Uv structure of n 8 supergravity
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Dave Dunbar, Swansea University. UV structure of N=8 Supergravity. Kasper Risager, NBI. Harald Ita , UCLA. Warren Perkins. Emil Bjerrum-Bohr, IAS. Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, ``The no-triangle hypothesis for N = 8 supergravity,'‘

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Uv structure of n 8 supergravity

Dave Dunbar, Swansea University

UV structure of N=8 Supergravity

Kasper Risager, NBI

Harald Ita, UCLA

Warren Perkins

Emil Bjerrum-Bohr, IAS

Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager,

``The no-triangle hypothesis for N = 8 supergravity,'‘

JHEP 0612 (2006) 072 , hep-th/0610043.

Windows on Quantum Gravity 18th June 08

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAA


  • Is N=8 Supergravity a self-consistent Quantum Field theory?

  • Does the theory have ultra-violet singularities or is it a ``finite’’ field theory.

N 8 maximal supergravity
N=8Maximal Supergravity?


Cremmer,Julia, Scherk

  • Field theory with N=8 supersymmetry

  • One graviton, eight gravitinos………….70 scalars

  • Maximal supersymmetry consistent with spins <=2

  • Field theory which couples gravity to all sorts of particles

  • Endless symmetries…

  • …really complicated Lagrangian

  • Descendant of N=1 in D=11

Uv structure of n 8 supergravity

``Finite for 8 loops but not beyond’’

2) Look at supergravity embedded within string theory

3) Find a dual theory which is solvable

Superstring Theory

Dual Theory

N=8 Supergravity

Green, Russo, Van Hove, Berkovitz, Chalmers

1) Approach problem within the theory

Abou-Zeid, Hull, Mason

Quantum problems renormalisability



Quantum Problems: Renormalisability

-calculate scattering amplitudes using Feynman vertices etc

Then we remove singularities by renormalising

- Only works if g is dimensionless


  • Gravity cannot be renormalised (in D=4)

  • Infinities must be removed by adding terms to lagrangian not present initially.

  • If we have to continually add terms then theory looses predictive power

  • Can avoid this if theory has no UV divergences (finite)

    (eg N=4 SYM in D=4 and String Theory)

-can we find a finite field theory of gravity???

Uv structure of n 8 supergravity

Using traditional techniques even the four-point tree amplitude for four graviton scattering is very difficult


Supergravity calculations where we must calculate using all particles in multiplet are even more difficult…..

however…… not terrible useful

N=8 supergravity has a lot of particles but it has enormous symmetry amongst them

Although computations are very difficult end results which must express this symmetry can be rather simple

New techniques which use symmetry to generate scattering amplitudes are particularly useful for supergravity

-return to S-matrix theory?

Try to derive behaviour from n 4 sym
-try to derive behaviour from N=4 SYM not terrible useful

  • N=4 SYM is a finite field theory

  • Try to exploit links to this for N=8 supergravity

  • In string theory,

    closed string ~ open string x open string


    N=8 ~ ( N=4) x (N=4)


Kawai lewellen tye relations
Kawai-Lewellen-Tye Relations not terrible useful

Kawai,Lewellen Tye, 86

-derived from string theory relations

-become complicated with increasing number of legs

-involves momenta prefactors

-applies to N=8/N=4 (and consequently pure YM/gravity)

Loop calculations in n 8 supergravity
Loop Calculations in N=8 Supergravity not terrible useful

  • Desperately complicated using Feynman diagrams

  • Pre strings revolution of 1984 people believed theory was finite. [Only candidate for quantum gravity…]

  • Post 1984 people believed theory was non-renormalisable and only appeared as a low energy effective theory [ of string theory]

  • In D=4 ``expect infinities’’ at 3-loops. [At this time no

    definite calculation of any infinity in D=4 in any supergravity theory]

  • In D > 4 they appear earlier ( …..s dDp »s |p|D-1d|p| )

One loop amplitudes

2 not terrible useful




One-Loop Amplitudes

  • Calculated by Green Schwarz and Brink using string theory

I4 (s,t) is scalar box integral

Remarkably similar to the N=4 Yang-Mills results

(colour ordered/leading in colour part/planar)

Two loop supergravity all d form
Two-Loop Supergravity, all D form, not terrible useful


-N=8 amplitudes very close to N=4

Bern, Rozowsky, Yan

(planar part)

Proof use unitarity methods
Proof: use unitarity methods not terrible useful

Bern, Dixon, Dunbar, Kosower 94,95..

-reconstruct amplitude using its unitary cuts:

Eg for 4pt two-loop amplitude

For n 4 sym n 8 sugra key identity

l not terrible useful2






For N=4 SYM/ N=8 SUGRA Key Identity


-pair of propagaters is exactly the cut in a scalar box integral

Equivalent identity for n 8
-equivalent identity for N=8 not terrible useful

-derivable using KLT relations

Using identity also works for 2 loop

4 not terrible useful


Using Identity also works for 2-loop


-which is

-this (plus other work) gives two-loop result

Uv structure of n 8 supergravity

-three loop cuts, YM not terrible useful







Loop momentum caught in integral



[ l1 . 4 ]

-gives ansatz for multiloop terms

Using identity for multiloop

4, N=8 not terrible useful

Using Identity for multiloop

-beyond 2 loops N=4 SYM and N=8 SUGRA have different functions

Uv behaviour of diagrams

Infinite if not terrible useful…..

Or Finite if

UV behaviour of diagrams

Worst behaved integral has integrand

Uv pattern of pattern 98 07

UV pattern of Pattern 98 not terrible useful

UV pattern of Pattern 98,07

Honest calculation/ conjecture (BDDPR)

N=8 Sugra

N=4 Yang-Mills

Based upon 4pt amplitudes

Caveats : not terrible useful

Caveats: 1) not all functions touched

2) assume no cancellations between diagrams

-gets N=4 SYM correct

Howe, Stelle

New results driven by progress in qcd
New Results: driven by progress in QCD not terrible useful

  • More loops

  • More legs

  • Formal Proofs

  • Start with more legs…

General decomposition of one loop n point amplitude

degree p in l not terrible useful


Vertices involve loop momentum

General Decomposition of One- loop n-point Amplitude

p=n : Yang-Mills

p=2n Gravity


Passarino veltman reduction
Passarino-Veltman reduction not terrible useful

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

Passarino veltman reduction1
Passarino-Veltman reduction not terrible useful

  • 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

N 4 susy yang mills
N=4 SUSY Yang-Mills not terrible useful

  • 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

    (Bern,Dixon,Dunbar,Kosower, 94??)

  • Determining rational ci determines amplitude

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

    Britto, Cachazo, Feng; RoibanSpradlin Volovich

    Bjerrum-Bohr, Ita, Bidder, Perkins, Risager, Brandhuber,Spence, Travaglini

Basis in n 4 theory
Basis in N=4 Theory not terrible useful

‘easy’ two-mass box

‘hard’ two-mass box

N 8 supergravity

r not terrible useful

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 and bubbles but

    only after reduction

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

No triangle hypothesis
No-Triangle Hypothesis not terrible useful

-against this expectation, it might be the case that…….


true for 4pt

5+6pt-point MHV

General feature

6+7pt pt NMHV

Green,Schwarz,Brink (no surprise)


Bern, Bjerrum-Bohr, Dunbar

Bjerrum-Bohr, Dunbar, Ita,Perkins Risager

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

Evidence??? not terrible useful

  • Attack different parts by different methods

  • Soft Divergences -one and two mass triangles

  • Unitary Cuts –bubbles and three mass triangles

  • Factorisation –rational terms

Soft divergences
Soft-Divergences not terrible useful

One-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.

(closely connected to BCFW recursion)

Soft divergences ii

[ not terrible useful









-form one-loop amplitude from boxes

-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

Triple cuts

[ not terrible useful




Triple Cuts

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

-if boxes reproduce C3 exactly (numerically)

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

Assuming no triangle is correct
-assuming no-triangle is correct.. not terrible useful

in loop momentum

n+4 powers cancel -8 powers by SUSY, (n-4) by ????

-look for where cancelation occurs

Large z shift on cuts
Large z shift on cuts not terrible useful

-use trick to look at the two-particle cuts

-normally s dLIPS doesn’t probe UV limit

-use analytic continuation to look at UV limit

Use spinor form of amplitudes twistor
Use Spinor Form of Amplitudes (Twistor) not terrible useful

  • Consider a massless particle with momenta

  • We can realise as

  • So we can express

    where are two component Weyl spinors

Probe uv by shifting cut legs bcf
-probe UV by shifting cut legs (BCF) not terrible useful

Analytic structure of tree amplitudes under this shift has led

to “on-shell recursion” Britto,Cachazo,Feng

-keeps legs onshell, effectively momentum becomes complex

-useful because the behaviour of tree amplitudes under this shift is known

Uv structure of n 8 supergravity

Look at large z behaviour not terrible useful

Use behaviour of trees

Bedford Brandhuber Spence Travaglini not terrible useful

Cachazo Svercek, BDIPR, Benincasa Boucher-Veronneau Cachazo












-use behaviour of trees

Valid for MHV and NMHV


- Consistent with boxes

Uv structure of n 8 supergravity

-supersymmetry not terrible useful

-any gravity amplitude

-much of cancelation already present in gravity theories

Does no triangle have implication beyond one loop
Does no-triangle have implication beyond one-loop? not terrible useful

-cancellation is stronger than expected

-cancellation is NOT diagram by diagram (unlike YM)

-cancellation is unexplained….

In general, for higher loops we expect,

M must satisfy a wide range of factorisation/unitary conditions –are integral functions with sub-triangles disallowed?

Implications beyond one loop e g
Implications beyond one-loop, e.g. not terrible useful

Beyond 2 loop , loop momenta get ``caught’’ within the integral functions

Generally, the resultant polynomial for maximal supergravity is 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)]


On the three particle cut
on the three particle cut.. not terrible useful

For Yang-Mills, we expect the loop to yield a linear pentagon integral

For 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

Three loops result
Three Loops Result not terrible useful

Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07


-actual for Sugra

SYM: K3D-18

Finite for D=4,5 , Infinite D=6

Sugra: K3D-16

-again N=8 Sugra looks like N=4 SYM

Large shifts on multiparticle cuts
Large Shifts on Multiparticle Cuts not terrible useful

-work in progress, Dunbar, Ita, Bjerrum-Bohr, Perkins

L+1 particle cut in L loop amplitude (sample)

Conclusions consequences
Conclusions/Consequences not terrible useful

  • Lots of recent progress in perturbation theory based upon analytic and physical properties.

    -the finiteness or otherwise of N=8 Supergravity is still unresolved although all explicit results favour finiteness

    -does it mean anything? Possible to quantise gravity with only finite degrees of freedom.

    -is N=8 supergravity the only finite field theory containing gravity? ….seems unlikely….N=6/gauged….

Rockall versus hawai
Rockall versus Hawai not terrible useful