On correlation functions in the AdS/CFT correspondence at strong coupling

Transcription

1 On correlation functions in the AdS/CFT correspondence at strong coupling Radu Roiban PennState University Based on work with A. Tseytlin ( )

2 Why correla+on func+ons? Interes+ng in many QFTs (e.g. DIS may be formulated in terms of a 2- point correla+on func+on of currents) In conformal field theories next more complicated quan++es aeer dimensions of operators may be used to define a CFT 3- point func+ons are in principle sufficient related to string theory effec+ve ac+on test the AdS/CFT correspondence The hope AdS/CFT methods may be generalized to other setups

3 AdS/CFT: Planar O 1 (x)o 2 (y)o 3 (z) = N =4SYM = tree- level string theory in AdS 5 S 5 Conformal invariance constraints and 2- and 3- point func+ons: O 1 (x)o 2 (y) = δ( 1 2 ) x y 2 What is known? C 123 (λ) x y 2( ) x z 2( ) y z 2( ) ½- BPS operators = lowest string modes (IIB supergravity) - and C 123 are protected (same at weak and strong coupling) - 3pf of chiral primaries are known C J1,J 2,J 3 = 1 N J1 J 2 J 3 Lee, Minwalla, Rangamani, Seiberg - examples of correlators of descendants; 4- point fcts, etc

4 AdS/CFT: Planar O 1 (x)o 2 (y)o 3 (z) = N =4SYM = tree- level string theory in AdS 5 S 5 Conformal invariance constraints and 2- and 3- point func+ons: O 1 (x)o 2 (y) = δ( 1 2 ) x y 2 C 123 (λ) x y 2( ) x z 2( ) y z 2( ) What is less/not known? correla+on func+ons of non- BPS operators (massive string modes) C 123 =C λ n c n n 1 ; conjecture: c 1 = γ (1) 1 + γ (1) 2 + γ (1) 3 Bianchi, Kovacs, Rossi, Stanev; Okuyama, Tseng; Grosardt, Pleda General use of integrability? Integrability as a tool = worldsheet energy, i.e. v. op. same as heavy sources RR, Volovich Escobedo, Gromov, Sever, Vieira hints at semiclassical expansion Janik, Surowka, Wereszczynski C ; similar to 2- point func+ons Buchbinder, Tseytlin 123 exp( a λ)

5 Outline Semi- classical approxima+on of correla+on func+ons General idea and previous work Closed string vertex operators 2- point func+on of operators with large dimension examples of HHL 3- point func+ons; limits Outlook

6 Flat space General structure of vertex operators: V flat space X X... S e ik X Correla+on func+ons: V 1...V N = [DX]V 1...V N e T R d 2 σl If k i k j T 1 vertex operators and ac+on are of the α same order; a saddle- point evalua+on is appropriate Gross, Mende Similar mechanism should work in AdS 5 S 5

7 AdS 5 S 5 General structure of vertex operators: V AdS5 S 5 something Q something else Correla+on func+ons: V 1...V N = [DX]V 1...V N e T R d 2 σl T = λ 2π If Q, λ the correla+on func+on is dominated by a classical solu+on with sources provided by vertex operators Similar ideas: Correla+on func+ons of Wilson loops 2- point func+ons Zarembo; Tsuji Polyakov; Tseytlin

8 Some more general cases of correla+on func+ons K n,m = V H1 (x 1 )...V Hn (x n )V L1 (x n+1 )...V Lm (x n+m ) large quantum numbers H Q L λ Expecta+on: governed by same classical solu+on as small quantum numbers H λ 1/4 Q L 1 H 1 Q L 1 (BPS) V H1 (x 1 )...V Hn (x n ) because the sources provided by the light v. ops. are subleading Strategy: 1) Find classical solu+on giving K n,0 = V H1 (x 1 )...V Hn (x n ) 2) Find K n,m by evalua+ng the V L1 (x n+1 )...V Lm (x n+m ) on this classical solu+on

9 Focus on 3- point func+ons No known e.g. of classical solu+on for Next best thing: use classical solu+on for Exis+ng examples: K 2,1 = V H1 (x 1 )V H2 (x 2 )V L1 (x 3 ) K 3,0 = V H1 (x 1 )V H2 (x 2 )V H3 (x 3 ) K 2,0 = V H1 (x 1 )V H2 (x 2 ) v. op. for state with large spin in BPS choices of V L - - CPO - - dilaton anempts by Janik et al V H (x) S 5 Tseytlin Buchbinder, Tseytlin Zarembo Costas, Monteiro, Santos, Zoakos In the following: - - other choices of V, twist ops and ops for 1 st H (x) excited states - - is a massive string state lighter than V H (x) V L RR, Tseytlin Yet more V H (x) : 3- spin (1+2) circular string Hernandez; Ryang

10 Examples of vertex operators in AdS 5 S 5 Ac+on + coordinates S = λ 4π Y M Y M = Y 2 0 Y Y Y 2 4 = 1 d 2 ξ Y M YM + X A XA + fermions X A X A = X X 2 6 =+1 Vertex operators: - V (Y,X,,, fermions) - e- vec of 2d dimension operator (dimension count to LO) - labels: (E; S 1,S 2 ; J 1,J 2,J 3 ) of SO(2, 4) SO(6)

11 Examples of vertex operators in AdS 5 S 5 Ac+on + coordinates S = λ 4π Y M Y M = Y 2 0 Y Y Y 2 4 = 1 A shortcut: iden+fy analog of e ik X d 2 ξ Y M YM + X A XA + fermions X A X A = X X 2 6 =+1 Vertex operators: - V (Y,X,,, fermions) Same as flat - e- vec of 2d dimension operator (dimension count to LO) space energy - labels: (E; S 1,S 2 ; J 1,J 2,J 3 ) of SO(2, 4) SO(6) Standard construc+ons are cumbersome Y 5 + iy 0 = cosh ρ e it Y 1 + iy 2 =sinhρ cos θ e iφ 1 Y 3 + iy 4 =sinhρ cos θ e iφ 2 V =(Y 5 + iy 0 ) E U = (cosh ρ) E e iet U

12 Examples of vertex operators in AdS 5 S 5 In Poincare patch Y m = x m z Y 4 = 1 2z ( 1+z2 + x m x m ) Y 5 = 1 2z (+1 + z2 + x m x m ) Map E to ; find familiar structure: (Y 5 + iy 0 ) E Y + K(x, z) =k z + z 1 x m x m ) bulk- boundary propagator Un- integrated vertex op: V Y + U[Y,X,...] Integrated vertex op at a point on the boundary of Poincare patch: V (x) = d 2 ξ V x(ξ) x;... = d 2 ξ K(x(ξ) x,z(ξ)) U[x(ξ) x,z(ξ),x(ξ),...]

13 V (x) = d 2 ξ V x(ξ) x;... = d 2 ξ K(x(ξ) x,z(ξ)) U[x(ξ) x,z(ξ),x(ξ),...] AdS boundary z worldsheet x K(x(ξ) x,z(ξ)) ξ

14 Examples of vertex operators in AdS 5 S 5 BPS states J -th KK mode of the dilaton - - dual to Tr[F µν F µν Z J +...] =Y+ X J x YM Y M + X k Xk + fermions V (dil) J X x = X 1 + ix 2 = cos ϑ e iϕ =4+J essen+ally the worldsheet ac+on Superconformal primary scalar dual to Tr[Z J ] V J =Y+ X J x z 2 x µ xµ z z X k Xk +fermions Representation [0,J,0] of SO(6) ; = J Berenstein, Corrado, Fishler, Maldacena A mixture of trace of S 5 graviton, AdS 5 graviton modes and RR 5- form; bosonic part is construc+ble also from the KK reduc+on of supergravity fields on S 5

15 Examples of vertex operators in AdS 5 S 5 Non- BPS states state on leading Regge trajectory - - flat space: V S =( x x xx ) S e ik X x x = x 1 + ix 2 E = 2 α (S 2) generaliza+on to AdS 5 S 5 Gloss over operator mixing; not relevant to leading order Other examples are also available, e.g. special unmixed scalars

16 2- point func+ons in semiclassical approxima+on The idea: x 0e z Massive AdS geodesics reaches boundary upon Euclidian con+nua+on States at = t = κτ z =(ch(κτ e )) 1,x 0e =th(κτ e ) ϕ = κτ x i =0, ϕ = iκτ e, τ e = iτ τ e ± x 0e = ±1, x i =0 τ e ± v. ops to cpx. plane by e τe+iσ = ξ ξ 2 ξ ξ 1 Mapped classical solu+on sourced by these states is the sta+onary point of the path integral describing the correla+on func+on of the vertex operators with same symmetry charges as the ini+al classical solu+on Tseytlin; Buchbinder, Tseytlin ξ ξ 1 and ξ 2 On the plane, v. ops at source for classical trajectory

17 Example: V S (x) V S (x ) Leading Regge trajectory state (dual to twist 2 operators) V S,J (0) = d 2 ξ (Y + ) (X x ) J S/2 Y x Yx Y x = 1 z (x 1 + ix 2 ) Finite spin S - - ellip+c solu+on; in the large spin limit - - ra+onal t e = κτ e, φ = iκτ e, ρ = µσ, ϕ = iντ e κ = µ 2 + ν 2, µ 1 π ln S 1, ν = J = J/ λ Solu+on in Euclidian Poincare patch x 0e = tanh(κτ e ) x ± x 1 ± ix 2 = tanh(µσ) cosh(κτ e ) e±κτ e z 2 + x 2 0e + x x 2 2 =1 + S 5 angle ϕ = iντ e z = (cosh(κτ e ) cosh(µσ)) 1 reaches boundary at τ e ± ; less innocent than massive geodesic - - light- like line intersec+on with boundary same follows from solving eom with sources in general solu+on is complex; typical for saddle points Buchbinder, Tseytlin

18 K 2,1 = V H1 (x 1 )V H2 (x 2 )V L1 (x 3 ) 3- point func+ons The setup: λ 1 ; H1 = H2 λ L aim to compute only The procedure: 1) find classical saddle point for V H1 (x 1 )V H2 (x 2 ) x 1, x 2 C 123 adjustable by rescaling and transla+ons; set them to x 1 =1= x 2 V H1 (x 1 )V H2 (x 2 ) e S cl 2) Conformal invariance: choose x 3 =0; compute C 123 = V H 1 (x 1 )V H2 (x 2 )V L (x 3 ) = V L (0) V H1 (x 1 )V H2 (x 2 ) on solution = d 2 ξ Y +,cl U[x cl(ξ),z cl (ξ),x cl (ξ)] map to complex plane can be undone by change of variables use directly cylinder coordinates (τ e, σ)

19 Higher twist heavy state V S,J : S λ, S = S + λ 1 V L is the dilaton 2π C 123 = c dτ e dσ z U c = c j+4 = 2 j/2 (j + 3) 2π2 0 U =(X x ) j z 2 ( x m x m + z z)+ X k Xk = j +4 Evaluated on the solu+on C 123 =4c dτ e π 2 0 dσ - - Analy+c evalua+on - - Result is combina+on of 2µ 2 e jντ e cosh(µσ) cosh(κτe ) - s 2F 1 κ 2 = µ 2 + ν 2 µ = 1 π ln S S = λs J = λν

20 An interes+ng limit: j =0, large S C 123 ln S J 2 + λ π ln 2 S 2 Obs: λ is constant part of dilaton λ λ C 123 λ λ S,J = λ ln 2 S 2π 2 J 2 + λ π 2 ln 2 S Inserts dilaton v. op. integrated over 4d space ln S mismatch related to divergence of integral over 4d space More limits & consistency checks: λ π ln S λ π ln S J : C i.e. no dilaton coupling with BMN states J : i.e. expected coupling w/ massive states C 123 const. Similar construc+on for CPO - reproduces 3 CPO correlator in appropriate limit - approaches a constant if J λ/π ln S

21 Higher twist heavy state V S,J : S λ, S = S + λ 1 V L is fixed spin operator on leading Regge trajectory V L : spin s fixed, s = 2(s 2)λ 1/ s S Spin- carrying part of vertex operator: U =( Y x Yx ) s/2 = e 2sκτ e Integrate; in the limit ν 0 : s/2 µ 2 cosh 2 (µσ)+κ 2 sinh 2 (µσ) C 123 µ s 2 (ln S) s 2 light string level (heavy anom. dim) large s limit: use saddle- point approxima+on λ) eh(s,s, s 2 C If s λ (formal limit) H(S, s) 123 = c s λ π contact with semiclassical eval. of correla+on fct. of 3 heavy operators? Similar structure for other excited string states

22 Summary and outlook semiclassical approxima+on provides new data involving 3- point func+ons of massive string states extension to more light operators is possible e.g. V H V H V L1 V L2 K 2,2 = V L1 V L2 V H V H + V L1 V L2 V H V H Buchbinder, Tseytlin If two dilaton v. ops. K 2,2 ( λ H ) 2 test against 4- point fct of CPO s Uruchurtu

23 Summary and outlook semiclassical approxima+on provides new data involving 3- point func+ons of massive string states extension to more light operators is possible e.g. V H V H V L1 V L2 test against 4- point fct of CPO s Buchbinder, Tseytlin Uruchurtu other light & heavy states; need vertex ops. & solu+ons quantum correc+on; need a bener handle on vertex operators what is the role of integrability? - - is it more than just a tool at weak coupling? - - higher charges: any reflec+on on the SFT 3- string vertex in AdS? - - any rela+on to open string correlator/wilson loop rela+on? Alday, Maldacena; + Eden, Korchemsky, Sokatchev - - can Landau- Lifshitz theory link weak and strong coupling integrability? Hernandez; Ryang

24 Extra slides

25 Light- like line boundary intersec+on τ e + : z 0, x 0e 1, x + 2 tanh(µσ), x 0 τ e : z 0, x 0e 1, x + 0, x 2 tanh(µσ)

26 An issue: - at 1- loop in the bosonic theory mixes with other operators: - true v. op. is eigenvector of 2d anomalous dimension op. - To leading order, classical solu+on is determined by energy and charges one may ignore mixing

27 Dilaton + (SJ) string

28 Higher twist heavy state V S,J : S λ, S = S + λ 1 V L is unmixed massive singlets V H : spin S λ, S = S + λ 1 V L : level r state, r = 2(r 2)λ 1/4 + S Spin- carrying part of vertex operator: U S 5 U AdS 5 =( X m X m Xp X p ) r/2 = ν 2r chiral components of stress tensor =( Y M Y M YP Y P ) r/2 = ν 2r 3pf coefficient comes our rela+vely simple: In the limit ln S 1, J 1, fixed C 123 2r (ln 1+ 2 S)2r 2 light string level (heavy anom. dim.)

29 Summary and outlook semiclassical approxima+on provides new data involving 3- point func+ons of massive string states extension to more light operators is possible e.g. V H V H V L1 V L2 K 2,2 = V L1 V L2 V H V H + V L1 V L2 V H V H If two dilaton v. ops. K 2,2 ( λ H ) 2 test against 4pf of CPO s Uruchurtu other light & heavy states; need v. ops. & solu+ons quantum correc+on; need a bener handle on vertex operators what is the role of integrability? - - higher charges: any reflec+on on the SFT 3- string vertex in AdS? - - any rela+on to open string correlator/wilson loop rela+on? Alday, Maldacena; + Eden, Korchemsky, Sokatchev - - how much can the Landau- Lifchitz theory be used? Hernandez