MeV, is reached. The exponentiated T 2 dependence inµQ leads to a significant sensitivity to the opacity of themedium, as well as to τ0 and Tc.

To understand the generic qualitative features of ournumerical results it is instructive to consider the simplest

• Fluctuations not included

γmax =M2

√λT 2

Much more to think about here

Quarkonia at Weak and Strong Coupling

• Collaborators:

– Clint Young, Kevin Dusling

– Johanna Erdmenger, Felix Rust, Matthias Gross

Work in progress!

Motivation:

27

“If the plasma is opaque enough to direct the J/! or its pre-

cursor cc into azimuthally asymmetric flow, then a v2

measurement of J/! could serve as a smoking gun.”

“Does the Charm Flow at RHIC?”

RHIC II White Paper

• Would be a great curve indeed!

• Bottomonium states seem to survive above TC

• Want to calculate the “no-coalescence” curve at strong coupling:

Will J/ψ and Υ follow the flow?

The Heavy Ion Swampland

4

shadowingor coherence

CGC - less charmat forward rapidity

absorptiond+Au constraint?

~40% feedownfrom χC, ψ’

(uncertain fraction)configuration of

ccbar state

Data – SPS, PHENIX,STAR, LHC…

Need high statistical& systematic accuracy

comoversmore mid-rapidity

suppression

lattice &dynamical screeningJ/ψ not destroyed?

large gluon densitydestroys J/ψ’s

Sequential screeningχC, ψ’ 1st, J/ψ later

Regeneration & destructionless suppression at mid-rapidity

narrowing of pT & yJ/ψ flow

large charmcross section

Regeneration(in medium?)

CharmdE/dx & flow

The J/ψ Puzzle

CNM

PHENIX J/ψ Suppression:• like SPS at mid-rapidity• stronger at forward rapiditywith forw/mid ~0.6 saturation• <pT

2> centrality indep.

Slide from Mike Leitch !

Quarkonia:

bb

Tπ1

oa

• Gluons probe the structure of the quarkonium system when:

(πT ) ao ∼ 1

• For bottomonium we have:

πTao = 0.7(

T

.240 GeV

) (a0

1.1 GeV−1

)Dipole approximation seems marginal – but (πTao)3 ' 0.33

But lattice differently – Hatsuda, Petreczky

• A. Jakovac, et al, hep-lat/061107. Bottomonium Data. 18

0

0.02

0.04

0.06

0.08

0.1

8 10 12 14 16 18 20 22

σ(ω

)/ω2

ω[GeV]

T=0, Ndata=16T=1.15Tc

FIG. 25: The pseudo-scalar bottomonium spectral functionat finite temperature.

ture as well as the number of data points decreases withincreasing temperature. We found that the spectral func-tions in the pseudo-scalar channel do not change up totemperatures 1.5Tc within systematic and statistical er-rors of our calculations. We have also learned that it isdifficult to make statements about the survival of char-monium states based solely on the shape of the spectralfunction as it is distorted by the limited time extent andstrongly depends on the default model. On the otherhand the small temperature dependence of the pseudo-scalar spectral function up to 1.5Tc holds for all defaultmodels. The correlation functions in all channels corre-sponding to 1P states show very different behavior. To-gether with the results for the scalar spectral functionsthis suggests the melting of the 1P charmonium statesat temperature T = 1.1 − 1.2Tc. In general, most of ourfindings in charmonium sector is in reasonable agreementwith results known from studies on fine isotropic lattices[17]. We also identify the transport contribution in thevector correlators for the first time.

Our investigations in the bottomonium sector are lessdetailed. We were able to reconstruct the bottomoniumspectral functions at zero temperature and show that thefirst peak is not affected by lattice artifacts. Nonethelessthe spectral functions are less accurate than for charmo-nium. We were not able to reconstruct reliably the bot-tomonium spectral functions at finite temperature. Thereason is that because of the large quark mass the corre-lators decay fast and the signal is poor. The temperaturedependence of the correlators suggests that the 1S bot-tomonium state can survive till much higher temperaturethan the charmonium states. On the other hand we see,unexpectedly, drastic changes in the 1P bottomoniumcorrelators right after deconfinement transition. To clar-ify the fate of the 1P bottomonium states further stud-ies are clearly needed. They presumably should rely onNRQCD where the large bottom quark mass is integratedout.

The use of anisotropic lattices and Fermilab approach

reduces discretization effects and allows to study theproblem of quarkonium dissolution on coarser lattices.In fact the analysis of the temperature dependence ofthe correlation functions shows that they can be reliablystudied already for lattice spacings a−1

s = 2.0GeV. Al-though the temperature dependence of the correlatorsturn out to be the same for all lattice spacings studied inthis paper, we clearly see quantitative differences. This isimportant when we want to extend the studies of quarko-nium spectral functions to full QCD, where the lat-tice spacing is main limiting factor. Preliminary resultsfrom calculations of charmonium spectral functions infull QCD have been reported in Ref. [56]. These resultsare in qualitative agreement with findings in quenchedQCD. However, no quantitative analysis has been per-formed so far to estimate the effect of see quarks. To getsufficiently close to the continuum limit in the analysis ofthe correlation function the spatial lattice spacing shouldpresumably be smaller than (4GeV)−1, i.e. it should bein the range used in studies with isotropic lattices [17].

Acknowledgments

This work has been supported by U.S. Departmentof Energy under Contract No. DE-AC02-98CH10886and by SciDAC project. A.V. was partially supportedby NSF-PHY-0309362. K.P. is supported by MarieCurie Excellence Grant under contract MEXT-CT-2004-01 3510. A.J. is supported by Hungarian Science FundOTKA (F043465). Support through LDRD funds atBrookhaven National Laboratory is also greatly appreci-ated. Most of numerical calculations have been done onQCDOC supercomputer at RIKEN-BNL Research Cen-ter. Authors used Columbia Physics System (CPS) withhigh-performance clover inverter by P. Boyle and otherparts by RBC collaboration. Special thanks to C. Jungfor his generous help with CPS. We would like to thank F.Karsch and A Mocsy for careful reading of the manuscriptand valuable comments. The final analysis of the latticedata has been done during the stay of one of the au-thors (P.P.) at the Institute for Nuclear Theory (INT) atWashington University. We are grateful to INT for theirsupport.

Appendix

In this appendix we present some technical details onthe calculations of the quarkonium spectral functions. Tostudy the reliability of MEM at zero temperature com-pare the masses and amplitudes of different states ob-tained from MEM and simple 2-exponential fit. Thiscomparison is shown in Table VI. We also tried to esti-mate the systematic errors in the fit by allowing higherexcited states (3 exponential fit). We show this system-atic uncertainty in the square brackets. One can see thatwe have a good agreement for the ground state in pseudo-

0.94

0.96

0.98

1

1.02

1.04

1.06

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

G(τ

,T)/

Gre

con(

τ)

τ [fm]

1.15Tc1.54Tc1.84Tc2.30Tc

• Bottomonia survives above TC until 1.5TC (Not totally settled.)

• No mass shift! Peter says:

δM < 50÷ 100 MeV

Quarkonia are weakly interacting =⇒ Dipole approximation good!

Dipole Interactions:

• Dipole moment∝ E

p = αEE UI = −12αEE2

• Effective Lagrangian:

– Write down the lowest dimension gauge invariant terms (Manohar ’94)

– higher dimension Ops suppressed by small size: (πTao)3

L = φv∂tφv +12αEE2φ†vφv +

12αBB2φ†vφv + . . .

– From Gravity perspective we will write this as

L = φv∂tφv + c1 T 00︸︷︷︸12(B2+E2)

+c2 F 2︸︷︷︸12(B2−E2)

Effective Lagrangian Virtues:

• Separates short distance physics from long

L = φv∂tφv +12αE︸︷︷︸

short

E2︸︷︷︸long

φ†vφv + . . .

• Polarizability “αE ” is short distance physics

– Not really something for AdS/CFT.

αE =28

3N2π a3

o ⇐= (Peskin ’80)

Perturbative Mass Shift:

• Leading order perturbation theory

δM = 〈HI〉

= −12

αE︸︷︷︸∼a3

B/N2

⟨E2⟩

︸ ︷︷ ︸∼N2T 4

• Numerically we have

δM =1445T (πTa0)3

' 22.5 MeV[

T

0.250

] [(πTao)3

0.3

]

• Mass shift is finite at large Nc

Mass shift is small enough to be consistent with Lattice.

Calculation of Drag and Diffusion

• κ is the mean squared momentum transfer per unit time

κ =∫dt 〈F(t,x)F(0,x)〉

• For the force on the dipole we have

F = −∇U =12αE∇E2

• Then the force force correlator is

κ = α2E ×

14

∫dt⟨∇E2(t,x)∇E2(t,x)

⟩︸ ︷︷ ︸

Computable in AdS

Computing κ – Kinetic Theory vs. Correlators

• κ is the mean squared momentum transfer per unit time:2E

=⇒

∣∣∣Mglue

∣∣∣2 ∝ α2Eω

4

Then

κ =∫p,q

q2 n(p)(1+n(p′))∣∣∣Mglue

∣∣∣2• κ is the αE∇E2 correlator:

Eα ×q Eα ×q =⇒

The Same Thing

Perturbation Theory Results and AdS/CFT Estimates

• The mean squared momentum per time

κ = α2E ×

14

∫dt⟨∇E2(t,x)∇E2(t,x)

⟩= α2

E︸︷︷︸(a3

0/N2)2

× 64π5

45N2T 9

• In terms of the Bohr Radius:

κ =41.29N2

πT 3(πTa0)6

=(300 MeV)2

fm

[T

250 MeV

]3 [(πTa0)3

0.3

]2

– Result does involve explicit coupling constant (like pressure)

– Suppressed by N2

– Coefficient is large because the phase space for σ ∝ ω4 is large

Relation to the mass shift

κ =(300 MeV)2

fm

[(δM)

22 MeV

]2 [ T

250 MeV

]The mass shift needs to be very small in order to get modest (not huge)

momentum relaxation rates

AdS/CFT and Mesons

• Insert a D7 brane into AdS5 × S5. Dual toN = 4 SYM + Heavy

N = 2 Quarks . (Karch, Katz)

SD7 = TD7

∫d8σ

√−detGMN

∂XM

∂σa

∂XN

∂σb

r =0

D7 Soup

Qr ~ M

r = L

r =0

D7 Soup

2 LJr =M

r = L

• The excitations of the D7 brane are mesons; can compute the mass

– The masses are:

MJ =2πMQ√

λ2√

2

– Meson mass is less than the quark mass. Relativistic Bound state!

Binding Energy �MQ

Interactions with the Medium: MJ � T

• The interactions are summarized by the effective Lagrangian

L = φv∂tφv + cT T 00︸︷︷︸12(B2+E2)+...

+ cF OF 2︸︷︷︸12(B2−E2)+...

• These are:

– The lowest dimension ops which are SU(4) singlets

– These operators have protected anomalous dimension

r =0

D7 Soup

2 LJr =M

r = L

2TLπ=or

Excitations of the D7 Soup at Finite Tempertature

• The excitations of the D7 brane are meson; one can compute the mass

• The masses are:

M = MJ + #T(T

MJ

)3

= MJ + cT︸︷︷︸∼ 1

M3J

N2

×⟨T 00

⟩︸ ︷︷ ︸∼N2T 4

• Still working on #.

• Basically involves replacement

αE ∝ a30

1N2

⇐⇒ cT =1M3

J

1N2

Putting it all together:

• The interactions are summarized by the effective Lagrangian

L = φv∂tφv + cTT00 + cF OF 2

• The Force is the gradient of the Potential

F = ∇U U = cTT00 + cFO2

F

• So

κ =∫dt 〈F(t)F(0)〉

∼ c2T

∫dt⟨T 00(t)T 00(0)

⟩∼ # × 1

M6JN

4× N2T 9

Summary

• For the Strong Coupling we have

κ = #︸︷︷︸∼1

41.29N2

πT 3 ×[πT

MJ

]6• Drag and diffusion Heavy Meson – Perturbative

κ =41.29N2

πT 3 × [πTa0]6

• For Both Weak and Strong Coupling we have

κ = (δM)2 πT12803N2

×#

κ =(300 MeV)2

fm

[(δM)

22 MeV

]2 [ T

250 MeV

]

Summary

• AdS/CFT is a useful foil to perturbation theory

• Its a wild theory!

• Many Questions (suitably posed) can be answered.

• My goal: find something computable in AdS/CFT in wild contradiction

with data.

– We haven’t found it yet.