results in n=8 supergravity
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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?????
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