Soft and Collinear Behaviour of Graviton Scattering Amplitudes

1. Soft and Collinear Behaviour of Graviton Scattering Amplitudes • David Dunbar, Swansea University

2. Soft theorems • Part of General exploration of singularities of scattering amplitude as route to computation and comprehension • singularity as a leg (n) becomes soft Weinberg,65 • soft factor is universal • receives no loop corrections • sub-leading terms are finite: for real momenta

3. s Sub-leading terms are singularities in complex momenta :engineering a cubic singularity

4. Soft Theorems Cachazo and Strominger White (subleading) Bern, Davies, Nohle

5. -Beyond the trees? =0 +other • Soft theorem consequence (Ward identity) of BMS symmetry • leading term protected Bondi, van der Burg Metzner Sachs

6. dependance upon Hodge, 2011 Berends Giele, Kuijf, 87; Mason Skinner, 2009 Gravity MHV amplitude

7. n=5 a b MHV “Twistor-link”-representation Nguyen, Spradlin,Volovich, Wen, 2010 connected tree diagrams involving positive helicity legs only n=6 n=7

8. a b = Alternate Formulation From a Seed

9. -soft lifting from three and four point tree

10. Alternate Formulation:2 From Seeds

11. Soft-Terms from diagrams n-1 -point diagram t-dependance lies purely on green line

12. -diagram with soft leg attached to outside -summing contributions gives leading soft factor

13. diagrams with soft leg between two legs are pure quadratic

14. { } B B B C A A A A C C C B -diagrams with trivalent vertex for soft leg are pure linear divergent -this matches

15. N=4 One-loop, MHV n-point

16. -softlifting rational term? a b =

17. Collinear limit : ansatz satisfies leading soft behaviour but fails collinear limit -need to add extra term -trivial when looked at the right way

18. a b = N=4 One-loop, MHV n-point Rn is obtained by summing all link diagrams with a single loop

19. -sub-leading soft gives “anomaly” • sub-leading soft can replace role of collinear limit in determining structure

20. Soft-Theorems for One-loop amplitudes • not many amplitudes available! • N=8 : all available • M(+++.....++++) • N=6,4 MHV • pure gravity 4pt+5pt completely • ..use what we have

21. Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

22. Finite Loop Amplitudes Bern Dixon Perelstein Rosowsky,98

23. Single Minus, double poles

24. double Poles • for real momenta amplitudes have single poles • double poles arise when we use complex momenta + a + + b

25. -double poles not intrinsically a problem • but we need a formula for sub-leading singularities

26. Augmented Recursion • need formalism to work a off-shell (partially) but still use helicity information: -light -cone gauge methods -carry out a BCFW shift

27. relies upon working off-shell , (a little as possible) • uses off-shell currents from Yang-Mills • assumes KLT , close to off-shell • produces very cumbersome but, usable, result • please, please trivialise Berends-Giele, Kosower, Mahlon Alston, Dunbar and Perkins http://pyweb.swan.ac.uk/~dunbar/graviton.html

28. Soft Theorems??? • all-plus satisfies theorem • single minus satisfies theorem when negative leg • single minus fails sub-sub-leading result Bern, Davies, Nohle He, Huang, and Wen

29. Soft-Limit is a coupled BCFW shift t • sub-sub-leading directly related to double poles

30. Conclusions • soft theorems seem good at sub-leading • fail at sub-sub-leading • sub-leading constraints equivalent to collinear • non-supersymmetric a long way from maximally

31. N=4