Aspects of strong coupled non-conformal gauge theories at finite - PowerPoint PPT Presentation

Aspects of strong coupled non-conformal gauge theories at finite temperature Alex Buchel (Perimeter Institute & University of Western Ontario) Based on: hep-th/0701142,arXiv:0708.3459, to appear with: Chris Pagnutti, Andre Blanchard, Patrick Kernel, · · · 1

Outline of the talk: • Motivation • CFT plasmas and why they can not answer the two questions raised in the motivation • N = 2 ∗ gauge theory as a toy model: = ⇒ Susy/non-susy mass deformations of N = 4 in QFT/supergravity ⇒ Thermodynamics of N = 2 ∗ for (non-)susy mass-deformations = ⇒ Bulk viscosity of N = 2 ∗ for (non)-susy mass-deformations = ⇒ “ N = 2 ∗ -based” thermometer for RHIC = • Conclusions and future directions 2

RHIC experiment at BNL collides bunches of Au ions at 99 . 995% of the speed of light. The ions ’melt’ and produce a new state of matter: the Quark-Gluon-Plasma (QGP). The typically temperature of the QGP is roughly estimated to be 1 . 5 T deconfinement ; thus the plasma is expected (and is observed) to be strongly coupled. We would like to study the properties of this QGP ’liquid’. Q.1: How we can measure the temperature of the QGP ball? Q.2: Hydrodynamic simulations of the QGP quite well agree with the experimental data; Kharzeev et.al proposed that fast equilibration is due to the large bulk viscosity of QGP near T deconfinement . Harvey Meyer’s lattice simulations suggests that ζ η ∼ 8 · · · 10 at T = 1 . 02 T deconfinement . Can we understand/see the grows of bulk viscosity from gauge/string duality? 3

In this talk we would like to answer Q.1 and Q.2 . A.1:’Rajagopal’s thermometer’. Assume that there is an ideal situation and we are able to extract as precise data as possible from the experiment to ’tune’ our hydrodynamic codes. From the hydrodynamic codes we expect to obtain the jet quenching parameter ˆ q as a function of the speed of sound c s , ˆ q = ˆ q ( c s ) . Now, using the static lattice simulations we can relate s = ∂P c 2 c 2 s = c 2 ⇒ = s ( T/T d ) ∂ǫ and thus obtain ˆ q = ˆ q ( T/T d ) . We would like to use toy models of gauge/string duality to obtain � � q s = ˆ ˆ q c 2 s s Is our thermometer universal? A.2: Use toy models of gauge/string duality to compute ζ η 4

Why a CFT plasma fails to answer Q.1 and Q.2 ? Q.1 In CFT at thermal equilibrium s = ∂P ∂ǫ = 1 T µ c 2 ⇒ ⇒ µ = 0 = ǫ = 3 P = 3 = constant! Q.2 For any fluid, to first order in velocity gradients, T µν = ǫ u µ u ν + P ∆ µν − ησ µν − ζ ∆ µν ( ∇ · u ) where { η, ζ } are the shear and the bulk viscosity, { ∆ µν , σ µν } are symmetric transverse tensor constructed from u µ (in case of ∆ ) and ∇ µ u ν (in case of σ ); also ∆ µ σ µ µ = 3 , µ = 0 Here again, the tracelessness of T µν (as required for the unbroken scale invariance), implies ζ CF T = 0 5

Introduce s − 1 δ ≡ c 2 3 a deviation from the conformality. δ � = 0 due to ւ ց � m � Λ � � T T explicit breaking spontaneous breaking by mass terms by a strong coupling scale ⇑ ⇑ mass deformed N = 4 cascading gauge theory In this talk we discuss in details explicit breaking of the scale invariance by mass terms. 6

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 7

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. 8

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 P = 1 � ∂W − 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. 9

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 10

Introduce x ≡ e − r/ 2 , ˆ then � x 2 � 1 9 ln 2 ( k ˆ 1 + k 2 ˆ 3 + 4 + k 4 ˆ x 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 + k 4 ˆ k 6 ˆ 3 + 2 x ) + 2 � � � � 3 ln( k ˆ x ) 18 + 2 ln( k ˆ x ) + O 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 11

⇒ 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. 12

= ⇒ 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 5 +Ω 2 4 � 1 � + 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. 13

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 � c 2 2 ) 2 ( x − 1) α ′′ + x − 1 α ′ − ∂α 2 − 3 c ′ 2 c 2 − 6( c ′ � ( x − 1) = 0 12 P c 2 2 ( x − 1) ∂ P � � 1 6( α ′ ) 2 + 2( χ ′ ) 2 � ∂χ χ ′′ + x − 1 χ ′ − c 2 2 − 3 c ′ 2 c 2 − 6( c ′ 2 ) 2 ( x − 1) � ( x − 1) = 0 4 P c 2 2 ( x − 1) 14

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 , x 1 / 4 b → 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 Schwartzchild horizon) = � � � � → c 2 ( x ) , α ( x ) , χ ( x ) constant , constant , constant 15

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 16