Twistors and Pertubative Gravity

1. Twistors and Pertubative Gravity including work (2005) with Z Bern, S Bidder, E Bjerrum-Bohr, H Ita, W Perkins, K Risager From Twistors to Amplitudes 2005

2. Summary • Review of Perturbative Gravity KLT approach • Recursive approach • MHV vertex approach • Loops • N=8, 1-loop comparison with gravity • beyond one-loop • Conclusions

3. Perturbative Quantum Gravity

4. Feynman diagram approach to 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

5. Kawai-Lewellen-Tye Relations KLT,86 -pre-twistors one of few useful techniques -derived from string theory relations -become complicated with increasing number of legs -contains unneccessary info -MHV amplitudes calculated using this Berends,Giele, Kuijf

6. Double-Poles • Naively, products of Yang-Mills amplitudes would contain double poles • A(1,2,3,4,5)xA(2,1,3,4,5) • Cancelled by momentum prefactors s34 s12 • Factorisation structure not manifest • Crossing Symmetric although not manifest

7. Twistor Structure Of Gravity Amplitudes • Look for Twistor inspired formalism • Not obvious such formalism exist (conformal gravity..) • Can we examine twistor structure by action of differential operators?

8. Collinearity of MHV amplitudes • For Yang-Mills FijkAn=0 trivially • This implies MHV amplitudes have collinear support when transforming to a function in twistor space • Independence upon implies has a  function

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

10. MHV amplitudes have suppport on line only -For Yang-Mills there is  function -For Gravity it is a derivative of a  function

11. 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

12. Coplanarity-MHV vertices -Points on one MHV vertex Two intersecting lines in twistor space define the plane

13. 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 P(z)

14. 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

15. 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 • Works for gravity……. Cachazo Svrcek Witten++ Wu,Zhu; Su,Wu; Georgiou Khoze Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Ozeren+Stirling Bjerrum-Bohr,DCD,Ita,Perkins, Risager

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

17. -problem 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 P(z) are on-shell -this fixes them uniquely Shift is the same as that used by Risager to derive MHV rules using analytic structure

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

19. applying momentum conservation gives -this a combination of three BCF shifts -demanding P(z)2=0 gives the condition on z -which fixes z and so determines prescription

20. Makes MHV apparent as a analytic shift • Has interpretation as contact terms since • and the P2 can cancel pole between MHV vertices • Construction ``expands’’ contact terms in a consistent manner

21. 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

22. Supersymmetric Decomposition Supersymmetric decomposition important for QCD amplitudes -this can be inverted

23. Decomposition of Graviton One-Loop Scattering Amplitude Known for Four-Point only -N=8 Green Schwarz & Brink ’ ! 0 limit of string theory, 1985 -N=0 Grisaru & Zak, 1980 -remainder Dunbar & Norridge, 1996 -focus upon N=8 for rest of talk

24. 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

25. l-k l k Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

26. -process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) • -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

27. 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 (BDDK) • 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 • UV finiteness of one-loop amplitudes trivial

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

29. 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

30. 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 Bern, Bjerrum-Bohr, Dunbar,Ita -factorisation suggests this is true for all one-loop amplitudes

31. consequences? • One-Loop amplitudes look just like N=4 SYM • UV finiteness obvious • …..as it is from field theory analysis • ..but no so for N<8 Dunbar,Julia,Seminara,Trigiante, 00

32. Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan IP planar double box integral Bern,Dixon,Dunbar,Perelstein,Rozowsky -N=8 amplitudes very close to N=4

33. Beyond 2-loops: UV pattern Honest calculation/ conjecture (BDDPR) N=8 Sugra N=4 Yang-Mills

34. 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.

35. Conclusions • Gravity calculations amenable to many of the new techniques • Both recursion and MHV– vertex formulations exist • Perturbative expansion of N=8 seems to be surprisingly simple. This may have consequences • Consequences for the duality?