1. Song He Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing

2. Motivations: QFT and Amplitudes

3. Quantum Field Theory (QFT) Our theoretical framework to describe Nature Essentially the consequence of two major principles

4. Perturbative QFT Feynman diagrams: pictures of particle interactions Perturbative expansion: trees, loops

5. Great success of QFT QFT has passed countless tests in last 70 years Example: Magnetic dipole moment of electron

6. Incomplete picture Our understanding of QFT is incomplete! Also, tension with gravity and cosmology Explicit evidence: scattering amplitudes If there is a new way of thinking about QFT, it must be seen even at weak coupling

7. Colliders at high energies Proton scattering at high energies Needed: amplitudes of gluons for higher multiplicities LHC - gluonic factory gg → gg … g

8. Early 80s Status of the art: gg → ggg Brute force calculation 24 pages of result

9. New collider 1983: Superconducting Super Collider approved Energy 40 TeV: many gluons! Demand for calculations, next on the list: gg→gggg

10. Parke-Taylor formula Process : gg→gggg 220 Feynman diagrams, ~ 100 pages of calculations 1985: Paper with 14 pages of result (Parke, Taylor 1985)

11. Parke-Taylor formula Within a year they realized

12. Parke-Taylor formula Within a year they realized

13. Birth of amplitudes

14. Spinor-Helicity Formalism

15. Lorentz group representations Spinors Infinitesimally with These are 6 generators of the Lorentz group: We see that the SO(1, 3) may be mapped to two commuting copies of SU(2)

16. Spinors Weyl spinors Chiral solutions of the massless Dirac equation Basis e.g.

17. Lower spin representations of 4-D Lorentz group Spinors

18. Spinor-helicity formalism for massless particles For a massless particle

19. An explicit realization Requiring the four-momentum to be real with Lorentz-signature translates into the relation The sign in this relation follows the sign of the energy of the associated four-momentum Spinor-helicity formalism for massless particles

20. Spinor invariants By definition, Lorentz invariant objects= functions of spinor invariants (“angle” and “square” brackets) Mandelstam invariants For real momenta, we have invariants of the two SU(2)’s

21. anti-symmetry Momentum conversation Schouten Identity Spinor invariants

22. Rescaling freedom (little group) for real momenta for complex momenta Little group

23. Under little group scaling, the amplitude transforms homogeneously. e.g. Little group

24. U(1) generator of helicity Helicity Lorentz invariant quantity for massless particles

25. Three point amplitudes which is a singular kinematic point for real momenta! Momentum conversation requires has two chirally conjugate solutions

26. solution 1: for solution 2 Up to an overall constant Three point amplitudes The 3-pt amplitude is non-singular, and completely fixed by Poincare invariance! Little group scaling fixes

27. Consider the special case of 3pt amps with identical spin (s) particles e.g. --+ or but the second case is obviously non-local! Poincare + locality → MHV three point amplitudes Three point amplitudes

28. Gluon polarizations Basic properties (null, transverse, etc.) are automatically satisfied

29. transforms like helicity +1 Gluon polarizations

30. pick Gluon polarizations: 3 pt

31. Graviton polarization ~ 200 terms

32. The on-shell amplitude is remarkable simple (the square of gluon 3pt amplitudes!) Graviton polarization: 3 pt

33. Unimaginable to calculate 4 or more gravitons amplitudes by brute force thousands of terms Graviton polarization: 4 pt

34. Gauge-theory Amps: Helicity, Color etc.

35. Helicity amplitudes Helicities of gluons (gravitons) +1 or -1 (+2 or -2) helicity amplitudes e.g. A(+++---), A(++----), A(--++++), … Huge simplifications, with different properties and structures! Number of negative-helicity gluons (gravitons): k k=0: A(+, +, +, …,+)=0 k=1: A(-, -, -, …, -)=0 k=n-1: A(-, +, +, …, +)=0 k=n: A(+, -, -, …, -)=0 both at tree level in general and For all loops with supersymmetries

36. Vanishing tree amplitudes

37. Similarly, the gluon tree-amplitude with one flipped helicity state vanishes which follows from the choice

38. Simplest amplitudes: MHV

39. MHV classification

40. Non-Abelian Gauge Theories invariant under the local gauge transformation

41. Feynman rules for non-abelian gauge theory

42. Color decompositions organize color d.o.f.→ colors x kinematics → pieces with simpler analytic properties

43. the SU(N c ) Fierz identity for simplifying the resulting traces photonphoton

44. Reducing the color factor to a single color trace

45. Different orderings related by permutations Gauge invariant This is a key object of our interest Color-summed cross section is indeed made up of color-ordered amplitudes Color-ordered amplitudes simpler pole structure

46. Color ordered Feynman rules

47. QCD amplitudes: e.g. similar decomposition for those with one fermion line

48. Properties of color-ordered amplitudes Cyclicity Reflection U(1) decoupling

49. Kleiss-Kuijf relations (fancy version of U(1) decoupling) Bern-Carrasco-Johansson relations

50. Some Examples

51. A simple four-point example little group check

52. exchanges labels 1 and 2 Other helicity structures two other terms can be obtained by parity

53. A five-point amplitude

54. four gluon amplitude

55. A nice choice of reference left only 2 diagrams vanish