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 Bohr International Academy Niels Bohr Institute TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AA A AA

Outline

Outline Lecture III We have considered how to compute tree and loop amplitudes in gravity We have seen how new efficient methods clearly simplifies computations In this lecture we would like to consider the new insights that we get into gravity amplitudes from this Especially we want to focus on new symmetries and what this might tell us on the high energy limit of gravity Gronningen 3-5 Dec 20093Playing with Gravity

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).

General 1-loop amplitudes Vertices carry factors of loop momentum n-pt amplitude p = 2n for gravity p=n for Yang-Mills Propagators Gronningen 3-5 Dec 20096Playing with Gravity

Gronningen 3-5 Dec 2009Playing with Gravity7 (Passarino-Veltman) reduction Collapse of a propagator General 1-loop amplitudes n=4: boxes n=5: triangles n=6: bubbles…

5pt cut revisited Gronningen 3-5 Dec 20098Playing with Gravity Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut)

Gronningen 3-5 Dec 20099Playing with Gravity 5pt cut revisited Using that We have

Gronningen 3-5 Dec Playing with Gravity 5pt cut revisited Surprice? Power counting seems to be seriously off? 5pt non-singlet shows similar behaviour… Part of a pattern..

No-Triangle Property

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) Gronningen 3-5 Dec 2009Playing with Gravity

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. (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 In double cuts: would scale like and Scaling properties of (massive) cuts. Gronningen 3-5 Dec 2009Playing with Gravity

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

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

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

Scaling behaviour of shifts 3-5 Dec 2009Playing with Gravity17 Supergravity amplitudes

Scaling behaviour 18 Yang-Mills Gravity QED (h i,h j ) : (+,+), (-,-), (+,-) (h i,h j ) : (-,+) (h i,h j ) : (+,+), (-,-), (+,-) (h i,h j ) : (-,+) (h i,h j ) : (+,-) (h i,h j ) : (-,+) (n-pt graviton amplitudes) (n-pt 2 photon amplitudes) (n-pt gluon amplitudes) Amazingly good behaviour 3-5 Dec 2009Playing with Gravity

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)

String based formalism

No-triangle hypothesis Generic loop amplitude (gravity / QED) Passarino-Veltman Naïve counting!! (NEJBB, Vanhove) Tensor integrals derivatives in Q n Gronningen 3-5 Dec Playing with Gravity

No-triangle hypothesis String based formalism natural basis of integrals is Constraint from SUSY Amplitude takes the form Gronningen 3-5 Dec Playing with Gravity

No-triangle hypothesis Now if we look at integrals Typical expressions Use + integration by parts Generalisation from 5 pts.. Gronningen 3-5 Dec Playing with Gravity

24 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) Gronningen 3-5 Dec 2009Playing with Gravity

No-triangle hypothesis QED (r = n, s = 0) Higher dimensional contributions – vanish by amplitude gauge invariance (NEJBB, Vanhove) (from n = 8) Gronningen 3-5 Dec Playing with Gravity

No-triangle hypothesis Generic gravity theories: Prediction N=4 SUGRA Prediction pure gravity N 3 theories constructable from cuts Gronningen 3-5 Dec Playing with Gravity

No-triangle at multi-loops

No-triangle for multi-loops 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) Gronningen 3-5 Dec Playing with Gravity

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 simplicity lacking…

Finiteness of N=8 SUGRA?

Finiteness Question Gronningen 3-5 Dec Playing with Gravity 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 present are 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)

Finiteness Question Gronningen 3-5 Dec Playing with Gravity 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 results at multi-loop level no-triangle must be a key element –Clues from string theory: Unorderness of amplitudes (and gauge invariance) –KEY: to get a better fundamental description of gravity

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 Gronningen 3-5 Dec 2009Playing with Gravity33

What can be new developments Recent years seen automated computations for QCD and Yang-Mills –Much of this should be simple to adapted to Gravity Recursion techniques for gravity (also at loop level) is something one thing one could consider.. Automated numerical cut techniques to fix the whole amplitude including rational parts (i.e. Blackhat programs etc) (Berger et al) Multi-loop need better tools esp integral basis.. Gronningen 3-5 Dec Playing with Gravity

Monodromy relations

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 Gronningen 3-5 Dec Playing with Gravity

Monodromy and KLT Double poles x x x x M...++= 1 2 1M s 12 s 1M s 123 (1) (2) (4) (s 124 ) 4pt Cyclicity and flip Gronningen 3-5 Dec Playing with Gravity

Monodromy and KLT Completely Left-Right symmetric formula Fantastic simplicity comparing to Lagrangian complexity…. N-3! basis functions 5pt N pt Gronningen 3-5 Dec Playing with Gravity

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 Gronningen 3-5 Dec 2009Playing with Gravity40

Observations Gravity amplitudes: Simpler than expected Lagrangian hides 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 Gronningen 3-5 Dec Playing with Gravity

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??? (Abou-Zeid, Hull and Mason) (Berkovits) (Green, Russo, Vanhove) (Schnitzer) Twistor-string theory for gravity?? Mass-less modes with non-perturbative origin?? (Green, Ooguri, Schwarz)

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 Diffeomorphism invariance 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.. Gronningen 3-5 Dec Playing with Gravity