QUASINORMAL MODES AND MESON DECAY RATES

Transcription

1 QUASINORMAL MODES AND MESON DECAY RATES Carlos Hoyos, Karl Landsteiner, Sergio Montero Physics Department University of Wales, Swansea Instituto de Física Teórica - CSIC Universidad Autónoma de Madrid Meson Melting p. 1

2 Overview AdS/CFT and Quark Gluon Plasma D7 Brane embeddings Holographic Meson Melting Quasinormal Modes Meson Masses Comments and Outlook Meson Melting p. 2

3 AdS/CFT and QGP Large N strong coupling limit of N=4 SYM = IIB on AdS 5 S 5 [Maldacena, hep-th/ ] ds 2 = r 2 (dτ 2 + d x 2 ) + dr2 r 2 + dω2 5 This metric describes the Coulomb phase Meson Melting p. 3

4 AdS/CFT and QGP But also plasma phase is possible T 0: AdS-Black Hole [Witten, hep-th/ ] ds 2 = r 2 (fdτ 2 + d x 2 ) + dr2 fr 2 + dω2 5 f = 1 r4 0 r 0 = πt Euclidean time τ τ + 1/T, Good for equilibrium thermodynamics (thermostatics) free energy phase structure glueball spectrum r 4 Meson Melting p. 3

5 AdS/CFT and QGP But also plasma phase is possible T 0: AdS-Black Hole [Witten, hep-th/ ] ds 2 = r 2 ( fdt 2 + d x 2 ) + dr2 fr 2 + dω2 5 f = 1 r4 0 r 0 = πt Lorentzian black hole with a horizon at r = r 0 Good for real time phenomena out of equilibrium r 4 transport theory (shear viscosity) energy loss of quarks relaxation time scales: quasinormal modes Meson Melting p. 3

6 D7 Brane Embeddings [Karch, Katz, arxiv:hep-th/ ] [Kruczenski, Mateos, Myers, Winters, arxiv:hep-th/ ] [Babington, Erdmenger, Evans, Guralnik, Kirsch, arxiv:hep-th/ ] [Mateos, Myers, Thomson, arxiv:hep-th/ ] [Albash, Filev, Johnson, Kundu, arxiv:hep-th/ ] [Karch, O Bannon, arxiv:hep-th/ ] D7 brane embedded in AdS (Black-hole) open strings = fundamental matter N=4 Theory + N f Hypermultiplets N=2 (at finite temperature) D7 as probe branes N c N f ("quenched approximation") Meson Melting p. 4

7 D7 Brane Embeddings holosym: AdS 5 S 5 D7 brane: R 4 u(partially) S 3 S D7 = T D7 d 8 ξ detp[g] x = 1 r2 0 r 2, dω 2 5 = dθ 2 + cos(θ) 2 dω sin(θ) 2 dφ 2 e.o.m : 3 tan(θ)l(θ) + ( 4[cos(Θ)] 6 ) (2 x)x (1 x) 4 Θ = 0 L(Θ) asymptotic behaviour Θ(x) x 0 = quark mass m and chiral condensate qq Meson Melting p. 5

8 D7 Brane Embeddings Two topologically different kinds of embeddings: D7-brane ends at some finite distance r > r 0 from Horizon S 3 caps off smoothly there! boundary conditions: Θ = π 2 and Θ H = D7-brane falls into the Black Hole in Euclidean regime: S 1 caps off smoothly boundary conditions: Θ H = Θ 0 and Θ H = 0 Important result: first order phase transition, chiral condensate jumps! Meson Melting p. 6

9 D7 Brane Embeddings Meson Melting p. 7

10 Meson Melting D7-End D7-brane Dirichlet or von Neumann infalling Horizon Meson Melting p. 8

11 Quasinormal Modes trivial background: Θ(x) = 0 fluctuations: ϑ(x,t) = e iωt h ω (x) infalling boundary conditions: ϑ Horizon e iω(t+r ) Heun equation (h ω x i ω 4 h ω ): d 2 [ h ω (x) γ dx 2 x + δ x 1 + ɛ ] dhω (x) 2 x dx + α 2 Q x(x 1)(x 2) h ω(x) = 0 boundary conditions h ω Horizon = 1 and h ω Boundary = 0 give quantization conditions for complex ω s ω n = Ω n i Γ n 2 Decay rate of n-th mode = meson melting Meson Melting p. 9

12 Quasinormal Modes ω 2(±1 i) lowest mode: i (lifetime τ = 1 2πT c Γ s) Meson Melting p. 10

13 Quasinormal Modes non-trivial embeddings: m > 0 lattice results: massive mesons are more stable [Karsch, Kharzeev, Satz, hep-lat/ ], [Wong, hep-ph/ ] numerically challenging three lowest quasinormal modes up to Θ H = Meson Melting p. 11

14 Meson Masses embeddings not falling into horizon: m > 0 stable mesons: ω real ω 1 = 0.96, ω 2 = 2.34, ω 3 = 3.44, ω 4 = 4.51 Meson Melting p. 12

15 Comments and Outlook We treated low spin mesons, high spin meson decay: [Peeters, Sonnenschein, Zamaklar, hep-th/ ] [Cotrone, Martucci, Troost, hep-th/ ] [Cotrone, previous talk!] Improve Numerics: "first shoot then relax" [Press, Teukolsky, Vetterling, Flannery, "Numerical Recipes"] Implementation of relaxation algorithm based on finite difference equations Apply to "realistic" models with chiral symmetry holographic QCD (AdS/QCD) [Erlich, Katz, Son, Stephanov, hep-ph/ ] [Da Rold, Pomarol, hep-ph/ ] Sakai-Sugimoto model [Sakai, Sugimoto, hep-th/ ] Meson Melting p. 13