1. Structure of Amplitudes in Gravity I Lagrangian Formulation of Gravity, Tree amplitudes, Helicity Formalism, Amplitudes in Twistor Space, New techniques 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.: AA A A A A A A

2. Outline

3. Quantum Gravity and General Relativity Lagrangian formulation of Gravity – Tree Amplitudes Helicity Formalism – Twistor Space New Techniques for tree Amplitudes Gronningen 3-5 Dec 2009Playing with Gravity3

4. Quantum Gravity

5. Gronningen 3-5 Dec 2009Playing with Gravity5 We desire a quantum theory with an interacting particle the graviton It should obey an attractive inverse square law (graviton mass-less) It should couple with equal strength to all matter sources (graviton tensor field) No observed or ‘ experimental ’ effects of a quantum theory for gravity so far …

6. Einstein-Hilbert Lagrangian Gronningen 3-5 Dec 2009Playing with Gravity6 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)

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 …. Gronningen 3-5 Dec 2009Playing with Gravity7

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 Gronningen 3-5 Dec 2009Playing with Gravity8

9. Effective Lagrangian Gronningen 3-5 Dec 2009Playing with Gravity9 Features: Derivative terms consistent with symmetry Action: valid till Planck scale by construction

10. Quantising Gravity Gronningen 3-5 Dec 2009Playing with Gravity10 Gravity with background field Scalar field coupling to Gravity

11. Features: Infinitely many and huge vertices! No manifest simplifications (Sannan) 45 terms + sym Gronningen 3-5 Dec 2009Playing with Gravity11 Pure graviton vertices Gravity with flat field Very messy!!

12. Perturbative amplitudes...++ 1 2 1M 1 2 3 s 12 s 1M s 123 Standard textbook way: Feynman rules 1) Lagrangian (easy) 2) Vertices (easy) (3-vertex gravity over 100 terms..) 3) Diagrams (increasing difficult) 4) Sum Diagrams over all contractions (hard) 5) Loops (integrations) more about this lecture II (close to impossible / impossible) Gronningen 3-5 Dec 200912Playing with Gravity ( )

13. Computation of perturbative amplitudes Gronningen 3-5 Dec 2009Playing with Gravity13 Complex expressions involving e.g. (no manifest symmetry or simplifications) Sum over topological different diagrams Generic Feynman amplitude # Feynman diagrams: Factorial Growth!

14. Amplitudes Gronningen 3-5 Dec 2009Playing with Gravity14 Simplifications Spinor-helicity formalism Recursion Specifying external polarisation tensors Loop amplitudes: (Unitarity, Supersymmetric decomposition) Colour ordering Inspiration from String theory

15. Helicity states formalism Spinor products : Momentum parts of amplitudes: Spin-2 polarisation tensors in terms of helicities, (squares of YM): (Xu, Zhang, Chang) Different representations of the Lorentz group Gronningen 3-5 Dec 2009 15 Playing with Gravity

16. Simplifications from Spinor- Helicity Vanish in spinor helicity formalism Gravity: Huge simplifications Contractions 45 terms + sym Gronningen 3-5 Dec 200916Playing with Gravity

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 Spinor Helicity formalism Gronningen 3-5 Dec 2009Playing with Gravity17

18. Note on notation We will use the notation: Traces... Gronningen 3-5 Dec 200918Playing with Gravity

19. Amplitudes via String Theory

20. Gravity Amplitudes Gronningen 3-5 Dec 2009Playing with Gravity20 Closed String Amplitude Left-moversRight-movers Sum over permutations Phase factor Open amplitudes: Sum over different factorisations (Link to individual Feynman diagrams lost..) Sum gauge invariant Certain vertex relations possible (Bern and Grant) (Kawai-Lewellen-Tye) Not Left-Right symmetric x x x x.. 1 2 3 M...++= 1 2 1M 1 2 3 s 12 s 1M s 123

21. Gravity Amplitudes Gronningen 3-5 Dec 2009Playing with Gravity21 KLT explicit representation:  ’ ! 0 e i  !  ,  ’  (n-3, ij) s ij = Polynomial ( s ij ) No manifest crossing symmetry Double poles x x x x.. 1 2 3 M...++= 1 2 1M 1 2 3 s 12 s 1M s 123 Sum gauge invariant (1) (2) (4) (s 124 ) Higher point expressions quite bulky.. Interesting remark: The KLT relations work independently of external polarisations

22. 22 Yang-Mills MHV-amplitudes (n) same helicities vanishes A tree (1 +,2 +,3 +,4 +,..) = 0 (n-1) same helicities vanishes A tree (1 +,2 +,..,j -,..) = 0 (n-2) same helicities: A tree (1 +,2 +,..,j -,..,k -,..) ¹ 0 A tree MHV Given by the formula (Parke and Taylor) and proven by (Berends and Giele) Tree amplitudes First non-trivial example, (M)aximally (H)elicity (V)iolating (MHV) amplitudes One single term!! Gronningen 3-5 Dec 200922Playing with Gravity

23. Examples of KLT relations Gronningen 3-5 Dec 2009Playing with Gravity23

24. Gravity MHV amplitudes Can be generated from KLT via YM MHV amplitudes. Berends-Giele-Kuijf recursion formula Gronningen 3-5 Dec 2009Playing with Gravity24 Anti holomorphic Contributions – feature in gravity Recent work: (Elvang, Freedman: Nguyen, Spradlin, Volovich, Wen)

25. KLT for NMHV KLT hold independent of helicity NMHV amplituder are more complicated but KLT can still be used NMHV amplitudes change much by Helicity structure In Lecture II we will see how KLT is very useful in cuts as well … Gronningen 3-5 Dec 200925Playing with Gravity

26. 26 Twistor space

27. Duality Proposal that N=4 super Yang-Mills is dual to a string theory in twistor space? (Witten) Topological String Theory with twistor target space CP 3 Perturbative N=4 super Yang-Mills Gronningen 3-5 Dec 200927Playing with Gravity

28. Twistor space Transformation of amplitudes into twistor space (Penrose) In metric signature ( + + - - ) : 2D Fourier transform In twistor space : plane wave function is a line: Tree amplitudes in YM on degenerate algebraic curves Degree : number of negative helicities (Witten) Degree : N-1+L Gronningen 3-5 Dec 200928Playing with Gravity

29. Review: CSW expansion of Yang- Mills amplitudes In the CSW-construction : off-shell MHV-amplitudes building blocks for more complicated amplitude expressions (Cachazo, Svrcek and Witten) MHV vertices: Gronningen 3-5 Dec 200929Playing with Gravity

30. Example of how this works Gronningen 3-5 Dec 200930Playing with Gravity Example of A 6 (1 -,2 -,3 -,4 +,5 +,6 + )

31. 31 Twistor space properties Twistor-space properties of gravity: More complicated! Derivatives of  - functions  -functions Signature of non-locality  typical in gravity N=4 Anti-holomorphic pieces in gravity amplitudes Gronningen 3-5 Dec 200931Playing with Gravity

32. 32 Collinear and Coplanar Operators Gronningen 3-5 Dec 200932Playing with Gravity

33. 33 Twistor space properties For gravity : Guaranteed that Five-point amplitude. (Giombi, Ricci, Rables-Llana and Trancanelli; Bern, NEJBB and Dunbar) Tree amplitudes : Acting with differential operators F and K Gronningen 3-5 Dec 200933Playing with Gravity (Bern, NEJBB and Dunbar)

34. 34 Recursion

35. 35 BCFW Recursion for trees Shift of the spinors : Amplitude transforms as We can now evaluate the contour integral over A(z) Complex momentum space!! a and b will remain on-shell even after shift (Britto, Cachazo, Feng, Witten) Gronningen 3-5 Dec 200935Playing with Gravity

36. 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) Gronningen 3-5 Dec 200936Playing with Gravity

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

38. 38 MHV vertex expansion for gravity tree amplitudes CSW expansion in gravity Shift (Risager) Reproduce CSW for Yang-Mills Shift : Correct factorisation CSW vertex (NEJBB, Dunbar, Ita, Perkins, Risager) Gronningen 3-5 Dec 200938Playing with Gravity

39. 39 MHV vertex expansion for gravity tree amplitudes Negative legs shifted in the following way Analytic continuation of amplitude into the complex plane If M n (z), 1) rational, 2) simple poles at points z, and 3) vanishes (justified assumption) : M n (0) = sum of residues (as in BCFW), Gronningen 3-5 Dec 200939Playing with Gravity

40. 40 All poles : Factorise as : vanishes linearly in z : Spinor products : not z dependent (normal CSW) MHV vertex expansion for gravity tree amplitudes Gronningen 3-5 Dec 200940Playing with Gravity

41. 41 For gravity : Substitutions MHV amplitudes on the pole MHV vertices! – MHV vertex expansion for gravity MHV vertex expansion for gravity tree amplitudes Contact term! non-locality Gronningen 3-5 Dec 200941Playing with Gravity Matter MHV expansion considered by (Bianchi, Elvang, Friedman) problem with expansion beyond 12pt..

42. Conclusions lecture 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 Gronningen 3-5 Dec 200942Playing with Gravity

43. Outline of lecture II Outline af lecture II – In Lecture II we will consider how the tree results can be used to derive results for loop amplitudes – We will see how simple results for tree amplitudes makes it possible to derive simple loop results – Also we will see how symmetries of trees are carried over to loop amplitudes Gronningen 3-5 Dec 200943Playing with Gravity

44. 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 200944Playing with Gravity