AdS/CFT and the Quark-Gluon Plasma (Beyond CFT & SUGRA) Alex Buchel (Perimeter Institute & University of Western Ontario) 1
Basic AdS/CFT correspondence: gauge theory string theory N = 4 SU ( N ) SYM ⇐ ⇒ N-units of 5-form flux in type IIB string theory g 2 ⇐ ⇒ g s Y M ⇒ In the simplest case, the SYM theory is in the ’t Hooft (planar limit), N → ∞ , g 2 Y M → 0 with Ng 2 Y M kept fixed. SUGRA is valid Ng s → ∞ . In which case the background geometry is AdS 5 × S 5 ⇒ The main message is that AdS/CFT sets up a framework that could be used in analyzing the dynamics of strongly coupled gauge theories, in particular, sQGP 2
To make a closer link to realistic systems we need to go beyond the basic AdS/CFT: • Beyond the conformal approximation (but still in a ’SUGRA’-land — non-conformal theories in the planar limit and infinite ’t Hooft coupling) ⇒ deform a conformal gauge theory by a relevant operator (introduce a mass term) ⇒ consider gauge theories without explicit mass terms, but with a non-vanishing β -function(s) for the gauge coupling(s) • Beyond the SUGRA approximation (but still in a ’CFT’-land — conformal theories with leading deviations from planar limit/infinite t’ Hooft coupling) 1 ⇐ ⇒ g s -corrections N -corrections 1 α ′ -corrections ⇐ ⇒ Y M -corrections Ng 2 ⇒ Ideally, we would like to combine the both beyonds... , but the technology is still not there yet. 3
Outline of the talk: • Thermodynamics of strongly coupled non-conformal plasma: Susy/non-susy mass deformations of N = 4 in QFT/supergravity ( N = 2 ∗ model) Gauge theories with β g Y M � = 0 in QFT/supergravity (Klebanov-Strassler model) • Bulk viscosity of gauge theory plasma at strong coupling: Sound modes in plasma and the corresponding quasinormal modes of the holographic dual — N = 4 and N = 2 ∗ gauge theory Bulk viscosity bound • Hydrodynamic relaxation time of holographic models: Why should be care about the relaxation time — fundamental & practical perspective Relaxation time bound 4
• Hydrodynamics of conformal plasma beyond SUGRA approximation: finite ’t Hooft coupling corrections — 1 / ( Ng 2 Y M ) finite 1 /N corrections is there a bound on η/s ? • sQGP as hCFT 5
N = 2 ∗ gauge theory (a QFT story) = ⇒ Start with N = 4 SU ( N ) SYM. In N = 1 4d susy language, it is a gauge theory of a vector multiplet V , an adjoint chiral superfield Φ (related by N = 2 susy to V ) and an adjoint pair { Q, ˜ Q } of chiral multiplets, forming an N = 2 hypermultiplet. The theory has a superpotential: √ W = 2 2 �� � � Q, ˜ Tr Q Φ g 2 Y M We can break susy down to N = 2 , by giving a mass for N = 2 hypermultiplet: √ W = 2 2 m �� � � � Tr Q 2 + Tr ˜ Q 2 � Q, ˜ Tr Q Φ + g 2 g 2 Y M Y M This theory is known as N = 2 ∗ gauge theory 6
When m � = 0 , the mass deformation lifts the { Q, ˜ Q } hypermultiplet moduli directions; we are left with the ( N − 1) complex dimensional Coulomb branch, parametrized by � Φ = diag (a 1 , a 2 , · · · , a N ) , a i = 0 i We will study N = 2 ∗ gauge theory at a particular point on the Coulomb branch moduli space: 0 = m 2 g 2 Y M N a 2 a i ∈ [ − a 0 , a 0 ] , π with the (continuous in the large N -limit) linear number density � a 0 2 � 0 − a 2 , a 2 ρ ( a ) = da ρ ( a ) = N m 2 g 2 − a 0 Y M Reason: we understand the dual supergravity solution only at this point on the moduli space. 7
N = 2 ∗ gauge theory (a supergravity story — a.k.a Pilch-Warner flow) Consider 5d gauged supergravity, dual to N = 2 ∗ gauge theory. The effective five-dimensional action is dξ 5 √− g 1 � � 1 4 R − ( ∂α ) 2 − ( ∂χ ) 2 − P � S = , 4 πG 5 M 5 where the potential P is �� ∂W � 2 � 2 � � ∂W P = 1 − 1 3 W 2 , + 16 ∂α ∂χ with the superpotential √ W = − 1 ρ 2 − 1 2 ρ 4 cosh(2 χ ) , α ≡ 3 ln ρ = ⇒ The 2 supergravity scalars { α, χ } are holographic dual to dim-2 and dim-3 operators which are nothing but (correspondingly) the bosonic and the fermionic mass terms of the N = 4 → N = 2 SYM mass deformation. 8
PW geometry ansatz: − dt 2 + d� ds 2 5 = e 2 A � x 2 � + dr 2 solving the Killing spinor equations, we find a susy flow: dA dr = − 1 dα dr = 1 ∂W dχ dr = 1 ∂W 3 W , ∂α , 4 4 ∂χ Solutions to above are characterized by a single parameter k : kρ 2 ρ 6 = cosh(2 χ ) + sinh 2 (2 χ ) ln sinh( χ ) e A = sinh(2 χ ) , cosh( χ ) In was found (Polchinski,Peet,AB) that k = 2 m 9
Introduce x ≡ e − r/ 2 , ˆ then � x 2 � 1 1 + k 2 ˆ + k 4 ˆ x 4 � 9 ln 2 ( k ˆ 3 + 4 − 7 90 + 10 x ) + 20 � � χ = k ˆ x 3 ln( k ˆ x ) 3 ln( k ˆ x ) �� x 6 ln 3 ( k ˆ k 6 ˆ � + O x ) , x 2 � 1 x 4 � 1 x 6 ln 3 ( k ˆ 3 ln 2 ( k ˆ ρ = 1+ k 2 ˆ 3 + 2 + k 4 ˆ x ) + 2 k 6 ˆ � � � � + O 3 ln( k ˆ x ) 18 + 2 ln( k ˆ x ) x ) , x 4 � 2 x 2 − k 4 ˆ x 6 ln 3 ( k ˆ 9 ln 2 ( k ˆ x ) − 1 3 k 2 ˆ 9 + 10 x ) + 4 k 6 ˆ � � � A = − ln(2ˆ + O 9 ln( k ˆ x ) x ) Or in standard Poincare-patch AdS 5 radial coordinate: α ∝ k 2 ln r χ ∝ k A ∝ ln r, r → ∞ , r , r 2 10
= ⇒ Notice that the nonnormalizable components of { α, χ } are related — this is holographic dual to N = 2 susy preserving condition on the gauge theory side: m b = m f But in general, we can keep m b � = m f : α ∝ m 2 b ln r χ ∝ m f A ∝ ln r, r → ∞ , r , r 2 The precise relation, including numerical coefficients can be works out. = ⇒ There are no singularity-free flows (geometries) with m b � = m f and at zero temperature T = 0 . However, one can study m b � = m f mass deformations of N = 4 SYM at finite temperature. 11
= ⇒ To study holographic duality in full details, we need the full ten-dimensional background of type IIB supergravity, i.e, we need the lift of 5-dimensional gauged SUGRA solutions. This will be obvious when we discuss jet quenching in N = 2 ∗ . Such a lift was constructed in J.Liu,AB. Specifically, for any 5d solution, the 5d background: 5 = g µν dx µ dx ν , ds 2 { α, χ } plus is uplifted to a solution of 10d type IIB supergravity: � σ 2 + σ 2 2 + σ 2 � 1 � � 5 +Ω 2 4 + sin 2 ( θ ) 1 c dθ 2 + ρ 6 cos 2 ( θ ) ds 2 10( E ) = Ω 2 ds 2 1 3 dφ 2 ρ 2 cX 2 X 1 X 2 Ω 2 = ( cX 1 X 2 ) 1 / 4 X 1 = cos 2 θ + c ( r ) ρ 6 sin 2 θ , X 2 = c cos 2 θ + ρ 6 sin 2 θ , ρ with c ≡ cosh 2 χ , plus dilaton-axion, various 3-form fluxes, various 5-form fluxes. 12
Thermodynamics of N = 2 ∗ for (non-)susy mass-deformations (with J.Liu,P .Kerner,...) Consider metric ansatz: 1 ( r ) dt 2 + c 2 ds 2 5 = − c 2 dx 2 1 + dx 2 2 + dx 2 + dr 2 � � 2 ( r ) 3 Introducing a new radial coordinate x ≡ 1 − c 1 , c 2 with x → 0 + being the boundary and x → 1 − being the horizon, we find: 1 2 − 5 2 ) 2 + 4 2 + 4 c 2 ( α ′ ) 2 − 3 c 2 ( χ ′ ) 2 = 0 c ′′ x − 1 c ′ ( c ′ c 2 ∂ P 1 � � 6( α ′ ) 2 + 2( χ ′ ) 2 � α ′′ + x − 1 α ′ − c 2 2 − 3 c ′ 2 c 2 − 6( c ′ 2 ) 2 ( x − 1) ∂α � ( x − 1) = 0 12 P c 2 2 ( x − 1) ∂ P 1 � � 6( α ′ ) 2 + 2( χ ′ ) 2 � ∂χ χ ′′ + x − 1 χ ′ − 2 − 3 c ′ 2 c 2 − 6( c ′ c 2 2 ) 2 ( x − 1) � ( x − 1) = 0 4 P c 2 2 ( x − 1) 13
We look for a solution to above subject to the following (fixed) boundary conditions: ⇒ near the boundary, x ∝ r − 4 → 0 + = m 2 � � � � m f T 2 x 1 / 2 ln x, x − 1 / 4 , b x 1 / 4 → c 2 ( x ) , α ( x ) , χ ( x ) T of course, we need a precise coefficients here relating the non-normalizable components of the sugra scalars to the gauge theory masses ⇒ near the horizon, x → 1 − (to have a regular, non-singular Schwarzschild horizon) = � � � � c 2 ( x ) , α ( x ) , χ ( x ) → constant , constant , constant 14
T ≪ 1 and m f System of above equations can be solved analytically when m b T ≪ 1 With the help of the holographic renormalization (in this model AB) we can independently compute the free energy density F = − P , the energy density E , and the entropy density s of the resulting black brane solution: −F = P = 1 � 1 − 192 1 − 8 � 8 π 2 N 2 T 4 π 2 ln( πT ) δ 2 π δ 2 2 E = 3 � 1 + 64 1 − 8 � 8 π 2 N 2 T 4 π 2 (ln( πT ) − 1) δ 2 3 π δ 2 2 � � s = 1 1 − 48 1 − 4 2 π 2 N 2 T 3 π 2 δ 2 π δ 2 2 with � 3 �� 2 � Γ δ 1 = − 1 � m b � 2 m f 4 , δ 2 = 2 π 3 / 2 24 π T T 15
A highly nontrivial consistency test on the analysis, as well as on the identification of gauge theory/supergravity parameters are the basic thermodynamics identities: F = E − sT d E = Tds ⇒ For finite (not small) m b /T and m f /T we need to do numerical analysis. However, we = always check the consistency of the thermodynamic relations. In our numerics d E − Tds ∼ 10 − 3 d E 16
The phase diagram of the model depends on m 2 f ∆ ≡ : m 2 b • when ∆ ≥ 1 there is no phase transition in the system; • when ∆ < 1 there is a critical point in the system with the divergent specific heat. The corresponding critical exponent is α = 0 . 5 : c V ∼ | 1 − T c /T | − α where T c = T c (∆) . For concreteness we discuss below 2 cases: (a) ∆ = 1 (’susy’ flows at finite temperature) (b) ∆ = 0 (’bosonic’ flows at finite temperature) 17
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