Wilson loops and amplitudes in N=4 Super Yang-Mills

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1 Wilson loops and amplitudes in N=4 Super Yang-Mills Vittorio Del Duca INFN Frascati 7 March 2011

2 N=4 Super Yang-Mills maximal supersymmetric theory (without gravity) conformally invariant, β fn. = 0 spin 1 gluon 4 spin 1/2 gluinos 6 spin 0 real scalars

3 N=4 Super Yang-Mills maximal supersymmetric theory (without gravity) conformally invariant, β fn. = 0 spin 1 gluon 4 spin 1/2 gluinos 6 spin 0 real scalars t Hooft limit: Nc with λ = g 2 Nc fixed only planar diagrams

4 N=4 Super Yang-Mills maximal supersymmetric theory (without gravity) conformally invariant, β fn. = 0 spin 1 gluon 4 spin 1/2 gluinos 6 spin 0 real scalars t Hooft limit: Nc with λ = g 2 Nc fixed only planar diagrams AdS/CFT duality Maldacena 97 large-λ limit of 4dim CFT weakly-coupled string theory (aka weak-strong duality)

5 AdS/CFT duality, amplitudes & Wilson loops planar scattering amplitude at strong coupling Alday Maldacena 07 M exp [ i ] λ 2π (Area) cl area of string world-sheet ( classical solution ) neglect O(1/ λ) corrections

6 AdS/CFT duality, amplitudes & Wilson loops planar scattering amplitude at strong coupling Alday Maldacena 07 M exp [ i ] λ 2π (Area) cl area of string world-sheet ( classical solution ) neglect O(1/ λ) corrections amplitude has same form as ansatz for MHV amplitudes at weak coupling M n = M (0) n exp [ l=1 ( ) ] a l f (l) (ɛ) m (1) n (lɛ)+const (l) + E n (l) (ɛ)

7 AdS/CFT duality, amplitudes & Wilson loops planar scattering amplitude at strong coupling Alday Maldacena 07 M exp [ i ] λ 2π (Area) cl area of string world-sheet ( classical solution ) neglect O(1/ λ) corrections amplitude has same form as ansatz for MHV amplitudes at weak coupling M n = M (0) n exp [ l=1 ( ) ] a l f (l) (ɛ) m (1) n (lɛ)+const (l) + E n (l) (ɛ) computation ``formally the same as... the expectation value of a Wilson loop given by a sequence of light-like segments

8 MHV amplitudes Wilson loops agreement between n-edged Wilson loop and n-point MHV amplitude at weak coupling (aka weak-weak duality)

9 MHV amplitudes Wilson loops agreement between n-edged Wilson loop and n-point MHV amplitude at weak coupling (aka weak-weak duality) verified for n-edged 1-loop Wilson loop up to 6-edged 2-loop Wilson loop Brandhuber Heslop Travaglini 07 Drummond Henn Korchemsky Sokatchev 07 Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08

10 MHV amplitudes Wilson loops agreement between n-edged Wilson loop and n-point MHV amplitude at weak coupling (aka weak-weak duality) verified for n-edged 1-loop Wilson loop up to 6-edged 2-loop Wilson loop Brandhuber Heslop Travaglini 07 Drummond Henn Korchemsky Sokatchev 07 Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 n-edged 2-loop Wilson loops also computed (numerically) Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09

11 MHV amplitudes Wilson loops agreement between n-edged Wilson loop and n-point MHV amplitude at weak coupling (aka weak-weak duality) verified for n-edged 1-loop Wilson loop up to 6-edged 2-loop Wilson loop Brandhuber Heslop Travaglini 07 Drummond Henn Korchemsky Sokatchev 07 Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 n-edged 2-loop Wilson loops also computed (numerically) Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 no amplitudes are known beyond the 6-point 2-loop amplitude

12 MHV amplitudes in planar N=4 SYM at any order in the coupling, colour-ordered MHV amplitude in N=4 SYM can be written as tree-level amplitude times helicity-free loop coefficient M n (L) = M n (0) m (L) n

13 MHV amplitudes in planar N=4 SYM at any order in the coupling, colour-ordered MHV amplitude in N=4 SYM can be written as tree-level amplitude times helicity-free loop coefficient M n (L) = M n (0) m (L) n at 1 loop m (1) n = pq F 2me (p, q, P, Q) n 6

14 MHV amplitudes in planar N=4 SYM at any order in the coupling, colour-ordered MHV amplitude in N=4 SYM can be written as tree-level amplitude times helicity-free loop coefficient M n (L) = M n (0) m (L) n at 1 loop m (1) n = pq F 2me (p, q, P, Q) n 6 at 2 loops, iteration formula for the n-pt amplitude m (2) n (ɛ) = 1 2 [ m (1) n ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n Anastasiou Bern Dixon Kosower 03

15 MHV amplitudes in planar N=4 SYM at any order in the coupling, colour-ordered MHV amplitude in N=4 SYM can be written as tree-level amplitude times helicity-free loop coefficient M n (L) = M n (0) m (L) n at 1 loop m (1) n = pq F 2me (p, q, P, Q) n 6 at 2 loops, iteration formula for the n-pt amplitude m (2) n (ɛ) = 1 2 [ m (1) n ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R at all loops, ansatz for a resummed exponent n Anastasiou Bern Dixon Kosower 03 m (L) n = exp [ l=1 ( a l f (l) (ɛ) m (1) n (lɛ)+const (l) + E n (ɛ)) ] (l) + R Bern Dixon Smirnov 05

16 MHV amplitudes in planar N=4 SYM at any order in the coupling, colour-ordered MHV amplitude in N=4 SYM can be written as tree-level amplitude times helicity-free loop coefficient M n (L) = M n (0) m (L) n at 1 loop m (1) n = pq F 2me (p, q, P, Q) n 6 at 2 loops, iteration formula for the n-pt amplitude m (2) n (ɛ) = 1 2 [ m (1) n ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R at all loops, ansatz for a resummed exponent n remainder function Anastasiou Bern Dixon Kosower 03 m (L) n = exp [ l=1 ( a l f (l) (ɛ) m (1) n (lɛ)+const (l) + E n (ɛ)) ] (l) + R Bern Dixon Smirnov 05

17 ansatz for MHV amplitudes in planar N=4 SYM M n = M (0) n = M (0) n coupling a = [ 1+ exp L=1 [ l=1 λ 8π 2 (4πe γ ) ɛ a L m (L) n (ɛ) ] ( ) ] a l f (l) (ɛ)m (1) n (lɛ)+const (l) + E n (l) (ɛ) λ = g 2 N Bern Dixon Smirnov 05 t Hooft parameter f (l) (ɛ) = ˆγ(l) K 4 + ɛ l 2 Ĝ(l) + ɛ 2 f (l) 2 E (l) n (ɛ) =O(ɛ) ˆγ (l) K Ĝ (l) cusp anomalous dimension, known to all orders of a collinear anomalous dimension, known through O(a 4 ) Korchemsky Radyuskin 86 Beisert Eden Staudacher 06 Bern Dixon Smirnov 05 Cachazo Spradlin Volovich 07 ansatz generalises the iteration formula for the 2-loop n-pt amplitude mn (2) m (2) n (ɛ) = 1 2 [ m (1) n ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + O(ɛ) n

18 Factorisation of a multi-leg amplitude in QCD Mueller 1981 Sen 1983 Botts Sterman 1987 Kidonakis Oderda Sterman 1998 Catani 1998 Tejeda-Yeomans Sterman 2002 Kosower 2003 Aybat Dixon Sterman 2006 Becher Neubert 2009 Gardi Magnea 2009 M N (p i /µ, ɛ) = L p i = β i Q 0 / 2 ( 2pi p j S NL (β i β j, ɛ) H L µ 2, (2p i n i ) 2 ) n 2 i µ2 i value of Q0 is immaterial in S, J J i ( (2pi n i ) 2 n 2 i µ2 J i ( 2(βi n i ) 2 n 2 i ), ɛ ), ɛ to avoid double counting of soft-collinear region (IR double poles), Ji removes eikonal part from Ji, which is already in S Ji/Ji contains only single collinear poles

19 N = 4 SYM in the planar limit colour-wise, the planar limit is trivial: can absorb S into Ji each slice is square root of Sudakov form factor M n = n i=1 [ ( )] 1/2 M [gg 1] si,i+1 µ 2, α s, ɛ h n ({p i },µ 2, α s, ɛ) β fn = 0 coupling runs only through dimension ᾱ s (µ 2 )µ 2ɛ =ᾱ s (λ 2 )λ 2ɛ Sudakov form factor has simple solution [ ( )] Q 2 ln Γ µ 2, α s(µ 2 ), ɛ = 1 2 ( αs (µ 2 ) n ( ) ) Q 2 nɛ [ ] γ (n) K π µ 2 2n 2 ɛ 2 + G(n) (ɛ) nɛ n=1 IR structure of N = 4 SUSY amplitudes Magnea Sterman 90 Bern Dixon Smirnov 05

20 Brief history of the ansatz the ansatz checked for the 3-loop 4-pt amplitude Bern Dixon Smirnov 05 2-loop 5-pt amplitude Cachazo Spradlin Volovich 06 Bern Czakon Kosower Roiban Smirnov 06

21 Brief history of the ansatz the ansatz checked for the 3-loop 4-pt amplitude Bern Dixon Smirnov 05 2-loop 5-pt amplitude Cachazo Spradlin Volovich 06 Bern Czakon Kosower Roiban Smirnov 06 the ansatz fails on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08

22 Brief history of the ansatz the ansatz checked for the 3-loop 4-pt amplitude Bern Dixon Smirnov 05 2-loop 5-pt amplitude Cachazo Spradlin Volovich 06 Bern Czakon Kosower Roiban Smirnov 06 the ansatz fails on 2-loop 6-pt amplitude at 2 loops, the remainder function characterises the deviation from the ansatz R (2) n = m (2) n (ɛ) 1 2 [ m (1) n Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08 ] 2 (ɛ) f (2) (ɛ) m (1) n (2ɛ) Const (2) R (2) 6 known numerically Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 Drummond Henn Korchemsky Sokatchev 08 Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 analitically Duhr Smirnov VDD 09

23 Wilson loops & Ward identities Drummond Henn Korchemsky Sokatchev 07 N=4 SYM is invariant under SO(2,4) conformal transformations

24 Wilson loops & Ward identities N=4 SYM is invariant under SO(2,4) conformal transformations the Wilson loops fulfill conformal Ward identities Drummond Henn Korchemsky Sokatchev 07

25 Wilson loops & Ward identities N=4 SYM is invariant under SO(2,4) conformal transformations the Wilson loops fulfill conformal Ward identities Drummond Henn Korchemsky Sokatchev 07 the solution of the Ward identity for special conformal boosts is given by the finite parts of the BDS ansatz + R

26 Wilson loops & Ward identities N=4 SYM is invariant under SO(2,4) conformal transformations the Wilson loops fulfill conformal Ward identities Drummond Henn Korchemsky Sokatchev 07 the solution of the Ward identity for special conformal boosts is given by the finite parts of the BDS ansatz + R for n = 4, 5, R is a constant for n 6, R is an unknown function of conformally invariant cross ratios

27 Wilson loops & Ward identities N=4 SYM is invariant under SO(2,4) conformal transformations the Wilson loops fulfill conformal Ward identities Drummond Henn Korchemsky Sokatchev 07 the solution of the Ward identity for special conformal boosts is given by the finite parts of the BDS ansatz + R for n = 4, 5, R is a constant for n 6, R is an unknown function of conformally invariant cross ratios for n = 6, the conformally invariant cross ratios are 2 u 1 = x2 13x 2 46 x 2 14 x2 36 u 2 = x2 24x 2 15 x 2 25 x2 14 u 3 = x2 35x 2 26 x 2 36 x xi are variables in a dual space s.t. p i = x i x i thus x 2 k,k+r =(p k p k+r 1 ) 2 5

28 W [C n ] = Tr P exp Wilson loops [ ig ] dτẋ µ (τ)a µ (x(τ)) closed contour C n made by light-like external momenta p i = x i x i+1 Alday Maldacena 07

29 W [C n ] = Tr P exp Wilson loops [ ig ] dτẋ µ (τ)a µ (x(τ)) closed contour C n made by light-like external momenta p i = x i x i+1 Alday Maldacena 07 non-abelian exponentiation theorem: vev of Wilson loop as an exponential, allows us to compute the log of W W [C n ] = 1 + L=1 a L W (L) n = exp L=1 a L w (L) n through 2 loops w (1) n = W (1) n w (2) n = W (2) n 1 2 ( W (1) n ) 2 Gatheral 83 Frenkel Taylor 84

30 W [C n ] = Tr P exp Wilson loops [ ig ] dτẋ µ (τ)a µ (x(τ)) closed contour C n made by light-like external momenta p i = x i x i+1 Alday Maldacena 07 non-abelian exponentiation theorem: vev of Wilson loop as an exponential, allows us to compute the log of W W [C n ] = 1 + L=1 a L W (L) n = exp L=1 a L w (L) n through 2 loops w (1) n = W (1) n w (2) n = W (2) n 1 2 ( W (1) n ) 2 Gatheral 83 Frenkel Taylor 84 relation between 1 loop amplitudes & Wilson loops w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n = m (1) n n ζ O(ɛ) Brandhuber Heslop Travaglini 07

31 Wilson loops Wilson loops fulfill a Ward identity for special conformal boosts the solution is the BDS ansatz + R

32 Wilson loops Wilson loops fulfill a Ward identity for special conformal boosts the solution is the BDS ansatz + R at 2 loops w n (2) (ɛ) =f (2) WL (ɛ) w(1) n (2ɛ)+C (2) WL + R(2) n,w L + O(ɛ) with f (2) WL (ɛ) = ζ 2 +7ζ 3 ɛ 5ζ 4 ɛ 2 (to be compared with f (2) (ɛ) = ζ 2 ζ 3 ɛ ζ 4 ɛ 2 for the amplitudes) R 4,W L = R 5,W L =0

33 Wilson loops Wilson loops fulfill a Ward identity for special conformal boosts the solution is the BDS ansatz + R at 2 loops w n (2) (ɛ) =f (2) WL (ɛ) w(1) n (2ɛ)+C (2) WL + R(2) n,w L + O(ɛ) with f (2) WL (ɛ) = ζ 2 +7ζ 3 ɛ 5ζ 4 ɛ 2 (to be compared with f (2) (ɛ) = ζ 2 ζ 3 ɛ ζ 4 ɛ 2 for the amplitudes) R 4,W L = R 5,W L =0 R (2) n,w L arbitrary function of conformally invariant cross ratios u ij = x2 ij+1 x2 i+1j x 2 ij x2 i+1j+1 with x 2 k,k+r =(p k p k+r 1 ) 2

34 Wilson loops Wilson loops fulfill a Ward identity for special conformal boosts the solution is the BDS ansatz + R at 2 loops w n (2) (ɛ) =f (2) WL (ɛ) w(1) n (2ɛ)+C (2) WL + R(2) n,w L + O(ɛ) with f (2) WL (ɛ) = ζ 2 +7ζ 3 ɛ 5ζ 4 ɛ 2 (to be compared with f (2) (ɛ) = ζ 2 ζ 3 ɛ ζ 4 ɛ 2 for the amplitudes) R 4,W L = R 5,W L =0 R (2) n,w L arbitrary function of conformally invariant cross ratios u ij = x2 ij+1 x2 i+1j x 2 ij x2 i+1j+1 with x 2 k,k+r =(p k p k+r 1 ) 2 duality Wilson loop MHV amplitude is expressed by R (2) n,w L = R(2) n

35 Brief history of 2-loop Wilson loops 4-edged Wilson loop 5-edged Wilson loop 6-edged Wilson loop (numeric) Drummond Henn Korchemsky Sokatchev 07 Drummond Henn Korchemsky Sokatchev 07 Drummond Henn Korchemsky Sokatchev 08 Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 6-edged Wilson loop (analytic) Duhr Smirnov VDD 09 n-edged Wilson loop (numeric) Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 checked that Rn = Rn(uij) checked multi-collinear limits

36 Collinear limits of Wilson loops collinear limit a b R6 0 R7 R6 Rn Rn-1 Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09

37 Collinear limits of Wilson loops collinear limit a b Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 R6 0 R7 R6 Rn Rn-1 triple collinear limit a b c R6 R6 R7 R6 R8 R6 + R6 Rn Rn-2 + R6

38 Collinear limits of Wilson loops collinear limit a b Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 R6 0 R7 R6 Rn Rn-1 triple collinear limit a b c R6 R6 R7 R6 R8 R6 + R6 Rn Rn-2 + R6 quadruple collinear limit a b c d R7 R7 R8 R7 R9 R6 + R7 Rn Rn-3 + R7

39 Collinear limits of Wilson loops collinear limit a b Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 R6 0 R7 R6 Rn Rn-1 triple collinear limit a b c R6 R6 R7 R6 R8 R6 + R6 Rn Rn-2 + R6 quadruple collinear limit a b c d R7 R7 R8 R7 R9 R6 + R7 Rn Rn-3 + R7 (k+1)-ple collinear limit i 1 i 2 i k+1 Rn Rn-k + Rk+4 (n-4)-ple collinear limit i 1 i 2 i n 4 Rn-1 Rn-1 Rn Rn-1 (n-3)-ple collinear limit Rn Rn i 1 i 2 i n 3

40 Collinear limits of Wilson loops collinear limit a b Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 R6 0 R7 R6 Rn Rn-1 triple collinear limit a b c R6 R6 R7 R6 R8 R6 + R6 Rn Rn-2 + R6 quadruple collinear limit a b c d R7 R7 R8 R7 R9 R6 + R7 Rn Rn-3 + R7 (k+1)-ple collinear limit i 1 i 2 i k+1 Rn Rn-k + Rk+4 (n-4)-ple collinear limit i 1 i 2 i n 4 Rn-1 Rn-1 Rn Rn-1 (n-3)-ple collinear limit Rn Rn i 1 i 2 i n 3 thus Rn is fixed by the (n-3)-ple collinear limit

41 Quasi-multi-Regge limit of hexagon Wilson loop 6-pt amplitude in the qmr limit of a pair along the ladder y 3 y 4 y 5 y 6 ; p 3 p 4 p 5 p 6 p 2 p 3 the conformally invariant cross ratios are q 2 u 36 = x2 13x 2 46 x 2 14 x2 36 = s 12s 45 s 123 s 345 p 4 p 5 u 14 = x2 24x 2 15 x 2 25 x2 14 = s 23s 56 s 234 s 123 p 1 q 1 p 6 u 25 = x2 35x 2 26 x 2 36 x2 25 = s 34s 61 s 234 s 345

42 Quasi-multi-Regge limit of hexagon Wilson loop 6-pt amplitude in the qmr limit of a pair along the ladder y 3 y 4 y 5 y 6 ; p 3 p 4 p 5 p 6 p 2 p 3 the conformally invariant cross ratios are q 2 u 36 = x2 13x 2 46 x 2 14 x2 36 = s 12s 45 s 123 s 345 p 4 p 5 u 14 = x2 24x 2 15 x 2 25 x2 14 = s 23s 56 s 234 s 123 p 1 q 1 p 6 u 25 = x2 35x 2 26 x 2 36 x2 25 = s 34s 61 s 234 s 345 the cross ratios are all O(1) R6 does not change its functional dependence on the u s

43 Quasi-multi-Regge limit of hexagon Wilson loop 6-pt amplitude in the qmr limit of a pair along the ladder y 3 y 4 y 5 y 6 ; p 3 p 4 p 5 p 6 p 2 p 3 the conformally invariant cross ratios are q 2 u 36 = x2 13x 2 46 x 2 14 x2 36 = s 12s 45 s 123 s 345 p 4 p 5 u 14 = x2 24x 2 15 x 2 25 x2 14 = s 23s 56 s 234 s 123 p 1 q 1 p 6 u 25 = x2 35x 2 26 x 2 36 x2 25 = s 34s 61 s 234 s 345 the cross ratios are all O(1) R6 does not change its functional dependence on the u s R6 is invariant under the qmr limit of a pair along the ladder Duhr Glover Smirnov VDD 08

44 Quasi-multi-Regge limit of n-sided Wilson loop 7-pt amplitude in the qmr limit of a triple along the ladder y 3 y 4 y 5 y 6 y 7 ; p 3 p 4 p 5 p 6 p 7 p 2 p 3 q 2 p 4 p 6 p 5 7 cross ratios, which are all O(1) R7 is invariant under the qmr limit of a triple along the ladder q 1 p 1 p 7

45 Quasi-multi-Regge limit of n-sided Wilson loop 7-pt amplitude in the qmr limit of a triple along the ladder y 3 y 4 y 5 y 6 y 7 ; p 3 p 4 p 5 p 6 p 7 p 2 p 3 q 2 p 4 p 6 p 5 7 cross ratios, which are all O(1) R7 is invariant under the qmr limit of a triple along the ladder q 1 p 1 p 7 can be generalised to the n-pt amplitude in the qmr limit of a (n-4)-ple along the ladder y 3 y 4... y n 1 y n ; p 3... p n Duhr Smirnov VDD 09

46 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09

47 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09 w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n

48 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09 w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n ln(s ij ) + Li 2 (1 u ij )

49 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09 w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n u s are invariant in the qmrk ln(s ij ) + Li 2 (1 u ij )

50 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09 w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n u s are invariant in the qmrk ln(s ij ) + Li 2 (1 u ij ) log s are not power suppressed

51 Quasi-multi-Regge limit of Wilson loops L-loop Wilson loops are Regge exact w n (L) (ɛ) =f (L) WL (ɛ) w(1) n (Lɛ)+C (L) WL + R(L) n,w L (u ij)+o(ɛ) Drummond Korchemsky Sokatchev 07 Duhr Smirnov VDD 09 w (1) n = Γ(1 2ɛ) Γ 2 (1 ɛ) m(1) n u s are invariant in the qmrk ln(s ij ) + Li 2 (1 u ij ) log s are not power suppressed we may compute the Wilson loop in qmrk the result will be correct in general kinematics!!!

52 hard diagram Diagrams of 2-loop Wilson loops curtain diagram cross diagram Y diagram Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 factorised cross diagram each diagram yields an integral, similar to a Feynman-parameter integral

53 Computing 2-loop Wilson loops cusp diagrams are given by cross and Y diagrams with gluons attaching to consecutive sides

54 Computing 2-loop Wilson loops cusp diagrams are given by cross and Y diagrams with gluons attaching to consecutive sides most difficult diagrams to compute are hard diagrams fh has 1/ε 2 singularities if Q1 = Q2 = 0, Q3 0 it has 1/ε singularities if Q1 = 0, Q2, Q3 0 it is finite if Q1, Q2, Q3 0 e.g. for n=6, the most difficult diagram is f H (p 1,p 3,p 5 ; p 4,p 6,p 2 ) which is finite

55 Computing 2-loop Wilson loops cusp diagrams are given by cross and Y diagrams with gluons attaching to consecutive sides most difficult diagrams to compute are hard diagrams fh has 1/ε 2 singularities if Q1 = Q2 = 0, Q3 0 it has 1/ε singularities if Q1 = 0, Q2, Q3 0 it is finite if Q1, Q2, Q3 0 e.g. for n=6, the most difficult diagram is f H (p 1,p 3,p 5 ; p 4,p 6,p 2 ) which is finite most general hard diagram has Q1 2, Q2 2, Q3 2 0; it occurs for n 9

56 Wilson loops: analytic calc 1. Use Mellin-Barnes (MB) representation of the Feynman-parameter integrals: replace each denominator by a contour integral 1 (A + B) λ = 1 Γ(λ) 1 2πi +i i dz Γ( z) Γ(λ + z) A z B λ+z integral turns into a sum of residues Res z= n Γ(z) = ( 1)n n!

57 Wilson loops: analytic calc 1. Use Mellin-Barnes (MB) representation of the Feynman-parameter integrals: replace each denominator by a contour integral 1 (A + B) λ = 1 Γ(λ) 1 2πi +i i dz Γ( z) Γ(λ + z) A z B λ+z integral turns into a sum of residues Res z= n Γ(z) = ( 1)n n! 2. Use Regge exactness in the qmr limit: retain only leading behaviour (i.e. leading residues) of the integral leading residue

58 Wilson loops: analytic calc 3. Use Regge exactness again: iterate the qmr limit n times, by taking the n cyclic permutations of the external legs

59 Wilson loops: analytic calc 3. Use Regge exactness again: iterate the qmr limit n times, by taking the n cyclic permutations of the external legs leading residue in step 2 leading residue in step 3

60 Wilson loops: analytic calc 3. Use Regge exactness again: iterate the qmr limit n times, by taking the n cyclic permutations of the external legs 4. Sum remaining towers of residues n=1 n=1 u n n = ln(1 u) u n n k = Li k(u) leading residue in step 2 leading residue in step 3

61 Wilson loops: analytic calc 3. Use Regge exactness again: iterate the qmr limit n times, by taking the n cyclic permutations of the external legs 4. Sum remaining towers of residues n=1 n=1 u n n = ln(1 u) u n n k = Li k(u) leading residue in step 2 leading residue in step 3 in general, get nested harmonic sums Goncharov polylogarithms n 1 =1 u n 1 1 n m 1 1 n 1 1 n 2 =1... n k 1 1 n k =1 u n k k n m k k =( 1) k G 0,..., 0, 1,..., 0,..., 0, }{{} u 1 }{{} m 1 1 m k 1 1 ;1 u 1... u k

62 Analytic 2-loop 6-edged Wilson loop compute 2-loop 6-edged Wilson loop in MB representation of the integrals in general kinematics, get up to 8-fold integrals

63 Analytic 2-loop 6-edged Wilson loop compute 2-loop 6-edged Wilson loop in MB representation of the integrals in general kinematics, get up to 8-fold integrals after procedure in qmr limit, at most 3-fold integrals in fact, only one 3-fold integral, which comes from f H (p 1,p 3,p 5 ; p 4,p 6,p 2 ) +i +i +i i i i dz 1 2πi dz 2 2πi dz 3 2πi (z 1 z 2 + z 2 z 3 + z 3 z 1 ) u z 1 1 uz 2 2 uz 3 3 Γ ( z 1 ) 2 Γ ( z 2 ) 2 Γ ( z 3 ) 2 Γ (z 1 + z 2 ) Γ (z 2 + z 3 ) Γ (z 3 + z 1 ) the result is in terms of Goncharov polylogarithms G(a, w; z) = z 0 dt (1 t a G( w; t), G(a; z) = ln z ) a

64 Analytic 2-loop 6-edged Wilson loop compute 2-loop 6-edged Wilson loop in MB representation of the integrals in general kinematics, get up to 8-fold integrals after procedure in qmr limit, at most 3-fold integrals in fact, only one 3-fold integral, which comes from f H (p 1,p 3,p 5 ; p 4,p 6,p 2 ) +i +i +i i i i dz 1 2πi dz 2 2πi dz 3 2πi (z 1 z 2 + z 2 z 3 + z 3 z 1 ) u z 1 1 uz 2 2 uz 3 3 Γ ( z 1 ) 2 Γ ( z 2 ) 2 Γ ( z 3 ) 2 Γ (z 1 + z 2 ) Γ (z 2 + z 3 ) Γ (z 3 + z 1 ) the result is in terms of Goncharov polylogarithms G(a, w; z) = z 0 dt (1 t a G( w; t), G(a; z) = ln z ) a the remainder function R6 (2) is given in terms of O(10 3 ) Goncharov polylogarithms G(u1, u2, u3) Duhr Smirnov VDD 09

65 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 Duhr Smirnov VDD 09

66 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) Duhr Smirnov VDD 09

67 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) it is of uniform, and intrinsic, transcendental weight 4 Duhr Smirnov VDD 09 transcendental weights: w(ln x) = w(π) = 1 w(li2(x)) = w(π 2 ) = 2

68 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) it is of uniform, and intrinsic, transcendental weight 4 Duhr Smirnov VDD 09 transcendental weights: w(ln x) = w(π) = 1 w(li2(x)) = w(π 2 ) = 2 it vanishes under collinear and multi-regge limits (in Euclidean space)

69 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) it is of uniform, and intrinsic, transcendental weight 4 Duhr Smirnov VDD 09 transcendental weights: w(ln x) = w(π) = 1 w(li2(x)) = w(π 2 ) = 2 it vanishes under collinear and multi-regge limits (in Euclidean space) it is in agreement with the numeric calculation by Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09

70 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) it is of uniform, and intrinsic, transcendental weight 4 Duhr Smirnov VDD 09 transcendental weights: w(ln x) = w(π) = 1 w(li2(x)) = w(π 2 ) = 2 it vanishes under collinear and multi-regge limits (in Euclidean space) it is in agreement with the numeric calculation by Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 straightforward computation qmr kinematics make it technically feasible

71 2-loop 6-edged remainder function R6 (2) the remainder function R6 (2) is explicitly dependent on the cross ratios u1, u2, u3 it is symmetric in all its arguments (in general it s symmetric under cyclic permutations and reflections) it is of uniform, and intrinsic, transcendental weight 4 Duhr Smirnov VDD 09 transcendental weights: w(ln x) = w(π) = 1 w(li2(x)) = w(π 2 ) = 2 it vanishes under collinear and multi-regge limits (in Euclidean space) it is in agreement with the numeric calculation by Anastasiou Brandhuber Heslop Khoze Spence Travaglini 09 straightforward computation qmr kinematics make it technically feasible finite answer, but in intermediate steps many divergences output is punishingly long

72 our result has been simplified and given in terms of polylogarithms R (2) 6,W L (u 1,u 2,u 3 ) = 1 8 Goncharov Spradlin Vergu Volovich 10 3 ( L 4 (x + i,x i ) 1 ) 2 Li 4(1 1/u i ) i=1 ( 3 ) 2 Li 2 (1 1/u i ) + J π2 12 J 2 + π4 72 i=1

73 where our result has been simplified and given in terms of polylogarithms x ± i = u i x ± R (2) 6,W L (u 1,u 2,u 3 ) = L 4 (x +,x )= Goncharov Spradlin Vergu Volovich 10 3 ( L 4 (x + i,x i ) 1 ) 2 Li 4(1 1/u i ) x ± = u 1 + u 2 + u 3 1 ± 2u 1 u 2 u 3 =( u 1 + u 2 + u 3 1) 2 4u 1 u 2 u 3 3 m=0 1 8 i=1 ( 3 ) 2 Li 2 (1 1/u i ) + J π2 12 J 2 + π4 72 i=1 ( 1) m (2m)!! log(x+ x ) m (l 4 m (x + )+l 4 m (x )) + 1 8!! log(x+ x ) 4 l n (x) = 1 2 (Li n(x) ( 1) n Li n (1/x)) J = 3 (l 1 (x + i ) l 1(x i )) i=1

74 where our result has been simplified and given in terms of polylogarithms x ± i = u i x ± R (2) 6,W L (u 1,u 2,u 3 ) = L 4 (x +,x )= Goncharov Spradlin Vergu Volovich 10 3 ( L 4 (x + i,x i ) 1 ) 2 Li 4(1 1/u i ) x ± = u 1 + u 2 + u 3 1 ± 2u 1 u 2 u 3 =( u 1 + u 2 + u 3 1) 2 4u 1 u 2 u 3 3 m=0 1 8 i=1 ( 3 ) 2 Li 2 (1 1/u i ) + J π2 12 J 2 + π4 72 i=1 ( 1) m (2m)!! log(x+ x ) m (l 4 m (x + )+l 4 m (x )) + 1 8!! log(x+ x ) 4 l n (x) = 1 2 (Li n(x) ( 1) n Li n (1/x)) J = 3 (l 1 (x + i ) l 1(x i )) i=1 not a new, independent, computation just a manipulation of our result

75 where our result has been simplified and given in terms of polylogarithms x ± i = u i x ± R (2) 6,W L (u 1,u 2,u 3 ) = L 4 (x +,x )= Goncharov Spradlin Vergu Volovich 10 3 ( L 4 (x + i,x i ) 1 ) 2 Li 4(1 1/u i ) x ± = u 1 + u 2 + u 3 1 ± 2u 1 u 2 u 3 =( u 1 + u 2 + u 3 1) 2 4u 1 u 2 u 3 3 m=0 1 8 i=1 ( 3 ) 2 Li 2 (1 1/u i ) + J π2 12 J 2 + π4 72 i=1 ( 1) m (2m)!! log(x+ x ) m (l 4 m (x + )+l 4 m (x )) + 1 8!! log(x+ x ) 4 l n (x) = 1 2 (Li n(x) ( 1) n Li n (1/x)) J = 3 (l 1 (x + i ) l 1(x i )) i=1 not a new, independent, computation just a manipulation of our result answer is short and simple introduces the theory of motives in TH physics

76 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1

77 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1 deg(const) = 0 deg(π) = 0 ln x : cut along [-, 0] with Disc = 2πi deg(ln x) = 1 Li2(x) : cut along [1, ] with Disc = -2πi ln x deg(li2(x)) = 2

78 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1 deg(const) = 0 deg(π) = 0 ln x : cut along [-, 0] with Disc = 2πi deg(ln x) = 1 Li2(x) : cut along [1, ] with Disc = -2πi ln x deg(li2(x)) = 2 take a fn. defined as an iterated integral Ri rational functions T k = b a d lnr 1 d lnr k the symbol is Sym[T k ]=R 1 R k defined on the tensor product of the group of rational functions, modulo constants R 1 R 2 = R 1 + R 2

79 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1 deg(const) = 0 deg(π) = 0 ln x : cut along [-, 0] with Disc = 2πi deg(ln x) = 1 Li2(x) : cut along [1, ] with Disc = -2πi ln x deg(li2(x)) = 2 take a fn. defined as an iterated integral Ri rational functions T k = b a d lnr 1 d lnr k the symbol is Sym[T k ]=R 1 R k defined on the tensor product of the group of rational functions, modulo constants R 1 R 2 = R 1 + R 2 Sym[ln x] =x Sym[Li 2 (x)] = (x 1) x

80 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1 deg(const) = 0 deg(π) = 0 ln x : cut along [-, 0] with Disc = 2πi deg(ln x) = 1 Li2(x) : cut along [1, ] with Disc = -2πi ln x deg(li2(x)) = 2 take a fn. defined as an iterated integral Ri rational functions T k = b a d lnr 1 d lnr k the symbol is Sym[T k ]=R 1 R k defined on the tensor product of the group of rational functions, modulo constants R 1 R 2 = R 1 + R 2 Sym[ln x] =x Sym[Li 2 (x)] = (x 1) x take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g] then f-g = h with deg(h) = n -1

81 Symbols Fn. F of deg(f) = n : fn. with log cuts, s.t. Disc = 2πi f, with w(f) = n-1 deg(const) = 0 deg(π) = 0 ln x : cut along [-, 0] with Disc = 2πi deg(ln x) = 1 Li2(x) : cut along [1, ] with Disc = -2πi ln x deg(li2(x)) = 2 take a fn. defined as an iterated integral Ri rational functions T k = b a d lnr 1 d lnr k the symbol is Sym[T k ]=R 1 R k defined on the tensor product of the group of rational functions, modulo constants R 1 R 2 = R 1 + R 2 Sym[ln x] =x Sym[Li 2 (x)] = (x 1) x take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g] then f-g = h with deg(h) = n -1 a symbol determines a polynomial of uniform degree up to a constant

82 Zn symmetric regular hexagons regular hexagons are characterised by 2 x 2 13 = x 2 24 = x 2 35 = x 2 46 = x 2 51 = x 2 62; x 2 14 = x 2 25 = x u 36 = x2 13x 2 46 x 2 14 x2 36 u 14 = x2 24x 2 15 x 2 25 x2 14 = s 12s 45 s 123 s 345 = s 23s 56 s 234 s 123 u 14 = u 25 = u 36 = u u 25 = x2 35x 2 26 x 2 36 x2 25 = s 34s 61 s 234 s 345

83 Zn symmetric regular hexagons regular hexagons are characterised by 2 x 2 13 = x 2 24 = x 2 35 = x 2 46 = x 2 51 = x 2 62; x 2 14 = x 2 25 = x u 36 = x2 13x 2 46 x 2 14 x2 36 u 14 = x2 24x 2 15 x 2 25 x2 14 = s 12s 45 s 123 s 345 = s 23s 56 s 234 s 123 u 14 = u 25 = u 36 = u u 25 = x2 35x 2 26 x 2 36 x2 25 = s 34s 61 s 234 s 345 At strong coupling, remainder function is obtained from ``minimal area surfaces in AdS5 which end on a null polygonal contour at the boundary. One gets ``integral equations which determine the area as a function of the shape of the polygon. The equations are identical to those of the Thermodynamics Bethe Ansatz. The area is given by the free energy of the TBA system. The high temperature limit of the TBA system can be exactly solved R strong 6 (u, u, u) = π 6 1 3π φ2 3 8 ( ln 2 (u) + 2 Li 2 (1 u) ) u = 1 4 cos 2 (φ/3) ( ( free energy BDS - BDSlike Alday Gaiotto Maldacena 09

84 Remainder function at weak and strong coupling compare remainder functions at weak and strong coupling introducing coefficients in the strong coupling result and try to curve fit the 2 results ( R strong π 6 (u, u, u) =c ) 3π φ2 + c 2 ( 3 8 ( ln 2 (u) + 2 Li 2 (1 u) )) + c 3 c 1 =0.263π 3 c 2 =0.860π 2 c 3 = π2 12 c 2 Alday Gaiotto Maldacena 09 Brandhuber Heslop Khoze Travaglini 09

85 Remainder function at weak and strong coupling compare remainder functions at weak and strong coupling introducing coefficients in the strong coupling result and try to curve fit the 2 results ( R strong π 6 (u, u, u) =c ) 3π φ2 + c 2 ( 3 8 ( ln 2 (u) + 2 Li 2 (1 u) )) + c 3 c 1 =0.263π 3 c 2 =0.860π 2 c 3 = π2 12 c 2 Alday Gaiotto Maldacena 09 Brandhuber Heslop Khoze Travaglini 09 the 2 curves are strikingly similar

86 Remainder function at weak and strong coupling ( R strong π 6 (u, u, u) =c ) ( 3 3π φ2 + c 2 8 ( ln 2 (u) + 2 Li 2 (1 u) )) + c 3 rendered uniform weight 4 by suitable coefficients ci

87 Remainder function at weak and strong coupling ( R strong π 6 (u, u, u) =c ) ( 3 3π φ2 + c 2 8 ( ln 2 (u) + 2 Li 2 (1 u) )) + c 3 rendered uniform weight 4 by suitable coefficients ci ( R (2) 1 6,W L 2, 1 2, 1 ) 2 = ζ 3 log log π2 log Li ( 1 ) π uniform, and intrinsic, weight 4

88 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that?

89 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 )

90 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 ) the dimensional regulator breaks conformal invariance and Regge exactness

91 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 ) the dimensional regulator breaks conformal invariance and Regge exactness Way out

92 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 ) the dimensional regulator breaks conformal invariance and Regge exactness Way out spontaneous-symmetry break N=4 SYM: switch on a vev for one of the scalars use the vev masses as regulators Alday Henn Plefka Schuster 09

93 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 ) the dimensional regulator breaks conformal invariance and Regge exactness Way out spontaneous-symmetry break N=4 SYM: switch on a vev for one of the scalars use the vev masses as regulators Alday Henn Plefka Schuster 09 preserves conformal invariance

94 the 2-loop n-pt amplitude is m (2) n (ɛ) = 1 2 [ m (1) n One-loop amplitude squared ] 2 (ɛ) + f (2) (ɛ) m (1) (2ɛ)+Const (2) + R n what about that? one-loop amplitude squared must be known at least through O(ɛ 2 ) the dimensional regulator breaks conformal invariance and Regge exactness Way out spontaneous-symmetry break N=4 SYM: switch on a vev for one of the scalars use the vev masses as regulators Alday Henn Plefka Schuster 09 preserves conformal invariance not practical for phenomenology (where DR rules the waves)

95 Amplitudes in twistor space twistors live in the fundamental irrep of SO(2,4) any point in dual space corresponds to a line in twistor space x a (Z a,z a+1 )

96 Amplitudes in twistor space twistors live in the fundamental irrep of SO(2,4) any point in dual space corresponds to a line in twistor space x a (Z a,z a+1 ) null separations in dual space correspond to intersections in twistor space

97 Amplitudes in twistor space twistors live in the fundamental irrep of SO(2,4) any point in dual space corresponds to a line in twistor space x a (Z a,z a+1 ) null separations in dual space correspond to intersections in twistor space 2-loop n-pt MHV amplitudes can be written as sum of pentaboxes in twistor space m (2) n = 1 2 i<j<k<l<i Arkani-Hamed Bourjaily Cachazo Trnka10

98 8-edged Wilson loop in AdS3 at strong coupling, Alday & Maldacena have considered 2n-sided polygons embedded into the boundary of AdS3 2n-sided remainder function depends on 2(n-3) variables

99 8-edged Wilson loop in AdS3 at strong coupling, Alday & Maldacena have considered 2n-sided polygons embedded into the boundary of AdS3 2n-sided remainder function depends on 2(n-3) variables for the octagon, the remainder function is 8,W L = 1 2 ln ( 1+χ ) ( ln 1+ 1 ) χ + + 7π 6 + m sinh t ( + dt tanh(2t +2iφ) ln 2π m cosh 1+e t) R strong Alday Maldacena 09 where χ + = e 2πIm m χ = e 2πRe m m = m e iφ

100 8-edged Wilson loop in AdS3 at strong coupling, Alday & Maldacena have considered 2n-sided polygons embedded into the boundary of AdS3 2n-sided remainder function depends on 2(n-3) variables for the octagon, the remainder function is 8,W L = 1 2 ln ( 1+χ ) ( ln 1+ 1 ) χ + + 7π 6 + m sinh t ( + dt tanh(2t +2iφ) ln 2π m cosh 1+e t) R strong Alday Maldacena 09 where χ + = e 2πIm m χ = e 2πRe m m = m e iφ at weak coupling, the 2-loop octagon remainder function is R (2) 8,W L (χ+, χ )= π ln ( 1+χ +) ( ln 1+ 1 ) χ + ln ( 1+χ ) ( ln 1+ 1 ) χ Duhr Smirnov VDD 10

101 8-edged Wilson loop in AdS3 at strong coupling, Alday & Maldacena have considered 2n-sided polygons embedded into the boundary of AdS3 2n-sided remainder function depends on 2(n-3) variables for the octagon, the remainder function is 8,W L = 1 2 ln ( 1+χ ) ( ln 1+ 1 ) χ + + 7π 6 + m sinh t ( + dt tanh(2t +2iφ) ln 2π m cosh 1+e t) R strong Alday Maldacena 09 where χ + = e 2πIm m χ = e 2πRe m m = m e iφ at weak coupling, the 2-loop octagon remainder function is R (2) 8,W L (χ+, χ )= π ln ( 1+χ +) ( ln 1+ 1 ) χ + ln ( 1+χ ) ( ln 1+ 1 ) χ 2-loop 2n-sided polygon R conjectured through collinear limits proven through OPE Duhr Smirnov VDD 10 Heslop Khoze 10 Gaiotto Maldacena Sever Vieira 10

102 Conclusions Planar N=4 SYM is a great lab where to test comparisons between strong and weak couplings features weak-strong duality and weak-weak duality Wilson loops are the ideal quantities to perform those comparisons

103 Conclusions Planar N=4 SYM is a great lab where to test comparisons between strong and weak couplings features weak-strong duality and weak-weak duality Wilson loops are the ideal quantities to perform those comparisons first (and so far only) analytic computation of a 2-loop remainder function analytic comparison between the 2-loop 6-edged Wilson loop at weak and strong couplings in a particular kinematic set-up, defined on the boundary of AdS3 first analytic computation of the 2-loop 8-edged Wilson loop at weak coupling; 2-loop Wilson loops in AdS3 are now completely solved

104 Conclusions Planar N=4 SYM is a great lab where to test comparisons between strong and weak couplings features weak-strong duality and weak-weak duality Wilson loops are the ideal quantities to perform those comparisons first (and so far only) analytic computation of a 2-loop remainder function analytic comparison between the 2-loop 6-edged Wilson loop at weak and strong couplings in a particular kinematic set-up, defined on the boundary of AdS3 first analytic computation of the 2-loop 8-edged Wilson loop at weak coupling; 2-loop Wilson loops in AdS3 are now completely solved more is to come... stay tuned!