Scaling and integrability from AdS 5 S 5

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1 Scaling and integrability from AdS 5 S 5 Riccardo Ricci Imperial College London DAMTP Cambridge 14th October Work in collaboration with S. Giombi, R.Roiban, A. Tseytlin and C.Vergu

2 Outline Introduction What are the cusp anomaly and generalized scaling function? Superstring in the light cone gauge Scaling function from string theory and comparison to gauge theory

3 Understand quantum gauge theories at any coupling. Make best use of symmetries by choosing simplest non trivial gauge theory. Ideal candidate: N = 4 SYM It has maximal symmetry It is conformal Infinite dimensional symmetry: Integrability It is dual to a superstring theory in a non-trivial background

4 AdS/CFT By the AdS/CFT correspondence, this gauge theory is believed to be dual to type IIB string theory on AdS 5 S 5. α expansion 1 λ genus expansion 1 N expansion This is a strong/weak duality which is in general very hard to test directly. How do we interpolate between weak (λ 1) and strong coupling (λ 1) regimes?

5 Can we bridge the gap between weak and strong coupling regimes? Riccardo Ricci Scaling and integrability from AdS5 S 5

6 ...sometime we have interpolating functions. Simplest example: Half-BPS circular Wilson loop x 2 x 1 ( ) W TrP exp A + Φ The VEV is non trivial and it is exactly computable for any λ! W = 2 λ I 1 ( λ), N Highly supersymmetric configuration

7 Non-supersymmetric example: cusp anomaly No simple formula, as for circular Wilson loop, but an integral equation exists. Beisert Eden Staudacher An exact solution is not known but one can extract a perturbative solution at any desired order. The cusp anomaly surprisingly appears in many different contexts: It governs the renormalization of a light-like Wilson loops with cusps Gluon scattering amplitudes It governs the logarithmic scaling of high spin twist operators.

8 The simplest case is twist two (i.e. only two Z fields) O = Tr ( ZD S +Z ), Z = Φ 1 + iφ 2 This is the N = 4 analogue of a QCD operator like q γ + D S + q, q = quark The conformal dimension can be read from the 2-point correlator 1 O(x)O(y) (x y) 2 (S) (S) = 2 + S + δ(s) For large spin we have a logarithmic scaling δ(s) = f(λ) log S, S

9 Comparing the spectra on the two sides of the duality: Gauge theory side: Compute the conformal dimension of local operators of planar (N ) N = 4 SYM AdS side: Compute the energy of free string in AdS 5 S 5 According to AdS/CFT duality it is the same computation Conformal dimension (λ) = String energy E(α ) What is the string in AdS 5 S 5 dual to the twist operator?

10 Folded spinning string It is a spinning closed string ( folded on itself) in AdS 3 Gubser Klebanov Polyakov AdS boundary It is pointlike in S 5

11 Why this is the correct string dual? E classical = S + λ π log S, Gubser Klebanov Polyakov This logarithmic behaviour persists after including quantum corrections E one loop = 3 log 2 π log S Frolov Tseytlin These results combined with the behaviour at weak coupling = S + (α 1 λ + α 2 λ 2 + ) log S support the idea that we have interpolation between weak and strong coupling = E string = S + f(λ) log S

12 Generalized scaling A new interpolating function appears when we additionally turn on one (large) angular momentum J in S 5. This corresponds in gauge theory to the operator O Tr(D S +Z J ) The conformal dimension is a non-trivial function (S, J, λ) It can be explored in various regimes of the parameters testing important features of gauge/string duality. Interpolation between BMN-like (large J, small S) and minimal twist (J = 2) operators.

13 Integrability allows to map the one-loop anomalous dimension into the energy of an SL(2) Heisenberg spin-chain. Tr (Z...Z...Z)......, ( ) Tr Z... D + Z...Z spin chain vacuum excitation (magnon)

14 The Bethe equations determine the allowed momenta for the S magnons. At one-loop (u k cot(p k /2)): ( ) uk + i/2 J = u k i/2 S j k u k u j i u k u j + i, S k=1 u k + i/2 u k i/2 = 1 δ(j, S) = g 2 S k=1 2 u 2 k + 1/4 An all loop generalization for the Bethe equations exists Beisert Eden Staudacher

15 Solve them at strong coupling and compare with string prediction Equations simplify in the semiclassical scaling limit λ 1 S λ 1 J λ 1 l πj λ log S = fixed The anomalous dimension scales logarithmically in the spin δ(l, S) = λ f(l, λ) log S π f(l, λ) is the generalized scaling dimension: new interpolating function Belitsky Gorsky Korchemsky; Freyhult, Rej, Staudacher

16 In the same regime also the string energy scales logarithmically: E S = λ f(l, λ) log S π It is a long string which lives in AdS 3 S 1 ds 2 = cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dθ 2 + dϕ 2, t = κτ, ρ = µσ, θ = κτ, ϕ = ντ, κ, µ, ν 1 µ 1 π log S, ν = J λ, κ 2 = µ 2 + ν 2 (Virasoro)

17 The string is a folded segment passing through AdS center up to the boundary and rotating in S 1 inside S 5 t ϕ θ ρ Length of the string is controlled by the spin: log S

18 Classically f 0 (l) = 1 + l 2 Quantum corrections 1/( λ) n modify the classical result f(l, λ) = f 0 (l) + 1 λ f 1 (l) + 1 λ f 2(l) + Up to one-loop string theory and Bethe-Ansatz agree Frolov-Tirziu-Tseytlin; Casteill-Kristjansen; Belitsky

19 Two-loops string theory computation f 2 (l, λ) = K + l 2 ( 8 log 2 l 6 log l + q 0 ) + O(l 4 ) q 0 = 3 2 log K, String (?) 4 Roiban-Tseytlin 07 Almost equal to the Bethe-Ansatz prediction q 0 = 3 2 log , Bethe Ansatz Gromov 08 K = k=0 ( 1) k (2k + 1)

20 Status so far: We compared the results at strong coupling for the conformal dimension of the operator O Tr(D S Φ J ) in gauge theory (Bethe-Ansatz) and in string theory. Disagreement emerges at 2-loops... Breakdown of integrability in string theory?? We will perform the string computation again, but this time using a different gauge.

21 A thermodynamical analogy We need to compute E S and J for our semiclassical string. Suppose we have a 2d σ-model with conserved charges Q i H 2d = H 2d + i µ i Q i, Z(µ i ) = e βσ(µ i) β H = Tre Averages Q i can be computed differentiating w.r.t. µ i. In our context it is natural to consider H 2d = H 2d + κ(e S) νj The parameters κ and ν explicitly appear in the string solution. They are not independent H 2d = 0 κ = κ(ν) Virasoro

22 Thermodynamics (grand-canonical ensemble) log Z grand.can. = U i µ i N i Here U = H 2d, N 1 = E S, N 2 = J Σ(ν) = H 2d + κ E S ν J, dσ(ν) dν = dκ(ν) E S J dν We can solve these two equations to extract E S and J.

23 We obtain E S = ( 1 + ν 2 Σ(ν) ν dσ(ν) ), dν Remember λ E S = f(l, λ) log S π The logarithmic scaling of E S is due to the effective action Σ which is proportional to the worldsheet volume: Σ Vol log S We can therefore read the generalized scaling f(l, λ) = ( 1 + ν 2 F(ν) ν df(ν) ), F Σ/Vol dν

24 AdS 5 S 5 superstring

25 It is based on the supercoset G H Metsaev, Tseytlin S λ = P SU(2, 2 4) SO(2, 3)SO(5) AdS 5 S 5 G MN X M X N + θ(d + F 5 )θ X + Classically integrable Lüscher, Pohlmeyer; Bena, Polchinski, Roiban Infinite tower of conserved charges: local (Noether)+non-local charges

26 Performing a conformal transformation on the folded string we can do all computations in the Poincare patch which is technically easier ds 2 = dxa dx a + dz M dz M z 2 M = 1,..., 6 Poincare patch The induced metric on the worldsheet, after Euclidean rotation, is just flat 2d space. We still need to fix the worldsheet symmetries...

27 Performing a conformal transformation on the folded string we can do all computations in the Poincare patch which is technically easier ds 2 = dxa dx a + dz M dz M z 2 M = 1,..., 6 Poincare patch The induced metric on the worldsheet, after Euclidean rotation, is just flat 2d space. We still need to fix the worldsheet symmetries...

28 Light-cone Superstring worldsheet symmetries 2d bosonic diffeomorphism σ σ(σ, τ), τ τ(σ, τ) fermionic κ symmetry Fix these symmetries by suitably choosing a gauge.

29 Let us consider κ symmetry: Θ I = (θ, η) θ Q, η S S-gauge η = 0 Conformal gauge and S-gauge lead to a simple quadratic fermionic action (after T-duality )....but bosonic propagator is not simple. Disagreement with Bethe-Ansatz...!

30 Let us consider κ symmetry: Θ I = (θ, η) θ Q, η S S-gauge η = 0 Conformal gauge and S-gauge lead to a simple quadratic fermionic action (after T-duality )....but bosonic propagator is not simple. Disagreement with Bethe-Ansatz...!

31 Other possibility: Γ + θ I = 0 as for Green-Schwarz in flat space. Left with 8 θ s and 8 η s. Combine this with bosonic light-cone gauge x 0 + x 1 = x + = τ, gg αβ = diag( z 2, 1/z 2 ) Metsaev Thorn Tseytlin This AdS light-cone action was never tested at the quantum level

32 Other possibility: Γ + θ I = 0 as for Green-Schwarz in flat space. Left with 8 θ s and 8 η s. Combine this with bosonic light-cone gauge x 0 + x 1 = x + = τ, gg αβ = diag( z 2, 1/z 2 ) Metsaev Thorn Tseytlin This AdS light-cone action was never tested at the quantum level

33 Very schematically L GS ẋẋ + ż M ż M + 1 z 4 ( x x + z M z M) +η η + θ θ + (η 2 ) 2 + η θ Fermionic action is quadratic in θ and quartic η. Bosonic propagator is simple (almost diagonal). This is an encouraging property for higher loop computations. We have a 2d QFT theory problem Compute its partition function ( Z string = D[x, z, θ, η] exp L GS )

34 Expand bosonic fields around the semiclassical string solution x = x 0 + δx, z M = z M 0 + δz M Read vertices and compute bosonic and fermionic Feynman diagrams at a given loop order At one loop we simply need the fluctuation spectrum F one loop = d 2 p log det K Z one loop = e V 2π F one loop Bosons and fermions conspire to give a finite answer

35 At two-loops for the computation of F 2loop log Z string we need to consider all connected Feynman diagrams

36 Possible integrals ( ) a I m 2 = d 2 p 1 (2π) 2 (p 2 + m 2 ) a This integral is UV divergent for a = 1 (and IR divergent for m = 0). ( ) a1 a 2 a 3 d 2 p d 2 q d 2 r δ (2) (p + q + r) I m 2 = 1 m2 2 m2 3 (2π) 4 (p 2 + m 2 1 )a 1 (q 2 + m 2 2 )a 2 (r 2 + m 2 3 )a 3 Catalan constant: ( ) I = K

37 Summing up: F 2loop (l) = K + l 2 ( ) + l 4 ( ) + O(l 6 ) All divergences cancel out! The quantum superstring in AdS light cone gauge is finite This proves the consistency of the light-cone superstring at the quantum level.

38 From the partition function we can extract the generalized scaling: f 2 = K ( + l 2 8 log 2 l 6 log l 3 2 log ) 4 + l ( 6 4 log 2 l 7 6 log l + 3 log 2 log l 9 8 log log K 233 ) + O(l 6 ) 576 In stupendous agreement with the Bethe-Ansatz prediction!

39 Leading logarithms and all loops results Summarising we have the following structure f 2 = h 2 (l) log 2 l + h 1 (l) log l + γ(l)k + δ(l) The coefficients h 1 (l), h 2 (l) and γ(l) are known exactly, i.e. to any order in l. At N-loop the expansion looks like f N = f (N) (l) log N l + f (N 1) (l) log N 1 l + It is possible to reconstruct the leading logarithm coefficient at any loop order...

40 log l arise from fields with mass l in one-loop integrals S 5 fields y a have mass l y a fields in the loops φ in the legs (z = e φ ) Conjecture: Leading Logs come from maximally disconnected graphs L = 1 4 cosh(2φ) + e2φ ( t y) 2 + e 2φ ( s y) l2 e 2φ y 2 The y a fields appear quadratically...integrate them out

41 This leads to an effective action for φ λ S eff (φ) = 2π F lead. log. = δs eff δφ = 0 e2φ = cosh(2φ) e2φ 2π l2 ln l 2. = λ 2π F l2 λ ln l λ F 1 lead. log., F 1 lead. log. = l 2 ln l 2 All loop-formula for the scaling function f(l, λ) l lead. log. = λ ln l. 2 Perfect agreement with asymptotic Bethe-Ansatz.

42 Large l expansion and non-renormalization An expansion at large l allows to make contact directly with perturbative gauge theory. E S ln S = j+ λ j j J log S = λ l 1 ( c 10 + c 11 j + c 12 )+ j λ2 j 3 ( c 20 + c 21 j + c ) 22(λ) j 2... E S = λ (E S) tree + (E S) 1 loop + 1 λ (E S) 2 loop λ c 12 j 3 = c λ l 3, c 12 = 1 3π 2 The same result was obtained in the one-loop spin-chain (Volin 08) confirming that c 12 is indeed protected.

43 Finite size corrections The worldsheet is no longer infinite: L = ln S < E = S + 1 π f(λ, J) ln S + h(λ, J) + u(λ, J) ln S +... In principle requires the use of the exact string solution with finite S... daunting!!! The worldsheet surface becomes a cylinder : R 2 R S 1 The S 1 momentum p 1 is quantized p 1 = 2π L, dp 1 2π L n Shortcut: keep folded string solution but on cylinder!

44 At one-loop for J = 0 this is enough Beccaria, Dunne, Forini, Pawellek Tseytlin Let us then use this prescription... Contributions come terms containing at least a massless field For example at one loop (L = ln S) π + p m 0 dp 0 dp 1 p p2 1 ( ) 1 dp 0 p m 1 0 coth 2 Lp 0 + dp 0 p 1 pm 0 p p2 1 = = Div. + 1 L m Finite

45 We obtain: u 1 (l) = π l 2 one loop u 2 (l = 0) = 0? two loop The two-loop result is seemingly in agreement with gauge theory expectations Fioravanti, Grinza, Rossi

46 Future directions All computations were actually done in a more general background ϕ = ντ + wσ, w = winding The solution with winding is a bended arc passing through AdS center

47 The partition function now depends on both momentum and winding: F(ν, w) It is again finite at two-loops The exchange of ν and w is a symmetry: T-duality! We can extract a generalized scaling function in presence of winding Prediction confirmed at leading order from Bethe-Ansatz. Kruczenski-Tirziu One-loop/two-loops?

48 Going up... q 2 q 1 q 3 q 1 q q 1 3 q 3 q 1 q 2 q 2 q 3 q 2 (a) (b) (c) (d) q 1 q 1 q 2 q 3 q 1 q 2 q q 3 1 q 2 q 3 q 2 (e) q 3 (f) (g) (h)

49 The most feasible computation is the cusp anomaly (J = 0) f(λ) = ( λ 1 + a 1 + a 2 λ λ + a 3 λ 3/2 + a ) 4 λ 2 + Remarkable transcendentality properties a 1 = 3 log 2, a 2 = K a 3 = 1 (27ζ(3) + 96 K log 2) 32 a 4 = 1 ( 84β(4) + 81ζ(3) log K K log 2 2 ) 16 Degree of transcedentality [log 2] = 1 [K] = 2 [ζ(n)] = n [β(n)] = n; β(2) = K

50 Conclusion The light-cone superstring computation solves a long standing discrepancy between string and Bethe-Ansatz. It provides a highly non trivial check of quantum integrability beyond one-loop string semiclassical level. Viceversa it provides an important consistency check of all-loop Bethe-Ansatz (and BES phase). We have initiated a study of 2-loop finite size corrections.