Holographic hydrodynamization Micha P . Heller m.p.heller@uva.nl - - PowerPoint PPT Presentation

Holographic hydrodynamization

Michał P . Heller m.p.heller@uva.nl

University of Amsterdam, The Netherlands

&

National Centre for Nuclear Research, Poland (on leave)

based on 1302.0697 [hep-th] MPH, R. A. Janik & P . Witaszczyk (PRL 110 (2013) 211602)

Introduction

Modern relativistic (uncharged) hydrodynamics

an EFT of the slow evolution of conserved currents in collective media „close to equilibrium” hydrodynamics is As any EFT it is based on the idea of the gradient expansion DOFs: always local energy density and local flow velocity ( ) EOMs: conservation eqns for systematically expanded in gradients

✏ uµ uνuν = −1 rµT µν = 0

T µν = ✏ uµuν + P(✏){ gµν + uµuν } ⌘(✏) µν ⇣(✏){ gµν + uµuν }(r · u) + . . .

T µν

perfect fluid stress tensor (famous) shear viscosity bulk viscosity (vanishes for CFTs) microscopic input: EoS terms carrying 2 and more gradients

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Applicability of hydrodynamics

T µν = ✏ uµuν + P(✏){ gµν + uµuν } ⌘(✏) µν ⇣(✏){ gµν + uµuν }(r · u) + . . .

perfect fluid stress tensor (famous) shear viscosity bulk viscosity (vanishes for CFTs) microscopic input: EoS terms carrying 2 and more gradients Naively one might be inclined to associate hydrodynamic regime with small gradients. But this is not how we should think about effective field theories! The correct way is to understand hydrodynamic modes as low energy DOFs. Of course, there are also other DOFs in fluid. The topic of my talk is to use holography to elucidate their imprint on hydro.

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Holographic plasmas and their degrees of freedom

Holography

From applicational perspective AdS/CFT is a tool for computing correlation functions in certain strongly coupled gauge theories, such as SYM at large and .

N = 4

For simplicity I will consider AdS1+4 / CFT1+3 and focus on pure gravity sector.

Nc λ Minkowski spacetime at the boundary bulk of AdS

Rab − 1 2Rgab − 6 L2 gab = 0

UV IR Different solutions correspond to states in a dual CFT with different .

hTµνi

???

ds2 = L2 z2 n dz2 + ηµνdxµdxν +2π2 N 2

c

hTµνi z4 + . . .

• 3/16

Excitations of strongly coupled plasmas

Kovtun & Starinets [hep-th/0506184] Tµν = 1 8π2N 2

c T 4 diag (3, 1, 1, 1)µν +δTµ⌫

(∼ e−i !(k) t+i~

k·~ x)

Consider small amplitude perturbations ( ) on top of a holographic plasma

δTµν/Nc

2 ⌧ T 4

Dissipation leads to modes with complex , which in the sound channel look like

ω(k)

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 3rd 1st 2nd 3rd

ω(k) → 0

as : slowly evolving and dissipating modes (hydrodynamic sound waves)

k → 0

all the rest: far from equilibrium (QNM) modes dampened over

∂ω ∂k

• k→0 = csound

ttherm = O(1)/T

There are two different kinds of modes:

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for hydrodynamics to work all the other DOFs need to relax.

Lesson 1:

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 3rd 1st 2nd 3rd

∂ω ∂k

• k→0 = csound

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0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 3rd 1st 2nd 3rd

∂ω ∂k

• k→0 = csound

Observation: No matter how long one waits, there will be always remnants of n-eq DOFs

Hydrodynamic gradient expansion cannot converge

Lesson 2:

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Dynamical model

Fantastic toy-model

x0

x1

x1

The simplest model in which one can test these ideas is the boost-invariant flow with no transverse expansion. In Bjorken scenario dynamics depends only on proper time

[Bjorken 1982]

pre-equilibrium stage QGP mixed phase hadronic gas

described by hydrodynamics

described by AdS/CFT in this scenario

and stress tensor (for a CFT) is entirely expressed in terms of local energy density with

τ = 0 τ = q (x0)2 − (x1)2

and pT (⇥) = (⇥) + 1

2⇥0(⇥) pL(⇥) = −(⇥) − ⇥0(⇥) hT µ

νi = diag{✏(⌧), pL(⌧), pT (⌧), pT (⌧)}

ds2 = −dτ 2 + τ 2dy2 + dx2

2 + dx2 3

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Hydrodynamization

6=

✏ − 3 pL ≈ 0.6 ✏ to 1.0 ✏

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4t THtL 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 e - 3 pL e

grey: full evolution red: 1st order hydrodynamics Large anisotropy at the onset of hydrodynamics Thus hydrodynamization

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 3rd 1st 2nd 3rd

∂ω ∂k

• k→0 = csound

For hydrodynamics to work all the other DOFs need to relax. Surprising consequence

thermalization isotropization MPH, R. A. Janik & P . Witaszczyk 1103.3452 [hep-th] PRL 108 (2012) 201602:

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Boost-invariant hydrodynamics

In hydrodynamics, the stress tensor is expressed in terms of , and their

T uµ rν

Key observation: in Bjorken flow is fixed by the symmetries and takes the form

uµ∂µ = ∂τ

uµ

Its gradients will come thus from Christoffel symbols ( )

ds2 = −dτ 2 + τ 2dy2 + dx2

2 + dx2 3

Lesson: in Bjorken flow hydrodynamic gradient expansion = late time power series At very late times pL = −✏ − ⌧✏0 = pT = ✏ + 1

2⌧✏0 ✏ ∼ 1 ⌧ 4/3 T ∼ 1 τ 1/3 Γy

τy = 1

τ

In holographic hydrodynamics gradient expansion parameter is

T µν = ✏ uµuµ + P(✏) {⌘µν + uµuν} − ⌘(✏) s(✏) ✏ + P(✏) T µν + . . . 1 T rµuν

For Bjorken flow is .

1 T rµuν 1

1 τ 1/3

1 τ = 1 τ 2/3

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High order hydrodynamics

Hydrodynamic series at high orders

MPH, R. A. Janik & P . Witaszczyk 1302.0697 [hep-th] PRL 110 (2013) 211602:

at large orders factorial growth of gradient contributions with order

T 00 = ✏(⌧) ⇠

∞

X

n=2

✏n(⌧ −2/3)n (T −1rµuν ⇠ ⌧ −2/3)

First evidence that hydrodynamic expansion has zero radius of convergence! at low orders behavior is different

✏ = 3 8N 2

c ⇡2

1 ⌧ 4/3 ✓ ✏2 + ✏3 1 ⌧ 2/3 + ✏4 1 ⌧ 4/3 + . . . ◆

(n!)1/n ∼ (2πn)1/2n e · n

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A standard method for asymptotic series is Borel transform and Borel summation

MPH, R. A. Janik & P . Witaszczyk

Singularities in the Borel plane

✏(u) ∼

∞

X

n=2

✏nun (u = ⌧ −2/3), B✏(˜ u) ∼

∞

X

n=2

1 n!✏n˜ un, Borel sum : ✏Bs(u) = Z ∞ 1 uB✏(t) exp (−t/u)dt

1302.0697 [hep-th] PRL 110 (2013) 211602:

This makes a difference only if we can find analytic continuation of .

B✏(˜ u)

Idea: use Pade approximant to reveal singularities of .

B✏(˜ u) = P120

m=0 cm˜

um P120

n=0 dn˜

un B✏(˜ u)

• 10

10 20 30 Re z0

• 30
• 20
• 10

10 20 30 Im z0

green dots: zeros numerator gray dots: zeros denominator those zeros cancel almost perfectly (up to 10-150) those are real singularities

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Hydrodynamic instantons and hydrodynamic gradient expansion

In Borel summation the outcome depends on the contour connect 0 with . Here there are two inequivalent contours (blue and orange).

MPH, R. A. Janik & P . Witaszczyk

Singularities of Borel transform and QNMs

is the frequency of the lowest non-hydrodynamic metric QNM at !

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 1st 2nd 3rd

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im ω/2πT Re ω/2πT k/2πT

1st 2nd 3rd

• 5

5 10 15 20 Re u é

• 20
• 10

10 20 Im u é

✏Bs(u) = Z ∞ 1 uB✏(t) exp (−t/u)dt

✏Bs(u) = Z ∞ 1 uB✏(t) exp (−t/u)dt

1302.0697 [hep-th] PRL 110 (2013) 211602:

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∞

✏ ∼ e−3/2 i ωBorelτ 2/3⇣ ⌧ αBorel + . . . ⌘

ωBorel = 3.1193 − 2.7471i αBorel = −1.5426 + 0.5192i ωBorel k = 0

MPH, R. A. Janik & P . Witaszczyk

Slow and fast modes

1302.0697 [hep-th] PRL 110 (2013) 211602:

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✏ ∼ e−3/2 i ωBorelτ 2/3⇣ ⌧ αBorel + . . . ⌘

Can we understand 3/2 in the exponent and the pre-exponential term in ? Yes, QNMs here are fast evolving modes on top of slowly evolving background. In local rest frame (as here), at the leading order they only care about T. But the temperature here is time-dependent. Imagine solving with slowly varying frequency. The leading order result is

−i Z ωQNM 2π Λ (Λτ)1/3 dτ + . . . = −i 3 2 ⇣ 2π Λ2/3ωQNM ⌘ τ 2/3 [hep-th/0606149] Janik & Peschanski ¨ x(t) = −ω(t)2x(t) x(t) ∼ e±i

R ω(t)dt

instanton-like dependence

• n the coupling “

1 g2

Y M

” = τ 2/3

How about pre-exponential term? Schematically

Z 1 τ 1/3 ✓ 1 + 1 τ 2/3 + . . . ◆ dτ ∼ τ 2/3 + log τ

Indeed agrees with !!!

!qnm = 3.11952.7467, , ↵qnm = 1.5422+0.5199 i. !Borel = 3.1193 − 2.7471 i ↵Borel = −1.5426 + 0.5192 i.

Interpretation and possible relevance

Famous examples of asymptotic expansions arise in pQFTs

MPH, R. A. Janik & P . Witaszczyk

There, the number of Feynman graphs grows ~order! at large orders* We suspect analogous mechanism might work also in the case of hydro series*

T µν = ✏ uµuν + P(✏){ gµν + uµuν } ⌘(✏) µν ⇣(✏){ gµν + uµuν }(r · u) + . . .

Πµν = −ησµν − τΠ

• DΠµν +

d d − 1Πµν(∇·u)

• + κ
• Rµν − (d − 2)uαRαµνβuβ
• + λ1

η2 ΠµλΠνλ − λ2 η ΠµλΩνλ + λ3ΩµλΩνλ . − τΠ

• DΠµν +

d d − 1Πµν(∇·u)

• −

+ κ

• Rµν − (d − 2)uαRαµνβuβ
• −

−

• + λ1

η2 ΠµλΠνλ − λ2 η ΠµλΩνλ + λ3ΩµλΩνλ

+ . . .

1st order hydro (1 transport coeff) 2nd order hydro (5 transport coeffs) ...

Why hydro series might be asymptotic?

+ . . .

+

1302.0697 [hep-th] PRL 110 (2013) 211602:

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Resummed hydrodynamics?

In boost-invariant hydrodynamics, hydro equations can be recast in the form

τ w d dτ w = Fhydro(w) w ,

=

2 3 + 1 9πw + 1 log 2 27π2w2 + 15 2π2 45 log 2 + 24 log2 2 972π3w3 +. . . (5)

perfect fluid 1st 2nd 3rd order hydro (⇤) = 3 8N 2

c ⇥2Teff(⇤)4

w = τ Teff

with and

0.25 0.30 0.35 0.40 0.45 w 0.72 0.74 0.76 0.78 0.80 0.82 0.84 Τ w d w d Τ

different data set work with M. Spaliński τ w d w d τ

convergence to a single curve!

MPH, R. A. Janik & P . Witaszczyk 1103.3452 [hep-th] PRL 108 (2012) 201602:

Idea: is it possible to obtain (part of) this curve from Borel resummation?

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Summary

Summary

Hydrodynamics is an asymptotic series

1302.0697 [hep-th] PRL 110 (2013) 211602.

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0.5 1 1.5 2 0.5 1 1.5 2 2.5 3

Re

0.5 1 1.5 2

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Im

Im ω/2πT Re ω/2πT k/2πT k/2πT

1st 2nd 3rd 1st 2nd 3rd

because in any fluid there are DOFs not captured by hydrodynamic approx.