1. UV structure of N=8 Supergravity Emil Bjerrum-Bohr, IAS Windows on Quantum Gravity 18 th June 08, UCLA Harald Ita, UCLA Warren Perkins Dave Dunbar, Swansea University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A Kasper Risager, NBI Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, ``The no-triangle hypothesis for N = 8 supergravity,'‘ JHEP 0612 (2006) 072, hep-th/0610043.

2. D Dunbar Windows on Quantum Gravity 2/44 Objective Is N=8 Supergravity a self-consistent Quantum Field theory? Does the theory have ultra-violet singularities or is it a ``finite’’ field theory.

3. D Dunbar Windows on Quantum Gravity 3/44 N=8 Maximal Supergravity? Field theory with N=8 supersymmetry One graviton, eight gravitinos………….70 scalars Maximal supersymmetry consistent with spins <=2 Field theory which couples gravity to all sorts of particles Endless symmetries… …really complicated Lagrangian Descendant of N=1 in D=11 gg 21  3/28 AA 128  ½56  070(35) Cremmer,Julia, Scherk (ungauged)

4. D Dunbar Windows on Quantum Gravity 4/44 Superstring Theory 2) Look at supergravity embedded within string theory N=8 Supergravity 1) Approach problem within the theory Dual Theory 3) Find a dual theory which is solvable Green, Russo, Van Hove, Berkovitz, Chalmers Abou-Zeid, Hull, Mason ``Finite for 8 loops but not beyond’’

5. D Dunbar Windows on Quantum Gravity 5/44 …..Perturbative Quantum Gravity

6. D Dunbar Windows on Quantum Gravity 6/44 Quantum Problems: Renormalisability -calculate scattering amplitudes using Feynman vertices etc - Only works if g is dimensionless = + Then we remove singularities by renormalising

7. D Dunbar Windows on Quantum Gravity 7/44 Gravity Gravity cannot be renormalised (in D=4) Infinities must be removed by adding terms to lagrangian not present initially. If we have to continually add terms then theory looses predictive power Can avoid this if theory has no UV divergences (finite) (eg N=4 SYM in D=4 and String Theory) -can we find a finite field theory of gravity???

8. D Dunbar Windows on Quantum Gravity 8/44 Feynman diagram approach to perturbative quantum gravity is not terrible useful Using traditional techniques even the four-point tree amplitude for four graviton scattering is very difficult Sannan,86 Supergravity calculations where we must calculate using all particles in multiplet are even more difficult…..

9. D Dunbar Windows on Quantum Gravity 9/44 however…… N=8 supergravity has a lot of particles but it has enormous symmetry amongst them Although computations are very difficult end results which must express this symmetry can be rather simple New techniques which use symmetry to generate scattering amplitudes are particularly useful for supergravity -return to S-matrix theory?

10. D Dunbar Windows on Quantum Gravity 10/44 -try to derive behaviour from N=4 SYM N=4 SYM is a finite field theory Try to exploit links to this for N=8 supergravity In string theory, closed string ~ open string x open string So… N=8 ~ ( N=4) x (N=4) Mandelstam

11. D Dunbar Windows on Quantum Gravity 11/44 Kawai-Lewellen-Tye Relations -derived from string theory relations -become complicated with increasing number of legs -involves momenta prefactors -applies to N=8/N=4 (and consequently pure YM/gravity) Kawai,Lewellen Tye, 86

12. D Dunbar Windows on Quantum Gravity 12/44 Loop Calculations in N=8 Supergravity Desperately complicated using Feynman diagrams Pre strings revolution of 1984 people believed theory was finite. [Only candidate for quantum gravity…] Post 1984 people believed theory was non-renormalisable and only appeared as a low energy effective theory [ of string theory] In D=4 ``expect infinities’’ at 3-loops. [At this time no definite calculation of any infinity in D=4 in any supergravity theory] In D > 4 they appear earlier ( ….. s d D p » s |p| D-1 d|p| )

13. D Dunbar Windows on Quantum Gravity 13/44 One-Loop Amplitudes Calculated by Green Schwarz and Brink using string theory I 4 (s,t) is scalar box integral Remarkably similar to the N=4 Yang-Mills results (colour ordered/leading in colour part/planar) 2 1 3 4

14. D Dunbar Windows on Quantum Gravity 14/44 Two-Loop Supergravity, all D form, Bern,Dixon,Dunbar,Perelstein,Rozowsky -N=8 amplitudes very close to N=4 (planar part) Bern, Rozowsky, Yan

15. D Dunbar Windows on Quantum Gravity 15/44 Proof: use unitarity methods Bern, Dixon, Dunbar, Kosower 94,95.. -reconstruct amplitude using its unitary cuts: Eg for 4pt two-loop amplitude

16. D Dunbar Windows on Quantum Gravity 16/44 For N=4 SYM/ N=8 SUGRA Key Identity propagators -pair of propagaters is exactly the cut in a scalar box integral 1 23 4 l1l1 l2l2

17. D Dunbar Windows on Quantum Gravity 17/44 -equivalent identity for N=8 -derivable using KLT relations

18. D Dunbar Windows on Quantum Gravity 18/44 Using Identity also works for 2-loop 3 4 -this (plus other work) gives two-loop result -consider -which is

19. D Dunbar Windows on Quantum Gravity 19/44 l l l l -three loop cuts, YM s2s2 s [ l 1. 4 ] 1 2 Loop momentum caught in integral -gives ansatz for multiloop terms

20. D Dunbar Windows on Quantum Gravity 20/44 Using Identity for multiloop 4, N=8 -beyond 2 loops N=4 SYM and N=8 SUGRA have different functions

21. D Dunbar Windows on Quantum Gravity 21/44 UV behaviour of diagrams Worst behaved integral has integrand Infinite if ….. Or Finite if

22. D Dunbar Windows on Quantum Gravity 22/44 UV pattern of Pattern 98,07 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) Based upon 4pt amplitudes N=8 Sugra UV pattern of Pattern 98

23. D Dunbar Windows on Quantum Gravity 23/44 Caveats : Caveats: 1) not all functions touched 2) assume no cancellations between diagrams -gets N=4 SYM correct Howe, Stelle

24. D Dunbar Windows on Quantum Gravity 24/44 New Results: driven by progress in QCD More loops More legs Formal Proofs Start with more legs…

25. D Dunbar Windows on Quantum Gravity 25/44 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

26. D Dunbar Windows on Quantum Gravity 26/44 Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

27. D Dunbar Windows on Quantum Gravity 27/44 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

28. D Dunbar Windows on Quantum Gravity 28/44 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 -Tremendous progress in last few years Green, Schwarz, Brink, Bern, Dixon, Del Duca, Dunbar, Kosower Britto, Cachazo, Feng; Roiban Spradlin Volovich Bjerrum-Bohr, Ita, Bidder, Perkins, Risager, Brandhuber,Spence, Travaglini

29. D Dunbar Windows on Quantum Gravity 29/44 Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

30. D Dunbar Windows on Quantum Gravity 30/44 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) or (2r-8) Beyond 4-point amplitude contains triangles and bubbles but only after reduction Expect triangles n > 4, bubbles n >5, rational n > 6 r

31. D Dunbar Windows on Quantum Gravity 31/44 No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4pt 5+6pt-point MHV General feature 6+7pt pt NMHV Bern,Dixon,Perelstein,Rozowsky Bern, Bjerrum-Bohr, Dunbar Green,Schwarz,Brink (no surprise) One-Loop amplitudes N=8 SUGRA look ``just like’’ N=4 SYM Bjerrum-Bohr, Dunbar, Ita,Perkins Risager

32. D Dunbar Windows on Quantum Gravity 32/44 Evidence??? Attack different parts by different methods Soft Divergences -one and two mass triangles Unitary Cuts –bubbles and three mass triangles Factorisation –rational terms

33. D Dunbar Windows on Quantum Gravity 33/44 Soft-Divergences One-loop graviton amplitude has soft divergences The divergences occur in both boxes and triangles (with at least one massless leg For no-triangle hypothesis to work the boxes alone must completely produce the expected soft divergence. (closely connected to BCFW recursion)

34. D Dunbar Windows on Quantum Gravity 34/44 Soft-Divergences-II = =0 [ ] ][ CC -form one-loop amplitude from boxes -check the soft singularities are correct -if so we can deduce one-mass and two-mass triangles are absent - this has been done for 5pt, 6pt and 7pt

35. D Dunbar Windows on Quantum Gravity 35/44 Triple Cuts [] C =0 -only boxes and a three-mass triangle contribute to this cut -if boxes reproduce C 3 exactly (numerically) -tested for 6pt +7pt (new to NMHV, not IR)

36. D Dunbar Windows on Quantum Gravity 36/44 -assuming no-triangle is correct.. in loop momentum n+4 powers cancel -8 powers by SUSY, (n-4) by ???? -look for where cancelation occurs

37. D Dunbar Windows on Quantum Gravity 37/44 Large z shift on cuts -use trick to look at the two- particle cuts -normally s dLIPS doesn’t probe UV limit -use analytic continuation to look at UV limit

38. D Dunbar Windows on Quantum Gravity 38/44 Use Spinor Form of Amplitudes (Twistor) Consider a massless particle with momenta We can realise as So we can express where are two component Weyl spinors

39. D Dunbar Windows on Quantum Gravity 39/44 -probe UV by shifting cut legs (BCF) -keeps legs onshell, effectively momentum becomes complex - useful because the behaviour of tree amplitudes under this shift is known Analytic structure of tree amplitudes under this shift has led to “on-shell recursion” Britto,Cachazo,Feng

40. D Dunbar Windows on Quantum Gravity 40/44 Look at large z behaviour

41. D Dunbar Windows on Quantum Gravity 41/44 -use behaviour of trees Valid for MHV and NMHV + + - -  x x  + - - + s s s - Consistent with boxes Bedford Brandhuber Spence Travaglini Cachazo Svercek, BDIPR, Benincasa Boucher-Veronneau Cachazo

42. D Dunbar Windows on Quantum Gravity 42/44 -supersymmetry - any gravity amplitude -much of cancelation already present in gravity theories

43. D Dunbar Windows on Quantum Gravity 43/44 Does no-triangle have implication beyond one-loop? -cancellation is stronger than expected -cancellation is NOT diagram by diagram (unlike YM) -cancellation is unexplained…. In general, for higher loops we expect, M must satisfy a wide range of factorisation/unitary conditions –are integral functions with sub-triangles disallowed?

44. D Dunbar Windows on Quantum Gravity 44/44 Implications beyond one-loop, e.g. Beyond 2 loop, loop momenta get ``caught’’ within the integral functions Generally, the resultant polynomial for maximal supergravity is 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 However…..

45. D Dunbar Windows on Quantum Gravity 45/44 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

46. D Dunbar Windows on Quantum Gravity 46/44 Three Loops Result  SYM: K 3D-18 Sugra: K 3D-16 Finite for D=4,5, Infinite D=6 -actual for Sugra -again N=8 Sugra looks like N=4 SYM Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07

47. D Dunbar Windows on Quantum Gravity 47/44 Large Shifts on Multiparticle Cuts L+1 particle cut in L loop amplitude (sample) -work in progress, Dunbar, Ita, Bjerrum- Bohr, Perkins

48. D Dunbar Windows on Quantum Gravity 48/44 Conclusions/Consequences -Lots of recent progress in perturbation theory based upon analytic and physical properties. -the finiteness or otherwise of N=8 Supergravity is still unresolved although all explicit results favour finiteness -does it mean anything? Possible to quantise gravity with only finite degrees of freedom. -is N=8 supergravity the only finite field theory containing gravity? ….seems unlikely….N=6/gauged….

49. D Dunbar Windows on Quantum Gravity 49/44 Rockall versus Hawai