Themodynamics at strong coupling from Holographic QCD

Themodynamics at strong coupling from Holographic QCD p. 1 Themodynamics at strong coupling from Holographic QCD Francesco Nitti APC, U. Paris VII Excited QCD Les Houches, February 23 2011 Work with E. Kiritsis, U. Gursoy, L. Mazzanti, G. Michalogiorgakis, 07-10

Themodynamics at strong coupling from Holographic QCD p. 2 Motivation: Thermodynamics of 4D YM 4D Pure YM theory exhibits a first order deconfining phase transition around T c 260MeV. Karsch, hep-lat/0106019

Themodynamics at strong coupling from Holographic QCD p. 2 Motivation: Thermodynamics of 4D YM 4D Pure YM theory exhibits a first order deconfining phase transition around T c 260MeV. Karsch, hep-lat/0106019 Can AdS/CFT techniques reproduce the phase transition and the YM equation of state found on the lattice?

Motivation: Thermodynamics of 4D YM 4D Pure YM theory exhibits a first order deconfining phase transition around T c 260MeV. Karsch, hep-lat/0106019 Can AdS/CFT techniques reproduce the phase transition and the YM equation of state found on the lattice? If yes, can it compute hydrodynamic and transport coefficient of the QCD plasma? Themodynamics at strong coupling from Holographic QCD p. 2

Themodynamics at strong coupling from Holographic QCD p. 3 Outline AdS/CFT 101 Holographic description of deconfinement transition AdS examples 5D Phenomenological models

Themodynamics at strong coupling from Holographic QCD p. 3 Outline AdS/CFT 101 Holographic description of deconfinement transition AdS examples 5D Phenomenological models I will take a phenomenological point of view: Type III according to the Nahrgang classification of the approaches to QCD

Themodynamics at strong coupling from Holographic QCD p. 3 Outline AdS/CFT 101 Holographic description of deconfinement transition AdS examples 5D Phenomenological models I will take a phenomenological point of view: Type III according to the Nahrgang classification of the approaches to QCD ( = being creative)

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime.

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5 ds 2 = ( ) l 2 ( dr 2 + η µνdx µ dx ν) + l 2 dω 2 5 r

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5 ds 2 = ( ) l 2 ( dr 2 + η µνdx µ dx ν) + l 2 dω 2 5 r AdS 5 S 5 has a boundary at r = 0 which maps to the R (3,1) where the gauge theory is defined.

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5 ds 2 = ( ) l 2 ( dr 2 + η µνdx µ dx ν) + l 2 dω 2 5 r AdS 5 S 5 has a boundary at r = 0 which maps to the R (3,1) where the gauge theory is defined. r RG direction/energy scale (UV r 0).

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5 ds 2 = ( ) l 2 ( dr 2 + η µνdx µ dx ν) + l 2 dω 2 5 r AdS 5 S 5 has a boundary at r = 0 which maps to the R (3,1) where the gauge theory is defined. r RG direction/energy scale (UV r 0). Isometry (r, x µ ) (αr,αx µ ) QFT is conformal

Themodynamics at strong coupling from Holographic QCD p. 4 Gauge/Gravity duality Equivalence between a gauge theory in D space-time dimensions and a gravitational theory in higher-dimensional curved spacetime. Archetipal example Maldacena 98: N = 4 Super YM on R (3,1) Type IIB string theory on AdS 5 S 5 ds 2 = ( ) l 2 ( dr 2 + η µνdx µ dx ν) + l 2 dω 2 5 r AdS 5 S 5 has a boundary at r = 0 which maps to the R (3,1) where the gauge theory is defined. r RG direction/energy scale (UV r 0). Isometry (r, x µ ) (αr,αx µ ) QFT is conformal I will restrict the discussion to the 5D (x µ, r) directions, and consider general non-conformal backgrounds.

Themodynamics at strong coupling from Holographic QCD p. 5 Field/Operator correspondence Gravity dual described by an action S grav = Mp 2 d 5 x gr ( Φ) 2 +...

Themodynamics at strong coupling from Holographic QCD p. 5 Field/Operator correspondence Gravity dual described by an action S grav = Mp 2 d 5 x gr ( Φ) 2 +... A bulk field Φ(x,r) acts as a source for a QFT operator O(x): d 4 x Φ(x)O(x), Φ(x) Φ(x, r = 0).

Themodynamics at strong coupling from Holographic QCD p. 5 Field/Operator correspondence Gravity dual described by an action S grav = Mp 2 d 5 x gr ( Φ) 2 +... A bulk field Φ(x,r) acts as a source for a QFT operator O(x): d 4 x Φ(x)O(x), Φ(x) Φ(x, r = 0). Generating Functional: Z QFT [ Φ] exp is grav [Φ cl (x, r)] Φcl Φ

Themodynamics at strong coupling from Holographic QCD p. 5 Field/Operator correspondence Gravity dual described by an action S grav = Mp 2 d 5 x gr ( Φ) 2 +... A bulk field Φ(x,r) acts as a source for a QFT operator O(x): d 4 x Φ(x)O(x), Φ(x) Φ(x, r = 0). Generating Functional: Z QFT [ Φ] exp is grav [Φ cl (x, r)] Φcl Φ Compute correlators of gauge-invariant operators from the gravity on-shell action by differentiating Z QFT [ Φ] w.r.t. Φ.

Themodynamics at strong coupling from Holographic QCD p. 5 Field/Operator correspondence Gravity dual described by an action S grav = Mp 2 d 5 x gr ( Φ) 2 +... A bulk field Φ(x,r) acts as a source for a QFT operator O(x): d 4 x Φ(x)O(x), Φ(x) Φ(x, r = 0). Generating Functional: Z QFT [ Φ] exp is grav [Φ cl (x, r)] Φcl Φ Compute correlators of gauge-invariant operators from the gravity on-shell action by differentiating Z QFT [ Φ] w.r.t. Φ. Gravity description valid at large N, large g 2 N

Themodynamics at strong coupling from Holographic QCD p. 6 Finite Temperature To describe thermodynamics @ T = 1/β, go to euclidean time τ, and identify τ τ + β.

Themodynamics at strong coupling from Holographic QCD p. 6 Finite Temperature To describe thermodynamics @ T = 1/β, go to euclidean time τ, and identify τ τ + β. The partition function is given by the gravity action evaluated at its extremum (solution of the of the gravity-side field equation) Z(β) = e S grav[g 0,Φ 0 ] where g 0 and Φ 0 are periodic with period β.

Themodynamics at strong coupling from Holographic QCD p. 7 Thermal phase transitions Different thermal equilibrium states Different classical solutions on the gravity side

Themodynamics at strong coupling from Holographic QCD p. 7 Thermal phase transitions Different thermal equilibrium states Different classical solutions on the gravity side Thermal partition function sum over stationary points: Z(T ) e βf 1 + e βf 2 βf i (T) = S grav soli

Themodynamics at strong coupling from Holographic QCD p. 7 Thermal phase transitions Different thermal equilibrium states Different classical solutions on the gravity side Thermal partition function sum over stationary points: Z(T ) e βf 1 + e βf 2 βf i (T) = S grav soli Phase transition happens at T c where F 1 (T c ) = F 2 (T c )

Themodynamics at strong coupling from Holographic QCD p. 7 Thermal phase transitions Different thermal equilibrium states Different classical solutions on the gravity side Thermal partition function sum over stationary points: Z(T ) e βf 1 + e βf 2 βf i (T) = S grav soli Phase transition happens at T c where F 1 (T c ) = F 2 (T c ) From F i (T) we can compute other equilibrium quantities (entropy, pressure, speed of sound) using standard thermodynamic formulae.

Themodynamics at strong coupling from Holographic QCD p. 8 Plasma/Black Hole correspondence Holographic description of a deconfined plasma in thermal equilibrium at temperature T 1/β: A Black Hole in the bulk with temperature T H = T.

Themodynamics at strong coupling from Holographic QCD p. 8 Plasma/Black Hole correspondence Holographic description of a deconfined plasma in thermal equilibrium at temperature T 1/β: A Black Hole in the bulk with temperature T H = T. Due to AdS boundary conditions, BH is stable. Polyakov loop gets a vev, signaling deconfinement. Spectrum of quasinormal modes.

Themodynamics at strong coupling from Holographic QCD p. 9 N = 4 SYM on R 3 S 1 Poincaré AdS 5 S = M 2 p d 5 x ( g R 12 ) l 2

Themodynamics at strong coupling from Holographic QCD p. 9 N = 4 SYM on R 3 S 1 Poincaré AdS 5 S = M 2 p d 5 x ( g R 12 ) l 2 Thermal gas in AdS AdS Black Hole ds 2 TG = l r 2 [ dr 2 + dτ 2 + d x 2] ds2 BH = l r 2 [ ] dr 2 f(r) + f(r)dτ2 + d x 2 f(r) = 1 (πtr) 4, r h = 1 πt

N = 4 SYM on R 3 S 1 Poincaré AdS 5 S = M 2 p d 5 x ( g R 12 ) l 2 Thermal gas in AdS AdS Black Hole ds 2 TG = l r 2 [ dr 2 + dτ 2 + d x 2] ds2 BH = l r 2 [ ] dr 2 f(r) + f(r)dτ2 + d x 2 f(r) = 1 (πtr) 4, r h = 1 πt F AdS > F BH for all T > 0: BH always dominates; Thermodynamics of a conformal gas, F = c T 4 Themodynamics at strong coupling from Holographic QCD p. 9

Themodynamics at strong coupling from Holographic QCD p. 10 N = 4 SYM on S 3 S 1 Global AdS 5 To get a phase transition at nonzero T we need to introduce a scale,, e.g. put N = 4 on an S 3.

Themodynamics at strong coupling from Holographic QCD p. 10 N = 4 SYM on S 3 S 1 Global AdS 5 To get a phase transition at nonzero T we need to introduce a scale,, e.g. put N = 4 on an S 3. T 2 BHs for T > T min, none for T < T min T min 2 l r h

Themodynamics at strong coupling from Holographic QCD p. 10 N = 4 SYM on S 3 S 1 Global AdS 5 To get a phase transition at nonzero T we need to introduce a scale,, e.g. put N = 4 on an S 3. T 2 BHs for T > T min, none for T < T min T min F 2 l r h Large BH dominant for T c > T min 2 l r h

Themodynamics at strong coupling from Holographic QCD p. 11 N = 4 SYM on S 3 S 1 Global AdS 5 To get a phase transition at nonzero T we need to introduce a scale,, e.g. put N = 4 on an S 3. F T c 3 2Πl T Hawking-Page transition.

Themodynamics at strong coupling from Holographic QCD p. 12 HP Transitions on R 3 Can we get a transition in R 3 rather than S 3? We need to introduce a scale in the theory

Themodynamics at strong coupling from Holographic QCD p. 12 HP Transitions on R 3 Can we get a transition in R 3 rather than S 3? We need to introduce a scale in the theory Black D4 background (Witten, 98) not conformal in the UV

Themodynamics at strong coupling from Holographic QCD p. 12 HP Transitions on R 3 Can we get a transition in R 3 rather than S 3? We need to introduce a scale in the theory Black D4 background (Witten, 98) not conformal in the UV Hard Wall AdS/QCD (Herzog, 06) F = F 0 ct 4, s T 3

Themodynamics at strong coupling from Holographic QCD p. 12 HP Transitions on R 3 Can we get a transition in R 3 rather than S 3? We need to introduce a scale in the theory Black D4 background (Witten, 98) not conformal in the UV Hard Wall AdS/QCD (Herzog, 06) F = F 0 ct 4, s T 3 still too conformal, except for the transition; issue of boundary conditions at the IR cutoff

Themodynamics at strong coupling from Holographic QCD p. 13 5D Einstein-Dilaton Theories A simple class of models that displays realistic thermodynamics S E = M 3 pn 2 c d 5 x g [ R 4 ] 3 ( Φ)2 V (Φ)

Themodynamics at strong coupling from Holographic QCD p. 13 5D Einstein-Dilaton Theories A simple class of models that displays realistic thermodynamics S E = M 3 pn 2 c d 5 x g Thermal gas (Confined phase) : [ R 4 ] 3 ( Φ)2 V (Φ) ds 2 = b 2 0(r) ( dr 2 + dτ 2 + dx 2 i ), Φ = Φ0 (r), 0 < r < +.

Themodynamics at strong coupling from Holographic QCD p. 13 5D Einstein-Dilaton Theories A simple class of models that displays realistic thermodynamics S E = M 3 pn 2 c d 5 x g Thermal gas (Confined phase) : [ R 4 ] 3 ( Φ)2 V (Φ) ds 2 = b 2 0(r) ( dr 2 + dτ 2 + dx 2 i ), Φ = Φ0 (r), 0 < r < +. Black holes (Deconfined phase - Gluon Plasma): ( ) dr ds 2 = b 2 2 (r) f(r) + f(r)dτ2 + dx 2 i, Φ = Φ(r), 0 < r < r h b(r) r 0 b 0 (r) l/r, f(r) r 0 1, f(r h ) = 0,

Themodynamics at strong coupling from Holographic QCD p. 14 Dilatonic Black holes Gürsoy, Kiritsis, Mazzanti, F.N. 08 Consider a potential that in the IR (large Φ) asymptotes: V (Φ) e 4/3Φ Φ (α 1)/α For α > 1 vacuum solution has a mass-gap and discrete spectrum (confinement). BHs have qualitatively different behavior in confining (α > 1) vs. non-confining (α < 1) cases:

Themodynamics at strong coupling from Holographic QCD p. 14 Dilatonic Black holes Gürsoy, Kiritsis, Mazzanti, F.N. 08 Consider a potential that in the IR (large Φ) asymptotes: V (Φ) e 4/3Φ Φ (α 1)/α For α > 1 vacuum solution has a mass-gap and discrete spectrum (confinement). BHs have qualitatively different behavior in confining (α > 1) vs. non-confining (α < 1) cases: confining case: a minimal BH temperature and 2 BH branches; non-confining case: only 1 BH for all temperatures

Situation as in the global AdS case (α > 1) or Poincaré AdS (α < 1) Themodynamics at strong coupling from Holographic QCD p. 14 Dilatonic Black holes Gürsoy, Kiritsis, Mazzanti, F.N. 08 Consider a potential that in the IR (large Φ) asymptotes: V (Φ) e 4/3Φ Φ (α 1)/α For α > 1 vacuum solution has a mass-gap and discrete spectrum (confinement). BHs have qualitatively different behavior in confining (α > 1) vs. non-confining (α < 1) cases: confining case: a minimal BH temperature and 2 BH branches; non-confining case: only 1 BH for all temperatures

Themodynamics at strong coupling from Holographic QCD p. 15 Dilatonic Black holes Gürsoy, Kiritsis, Mazzanti, F.N. 08 Consider a potential that in the IR (large Φ) asymptotes: V (Φ) e 4/3Φ Φ (α 1)/α For α > 1 vacuum solution has a mass-gap and discrete spectrum (confinement). BHs have qualitatively different behavior in confining (α > 1) vs. non-confining (α < 1) cases: confining case: Hawking-Page transition

Themodynamics at strong coupling from Holographic QCD p. 16 Explicit Model Gürsoy, Kiritsis, Mazzanti, F.N. 09 Simple parametrization for V (φ) good numerical agreement with lattice Yang-Mills thermodynamics

Themodynamics at strong coupling from Holographic QCD p. 17 Explicit Model Gürsoy, Kiritsis, Mazzanti, F.N. 09 Simple parametrization for V (φ) good numerical agreement with lattice Yang-Mills thermodynamics

Being Phenomenological Themodynamics at strong coupling from Holographic QCD p. 18

Themodynamics at strong coupling from Holographic QCD p. 18 Being Phenomenological The setups I described (including AdS/QCD) are examples of phenomenological holographic models: they are designed to capture some features of 4D QCD, and at the same time stay simple and calculable.

Themodynamics at strong coupling from Holographic QCD p. 18 Being Phenomenological The setups I described (including AdS/QCD) are examples of phenomenological holographic models: they are designed to capture some features of 4D QCD, and at the same time stay simple and calculable. The true holographic dual of QCD is likely a full non-critical string theory (not just 2-derivative gravity). True reason for this is the existence of a single scale in QCD stringy states have masses comparable to gravity states. *if it exists

Themodynamics at strong coupling from Holographic QCD p. 18 Being Phenomenological The setups I described (including AdS/QCD) are examples of phenomenological holographic models: they are designed to capture some features of 4D QCD, and at the same time stay simple and calculable. The true holographic dual of QCD is likely a full non-critical string theory (not just 2-derivative gravity). True reason for this is the existence of a single scale in QCD stringy states have masses comparable to gravity states. The potential V (Φ) is supposed to be fixed phenomenologically as a way of parametrizing some of the unknown features of the full holographic dual. *if it exists

Conclusion Themodynamics at strong coupling from Holographic QCD p. 19

Themodynamics at strong coupling from Holographic QCD p. 19 Conclusion Holography seems to naturally capture the thermodynamics of the deconfined phase of 4D gauge theories

Themodynamics at strong coupling from Holographic QCD p. 19 Conclusion Holography seems to naturally capture the thermodynamics of the deconfined phase of 4D gauge theories Phenomenological 5D models can be used obtain quantitatively accurate results

Themodynamics at strong coupling from Holographic QCD p. 19 Conclusion Holography seems to naturally capture the thermodynamics of the deconfined phase of 4D gauge theories Phenomenological 5D models can be used obtain quantitatively accurate results Natural to extend the description beyond equilibrium (e.g. diffusion)