1 / 35

Twistors and Pertubative Gravity

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. Summary. Review of Perturbative Gravity KLT approach Recursive approach MHV vertex approach Loops N=8, 1-loop

jackie
Download Presentation

Twistors and Pertubative Gravity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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?

More Related