Twistors and Perturbative Gravity

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

2. 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. 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. Featutures of Duality Topological String Theory with twistor target space CP3 -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. Topological String Theory: harder, uninteresting Perturbative Gauge Theories, hard, interesting Is the duality useful? Theory A : hard, interesting Theory B: easy -duality may be useful indirectly

6. Twistor Definitions • Consider a massless particle with momenta • We can realise as • So we can express where are two component Weyl spinors

7. 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. Transform to Twistor Space Penrose+ -note we make a choice which to transform new coordinates Twistor Space is a complex projective (CP3) space n-point amplitude is defined on (CP3)n

9. 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. We test collinearity and coplanarity by acting with differential operators Fijkand Kijkl -action of F is obtained using fact that points Zi collinear if Allows us to test without determining

11. Collinearity of MHV amplitudes • We organise gluon scattering amplitudes according to the number of negative helicities • Amplitude withno 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. Collinearity of MHV amplitudes • MHV amplitudes only depend upon • So, for Yang-Mills, FijkAn=0 trivially • MHV amplitudes have collinear support when transforming to a function in twistor space since Penrose transform yields a function after integration .

13. MHV amplitudes have suppport on line only Curve of degree 1 (= 0+2-1)

14. 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. Techniques:I MHV-vertex construction • Promotes MHV amplitude to fundamental object by off-shell continuation • 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) Cachazo Svrcek Witten, Nair Wu,Zhu; Su,Wu; Georgiou Khoze Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Ozeren+Stirling

16. + A MHV diagram _ _ + _ + _ _ + _ + _ + + _ -three point vertices allowed -number of vertices = (number of -) -1

17. eg for NMHV amplitudes k+ k+1+ q + - + 1- 3- 2- 2(n-3) diagrams Topology determined by number of –ve helicity gluons

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

19. Techniques:2 Recursion Relations Britto,Cachazo,Feng and Witten • 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 Within the amplitude momenta containing only one of the pair are z-dependant q(z)

20. 1 2 -results in recursive on-shell relation ( cf Berends-Giele off-shell recursive technique ) q (three-point amplitudes must be included) Amplitude has poles  Amplitude is poles

21. 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. 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. …..Perturbative Quantum Gravity…first some review

24. 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. Kawai-Lewellen-Tye Relations Kawai,Lewellen Tye, 86 -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 Berends,Giele, Kuijf

26. 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. Gravity MHV amplitudes • For Gravity Mn is polynomial in with degree (2n-6), eg • Consequently • In fact….. • Upon transforming Mnhas a derivative of  function support

28. 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. MHV construction for gravity • Need the correct off-shell continuation • Proved to be difficult • Resolution involves continuing the of the negative helicity legs • The ri are chosen so that a) momentum is conserved b) multi-particle poles q2(ri) are on-shell -this fixes them uniquely Shift is the same as that used by Risager to derive MHV rules using analytic structure

30. Eg NMHV amplitudes k+ k+1+ + - + 1- 3- 2-

31. 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. Tale of two theories, N=4 SYM vs N=8 Supergravity Cremmer, Julia, Scherk 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

33. degree p in l p Vertices involve loop momentum propagators General Decomposition of One- loop n-point Amplitude p=n : Yang-Mills p=2n Gravity

34. 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. 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 ci determines amplitude • 4pt…. Green, Schwarz, Brink • MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca Dunbar, Kosower Britto, Cachazo, Feng; RoibanSpradlin Volovich Bidder, Perkins, Risager

36. Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

37. Box Coefficients-Twistor Structure • Box coefficients has coplanar support for NMHV 1-loop • amplitudes -true for both N=4 and QCD!!!

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

40. Beyond one-loops Two-Loop Result obtained by reconstructing amplitude from cuts

41. Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan IPs,tplanar double box integral Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR) -N=8 amplitudes very close to N=4

42. Beyond 2-loops: UV pattern (98) Honest calculation/ conjecture (BDDPR) N=8 Sugra N=4 Yang-Mills Based upon 4pt amplitudes

43. 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(li)=(l1+l2)2 SUGRA :P(li)=((l1+l2)2)2 l1 l2 I[ P(li)] BUT…………..

44. 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. Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4SYM 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. 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?