.
. Flavour and Mesons in the AdS/CFT correspondence
Johanna Erdmenger
Max Planck-Institut fur¨ Physik, Munchen¨
1 Outline
I. Introduction to AdS/CFT
II. Adding Flavour
III. Chiral Symmetry Breaking and Mesons
IV. Finite Temperature and Density
2 AdS/CFT Correspondence
(Maldacena 1997, AdS: Anti de Sitter space, CFT: conformal field theory)
Duality Quantum Field Theory ⇔ Gravity Theory
Arises from String Theory in a particular low-energy limit
Duality: Quantum field theory at strong coupling ⇔ Gravity theory at weak coupling
Works for large N gauge theories at large ’t Hooft coupling λ
Conformal field theory in four dimensions 5 ⇔ Supergravity Theory on AdS5 × S
3 AdS/CFT correspondence
Anti-de Sitter space is a curved space with constant negative curvature. It has a boundary.
2 2r/R µ ν 2 Metric: ds = e ηµνdx dx − dr
Isometry group of (d + 1)-dimensional AdS space coincides with conformal group in d dimensions (SO(d, 2)).
SO(6) ' SU(4): Isometry of S5 ⇔ N = 4 Supersymmetry
Dictionary:
field theory operators ⇔ supergravity fields q d d2 2 2 O∆ ↔ φm , ∆ = 2 + 4 + R m
4 AdS/CFT correspondence
field-operator correspondence:
R d d x φ0(~x)O(~x) he iCFT = Zsugra φ(0,~x)=φ0(~x)
Generating functional for correlation functions of particular composite operators in the quantum field theory
coincides with
Classical tree diagram generating functional in supergravity
5 String theory origin of AdS/CFT correspondence
D3 branes in 10d
duality
5 AdS 5 x S near-horizon geometry
⇓ Low-energy limit
N = 4 SU(N) theory in four Supergravity on AdS × S5 dimensions (N → ∞) 5
6 Generalizations of the AdS/CFT correspondence
N = 4 SU(N) SUSY Gauge theory: QCD:
N = 3 N → ∞ No supersymmetry Supersymmetry Confinement Conformal symmetry Quarks in fundamental represen- All fields in the adjoint representa- tation of the gauge group tion of the gauge group
Desirable extensions of AdS/CFT:
Relax N → ∞ limit (1/N corrections) ⇔ String theory instead of supergravity Break SUSY and conformal symmetry ⇔ Deformation of AdS space Add quarks in fundamental representation of gauge group
7 Deformations of AdS space
Minkowski− spacetime Anti−De Sitter− spacetime
deformation
Fifth Dimension ⇔ Energy scale Renormalization group flow from supergravity ⇒ ‘holographic’ Renormalization Group flow
SUSY broken by deformation of S5
8 Running gauge coupling
g
Coupling
deformed N=4 Theory
QCD
Energy IR UV
9 Quarks (fundamental fields) within the AdS/CFT correspondence
D7 brane probe:
0 1 2 3 4 5 6 7 8 9 D3 X X X X D7 X X X X X X X X
D7 brane
AdS S5
S 3 fluctuation
horizon
10 Quarks (fundamental fields) from brane probes
N D3 0123 conventional open/closed string duality 4567 SYM AdS 89 3−3 5 quarks 3−7 7−7 flavour open/open string duality N f probe D7 AdS R4 5 brane N → ∞ (standard Maldacena limit), Nf small (probe approximation)
duality acts twice:
N = 4 SU(N) Super Yang-Mills theory 5 IIB supergravity on AdS5 × S coupled to ←→ + 3 N = 2 fundamental hypermultiplet Probe brane DBI on AdS5 × S
Karch, Katz 2002
11 Chiral symmetry breaking within generalized AdS/CFT
Combine the deformation of the supergravity metric with the addition of brane probes:
Dual gravity description of chiral symmetry breaking and Goldstone bosons
J. Babington, J. E., N. Evans, Z. Guralnik and I. Kirsch,
“Chiral symmetry breaking and pions in non-SUSY gauge/gravity duals”
hep-th/0306018
12 D7 brane probe in deformed backgrounds
D7 brane probe in gravity backgrounds dual to
confining gauge theories without supersymmetry.
Example:
5 Constable-Myers background (particular deformation of AdS5 × S metric)
The deformation introduces a new scale into the metric.
5 3 In UV limit, geometry returns to AdS5 ×S with D7 probe wrapping AdS5 ×S .
13 General strategy
1. Start from Dirac-Born-Infeld action for a D7-brane embedded in deformed background
2. Derive equations of motion for transverse scalars (w5, w6)
3. Solve equations of motion numerically using shooting techniques
Solution determines embedding of D7-brane (e.g. w5 = 0, w6 = w6(ρ))
4. Meson spectrum:
Consider fluctuations δw5, δw6 around a background solution obtained in 3.
Solve equations of motion linearized in δw5, δw6
14 Asymptotic behaviour of supergravity solutions
UV asymptotic behaviour of solutions to equation of motion:
−r −3r w6 ∝ m e + c e
Identification of the coefficients as in the standard AdS/CFT correspondence: m quark mass, c = hqq¯ i quark condensate
Here: m 6= 0: explicit breaking of U(1)A symmetry c 6= 0: spontaneous breaking of U(1)A symmetry
15 The Constable-Myers deformation
µν N = 4 super Yang-Mills theory deformed by VEV for tr F Fµν (R-singlet operator with D = 4) → non-supersymmetric QCD-like field theory The Constable-Myers background is given by the metric
δ/4 (2−δ)/4 6 w4 + b4 w4 + b4 w4 − b4 X ds2 = H−1/2 dx2 + H1/2 dw2, w4 − b4 4 w4 − b4 w4 i i=1 where w4 + b4δ H = − 1 (∆2 + δ2 = 10) w4 − b4 and the dilaton and four-form
w4 + b4∆ 1 e2φ = e2φ0 ,C = − H−1dt ∧ dx ∧ dy ∧ dz w4 − b4 (4) 4
This background has a singularity at w = b
16 Chiral symmetry breaking
Solution of equation of motion for Numerical Result: w probe brane 2 6
w6 1.75 m=1.5, c=0.90 1.5 m=1.25, c=1.03 1.25 m=1.0, c=1.18 1 m=0.8, c=1.31 s i n 0.75 g u m=0.6, c=1.45 la 0.5 r m=0.4, c=1.60 i m=0.2, c=1.73 bad 0.25 t good y m=10^−6, c=1.85 ρ ugly ρ 0.5 1 1.5 2 2.5 3 singularity
c Result:
1.5 Screening effect: Regular solutions do not reach the singularity 1.0
0.5 Spontaneous breaking of U(1)A symmetry: For m → 0 we have 0 ¯ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 m c ≡ hψψi= 6 0
17 Meson spectrum
From fluctuations of the probe brane
2 2 Ansatz: δwi(x, ρ) = fi(ρ)sin(k · x) ,M = −k
meson mass
6
5
4
3
2
1
quark mass 0.5 1 1.5 2
Goldstone boson (η0) √ Gell-Mann-Oakes-Renner relation: MMeson ∝ mQuark
18 Comparison to lattice gauge theory
J.E., Evans, Kirsch, Threlfall 0711.4467, EPJA
2 mρ vs. mπ (Lattice: Lucini, Del Debbio, Patella, Pica 0712.3036)
0.9 mΡ 1 0.8
0.8 0.7
ρ 0.6 0.6 m
0.5 SU(2) 0.4 SU(3) SU(4) 0.4 SU(6) 0.2 N = inf.
0.3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 mΠ 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 mπ Slope: 0.57 Slope: 0.52 Normalized to scale in metric Normalized to lattice spacing
(Similar results by Bali and Bursa)
19 Finite Temperature
N = 4 Super Yang-Mills theory at finite temperature is dual to AdS black hole
1 % 2 f 2 R2 ds2 = − dt2 + f˜d~x2 + (d%2 + %2dΩ2) 2 R f˜ % 5
%4 %4 f(%) = 1 − H , f˜(%) = 1 + H %4 %4
Temperature and horizon related by
% T = H πR2
R AdS radius
20 Condensate in field theory at finite temperature
D7 brane embedding in black hole Condensate c versus quark mass m background (c, m normalized to T )
w6 2 -c 1.75
1.5 m=1.5 Karch−Katz 0.1 1.25 m=1.25 0.08 m=1.0 phase 1 m=0.92 m=0.91 transition 0.75 m=0.8 0.06 h o m=0.6 0.5 r i z 0.04 o n m=0.4 Black hole 0.25 m=0.2 0.02 ρ 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1.0 1.2 m
Phase transition at m/T ≈ 0.92
Babington, J.E., Evans, Guralnik, Kirsch 0306018
21 Phase transition
0.035
0.0325
0.03 A 0.0275
0.025 B 0.0225 C D 0.02 E 0.9160.918 0.92 0.9220.9240.9260.928 0.93
First order phase transition in type II B AdS black hole background
Ingo Kirsch, PhD thesis 2004
22 Finite U(1) baryon density
Baryon density nB and U(1) chemical potential µ from VEV for gauge field time component:
˜ 5/2 d ˜ 2 A¯0(ρ) ∼ µ + , d = √ nB 2 3 ρ Nf λT Finite U(1) baryon density
Baryon density nB and U(1) chemical potential µ from VEV for gauge field time component:
˜ 5/2 d ˜ 2 A¯0(ρ) ∼ µ + , d = √ nB 2 3 ρ Nf λT
At finite baryon density, all embeddings are black hole embeddings
Mateos, Myers et al
23 Spectral functions for vector mesons
J.E., Kaminski, Kerner, Rust 0807.2663
From fluctuations of gauge field on the D7 brane
Meson melting for high densities
2000 m5 dt1 1500 dt2 dt20 dt200 R 1000 dt2000
500
0 0 2 4 6 8 10 12 w
24 Spectral functions for mesons at finite isospin density
1200
m3 1000 dt8 800 dt14
R 600
400
200
0 0 1 2 3 4 5 6 w
Vector meson condensation
25 Quarkonium transport in thermal AdS/CFT
Dusling, J.E., Kaminski, Rust, Teaney, Young 0808.0957
Consider heavy meson moving slowly through the plasma
Interaction with gluons Quarkonium transport in thermal AdS/CFT
Dusling, J.E., Kaminski, Rust, Teaney, Young 0808.0957
Consider heavy meson moving slowly through the plasma
Interaction with gluons
Calculation of ratio κ/δM 2
κ: momentum diffusion, δM: in-medium mass shift both at strong and at weak coupling in N = 4 theory Quarkonium transport in thermal AdS/CFT
Dusling, J.E., Kaminski, Rust, Teaney, Young 0808.0957
Consider heavy meson moving slowly through the plasma
Interaction with gluons
Calculation of ratio κ/δM 2
κ: momentum diffusion, δM: in-medium mass shift both at strong and at weak coupling in N = 4 theory
AdS/CFT result five times smaller than effective field theory result
26 Conclusion Conclusion
Generalized AdS/CFT provides new tools for strongly coupled gauge theories Conclusion
Generalized AdS/CFT provides new tools for strongly coupled gauge theories
New relation: String theory ⇔ QFT, strong interactions Conclusion
Generalized AdS/CFT provides new tools for strongly coupled gauge theories
New relation: String theory ⇔ QFT, strong interactions
Numerous applications for specific problems: Chiral symmetry breaking Spectral functions and transport processes in quark-gluon plasma
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