Holographic Mesons Johanna Erdmenger Max-Planck-Institut f ur - PowerPoint PPT Presentation

. . Holographic Mesons Johanna Erdmenger Max-Planck-Institut f¨ ur Physik, M¨ unchen work in collaboration with Z. Guralnik, I. Kirsch, J. Babington, R. Apreda, J. Große, N. Evans, R. Meyer, D. L¨ ust, M. Kaminski, F . Rust, K. Ghoroku 1

Motivation + Aim To generalize the AdS/CFT correspondence such that it describes realistic field theories

Motivation + Aim To generalize the AdS/CFT correspondence such that it describes realistic field theories Holographic description of fields in the fundamental representation of the gauge group (quarks) chiral symmetry breaking and meson spectra finite temperature field theories (+ finite density) 2

Outline 1. Adding flavour to AdS/CFT 2. Chiral symmetry breaking 3. Mesons 4. Finite temperature and finite density 3

Generalizations of the AdS/CFT correspondence N = 4 SU ( N ) 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: Break SUSY and conformal symmetry ⇔ Deformation of AdS space Add quarks in fundamental representation of gauge group Relax N → ∞ limit ( 1 /N corrections) ⇔ String theory instead of supergravity 4

Models of generalized AdS/CFT with flavour

Models of generalized AdS/CFT with flavour D3/D7 model: Embed D 7 brane probes in (deformed versions of) AdS 5 × S 5 U (1) axial (chiral) symmetry In UV, theory returns to d = 4 N = 2 theory ( N = 4 + fundamental hypermultiplet)

Models of generalized AdS/CFT with flavour D3/D7 model: Embed D 7 brane probes in (deformed versions of) AdS 5 × S 5 U (1) axial (chiral) symmetry In UV, theory returns to d = 4 N = 2 theory ( N = 4 + fundamental hypermultiplet) Sakai-Sugimoto model D 8 and D 8 in D 4 background with compactified space direction SU ( N f ) × SU ( N f ) chiral symmetry UV theory five-dimensional 5

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 S 5 fluctuation S 3 horizon 6

Quarks (fundamental fields) from brane probes D3 N ������������ ������������ conventional 0123 ������������ ������������ ��������� ��������� open/closed string duality ������������ ������������ ��������� ��������� SYM 4567 ������������ ������������ ��������� ��������� AdS 5 89 ������������ ������������ ��������� ��������� 3−3 ������������ ������������ ��������� ��������� ������������ ������������ ��������� ��������� quarks ������������ ������������ ��������� ��������� 3−7 7−7 ������������ ������������ ��������� ��������� flavour open/open ������������ ������������ ��������� ��������� string duality N probe D7 ������������ ������������ AdS ��������� ��������� f R 4 5 ������������ ������������ ��������� ��������� brane N → ∞ (standard Maldacena limit), N f small (probe approximation) duality acts twice: N = 4 SU(N) Super Yang-Mills theory IIB supergravity on AdS 5 × S 5 ← → coupled to + Probe brane DBI on AdS 5 × S 3 N = 2 fundamental hypermultiplet Karch, Katz 2002 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 S 5 8

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, hep-th/0306018 9

D7 brane probe in deformed backgrounds D7 brane probe in gravity backgrounds dual to confining gauge theories without supersymmetry. Example: (particular deformation of AdS 5 × S 5 metric) Constable-Myers background non-constant dilaton non-constant S 5 radius naked singularity in IR dual field theory confining The deformation introduces a new scale into the metric. In UV limit, geometry returns to AdS 5 × S 5 with D7 probe wrapping AdS 5 × S 3 . 10

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 ( w 5 , w 6 ) 3. solve equations of motion numerically using shooting techniques solution determines embedding of D7-brane (e.g. w 5 = 0 , w 6 = w 6 ( ρ ) ) 4. meson spectrum: consider fluctuations δw 5 , δw 6 around a background solution obtained in 3. solve equations of motion linearized in δw 5 , δw 6 11

Asymptotic behaviour of supergravity solutions UV asymptotic behaviour of solutions to equation of motion: w 6 ∝ m e − r + c e − 3 r Identification of the coefficients as in the standard AdS/CFT correspondence: m quark mass, c = � ¯ qq � quark condensate Here: m � = 0 : explicit breaking of U (1) A symmetry c � = 0 : spontaneous breaking of U (1) A symmetry 12

The Constable-Myers deformation N = 4 super Yang-Mills theory deformed by VEV for tr F µν F µν → non-supersymmetric QCD-like field theory (R-singlet operator with D = 4 ) The Constable-Myers background is given by the metric � (2 − δ ) / 4 w 4 − b 4 � w 4 + b 4 � δ/ 4 � w 4 + b 4 6 ds 2 = H − 1 / 2 � dx 2 4 + H 1 / 2 dw 2 i , w 4 − b 4 w 4 − b 4 w 4 i =1 where � w 4 + b 4 � δ (∆ 2 + δ 2 = 10) H = − 1 w 4 − b 4 and the dilaton and four-form � w 4 + b 4 � ∆ C (4) = − 1 e 2 φ = e 2 φ 0 4 H − 1 dt ∧ dx ∧ dy ∧ dz , w 4 − b 4 This background has a singularity at w = b 13

Chiral symmetry breaking Solution of equation of motion for Numerical Result: w probe brane 6 2 w 6 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 g u l m=0.6, c=1.45 0.75 a m=0.4, c=1.60 r 0.5 i m=0.2, c=1.73 bad t 0.25 y m=10^−6, c=1.85 good ugly ρ ρ 0.5 1 1.5 2 2.5 3 singularity c Result: Screening effect: Regular solutions 1.5 do not reach the singularity 1.0 Spontaneous breaking of U (1) A 0.5 symmetry: For m → 0 we have c ≡ � ¯ ψψ � � = 0 0 m 4.0 3.5 0.5 1.0 1.5 2.0 2.5 3.0 14

Meson spectrum From fluctuations of the probe brane M 2 = − k 2 δw i ( x, ρ ) = f i ( ρ ) sin ( k · x ) , Ansatz: meson mass 6 5 4 3 2 1 quark mass 0.5 1 1.5 2 Goldstone boson ( η ′ ) Gell-Mann-Oakes-Renner relation: M Meson ∝ √ m Quark 15

Comparison with lattice gauge theory J.E., Evans, Kirsch, Threlfall 0711.4467, EPJA m ρ vs. m π 2 (Lattice: Lucini, Del Debbio, Patella, Pica 0712.3036) 0.9 m Ρ 1 0.8 0.7 0.8 m ρ 0.6 0.6 0.5 SU(2) 0.4 SU(3) SU(4) 0.4 SU(6) 0.2 N = inf. 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m Π 2 m π 2 0.1 0.2 0.3 0.4 0.5 0.6 Slope: 0.52 Slope: 0.57 Normalized to lattice spacing Normalized to scale in metric (Similar results by Bali and Bursa) 16

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 (4 w 4 − b 4 ) 2 6 w 2 + b 4 � � 4 w 2 (4 w 4 + b 4 ) dτ 2 + 1 ds 2 = x 2 + � dw 2 d� i 4 w 2 w 2 i =1 with radial coordinate w 2 = ρ 2 + w 2 5 + w 2 6 b deformation parameter, τ periodic (period πb = T − 1 ) horizon: S 1 collapses at w = 1 2 b 17

Condensate in field theory at finite temperature Condensate c versus quark mass m D7 brane embedding in black hole ( c , m normalized to T ) background w 6 -c 2 1.75 m=1.5 1.5 0.1 Karch−Katz m=1.25 1.25 0.08 phase m=1.0 m=0.92 1 transition m=0.91 0.06 m=0.8 0.75 h o m=0.6 r 0.04 i z 0.5 condensate o m=0.4 n 0.25 0.02 m=0.2 ρ 0.5 1 1.5 2 0 m 0.2 0.4 0.6 0.8 1.0 1.2 Phase transition at m c ≈ 0 . 92 No condensate for m = 0 (no spontaneous chiral symmetry breaking) BEEGK 0306018 18

Phase transition 0.035 0.0325 0.03 A 0.0275 0.025 B C 0.0225 D E 0.02 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 (Related work by Mateos, Myers et al) 19

Quarkonium transport in AdS/CFT Dusling, J.E., Kaminski, Rust, Teaney, Young in progress Diffusion and momentum broadening of heavy mesons