1. Structure of Amplitudes in Gravity II Unitarity cuts, Loops, Inherited properties from Trees, Symmetries Playing with Gravity - 24 th Nordic Meeting Gronningen 2009 Niels Emil Jannik Bjerrum-Bohr Niels Bohr International Academy Niels Bohr Institute TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAAA A

2. Outline

3. Outline of lecture II Summery of lecture I Tree amplitudes and Helicity formalism How to compute and New Techniques – In this lecture we will consider loop amplitudes in gravity Traditional methods vs. Unitarity Supersymmetry and matter amplitudes Organisation of amplitudes Twistor Space and amplitudes beyond one-loop Gronningen 3-5 Dec 20093Playing with Gravity

4. Simplicity … SUSY N=4, N=1, QCD, Gravity.. Loops simple and symmetric Unitarity Cuts Trees (Witten) Twistors Trees simple and symmetric Hidden Beauty! New simple analytic expressions Gronningen 3-5 Dec 20094Playing with Gravity

5. One-loop amplitudes

6. 3-5 Dec 2009Playing with Gravity6 Loop amplitudes in field theory 1 n Standard way: Choose gauge Expand Lagrangian Features: 3pt vertex: approx 100 terms 4pt vertex much worse Propagator: 3 terms Number of topologies grows as n! Problems: off-shell formalism Not directly usable with spinor- helicity Much worse than tree level – one have to do integrations In sums of contributions to loop amplitudes cancellations appear (but only after doing horrible integrals … )

7. Unitarity cuts Unitarity methods are building on the cut equation SingletNon-Singlet Gronningen 3-5 Dec 20097Playing with Gravity

8. General 1-loop amplitudes Vertices carry factors of loop momentum n-pt amplitude p = 2n for gravity p=n for Yang-Mills Propagators Gronningen 3-5 Dec 20098Playing with Gravity (Passarino-Veltman) reduction Collapse of a propagator (Maximal graph)

9. Passarino-Veltman Gronningen 3-5 Dec 20099Playing with Gravity Due to this generic loop amplitudes have the form: Illustrative Passarino-Veltman

10. Unitarity cuts Gronningen 3-5 Dec 200910Playing with Gravity Generic one-loop amplitude (without R term): Relate kinematic discontinuity of the one loop amplitude. This imposes constraints on the coefficients Early problems in 60ties with cutting techniques is related to not having a integral basis (dimensionally regularised).

11. Quadruple Cut Gronningen 3-5 Dec 200911Playing with Gravity In 4D an algebraic expression! Boxes only! (Britto, Cachazo and Feng) Having complex momentum Crucial for mass-less corners

12. Triple Cut Gronningen 3-5 Dec 200912Playing with Gravity In 4D still one integral left! Scalar Boxes and Scalar Triangles

13. Double Cut Gronningen 3-5 Dec 200913Playing with Gravity In 4D still two integrals left! Scalar Boxes and Scalar Triangles and Bubbles

14. Supersymmetry

15. Unitarity Cuts for different theories Have to sum over multiplet to compute supersymmetric amplitudes Hence we need tree amplitudes with matter lines.. Gronningen 3-5 Dec 200915Playing with Gravity Sum over particles in multiplet (singlet) Sum over particles in multiplet (non-singlet states)

16. N=8 Supergravity Gronningen 3-5 Dec 200916Playing with Gravity DeWit, Freedman; Cremmer, Julia, Scherk; Cremmer, Julia 2 8 = 256 massless states ( helicity) 1+1=2 graviton states (+2,-2) 8+8=16 gravitino states (+3/2, -3/2) 28+28 = 56 vector states (-1,1) 56+56 = 112 fermion states (-1/2,1/2) 70 scalars(0) Maximal theory of supergravity Features: Need to sum over multiplet of all 256 states … in cuts

17. KLT and N=4 Yang-Mills Gronningen 3-5 Dec 200917Playing with Gravity 2 4 = 16 massless states ( helicity) 1+1=2 vector states (+1,-1) 4+4=8 fermion states (+1/2, -1/2) 6 scalars(0) Maximal theory of super Yang-Mills Features: Uses two things: KLT writes N=8 amplitudes as products of N=4 amplitudes. [Spectrum of N=8] = [Spectrum of N=4] x [Spectrum of N=4]

18. Supersymmetric Ward Identities Gronningen 3-5 Dec 200918Playing with Gravity Need a method to sum over states in cut Possibilities: Use CSW, BCFW, other recursive techniques to generate amplitudes Use SUSY ward identities to sum over terms in Cut. Very useful for MHV amplitudes Helps for N k MHV amplitudes but much more work... Sum over particles in multiplet (singlet) Sum over particles in multiplet (non-singlet states)

19. SUSY Ward identities Gronningen 3-5 Dec 200919Playing with Gravity MHV N=4

20. Ward identities Gronningen 3-5 Dec 200920Playing with Gravity Needed to work out For N=8 6pt SUGRA amplitudes NMHV

21. Recipe for computations in N=8 SUGRA Gronningen 3-5 Dec 200921Playing with Gravity 1. Write down 1-loop amplitude 2. Write down all helicity configurations 3. Write down all possible cuts (consider various cut channels) 4. Write down cut trees (including all trees with internal SUSY particles) 5. Fix box coefficients from quadruple cuts 6. Fix triangles and bubbles from triple and double cuts 7. Finally check that amplitude does not have rational parts: 1. If rational parts exist either compute using cuts in 2. Or use new recursive techniques (will be discussed in lecture III)

22. Examples of cuts

23. Example of quadruple cut 3-5 Dec 2009Playing with Gravity23 Have to solve … If corners is massive we can just solve constraints If one corner is massless we have to assume complex momenta of say Thereby we can write Where either

24. Examples of cuts Gronningen 3-5 Dec 200924Playing with Gravity Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut) We have

25. Examples of cuts Gronningen 3-5 Dec 200925Playing with Gravity In this example we have 4 terms (after some algebra … )

26. Examples of cuts Gronningen 3-5 Dec 200926Playing with Gravity Using that We have

27. Supergravity boxes (Bern, NEJBB, Dunbar) KLT N=4 YM results can be recycled into results for N=8 supergravity Gronningen 3-5 Dec 200927Playing with Gravity

28. Supergravity amplitudes (Bern, NEJBB, Dunbar) Box Coefficients Gronningen 3-5 Dec 200928Playing with Gravity

29. A way to organise cuts is through use the scaling behaviour of shifts 3-5 Dec 2009Playing with Gravity29 Supergravity amplitudes

30. 3-5 Dec 2009Playing with Gravity30 Supergravity amplitudes This can serve as a way to organise the amplitude. Especially if the large-z limit is zero then bubbles will be vanishing Terms corresponding to box terms will go as While triangles goes as We will discuss this in more details in Lecture III

31. Factorisation of amplitudes

32. Singularity structure of amplitude Tree amplitude has factorisations: Loop amplitudes has the following generic factorisation structure: (Bern and Chalmers) 3-5 Dec 200932Playing with Gravity

33. IR singularities of gravity 3-5 Dec 200933Playing with Gravity Gravity amplitudes have IR singularities of the form IR singularities can arise from both box and triangle integral functions

34. Singularity structure of amplitude Singularity structure can be used to check validity of amplitude expressions Looking at IR singularities can be used to determine if certain terms are in amplitude Complete control of singularity structure can be used to do recursive computations – Will discuss more in Lecture III … 3-5 Dec 200934Playing with Gravity

35. Twistor space symmetry

36. Twistor space properties of gravity loop amplitudes Unitarity : loop behaviour from trees – Cuts of the MHV box – Consider the cut C123, where the gravity tree amplitude is M tree (l 5, 1, 2, 3, l 3 ). – This tree is annihilated by F 3 (123) Hence F 3 (123)c N=8 (45)123 = 0 Similarly F 3 (145)c N=8 (45)123 = F 3 (345)c N=8 (45)123 = 0. Remaining choices of F ijk : consider more generalised cuts, e.g., C(4512) and hence F 4 (124)c N=8 (45)123 = 0. Summarising: Gronningen 3-5 Dec 200936Playing with Gravity

37. Twistor space properties of gravity loop amplitudes Inspecting the general n-point case, we can now predict Similarly we can deduce that (consistent with the YM picture), Topology : As N=4 super-Yang-Mills ) Points lie on three intersecting lines (Bern, Dixon and Kosower) Gronningen 3-5 Dec 200937Playing with Gravity

38. Multi-loop amplitudes

39. Multi-loop amplitude Most of the cut techniques we have discussed can be applied also at multi-loop level Difficulties: more difficult factorisations + no set basis of integral functions Gronningen 3-5 Dec 200939Playing with Gravity

40. Conclusions

41. We have seen how it possible to deal with loop amplitudes in new and efficient ways On-shell tree amplitudes can be used as input for cuts. – Calculating all cuts we can compute the amplitude – Feature: Symmetries for tree amplitudes leads to symmetries for loop amplitudes Gronningen 3-5 Dec 200941Playing with Gravity

42. Outline af III In Lecture III – we will discuss how new techniques for gravity amplitudes can be used learn new aspects of gravity amplitudes Among other things we will discuss – Additional symmetry for gravity – No-triangle Property of N=8 Supergravity Possible Finiteness of N=8 Supergravity Gronningen 3-5 Dec 200942Playing with Gravity