Niels Emil Jannik Bjerrum-Bohr Niels Bohr International Academy Niels Bohr Institute

1. Structure of Amplitudes in Gravity IIUnitarity cuts, Loops, Inherited properties fromTrees, Symmetries Playing with Gravity - 24th 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.: AAAAAAA

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 PlayingwithGravity

4. Simplicity… Twistors Trees SUSY N=4, N=1, QCD, Gravity.. (Witten) Hidden Beauty! New simple analytic expressions Trees simple and symmetric Cuts Loops simple and symmetric Unitarity PlayingwithGravity

5. One-loop amplitudes

6. Loop amplitudes in field theory • Standard way: • Choose gauge • ExpandLagrangian • 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 • n Much worse than tree level – one have to do integrations • 1 In sums of contributions to loop amplitudes cancellations appear (but only after doing horrible integrals…) PlayingwithGravity

7. Unitarity cuts Unitarity methods are building on the cut equation Singlet Non-Singlet PlayingwithGravity

8. General 1-loop amplitudes n-pt amplitude (Maximal graph) Vertices carry factors of loop momentum p = 2n for gravity p=n for Yang-Mills Propagators (Passarino-Veltman)reduction Collapse of a propagator PlayingwithGravity

9. Passarino-Veltman Illustrative Passarino-Veltman Due to this generic loop amplitudes have the form: PlayingwithGravity

10. Unitarity cuts 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 isrelated to not having a integral basis(dimensionally regularised). PlayingwithGravity

11. Quadruple Cut Boxes only! Having complex momentum Crucial for mass-less corners In 4D an algebraic expression! (Britto, Cachazo and Feng) PlayingwithGravity

12. Triple Cut Scalar Boxes and Scalar Triangles In 4D still one integral left! PlayingwithGravity

13. Double Cut Scalar Boxes and Scalar Triangles and Bubbles In 4D still two integrals left! PlayingwithGravity

14. Supersymmetry

15. UnitarityCuts for differenttheories Sum over particles in multiplet (singlet) Sum over particles in multiplet (non-singlet states) • Have to sum over multiplet to compute supersymmetric amplitudes • Hence we need tree amplitudes with matter lines.. PlayingwithGravity

16. N=8 Supergravity Maximal theory of supergravity DeWit, Freedman; Cremmer, Julia, Scherk; Cremmer, Julia Features: 28 = 256 masslessstates (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) Need to sum over multiplet of all 256 states… in cuts PlayingwithGravity

17. KLT and N=4 Yang-Mills Maximal theory of super Yang-Mills Features: 24 = 16 masslessstates (helicity) 1+1=2 vector states (+1,-1) 4+4=8 fermion states (+1/2, -1/2) 6 scalars (0) • 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] PlayingwithGravity

18. Supersymmetric Ward Identities Sum over particles in multiplet (singlet) Sum over particles in multiplet (non-singlet states) • 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 NkMHV amplitudes but much more work... PlayingwithGravity

19. SUSY Ward identities N=4 MHV PlayingwithGravity

20. Ward identities NMHV Needed to work out For N=8 6pt SUGRA amplitudes PlayingwithGravity

21. Recipe for computations in N=8 SUGRA • Write down 1-loop amplitude • Write down all helicity configurations • Write down all possible cuts (consider various cut channels) • Write down cut trees (including all trees with internal SUSY particles) • Fix box coefficients from quadruple cuts • Fix triangles and bubbles from triple and double cuts • Finally check that amplitudedoes not have rational parts: • If rational partsexist either compute using cuts in • Or use new recursive techniques (will be discussed in lecture III) PlayingwithGravity

22. Examples of cuts

23. Example of quadruple cut 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 Playing with Gravity

24. Examples of cuts Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut) We have PlayingwithGravity

25. Examples of cuts In this example we have 4 terms (after some algebra…) PlayingwithGravity

26. Examples of cuts Using that We have PlayingwithGravity

27. Supergravityboxes KLT N=4 YM results can be recycled into results for N=8 supergravity (Bern, NEJBB, Dunbar) PlayingwithGravity

28. Supergravity amplitudes Box Coefficients (Bern, NEJBB, Dunbar) PlayingwithGravity

29. Supergravity amplitudes • A way to organise cuts is through use the scaling behaviour of shifts Playing with Gravity

30. 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 Playing with Gravity

31. Factorisationof amplitudes

32. Singularitystructure of amplitude Tree amplitude has factorisations: Loop amplitudes has the followinggenericfactorisationstructure: (Bern and Chalmers) Playing with Gravity

33. IR singularities of gravity Gravity amplitudes have IR singularities of the form IR singularities can arise from both box and triangle integral functions PlayingwithGravity

34. Singularitystructure of amplitude • Singularitystructurecanbeused to check validity of amplitude expressions • Looking at IR singularitiescanbeused to determineifcertain terms are in amplitude • Completecontrol of singularitystructurecanbeused to do recursivecomputations • Will discuss more in Lecture III… Playing with Gravity

35. Twistor spacesymmetry

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 Mtree(l5, 1, 2, 3, l3). • This tree is annihilated by F3(123) • Hence F3(123)cN=8(45)123 = 0 • Similarly F3(145)cN=8(45)123 = F3(345)cN=8(45)123 = 0. • Remaining choices of Fijk : consider more generalised cuts, e.g., C(4512) and hence F4(124)cN=8(45)123 = 0. • Summarising: PlayingwithGravity

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) Pointslie on three intersecting lines(Bern, Dixon and Kosower) PlayingwithGravity

38. Multi-loopamplitudes

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 PlayingwithGravity

40. Conclusions

41. Conclusions • 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 PlayingwithGravity

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 PlayingwithGravity