Electromagnetic response in strongly coupled super-Yang-Mills fluidLessons from AdS/CFT

Pavel Kovtun

KITP, University of California, Santa Barbara

work done with

S.Caron-Huot, (McGill University, Montreal, Canada)

G.Moore, (McGill University, Montreal, Canada)

A.Starinets, (Perimeter Institute, Waterloo, Canada)

L.Yaffe, (University of Washington, Seattle, USA)

Talk at Non-equilibrium quark-gluon plasma workshop, INT, Seattle

September 28, 2006

Problem: Understand real-time response in thermal QCD at T & Tc

− Momentum fluctuations

− Electromagnetic fluctuations

− Heavy particle moving through the medium

Address these problems not in QCD, but in another gauge theory:

strongly coupled SU(Nc), N=4 supersymmetric Yang-Mills theory (SYM)

Why study SYM instead of QCD?

• SYM is simpler than QCD

• SYM is an example of a system which can be very strongly coupled, but still

have an ideal-gas equation of state ǫ = 3P

• At scales ≫ 1/T , strongly coupled SYM behaves as a near-ideal fluid

• Strongly coupled SYM is a 4d gauge theory where analytic results can be

derived at strong coupling (using AdS/CFT) in real time and at non-zero

chemical potential

AdS/CFT correspondence FAQ

What is AdS/CFT correspondence?

A1: Exact equivalence between a field theory and a string theory

A2: A tool to perform calculations in quantum field theory

When are AdS/CFT methods useful?

When the field theory is strongly coupled, and perturbation theory is in-

valid. In this sense, AdS/CFT is similar to lattice

Does this mean AdS/CFT is Euclidean?

No. Both real-time and µ 6= 0 are naturally built into AdS/CFT

To what field theories can AdS/CFT be applied?

Typically conformal in the UV and large Nc; various degree of susy

Can QCD be analyzed by AdS/CFT methods?

No (at least at the moment) — duality is not developed for AF theories

If QCD can not be addressed, why should one care about AdS/CFT?

– Field theories which can be analyzed within AdS/CFT are close cousins of

QCD. AdS/CFT can treat field theories which exhibit confinement, mass

gap, global symmetries, Goldstone bosons, and χSB.

– One can try to use AdS/CFT-based models to analyze parts of QCD dy-

manics which are not sensitive to short-distance physics

– Many important questions in QCD can not be treated by either perturba-

tion theory or lattice methods. For such questions, it is valuable to gain

insight from QCD-related models, which can be analyzed by AdS/CFT.

– For theories which can be treated by AdS/CFT methods, analytic results

can be derived at both strong and weak coupling. Thus one can quantita-

tively compare perturbative and non-perturbative regimes. In fact, this is

precisely what I am going to do in this talk

Simplest field theory whose strong coupling regime can be treated by AdS/CFT:

N=4 supersymmetric Yang-Mills theory, SU(Nc), λ = g2Nc

(in the limit of large Nc and large λ)

What is N=4 SYM?

• Gauge fields + 4 fermions + 6 scalars in adjoint of SU(Nc)

• Conformal theory, λ is a tunable parameter (does not run)

• Supersymmetric, but SUSY not essential at finite temperature

• ǫ = 3P , vs = 1√3, ζ = 0, at any non-zero temperature

• ǫ, P , η are finite in the limit λ→∞

• η

s=

1

4πin the limit λ→∞ (P.K., D.Son, A.Starinets, hep-th/0309213)

Use strongly coupled N=4 SYM medium at T 6=0 as a model of

strongly interacting QCD medium at T 6=0

Collective behavior in SYM medium(P.K., A.Starinets, hep-th/0602059)

1w

(

χ(w) − χT=0(w)) [

1π2N2

cT 4

]

1 2 3

-0.2

0.2

0.4

0.6

0.8

w ≡ ω2πT

χ(ω, k) = −2 Im Gretxy,xy(ω, k)

χ(ω) ∼ ω, ω ≪ 2πT

χ(ω)−χT=0(ω) ∼ e−γω, ω ≫ 2πT

η = π8 N2

c T 3

T 3 by conformal invariance, N2c counts d.o.f.

Spectral function for conserved energy-momentum

0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

1.2

w

χtx,tx(w)

0.5 1 1.5

2

4

6

8

10

w

χtt,tt(w)

Hydrodynamic peaks are clearly visible in dual classical gravity

Production of real and virtual photons(e.g. L.D.McLerran, T.Toimela, PRD 31, 545 (1985), H.A.Weldon, PRD 42, 2384 (1990))

Γ — number of photons per unit time per unit volume

Photon interaction: e JEMµ Aµ; el. charge e small ⇒ photons do not thermalize

dΓ =d3k

(2π)3e2

2|k| ηµνC<µν(k)

∣

∣

∣

lightlike kwhere C<

µν(x) = 〈JEMµ (0)JEM

ν (x)〉

Virtual photon can decay into a lepton pair: dΓ =d4k

(2π)4e2e2

ℓ

6πk2ηµνC<

µν(k)∣

∣

∣

timelike k

Emission spectra are determined by EM current-current spectral function

ee

ee=

∑

. . .true to leading order in e,

but to all orders in g

In SYM, can evaluate the whole “blob” at large λ ≡ g2Nc (using AdS/CFT)

Production of real and virtual photons from N=4 plasma(S.Caron-Huot, P.K., G.Moore, A.Starinets, L.Yaffe, hep-th/0607237)

But wait... N=4 SYM does not have a photon[U(1) gauge field coupled to aconserved current]

Let’s introduce one! To do so:

Gauge a U(1) subgroup of SU(4) R-symmetry with small coupling e

The symmetry is not anomalous

EM current JEMµ (x) ≡ JR

µ (x)conservednon-anomalouscouples to a U(1) gauge field

Photon and dilepton emission rate is given by R-current spectral function

In SYM can compute χµµ(ω) ≡ −2 Im ηµνCRET

µν (ω, q)∣

∣

∣

ω=qat strong coupling, then

emission rate is

dΓ

d3k∝ nB(ω)

χµµ(ω)

ω

On-shell photons at strong coupling

2 4 6 8

0.2

0.4

0.6

0.8

1

w ≡ ω2πT

χµµ(w)

w

[

2

N2c T 2

]

χµµ(ω) = −2 Im ηµνCRET

µν (ω, q)∣

∣

∣

ω=q

Small frequency:

χµµ(w)∼ 1

2N2c T 2w, in accord with hydro

High frequency:

χµµ(w) ∼ N2

cT 2

4 w2/3[

35/6 Γ(2/3)Γ(1/3)

]

Spectral function for on-shell photons can be computed in closed form!

χµµ(w) =

N2c T 2w

8

∣

∣

∣

∣

2F1

(

1 − 1

2(1+i)w, 1 +

1

2(1−i)w; 1−iw;−1

)∣

∣

∣

∣

−2

Emission rate is finite and λ-independent in the limit of large λ

Off-shell photons at strong coupling

Plot spectral function for several values of q ≡ k/(2πT )

0.5 1 1.5 2

2

4

6

8

10

12

14

q=0,1,1.5

χµ

µ(w

,q)

w = k0/(2πT )

0.5 1 1.5 2 2.5 3 3.5 4

-0.5

-0.25

0

0.25

0.5

0.75

1q=0,1,1.5,2

∆χ

µ

µ(w

,q)

w = k0/(2πT )

Spectral function oscillates around the zero-temperature value... where is hydro?

1 2 3 4 5

1

2

3

4

5

6

q=0.2,0.3,0.5,1,2,3,4

w = k0/(2πT )

χzz(w

,q)/

w2

In the hydro limit:χzz(ω, k)

ω2=

ω 2DΞ

ω2 + (Dk2)2

Diffusion bump clearly visible at

small ω and k

Electrical conductivity

Kubo formula: σ = e2 limω→0

χii(ω, q=0)

6ω

In strongly coupled SYM: σ = e2 N2c T

16π, does not depend on λ

In weakly coupled SYM: σ = e2 N2c T

λ2 ln(1/λ)C, where C=O(1)

Normalize to the number of degrees of freedom:σ

e2Ξ, where Ξ is charge

susceptibility. In strongly coupled SYM,

σ

e2Ξ=

1

2πT

Einstein relationσ

e2Ξ= D is satisfied, as it should.

On-shell photons at weak coupling

N = 4 SYM is simply a non-abelian plasma with adjoint representation matter.

Follow approach of Arnold-Moore-Yaffe (hep-ph/0111107) for QCD...

χµµ(k) =

λ(N2c −1)T 2 nF (k)

4π nB(k)

[

lnλ−1/2 + Ctot(k/T ) + O(√

λ)

]

, k ≫ λ2T

Momentum dependent Ctot(k/T ) is not too different between QCD and SYM

Off-shell photons at weak coupling

For time-like momenta, χµν(k) is non-zero already at O(λ0). One-loop diagram:

χµµ(K) =

(N2c −1)

16π(−K2)

[

3 − 2T

kln

1+e−(k0−k)/2T

1+e−(k0+k)/2T+

T

kln

1−e−(k0+k)/2T

1−e−(k0−k)/2T

]

LPM resummation for dileptons in N = 4 SYM: needs to be done...

Electrical conductivity

Evaluated at leading log order, following Arnold-Moore-Yaffe, hep-ph/0010177

σ = 1.28349e2(N2

c −1)T

λ2 [ln(λ−1/2) + O(1)].

The O(1) constant: needs to be done...

Now can compare weakly coupled SYM andstrongly coupled SYM

Photon emission

0 2 4 6 8 10k�T

0.0025

0.005

0.0075

0.01

0.0125

0.015

dG�dk@ΑEMNc2T3D-1

SYM, Λ=0.5

SYM, Λ®¥

QCD, Nf=2, Αs=0.2 At weak coupling:

k ≫ λ2T , follow AMY

k ≪ λ2T , hydro regime

k ∼ λ2T : technically hard.

Height of the peak is unknown

at weak coupling in either

SYM or QCD.

At strong coupling:

dΓγ

dk=

αEMN2c T 3

16π2

(k/T )2

ek/T−1

∣

∣

∣

∣

2F1

(

1 − (1+i)k

4πT, 1 +

(1−i)k

4πT; 1− ik

2πT;−1

)

∣

∣

∣

∣

−2

− emission rate for all k is given by a simple analytic expression

− hydro limit is naturally reproduced when k ≪ T

− as coupling grows, the rate becomes finite and coupling-independent

At small k/T , photon emission rate is greater in weakly-coupled theory

At large k/T , photon emission rate is greater in strongly-coupled theory

Dilepton emission

Plot relative thermal correction to the trace of the spectral function, q ≡ k/(2πT ), w ≡ k0/(2πT )

0.5 1 1.5 2 2.5 3

-0.02

0

0.02

0.04

0.06

0.08

0.1

q=0,1,2,3

p

w2− q2

SYM, λ=0

∆χ

µ

µ/χ

µ

µ(T

=0)

0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

q=0,1,2,3

p

w2− q2

SYM, λ=∞

∆χ

µ

µ/χ

µ

µ(T

=0)

For√−K2 & 2πT , thermal correction to the spectral function are < 2% at weak

coupling, and < 15% at strong coupling.

Dilepton spectrum is nearly identical at weak and strong coupling, for large

invariant mass of the pair

Conclusions

• AdS/CFT is a powerful tool to compute real-time thermal cor-

relation functions at strong coupling

• At strong coupling, photon emission rate in SYM is given by

an explicit analytic formula; dilepton rate is easily computed

by solving a simple ODE

• At weak coupling, photon rate in SYM mimics the rate in QCD

pretty well, when the two theories are compared at the same

value of thermal fermion mass

• In the limit of infinite coupling, kinetic coefficients and emis-

sion rates in SYM are well-defined and finite

What’s next?

• Try SYM spectral functions instead of AMY spectral functions in hydro

simulations of thermal photon production at RHIC?

• These same questions about photon production can be addressed in

theories which are more similar to QCD than N=4 SYM. Simplest

modifications: add mass terms to SYM; add non-zero chemical poten-

tial. Volunteers?

• How does photon spectrum in SYM change when coupling is taken to

be finite (not infinite)?

• Photons from fundamental representation matter fields (SYM has ad-

joints only)?

• Now that one has the method to compute spectral functions: what

does AdS/CFT say about heavy-quark resonances? Lattice studies for

cc̄ are available...

How can AdS/CFT results be useful in understanding QCD?

BACKUP SLIDES

SYM vs QCD

0 1 2 3 4 5 6 7k�T

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

HdGΓ�dkL�HΑEMT3L

0 1 2 3 4 5 6 7k�T

0.02

0.04

0.06

0.08

0.1

HdGΓ�dkL�HΑEMT3L

αs = αSYM = 0.1 αs = 9αSYM = 0.1

Singularities of Gret(ω, k)(A.Nunez, A.Starinets, hep-th/0302026, P.K., A.Starinets, hep-th/0506184)

-4 -2 2 4

-5

-4

-3

-2

-1

Gretxy,xy(ω)

Re(ω̃)

Im(ω̃)

• Infinite series of poles

• ωn = 2πnT (±1 − i) as n→∞• For conserved densities, ω0→0 as k→0

• Hydro poles agree with Kubo formula

Grettx,tx(ω) Gret

tt,tt(ω)-4 -2 2 4

-5

-4

-3

-2

-1

Re(ω̃)

Im(ω̃)

-4 -2 2 4

-5

-4

-3

-2

-1

Re(ω̃)

Im(ω̃)

Singularities of Gret(ω, k) are (quasi)normal modes of the dual gravity background