Gluon scattering amplitudes and two-dimensional integrable systems

Transcription

1 Gluon scattering amplitudes and two-dimensional integrable systems Yuji Satoh (University of Tsukuba) Based on Y. Hatsuda (DESY), K. Ito (TIT), K. Sakai (YITP) and Y.S. JHEP 004(200)08; 009(200)064; 04(20)00 ; Y. Hatsuda (DESY), K. Ito (TIT) and Y.S. JHEP 202(202)003; arxiv:2.6225, to appear in JHEP

2 . Introduction gauge/string duality AdS/CFT correspondence string theory in 4D maximal AdS space super Yang-Mills dual strong/weak (λ: coupling) [AdS: anti-de Sitter] [CFT: conformal field theory] discovery of integrability opened up new dimensions

3 gauge/string duality beyond susy sectors quantitative analysis of gauge theory dynamics at strong coupling

4 gauge/string duality beyond susy sectors quantitative analysis of gauge theory dynamics at strong coupling stimulating study of SYM insights into applications.

5 In fact, spectrum of planar AdS/CFT for arbitrary coupling [Gromov-Kazakov-Vieira 09, Bombardelli-Fioravanti-Tateo 09, Arutyunov-Frolov 09,...] given by thermodynamic Bethe ansatz (TBA) eqs. ( finite size effects of 2D integrable models ) checked up to 5/6-loops of SYM [Bajnok-Hegedus-Janik-Lukowski 09,... ] ( 4 loops Feynman diagrams )

6 In fact, spectrum of planar AdS/CFT for arbitrary coupling Also, [Gromov-Kazakov-Vieira 09, Bombardelli-Fioravanti-Tateo 09, Arutyunov-Frolov 09,...] given by thermodynamic Bethe ansatz (TBA) eqs. ( finite size effects of 2D integrable models ) checked up to 5/6-loops of SYM other aspects? [Bajnok-Hegedus-Janik-Lukowski 09,... ] ( 4 loops Feynman diagrams ) gluon scattering amplitude/wilson loops [this talk] correlation fn. quark - anti-quark potential/cusp anomalous dim..

7 gluon scatt. amplitudes at strong coupling minimal surfaces in AdS5 x S5 thermodynamic Bethe ansatz (TBA) equations [ AdS/CFT ] [ integrability ]

8 In this talk, discuss maximally helicity violating (MHV) amplitude/ Wilson loops of 4D maximal ( N =4 ) SYM at strong coupling underlying 2D integrable models and CFTs derive analytic expansions around certain kinematic points [ regular polygonal Wilson loops ]

9 Plan of talk. Introduction 2. Gluon scattering amplitudes at strong coupling 3. Minimal surfaces in AdS and integrability 4. Analytic expansion of amplitudes at strong coupling 5. Summary [ amplitude min. surface ] [ min. surface Hitchin system, TBA ] [ AdS3 case, AdS4 case ]

10 k4 2. Gluon scattering amplitudes at strong coupling x4 amplitudes of SYM at strong coupling x x3 x2 k2 N =4 M = AdS/CFT [Alday-Maldacena 07] minimal surfaces in AdS M : scalar part of MHV amplitudes λ : t Hooft coupling null boundary at AdS boundary x µ i+ x µ i =2 kµ i n-pt. amplitude e p 2 (Area) (momentum of particle) n-cusp min. surface

11 4-cusp solution (4-pt. amplitude) AdS5 is parametrized in R 2,4 as Y Y := Y 2 Y Y 2 + Y Y Y 2 4 = string e.o.m. eq. of min. surface Y ( Y Y ) Y =0 ( Y ) 2 =( Y ) 2 =0 ( = z, = z ) simple solution AdS3 Y + Y 4 Y + Y 0 Y Y 0 Y Y 4 Y 2 = Y 3 = 0, z = + i = p 2 e + e e + e

12 Poincare coordinates Y µ = xµ r (µ =0,, 2, 3) Y + Y 4 = r, Y Y 4 = r2 + x µ x µ r ) ds 2 = r 2 dr2 + dx µ dx µ [ r 0 : AdS boundary] boundary coord. of AdS3 : x ± = Y 0 ± Y Y + Y 4 2 Z 3 4

13 regularizing area, e.g., by 4! 4 2 M 4 apple M div 4 exp 8 f( ) ln s t 2 + const. f( )= p / [cusp anom. dim] s, t : Mandelstam variables precisely matches BDS conjecture [all orders in perturbation] f( ) : exactly same as in spectral problem [Bern-Dixon-Smirnov 05]

14 Insights into SYM amplitudes min. surface Wilson loop Remainder fn. : deviation from BDS formula dual conformal symm. remainder fn. = fn. of cross-ratios of cusp. coord. x µ a

15 Insights into SYM amplitudes min. surface Wilson loop Remainder fn. : deviation from BDS formula dual conformal symm. remainder fn. = fn. of cross-ratios of cusp. coord. x µ a amplitude at strong coupling geometrical object

16 cf. 2-loop 6-pt. remainder fn. [Del Duca-Duhr-Smirnov 09] R (2) 6,W L (u,u 2,u 3 )= (H.) ( ) u 2 24 π2 G, u u + u 2 ; + ( ) 24 π2 G, ; + ( ) u u + u 2 24 π2 G, ; + ( ) u u + u 3 u 3 24 π2 G, u 2 u 2 + u 3 ; + ( ) 24 π2 G, ; + ( ) u 2 u + u 2 24 π2 G, ; + ( ) u 2 u 2 + u 3 u 24 π2 G, u 3 u + u 3 ; + ( ) 24 π2 G, ; + ( ) u 3 u + u 3 24 π2 G, ; + ( u 3 u 2 + u G 0, 0, ), ; + 32 ( u u + u G 0, 0, ), ; + 32 ( 2 u u + u G 0, 0, ), ; + ( 3 u 2 u + u G 0, 0, ), ; + 32 ( u 2 u 2 + u G 0, 0, ), ; + 32 ( 3 u 3 u + u G 0, 0, ), ; ( 3 u 3 u 2 + u 3 2 G 0,, 0, ) ( ; + G 0, ), 0, ; 2 ( u u 2 u u + u G 0,, 0, ) ; + ( 2 u u 3 G 0, ), 0, ; 2 ( u u + u G 0,, ), ; 2 ( 3 u u u + u G 0,, ), ; ( 2 u u u + u 3 2 G 0,, ), ; 2 ( u u 2 u + u G 0,, ), ; 2 ( 2 u u 3 u + u G 0,, 0, ) ; + ( 3 u 2 u G 0, ), 0, ; 2 ( u 2 u + u G 0,, 0, ) ( ; + G 0, ), 0, ; ( 2 u 2 u 3 u 2 u 2 + u 3 2 G 0,, ), ; 2 ( u 2 u u + u G 0,, ), ; 2 ( 2 u 2 u 2 u + u G 0,, ), ; ( 2 u 2 u 2 u 2 + u 3 2 G 0,, ), ; + 4 ( ) u 2 u 3 u 2 + u G u 2 0, 3 u + u 2, 0, ; + ( ) u 4 G u 2 0, u + u 2,, 0; 4 ( ) u G u 2 0, u + u 2,, ; + ( ) u 4 G u 2 0, u + u 2,, ; 4 ( ) u u G u 2 0, u + u 2, u 2 u + u 2, ; ( u 2 G 0,, 0, ) ; 2 ( u 3 u G 0,, 0, ) ( ; + G 0, ), 0, ; + ( u 3 u 2 u 3 u + u 3 G 0, ), 0, ; 2 ( u 3 u 2 + u G 0,, ), ; 2 ( 3 u 3 u u + u G 0,, ), ; ( 3 u 3 u 2 u 2 + u 3 2 G 0,, ), ; 2 ( u 3 u 3 u + u G 0,, ), ; + ( 3 ) u 3 u 3 u 2 + u 3 4 G u 0, u + u 3, 0, ; + 4 ( ) u G u 0, 3 u + u 3,, 0; ( ) u 3 4 G u 0, u + u 3,, ; + 4 ( ) u G u 0, 3 u + u 3,, ; ( ) u 3 u 3 4 G u 0, u + u 3, u u + u 3, ; + 4 ( ) u G u 3 0, 3 u 2 + u 3, 0, ; + ( ) u 2 4 G u 3 0, u 2 + u 3,, 0; 4 ( ) u G u 3 0, 2 u 2 + u 3,, ; + ( ) u 2 u 3 G 0,,, ; ( ) u 3 u 3 G 0,,, ; [ heroic effort ] continues 7 pages...

17 w/ a little more heroic effort... [Goncharov-Spradlin-Vergu-Voloich 0] R 2-loop 6 = 3 i= L 4 (x + i,x i ) 2 Li 4( /u i ) 2 Li 2 ( /u i ) J J amplitude at strong coupling sum up all-orders

18 3. Minimal surfaces and integrability difficult to construct min. surfaces w/ null bound. for n 5 [cf. special 6-cusp sol., Sakai-Satoh 09] but possible to obtain A(area) w/o explicit solutions [Alday-Maldacena 09] string e.o.m. Hitchin system patching 4-cusp sol. amplitudes TBA eq. [Alday-Gaiotto-Maldacena 09, Hatsuda-Ito-Sakai-YS 0 Alday-Maldacena-Sever-Vieira 0]

19 Let us see this for 2n-cusp min. surface in AdS3 4D external momenta in # of cusps : even in AdS3 R, [mom. conservation] 2 light-cone coord. x ± at (AdS)

20 Pohlmeyer reduction [evolution of moving frame] basis in R 2,2 AdS 3 : q =( Y, Y, Y, N) T N a = 2 e abcdy b Y c Y d, e 2 = 2 Y Y string e.o.m. evolution eq. of q : (d + U)q =0 so(4) su(2) su(2) introducing spectral parameter ζ 0= d + B( ), B z = B z = linearized eq. 2 e e p 2 p = p(z) := 2 e p e Y N

21 compatibility/flatness cond. set B z = A z + z, B z = A z + z ) D z z = D z z = 0, F z z +[ z, z] =0 [ su(2) Hitchin system ] e 2 + p(z) 2 e 2 =0 [ generalized sinh-gordon ] original solution Y Y aȧ = + Y 2 Y + Y 0 Y Y 0 Y Y 2 = ( = )M ( = i) =(, 2) ; M : certain matrix

22 2n-cusp solution change of variables dw = p pdz, ˆ = 4 ln p ˆ 2 sinh ˆ =0 [ sinh-gordon ] ˆ =0 4-cusp sol. in w-plane take p(z) =z n 2 + [polynomial] ) w z n 2 ( z ) solution w/ ˆ! 0 ( w ) : 2n-cusp sol. in z-plane Z 2 W n/2 times 3 4 [no explicit form]

23 Cross-ratios [ how to obtain physical info. w/o explicit sol. ] in each (i-th) quadrant in w-plane ( ; w) =b i ( ; w)+s i ( ; w) b i /s i : big/small sol. ew/ + w 0, e 0 (w/ + w ) can show cross-ratios are given by Stokes data x ± ij x± kl x ± ik x± jl = (s i ^ s j )(s k ^ s l ) ( ) [ =, ±i for +, ] (s i ^ s k )(s j ^ s l ) s i ^ s j := det (s i,s j ); x µ i,i+ := xµ i+ x µ i =2 kµ i shape of min. surface cross-ratios/momenta

24 Finding cross-ratios define T- and Y- fn. by T 2k+ =(s k ^ s k+ ), T 2k =(s k ^ s k ) [+] Y s = T s T s+ [ cross-ratios] where f [a] ( ):=f(e a i/2 ) identities among det (s i ^ s j )(s k ^ s l )=(s i ^ s k )(s j ^ s l )+(s i ^ s l )(s k ^ s j ) T [+] s T [ ] s = T s+ T s + Y [+] s Y [ ] s =(+Y s )( + Y s+ ) (s =,..., n-3 ) [T-system] [Y-system]

25 asymptotics for ζ 0, by WKB for Hitchin system log Y s Z s (! 0) etc. Z s = H s p pdz [ s : cycle on alg. curve] (asymptotics) + (analyticity) log Y s = m s cosh + X r K sr log( + Y r ) = e,m s =2Z s [for real Zs] K sr = I sr / cosh, I sr = s,r+ + s,r Y 0 = Y n 2 =0

26 Computing area amplitudes regularization of area (Area) =4 Z d 2 ze 2! 4 Z d 2 z (e 2 p p p)+4 Z r d 2 z p p p A free = X s Z =: A free + A reg d 2 m s cosh log( + Y s ). remainder fn. R (remainder fn.) = A A BDS = A free +

27 Summarizing so far, TBA eqs. to compute AdS3 amplitudes (2n-pt.), need to solve log Y s ( )= m s cosh + r K sr log( + Y r ) Y Y Y 4 Y 7 7 n = 7 Ys : (extended) cross-ratios of e.g. ) Y i 2 = x + 5 x+ 67 x + 56 x+ 7 x ± a, Y 0) = x 5 x 67 x 56 x 7 m s : complex param. shape of surface momenta Ksr Isr (incidence mat. of An-3)

28 Summarizing so far, TBA eqs. to compute AdS3 amplitudes (2n-pt.), need to solve log Y s ( )= m s cosh Y Y Y 4 Y 7 7 n = 7 r K sr log( + Y r ) TBA eq. of hom. sine-gordon model Ys : (extended) cross-ratios of e.g. ) Y i 2 = x + 5 x+ 67 [Hatsuda-Ito-Sakai-YS 0] x + 56 x+ 7 x ± a, Y 0) = x 5 x 67 x 56 x 7 m s : complex (mass) param. shape of surface momenta Ksr Isr (incidence mat. of An-3)

29 Remainder function [overall coupling dependence : omitted] Once Y-fn. are obtained R 2n := A [amp.] - A BDS [ n odd ] [BDS formula] = 7 2 (n 2) + A periods + A BDS + A free 7 A periods = 4 m r I rs m s c 6 = x 23 x 45 x 6 x 2 x 34 x 56 A BDS = 4 n i,j= log c+ i,j c + i,j+ log c i,j c i,j, c ± i,j = x± i+2,i+ x± i+4,i+3 x± j,i x ± i+,i x± i+3,i+2 x± j,j A free = n 3 s= d 2 m s cosh log( + Y s ( )) free energy of TBA system non-trivial part : A free, A BDS for given ms

30 4D amplitude at strong coupling 2D integrable systems 0D superstring

31 Solving TBA eqs. exact solutions { m s m s 0 : CFT limit regular polygon TBA system : solved numerically

32 Solving TBA eqs. exact solutions { m s m s 0 : CFT limit regular polygon TBA system : solved numerically [but sometimes simple iteration fails] solutions to TBA system : not fully investigated momentum dependence : ms momentum

33 Solving TBA eqs. exact solutions { m s m s 0 : CFT limit regular polygon TBA system : solved numerically [but sometimes simple iteration fails] solutions to TBA system : not fully investigated momentum dependence : ms momentum Analytic results at strong coupling except m s! 0,?? we discuss expansions around CFT lim. l = ML 0 ( m s = M s L = M s ML) [ M: mass scale, L: length scale ]

34 4. Analytic expansion at strong coupling [Hatsuda-Ito-Sakai-YS, Hatsuda-Ito-YS ] Basis of expansion TBA for AdS3 2n-pt. amplitude HSG from bsu(n 2) 2 [bu()] n 3 [w/ imaginary resonance param.] [Hatsuda-Ito-Sakai-YS 0] relation between T-fn. and g-function (boundary entropy) [Bazhanov-Lukyanov-Zamolodchikov 94; Dorey-Runkel-Tateo-Watts 99; Dorey-Lishman-Rim-Tateo 05]

35 Homogenous sine-gordon (HSG) model start w/ coset CFT bsu(n) k /[bu()] N [Fernandez Pousa-Gallas-Hollowood-Miramontes 96] [ for 2n-cusp AdS3 min. surface N = n-2, k=2 ] HSG model : integrable deformation of coset CFT S HSG = S gwznw + d 2 x Φ : comb. of weight 0 adjoint op. = = N N + k, = M 2( ) [multi-param. deformation] simplest case from bsu(2) k /bu() complex sine-gordon/minimal A k affine Toda

36 spectrum [copies of min. affine Toda] k- a k Ms a = sin M s, sin k M s = M M s a N- M M 2 s M N min. surface 2D integrable model CFT

37 Expansion of Afree : [ R 2n A free + A BDS ] free energy around CFT limit CFT perturbation for the case of 2n-cusp min. surface in AdS3 A free = 6 c n + f bulk n + c n = k=2 f (k) n (n 2)(n 5) n fn bulk = 4 m r Irs f (k) n k-pt. fn.of m s l 4k n [central charge]

38 A BDS : ( c ± ij ) can show cross-ratios c ± ij : nothing but T-fn. [ trans. matrix] c + ij = T [i+j] i j (0), c ij = T [i+j+] i j (0) where T [k] s ( ):=T s + i 2 k ) A BDS = A BDS (T s ) everything fits into language of 2D integrable model

39 T-function g-function g-fn. log G (0) Z = e RH = (boundary entropy) R integral eq. for g-fn. is known : similar to TBA eq., including boundary contributions k=0 [Dorey-Lishman-Rim-Tateo 05; Poszgay 0; Woynarovich 0] comparing this w/ TBA eq. following G (k) (l) 2 e RE k [Dorey-Runkel-Tateo-Watts 99; Dorey-Lishman-Rim-Tateo 05] [counts ground state degeneracy] L G (0) s,c. G (0) = T s i 2 C

40 boundaries reflection factors : trivial boundary R s,c > r s, C : ( R r s,c ( ) = R r ( ) Z s,c r ( ) [deforming factor] [Sasaki 93] need to find Z s,c r satisfying boundary bootstrap, unitarity, crossing symm. corresponding precisely to T s ) Z s,c r ( ) := ( + C) ( C) sr (x) := sinh 2 ( + i 2 x) sinh 2 ( i 2 x)

41 Expansion of T-function periodicity [ T-system ] T s ( )= t (p,q) s l ( )(p+q) cosh 2p n p,q=0 boundary CFT perturbation of g-fn. t (2,0) s t (0,0) s t (0,0) s = = ng( M j )B( 2, ) sin( 3(s+) 2(2 ) 2 sin( (s+) n ) sin (s+) n sin( n ), [Dorey-Runkel-Tateo-Watts 99; Dorey-Lishman-Rim-Tateo 05] n ) s sin( n ) sin( 3 n ) (z) (0) = G2 ( M s ) z 4 s sin( 3 n )! sin( n ) [ given by modular S-matrix ] t (2,0) s,t (0,4) t (2,0) s T-system ng( M s )

42 Expansion of 2n-pt. remiander function Combining all, R 2n = R (0) 2n + l 8 (4) n R 2n fn. of := t (2,0) s + O(l 2 n ) R (0) 2n = 4n (n 2)(3n 2) n 2 2 R (4) 2n = 6 C(2) n t (2,0) A n,s + t (0,0) s t (2,0) s t (0,0) s 2 + t(2,0) s t (0,0) s 2 2 t (2,0) s t (0,0) s C (2) n = 3(2 ) 2(n 4) n 2 n 2 n 2 ng 2 ( M j ) ng( M s ) n 4 4 (n 3)/2 X s= (n 3)/2 X s= sin( (s+) log 2 n ) sin( s n ) cos regular polygon (2,0) t (n 3)/2 A n,s 2 t (0,0) (n 3)/2 2 n 2 4 t(0,4) s t (0,0) s 4 n n (2,0) 2t s t(2,0) s, t (0,0) s t(0,0) s t (0,4) s t (0,0) s 2 sin n log t(0,0) s t (0,0) s (x) = (x)/ ( x)

43 To further express amplitudes by momenta need ng, n, m s [mass-coupling/relevant op. relation] invert relation btw Y-fn. (cross-ratio) and m s ) m s = m s (k a ) [k a : momenta]

44 To further express amplitudes by momenta need ng, n, m s [mass-coupling/relevant op. relation] invert relation btw Y-fn. (cross-ratio) and m s ) m s = m s (k a ) [k a : momenta] this can be done for single mass cases simpler TBA of perturbed coset [Zamolodchikov 95; Fateev 94] e.g.) M s = s, M ) su(2) su(2) n 3 su(2) n 2 = M n,n M s =( s, + s,n 3 )M [minimal models] ) su(2) su(2) n/2 2 su(2) n/2 = M n 2,n

45 amplitudes along certain trajectories in momentum space CFT/regular polygon-lim.

46 8-pt. remainder function [simplest for AdS3] A integral representation of 8-pt. remainder fn. HSG model Ising model all order expansion in l 2 [Alday-Maldacena 09] R 8 = k=0 R (2k) 8 ( )l 2k R (0) 8 ( ) =5 4 R (2) 8 ( ) = 8 log log 2 6 R (4) log 2 8 ( ) = [ l = ML, m = le i ] + 2 log (3) 92 3 cos 4 [Hatsuda-Ito-Sakai-YS 0]

47 0 pt. remainder function For 0-pt., these completely fix ng, m s R 0 = R (0) 0 + R(4) 0 l8/5 + O(l 2/5 ) R (0) 0 = log2 2 cos 5 R (4) 0 = 5 tan 5 + C t (2,0) = C 2 ( M 4/5 + C = 20 cos t (2,0) 2 4/5 2/5 M 2 C 3 M [ l = ML, m s = M s l ] complex regular polygon M 2/5 2 ) 5 /2 log 2 cos 5 C 2 = /5 0 cos 4 6 /5 5 3 /5 C 3 =2 2 /4 4/5 3/5 4/5 /2

48 M - =M- 2 = 2M - =M- 2 = 0M - =M- 2 = R CFT lim. l ( = /20, 2 = /5 ) l- dependence of 0-pt. remainder function [ m s = M s le i s ] dashed lines : R (0) 0 + R(4) 0 l8/5 good agreement w/ numerics

49 comparison w/ analytic results at two loops [Heslop-Khoze 0] rescaled remainder fn. : close for all 2n-pt. amplitudes

50 5. AdS4 minimal surfaces and W minimal models [ Hatsuda-Ito-YS, to appear ] Amplitudes at strong coupling for momenta R,2 min. surface in AdS4 Y-system for AdS4 m-cusp sol. [Alday-Maldacena-Sever-Vieira 0] Y 2,s Y 2,s + = ( + Y 2,s+)( + Y 2,s ) Y,s Y 3,s ( + Y,s )( + Y 3,s ) Y 3,s Y,s + = ( + Y 3,s+)( + Y,s Y 2,s +Y 2,s ) Y,s Y 3,s + = ( + Y,s+)( + Y 3,s Y 2,s +Y 2,s ) Y,s = Y 3,s Y ± a,s ( )=Y a,s ( ± m - 5 i), s =,...,m 5

51 TBA for AdS4 m-pt. amplitude HSG from bsu(m 4) 4 [bu()] m 5 simple iteration for numerics near CFT lim. does not seem to work, generally analytic expansion near CFT lim. : possible [cf. Castro Alvaredo-Fring 04]

52 single mass cases AdS3 case: su(2) cosets n-3 AdS4 case: su(4) cosets in particular, 2 3 m - 5 M s = s, M ) su(4) su(4) m 5 su(4) m 4 = WA (m,m) 3 M s =( s, + s,m 5 )M [W-minimal models] su(4) su(4) m/2 5 su(4) m/2 4 ) = WA (m 2,m) 3

53 6-pt. remainder function [simplest for AdS4/5] [cf. Alday-Gaiotto-Maldacena 09] HSG model Z4 parafermion CFT deformed by one relevant op. [no mixing] expansion near CFT limit agrees w/ partially numerical results in [Hatsuda-Ito-Sakai-YS ] rescaled remainder fn. : close to 2-loop result

54 7-pt. remainder function similarly to 0-pt. for AdS3, mass-coupling relation is completely determined R 7 = R (0) 7 + R (4) 7 l 6/7 + O(l 24/7 ) R (0) 7 = R (4) 7 = C + C2 2 G 2 applelog 2 2 cos 7 2 cos Li 2 2 cos [ l = ML, m s = M s l ] regular polygon G = M 8/7 + M 8/7 2 + C 3 M 4/7 M 4/7 2 C = C 2 = 2(2 ) 2/7 2 (3/7) (/7) B(/7, 3/7) 2(2 ) /7 sin 3 7 sin 3 7 s sin 4 sin 5 4 s sin 5 4 sin 4! C 3 = /4 apple 4 6 /2 apple ( 7 8 ) ( 3 4 ) ( 5 8 ) 8/7 2

55 CFT lim l = 40, 2 = 20 l- dependence of 7-pt. remainder function dashed lines : R (0) 7 + R (4) 7 l6/7 good agreement w/ numerics (points) [ m s = M s e i s l ]

56 AdS5 case [Alday-Maldacena-Sever-Vieira 0] Y-system for AdS5 m-cusp sol. [ momenta R,3 ] Y 2,s Y 2,s + = ( + Y 2,s+)( + Y 2,s ) Y,s Y 3,s ( + Y,s )( + Y 3,s ) Y 3,s Y,s + = ( + Y 3,s+)( + Y,s Y 2,s +Y 2,s ) Y,s Y 3,s + = ( + Y,s+)( + Y 3,s Y 2,s +Y 2,s ) 2 3 m - 5 Y,s 6= Y 3,s Y couples to on l.h.s.,s Y 3,s [ non-standard Y-system] underlying integrable model: not identified yet

57 7. Summary N =4 Gluon scatt. amplitudes of SYM at strong coupling minimal surfaces in AdS [ AdS/CFT ] TBA equations [ integrability ]

58 7. Summary N =4 Gluon scatt. amplitudes of SYM at strong coupling minimal surfaces in AdS TBA equations [ AdS/CFT ] [ integrability ] Underlying 2D integrable (HSG) model analytic expansions of amplitudes/wilson loops around CFT lim. [regular polygon] A free bulk CFT perturbation A BDS T-function T-/Y-fn. g-fn. boundary CFT perturbation

59 Why minimal surface TBA? [cf. ODE/IM] T-function g-function (boundary entropy)? Why strong coupling 2 loops : close [ full structure of amplitudes ] General AdS5 case? Strong coupling corrections? [cf. Alday-Gaiotto-Maldacena-Sever-Vieira 0].

60 4D N =4 SYM 0D string on AdS5 x S5 2D integrable system CFT [ λ: t Hooft coupling ]