SQG4 - Perturbative and Non-Perturbative Aspects of String Theory and Supergravity Marcel Grossmann -- Paris Niels Emil Jannik Bjerrum-Bohr Niels Bohr International Academy, Niels Bohr Institute Research collaborations with S. Badger, Z. Bern, D. Dunbar, H. Ita, W. Perkins and K. Risager, P. Vanhove, (hep-th/ , hep-th/ , [hep-th], [hep-th]) On the Structure of Amplitudes in N=8 Maximal Supergravity

2 Gravity Amplitudes Expand Einstein-Hilbert Lagrangian : Features: Infinitely many huge vertices! No manifest simplifications (Sannan) 45 terms + sym Simplifications from Spinor-Helicity Gravity:

3 Gravity Amplitudes Closed String Amplitude Left-moversRight-movers Sum over permutations Phase factor x x x x M...++= 1 2 1M s 12 s 1M s 123 Open amplitudes: Sum over different factorisations (Link to individual Feynman diagrams lost..) Sum gauge invariant Certain vertex relations possible (Bern and Grant) (Kawai-Lewellen-Tye) Not Left-Right symmetric

4 Gravity MHV amplitudes Can be generated from KLT via YM MHV amplitudes. (Berends-Giele-Kuijf) recursion formula Anti holomorphic Contributions – feature in gravity

5 Making KLT more symmetric.. Rewriting: KLT in a manifest Left – Right symmetric form possible (NEJBB, Damgaard, Vanhove) Monodromy invariance of KLT nessecary ® ’ ! 0

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

7 Monodromy invariance for KLT

8 Gravity Trees (Britto, Cachazo, Feng, Witten, Bedford, Brandhuber,Spence, Travaglini; Cachazo, Svrtec; NEJBB, Dunbar, Ita; Ozeren, Stirling, Arkani-Hamed, Kaplan; Hall; Cheung, Arkani- Hamed, Cachazo, Kaplan) Tree properties only 3pt amplitudes needed Amplitudes in Yang-Mills, QED and gravity can all be generated from BCFW recursion Features: helicity independent scaling behaviour

Scaling behaviour 99 Yang-Mills Gravity QED ( h i,h j ) : (+,+), (-,-), (+,-) » 1/z ( h i,h j ) : (-,+) » z 3 ( h i,h j ) : (+,+), (-,-), (+,-) » 1/z 2 » (1/z) 2 ( h i,h j ) : (-,+) » z 6 » (z 3 ) 2 ( h i,h j ) : (+,-) » z (3-n) ( h i,h j ) : (-,+) » z (5-n) (n-pt graviton amplitudes) (n-pt 2 photon amplitudes) (n-pt gluon amplitudes) Amazingly good behaviour KLT??

10 General 1-loop amplitudes Vertices carry factors of loop momentum n-pt amplitude (Passarino-Veltman) reduction Collapse of a propagator p = 2n for gravity p=n for YM Propagators

11 Unitarity cuts Unitarity methods are building on the cut equation SingletNon-Singlet

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

13 No-Triangle Hypothesis by Cuts 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. (NEJBB, Dunbar,Ita, Perkins, Risager; Arkani-Hamed, Cachazo, Kaplan; Badger, NEJBB, Vanhove) Check that boxes gives the correct IR divergences In double cuts: would scale like » 1/z In double cuts: would scale like » z 0 and 1/z Scaling properties of (massive) cuts.

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

15 No-triangle hypothesis String based formalism natural basis of integrals is Constraint from SUSY Gravity Amplitude takes the form

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

17 No-triangle hypothesis Generic gravity theories: Prediction N=4 SUGRA Prediction pure gravity N · 3 theories constructable from cuts

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

19 Two-Loop SYM/ Supergravity (Bern,Rozowsky,Yan) (Bern,Dixon,Dunbar,Perelstein,Rozowsky) Explicit at two loops : ‘No-triangle hypothesis’ holds at two-loops 4pt Two-loop 5pt would be interesting to know

Three and Four-Loop SYM/ Supergravity Three and Four -loop four-point amplitude of N=8 supergravity directly constructed via unitarity. Divergences in D dimensions at three and four loop: NO WORSE than N=4 super-Yang-Mills theory. Amplitude UV finite in four dimensions. Confirms ‘no-triangle hypothesis’ for three and four loops. (Bern, Carrasco, Dixon, Johansson, Kosower, Roiban)

Observations 21 Magical properties for amplitudes Monodromy relations for tree amplitudes in Yang-Mills and possibility of left-right symmetric KLT relation. SURPRISE: Gravity and QED: No-triangle property Unorderedness (+ gauge invariance) of amplitudes: Better behaviour (Gravity simpler than YM.) Helicity: NO ROLE for scaling behaviour of amplitudes Lower loop simplifications links to higher loop simplification (Link to KLT? -- Enough for finiteness of N=8 SUGRA??)