Holographic Mesons at Finite Temperature and Density Johanna Erdmenger
Transcript of 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
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Motivation + Aim
Combine
AdS/CFT with added flavour
with
AdS/CFT applied to hydrodynamics
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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
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Quarks (fundamental fields) from brane probes
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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
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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
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b deformation parameter, τ periodic (period πb = T−1)horizon: S1 collapses at w = 1
2b
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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
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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)
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Hydrodynamics from AdS/CFT - Spectral functions
Hydrodynamics from AdS/CFT Son, Starinets et al
Transport processes in the quark-gluon plasma
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
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
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
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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
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
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
Identify mesons with quasiparticle resonances
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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
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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; . . .
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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
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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
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)
For increasing m/T , peaks first move to smaller, then to larger frequencies
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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)
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Isospin spectral function in thermal AdS/CFT with flavour
J.E., Kaminski, Rust 0710.0334
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
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Isospin spectral function in thermal AdS/CFT with flavour
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
Isospin spectral function in thermal AdS/CFT with flavour
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
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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
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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
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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)
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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)
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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
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AdS/CFT calculation - result
This givesκ
(δM)2=
πT
N28.37
Result five times smaller than perturbative N = 4 SYM result!
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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
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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
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