Holographic thermalization patterns Stefan Stricker TU Vienna - - PowerPoint PPT Presentation

Holographic thermalization patterns

Stefan Stricker

• R. Baier, SS, O. Taanila, A. Vuorinen, 1207.116 (PRD)
• D. Steineder, SS, A. Vuorinen, 1209.0291 (PRL), 1304.3404 (JHEP)
• S. Stricker, arxive:1307.2736

TU Vienna

EPS-HEP, Stockholm

July 22 2013

THEORETICAL INSTITUTE for Vienna University of Technology PHYSICS

Saturday, 20 July, 13

Motivation

Quark gluon plasma

Produced in heavy collisions at RHIC and LHC Behaves as a strongly coupled liquid Thermalization process not well understood:

Goals

Gain insight into the thermalization process Modification of production rates of photons Modification of energy momentum tensor correlators Which modes thermalize first: top-down or bottom-up ? Dependence on coupling strength

Strategy

SYM where strong and weak coupling regimes are accessible AdS/CFT: strongly coupled N=4 SYM is dual to classical SUGRA in AdS space

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Outline

Photons in N=4 SYM plasma

Motivation Equilibrium properties at infinite and finite coupling Out of equilibrium at Out of equilibrium at

Plasma constituents

Properties of energy momentum tensor correlators } thermalization scenarios

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Thermalization scenarios

Bottom up scenario

At weak coupling Scattering processes

In the early stages many soft gluons are emitted which then thermalize the system (Baier et al (2001)) Supported by classical Yang-Mills simulations (Berges et al (2013))

Driven by instabilities

Instabilities isotropize the momentum distributions more rapidly than scattering processes (Kurkela, Moore (2011)) Top down scenario

At strong coupling UV modes thermalize first In AdS calculations, follows naturally from causality

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Photon emission in heavy ion collisions

Photons are emitted at all stages of the collision

Initial hard scattering processes: quark anti-quark annihilation:

• n-shell photon or virtual photon → dilepton pair

Strongly coupled out of equilibrium phase: no quasiparticle picture Additional (uninteresting) emissions from charged hadron decays

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Probing the plasma

Probing the plasma

Once produced photons stream through the plasma almost unaltered Provide observational window in the thermalization process of the plasma

Fluctuation dissipation theorem Production rate Quantity of interest

Spectral density : Number of emitted photons χµ

µ = −2Im(Πret)µ µ(k0)

ηµνΠ<

µν(ω) = −2nB(ω)Im(Πret)µ µ(ω) = nB(ω)χ(ω)

k0 dΓγ d3k = α 4π2 ηµνΠ<

µν(ω = k0)

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Photon emission in equilibrium SYM plasma

Huot et al (2006) Hassanain, Schvellinger (2012)

Perturbative result

Increasing the coupling: slope at k=0 decreases, hydro peak broadens and moves right

Strong coupling result

Decreasing coupling from : peak sharpens and moves left λ = ∞

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Equilibrium summary

Equilibrium picture in SYM fairly complete

How does photon/dilepton production get modified out of equilibrium Can one access thermalization at finite coupling?

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The falling shell setup

Outside and inside spacetime

metric:

u = r2

h

r2

ds2 = (πTL)2 u

• f(u)dt2 + dx2 + dy2 + dz2

+ L2 4u2f(u)du2

f(u) = ⇢ f+(u) = 1 − u2 , for u > 1 f−(u) = 1 , for u < 1 ,

r = ∞ rh r = 0 rs

AdS AdS-bh

Danielsson, Keski-Vakkuri, Kruczenski (1999)

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The falling shell setup

Outside and inside spacetime

metric: AdS-bh

u = r2

h

r2

ds2 = (πTL)2 u

• f(u)dt2 + dx2 + dy2 + dz2

+ L2 4u2f(u)du2

f(u) = ⇢ f+(u) = 1 − u2 , for u > 1 f−(u) = 1 , for u < 1 ,

r = ∞ rh r = 0 rs

AdS

Thermalization from geometric probes:

Entanglement entropy and Wilson loop: always top down thermalization

Danielsson, Keski-Vakkuri, Kruczenski (1999)

Saturday, 20 July, 13

The falling shell setup

Outside and inside spacetime

metric: AdS-bh

u = r2

h

r2

ds2 = (πTL)2 u

• f(u)dt2 + dx2 + dy2 + dz2

+ L2 4u2f(u)du2

f(u) = ⇢ f+(u) = 1 − u2 , for u > 1 f−(u) = 1 , for u < 1 ,

r = ∞ rh r = 0 rs

AdS

Thermalization from geometric probes:

Entanglement entropy and Wilson loop: always top down thermalization

Danielsson, Keski-Vakkuri, Kruczenski (1999)

Saturday, 20 July, 13

The falling shell setup

Outside and inside spacetime

metric:

Outside solution

Ein

r = ∞ rh r = 0 Eout E− rs E+ = c+Ein + c−Eout

u = r2

h

r2

ds2 = (πTL)2 u

• f(u)dt2 + dx2 + dy2 + dz2

+ L2 4u2f(u)du2

f(u) = ⇢ f+(u) = 1 − u2 , for u > 1 f−(u) = 1 , for u < 1 ,

AdS AdS-bh

Danielsson, Keski-Vakkuri, Kruczenski (1999)

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Holographic Green’s functions

Off-equilibrium correlators offer a useful window to thermalization:

Probe how different energy scales approach equilibrium Related to measurable quantites, e.g. production rates

Some computational details

Solve classical EoM for bulk electric field E inside and outside the shell Matching conditions: Use conventional methods to obtain retarded correlator Behaviour of crucial for out of equilibrium dynamics

Quasistatic approximation:

Energy scale of interest >> characteristic time scale of shell’s motion

E(ω)|us = p fmE+(ω+)|us, E0

(ω)|us =

fmE0

+(ω+)|us.

Π(ω, q) = −N 2

c T 2

8 lim

u!0

E0(u, Q) E(u, Q) = −N 2

c T 2

8 Πtherm 1 + c

c+ E0

• ut

E0

in

1 + c

c+ Eout Ein

c−/c+

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Photon spectral density

spectral density for rs /rh =1.1 for different virtualities

Out of equilibrium effect: oscillations around thermal value As the shell approaches the horizon equilibrium is reached

virtuality

v = ˆ ω2 − ˆ q2 ˆ ω2

parametrize

q = c ˆ ω

c=0.8 c=0 c=1

5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 ΩêT ΧΜ

Μ

Nc

2 T Ω

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Relative deviation of spectral density

Relative deviation from thermal equilibrium

R(ˆ ω) = χ(ˆ ω) − χth(ˆ ω) χth(ˆ ω)

relative deviation R for rs=1.1 and c=1, 0.8, 0

Top down thermalization: highly energetic modes are closer to equ. value Highly virtual field modes thermalize first

χ(ˆ ω) ≈ ˆ ω

2 3

✓ 1 + f1(us) ˆ ω ◆ , R ≈ 1 ˆ ω

20 40 60 80

• 0.10
• 0.05

0.00 0.05 0.10 wêT R

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Photon production rate at infinite coupling

photon production rate for rs/rh=1.1, 1.01, 1.001

Enhancement of production rate Hydro peak broadens and moves right Apparently no dramatic observable signature in off-equilibrium photon production

1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 wêT 100 dG ê dk0 a Hp NcL2 T3

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Combining the two allows to study thermalization at finite coupling!

Photon production rate at infinite coupling

photon production rate for rs/rh=1.1, 1.01, 1.001

Enhancement of production rate Hydro peak broadens and moves right Apparently no dramatic observable signature in off-equilibrium photon production

1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 wêT 100 dG ê dk0 a Hp NcL2 T3

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Finite coupling corrections

Key relation in AdS/CFT:

Go beyond : add terms to SUGRA action, i.e. first non trivial terms in a small curvature expansion Leading order corrections:

Improved type IIB SUGRA action:

Paulos (2008)

γ ≡ 1 8ζ(3)λ− 3

2

Tabcdef = iraF +

bcdef + 1

16 ⇣ F +

abcmnF + mn def

3F +

abfmnF + mn dec

⌘ ,

Gubser et al; Pawelczyk, Theisen (1998)

Leads to -corrected metric

EoM for different fields

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Quasinormal modes infinite coupling

Strong coupling equivalent to quasiparticle picture at weak coupling Characterize the response of the system to inf. perturbation Appear as poles in the retarded correlator: First indication of top down thermalization at strong coupling

à æ ì ò à æ ì ò

• 4
• 3
• 2
• 1
• 4
• 2

2 4 Im w ` Re w `

ωn|q=0 = n(±1 − i)

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Quasinormal modes finite coupling

Effect of decreasing coupling: Imaginary part decreases Outside the limit, response of the plasma appears to change, moving towards a quasiparticle picture Larger impact on higher energetic modes Convergence of strong coupling expansion not guaranteed when shift is of What happens if we take system further away from equilibrium?

Ê Ê Ê Ê Ê Ê ‡ ‡ ‡‡ ‡ ‡ ÏÏ Ï Ï Ï Ï Ú Ú Ú Ú Ú Ù

l=1000 l=3000 l=2000 l=• l=10000 l=5000

• 4
• 3
• 2
• 1

00 1 2 3 4 5 6 7 Im w ` Re w `

λ = ∞

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Photon production rate at intermediate coupling

Behaviour qualitatively similar to equilibrium case: in particular the result is much less sensitive to finite coupling corrections than QNM spectrum

2 4 6 8 0.000 0.005 0.010 0.015 0.020 wêT dG ê dk0 a Nc2 T3

emission rate for photons rs/rh=1.01 and

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Thermalization at finite coupling

R for rs/rh=1.1 and λ = ∞, 500, 300

Relative deviation from thermal limit for on shell photons

20 40 60 80 100 120 140 160

• 0.05

0.00 0.05 wêT R 20 40 60 80 100 120

• 0.4
• 0.2

0.0 0.2 0.4 0.6 wêT R

R for rs/rh=1.1 and

λ = 150, 100, 75

Behaviour of relative deviation changes at large frequency UV modes are no longer first to thermalize Decreasing the coupling: change happens at lower frequency

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Thermalization at finite coupling

Virtuality dependence of the relative deviation

20 40 60 80 100

• 0.3
• 0.2
• 0.1

0.0 0.1 0.2 0.3 wêT R¶

R for rs/rh=1.1 and c=1, 0.8, 0 for

For maximally virtual photons (c=0), R approaches a constant at For on-shell photons (c=1): amplitude of R rises linearly with Indication that thermalization pattern changes from top-down towards bottom-up

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Interpretation

What to make of all this? Evidence for the holographic plasma starting to behave like a system of weakly coupled quasiparticles Or is this simply a peculiarity of photon production but not a fundamental feature of the collective behaviour of the plasma constituents?

Next: Repeat the same analysis for energy momentum tensor correlators

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Energy momentum tensor correlators

Metric fluctuations

corresponds to the energy momentum tensor of the field theory

Construct gauge invariants

Scalar channel: Shear channel: Sound channel:

Repeat analysis from photons

Straight forward but more involved

hTµνTαβi

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l=•

Ê Ê Ê Ê ‡ ‡ ‡ ‡ Ï Ï Ï Ï Ú Ú Ú Ú

l=500 l=1000 l=2000

1 2 3 4 5 6

• 4
• 3
• 2
• 1

Re w ` Im w `

Quasinormal modes at finite coupling

QNM spectrum for the shear and shear channel for q=2πT All three channels show the same behaviour Qualitatively identical to virtual photons

Scalar channel Shear channel

l=•

Ê Ê Ê Ê ‡ ‡ ‡‡ Ï ÏÏ Ï Ú Ú Ú Ú

l=500 l=1000 l=2000

1 2 3 4 5 6

• 4
• 3
• 2
• 1

Re w ` Im w `

Saturday, 20 July, 13

Relative deviation at finite coupling

Scalar channel

Relative deviation for the scalar/shear channel for rs/rh=1.1, c=0, 6/9, 8/9 and All three channels show the same behaviour Again similar to photons with the same dependence on c shift from top-down towards bottom-up

Shear channel

20 40 60 80 100 120

• 0.4
• 0.2

0.0 0.2 0.4 wêT R1

20 40 60 80 100 120

• 0.4
• 0.2

0.0 0.2 0.4 wêT R2

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Conclusions

Holographic (thermalization) calculations at finite coupling are possible and potentially a very fruitful exercise All calculations suggest that the system starts to behave like a weakly coupled system in the realm of the strong coupling expansion QNM modes: flow towards quasiparticle picture, independent of the thermalization model Thermalization pattern shifts from top-down towards bottom-up

Open questions How universal is the shift from top-down towards bottom-up Go beyond quasistatic approximation

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