New insights in the amplitude/wilson loop duality

Transcription

1 New insights in the amplitude/wilson loop duality Paul Heslop Amplitudes th May based on : Brandhuber, Khoze, Travaglini, PH : Brandhuber, Katsaroumpas, Nguyen, Spence, Spradlin, Travaglini, PH Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

2 Outline of talk Introduction Duality between two objects in N =4 SYM: p 6 p 1 p 1 p 2 p 5 p p 2 4 p 3 = 6 p 4 p 5 Gluon amplitudes = Wilson loops p p 3 compute all MHV 2-loop gluon amplitudes for any n. 2 unexpected matches 1 strong-weak matching 2 amplitude/wl match away from finite order in dim reg Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

3 Introduction MHV Amplitudes in N = 4 SYM planar (N ) Maximally Helicity Violating (MHV) gluon amplitudes j + i We will focus on M (L) n A n = A tree n M n Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

4 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

5 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

6 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

7 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

8 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

9 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

10 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

11 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

12 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

13 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

14 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

15 The WL/amplitude duality The basic statement Amplitude/Wilson loop duality [Alday Maldacena, Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] amplitude M n = W [C n ] (D = 4 2ɛ) (D = 4 + 2ɛ) Wilson loop over the polygonal contour C n W [C] := Tr P exp [ig ( )] C dτ A µ(x(τ))ẋ µ (τ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

16 The WL/amplitude duality Where is the idea from? At strong coupling String theory: gauge string duality At λ (thooft coupling constant) amplitudes given in string theory as the area of minimal surfaces in AdS ending on a polygon at the boundary [Alday Maldacena] = Wilson loop of polygon at λ (AdS/CFT) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

17 The WL/amplitude duality At 1 loop The duality in perturbation theory 1-loop O(λ), [Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] The expressions for the n point amplitude and for the WL are very closely related: amplitude = p,q = +O(ɛ) [Bern Dixon Dunbar Kosower 1994] Wilson loop = p,q = 1 0 dτ pdτ q u [ ( P 2 +(s P 2 )τp +(t P 2 )τq +uτp τq) ] 1+ɛ P = q 1 k=p+1 k Q = p 1 k=q+1 k Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

18 The WL/amplitude duality At 1 loop The duality in perturbation theory 1-loop O(λ), [Drummond Korchemsky Sokatchev, Brandhuber Travaglini PH] The expressions for the n point amplitude and for the WL are very closely related: amplitude = p,q = +O(ɛ) [Bern Dixon Dunbar Kosower 1994] Wilson loop = p,q = 1 0 dτ pdτ q u [ ( P 2 +(s P 2 )τp +(t P 2 )τq +uτp τq) ] 1+ɛ P = q 1 k=p+1 k Q = p 1 k=q+1 k Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

19 The WL/amplitude duality Beyond 1 loop L-loop amplitude/wl BDS [Anastasiou Bern Dixon Kosower, Bern Dixon Smirnov] The BDS formula: an all-loop expression for n = 4, 5 log M n (ɛ) = (BDS) n = L=1 λl f (L) A (ɛ)m(1) (Lɛ) + C(λ) + O(ɛ) n f (L) A (L) (ɛ) = f 0 + f (L) 1 ɛ + f (L) 2 ɛ2 where f (L) is a number. Similarly for the WL: all-loop expression for n = 4, 5 i log W n (ɛ) = (BDS) WL n := L=1 λl f (L) W (1) (ɛ)w (Lɛ) + C W (λ) + O(ɛ) n [Drummond Henn Korchemsky Sokatchev] Fixed by (dual) conformal invariance Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

20 The WL/amplitude duality Beyond 1 loop The duality for general n The remainder function log M n (ɛ) = (BDS) n + R n (p i ; λ; ɛ) log W n (ɛ) = (BDS) WL n + R W n (p i ; λ; ɛ) (R = O(ɛ) at 4,5 points) non-zero remainder at two-loop six-points, ɛ = 0, but Wilson loop/amplitude duality at two loops R (2) 6 = R (2),W 6 + O(ɛ) [Drummond Henn Korchemsky Sokatchev, Bern Dixon Kosower Roiban Spradlin Vergu Volovich] R = R W + O(ɛ) is the traditional statement of the duality (see Valya Khoze s talk for an alternative equivalent statement of the duality in terms of the WL/amplitude ratio) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

21 entire S-matrix, Yangian, integrability Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35 The WL/amplitude duality Beyond 1 loop (Dual) conformal invariance [Drummond Henn Korchemsky Sokatchev] A smooth Wilson loop is conformally invariant but polygonal Wilson loop is divergent in four-dimensions However R W n (p i ; λ; ɛ = 0) = log W n (BDS) WL n is finite and (dual) conformally invariant ( dual for amplitude) R W n (p i ) = R W n (u ij ), u ij are conformal cross-ratios: u ij = x 2 ij+1 x 2 ji+1 x 2 ij x 2 i+1j+1

22 The WL/amplitude duality At two loops n-point Wilson loop at O(λ 2 ) Wilson loop at 2 loops much harder than 1 loop nevertheless much easier than the corresponding amplitude computation Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

23 The WL/amplitude duality At two loops n-point Wilson loop at O(λ 2 ) Wilson loops/mhv amplitudes at two loops O(λ 2 ) we have constructed a computer algorithm to compute all polygonal Wilson loops at O(λ 2 ) [Anastasiou Brandhuber Khoze Spence Travaglini P. H.] recent analytic result for the six-point WL at two loops [Del Duca Duhr Smirnov] Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

24 Strong-weak coupling invariance Wilson loop at strong coupling [Alday Maldacena, Alday Gaiotto Maldacena, Alday Maldacena Sever Vieira] Parallel developments at strong coupling λ arbitrary n-point Wilson loops at strong coupling in terms of integral equations (Y-system) found in integrability Area = free energy new data to compare with weak coupling integrability at weak coupling? Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

25 Strong-weak coupling invariance Comparison with Alday-Maldacena Alday and Maldacena have considered the 8-point amplitude at strong coupling via string theory special kinematics lying in dimensions depends on only two parameters χ +, χ (up to conformal transformations) contour is closed by sending three points to infinity (light-cone coordinates) x 6 = (, 1), x 7 = (, ), x 8 = (0, ). Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

26 Strong-weak coupling invariance χ + and χ cross-ratios at n = 8 (n = 4k ) there exist 2 special cross-ratios defined purely in terms of two-particle invariants χ + := x 2 24 x 2 68 x 2 46 x 2 28 = u 14u 58 u 15 u 38 u 47 u 37 χ := x 2 13 x 2 57 x 2 35 x 2 17 = u 38u 47 u 48 u 27 u 36 u 26 Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

27 Strong-weak coupling invariance corresponding (12) cross-ratios: χ + u 15 = 1 + χ, u χ + 26 = 1 + χ, 1 u 37 = 1 + χ, u = 1 + χ, u i i+3 = 1, i = 1,..., 8. direct use of this kinematics is awkward: infinities use conformally related kinematics A, B and C kinematics A: apply conformal transformation to zig-zag kinematics B: invariants without specific region momenta x 2 i+5 i+8 = 1 = x 2 i i+4, i = 1,..., 4 remaining invariants determined so that u s are as above kinematics C: circular kinematics... Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

28 Strong-weak coupling invariance Circular kinematics x 3 2 t y x x x x 2 z Π 4 Π 4 Π 4 Π 4 2 y Π 4 x x Momenta in dimensions circumscribing a circle Cross-ratios as before if x and y are related to χ ± in the following way, χ + = tan x 2 + 1, χ = tan y tan x tan x 1 2 tan y tan y 1 Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

29 Strong-weak coupling invariance Strong coupling remainder function Alday and Maldacena find the Wilson loop remainder at strong coupling R AM 8 (m) = 1 2 log(1 + χ+ ) log(1 + χ ) + 7π 6 + m sinht dt tanh(2t + 2iφ) log ( 1 + e 2π m cosht), χ + := e 2πIm(m) χ := e 2πRe(m) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

30 Strong-weak coupling invariance Corresponding weak coupling remainder the two remainder functions have an identical shape! [Brandhuber Khoze Travaglini PH] Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

31 Strong-weak coupling invariance shifted and rescaled remainder, R 8 varies between 0 and -1 R(m = 0) = R reg 8 R(m ) = 2R reg 6 R 8 (m) := R 8(m) R reg 8 R reg 8 2R reg 6 strong coupling remainder function R 8 (m) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

32 Strong-weak coupling invariance shifted and rescaled remainder, R 8 varies between 0 and -1 R(m = 0) = R reg 8 R(m ) = 2R reg 6 R 8 (m) := R 8(m) R reg 8 R reg 8 2R reg 6 weak coupling remainder function R 8 (m) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

33 Strong-weak coupling invariance shifted and rescaled remainder, R 8 varies between 0 and -1 R(m = 0) = R reg 8 R(m ) = 2R reg 6 R 8 (m) := R 8(m) R reg 8 R reg 8 2R reg 6 weak and strong coupling superimposed R 8 (m) Rescaled weak and strong coupling remainders coincide! Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

34 2d plots Strong-weak coupling invariance arg(m) = 0 arg(m) = π Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

35 Strong-weak coupling invariance linear relation between strong and weak coupling R (2) 8 (u ij) = A (2) R AM 8 (u ij ) + B (2) A (2) = ± B (2) = ± strong- (horizontal axis) to weak-coupling (vertical axis) blue line is the least squares linear fit Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

36 Strong-weak coupling invariance General n universality evidence of a universal form for the octagon remainder for special kinematics what about other amplitudes? no strong-weak matching for regular polygons at n = 6, 8, 10 (see Valya Khoze s talk) for AdS 3 only? (n 6 cross-ratios) check at n = 10 or is it to do with the even more specialised case when n = 4k with just χ + and χ in this case the physical WL ratio vanishes just as at 4 and 5 points [Khoze PH] no integral equations at strong coupling? n = 4k case derived recently in [Yang] Clearly there is much still to be understood Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

37 Higher orders in dim reg Moving away from 4 dimensions [Brandhuber Nguyen Katsaroumpas Spence Spradlin Travaglini PH] the WL/amplitude duality at 1 loop matches Wilson loop diagrams to certain scalar integrals (2me boxes) to all orders in ɛ in general the amplitude contains other integrals which vanish as ɛ 0 (pentagons/hexagons) WL does not reproduce these two loops: technically much harder to check higher orders standard lore: WL/amplitude duality in 4 dimensions only but higher orders not previously checked beyond 1 loop... Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

38 Higher orders in dim reg One loop amplitude/wl higher orders in ɛ four-points: duality to all orders in ɛ M (1) (1) 4 (ɛ)= = C(ɛ)W 4 (ɛ) five-points: duality at O(ɛ 0 ) only M (1) 5 (ɛ)= = C(ɛ)W (1) (ɛ) + O(ɛ) 5 Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

39 Higher orders in dim reg One loop amplitude/wl higher orders in ɛ four-points: duality to all orders in ɛ M (1) (1) 4 (ɛ)= = C(ɛ)W 4 (ɛ) five-points parity even part: duality to all orders in ɛ M (1) 5+ (ɛ)= = C(ɛ)W (1) 5 (ɛ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

40 Higher orders in dim reg One loop all orders boxes various all orders in ɛ expressions for the 2me boxes (needed for 1 loop MHV amplitude) [Duplancic Nizic 2000, Brandhuber Spence Travaglini] new expression: (finite part of the) 2me box function where f (x) = 1 ɛ x = ( a)ɛ [ ] f (1 ap 2 ) + f (1 aq 2 ) f (1 as) f (1 at), 2 dσ (1 σ) ɛ 1 σ = x 3 F 2 (1, 1, 1 + ɛ; 2, 2; x) = 0 ( = 1 [ ( x) ɛ ɛ 2 2 F 1 (ɛ, ɛ; 1 + ɛ; 1/x) + ɛ log x ] + C (1 ap 2 )(1 aq 2 ) (1 as)(1 at) = 1 ) n=1 ɛ n S 1 n+1 (x) ( s = (P + p) 2 t = (P + q) 2 a = P2 + Q 2 ) s t P 2 Q 2. st manifestly finite, good analytic properties, nice expansion F ɛ=0 = Li 2 (1 ap 2 ) + Li 2 (1 aq 2 ) Li 2 (1 as) Li 2 (1 at) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

41 Higher orders in dim reg Two loops higher orders in ɛ? Four points The 2-loop amplitude M (2) 4 [Bern Rozowsky Yan] M (2) 4 = 1 4 s2 ti (2) 4 + (s t), I (2) 4 = given analytically to O(ɛ 2 ) in [Bern Dixon Smirnov] fairly simple remainder (no HPLs): R (2) 4 (ɛ) = ɛ E 4 + O(ɛ 2 ) E 4 = 5 Li 5 ( x) 4L Li 4 ( x) (3L2 + π 2 ) Li 3 ( x) L 3 (L2 + π 2 ) Li 2 ( x) 1 24 (L2 + π 2 ) 2 log(1 + x) π4 L 39 2 ζ π2 ζ 3, (x = s/t L = log(s/t)) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

42 Higher orders in dim reg Four-point Wilson loop At four-points we have found analytic expressions to all orders in ɛ for all diagrams (except the hard diagram) Example = 1 1 τ1 σ1 dτ 1 dσ 1 dτ 2 dσ u u ( sσ 1 tτ 2 uσ 1 τ 2 ) 1+ɛ ( sσ 2 tτ 1 uσ 2 τ 1 ) 1+ɛ = 1 8ɛ 4 ( ) st 2ɛ { 14 3 F 2 (2ɛ, 2ɛ, 2ɛ; 1 + 2ɛ, 1 + 2ɛ; x)x 2ɛ + Γ( ɛ + 1)2 3F 2 (2ɛ, ɛ, ɛ; 1 + ɛ, 1 + ɛ; x)x ɛ u Γ( 2ɛ + 1) + 1 [ 2F 1 (ɛ, ɛ; 1 + ɛ; x)x ɛ ( + 2 F 1 ɛ, ɛ; 1 + ɛ; 1 ) ] 2 x ɛ 2πɛ cot(ɛπ) 4 x ɛ 2 π cot(2πɛ) (ψ(2ɛ) + γ) π 2 ɛ 2 cot(πɛ) 2 ( + x 1 )} x Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

43 Higher orders in dim reg WL/amplitude duality at O(ɛ) four-points hard diagram: analytic expression up to O(ɛ) these results WL remainder: R WL,(2) 4 (ɛ) = ɛ E 4,WL + O(ɛ 2 ) E 4,WL = 5 Li 5 ( x) 4L Li 4 ( x) (3L2 + π 2 ) Li 3 ( x) L 3 (L2 + π 2 ) Li 2 ( x) 1 24 (L2 + π 2 ) 2 log(1 + x) π4 L 33 2 ζ π2 ζ 3, Remarkably this is completely identical to the amplitude remainder up to a constant! E 4 = E 4,WL 3 ζ 5 Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

44 Five points? Higher orders in dim reg (parity even part of the) amplitude M (2) is built from: 5+ [Cachazo Spradlin Volovich, Bern Czakon Kosower Roiban Smirnov 2006, Bern Rozowsky Yan] We compute these numerically to O(ɛ) using MB [Czakon] We also compute all pentagon Wilson loop diagrams numerically using MB (with the help of the LHC grid) Find E 5 and E 5,WL numerically Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

45 5 point results Higher orders in dim reg Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

46 Higher orders in dim reg 5 point results: remainders at O(ɛ) # (s 12, s 23, s 34, s 45, s 51 ) E (2) E (2) 5 5,WL 1 ( 1, 1, 1, 1, 1) ± ( 1, 1, 2, 1, 1) ± ( 1, 2, 2, 1, 1) ± ( 1, 2, 3, 4, 5) ± ( 1, 1, 3, 1, 1) ± ( 1, 2, 1, 2, 1) ± ( 1, 3, 3, 1, 1) ± ( 1, 2, 3, 2, 1) ± ( 1, 3, 2, 5, 4) ± ( 1, 3, 1, 3, 1) ± ( 1, 4, 8, 16, 32) ± ( 1, 8, 4, 32, 16) ± ( 1, 10, 100, 10, 1) ± ( 1, 100, 10, 100, 1) ± ( 1, 1, 100, 1, 1) ± ( 1, 100, 1, 100, 1) ± ( 1, 100, 100, 1, 1) 2.60 ± ( 1, ( 100, 10, 100, 10) ) ± , 1 4, 1 9, 16 1, 25 1 ( ) ± , 1 9, 1 4, 25 1, ± ) 21 ( 1, 1, 1 4, 1, ± ) 22 ( 1, 1 4, 4 1, 1, ± ) 23 ( 1, 1 4, 1, 4 1, ± ) 24 ( 1, 1 4, 1 9, 1 4, ± ) 25 ( 1, 1 9, 1 4, 1 9, ± Consistent with E (2) 5 E (2) 5,WL = 5 2 ζ Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

47 Higher orders in dim reg 5 point results: remainders at O(ɛ) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

48 Going further... Higher orders in dim reg O(ɛ 2 ) four and five-points? Work in progress O(ɛ) six-points and higher? at six-points and higher we no longer expect a match with the parity even part of the amplitude 6pnt 1 loop parity even integrals (p 2 = p 2 [4] µ2 p) I 1m and I 2me are pseudo-conformal and determined to all orders in ɛ by the WL I hex is O(ɛ), non-conformal, missing from the WL Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

49 Higher orders in dim reg Two loops 6-point MHV amplitude 6 point two loop parity even integrals [Bern Dixon Kosower Roiban Spradlin Vergu Volovich] all contributing integrals Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

50 Higher orders in dim reg Two loops 6-point MHV amplitude 6 point two loop parity even integrals [Bern Dixon Kosower Roiban Spradlin Vergu Volovich] pseudo-conformal integrals Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

51 Higher orders in dim reg Two loops 6-point MHV amplitude 6 point two loop parity even integrals [Bern Dixon Kosower Roiban Spradlin Vergu Volovich] non-conformal integrals O(ɛ) (in log M) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

52 Higher orders in dim reg Two loops 6-point MHV amplitude 6 point two loop parity even integrals [Bern Dixon Kosower Roiban Spradlin Vergu Volovich] also parity odd integrals: [Cachazo Spradlin Volovich] non-conformal O(ɛ) (in log M) Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

53 Higher orders in dim reg Two loops 6-point MHV amplitude 6 point two loop parity even integrals [Bern Dixon Kosower Roiban Spradlin Vergu Volovich] WL/amplitude duality at higher orders in ɛ n-points? WL determines the pseudo conformal part of the amplitude Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35

54 Conclusions Summary Wilson loops = MHV amplitudes in N = 4 SYM compute all n-point two loop WL/amplitudes Two unexpected matches strong weak match for dimensional 8 point kinematics all orders expressions for MHV amplitudes? WL/amplitude duality beyond four dimensions the WL/amplitude duality is more fundamental than symmetry? Much more to be understood Paul Heslop ( Durham University) New insights in amplitude/wl duality Amplitudes / 35