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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??)