Dave Dunbar, Swansea University. UV structure of N=8 Supergravity. Kasper Risager, NBI. Harald Ita , UCLA. Warren Perkins. Emil BjerrumBohr, IAS. BjerrumBohr, Dunbar, Ita, Perkins and Risager, ``The notriangle hypothesis for N = 8 supergravity,'‘
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Dave Dunbar, Swansea University
Kasper Risager, NBI
Harald Ita, UCLA
Warren Perkins
Emil BjerrumBohr, IAS
BjerrumBohr, Dunbar, Ita, Perkins and Risager,
``The notriangle hypothesis for N = 8 supergravity,'‘
JHEP 0612 (2006) 072 , hepth/0610043.
Windows on Quantum Gravity 18th June 08
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAA
(ungauged)
Cremmer,Julia, Scherk
``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
AbouZeid, Hull, Mason
=
Quantum Problems: Renormalisabilitycalculate scattering amplitudes using Feynman vertices etc
Then we remove singularities by renormalising
 Only works if g is dimensionless
(eg N=4 SYM in D=4 and String Theory)
can we find a finite field theory of gravity???
Using traditional techniques even the fourpoint tree amplitude for four graviton scattering is very difficult
Sannan,86
Supergravity calculations where we must calculate using all particles in multiplet are even more difficult…..
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 Smatrix theory?
closed string ~ open string x open string
So…
N=8 ~ ( N=4) x (N=4)
Mandelstam
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)
definite calculation of any infinity in D=4 in any supergravity theory]
2 not terrible useful
3
4
1
OneLoop AmplitudesI4 (s,t) is scalar box integral
Remarkably similar to the N=4 YangMills results
(colour ordered/leading in colour part/planar)
Bern,Dixon,Dunbar,Perelstein,Rozowsky
N=8 amplitudes very close to N=4
Bern, Rozowsky, Yan
(planar part)
Bern, Dixon, Dunbar, Kosower 94,95..
reconstruct amplitude using its unitary cuts:
Eg for 4pt twoloop amplitude
l not terrible useful2
2
3
1
4
l1
For N=4 SYM/ N=8 SUGRA Key Identitypropagators
pair of propagaters is exactly the cut in a scalar box integral
derivable using KLT relations
4 not terrible useful
3
Using Identity also works for 2loopconsider
which is
this (plus other work) gives twoloop result
three loop cuts, YM not terrible useful
l
1
l
l
l
2
Loop momentum caught in integral
s2
s
[ l1 . 4 ]
gives ansatz for multiloop terms
4, N=8 not terrible useful
Using Identity for multiloopbeyond 2 loops N=4 SYM and N=8 SUGRA have different functions
Infinite if not terrible useful…..
Or Finite if
UV behaviour of diagramsWorst behaved integral has integrand
UV pattern of Pattern 98 not terrible useful
UV pattern of Pattern 98,07Honest calculation/ conjecture (BDDPR)
N=8 Sugra
N=4 YangMills
Based upon 4pt amplitudes
Caveats: 1) not all functions touched
2) assume no cancellations between diagrams
gets N=4 SYM correct
Howe, Stelle
degree p in l not terrible useful
p
Vertices involve loop momentum
General Decomposition of One loop npoint Amplitudep=n : YangMills
p=2n Gravity
propagators
Decomposes a npoint integral into a sum of (n1) integral functions obtained by collapsing a propagator
Decomposes a npoint integral into a sum of (n1) integral functions obtained by collapsing a propagator
(Bern,Dixon,Dunbar,Kosower, 94??)
Britto, Cachazo, Feng; RoibanSpradlin Volovich
BjerrumBohr, Ita, Bidder, Perkins, Risager, Brandhuber,Spence, Travaglini
r not terrible useful
N=8 Supergravityonly after reduction
against this expectation, it might be the case that…….
Evidence?
true for 4pt
5+6ptpoint MHV
General feature
6+7pt pt NMHV
Green,Schwarz,Brink (no surprise)
Bern,Dixon,Perelstein,Rozowsky
Bern, BjerrumBohr, Dunbar
BjerrumBohr, Dunbar, Ita,Perkins Risager
Oneloop graviton amplitude has soft divergences
The divergences occur in both boxes and triangles (with
at least one massless leg
For notriangle hypothesis to work the boxes alone must completely
produce the expected soft divergence.
(closely connected to BCFW recursion)
[ not terrible useful
]
[
]
=0
=
C
C
SoftDivergencesIIform oneloop amplitude from boxes
check the soft singularities are correct
if so we can deduce onemass and twomass
triangles are absent
this has been done for 5pt, 6pt and 7pt
[ not terrible useful
]
=0
C
Triple Cutsonly boxes and a threemass triangle contribute to this cut
if boxes reproduce C3 exactly (numerically)
tested for 6pt +7pt (new to NMHV, not IR)
in loop momentum
n+4 powers cancel 8 powers by SUSY, (n4) by ????
look for where cancelation occurs
use trick to look at the twoparticle cuts
normally s dLIPS doesn’t probe UV limit
use analytic continuation to look at UV limit
where are two component Weyl spinors
Analytic structure of tree amplitudes under this shift has led
to “onshell recursion” Britto,Cachazo,Feng
keeps legs onshell, effectively momentum becomes complex
useful because the behaviour of tree amplitudes under this shift is known
Look at large z behaviour not terrible useful
Bedford Brandhuber Spence Travaglini not terrible useful
Cachazo Svercek, BDIPR, Benincasa BoucherVeronneau Cachazo
s
+
+


x

+
s
+

use behaviour of treesValid for MHV and NMHV
s
 Consistent with boxes
supersymmetry not terrible useful
any gravity amplitude
much of cancelation already present in gravity theories
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 subtriangles disallowed?
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 yangmills
eg in this case YM :P(li)=(l1+l2)2
SUGRA :P(li)=((l1+l2)2)2
l1
l2
I[ P(li)]
However…..
For YangMills, 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 onshell amplitude
indication of better behaviour in entire amplitude
Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07
S
actual for Sugra
SYM: K3D18
Finite for D=4,5 , Infinite D=6
Sugra: K3D16
again N=8 Sugra looks like N=4 SYM
work in progress, Dunbar, Ita, BjerrumBohr, Perkins
L+1 particle cut in L loop amplitude (sample)
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….