QCD-like properties of anomalous dimensions in ADS/CFT

Transcription

1 QCD-like properties of anomalous dimensions in ADS/CFT Valentina Forini AEI Potsdam in collaboration with M. Beccaria (Lecce U.), A. Tirziu (Purdue U.) and A. A. Tseytlin References: V.F., M. Beccaria arxiv: M. Beccaria, V.F., A. Tirziu, A. A. Tseytlin arxiv: Imperial College, November

2 Anomalous dimensions γ in N=4 SYM play a crucial role in the AdS/CFT duality, and integrability offers new exceptional tools to calculate them at many loops. In the case of the operators Tr(Φ D S Φ...Φ) the γ(s) are expected to contain important information encoded in their dependence on the Lorentz spin S. The physical content of this information can be extracted by exploiting facts known for similar twist operators, arising in the QCD analysis of DIS. For twist two, γ(s) splitting functions P(x). Not surprisingly, String Theory configurations contain this information and exhibit it in perturbation theory.

3 The central role of N=4 SYM from string theory to strong interactions energies anomalous dimensions splitting functions AdS 5 S 5 N = 4 SYM N = 1, 2 SYM QCD I II I. AdS/CFT duality conjecture type IIB strings on AdS 5 xs 5 N=4 Super Yang Mills in d=31 [Maldacena, 97] Agreement of the underlying symmetry supergroup PSU(2,2 4) 4πλ Weak/strong coupling duality N Prediction E string = = g R 2 s, λ = α, (λ = N g2 YM) CF T Planar limit E string = CFT N = g s = 0 N g s = 0 free string. Integrability! II. Superconformal (β=0) vs. confined (β<0), SU(N) vs SU(3), adjoint vs fundam.

4 Integrability Quantitative understanding of the duality: remarkable boost with data coming from perturbative gauge theory (4-loop in λ) and perturbative string theory (2-loop in 1/ λ ). ( long gauge operators semiclassical string states with large quantum numbers) Framework: integrable structures discovered on both sides of AdS/CFT (planar limit!) Integrable CFT: not in the sense of factorised space-time scattering! Observables of the theory: correlation functions of gauge invariant local composite operators O = Tr(X YZF µν Ψ(D µ Z)...) It is integrable the evolution of the composite operators with the RG scale. D The planar dilatation operator, measuring the scaling dimensions, maps to a spin chain Hamiltonian, D(λ) = D 0 integrable - thus solvable by means of a Bethe Ansatz. l 1 λ l H (l) integrable

5 Solvability Spectral problem: diagonalization of a suitable Hamiltonian. Easier with S-matrices (scattering of the elementary excitations on the lattice hidden inside the trace operators!) constrained by the symmetry. All-loop formulation: psu(2,2 4) S-matrix [Beisert, Staudacher 05] CAVEAT: wrapping! Bethe equations are asymptotic! Correct, for length L operators, up to O(λ L ) [Kotikov, Lipatov, Rej, Staudacher, Velizhanin, 07]

6 Remarkable outcomes The large spin limit Spectacular interpolation weak/strong coupling: the scaling function/cusp anomaly QCD: logarithmic scaling in leading twist operators at large spin S (x 1) γ(s) = 2 Γ cusp (α) log S O(S 0 ) cusp governs the renormalization of a Wilson loops evaluated over a closed contour with a cusp. SYM: Twist two operators in sl(2) psu(2,2 4) via Bethe Ansatz γ(s) = f(λ) log S O(S 0 ) At weak coupling f(λ) = λ 2π 2 λ2 96π 2 O S = q(γ D ) S q O S = Tr(ϕ D S ϕ) Agreement with MHV 4-point gluon amplitudes of N=4 SYM at four loops! [Bern, Czakon, Dixon, Kosower, Smirnov, 06] At strong coupling f(λ) = numerically analytically λ 2π 3 log 2 2π [Benvenuti et al 07, Alday et al 07, Kostov et al 07, Beccaria VF 0703] [Basso, Korchemsky 07] STRINGS: Agreement with two loops string calculations! [Roiban, Tseytlin 07]

7 Remarkable byproducts Wilson loops vs gluon scattering amplitudes at weak and strong coupling [Alday, Maldacena 07] Generalised scaling function f(λ, J) [Korchemsky et al. 07, Bern et al. 08,...] [Freyult Rej Staudacher 07, Roiban Tseytlin 07, Gromov 08, Fioravanti et al. 1& 2 08, Beccaria 08] and natural outlook! How this triality survives in the large spin expansion beyond the leading order? Sectors other than sl(2)? Logarithmic scaling also at higher twist? Strings?

8 N=4 SYM and QCD interplay: I QCD-inspired closed formulas Crucial to explore the different regimes x 0, x 1. Bethe eqs: highly non linear, complicated. Appropriate Ansatz for coefficients needed! [ e.g. Operators with variable length Tr F L ] [Beccaria Forini, 0710] Maximum transcendentality principle, KLOV [Kotikov, Lipatov, Onishchenko, Velizhanin, 04] The N=4 universal twist two anomalous dimension at n loops is a linear combination of harmonic sums of transcendentality. 2n 1 3-loop γ(s) uni extracted from the most transcendental terms of the QCD result. [Moch, Vermaresen, Vogt, 04] Applied to the numbers coming from the Bethe eqs it works! γ (1) (S) = c τ S τ (S) = c S 1 (S) (Nested) harmonic sums τ =1 γ (2) (S) = τ =3 γ (3) (S) = τ =5 c τ S τ (S) c τ S τ (S) S a (S) = S a,b (S) = Recent development and confirmation of max transc: analytic solutions to the Baxter equation for operator of twist two and three! S n=1 S n=1 (sign(a)) n n a (sign(a)) n S b (n) n a [Kotikov, Rej, Zieme, 08]

9 Generalising KLOV: Twist-3 sl(2), three scalars Four-loop closed formula for O S = TrD(ϕ) S 3 [Beccaria 07] γ(s) in terms of S a1...a n (S/2) [Kotikov et al 07] confirmed by [Kotikov, Rej, Zieme, 08] Twist-3 sl(2 1), three gauginos O S = TrD(λ) S 3 [Beccaria 07] Theorem γ λλλ (S) = γ twist 2 (S 2) The most complete (and complicated) test of universality is the case of three gauge fields, closed only at one loop representing the full psu(2,2 4). O S = TrD S (A) 3 [Beccaria 07] [Beccaria, Forini 08] Identify the superconformal primary state to which apply the Bethe eqs. Generalization of the KLOV principle γ n = 2 n 1 τ=0 γ (τ) n, γ (τ) n = kl=τ H τ,l (n) (n 1) k γ 1 = 4 S 1 2 n 1 4, γ 2 = 2 S 3 4 S 1 S 2 2 S 2 n 1 2 S 1 (n 1) 2 2 (n 1) 3 4 S 2 Harmonic sums evaluated in S/21. 2 (n 1) 2 8,

10 N=4 SYM and QCD interplay: II x 0, negative spin S (BFKL equation) The (analitically continued) anomalous dimension for twist two sl(2) operators maximally violates, at four loops, the BFKL prediction for the (Regge) poles Breakdown of the Bethe eqs. at four loops [Kotikov, Lipatov, Rej, Staudacher, Velizhanin, 07] N=4 SYM and QCD interplay: III x 1, large spin S (quasi elastic limit) Leading logarithmic behavior: universal in twist and flavors for all gauge theories! γ(s) = 2 Γ cusp (α) log S O(S 0 ) [Belitsky, Gorsky, Korchemsky, 06] Subleading logarithmic behavior: much less clear! MVV relations for twist two operators [Moch, Vermaresen, Vogt, 04] γ σ (S) = A log S B C σ log S S C σ = 1 2 σ A2 Highly non trivial constraints, since A, B, C are functions of the coupling. Structural explanation: revisiting the parton evolution in the x 1 regime

11 Anomalous dimensions vs. splitting functions df σ (x, Q 2 ) d log Q 2 = 1 x f σ DGLAP evolution equations for, parton distribution function (DIS) or fragmentation functions (ee- annihilation into hadrons) dz z P σ(z, α s )f σ ( x z, Q2) df σ (S, Q 2 ) d log Q 2 = γσ(s)f σ (S, Q 2 ) γ σ (S) = dx x S 1 P σ (x) 0 The Mellin transforms of the space or time-like ( σ = 1) splitting function P σ are anomalous dimensions in the space-like case, those of twist-2 operators. 1 Proposal by Gribov-Lipatov: the two splitting functions coincide and ( ) 1 P (x) = x P GL reciprocity x Broken in QCD beyond 1-loop [Gribov, Lipatov, 72] [Curci, Furmansky, Petronzio, 80] Proposal by Marchesini-Dokshitzer: Reciprocity Respecting Evolution Equations df σ (x, Q 2 ) d log Q 2 = P with a universal kernel. 1 x dz z P(z, α s)f σ ( x z, zσ Q 2) RREE [Marchesini, Dokshitzer, Salam, 05]

12 RREE are solved by the non-linear relation ( γ σ = P S 1 ) 2 σγ σ(s) If the kernel satisfies the GL reciprocity ( ) 1 P(x) = x P or P(S) = f(s(s 1)) x MVV relations for subleading terms are satisfied! Verified at three loops for non singlet twist two [Mitov, Moch, Vogt 06] Generalized Reciprocity in N=4 SYM f γ = f ( S 1 2 γ) [Basso, Korchemsky 06] The function defined by is a reciprocity respecting (RR) function if its large spin expansion takes the form f ( S ) = n a n (ln C) C 2n C 2 = (S J l)(s J l 1) O = Tr{D k 1 X...D k J X} k 1... k J = S J : twist l: ϕ λ A

13 Checks of Reciprocity Step 1: solve for f(s) = k=1 ( 1 2 S ) k 1 [γ(s)] k = γ 1 4 (γ2 ) 1 24 (γ3 )... As the anomalous dimension, f will be a perturbative series (weak, or strong!, coupling) Step 2: Expand in large S, rewrite in terms of C and check the parity invariance wrt (C 2 ) -1 Ex. twist two f (1) = 8 ln C C C 4... f (2) = 8 3 π2 ln C 24 ζ 3 (8 4π2 9 ) 1 C 2 (4 4π2 45 ) 1 C 4... All γ S for twist-2 in QCD [Basso, Korchemsky 06] Universal twist-2 in N=4 SYM [Basso, Korchemsky 06] Twist-3 scalars, gauginos, gluons (in closed form!) in N=4 SYM [Beccaria, Marchesini, Dokshitzer 07] [Beccaria 07] [Beccaria Forini 08] It is an (empirical) evidence, but what is its origin?

14 (Partial) origin: conformal symmetry Operators as O = Tr{D κ 1 X...D κ J X} can be classified according to representations of the collinear SL(2; R) subgroup of the full SO(2, 4) conformal group. [Ohrndorf 82] Conformal invariance: different SL(2; R) multiplets cannot mix under renormalization and their anomalous dimension depends on the conformal SL(2; R) spin s s = (S ) 2 The scaling dimension receives anomalous contribution due to interaction S = Lorentz spin = scaling dimension (λ) = S J γ λ (S) the conformal spin gets modified in higher loops (e.g. scalars) s(λ = 0) = S J 2 s(λ) = S J γ(s) The anomalous dimension is actually a (twist-dep) function of the conformal spin γ(s) = f (S 12 ) γ(s) But: Do we have an explanation for the parity invarince of the f? NO So what! Just a change of variable. NOT REALLY...

15 (Physical) assumption: f is simpler In all known cases (also at strong coupling..) the expansion for f shows a remarkable reduction of singularities! In particular, it does not contain (ln S/S) p terms - sharp constrast with γ(s) With this assumption, and considering that f starts as the anomalous dimension, the leading (ln S/S) p power corrections can be resummed into γ(s) = f ln ( S 1 2 f ln S)... = f ln S f 2 The leading logarithms (ln S/S) p are governed at all loops by the cusp anomalous dimension! 2 ln S S f 3 8 ln 2 S S 2 f 4 24 they should be universal in twist (and flavor) ln 3 S S Verified up to three loops for all the known anomalous dimensions! (twist two and three, all flavors)

16 Twist 2 scalar γ ϕ L=2 (ˆλ) = ) ] [8ˆλ 8π2 3 ˆλ 2 88π4 45 ˆλ 3 ( 584π ζ2 3 ˆλ 4 ln S ( ) ( 16 24ζ 3ˆλ2 3 π2 ζ 3 160ζ 5 ˆλ π4 ζ 3 80 ) 3 π2 ζ ζ 7 ˆλ 4 ] [32ˆλ 2 64 π2 3 ˆλ 3 96 π4 5 ˆλ ln S 4 S ( ) [4ˆλ 4π2 3 ˆλ 44π ζ 3 ˆλ 3 ( 292π ) ] 1 3 π2 ζ 3 32ζ ζ 5 ˆλ 4 S [ 64ˆλ 3 (64π 2 128ζ 3 )ˆλ 4] ln 2 S [ ) S 2 ] 16ˆλ 2 16 π2 (128 ˆλ 3 ( 128π 2 32 π4 ln ζ S 3)ˆλ 4 S 2 [ 2 ) ) 3 ˆλ (24 2π2 ˆλ 2 ( 32π2 22π ζ 3 ˆλ 3 ( 136π 4 146π ζ 3 32 ) ] 3 π2 ζ 3 16ζ ζ 5 ˆλ 4 3 S 2 [ ] ˆλ ln 3 S 4 S 3 [ ) ] 64 ˆλ 3 64 π2 ln 2 S ( ζ 3 ˆλ 4 3 S 3 [ ) 16 ( 256 (512 16π2 512π2 3 ˆλ 2 [ 56 3 ˆλ 2 9 (96 40π2 9 ˆλ 3 3 ) 16ζ 3 ˆλ 3 ( 224π2 3 ) ] 64π4 ln ζ S 3 ˆλ 4 S 3 32π ζ π2 ζ 3 320ζ 5 3 ) ˆλ 4 ] 1 S 3 Where S = e γ E λ S and ˆλ = 16π 2

17 Twist 3 scalar γ ϕ L=3 = ) ] ( ) [8ˆλ 8π2 3 ˆλ 2 88π4 45 ˆλ 3 ( 584π ζ2 3 ˆλ 4 ln S 8 ln 2ˆλ 3 π2 ln 2 8ζ 3 ˆλ 2 ( π4 ln 2 8 ) 3 π2 ζ 3 8ζ 5 ˆλ 3 8 ( 73π 6 ln 2 84π 4 ζ ln 2ζ π 2 ) ζ ζ 7 ˆλ4 315 ] [ ) ( [32ˆλ 2 64 π2 3 ˆλ 3 96 π4 5 ˆλ ln S 4 S 8ˆλ ( 8π2 88π 32 ln 2 ˆλ ) 3 π2 ln 2 32ζ 3 ˆλ 8 ( 73π 6 756π 4 ln 2 840π 2 ζ ζ 2 ) ζ 5 ˆλ4] S [ 64ˆλ 3 64π 2ˆλ 4] ln 2 S [ S 2 32ˆλ 2 (128 64π2 256ζ 3 ) ˆλ 4] ln S S 2 64 ln ζ 3 ) ˆλ3 ) 128 ln 2 ˆλ 3 ( π 2 96π4 3 5 [ 8λ3 ) (48 8π2 32 ln 2 ˆλ 2 (32 80π2 9 3 ( π π π4 ln 2 64π 2 ln ζ π2 ζ ln 2ζ 3 64ζ2 3 [ 512 [ ] 3 ˆλ ln 3 S 4 S 3 128ˆλ 3 ( π ln 2 ) ] ln 2 S ˆλ4 [ ) S ˆλ 2 ( π2 256 ln 2 ˆλ 3 ( π π 2 ln 2 88π π ln 2 128π2 ln 2 ) ] 1 32ζ 5 ˆλ 4 3 S 2 64π ln ln π2 ln 256π 2 ln ln 2 ) ] ln 2 512ζ S 3 ˆλ4 [( S ) 64 ln 2 ˆλ 2 ( π2 512 ln π2 ln ln ζ ) 3 ˆλ 3 3 ( π2 512π4 768 ln π2 ln π4 ln ln π 2 ln ln ζ π2 ζ ln 2ζ 3 64ζ ) ] 5 1 ˆλ 4 3 S 3

18 Twist 3 gauge sector γ A (S) = ) ] [8ˆλ 8π2 3 ˆλ 2 88π4 45 ˆλ 3 ( 584π ζ2 3 ˆλ 4 ln S c A ] [32ˆλ 2 64 π2 3 ˆλ 3 96 π4 5 ˆλ ln S 4 S 1 LA 10 S [ 64ˆλ 3 64π 2ˆλ 4] ln 2 S S 2 ln S LA 21 S 2 LA 20 [ ] ˆλ ln 3 S 4 S 3 ln 2 S LA 32 S 3 L A 31 1 S 2 ln S S 3 LA 30 1 S 3 Also twist 2 gauginos and gauge, and twist 3 gauginos! Their closed formulas are obtained by the twist two scalar case by just shifting the argument of the harmonic sums and shifts do not affect (ln S/S) p coefficients

19 What we learn at weak coupling about the large Lorentz spin expansion of anomalous dimensions Structure (for twist J<3) γ(s) S 1 = f ln S f c f 11 ln S f 10 S f 22 ln 2 S f 21 ln S f 20 S 2 ( ln 3 S ) O S 3 Functional/inheritance relation (f simpler): all-orders description for the leading logs! The function f derived by the anomalous dimension, assumed to be simpler, implies alle leading logs in terms of the cusp anomaly f ( S ) = n γ = f ( S 1 2 γ) Reciprocity, or parity invariance (for minimal anomalous dimensions!) a n (ln C) C 2n C 2 = (S J l)(s J l 1) For twist-j higher than 2, anomalous dimensions, as functions of S, occupy a band! Our discussion was restricted to the minimal energy, lower edge of the band. Such (empirical) evidence at strong coupling?

20 Reciprocity and AdS/CFT? Anomalous dimensions of operators in N=4 SYM Energies of semiclassical strings in AdS 5 S 5 Classical sols of the AdS 5 S 5 σ-model: states belonging to reprs of SO(2, 4) x SO(6). Classified by 6 charges (E, S 1, S 2 ; J 1, J 2, J 3 ). Operators with large Lorentz spin and minimal energy correspond to folded strings rotating in AdS : 3 the classical result for the energy reproduces the logarithmic behavior! λ E = S [Gubser, Klebanov, Polyakov 02] π log S... S λ λ i.e. the behavior of the (classical anomalous) dimension of twist operators with minimal energy. Confirmed at one loop! [Frolov, Tseytlin 02] The energy organizes in the semiclassical expansion E = [ λ E 0 E 1 E ] 2 λ ( λ)... 2 and its relation to the anomalous dimension is (supposed to be ) γ(s) = E S L

21 Folded string in AdS 3: I The S 5 momentum J of the string state can be ignored theory operator small compared to Lorentz spin. twist of the gauge Folded string: closed string folded on itself to form a segment line rotating in AdS Long rigid QCD-like rod! Large spin limit: the cusps approach the boundary of AdS, responsible for logs Ansatz for a stationary solution rotated and boosted ρ = ρ(σ) t = κ τ φ = ω τ where ρ(σ) varies from 0 to ρ 0 (~string length!) defined via coth 2 ρ 0 = 1 η Exact solution sinh ρ = 1 sn [ κ η σ, 1 ] π, 0 σ η η 2 To construct the fully periodic solution: 4 such functions glue together (1-fold).

22 Folded string in AdS 3 : II Integrals of motion: energy and spin E = P t = λ κ 2π In parametric form 0 dσ 2π cosh2 ρ λ E S = P φ = 2π λ ω 0 E = 2 ( π η E 1 ) η S = 2 π η [ ( E 1 ) η K dσ 2π sinh2 ρ λ S ( 1 )] η In the long string limit η 0 Same structure! Leading logs! E = S ln S 1 E S = ln S 1 2 π 2 S 2 ln2 S 9 ln S 5 16 π 3 S 2... S 8 π S π λ [S π ln 1 2 λ π ln S... ]... Reciprocity! f(s) the function f runs in even negative powers of the Casimir f = 1 [ ln λ π S 1 ln S 1 16π 2 S 2 O ( ) 1 ] S 4 O( 1 ) λ C S

23 Folded string in AdS 3 x S 1 Integrals of motion: E = λ E and two angular momenta S = λ S and J = λ J We are interested in the limit S J Slow long strings Leading logs! J ln S Same structure! E S J 1 π (ln S 1) π J 2 2 ln S π3 J 4 4 S [ 1 π (ln S 1) π J 2 2 ln 2 3π3 J 4 S 4 ln 4 S 8 ln 3 S ( 1 ) 1 ln S... ) ] ( ln S Reciprocity! f runs in even negative powers of the Casimir C S 1 2 J It can be seen systematically by using an integral representation for the (inverse) functional relation and using parametric expressions for γ and S 1 s f(c) (η) = dη γ(η) 2π i Γ s(η) C, s(η) = S(η) 1 2 γ(η) The anomalous dimension is a series in even powers of η, and s in odd powers. f(c), which is γ evaluated at the pole, will be in even (negative) powers of C. Fast long strings ln S J S : again reciprocity respecting.

24 Spiky strings vs. higher twist on excited trajectories Rigidly rotating, n cusps or spikes. Same logarithmic asymptotics of anomalous dimensions BUT proportional to the number of spikes, n >2 higher energy E S = n 2 π ln 16 π S n... [Kruczenski. 04] Proper correspondent to twist J operators with NON MINIMAL anomalous dimensions Beyond the leading large spin limit: the ends of the spikes do not approach the bndry Leading logs: YES! E S = n 2π ln S n2 8 π 2 S ln S n3 64 π 3 S 2 ln2 S... Reciprocity: NO! n [ f(s) = ln 2π S q 1 q...] 2 S Exactly as it happens at weak coupling! [Belitsky, Korchemsky, Pasechnik 08]

25 String perturbation theory: I With the 1-loop corrections included: the structure of the large spin expansion remains the same? are the MVV constraints still satisfied? procedure for leading (1 loop) Gauge and string perturbative expansions are different! Gauge theory String theory λ 1 S = fixed and then S 1 λ 1 S = λ fixed and then S λ 1 Quantization of folded string: doable via semiclassical considerations and in the special limits of short and long string! Standard procedure for leading (1 loop) quantum corrections: [Frolov, Tseytlin 02] Fluctuation lagrangean L Euclidean E 1 = Γ 1 κt, 2-d Effective action Γ 1 T dτ

26 String perturbation theory II: Details Fluctuation Lagrangeans L B = a t a t µ 2 t t 2 a φ a φ µ 2 φ φ2 4 ρ(κ sinh ρ 0 t w cosh ρ 0 φ) a ρ a ρ µ 2 ρ ρ 2 a β u a β u µ 2 ββ 2 u a ϕ a ϕ a ζ s a ζ s t, ρ, φ β u ϕ, ζ s 3 flucts in AdS 3 2 flucts transverse to AdS 3 5 flucts in S 5 L F = 2i( Ψγ a a Ψ µ F ΨΓ234 Ψ) 44 2d Majorana fermions Coefficients, masses are sigma dependent in a non trivial way! E.g. µ 2 β = ρ 2 Stationary solution: determinants are only 1-dimensional dω det[ m 2 ] = T 2π det[ 2 1 ω 2 m 2 ] Small η (large S) expansion LB = L 0 η L 1 achieved expanding the solution and the parameters, in particular κ = κ 0 η κ 1 where κ 0 ln η

27 String perturbation theory III: Details 1-loop correction to the effective action [ Γ 1 = dω Q ω T 4π ln 8 ln det[ 2 1 ω 2 ρ 2 ] det[ 1 2 ω2 κ 2 0 ] 2 ln det[ 2 1 ω 2 2ρ 2 ] det[ 1 2 ω2 2κ 2 0 ] det 8 [ 2 1 ω 2 κ 2 0] det 2 [ 2 1 ω2 2κ 2 0 ] det6 [ 2 1 ω2 ] ln det Q ω det Q (0) ω : quadratic fluctuation operator Q ω = Q (0) ω η Q (1) ω... ln det P ω det Q (0) ω ] We calculate the correction O(η) Γ 1 = Γ (0) 1 η Γ (1) 1 O(η 2 ) Order zero contribution (constant mass relativistic fields on the cilinder!) Γ (0) 1 = 1 [ ] 2 n 2 2 2κ 20 n 2 4κ n 2 8 n 2 κ 2 0 n= [Frolov, Tseytlin 02] Order eta contribution Γ (1) 1 = T η 4π n= [ 8κ 0 n2 κ 2 0 4κ 0 n2 2κ 2 0 4κ ] 0 n2 4κ 2 0

28 String perturbation theory: IV Final result The 1-loop correction to the energy up to order 1/S is E 1 = b 0 ln S b c b 11 ln S b 10 S O ( ln 2 S S 2 ) b 0 = 3 ln 2 π, b c = 3 ln 2 π ln 8π, b 11 = 3 ln 2 π 2, b 10 = 3 ln 2 ( 1) ln 8π π 2 2 The structure is identical to the one at weak coupling! Despite the different order of limits... MVV-like relations (reciprocity) satisfied! b 11 = a 0 b 0, b 10 = 1 2 (a 0b c b 0 a c ) Strong indication that reciprocity holds also at strong coupling

29 Conclusions CFT/ Long range Bethe equations multiloop anomalous dimensions. Closed formulas QCD-inspired Physical properties: BFKL singularities, reciprocity. Why? /AdS Anomalous dimensions at strong coupling Reciprocity in AdS. Why? QCD Complete solution of N =4 SYM (and thus string theory!) should provide a oneline-all-orders description of the major part of QCD parton dynamics. [Dokshitzer, DIS 2007]

30 Outlook 1, Strings Further orders in 1/S, generalization to (S,J) Outlook 1, N=4 and QCD The 4-loop anomalous dimension for twist two operators at generic spin is unknown, γ (4) [ Konishi anomalous dimension [Janik, 08] Kon = ζ ζ 5 ] [Zanon et al. 07, 08] but known to be a combination of harmonic sums with max transcendentality 7. Reciprocity (and BFKL) constrain the form of these coefficients. Prediction that for the first time might go from N=4 SYM to QCD! Outlook 3: Universalities at the level of the amplitudes in N=4 SYM The first two coefficients are connected with singularities in the amplitudes. What about subleading? [Bern, Czakon, Dixon, Kosower, Smirnov, 06] [Dixon, Magnea, Sterman 08]