T O W A R D S W E A K C O U P L I N G I N H O L O G R A P H Y

I N S T I T U U T- L O R E N T Z F O R T H E O R E T I C A L P H Y S I C S L E I D E N U N I V E R S I T Y

O X F O R D, 6 . 3 . 2 0 1 7

S AŠO G R O Z D A N O V

W O R K I N C O L L A B O R AT I O N W I T H J . C A S A L D E R R E Y- S O L A N A , N . K A P L I S , N . P O O V U T T I K U L , A . S TA R I N E T S A N D W. V A N D E R S C H E E

O U T L I N E

• from holography to experiment

• coupling constant dependence and universality in hydrodynamics

• thermalisation and higher-energy spectrum

• heavy ion collisions

• conclusion and future directions

2

F R O M H O L O G R A P H Y T O E X P E R I M E N T

• some strongly coupled field theories have a dual gravitational description (AdS/CFT correspondence)

• originally a duality in type IIB string theory

• so far: universality at strong coupling

• challenges

• field content (no supersymmetry)

• away from infinite N

• away from infinite coupling

Z[field theory] = Z[string theory]

3

↵0 / 1/�1/2

� ⌘ g2YMN

T W O C L A S S E S O F T H E O R I E S• top-down dual of N=4 theory with ’t Hooft coupling corrections from

type IIB string theory

SIIB =1

2210

Zd

10x

p�g

✓R� 1

2(@�)2 � 1

4 · 5!F25 + �e

� 32�W + . . .

◆

� = ↵03⇣(3)/8 ↵0/L2 = ��1/2

W = C↵���Cµ��⌫C⇢�µ

↵ C⌫⇢�� +

1

2C↵���Cµ⌫��C

⇢�µ↵ C⌫

⇢��

4

• bottom-up curvature-squared theory with a special, non-perturbative case of Gauss-Bonnet gravity

SR2 =1

225

Zd

5x

p�g

⇥R� 2⇤+ L

2�↵1R

2 + ↵2Rµ⌫Rµ⌫ + ↵3Rµ⌫⇢�R

µ⌫⇢��⇤

SGB =1

225

Zd

5x

p�g

R� 2⇤+

�GB

2L

2�R

2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R

µ⌫⇢���

H Y D R O DY N A M I C S

H Y D R O DY N A M I C S

• QCD and quark-gluon plasma

• low-energy limit of QFTs (effective field theory)

• tensor structures (phenomenological gradient expansions) with transport coefficients (microscopic)

rµTµ⌫ = 0

Tµ⌫�u�, T, µ

�= ("+ P )uµu⌫ + Pgµ⌫ � ⌘�µ⌫ � ⇣r · u�µ⌫ + . . .

6

H Y D R O DY N A M I C S

• conformal (Weyl-covariant) hydrodynamics

• infinite-order asymptotic expansion

• classification of tensors beyond Navier-Stokes

first order: 2 (1 in CFT) - shear and bulk viscosities

second order: 15 (5 in CFT) - relaxation time, … [Israel-Stewart and extensions]

third order: 68 (20 in CFT) - [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461]

Tµµ = 0

gµ⌫

! e�2!(x)gµ⌫ Tµ⌫ ! e6!(x) Tµ⌫

! =OHX

n=0

↵nkn+1Tµ⌫ =

1X

n=0

Tµ⌫(n)

7

H Y D R O DY N A M I C S• diffusion and sound dispersion relations in CFT

• loop corrections break analyticity of the gradient expansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (2003)]

• entropy current, constraints on transport and new transport coefficients (anomalies, broken parity)

• non-relativistic hydrodynamics

• hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]

shear: ! = �i⌘

"+ Pk2 � i

"⌘2⌧⇧

("+ P )

2 � 1

2

✓1"+ P

#k4 +O �

k5�

sound: ! = ±csk � i�ck2 ⌥ �c

2cs

��c � 2c2s⌧⇧

�k3 � i

"8

9

⌘2⌧⇧

("+ P )

2 � 1

3

✓1 + ✓2"+ P

#k4 +O �

k5�

8

H Y D R O DY N A M I C S F R O M H O L O G R A P H Y

• holography can compute microscopic transport coefficients [Policastro, Son, Starinets (2001)]

• low-energy limit of QFTs / low-energy gravitational perturbations in backgrounds with black holes

• Green’s functions and Kubo formulae

• quasi-normal modes [Kovtun, Starinets (2005)]

• fluid-gravity [Bhattacharyya, Hubeny, Minwalla, Rangamani (2007)]

⌦T

abR (0)

↵=G

abR (0)� 1

2

Zd

4xG

ab,cdRA (0, x)hcd(x) +

1

8

Zd

4xd

4yG

ab,cd,efRAA (0, x, y)hcd(x)hef (y) + . . .

! =1X

n=0

↵nkn+1

9

N = 4 AT I N F I N I T E C O U P L I N G• type IIB theory on S5 dual to N=4 supersymmetric Yang-Mills at

infinite ’t Hooft coupling and infinite Nc

• black brane

• use to find field theory stress-energy tensor to third order

S =1

225

Zd

5x

p�g

✓R+

12

L

2

◆

5 = 2⇡/Nc

ds

2 =r

20

u

��f(u)dt2 + dx

2 + dy

2 + dz

2�+

du

2

4u2f(u)

f(u) = 1� u2

T ab = "uaub + P�ab � ⌘�ab + ⌘⌧⇧

hD�abi +

1

3�ab (r · u)

�+

hRhabi � 2ucR

chabidud

i

+ �1�hac�

bic + �2�hac⌦

bic + �3⌦hac⌦

bic +20X

n=1

�(3)n On

10

N = 4 AT I N F I N I T E C O U P L I N G• transport coefficients [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461]

⌘

s=

1

4⇡

�(3)1 + �(3)

2 + �(3)4 ⌘ �✓1 = �N2

c T

32⇡

�(3)3 + �(3)

5 + �(3)6 ⌘ �✓2 =

N2c T

384⇡

✓⇡2

12+ 18 ln 2� ln2 2� 22

◆

�(3)1 � �(3)

16 =N2

c T

16⇡

✓⇡2

12+ 4 ln 2� ln2 2

◆

�(3)17 =

N2c T

16⇡

✓⇡2

12+ 2 ln 2� ln2 2

◆

�(3)1

6+

4�(3)2

3+

4�(3)3

3+

5�(3)4

6+

5�(3)5

6+

4�(3)6

3� �(3)

7

2

+3�(3)

8

2+

�(3)9

2� 2�(3)

10

3� 11�(3)

11

6� �(3)

12

3+

�(3)13

6� �(3)

15 =N2

c T

648⇡

�15� 2⇡2 � 45 ln 2 + 24 ln2 2

�

⌘ =⇡

8N2

c T3

⌧⇧ =(2� ln 2)

2⇡T =

N2c T

2

8�1 =

N2c T

2

16�2 = �N2

c T2

8ln 2 �3 = 0

11

T O P- D O W N C O N S T R U C T I O N• type IIB action with ’t Hooft coupling corrections

• dimensional reduction

• black brane

SIIB =1

2210

Zd

10x

p�g

✓R� 1

2(@�)2 � 1

4 · 5!F25 + �e

� 32�W + . . .

◆

� = ↵03⇣(3)/8 ↵0/L2 = ��1/2

W = C↵���Cµ��⌫C⇢�µ

↵ C⌫⇢�� +

1

2C↵���Cµ⌫��C

⇢�µ↵ C⌫

⇢��

S =1

225

Zd

5x

p�g

✓R+

12

L

2+ �W

◆

ds

2 =r

20

u

��f(u)Ztdt

2 + dx

2 + dy

2 + dz

2�+ Zu

du

2

4u2f

f(u) = 1� u2

Zt = 1� 15��5u2 + 5u4 � 3u6

�Zu = 1 + 15�

�5u2 + 5u4 � 19u6

�

12

T O P- D O W N C O N S T R U C T I O N• N=4 transport coefficients to second order [S. G., Starinets, JHEP 1503 (2015)

007 arXiv:1412.5685]

⌘ =⇡

8N2

c T3 (1 + 135� + . . . )

⌧⇧ =(2� ln 2)

2⇡T+

375�

4⇡T+ . . .

=N2

c T2

8(1� 10� + . . . )

�1 =N2

c T2

16(1 + 350� + . . . )

�2 = �N2c T

2

16(2 ln 2 + 5 (97 + 54 ln 2) � + . . . )

�3 =25N2

c T2

2� + . . .

⌘

s=

1

4⇡

⇣1 + 15⇣(3)��3/2 + . . .

⌘

0

h̄

4πkB

η

s

g2Nc

[Kovtun, Son, Starinets (2005)]

13

B O T T O M - U P C O N S T R U C T I O N• curvature-squared theory [S. G., Starinets, JHEP 1503 (2015) 007 arXiv:1412.5685]

SR2 =1

225

Zd

5x

p�g

⇥R� 2⇤+ L

2�↵1R

2 + ↵2Rµ⌫Rµ⌫ + ↵3Rµ⌫⇢�R

µ⌫⇢��⇤

⌘ =r3+22

5

(1� 8 (5↵1 + ↵2)) +O(↵2i )

⌘⌧⇧ =r2+ (2� ln 2)

425

✓1� 26

3(5↵1 + ↵2)

◆� r2+ (23 + 5 ln 2)

1225

↵3 +O(↵2i )

=r2+22

5

✓1� 26

3(5↵1 + ↵2)

◆� 25r2+

625

↵3 +O(↵2i )

�1 =r2+42

5

✓1� 26

3(5↵1 + ↵2)

◆� r2+

1225

↵3 +O(↵2i )

�2 = �r2+ ln 2

225

✓1� 26

3(5↵1 + ↵2)

◆� r2+ (21 + 5 ln 2)

625

↵3 +O(↵2i )

�3 = �28r2+25

↵3 +O(↵2i )

14

B O T T O M - U P C O N S T R U C T I O N• Gauss-Bonnet theory [S. G., Starinets, Theor. Math. Phys. 182 (2015) 1, 61-73]

SGB =1

225

Zd

5x

p�g

R� 2⇤+

�GB

2L

2�R

2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R

µ⌫⇢���

⌘ = s�2/4⇡

⌧⇧ =

1

2⇡T

✓1

4

(1 + �)

✓5 + � � 2

�

◆� 1

2

log

2 (1 + �)

�

�◆

�1 =

⌘

2⇡T

(1 + �)

�3� 4� + 2�3

�

2�2

!

�2 = � ⌘

⇡T

✓�1

4

(1 + �)

✓1 + � � 2

�

◆+

1

2

log

2 (1 + �)

�

�◆

�3 = � ⌘

⇡T

(1 + �)

�3 + � � 4�2

�

�2

!

=

⌘

⇡T

(1 + �)

�2�2 � 1

�

2�2

!

✓1 =

⌘

8⇡2T 2��2�2

+ � � 1

�

� =p1� 4�GB

⌘

s=

1

4⇡(1� 4�GB)

15

L I M I T S O F T H E G A U S S - B O N N E T T H E O R Y

• exact spectrum in the extreme (anomalous) limit of

16

�GB = 1/4

Scalar: w = �i⇣4 + 2n1 �

p4� 3q2

⌘, w = �i

⇣4 + 2n2 +

p4� 3q2

⌘

Shear: w = �2i (1 + n1) , w = �2i (3 + n2)

Sound: w = �i⇣4 + 2n1 �

p4 + q2

⌘, w = �i

⇣4 + 2n2 +

p4 + q2

⌘

• in the extreme ``weak” limit there is a curvature singularity, which needs a stringy resolution

lim�GB!�1

SGB =�GBL

2

425

Zd

5x

p�g

R

2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R

µ⌫⇢� � 4⇤

�GBL2

�

ds

2 =p

��GB

2

4� r̃

2

L

2

r1�

r̃

4+

r̃

4dt

2 +L

2

r̃

2

q1� r̃4+

r̃4

dr̃

2 +r̃

2

L

2

�dx

2 + dy

2 + dz

2�3

5

�GB ! �1

• how can we interpret the extreme limits of the Gauss-Bonnet coupling?

C H A R G E D I F F U S I O N• construct the most general four-derivative action [S. G., Starinets (2016)

arXiv:1611.07053]

17

• make it such that the equations of motion are only second order

S =1

225

Zd

5x

p�g [R� 2⇤+ LGB ] +

Zd

5x

p�gLA

LA = �1

4Fµ⌫F

µ⌫ + ↵4RFµ⌫Fµ⌫ + ↵5R

µ⌫Fµ⇢F⇢

⌫ + ↵6Rµ⌫⇢�Fµ⌫F⇢� + ↵7 (Fµ⌫F

µ⌫)2

+ ↵8rµF⇢�rµF ⇢� + ↵9rµF⇢�r⇢Fµ� + ↵10rµFµ⌫r⇢F⇢⌫ + ↵11F

µ⌫F⌫⇢F⇢�F�µ

LA =� 1

4Fµ⌫F

µ⌫ + �1L2 (RFµ⌫F

µ⌫ � 4Rµ⌫Fµ⇢F⇢

⌫ +Rµ⌫⇢�Fµ⌫F⇢�)

+ �2L2 (Fµ⌫F

µ⌫)2 + �3L2Fµ⌫F⌫⇢F

⇢�F�µ

D =(1 + �GB)(1 + 2�)

⇣� +

p�2 � �2

GB

⌘

6(� � 1)h�⇣� +

p�2 � �2

GB

⌘� �2

GB

i(p(1� �2

GB) (�2 � �2

GB) ln

"�GB

1 +p1� �2

GB

#

��� � �2

GB

�ln

"�GB

� +p�2 � �2

GB

#)

• diffusion

�GB =p

1� 4�GB

� = 1 + 48�1

D 6= D = 0

D 6= D = 0

U N I V E R S A L I T Y• membrane paradigm (conserved current)

• first-order hydrodynamics [Kovtun, Policastro, Son, Starinets]

• second-order hydrodynamics [Haack, Yarom (2009); S. G., Starinets, JHEP 1503 (2015) 007 arXiv:1412.5685]

⌘

s=

1

4⇡

@rJ = 0

2⌘⌧⇧ � 4�1 � �2 = O ��2

�

2⌘⌧⇧ � 4�1 � �2 = O �↵2i

�

2⌘⌧⇧ � 4�1 � �2 = � ⌘

⇡T

(1� �GB)�1� �2

GB

�(3 + 2�GB)

�2GB

= �40�2GB⌘

⇡T+O �

�3GB

�

18

( M O R E ) U N I V E R S A L I T Y• non-renormalisation of anomalous conductivities

[Gursoy, Tarrio (2015); S. G., Poovuttikul, JHEP 1609 (2016) 046 arXiv:1603.08770]

• universal values

• bulk with arbitrarily high derivatives (gauge, diffeomorphism)

• proof to all orders in the coupling constant expansion

✓h�Jµih�Jµ

5 i

◆=

✓�JB �J!

�J5B �J5!

◆✓Bµ

!µ

◆

�J5B = �2�µ �JB = �2�µ5,

�J5! = µ25 + �µ2 + 2�(2⇡T )2 �J! = 2�µ5µ

S =

Zd

5x

p�g {L [Aa, Va, gab,�i] + LCS [Aa, Va, gab]}

LCS [Aa, Va, gab] = ✏abcdeAa

⇣3FA,bcFA,de + �FV,bcFV,de + �Rp

qbcRqpde

⌘

19

B E Y O N D H Y D R O DY N A M I C S

W E A K C O U P L I N G ( K I N E T I C T H E O R Y )

• coupling constant dependence of non-hydrodynamic transport [S. G., Kaplis, Starinets, JHEP 1607 (2016) 151 arXiv:1605.02173]

• instead of hydrodynamics, start with (weakly coupled) kinetic theory

• essential concept: quasi-particles

• Boltzmann equation

• close to equilibrium

• resulting equation

• solution

21

@F

@t+

pim

@F

@ri� @U(r)

@ri@F

@pi= C[F ]

F (t, r,p) = F0(r,p) [1 + '(t, r,p)]

@'

@t= �pi

m

@'

@ri+

@U(r)

@ri@'

@pi+ L0[']

'(t, r,p) = etL '0(r,p) =1

2⇡i

�+i1Z

��i1

est Rsds'0(r,p)

Rs = (sI � L)�1

W E A K C O U P L I N G ( K I N E T I C T H E O R Y )

• ansatz for homogeneous eq. distribution

• eigenvalue equation for lin. coll. operator

• hierarchy of relaxation times

22

'(t,p) = e�⌫th(p)

�⌫h = L0[h]

'(t,p) =X

n

Cne�⌫nthn(p)

dominant:

⌧R = 1/⌫min

T H E R M A L I S AT I O N ( R E L A X AT I O N )

• KGB equation

• kinetic theory predicts

• relaxation time bound [Sachdev]

• Ising model (BTZ)

⌘ = ⌧R s T

⌧R � C ~kBT

GR(!, q) =

C�⇡ �

2(�� 1) sin⇡�

�����✓�

2

+

i(! � q)

4⇡T

◆�

✓�

2

+

i(! + q)

4⇡T

◆����2

⇥"cosh

q

2T� cos⇡� cosh

!

2T+ i sin⇡� sinh

!

2T

#

! = ±q � i4⇡T

✓n+

�

2

◆

⌧R =1

2⇡�

~kBT

23

@F

@t+

pim

@F

@ri� @U(r)

@ri@F

@pi= �F � F0

⌧R

cute: � = 2 ) “⌘/s” = 1/4⇡

F U L L Q U A S I N O R M A L S P E C T R U M • weak/strong coupling (perturbative/holographic quasi-normal

mode calculations) [Hartnoll, Kumar (2005)]

qq-q-q

-i4πT-i4πT

-i8πT-i8πT

-i12πT-i12πT

-i16πT-i16πT

Reω

Imω

Reω

Imω

� ! 0 � ! 1

24

R E S U LT S : Q N M S T R U C T U R E

• different trends depending on

• poles become denser (branch cut)

• new poles on imaginary axis

x xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxxx

◆

◆

◆

ReωImω

x xx

x xx

x xx

x xx

x xx

x xx

x xx

x xx

x xx

x xx

x xx

xx

◆

◆

◆

ReωImω

⌘/s

⌘/s > ~/4⇡kB ⌘/s < ~/4⇡kB

``weak limit”

� ! 1�GB ! �1

``anomalous limit”�GB ! 1/4

25

Q N M S T R U C T U R E• shear channel QNMs in N=4

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◆

◆

◆

ReωImω

x xx

x xx

x xx

x xx

x xx

x xx

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x xx

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xx

◆

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26

Q N M S T R U C T U R E• shear channel QNMs in Gauss-Bonnet

x xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxxx

◆

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x xx

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27

K I N E T I C T H E O R Y R E S U LT• kinetic theory behaviour quickly approached

x xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxxx

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ReωImω

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ReωImω

⌘/s ⇠ const ⌧RT

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28

H Y D R O DY N A M I C S• breakdown of hydrodynamics (diffusion) qc ⇠ �3/4

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H Y D R O DY N A M I C S• breakdown of hydrodynamics (diffusion)

⟩

⟩

qc ⇠ �3/4

30

H Y D R O DY N A M I C S• breakdown of hydrodynamics (sound)

⟩

⟩

qc ⇠ �3/4

31

H Y D R O DY N A M I C S• breakdown of hydrodynamics due to new poles

• shear and sound correlators in Gauss-Bonnet [S. G., Starinets (2016) arXiv:

1611.07053]

32

Gtt,tt(!, q) =3p2⇡4T 4

(1 + �GB)3/225

�5q2 � 3!2

�(1� !/!g)� i�2

GB!q2/⇡T

(3!2 � q2) (1� !/!g) + i�2GB!q

2/⇡T

!

Gxz,xz

(!, q) =

p2⇡3T 3�2

GB

(1 + �GB)3/225

✓!2

i! � i!2/!g � �2GBq

2/4⇡T

◆

• dispersion relations

!1 = �i�2

GB

4⇡Tq2 !2 = !g + i

�2GB

4⇡Tq2

!1,2 = ± 1p3q � i

�2GB

6⇡Tq2 !3 = !g + i

�2GB

3⇡Tq2

!g = � 8⇡Ti

�GB (�GB + 2)� 3 + 2 ln⇣

2�GB+1

⌘ ⇡ �8⇡Ti

�2GB

• gap

Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )

• quasi-particles appear in the spectrum [Casalderrey-Solana, S. G., Starinets (2017)]

• weak coupling (Boltzmann equation)

33

• holographic results in N=4 theory (scalar channel)

34

Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )

• holographic results in a dual of the Gauss-Bonnet theory (scalar channel)

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Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )

Q U A S I - PA R T I C L E S I N N = 436

Q U A S I - PA R T I C L E S I N G A U S S - B O N N E T37

• quasi-particles appear also in other channels (shear)

38

• how do these peaks manifest themselves (if at all) in a weakly coupled field theory like QCD?

Q U A S I - PA R T I C L E S I N G A U S S - B O N N E T

H E A V Y I O N C O L L I S I O N S

H E A V Y I O N C O L L I S I O N S• R2 coupling constant corrections to shockwave collisions

[S. G., van der Schee (2016) arXiv:1610.08976]

• energy density along the longitudinal direction after collision: less stopping of narrow shocks (88% higher energy density on lightcone) and decreased energy density of wide shocks

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H E A V Y I O N C O L L I S I O N S• delayed hydrodynamisation

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thydThyd = {0.41� 0.52�GB,

0.43� 6.3�GB}

at �GB = �0.2 :

{25%, 290%}

H E A V Y I O N C O L L I S I O N S

• rapidity profile: start wider and smaller but later become comparable

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H E A V Y I O N C O L L I S I O N S• change in entropy density: enhanced on lightcone, negative in plasma

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• total entropy: reduced at intermediate coupling

C O N C L U S I O N A N D F U T U R E D I R E C T I O N S

• various weakly coupled properties are recovered remarkably quickly

• the topic of this workshop: how precisely do they match onto those

computed from weakly coupled, perturbative physics?

• use interpolations between weakly coupled physics and coupling-

dependent holography to understand intermediate coupling

• can we see a signature of higher-QNMs (quasi-particles) in

experiments?

• a harder task for the future: understand 1/N corrections

44

T H A N K Y O U !