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

1. Structure of Amplitudes in Gravity ILagrangian Formulation of Gravity, Tree amplitudes, Helicity Formalism, Amplitudes in Twistor Space, New techniquesPlaying 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.: AAAAAAAA

2. Outline

3. Outline • Quantum Gravity and General Relativity • Lagrangian formulation of Gravity • Tree Amplitudes • HelicityFormalism • Twistor Space • New Techniques for treeAmplitudes Playing with Gravity

4. Quantum Gravity

5. Quantum Gravity We desire a quantum theory with an interacting particle the graviton • It should obey an attractiveinverse square law(graviton mass-less) • It should couple with equal strength to all matter sources (graviton tensor field) No observedor ‘experimental’ effects of a quantum theory for gravity so far… PlayingwithGravity

6. Einstein-HilbertLagrangian Features: • Consistent with General Relativity(gives trees) • Action: Non-renormalisable! • Not valid beyond tree-level / one-loop • Explicit one-loop divergence with matter (t’ Hooft and Veltman) • Explicit two-loop divergence!(Goroff, Sagnotti; van de Van) PlayingwithGravity

7. Quantum Gravity • Still waiting on a fundamental theory for Gravity.. • String theory: • a natural candidate • not point like theory • however still not a string theory model fully consistent with field theory…. PlayingwithGravity

8. Quantum Gravity • Effective field theory description • Consistent with String theory • Low energy predictions unique and fit General Relativity • The simplest extension of Einstein-Hilbert we can think of • Including supersymmetry: easy and excludes certain higher derivative terms PlayingwithGravity

9. EffectiveLagrangian Features: • Derivative terms consistent with symmetry • Action: validtill Planck scale by construction PlayingwithGravity

10. Quantising Gravity Gravitywithbackground field Scalarfieldcoupling to Gravity PlayingwithGravity

11. Pure graviton vertices Gravitywithflatfield (Sannan) 45 terms + sym Features: • Infinitelymany and huge vertices! • No manifest simplifications Verymessy!! Gronningen 3-5 Dec 2009 PlayingwithGravity 11

12. Perturbative amplitudes Standard textbookway: Feynmanrules 1) Lagrangian(easy) 2) Vertices(easy) (3-vertex gravity over 100 terms..) 3) Diagrams(increasingdifficult) 4) Sum Diagrams over all contractions (hard) 5) Loops (integrations) more aboutthislecture II (close to impossible / impossible) 1 M 1 1 2 s12 s1M s123 + + ... 2 3 ( ) PlayingwithGravity

13. Computation of perturbative amplitudes Complex expressions involving e.g. (no manifest symmetry or simplifications) # Feynman diagrams: Factorial Growth! Sum over topological different diagrams Generic Feynman amplitude PlayingwithGravity

14. Amplitudes Colour ordering Specifying external polarisation tensors Simplifications Recursion Inspiration from String theory Loop amplitudes: (Unitarity, Supersymmetric decomposition) Spinor-helicity formalism PlayingwithGravity

15. Helicity states formalism Different representations of the Lorentz group Spinor products : Momentum parts of amplitudes: Spin-2 polarisation tensors in terms of helicities, (squares of YM): (Xu, Zhang, Chang) PlayingwithGravity

16. Simplifications from Spinor-Helicity Huge simplifications 45 terms + sym Vanish in spinor helicity formalism Gravity: Contractions PlayingwithGravity

17. Scattering amplitudes in D=4 • Amplitudes in gravity theories as well as Yang-Mills can hence be expressed completely specifying • The external helicies e.g. : A(1+,2-,3+,4+, .. ) • The spinor variables SpinorHelicity formalism PlayingwithGravity

18. Note on notation We will use the notation: Traces... PlayingwithGravity

19. Amplitudes via String Theory

20. Gravity Amplitudes NotLeft-Rightsymmetric Closed String Amplitude Phase factor Left-movers Right-movers Sum over permutations (Kawai-Lewellen-Tye) Sum gauge invariant 2 1 M 1 x 3 1 x x 1 2 s12 s1M s123 x = + + ... . 2 3 . M Open amplitudes: Sum over different factorisations (Link to individual Feynman diagrams lost..) Certain vertex relations possible (Bern and Grant) PlayingwithGravity

21. Gravity Amplitudes KLT explicit representation: ’ ! 0 ei ! ,’ (n-3, ij) sij = Polynomial (sij) No manifest crossing symmetry Higher point expressions quite bulky .. (2) Double poles Sum gauge invariant 2 (4) 1 M (1) 1 x 3 1 (s124) x x 1 2 s12 s1M s123 x = + + ... . 2 3 . M (4) Interesting remark: The KLT relations work independently of external polarisations PlayingwithGravity

22. Yang-Mills MHV-amplitudes Tree amplitudes (n) samehelicities vanishes Atree(1+,2+,3+,4+,..) = 0 (n-1) samehelicities vanishes Atree(1+,2+,..,j-,..) = 0 (n-2) samehelicities: Atree(1+,2+,..,j-,..,k-,..) ¹ 0 Atree MHV Given by the formula (Parke and Taylor) and proven by (Berends and Giele) First non-trivial example, (M)aximally (H)elicity (V)iolating (MHV) amplitudes One single term!! PlayingwithGravity 22

23. Examples of KLT relations Playing with Gravity

24. Gravity MHV amplitudes Can be generated from KLT via YM MHV amplitudes. Berends-Giele-Kuijf recursion formula Anti holomorphic Contributions – feature in gravity Recent work: (Elvang, Freedman: Nguyen, Spradlin, Volovich, Wen) PlayingwithGravity

25. KLT for NMHV • KLT hold independent of helicity • NMHV amplituder are morecomplicated but KLTcanstill beused • NMHV amplitudes changemuch by Helicitystructure • In Lecture IIwewillseehowKLT is veryuseful in cuts as well… PlayingwithGravity

26. Twistor space

27. Duality Proposal that N=4 super Yang-Mills is dual to a string theory in twistor space?(Witten) Topological StringTheory withtwistor targetspace CP3 Perturbative N=4 super Yang-Mills PlayingwithGravity

28. Twistor space • Tree amplitudes in YM on degenerate algebraic curves Degree : number of negative helicities Degree : N-1+L • Transformation of amplitudes into twistor space(Penrose) • In metric signature ( + + - - ) : 2D Fourier transform • In twistor space : plane wave function is a line: (Witten) PlayingwithGravity

29. Review: CSW expansion of Yang-Millsamplitudes • In the CSW-construction : off-shell MHV-amplitudes building blocks for more complicated amplitude expressions (Cachazo, Svrcek and Witten) • MHV vertices: PlayingwithGravity

30. Example of A6(1-,2-,3-,4+,5+,6+) Example of how this works PlayingwithGravity

31. Twistor space properties • Twistor-spaceproperties of gravity: More complicated! N=4 -functions Signature of non-locality  typical in gravity Anti-holomorphic pieces in gravity amplitudes Derivatives of -functions PlayingwithGravity 31

32. Collinear and Coplanar Operators PlayingwithGravity 32

33. Twistor space properties Acting with differential operators F and K • For gravity : Guaranteed that • Five-point amplitude. (Giombi, Ricci, Rables-Llana and Trancanelli; Bern, NEJBB and Dunbar) Tree amplitudes : (Bern, NEJBB and Dunbar) PlayingwithGravity 33

34. Recursion

35. BCFW Recursion for trees Complex momentum space!! Shift of the spinors : Amplitude transforms as We can now evaluate the contour integral over A(z) a and b will remain on-shell even after shift (Britto, Cachazo, Feng, Witten) PlayingwithGravity 35

36. BCWF Recursion for trees Given that • A(z) vanish for • A(z) is a rational function • A(z) has simple poles Residues : Determined by factorization properties Tree amplitude : Factorise in product of tree amplitudes • in z (Britto, Cachazo, Feng, Witten) PlayingwithGravity 36

37. A B 4pt Example [2 p] unaffected by shift so non-zero so <2p> must vanish! 3pt vertex defined in complex momentum

38. MHV vertex expansion for gravity tree amplitudes • CSW expansion in gravity • Shift (Risager) Reproduce CSW for Yang-Mills (NEJBB, Dunbar, Ita, Perkins, Risager) Shift : Correct factorisation CSW vertex PlayingwithGravity 38

39. MHV vertex expansion for gravity tree amplitudes • Negative legs shifted in the following way • Analytic continuation of amplitude into the complex plane • If Mn(z), 1) rational, 2) simple poles at points z, and 3) vanishes (justified assumption) : Mn(0) = sum of residues (as in BCFW), PlayingwithGravity 39

40. MHV vertex expansion for gravity tree amplitudes • All poles : Factorise as : • vanishes linearly in z : • Spinorproducts : not z dependent (normal CSW) PlayingwithGravity 40

41. MHV vertex expansion for gravity tree amplitudes • For gravity : Substitutions MHV amplitudes on the pole MHV vertices! • MHV vertex expansion for gravity non-locality Contact term! Matter MHV expansion considered by (Bianchi, Elvang, Friedman) problem with expansion beyond 12pt.. PlayingwithGravity 41

42. Conclusionslecture I • Considered Lagrangian Formulation of Quantum Gravity • Einstein-Hilbert / Effective Lagrangian • Tree amplitudes • Helicity Formalism • Amplitudes in Twistor Space, • New techniques • Amplitudes via KLT • Amplitudes via Recursion, BCFW and CSW PlayingwithGravity

43. Outline of lecture II • Outline af lecture II • In Lecture IIwewillconsiderhow the treeresultscanbeused to deriveresults for loop amplitudes • Wewillseehowsimple results for tree amplitudesmakes it possible to derive simple loop results • Alsowewillseehowsymmetries of treesarecarried over to loop amplitudes PlayingwithGravity

44. 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