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Niels Emil Jannik Bjerrum-Bohr

Structure of Amplitudes in Gravity III Symmetries of Loop and Tree amplitudes, No-Triangle Property, Gravity amplitudes from String Theory Playing with Gravity - 24 th Nordic Meeting Gronningen 2009. Niels Emil Jannik Bjerrum-Bohr. Niels Emil Jannik Bjerrum-Bohr

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Niels Emil Jannik Bjerrum-Bohr

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  1. Structure of Amplitudes in Gravity IIISymmetries of Loop and Tree amplitudes, No-Triangle Property, Gravity amplitudes from String Theory Playing with Gravity - 24th Nordic Meeting Gronningen 2009 Niels Emil Jannik Bjerrum-Bohr Niels Emil Jannik Bjerrum-Bohr Niels Bohr International Academy Niels Bohr Institute TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

  2. Outline

  3. OutlineLecture III • We haveconsideredhow to computetree and loop amplitudes in gravity • We have seenhownew efficientmethodsclearlysimplifiescomputations • In thislecturewewouldlike to consider the new insightsthatwegetintogravity amplitudes from this • Especiallywewant to focusonnew symmetries and whatthismighttelluson the highenergy limit of gravity PlayingwithGravity

  4. Generic loop amplitudes

  5. Supersymmetric decomposition in YM • Super-symmetry imposes a simplicity of the expressions for loop amplitudes. • For N=4 YM only scalar boxes appear. • For N=1 YM scalar boxes, triangles and bubbles appear. • One-loop amplitudes are built up from a linear combination of terms(Bern, Dixon, Dunbar, Kosower).

  6. General 1-loop amplitudes n-pt amplitude p = 2n for gravity p=n for Yang-Mills Vertices carry factors of loop momentum Propagators PlayingwithGravity

  7. General 1-loop amplitudes (Passarino-Veltman)reduction Collapse of a propagator n=4: boxes n=5: triangles n=6: bubbles… Playing with Gravity

  8. 5pt cut revisited Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut) PlayingwithGravity

  9. 5pt cut revisited Using that We have PlayingwithGravity

  10. 5pt cut revisited • Surprice? • Power counting seems to be seriously off? • 5pt non-singlet shows similar behaviour… • Part of a pattern.. PlayingwithGravity

  11. No-TriangleProperty

  12. No-Triangle Hypothesis Consequence: N=8 supergravity same one-loop structure as N=4 SYM History True for 4pt n-point MHV 6pt NMHV (IR) 6pt Proof 7pt evidence n-pt proof (Green,Schwarz,Brink) (Bern,Dixon,Perelstein,Rozowsky) (Bern, NEJBB, Dunbar,Ita) Direct evaluation of cuts (NEJBB, Dunbar,Ita, Perkins, Risager; Bern, Carrasco, Forde, Ita, Johansson) (NEJBB, Vanhove; Arkani-Hamed, Cachazo, Kaplan) PlayingwithGravity

  13. No-Triangle Hypothesis: Cuts by cut… Attack different parts of amplitudes 1) .. 2) .. 3) .. (1) Look at soft divergences (IR) 1m and 2m triangles (2)Explicit unitary cuts bubble and 3m triangles (3) Factorisation rational terms. Check that boxes gives the correct IR divergences In double cuts: would scale like In double cuts: would scale like and (NEJBB, Dunbar,Ita, Perkins, Risager; Arkani-Hamed, Cachazo, Kaplan; Badger, NEJBB, Vanhove) Scaling properties of (massive) cuts. PlayingwithGravity

  14. No-Triangle Hypothesis Gravity IR loop relation : Compact result for SYM tree amplitudes (Bern, Dixon and Kosower; RoibanSpradlin and Volovich) Check that boxes gives the correct IR divergences No one mass and two mass triangles (no statement about three mass triangles Checked until 7pt! x C(1m) = 0 x C(2m) = 0

  15. No-Triangle Hypothesis Three mass triangles x C(3m) = 0

  16. No-Triangle Hypothesis Evaluate double cuts Directly using various methods, Identify singularities. x C(bubble) = 0 (e.g. Buchbinder, Britto,CachazoFeng,Mastrolia)

  17. Supergravity amplitudes • Scaling behaviour of shifts Playing with Gravity

  18. Scaling behaviour Amazingly good behaviour Yang-Mills (n-pt gluon amplitudes) (hi,hj) : (+,+), (-,-), (+,-) (hi,hj) : (-,+) Gravity (n-pt graviton amplitudes) (hi,hj) : (+,+), (-,-), (+,-) (hi,hj) : (-,+) QED (n-pt 2 photon amplitudes) (hi,hj) : (+,-) (hi,hj) : (-,+) Playing with Gravity

  19. No-Triangle Hypothesis N=4 SUSY Yang-Mills No-triangle property: YES Expected from power-counting and z-scaling properties N=8 SUGRA No-triangle property: YES NOT expected from naïve power-counting (consistent with string based rules) QED (and sQED) No-triangle property: from 8pt NOT as expected from naive power-counting (consistent with string based rules)

  20. String based formalism

  21. No-triangle hypothesis (NEJBB, Vanhove) Generic loopamplitude (gravity / QED) Passarino-Veltman Naïve counting!! Tensor integrals derivatives in Qn PlayingwithGravity

  22. No-triangle hypothesis String based formalism natural basis of integrals is Amplitude takes the form Constraint from SUSY PlayingwithGravity

  23. No-triangle hypothesis Now if we look at integrals Typical expressions Use + integration by parts Generalisation from 5 pts.. PlayingwithGravity

  24. No-triangle hypothesis N=8 Maximal Supergravity (r = 2 (n – 4), s = 0) (NEJBB, Vanhove) (r = 2 (n – 4) - s, s >0) Higher dimensional contributions – vanish by amplitude gauge invariance Proof of No-triangle hypothesis Playing with Gravity

  25. No-triangle hypothesis QED (NEJBB, Vanhove) (r = n, s = 0) (from n = 8) Higher dimensional contributions – vanish by amplitude gauge invariance PlayingwithGravity

  26. No-triangle hypothesis N 3 theories constructable from cuts Generic gravity theories: • Prediction N=4 SUGRA • Prediction pure gravity PlayingwithGravity

  27. No-triangle at multi-loops

  28. No-triangle for multi-loops No-triangle hypothesis 1-loop Consequences for powercounting arguments above one-loop.. Possible to obtain YM bound?? D = 6/L + 4 for gravity??? Two-particle cut might miss certain cancellations Iterated two-particle cut Three/N-particle cut Explicitly possible to see extra cancellations! (Bern, Dixon, Perelstein, Rozowsky; Bern, Dixon, Roiban) PlayingwithGravity

  29. No-triangle for multiloops (Bern,Rozowsky,Yan) (Bern,Dixon,Dunbar, Perelstein,Rozowsky) Explicit at two loops : ‘No-triangle hypothesis’ holds at two-loops 4pt (Bern, Carrasco, Dixon, Johansson, Kosower, Roiban) …and even higher loops. Still general principle for simplicitylacking…

  30. Finiteness of N=8 SUGRA?

  31. FinitenessQuestion • For finiteness of N=8 supergravity we need a strong symmetry to remove the possible UV divergences that can be encountered at n-loop order. • We know that SUSY limits the possibilities for UV divergences in supergravity considerably • 4-loop computation explicit shows that particular divergences which could be presentare in fact not • Still however such divergences are not in conflict with SUSY – they can be adapted within formalism • There will be a make or break point around 7-9 loops however…(this is far beyond present capabilities) PlayingwithGravity

  32. FinitenessQuestion • The no-triangle property is not related to SUSY it is a symmetry of the amplitude which is also present in pure gravity • Combined with SUSY we get a temendous simplification of the N=8 one-loop amplitudes • This is related to scaling behaviour at tree-level • Origin is however still not understood.. • To understand resultsat multi-loop level no-triangle must be a key element • Clues from string theory: Unordernessof amplitudes (and gauge invariance) • KEY:to get a better fundamental description of gravity PlayingwithGravity

  33. Summery of cookbook • We use cut techniques for gravity • Problems: cuts with many legs get more and more cumbersome • Problem but can be dealt with using more numerical techniques • Solution (maybe) • Recursive inspired techniques • String based techniques Playing with Gravity

  34. Whatcanbe new developments • Recent yearsseenautomatedcomputations for QCD and Yang-Mills • Much of thisshouldbesimple to adapted to Gravity • Recursiontechniques for gravity (also at loop level) is somethingonethingonecouldconsider.. • Automatednumerical cut techniques to fix the whole amplitudeincludingrational parts (i.e. Blackhat programs etc) (Berger et al) • Multi-loop needbettertoolsesp integral basis.. PlayingwithGravity

  35. Monodromyrelations

  36. Monodromy relations for Yang-Mills amplitudes (n-3)! functions in basis Monodromyrelated Real part : (Kleiss – Kuijf) relations Imaginary part : New relations (Bern, Carrasco, Johansson) PlayingwithGravity

  37. Monodromy and KLT Cyclicity and flip (2) Double poles 2 (4) 1 M (1) 1 x 3 1 (s124) x x 1 2 s12 s1M s123 x = + + ... . 2 3 . M 4pt (4) PlayingwithGravity

  38. Monodromy and KLT 5pt CompletelyLeft-Rightsymmetricformula Fantasticsimplicitycomparing to Lagrangiancomplexity…. N-3! basis functions N pt PlayingwithGravity

  39. Summery

  40. Summery • Good news • Today we can do many more computations than 10 years ago • This opens a window to further push limits for our understanding of gravity • We have seen how to do tree and loops with great efficiency • Need better understanding and techniques still multi-loop level This is important for finiteness question Playing with Gravity

  41. Observations • Gravity amplitudes: Simpler than expected • Lagrangianhides simplicity • Amplitudes satisfy KLT squaring relation • KLT can be made more symmetric due to monodromy • Amplitude has simplicity due to unorderedness/diffeomorphism invariance. • Lead to no-triangle property • Simplicity already present in trees.. • Amplitude has many properties inherited from Yang-Mills : e.g.twistor space structure PlayingwithGravity

  42. Conclusions

  43. Conclusions • The calculation of gravity amplitudes benefit hugely from the use of new techniques. • More perturbative calculations of loop amplitudes from unitarity will be helpful to understand the symmetry that we see… • Importance of supersymmetry for cancellations not completely understood. • Will theories with less supersymmetry have similar surprising cancellations?? N=6 (string theory says: YES) • KLT seems to play an important role Gravity =(Yang Mills) x (Yang Mills’) • ‘No-triangle cancellations’ needs to be understood at 1-loop • Calculations beyond 4pt could be important : 5pt 2-loop maybe?

  44. Conclusions • The perturbative expansion of N=8 seems to be surprisingly simple and very similar to N=4 at one-loop. At three loop no worse UV-divergences than N=4! • This may have important consequences .. • Hints from String theory?? Explaination ??? • Perturbative finite / Non-perturbative completion??? (Berkovits) (Green, Russo, Vanhove) (Abou-Zeid, Hull and Mason) Twistor-string theory for gravity?? (Schnitzer) Mass-less modes with non-perturbative origin?? (Green, Ooguri, Schwarz)

  45. Conclusions • Clear no-triangle property at one-loop leads to constrains for amplitude at higher loops. • Enough for finiteness… open question still • Important to understand in full details : • KLT squaring relation for gravity • Diffeomorphisminvariance and unorderedness • of gravity • KEY: We need better way to express this better • in order to understand symmetry • Possible twistor space construction of gravity (Arkani-Hamed, Cachazo, Cheung, Kaplan) • Development of new and even better techniques for computations important.. PlayingwithGravity

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