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 withS. Badger, Z. Bern, D. Dunbar, H. Ita, W. Perkins and K. Risager, P. Vanhove,

(hep-th/0501137, hep-th/0610043, 0805.3682 [hep-th], 0811.3405 [hep-th])

On the Structure of Amplitudes in N=8 Maximal Supergravity

2

Gravity AmplitudesExpand 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-movers Right-movers

Sum over

permutations

Phase factor

xx

x

x

. .

1

2

3

M

...+ +=

1

2

1 M 12

3

s12 s1M s123

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

(hi,hj) : (+,+), (-,-), (+,-) » 1/z

(hi,hj) : (-,+) » z3

(hi,hj) : (+,+), (-,-), (+,-) » 1/z2 »(1/z)2

(hi,hj) : (-,+) » z6 »(z3)2

(hi,hj) : (+,-) » z(3-n)

(hi,hj) : (-,+) » 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

Singlet Non-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 divergencesIn double cuts:

would scale like » 1/z

In double cuts:

would scale like » z0 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 hypothesisString based formalism natural basis of integrals is

Constraint from SUSY

Gravity

Amplitude takes the form

16

No-triangle hypothesisN=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)

21

ObservationsMagical properties for amplitudesMonodromy 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??)