1. Twistors and Perturbative Gravity Emil Bjerrum-Bohr UK Theory Institute 20/12/05 Steve Bidder Harald Ita Warren Perkins +Zvi Bern (UCLA) and Kasper Risager (NBI) Dave Dunbar, Swansea University

2. D Dunbar UK Inst 05 2/46 Plan Recently a duality between Yang-Mills and twistor string theory has inspired a variety of new techniques in perturbative Yang-Mills theories. First part of talk will review these Look at Gravity Amplitudes -which, if any, features apply to gravity Application: Loop Amplitudes N=4 Yang –Mills N=8 Supergravity Consequences and Conclusions

3. D Dunbar UK Inst 05 3/46 Duality with String Theory Witten (2003) proposed a Weak-Weak duality between A) Yang-Mills theory ( N=4 ) B) Topological String Theory with twistor target space -Since this is a `weak-weak` duality perturbative S-matrix of two theories should be identical order by order - True for tree level scattering Rioban, Spradlin,Volovich

4. D Dunbar UK Inst 05 4/46 Featutures of Duality Topological String Theory with twistor target space CP 3 -open string instantons correspond to Yang-Mills states -theory has conformal symmetry, N=4 SYM -closed string states correspond to N=4 superconformal gravity - N < 4 ?? Berkovits+Witten, Berkovits

5. D Dunbar UK Inst 05 5/46 Is the duality useful? Theory A : hard, interesting hard, interesting Theory B: easy Perturbative Gauge Theories, hard, interesting Topological String Theory : harder, uninteresting -duality may be useful indirectly

6. D Dunbar UK Inst 05 6/46 Twistor Definitions Consider a massless particle with momenta We can realise as So we can express where are two component Weyl spinors

7. D Dunbar UK Inst 05 7/46 This decomposition is not unique but We can also turn polarisation vector into fermionic objects, ``Spinor Helicity`` formalism Xu, Zhang,Chang 87 -Amplitude now a function of spinor variables

8. D Dunbar UK Inst 05 8/46 Transform to Twistor Space Twistor Space is a complex projective (CP 3 ) space n-point amplitude is defined on (CP 3 ) n new coordinates -note we make a choice which to transform Penrose+

9. D Dunbar UK Inst 05 9/46 Twistor Structure Conjecture (Witten) : amplitudes have non-zero support on curves in twistor space support should be a curve of degree (number of –ve helicities)+(loops) -1 Carrying out the transform is problematic, instead we can test structure by acting with differential operators

10. D Dunbar UK Inst 05 10/46 We test collinearity and coplanarity by acting with differential operators F ijk and K ijkl -action of F is obtained using fact that points Z i collinear if Allows us to test without determining

11. D Dunbar UK Inst 05 11/46 Collinearity of MHV amplitudes We organise gluon scattering amplitudes according to the number of negative helicities Amplitude with no or one negative helicities vanish [ for supersymmetric theories to all order; for non-supersymmetric true for tree amplitudes] Amplitudes with exactly two negative helicities are refered to as `MHV` amplitudes Parke-Taylor, Berends-Giele (amplitudes are color-ordered)

12. D Dunbar UK Inst 05 12/46 Collinearity of MHV amplitudes MHV amplitudes only depend upon So, for Yang-Mills, F ijk A n =0 trivially MHV amplitudes have collinear support when transforming to a function in twistor space since Penrose transform yields a  function after integration.

13. D Dunbar UK Inst 05 13/46 MHV amplitudes have suppport on line only Curve of degree 1 (= 0+2-1)

14. D Dunbar UK Inst 05 14/46 NMHV amplitudes in twistor space amplitudes with three –ve helicity known as NMHV amplitudes remarkably NMHV amplitudes have coplanar support in twistor space prove this not directly but by showing - time to look at techniques motivated by duality

15. D Dunbar UK Inst 05 15/46 Techniques:I MHV-vertex construction Works for gluon scattering tree amplitudes Works for (massless) quarks Works for Higgs and W’s Works for photons -No known derivation from a Lagrangian (but…… Khoze, Mason, Mansfield) Ozeren+Stirling Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Wu,Zhu; Su,Wu; Georgiou Khoze Cachazo Svrcek Witten, Nair Promotes MHV amplitude to fundamental object by off-shell continuation

16. D Dunbar UK Inst 05 16/46 + _ _ _ _ _ + + + + + + _ _ _ -three point vertices allowed -number of vertices = (number of -) -1 A MHV diagram

17. D Dunbar UK Inst 05 17/46 eg for NMHV amplitudes 3-3- 1-1- k+k+ 2-2- k+1 + 2(n-3) diagrams + Topology determined by number of –ve helicity gluons -+ q

18. D Dunbar UK Inst 05 18/46 Coplanarity-byproduct of MHV vertices Two intersecting lines in twistor space define the plane -NMHV amplitudes is sum of two MHV vertices Curve is a degenerate curve of degree 2

19. D Dunbar UK Inst 05 19/46 Techniques:2 Recursion Relations Return of the analytic S-matrix! Shift amplitude so it is a complex function of z Amplitude becomes an analytic function of z, A(z) Full amplitude can be reconstructed from analytic properties Britto,Cachazo,Feng and Witten Within the amplitude momenta containing only one of the pair are z-dependant q(z)

20. D Dunbar UK Inst 05 20/46 -results in recursive on-shell relation (three-point amplitudes must be included) 12 ( cf Berends-Giele off-shell recursive technique ) q Amplitude has poles  Amplitude is poles

21. D Dunbar UK Inst 05 21/46 MHV vs BCF recursion Difference MHV asymmetric between helicity sign BCF chooses two special legs For NMHV : MHV expresses as a product of two MHV : BCF uses (n-1)-pt NMHV Similarities- both rely upon analytic structure both for trees but… Loops: MHV: Bedford, Brandhuber,Spence, Travaglini Recursive: Bern,Dixon Kosower; Bern, Bjerrum-Bohr, Dunbar, Ita, Perkins

22. D Dunbar UK Inst 05 22/46 Gravity-Strategy 1) Try to understand twistor structure 2) Develop formalisms - a priori we might expect Einstein gravity to contain no knowledge of twistor structure since duality contains conformal gravity

23. D Dunbar UK Inst 05 23/46 …..Perturbative Quantum Gravity …first some review

24. D Dunbar UK Inst 05 24/46 Feynman diagram approach to perturbative quantum gravity is extremely complicated Gravity = (Yang-Mills) 2 Feynman diagrams for Yang-Mills = horrible mess How do we deal with (horrible mess) 2 Using traditional techniques even the four-point tree amplitude is very difficult Sannan,86

25. D Dunbar UK Inst 05 25/46 Kawai-Lewellen-Tye Relations -pre-twistors one of few useful techniques -derived from string theory relations -become complicated with increasing number of legs -involves momenta prefactors -MHV amplitudes calculated using this Kawai,Lewellen Tye, 86 Berends,Giele, Kuijf

26. D Dunbar UK Inst 05 26/46 Recursion for Gravity Gravity, seems to satisfy the conditions to use recursion relations Allows (re)calculation of MHV gravity tree amps Expression for six-point NMHV tree Bedford, Brandhuber, Spence, Travaglini Cachazo,Svrcek Bedford, Brandhuber, Spence, Travaglini Cachazo,Svrcek

27. D Dunbar UK Inst 05 27/46 Gravity MHV amplitudes For Gravity M n is polynomial in with degree (2n-6), eg Consequently In fact….. Upon transforming M n has a derivative of  function support

28. D Dunbar UK Inst 05 28/46 Coplanarity NMHV amplitudes in Yang-Mills have coplanar support For Gravity we have verified n=5 by Giombi, Ricci, Robles-Llana Trancanelli n=6,7,8 Bern, Bjerrum-Bohr,Dunbar

29. D Dunbar UK Inst 05 29/46 MHV construction for gravity Need the correct off-shell continuation Proved to be difficult Resolution involves continuing the of the negative helicity legs The r i are chosen so that a) momentum is conserved b) multi-particle poles q 2 (r i ) are on-shell -this fixes them uniquely Shift is the same as that used by Risager to derive MHV rules using analytic structure

30. D Dunbar UK Inst 05 30/46 Eg NMHV amplitudes 3-3- 1-1- k+k+ 2-2- k+1 + + -+

31. D Dunbar UK Inst 05 31/46 Loop Amplitudes Loop amplitudes perhaps the most interesting aspect of gravity calculations UV structure always interesting Chance to prove/disprove our prejudices Studying Amplitudes may uncover symmetries not obvious in Lagrangian Loop amplitudes are sensitive to the entire theory For loops we must be specific about which theory we are studying

32. D Dunbar UK Inst 05 32/46 Tale of two theories, N=4 SYM vs N=8 Supergravity N=4 SYM is maximally supersymmetric gauge theory (spin · 1 ) N=8 Supergravity is maximal theory with gauged supersymmetry (spin · 2 ) -both appear in low energy limit of superstring theory -S-matrix of both theories is constrained by a rich set of symmetries -N=4 key in Weak-Weak duality -in D=4 YM has dimensionless coupling constant wheras gravity has a dimensionful coupling constant -both theories are extremelly important models: toy or otherwise Cremmer, Julia, Scherk

33. D Dunbar UK Inst 05 33/46 General Decomposition of One- loop n-point Amplitude Vertices involve loop momentum propagators p degree p in l p=n : Yang-Mills p=2n Gravity

34. D Dunbar UK Inst 05 34/46 Passarino-Veltman reduction process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) In going from 4-> 3 scalar boxes are generated similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. so in general, for massless particles Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

35. D Dunbar UK Inst 05 35/46 N=4 Susy Yang-Mills In N=4 Susy there are cancellations between the states of different spin circulating in the loop. Leading four powers of loop momentum cancel (in well chosen gauges..) N=4 lie in a subspace of the allowed amplitudes (Bern,Dixon,Dunbar,Kosower, 94) Determining rational c i determines amplitude -4pt…. Green, Schwarz, Brink -MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca Dunbar, Kosower Britto, Cachazo, Feng; Roiban Spradlin Volovich Bidder, Perkins, Risager

36. D Dunbar UK Inst 05 36/46 Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

37. D Dunbar UK Inst 05 37/46 Box Coefficients-Twistor Structure Box coefficients has coplanar support for NMHV 1-loop amplitudes -true for both N=4 and QCD!!!

38. D Dunbar UK Inst 05 38/46 N=8 Supergravity Loop polynomial of n-point amplitude of degree 2n. Leading eight-powers of loop momentum cancel (in well chosen gauges..) leaving (2n-8) Beyond 4-point amplitude contains triangles..bubbles Beyond 6-point amplitude is not cut-constructible

39. D Dunbar UK Inst 05 39/46 No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4pt n-point MHV 6pt NMHV -factorisation suggests this is true for all one-loop amplitudes Bern,Dixon,Perelstein,Rozowsky Bjerrum-Bohr, Dunbar,Ita Green,Schwarz,Brink consequences? One-Loop amplitudes N=8 SUGRA look just like N=4 SYM

40. D Dunbar UK Inst 05 40/46 Beyond one-loops Two-Loop Result obtained by reconstructing amplitude from cuts

41. D Dunbar UK Inst 05 41/46 Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR) -N=8 amplitudes very close to N=4 I P s,t planar double box integral

42. D Dunbar UK Inst 05 42/46 Beyond 2-loops: UV pattern (98) D=110 #/  D=100(!) #/  D=90 #/  D=8 #/  #’/   +#”/  D=70 #/  D=600 D=5000 D=40000 L=1L=2L=3L=4L=5L=6 N=4 Yang-Mills Honest calculation/ conjecture (BDDPR) N=8 Sugra Based upon 4pt amplitudes

43. D Dunbar UK Inst 05 43/46 Pattern obtained by cutting Beyond 2 loop, loop momenta get ``caught’’ within the integral functions Generally, the resultant polynomial for maximal supergravity of the square of that for maximal super yang-mills Eg in this case YM :P(l i )=(l 1 +l 2 ) 2 SUGRA :P(l i )=((l 1 +l 2 ) 2 ) 2 I[ P(l i )] l1l1 l2l2 BUT…………..

44. D Dunbar UK Inst 05 44/46 on the three particle cut.. For Yang-Mills, we expect the loop to yield a linear pentagon integral For Gravity, we thus expect a quadratic pentagon However, a quadratic pentagon would give triangles which are not present in an on-shell amplitude -indication of better behaviour in entire amplitude ? relations to work of Green and Van Hove

45. D Dunbar UK Inst 05 45/46 Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4 SYM than power counting/field theory arguments suggest???? If factorisation is the key then perhaps yes. Four point amplitudes very similar Is N=8 SUGRA perturbatively finite?????

46. D Dunbar UK Inst 05 46/46 Conclusions Perturbation theory is interesting and still contains many surprises Recent “discoveries” are interesting and useful Studying on-shell amplitudes can give information not obvious in the Lagrangian Gravity calculations amenable to many of the new twistor inspired techniques -both recursion and MHV– vertex formulations exist -perturbative expansion of N=8 seems to be surprisingly simple. This may have consequences for the UV behaviour Consequences for the duality?