Holographic Mesons at Finite Temperature and Density Johanna Erdmenger

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Holographic Mesons at Finite Temperature and Density

Johanna Erdmenger

Max-Planck-Institut fur Physik, Munchen

1

Motivation + Aim

Combine

AdS/CFT with added flavour

with

AdS/CFT applied to hydrodynamics

2

Quarks (fundamental fields) within the AdS/CFT correspondence

D7 brane probe:

0 1 2 3 4 5 6 7 8 9D3 X X X XD7 X X X X X X X X

D7 brane

AdS

fluc

tuat

ion

S5

S3

3

Quarks (fundamental fields) from brane probes

��

��

��

��

89

0123

4567

D3N

4R

AdS5

open/closed string duality

7−7

AdS5brane

flavour open/open string duality

conventional

3−7quarks

3−3

SYM

N probe D7f

N →∞ (standard Maldacena limit), Nf small (probe approximation)

duality acts twice:

N = 4 SU(N) Super Yang-Mills theory

coupled to

N = 2 fundamental hypermultiplet

←→IIB supergravity on AdS5 × S5

+Probe brane DBI on AdS5 × S3

Karch, Katz 2002

4

Flavour in the AdS Black Hole geometry

Consider N = 4 SU(N) SYM at finite temperature (Witten, 1998)

Dual string theory background: Euclidean AdS-Schwarzschild solution

ds2 =(

w2 +b4

4w2

)d~x2 +

(4w4 − b4)2

4w2(4w4 + b4)dτ2 +

1w2

6∑i=1

dw2i

with radial coordinate w2 = ρ2 + w25 + w2

6

b deformation parameter, τ periodic (period πb = T−1)horizon: S1 collapses at w = 1

2b

5

Condensate in field theory at finite temperature

D7 brane embedding in black holebackground

0.5 1 1.5 2

0.25

0.5

0.75

1

1.25

1.5

1.75

2

nozirh o

m=0.2

m=0.4

m=0.6m=0.8

m=1.5

m=1.25

m=1.0 m=0.92m=0.91

Karch−Katz

Black hole

w6

ρ

Condensate c versus quark mass m(c, m normalized to T )

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1.0

-c

m1.2

phasetransition

Phase transition at m/T ≈ 0.92

Babington, J.E., Evans, Guralnik, Kirsch 0306018

6

Phase transition

0.9160.918 0.92 0.9220.9240.9260.928 0.93

0.02

0.0225

0.025

0.0275

0.03

0.0325

0.035

CD

E

A

B

First order phase transition in type II B AdS black hole background

Ingo Kirsch, PhD thesis 2004

(Related work by Mateos, Myers et al)

7

Hydrodynamics from AdS/CFT - Spectral functions

Hydrodynamics from AdS/CFT Son, Starinets et al

Transport processes in the quark-gluon plasma




Time dependence requires Minkowski signature AdS black hole

Infalling boundary condition required at black hole horizon




Time dependence requires Minkowski signature AdS black hole

Infalling boundary condition required at black hole horizon

Spectral function from imaginary part of retarded Green function

GR(ω, k) = −i

∫d4x ei ~k~x θ(x0) 〈[J(~x), J(0)]〉

Correlator calculated from propagation through AdS black hole space

J : flavour current

Spectral function⇒ Quasiparticle spectrum

8

Spectral functions - Masses and decay widths of mesons

Standard procedure in D3/D7:

Meson masses calculated from linearized fluctuations of D7 embedding

Fluctuations: δw(x, ρ) = f(ρ)ei(ωt−~k·~x), M2 = −k2





Landsteiner, Hoyos, Montero 2006:

For black hole embeddings, ω develops an imaginary part

⇒ damping⇒ decay width

Mesons unstable⇒ quasinormal modes





Landsteiner, Hoyos, Montero 2006:

For black hole embeddings, ω develops an imaginary part

⇒ damping⇒ decay width

Mesons unstable⇒ quasinormal modes

Identify mesons with quasiparticle resonances

9

Finite U(1) baryon density

Baryon density nB and U(1) chemical potential µfrom VEV for gauge field time component:

A0(ρ) ∼ µ +d

ρ2, d =

25/2

Nf

√λT 3

nB

At finite baryon density, all embeddings are black hole embeddings

Mateos, Myers et al

10

Phase diagram with finite U(1) baryon density

Phase diagram:

grey region: nB = 0white region: nB 6= 0

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1

1

1

1

µq/m

qµ

q/m

q

T/MT/M

d = 0

d = 0.00315

d = 4

d = 0.25

Mateos Myers et al; Karch, O’Bannon; . . .

11

Spectral Functions for Vector Mesons

Gauge field on D7: background + fluctuations

Aµ(ρ, ~x) = δ0µA0(ρ) + Aµ(~x, ρ)

Insert this into equations of motion from DBI action of D7 brane

In Fourier space, use gauge invariant quantities

Ex = ωAx + qA0, Ey,z = ωAy,z (q = 0)

Green functions

GR = GRxx = GR

yy = GRzz =

NfNcT2

8lim

ρ→∞

(ρ3∂ρE(ρ)

E(ρ)

)

At vanishing density: Myers, Starinets, Thompson

12

Spectral function for vector mesons with U(1) quark chemical potential

J.E., Kaminski, Rust 0710.0334

Spectral functions - temperature dominated regime

0.0 0.5 1.0 1.5 2.0 2.5-6

-4

-2

0

2

4

w

R(w

,0)−

R0 d = 0.25

χ0 = 0.1χ0 = 0.5χ0 = 0.7χ0 = 0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-50

0

50

100

150

200

w

R(w

,0)−

R0

d = 0.25

χ0 = 0.8χ0 = 0.94

χ0 = 0.962

(R0 = NfNcT2 πw2)

Spectral function for vector mesons with U(1) quark chemical potential


Spectral functions - temperature dominated regime

0.0 0.5 1.0 1.5 2.0 2.5-6

-4

-2

0

2

4

w

R(w

,0)−

R0 d = 0.25

χ0 = 0.1χ0 = 0.5χ0 = 0.7χ0 = 0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-50

0

50

100

150

200

w

R(w

,0)−

R0

d = 0.25

χ0 = 0.8χ0 = 0.94

χ0 = 0.962

(R0 = NfNcT2 πw2)

For increasing m/T , peaks first move to smaller, then to larger frequencies

13

Spectral function

Spectral functions - potential-dominated regime

0 2 4 6 8 10

-100

-50

0

50

100

150

200

250

w

R(w

,0)−

R0

d = 0.25

χ0 = 0.962

n = 0 n = 1 n = 2 n = 3

0 5 10 15 20 25 30 350

20 000

40 000

60 000

80 000

100 000

120 000

140 000

w

R(w

,0)

d = 0.25

χ0 = 0.999

n = 0 n = 1 n = 2 n = 3

Spectral function

Spectral functions - potential-dominated regime

0 2 4 6 8 10

-100

-50

0

50

100

150

200

250

w

R(w

,0)−

R0

d = 0.25

χ0 = 0.962

n = 0 n = 1 n = 2 n = 3

0 5 10 15 20 25 30 350

20 000

40 000

60 000

80 000

100 000

120 000

140 000

w

R(w

,0)

d = 0.25

χ0 = 0.999

n = 0 n = 1 n = 2 n = 3

Agrees with supersymmetric meson spectrum

(Calculated analytically by Kruczenski, Mateos, Myers, Winters 0304032)

14

Isospin spectral function in thermal AdS/CFT with flavour


Two coincident D7 brane probes in AdS black hole background

SU(2) background field

A30σ

3 = A0(ρ)(

1 00 −1

)

Background:

Fρ0 σ3 =(

∂ρA0 00 −∂ρA0

)Non-abelian structure⇒

Coupled equations of motion for different flavour components ET1, ET

2

Decoupling achieved for X = ET1 + iET

2, Y = ET1 − iET

2

15


0

2 4 6 8 10

1000

2000

3000

4000

XY

XY

Y X

Y X

E3E3

E3E3

n = 0n = 0n = 0

n = 1n = 1n = 1R−R0 wBlack line: G33, green lines: GXY and GY X

Line splitting observed (Isospin triplet): Energy shift by ±µ

Larmor precession in flavour space Son


0

2 4 6 8 10

1000

2000

3000

4000

XY

XY

Y X

Y X

E3E3

E3E3

n = 0n = 0n = 0

n = 1n = 1n = 1R−R0 wBlack line: G33, green lines: GXY and GY X

Line splitting observed (Isospin triplet): Energy shift by ±µ

Larmor precession in flavour space Son

Old ground state becomes unstable cf. Apreda, J.E., Evans, Guralnik ’05

16

Quarkonium transport in AdS/CFT

Dusling, J.E., Kaminski, Rust, Teaney, Young in progress

Diffusion and momentum broadening of heavy mesons

Quarkonium transport in AdS/CFT

Dusling, J.E., Kaminski, Rust, Teaney, Young in progress

Diffusion and momentum broadening of heavy mesons

Perturbative effective field theory results: Manohar et al; Peskin

L = +φ†viv · ∂φv +cE

N2φ†vOEφv +

cB

N2φ†vOBφv

φv: heavy scalar meson with velocity vµ (use rest frame vµ = (1, 0, 0, 0))

OE = EA · EA, OB = BA ·BA

Non-relativistic polarizabilities: cE = 28π3 a0

3, cB = 0

Bohr radius: a0 = (mqN2 αs)−1

In-medium mass shift: δM = −〈Lint〉 = T (πTa0)3 14

45

17

Kinetics of heavy meson in medium

κ: Drag coefficient describing momentum broadening in Langevin theory

Microscopically, with dipole force ~F = −12~∇(Ea · Ea):

κ =13

c2E

N4

∫d3q

(2π)3q2

[−2T

ωIm GE2E2

R (ω, q)]

From perturbative calculation

κQCD 'T 3

N2(πTa0)6 130

To compare with strong coupling calculation consider

κ

δM2' πT

N2426

18

AdS/CFT calculation

N = 4 SYM:

Leff = φv(x, t)iv·∂φv(x, t) +cT

N2φ†v(x, t)Tµνvµvνφv(x, t) +

cF

N2φv(x, t)†(trF 2)φv(x, t)

Perturbative result for N = 4 SYM:

κ

δM2' πT

N237

AdS/CFT calculation

N = 4 SYM:

Leff = φv(x, t)iv·∂φv(x, t) +cT

N2φ†v(x, t)Tµνvµvνφv(x, t) +

cF

N2φv(x, t)†(trF 2)φv(x, t)

Perturbative result for N = 4 SYM:

κ

δM2' πT

N237

Strong coupling calculation from gauge/gravity duality

Polarization coefficients to be determined from mass shifts

δMT =cT

N2〈T 00〉 , δMF =

cF

N2〈trF 2〉

(Meson mass: Lowest mode M = mq√λ2√

2 in AdS5 × S5)

19

AdS/CFT calculation

To obtain the polarizabilities, we calculate

δMT from linear response to switching on black hole background

δMF from linear response to switching on dilaton flow background

Dilaton background of Liu, Tseytlin 1999:

eφ = gs(1 +q4

r4) , q4 =

2π2R8

N2〈trF 2〉

δM is obtained analyticallyby expanding new eigenfunctions in basis of solutions of the unperturbed case

−∂ρρ3∂ρφ(ρ) = M2 ρ3

(ρ + 1)2φ(ρ) + ∆(ρ)φ(ρ)

20

AdS/CFT calculation

Drag coefficient

κ = limω→0

∫d3q

(2π)3q2

3

[( cF

N2

)2 −2T

ωIm GF 2F 2

R (ω, q) +( cT

N2

)2 −2T

ωIm GTT

R (ω, q)]

Green functions calculatedfrom propagation through AdS black hole background

Putting everything together:

κ =T 3

N2

(2πT

M

)6((

85π

)2

67.258 +(

125π

)2

355.1

)

=T 3

N2

(2πT

M

)6

224.7

Temperature, scale and N dependence agree with perturbative result

21

AdS/CFT calculation - result

This givesκ

(δM)2=

πT

N28.37

Result five times smaller than perturbative N = 4 SYM result!

22

External B field on D7 brane probe - in (0,1) or (2,3) directions

Albash, Filev, Johnson, Kundu 2007; Karch, O’BannonJ.E., Meyer, Shock 0709.1551

cf. Oren Bergman’s talk

External magnetic field induces chiral symmetry breaking

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B�=0

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B�=5

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B�=10

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B�=17

23

External B field in D3/D7 system

External electric field: Singular shell; Compensating finite baryon number den-sity

Insulator-metal transition

-1 1 2 3 4 5 6Ρ�

-1.0

-0.5

0.5

1.0

1.5

2.0

L�

Mesons display Zeeman and Stark effects

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Conclusion

First order transition at high temperature in deconfined phase

– corresponds to meson melting

Spectral functions for baryon or isospin chemical potential

For increasing m/T , resonance peaks move first to the left, then to the right

Isospin triplet splitting

Meson diffusion in N = 4 plasma

κ/(δM)2 smaller in strongly coupled case

External B field in (0, 1) or (2, 3) directions for D7 probe

25

Holographic Mesons at Finite Temperature and Density Johanna Erdmenger

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