Results in N=8 Supergravity Emil Bjerrum-Bohr HP 2 Zurich 9/9/06 Harald Ita Warren Perkins Dave Dunbar, Swansea University hep-th/0609??? Kasper Risager

D Dunbar HP /22 Plan One-Loop Amplitudes in N=8 Supergravity No Triangle Hypothesis Evidence for No-triangle hypothesis Consequences and Conclusions

D Dunbar HP /22 General Decomposition of One- loop n-point Amplitude Vertices involve loop momentum propagators p degree p in l p=n : Yang-Mills p=2n: Gravity ( massless particles)

D Dunbar HP /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 HP /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 c i 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 HP /22 Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

D Dunbar HP /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 HP /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 HP /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 HP /22 Soft-Divergences 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.

D Dunbar HP /22 Soft-Divergences-II   [ ] ][ CC -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 HP /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 C 3 exactly (numerically)  c box c 3m = + = -

D Dunbar HP /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 (l 1,l 2 ) and look at large z behaviour Britto,Cachazo,Feng

D Dunbar HP /22 Bubbles -II

D Dunbar HP /22 Bubbles –III Valid for MHV and NMHV  x x  s s s - No bubbles (MHV, 6+7pt NMHV )

D Dunbar HP /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 HP /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 HP /22 Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan Bern,Dixon,Dunbar,Perelstein,Rozowsky -N=8 amplitudes very close to N=4 I P s,t planar double box integral

D Dunbar HP /22 Beyond 2-loops: UV pattern (98) D=110 #/  D=100(!) #/  D=90 #/  D=8 #/  #’/   +#”/  D=70 #/  D=600 D=5000 D=40000 L=1L=2L=3L=4L=5L=6 N=4 Yang-Mills Honest calculation/ conjecture (BDDPR) N=8 Sugra Based upon 4pt amplitudes

D Dunbar HP /22 Pattern obtained by cutting Beyond 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(l i )=(l 1 +l 2 ) 2 SUGRA :P(l i )=((l 1 +l 2 ) 2 ) 2 I[ P(l i )] l1l1 l2l2 Caveats: not everything touched and assume no cancelations between diagrams (good for N=4 YM) However…..

D Dunbar HP /22 on the three particle cut.. 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

D Dunbar HP /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?????