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UV structure of N=8 Supergravity

Dave Dunbar, Swansea University. UV structure of N=8 Supergravity. Kasper Risager, NBI. Harald Ita , UCLA. Warren Perkins. Emil Bjerrum-Bohr, IAS. Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, ``The no-triangle hypothesis for N = 8 supergravity,'‘

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UV structure of N=8 Supergravity

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

  2. 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. N=8Maximal Supergravity? (ungauged) Cremmer,Julia, Scherk • 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

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

  5. …..Perturbative Quantum Gravity

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

  7. 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. 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. 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. -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. Kawai-Lewellen-Tye Relations Kawai,Lewellen Tye, 86 -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)

  12. 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 dDp »s |p|D-1d|p| )

  13. 2 3 4 1 One-Loop Amplitudes • Calculated by Green Schwarz and Brink using string theory I4 (s,t) is scalar box integral Remarkably similar to the N=4 Yang-Mills results (colour ordered/leading in colour part/planar)

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

  15. Proof: use unitarity methods Bern, Dixon, Dunbar, Kosower 94,95.. -reconstruct amplitude using its unitary cuts: Eg for 4pt two-loop amplitude

  16. l2 2 3 1 4 l1 For N=4 SYM/ N=8 SUGRA Key Identity propagators -pair of propagaters is exactly the cut in a scalar box integral

  17. -equivalent identity for N=8 -derivable using KLT relations

  18. 4 3 Using Identity also works for 2-loop -consider -which is -this (plus other work) gives two-loop result

  19. -three loop cuts, YM l 1 l l l 2 Loop momentum caught in integral s2 s [ l1 . 4 ] -gives ansatz for multiloop terms

  20. 4, N=8 Using Identity for multiloop -beyond 2 loops N=4 SYM and N=8 SUGRA have different functions

  21. Infinite if….. Or Finite if UV behaviour of diagrams Worst behaved integral has integrand

  22. UV pattern of Pattern 98 UV pattern of Pattern 98,07 Honest calculation/ conjecture (BDDPR) N=8 Sugra N=4 Yang-Mills Based upon 4pt amplitudes

  23. Caveats : Caveats: 1) not all functions touched 2) assume no cancellations between diagrams -gets N=4 SYM correct Howe, Stelle

  24. New Results: driven by progress in QCD • More loops • More legs • Formal Proofs • Start with more legs…

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

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

  27. 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. 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 • Tremendous progress in last few years Green, Schwarz, Brink,Bern, Dixon, Del Duca, Dunbar, Kosower Britto, Cachazo, Feng; RoibanSpradlin Volovich Bjerrum-Bohr, Ita, Bidder, Perkins, Risager, Brandhuber,Spence, Travaglini

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

  30. r 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

  31. 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 Green,Schwarz,Brink (no surprise) Bern,Dixon,Perelstein,Rozowsky Bern, Bjerrum-Bohr, Dunbar Bjerrum-Bohr, Dunbar, Ita,Perkins Risager • One-Loop amplitudes N=8 SUGRA look ``just like’’ N=4 SYM

  32. 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. 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. [ ] [ ] =0 = C C Soft-Divergences-II -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. [ ] =0 C Triple Cuts -only boxes and a three-mass triangle contribute to this cut -if boxes reproduce C3 exactly (numerically) -tested for 6pt +7pt (new to NMHV, not IR)

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

  40. Look at large z behaviour

  41. Bedford Brandhuber Spence Travaglini Cachazo Svercek, BDIPR, Benincasa Boucher-Veronneau Cachazo s + + - -  x - + s + - -use behaviour of trees Valid for MHV and NMHV s - Consistent with boxes

  42. -supersymmetry -any gravity amplitude -much of cancelation already present in gravity theories

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

  45. 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. Three Loops Result Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07 S -actual for Sugra SYM: K3D-18 Finite for D=4,5 , Infinite D=6 Sugra: K3D-16 -again N=8 Sugra looks like N=4 SYM

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

  48. 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. Rockall versus Hawai

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