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. Dave Dunbar QMUL, Nov 05 2/36 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. Dave Dunbar QMUL, Nov 05 3/36 Perturbative Quantum Gravity

4. Dave Dunbar QMUL, Nov 05 4/36 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. Dave Dunbar QMUL, Nov 05 5/36 Kawai-Lewellen-Tye Relations -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 KLT,86 Berends,Giele, Kuijf

6. Dave Dunbar QMUL, Nov 05 6/36 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 s 34 s 12 Factorisation structure not manifest Crossing Symmetric although not manifest

7. Dave Dunbar QMUL, Nov 05 7/36 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. Dave Dunbar QMUL, Nov 05 8/36 Collinearity of MHV amplitudes For Yang-Mills F ijk A n =0 trivially This implies MHV amplitudes have collinear support when transforming to a function in twistor space Independence upon implies has a  function

9. Dave Dunbar QMUL, Nov 05 9/36 Gravity MHV amplitudes For Gravity M n is polynomial in with degree (2n-6), eg Consequently In fact….. Upon transforming M has a derivative of  function support

10. Dave Dunbar QMUL, Nov 05 10/36 MHV amplitudes have suppport on line only -For Yang-Mills there is  function -For Gravity it is a derivative of a  function

11. Dave Dunbar QMUL, Nov 05 11/36 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. Dave Dunbar QMUL, Nov 05 12/36 Coplanarity-MHV vertices Two intersecting lines in twistor space define the plane -Points on one MHV vertex

13. Dave Dunbar QMUL, Nov 05 13/36 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 P(z)

14. Dave Dunbar QMUL, Nov 05 14/36 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. Dave Dunbar QMUL, Nov 05 15/36 MHV-vertex construction Works for gluon scattering tree amplitudes Works for (massless) quarks Works for Higgs and W’s Works for photons Works for gravity……. Bjerrum-Bohr,DCD,Ita,Perkins, Risager Ozeren+Stirling Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Wu,Zhu; Su,Wu; Georgiou Khoze Cachazo Svrcek Witten++ Promotes MHV amplitude to fundamental object by off-shell continuation

16. Dave Dunbar QMUL, Nov 05 16/36 + _ _ _ _ _ + + + + + + _ _ _ -three point vertices allowed -number of vertices = (number of -) -1

17. Dave Dunbar QMUL, Nov 05 17/36 -problem 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 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. Dave Dunbar QMUL, Nov 05 18/36 Eg NMHV amplitudes 3-3- 1-1- k+k+ 2-2- k+1 + + -+

19. Dave Dunbar QMUL, Nov 05 19/36 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. Dave Dunbar QMUL, Nov 05 20/36 Makes MHV apparent as a analytic shift Has interpretation as contact terms since and the P 2 can cancel pole between MHV vertices Construction ``expands’’ contact terms in a consistent manner

21. Dave Dunbar QMUL, Nov 05 21/36 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. Dave Dunbar QMUL, Nov 05 22/36 Supersymmetric Decomposition Supersymmetric decomposition important for QCD amplitudes -this can be inverted

23. Dave Dunbar QMUL, Nov 05 23/36 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. Dave Dunbar QMUL, Nov 05 24/36 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

25. Dave Dunbar QMUL, Nov 05 25/36 Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator k l l-kl-k

26. Dave Dunbar QMUL, Nov 05 26/36 -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. Dave Dunbar QMUL, Nov 05 27/36 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 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 UV finiteness of one-loop amplitudes trivial

28. Dave Dunbar QMUL, Nov 05 28/36 Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

29. Dave Dunbar QMUL, Nov 05 29/36 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. Dave Dunbar QMUL, Nov 05 30/36 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 Bern, Bjerrum-Bohr, Dunbar,Ita Green,Schwarz,Brink

31. Dave Dunbar QMUL, Nov 05 31/36 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. Dave Dunbar QMUL, Nov 05 32/36 Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan Bern,Dixon,Dunbar,Perelstein,Rozowsky -N=8 amplitudes very close to N=4 I P planar double box integral

33. Dave Dunbar QMUL, Nov 05 33/36 Beyond 2-loops: UV pattern D=110 #/  D=100(!) #/  D=90 #/  D=8 #/  #’/  2 +#”/  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

34. Dave Dunbar QMUL, Nov 05 34/36 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.

35. Dave Dunbar QMUL, Nov 05 35/36 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?