SAŠO GROZDANOV INSTITUUT-LORENTZ FOR THEORETICAL PHYSICS LEIDEN UNIVERSITY TOWARDS WEAK COUPLING IN HOLOGRAPHY WORK IN COLLABORATION WITH J. CASALDERREY-SOLANA, N. KAPLIS, N. POOVUTTIKUL, A. STARINETS AND W. VAN DER SCHEE OXFORD, 6.3.017
OUTLINE from holography to eperiment coupling constant dependence and universality in hydrodynamics thermalisation and higher-energy spectrum heavy ion collisions conclusion and future directions
3 FROM HOLOGRAPHY TO EXPERIMENT some strongly coupled field theories have a dual gravitational description (AdS/CFT correspondence) Z[field theory] = Z[string theory] originally a duality in type IIB string theory so far: universality at strong coupling challenges field content (no supersymmetry) away from infinite N away from infinite coupling 0 / 1/ 1/ g YMN
4 TWO CLASSES OF THEORIES top-down dual of N=4 theory with t Hooft coupling corrections from type IIB string theory S IIB = 1 apple 10 Z d 10 p g R 1 (@ 1 ) 4 5! F 5 + e 3 W +... = 03 (3)/8 0 /L = 1/ W = C C µ C µ C + 1 C C µ C µ bottom-up curvature-squared theory with a special, non-perturbative case of Gauss-Bonnet gravity S R = 1 Z apple d 5 p g R + L 1 R + R µ R µ + 3 R µ R µ 5 S GB = 1 apple 5 Z d 5 p g C apple R + GB L R 4R µ R µ + R µ R µ
HYDRODYNAMICS
6 HYDRODYNAMICS QCD and quark-gluon plasma low-energy limit of QFTs (effective field theory) T µ u,t,µ =(" + P ) u µ u + Pg µ µ r u µ +... r µ T µ =0 tensor structures (phenomenological gradient epansions) with transport coefficients (microscopic)
7 HYDRODYNAMICS conformal (Weyl-covariant) hydrodynamics T µ µ =0 g µ! e!() g µ T µ! e 6!() T µ infinite-order asymptotic epansion T µ = 1X n=0 T µ (n)! = O H X n=0 n k n+1 classification of tensors beyond Navier-Stokes first order: (1 in CFT) - shear and bulk viscosities second order: 15 (5 in CFT) - relaation time, [Israel-Stewart and etensions] third order: 68 (0 in CFT) - [S. G., Kaplis, PRD 93 (016) 6, 06601, arxiv:1507.0461]
8 HYDRODYNAMICS diffusion and sound dispersion relations in CFT shear:! = i " " + P k i 1 (" + P ) 1 " + P sound:! = ±c s k i c k c c s c c s k 3 i # k 4 + O k 5 " 8 9 1 (" + P ) 3 1 + " + P # k 4 + O k 5 loop corrections break analyticity of the gradient epansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (003)] entropy current, constraints on transport and new transport coefficients (anomalies, broken parity) non-relativistic hydrodynamics hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]
9 HYDRODYNAMICS FROM HOLOGRAPHY holography can compute microscopic transport coefficients [Policastro, Son, Starinets (001)] low-energy limit of QFTs / low-energy gravitational perturbations in backgrounds with black holes Green s functions and Kubo formulae T ab R (0) =G ab R (0) 1 Z d 4 G ab,cd RA (0,)h cd()+ 1 8 Z d 4 d 4 yg ab,cd,ef RAA (0,,y)h cd()h ef (y)+... quasi-normal modes [Kovtun, Starinets (005)]! = 1X n=0 n k n+1 fluid-gravity [Bhattacharyya, Hubeny, Minwalla, Rangamani (007)]
10 N=4 AT INFINITE COUPLING type IIB theory on S 5 dual to N=4 supersymmetric Yang-Mills at infinite t Hooft coupling and infinite N c S = 1 Z apple d 5 p g R + 1 5 L black brane apple 5 = /N c ds = r 0 u f(u)dt + d + dy + dz + du 4u f(u) f(u) =1 u use to find field theory stress-energy tensor to third order T ab = "u a u b + P ab ab + apple h D abi + 1 3 ab (r u) i + apple hr habi u c R chabid u d + 1 ha c bic + ha c bic + 3 ha c bic + X0 n=1 (3) n O n
11 N=4 AT INFINITE COUPLING transport coefficients [S. G., Kaplis, PRD 93 (016) 6, 06601, arxiv:1507.0461] = 8 N c T 3 = ( ln ) T apple = N c T 8 1 = N c T 16 = N c T 8 ln 3 =0 s = 1 4 (3) 1 + (3) + (3) 4 1 = N c T 3 (3) 3 + (3) 5 + (3) 6 = N c T 384 1 + 18 ln ln (3) 1 (3) 16 = N c T 16 (3) 17 = N c T 16 1 +4ln ln 1 +ln ln (3) 1 6 + 4 (3) 3 + 3 (3) 8 + + 4 (3) 3 3 (3) 9 (3) 10 3 + 5 (3) 4 6 + 5 (3) 5 6 11 (3) 11 6 + 4 (3) 6 3 (3) 1 3 + 13 6 (3) (3) 7 (3) 15 = N c T 648 15 45 ln + 4 ln
1 TOP-DOWN CONSTRUCTION type IIB action with t Hooft coupling corrections S IIB = 1 apple 10 Z d 10 p g R 1 (@ 1 ) 4 5! F 5 + e 3 W +... = 03 (3)/8 0 /L = 1/ W = C C µ C µ C + 1 C C µ C µ C dimensional reduction S = 1 apple 5 Z d 5 p g R + 1 L + W black brane ds = r 0 u f(u)z t dt + d + dy + dz + Z u du 4u f Z t =1 15 5u +5u 4 3u 6 Z u = 1 + 15 5u +5u 4 19u 6 f(u) =1 u
13 TOP-DOWN CONSTRUCTION N=4 transport coefficients to second order [S. G., Starinets, JHEP 1503 (015) 007 arxiv:141.5685] = 8 N c T 3 (1 + 135 +...) = ( ln ) T apple = N c T 8 1 = N c T 16 = N c T 16 3 = 5N c T + 375 4 T +... (1 10 +...) (1 + 350 +...) ( ln + 5 (97 + 54 ln ) +...) +... h η s s = 1 4 1 + 15 (3) 3/ +... 4πk B 0 g N c [Kovtun, Son, Starinets (005)]
14 BOTTOM-UP CONSTRUCTION curvature-squared theory [S. G., Starinets, JHEP 1503 (015) 007 arxiv:141.5685] S R = 1 apple 5 Z d 5 p g R + L 1 R + R µ R µ + 3 R µ R µ = r3 + apple 5 (1 8(5 1 + )) + O( i ) = r + ( ln ) apple = r + apple 5 1 = r + 4apple 5 1 4apple 5 1 = r + ln apple 5 1 6 3 (5 1 + ) 6 3 (5 1 + ) 1 6 r 3 (5 1 + ) + (3 + 5 ln ) 1apple 3 + O( i ) 5 5r + 3 + O( i ) 6 3 (5 1 + ) 6apple 5 r+ 1apple 3 + O( i ) 5 r + (1 + 5 ln ) 3 + O( i ) 6apple 5 3 = 8r + apple 5 3 + O( i )
15 BOTTOM-UP CONSTRUCTION Gauss-Bonnet theory [S. G., Starinets, Theor. Math. Phys. 18 (015) 1, 61-73] S GB = 1 apple 5 Z d 5 p g apple R + GB L R 4R µ R µ + R µ R µ = p 1 4 GB = s /4 = 1 1 T 4 (1 + ) apple 1 (1+ ) 5+ log! 1 = (1 + ) 3 4 + 3 T + 1 apple (1+ ) log = 3 = T T 1 4 (1 + ) 1+ (1 + ) 3+ 4 apple = (1 + ) 1 T 1 = 8 T + 1!! s = 1 4 (1 4 GB)
16 LIMITS OF THE GAUSS-BONNET THEORY how can we interpret the etreme limits of the Gauss-Bonnet coupling? eact spectrum in the etreme (anomalous) limit of Scalar: w = i GB =1/4 p 4+n 1 4 3q, w = i 4+n + p 4 3q Shear: w = i (1 + n 1 ), w = i (3 + n ) p Sound: w = i 4+n 1 4+q, w = i 4+n + p 4+q in the etreme ``weak limit GB! 1 there is a curvature singularity, which needs a stringy resolution lim S GB = GBL GB! 1 4apple 5 Z d 5 p g apple R 4R µ R µ + R µ R µ 4 GBL ds = p GB 4 r L r 1 r 4 + r 4 dt + L r q1 d r + r r + 4 L d + dy + dz r 4 3 5
17 CHARGE DIFFUSION construct the most general four-derivative action [S. G., Starinets (016) arxiv:1611.07053] S = 1 apple 5 Z d 5 p g [R + L GB ]+ Z d 5 p gl A L A = 1 4 F µ F µ + 4 RF µ F µ + 5 R µ F µ F + 6 R µ F µ F + 7 (F µ F µ ) + 8 r µ F r µ F + 9 r µ F r F µ + 10 r µ F µ r F + 11 F µ F F F µ make it such that the equations of motion are only second order L A = 1 4 F µ F µ + 1 L (RF µ F µ 4R µ F µ F + R µ F µ F ) + L (F µ F µ ) + 3 L F µ F F F µ diffusion D 6= D =0 D 6= D =0 D = (1 + GB )(1 + ) h 6( 1) " GB ln + p GB + p GB GB + p GB GB #) i( p(1 GB)( GB) ln " GB 1+ p 1 GB GB = p 1 = 1 + 48 1 4 GB #
18 UNIVERSALITY membrane paradigm (conserved current) @ r J =0 first-order hydrodynamics [Kovtun, Policastro, Son, Starinets] second-order hydrodynamics [Haack, Yarom (009); S. G., Starinets, JHEP 1503 (015) 007 arxiv:141.5685] s = 1 4 4 1 = O 4 1 = O i 4 1 = T (1 GB) 1 GB (3 + GB ) GB = 40 GB T + O 3 GB
19 (MORE) UNIVERSALITY non-renormalisation of anomalous conductivities [Gursoy, Tarrio (015); S. G., Poovuttikul, JHEP 1609 (016) 046 arxiv: 1603.08770] universal values h J µ i h J µ 5 i = JB J! J 5 B J 5! B µ! µ J 5 B = µ JB = µ 5, J 5! = appleµ 5 + µ + ( T ) J! = µ 5 µ bulk with arbitrarily high derivatives (gauge, diffeomorphism) Z S = d 5 p g {L [A a,v a,g ab, i]+l CS [A a,v a,g ab ]} apple L CS [A a,v a,g ab ]= abcde A a 3 F A,bcF A,de + F V,bc F V,de + R p qbc Rq pde proof to all orders in the coupling constant epansion
BEYOND HYDRODYNAMICS
1 WEAK COUPLING (KINETIC THEORY) coupling constant dependence of non-hydrodynamic transport [S. G., Kaplis, Starinets, JHEP 1607 (016) 151 arxiv:1605.0173] instead of hydrodynamics, start with (weakly coupled) kinetic theory essential concept: quasi-particles Boltzmann equation @F @t + p i m @F @r i @U(r) @r i @F @p i = C[F ] close to equilibrium F (t, r, p) =F 0 (r, p)[1+'(t, r, p)] resulting equation @' @t = p i m @' @r i + @U(r) @r i @' @p i + L 0 ['] solution '(t, r, p) =e tl ' 0 (r, p) = 1 i Z+i1 i1 R s =(si L) 1 e st R s ds ' 0 (r, p)
WEAK COUPLING (KINETIC THEORY) ansatz for homogeneous eq. distribution eigenvalue equation for lin. coll. operator '(t, p) =e t h(p) h = L 0 [h] hierarchy of relaation times '(t, p) = X n C n e nt h n (p) dominant: R =1/ min
3 THERMALISATION (RELAXATION) KGB equation @F @t + p i m @F @r i @U(r) @r i @F @p i = F F 0 R kinetic theory predicts = R st relaation time bound [Sachdev] Ising model (BTZ) G R (!,q)= C ( 1) sin " cosh q T R C ~ k B T cos cosh! T i(! q) + 4 T + i sin sinh! T i(! + q) + 4 T #! = ±q i4 T n + R = 1 ~ k B T cute: =) /s =1/4
4 FULL QUASINORMAL SPECTRUM weak/strong coupling (perturbative/holographic quasi-normal mode calculations) [Hartnoll, Kumar (005)] -q Im ω q Re ω Im ω Re ω -i4πt -i8πt -i1πt -i16πt! 0!1
5 RESULTS: QNM STRUCTURE different trends depending on poles become denser (branch cut) new poles on imaginary ais /s ``weak limit!1 GB! 1 ``anomalous limit GB! 1/4 Im ω Im ω Re ω Re ω /s > ~/4 k B /s < ~/4 k B
QNM STRUCTURE shear channel QNMs in N=4 Re ω Im ω Re ω Im ω 6
QNM STRUCTURE shear channel QNMs in Gauss-Bonnet Re ω Im ω Re ω Im ω 7
8 KINETIC THEORY RESULT kinetic theory behaviour quickly approached /s const R T Im ω Im ω Re ω Re ω
9 HYDRODYNAMICS breakdown of hydrodynamics (diffusion) q c 3/4
30 HYDRODYNAMICS breakdown of hydrodynamics (diffusion) q c 3/4
31 HYDRODYNAMICS breakdown of hydrodynamics (sound) q c 3/4
3 HYDRODYNAMICS breakdown of hydrodynamics due to new poles shear and sound correlators in Gauss-Bonnet [S. G., Starinets (016) arxiv: 1611.07053] G z,z (!,q)= G tt,tt (!,q)= dispersion relations p 3 T 3 3 p 4 T 4 (1 + GB ) 3/ apple 5 GB (1 + GB ) 3/ apple 5! i! i! /! g GBq /4 T 5q 3! (1!/! g ) i GB!q / T (3! q )(1!/! g )+i GB!q / T!! 1 = i! 1, = ± 1 p 3 q GB 4 T q i GB 6 T q! =! g + i GB 4 T q! 3 =! g + i GB 3 T q gap! g = GB ( GB + ) 8 Ti 3+ln GB+1 8 Ti GB
33 QUASI-PARTICLES (TRANSPORT PEAK) quasi-particles appear in the spectrum [Casalderrey-Solana, S. G., Starinets (017)] weak coupling (Boltzmann equation)
34 QUASI-PARTICLES (TRANSPORT PEAK) holographic results in N=4 theory (scalar channel)
35 QUASI-PARTICLES (TRANSPORT PEAK) holographic results in a dual of the Gauss-Bonnet theory (scalar channel)
QUASI-PARTICLES IN N=4 36
QUASI-PARTICLES IN GAUSS-BONNET 37
38 QUASI-PARTICLES IN GAUSS-BONNET quasi-particles appear also in other channels (shear) how do these peaks manifest themselves (if at all) in a weakly coupled field theory like QCD?
HEAVY ION COLLISIONS
40 HEAVY ION COLLISIONS R coupling constant corrections to shockwave collisions [S. G., van der Schee (016) arxiv:1610.08976] energy density along the longitudinal direction after collision: less stopping of narrow shocks (88% higher energy density on lightcone) and decreased energy density of wide shocks
41 HEAVY ION COLLISIONS delayed hydrodynamisation t hyd T hyd = {0.41 0.5 GB, 0.43 6.3 GB } at GB = 0. : {5%, 90%}
4 HEAVY ION COLLISIONS rapidity profile: start wider and smaller but later become comparable
43 HEAVY ION COLLISIONS change in entropy density: enhanced on lightcone, negative in plasma total entropy: reduced at intermediate coupling
44 CONCLUSION AND FUTURE DIRECTIONS various weakly coupled properties are recovered remarkably quickly the topic of this workshop: how precisely do they match onto those computed from weakly coupled, perturbative physics? use interpolations between weakly coupled physics and couplingdependent holography to understand intermediate coupling can we see a signature of higher-qnms (quasi-particles) in eperiments? a harder task for the future: understand 1/N corrections
THANK YOU!