Computing amplitudes using Wilson loops

Transcription

1 Computing amplitudes using Wilson loops Paul Heslop Queen Mary, University of London NBIA, Copenhagen 13th August 2009 based on: Brandhuber, Travaglini, P.H Anastasiou, Brandhuber, Khoze, Spence, Travaglini, P.H. Work in progress

2 Brief introduction to amplitudes in N = 4 Duality between two objects in N =4 Super Yang-Mills: 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 Vast simplification of the computation of amplitudes Example We compute all MHV 2-loop gluon scattering amplitudes (assuming the conjectured duality) for any n.

3 1 Introduction 2 The duality The evidence so far... Wilson loop calculations - 1 loop Wilson loop calculations - 2 loop 3 Results of two-loop computations, n = 6, 7, 8 4 Higher n

4 Motivation Theoretical Hidden structures/symmetries (eg twistor string theory, no triangle hypothesis, dual conformal symmetry, integrability...) Practical Simpler/faster ways to compute amplitudes (recent advances include generalised unitarity/bcfw recursion relations etc.) Background processes at LHC

5 MHV Amplitudes in N = 4 SYM Colour-stripped, planar Maximally Helicity Violating (MHV) amplitudes j + i We will focus on M (L) n A n = A tree n M n

6 L-loop amplitude The BDS conjecture [Anastasiou Bern Dixon Kosower 2003, Bern Dixon Smirnov 2005] IR divergences: dimensional regularisation d = 4 2ɛ The BDS formula: an all-loop expression for any n ( ) log M n (ɛ) = ( a L L=1 f (L) A ) (ɛ)m(1) n (Lɛ) + C (L) + O(ɛ) a is the t Hooft coupling constant Here f (L) (L) A (ɛ) = f 0 + f (L) 1 ɛ + f (L) 2 ɛ2 where f (L) is a number. needs modification from n = 6 points... i

7 Outline 1 Introduction 2 The duality The evidence so far... Wilson loop calculations - 1 loop Wilson loop calculations - 2 loop 3 Results of two-loop computations, n = 6, 7, 8 4 Higher n

8 Amplitude/Wilson loop duality p 5 p 6 p 1 = x p 2 p 1 x p 4 p 2 p 3 amplitude M n = W [C n ] (d = 4 2ɛ) (d = 4 + 2ɛ) p 6 p 5 5 p 4 p 3 4 Wilson loop over the [ polygonal contour C ( n ) ] W [C] := Tr P exp ig dτ A µ (x(τ))ẋ µ (τ) C

9 Amplitude/Wilson loop duality p 6 p 1 p 5 = p 4 p 2 p 3 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 dτ A µ (x(τ))ẋ µ (τ) C

10 Evidence so far... Number of loops infinity 4 3 amplitude calculated at strong coupling (using string theory via AdS/CFT correspondence) amplitude=wilson loop [Alday, Maldacena 2007] infinity Number of points n

11 Evidence so far... Number of loops infinity points 1 loop Wilson loop = amplitude [Green, Schwarz, Brink 1982] [Drummond Korchemsky Sokatchev 2007] infinity Number of points n

12 Evidence so far... Number of loops infinity loop, all n, Wilson loop = amplitude [Brandhuber Travaglini PH 2007] [Bern Dixon Dunbar Kosower 1994] infinity Number of points n

13 Evidence so far... Number of loops infinity loops, 4 points, Wilson loop = amplitude [Drummond, Henn, Korchemsky, Sokatchev 2007] [Bern Rosowsky Yan 1997] infinity Number of points n

14 Evidence so far... Number of loops infinity } Dual conformal symmmetry (4,5 points all loops) (2 loop, 5 pt) Wilson loop = amplitude [Drummond, Henn, Korchemsky, Sokatchev 2007] [Cachazo Spradlin Volovich 2006, Bern Czakon Kosower Roiban Smirnov 2006] infinity Number of points n

15 Evidence so far... Number of loops infinity All in agreement with BDS } Dual conformal symmmetry (4,5 points all loops) (2 loop, 5 pt) Wilson loop = amplitude [Drummond, Henn, Korchemsky, Sokatchev 2007] [Cachazo Spradlin Volovich 2006, Bern Czakon Kosower Roiban Smirnov 2006] infinity Number of points n

16 Remainder function n 6 ( ) log M n (ɛ) ( ) log W n (ɛ) = = a l f (l) A (ɛ)m(1) n (lɛ) + C A (a) + R A n (p i ; a) + O(ɛ) l=1 l=1 a l f (l) W (1) (ɛ)w n (lɛ) + C w (a) + R W n (p i ; a) + O(ɛ) non-zero remainder function found for the two-loop six-point amplitude and the Wilson loop [Drummond Henn Korchemsky Sokatchev 2008, Bern Dixon Kosower Roiban Spradlin Vergu Volovich 2008]

17 Evidence so far... Number of loops infinity 4 3 (2 loop, 6 pt) Wilson loop = amplitude R W =R A 0 [Drummond Henn Korchemsky Sokatchev 2008] [Bern Dixon Kosower Roiban Spradlin Vergu Volovich 2008] infinity Number of points n

18 Wilson loop calculations, 1-loop the expression for the n point amplitude and for the WL are very closely related: Amplitude = p,q +O(ɛ) = Wilson loop = p,q P = q 1 k=p+1 k Q = p 1 k=q+1 k WL

19 Wilson loop calculations, 1-loop the expression for the n point amplitude and for the WL are very closely related: Amplitude = p,q +O(ɛ) = Wilson loop = p,q P = q 1 k=p+1 k Q = p 1 k=q+1 k WL

20 2-loop n-point Wilson loop (log of) Only four new master integrals to be computed for all n f H (p 1, p 2, p 3 ; Q 1, Q 2, Q 3 ) f Y (p 1, p 2 ; Q 1, Q 2 ) f X (p 1, p 2 ; Q 1, Q 2 ) f C (p 1, p 2, p 3 ; Q 1, Q 2, Q 3 )

21 Also factorised cross diagram This is given by the product of two one loop diagrams 1/2f P (p i, p j ; Q ji, Q ij )f P (p k, p l ; Q lk, Q kl )

22 ( Compare with amplitude (parity even part)) n = 4 n = 5 n = 6

23 n = 7 [Vergu]

24 Complete 2-loop Wilson loop The logarithm of the complete n-sided Wilson loop is given in terms of the four new master diagrams together with the one loop diagram f P (p i, p j ; Q ji, Q ij ) as 1 i<j<k n i<j n 1 i<k<j<l n [ f H (p i, p j, p k ; Q jk, Q ki, Q ij ) + f C (p i, p j, p k ; Q jk, Q ki, Q ij ) ] + f C (p j, p k, p i ; Q ki, Q ij, Q jk ) + f C (p k, p i, p j ; Q ij, Q jk, Q ki ) [ ] f X (p i, p j ; Q ji, Q ij ) + f Y (p i, p j ; Q ji, Q ij ) + f Y (p j, p i ; Q ij, Q ji ) ( 1/2)f P (p i, p j ; Q ji, Q ij )f P (p k, p l ; Q lk, Q kl )

25 Outline 1 Introduction 2 The duality The evidence so far... Wilson loop calculations - 1 loop Wilson loop calculations - 2 loop 3 Results of two-loop computations, n = 6, 7, 8 4 Higher n

26 Computations at n = 6, 7, 8... Using sector decomposition and the numerical techniques of [Anastasiou Beerli Daleo (2007,2008), Lazopoulos Melnikov Petriello (2007), Anastasiou Melnikov Petriello (2005)] we compute the 2-loop master integrals Computations of WL performed for n = 4, 5, 6, 7, 8 considerable amount of data collected. Verified that the remainder function is conformally invariant Verified cyclic and parity (dihedral) symmetry Collinear limits

27 Conformal invariants: cross-ratios Number of independent cross-ratios is n(n 5)/2 Basis: u ij = x 2 ij+1 x 2 ji+1 x 2 ij x 2 i+1j+1 This ignores the Gram determinant n(n 5)/2 > 3n 15 physical kinematics will form a 3n 15 dimensional slice of this space of cross-ratios

28 Hexagon computations 3 cross-ratios u 36 = x 2 31 x 2 46 x 2 36 x 2 41 := u 1, u 14 = x 2 15 x 2 24 x 2 14 x 2 25 := u 2, u 25 = x 2 26 x 2 35 x 2 25 x 2 36 := u 3 remainder function R(u 1, u 2, u 3 )

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30 6-pnt Wilson loop R W 6 with u 1 = u, u 2 = v, u 3 = w w = 1 blue, w = 10 green, w = 100 yellow, w = 1000 orange, w = red

31 Plot of R 6 (u, u, u)

32 Plot of R 6 (u, u, u)

33 Plot of R 7 (u, u, u, u, u, u, u)

34 Plot of R 7 (u, u, u, u, u, u, u)

35 Plot of R 7 (u, u, u, u, u, u, u)

36 8 points Conformal invariance Cyclicity and parity (also checked at 6, 7 points)

37 Collinear limits R n (u) should have trivial simple collinear limits R n R n 1 We verify this for n = 6, 7, 8 (with no constant shifts)

38 Outline 1 Introduction 2 The duality The evidence so far... Wilson loop calculations - 1 loop Wilson loop calculations - 2 loop 3 Results of two-loop computations, n = 6, 7, 8 4 Higher n

39 Higher n We can compute for arbitrarily large n Alday and Maldacena recently considered special n-point amplitude kinematics at strong coupling via string theory t y y xx x Momenta in dimensions in notation (t, z) x 2k = ( 2 sin π 2k+1 ), eiπ n, x2k+1 = ( ) 0, e iπ 2k n 2n space projection = regular polygon, zig-zags in time

40 this kinematics leads to the cross-ratios u ij = 1, i j = odd, ( sin π ) 2 n u ij = 1 sin πa, i j = 2a, n ( 3 At strong coupling: A n = π 8 n ) [Alday Maldacena] n does the weak coupling result share any features with this? eg naive counting of two loop diagrams n 4 growth put the above kinematics in our program...

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42 Results: no.points R n n = n = n = n = n = n = n = n = n = n = n =

43 Plot of two-loop data versus linear fit Best linear fit: R n n Error /n term: R n n /n Error 0.01 Including 1/n 2 term R n n /n /n 2 Error = numerical error

44 Summary of results Summary: the number of distinct integrals for the 2-loop n-gon WL is independent of n We compute all n-sided polygonal light-like Wilson loops at two loops (eg recent computation of an n = 30 WL) no additional complexity as n increases: the number of diagrams increases but the type of integral is n-independent Assuming the amplitude/wilson loop duality we compute two-loop planar MHV amplitudes for any number of points

45 Number of loops infinity 4 two loop Wilson loop computed for all n [Anastasiou Brandhuber Khoze Spence Travaglini PH] 3 2 W W infinity Number of points n

46 Future directions amplitude calculation at n 7-points needed! [Vergu] analytic determination of 6-pnt amplitude/wilson loop Proof of WL/amplitude duality Generalisations of WL to NMHV amplitudes etc. [ Berkovits Maldacena] Generalisations to other theories Understanding the role of standard (super)conformal symetry Yangian, infinite new symmetries ( integrability) [Beisert Ricci Tseytlin Wolf, Berkovits Maldacena, Drummond Henn Plefka, Bargheer Beisert Galleas Loebbert McLoughlin]