Results in N=8 Supergravity

1. Dave Dunbar, Swansea University Results in N=8 Supergravity Kasper Risager Harald Ita Warren Perkins Emil Bjerrum-Bohr hep-th/0609??? HP2 Zurich 9/9/06

2. Plan • One-Loop Amplitudes in N=8 Supergravity • No Triangle Hypothesis • Evidence for No-triangle hypothesis • Consequences and Conclusions

3. degree p in l p Vertices involve loop momentum propagators General Decomposition of One- loop n-point Amplitude (massless particles) p=n : Yang-Mills p=2n: Gravity

4. 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()

5. 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; RoibanSpradlin Volovich Bjerrum-Bohr, Ita, Bidder, Perkins, Risager

6. Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

7. r 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

8. 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 Green,Schwarz,Brink (no surprise) Bern,Dixon,Perelstein,Rozowsky Bern, Bjerrum-Bohr, Dunbar Bjerrum-Bohr, Dunbar, Ita,Perkins Risager • One-Loop amplitudes N=8 SUGRA look just like N=4 SYM

9. Evidence??? • Attack different parts by different methods • Soft Divergences -one and two mass triangles • Unitary Cuts –bubbles and three mass triangles • Factorisation –rational terms

10. 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.

11. [ ] [ ] = 0 C = C Soft-Divergences-II -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

12. = + [ ] = 0 C Triple Cuts (real) -only boxes and a three-mass triangle contribute to this cut  cbox - c3m = -if boxes reproduce C3 exactly (numerically) -tested for 6pt +7pt (new to NMHV, not IR)

13. 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

14. Bubbles -II

15. + + - - s  x - + + - s Bubbles –III Valid for MHV and NMHV s - No bubbles (MHV, 6+7pt NMHV )

16. 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

17. 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

18. Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan IPs,tplanar double box integral Bern,Dixon,Dunbar,Perelstein,Rozowsky -N=8 amplitudes very close to N=4

19. Beyond 2-loops: UV pattern (98) Honest calculation/ conjecture (BDDPR) N=8 Sugra N=4 Yang-Mills Based upon 4pt amplitudes

20. 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(li)=(l1+l2)2 SUGRA :P(li)=((l1+l2)2)2 l1 l2 I[ P(li)] Caveats: not everything touched and assume no cancelations between diagrams (good for N=4 YM) However…..

21. 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

22. Conclusions • Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4SYM 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?????