Results in N=8 Supergravity

Emil Bjerrum-Bohr

HPHP22 Zurich 9/9/06 Zurich 9/9/06

Harald Ita

Warren Perkins

Dave Dunbar, Swansea University

hep-th/0609???

Kasper Risager

D Dunbar HP2 2006 2/22

Plan• One-Loop Amplitudes in N=8 Supergravity

• No Triangle Hypothesis

• Evidence for No-triangle hypothesis

• Consequences and Conclusions

D Dunbar HP2 2006 3/22

General Decomposition of One- loop n-point Amplitude

Vertices involve loop momentumpropagators

p

degree p in l

p=n : Yang-Mills

p=2n: Gravity

(massless particles)

D Dunbar HP2 2006 4/22

Passarino-Veltman reduction

•process continues until we reach four-point integral functions with (in Yang-Mills up to quartic numerators) In going from 4-> 3 scalar boxes are generated•similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. •so in general, for massless particles

Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

+O()

D Dunbar HP2 2006 5/22

N=4 SUSY Yang-Mills• In N=4 Susy there are cancellations between the

states of different spin circulating in the loop.• Leading four powers of loop momentum cancel (in

well chosen gauges..)

• N=4 lie in a subspace of the allowed amplitudes

• Determining rational ci determines amplitude

- Tremendous progress in last few years Green, Schwarz, Brink, Bern, Dixon, Del Duca, Dunbar, Kosower

Britto, Cachazo, Feng; Roiban Spradlin Volovich

Bjerrum-Bohr, Ita, Bidder, Perkins, Risager

D Dunbar HP2 2006 6/22

Basis in N=4 Theory‘‘easy’ two-mass easy’ two-mass boxbox

‘‘hard’ two-mass hard’ two-mass boxbox

D Dunbar HP2 2006 7/22

N=8 Supergravity • Loop polynomial of n-point amplitude of degree 2n.

• Leading eight-powers of loop momentum cancel (in well chosen gauges..) leaving (2n-8) or (2r-8)

• Beyond 4-point amplitude contains triangles..bubbles but

only after reduction

• Expect triangles n > 4 , bubbles n >5 , rational n > 6

r

D Dunbar HP2 2006 8/22

No-Triangle Hypothesis-against this expectation, it might be the case that…….

Evidence?true for 4pt

5+6pt-point MHV

General feature

6+7pt pt NMHV

Bern,Dixon,Perelstein,Rozowsky

Bern, Bjerrum-Bohr, Dunbar

Green,Schwarz,Brink (no surprise)

• One-Loop amplitudes N=8 SUGRA look just like N=4 SYM

Bjerrum-Bohr, Dunbar, Ita,Perkins Risager

D Dunbar HP2 2006 9/22

Evidence???

• Attack different parts by different methods

• Soft Divergences -one and two mass triangles

• Unitary Cuts –bubbles and three mass triangles

• Factorisation –rational terms

D Dunbar HP2 2006 10/22

Soft-DivergencesOne-loop graviton amplitude has soft divergences

The divergences occur in both boxes and triangles -with at least one massless leg

For no-triangle hypothesis to work the boxes alone must completely produce the expected soft divergence.

D Dunbar HP2 2006 11/22

Soft-Divergences-II

[ ] ][C C

-form one-loop amplitude from boxes using quadruple cuts Britto,Cachazo Feng-check the soft singularities are correct

-if so we can deduce one-mass and two-mass triangles are absent

-this has been done for 5pt, 6pt and 7pt

-three mass triangle IR finite so no info here

D Dunbar HP2 2006 12/22

Triple Cuts (real)

[ ]C

-only boxes and a three-mass triangle contribute to this cut

-tested for 6pt +7pt (new to NMHV, not IR)

-if boxes reproduce C3 exactly (numerically)

cbox c3m= +=-

D Dunbar HP2 2006 13/22

Bubbles

• Two Approaches both looking at two-particle cuts

-one is by identifying bubbles in cuts, by reduction (see Buchbinder, Britto,Cachazo Feng,Mastrolia)

-other is to shift cut legs (l1,l2)

and look at large z behaviour Britto,Cachazo,Feng

D Dunbar HP2 2006 14/22

Bubbles -II

D Dunbar HP2 2006 15/22

Bubbles –IIIValid for MHV and NMHV

+

+

-

-

+

-

-

+

x

s

ss

- No bubbles (MHV, 6+7pt NMHV )

D Dunbar HP2 2006 16/22

Rational Parts (n > 6)

4,5,6,……. infinity !

-If any form of bootstrap works for gravity rational terms then rational parts of N=8 will automatically vanish

-very difficult to accomadate rational pieces for n > 6 and satisfy factorisation,soft, collinear limits

D Dunbar HP2 2006 17/22

Comments

• No triangle hypothesis is unexplained – presumably we are seeing a symmetry

• Simplification is like 2n-8 - n-4 in loop momentum

• Simplification is NOT diagram by diagram

• …..look beyond one-loop

D Dunbar HP2 2006 18/22

Two-Loop SYM/ Supergravity

Bern,Rozowsky,Yan

Bern,Dixon,Dunbar,Perelstein,Rozowsky

-N=8 amplitudes very close to N=4

IPs,t planar double box integral

D Dunbar HP2 2006 19/22

Beyond 2-loops: UV pattern (98)

D=11

0 #/

D=10

0(!) #/

D=9 0 #/

D=8 #/ #’/+#”/

D=7 0 #/

D=6 0 0

D=5 0 0 0

D=4 0 0 0 0

L=1 L=2 L=3 L=4 L=5 L=6

N=4 Yang-Mills

Honest calculation/ conjecture (BDDPR)

N=8 Sugra

Based upon 4pt amplitudes

D Dunbar HP2 2006 20/22

Pattern obtained by cuttingBeyond 2 loop , loop momenta get ``caught’’ within the integral functions

Generally, the resultant polynomial for maximal supergravity of the square of that for maximal super yang-mills

eg in this case YM :P(li)=(l1+l2)2

SUGRA :P(li)=((l1+l2)2)2

I[ P(li)]

l1

l2

Caveats: not everything touched and assume no cancelations between diagrams (good for N=4 YM)

However…..

D Dunbar HP2 2006 21/22

on the three particle cut..

For Yang-Mills, we expect the loop to yield a linear pentagon integralFor Gravity, we thus expect a quadratic pentagon

However, a quadratic pentagon would give triangles which are not present in an on-shell amplitude -indication of better behaviour in entire

amplitude

D Dunbar HP2 2006 22/22

Conclusions

• Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4 SYM than power counting/field theory arguments suggest????

• If factorisation is the key then perhaps yes. Four point amplitudes very similar

• Is N=8 SUGRA perturbatively finite?????