1. Computational Methods in Particle Physics: On-Shell Methods in Field Theory David A. Kosower University of Zurich, January 31–February 14, 2007 Lecture I

2. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Tools for Computing Amplitudes New tools for computing in gauge theories — the core of the Standard Model (but they are useful for gravity too) Motivations and connections – Particle physics: SU (3)  SU (2)  U (1) – N =4 supersymmetric gauge theories and AdS/CFT – Witten’s twistor string

3. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Outline Review: motivations; jets; QCD and parton model; radiative corrections; Color decomposition and color ordering; spinor product and spinor helicity Twistors and spinors; CSW rules; Factorization, collinear and soft limits Off-shell and on-shell recursion relations Supersymmetry Ward identities Unitarity method and one-loop amplitudes Loop-level on-shell recursion relations and the bootstrap Infrared cancellations Higher loops

4. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Particle Physics Why do we compute in field theory? Why do we do hard computations? What quantities should we compute in field theory? The LHC is coming, the LHC is coming! Now 275 to 490 days away…

5. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007

7. D0 event

8. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 SU(3)  SU(2)  U(1) Standard Model Known physics, and background to new physics Hunting for new physics beyond the Standard Model Discovery of new physics Compare measurements to predictions — need to calculate signals Expect to confront backgrounds Backgrounds are large

9. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Guenther Dissertori (Jan ’04)

10. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Hunting for New Physics Yesterday’s new physics is tomorrow’s background To measure new physics, need to understand backgrounds in detail Heavy particles decaying into SM or invisible states – Often high-multiplicity events – Low multiplicity signals overwhelmed by SM: Higgs → → 2 jets Predicting backgrounds requires precision calculations of known Standard Model physics

11. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Complexity is due to QCD Perturbative QCD: Gluons & quarks → gluons & quarks Real world: Hadrons → hadrons with hard physics described by pQCD Hadrons → jetsnarrow nearly collimated streams of hadrons

12. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Jets Defined by an experimental resolution parameter – invariant mass in e + e − – cone algorithm in hadron colliders: cone size in and minimum E T – k T algorithm: essentially by a relative transverse momentum CDF (Lefevre 2004) 1374 GeV

13. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 In theory, theory and practice are the same. In practice, they are different — Yogi Berra

14. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 QCD-Improved Parton Model

15. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 The Challenge Everything at a hadron collider (signals, backgrounds, luminosity measurement) involves QCD Strong coupling is not small:  s (M Z )  0.12 and running is important  events have high multiplicity of hard clusters (jets)  each jet has a high multiplicity of hadrons  higher-order perturbative corrections are important Processes can involve multiple scales: p T (W) & M W  need resummation of logarithms Confinement introduces further issues of mapping partons to hadrons, but for suitably-averaged quantities (infrared-safe) avoiding small E scales, this is not a problem (power corrections)

16. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Approaches General parton-level fixed-order calculations – Numerical jet programs: general observables – Systematic to higher order/high multiplicity in perturbation theory – Parton-level, approximate jet algorithm; match detector events only statistically Parton showers – General observables – Leading- or next-to-leading logs only, approximate for higher order/high multiplicity – Can hadronize & look at detector response event-by-event Semi-analytic calculations/resummations – Specific observable, for high-value targets – Checks on general fixed-order calculations

17. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Precision Perturbative QCD Predictions of signals, signals+jets Predictions of backgrounds Measurement of luminosity Measurement of fundamental parameters (  s, m t ) Measurement of electroweak parameters Extraction of parton distributions — ingredients in any theoretical prediction Everything at a hadron collider involves QCD

18. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Leading-Order, Next-to-Leading Order LO: Basic shapes of distributions but: no quantitative prediction — large scale dependence missing sensitivity to jet structure & energy flow NLO: First quantitative prediction improved scale dependence — inclusion of virtual corrections basic approximation to jet structure — jet = 2 partons NNLO: Precision predictions small scale dependence better correspondence to experimental jet algorithms understanding of theoretical uncertainties Anastasiou, Dixon, Melnikov, & Petriello

19. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 What Contributions Do We Need? Short-distance matrix elements to 2-jet production at leading order: tree level

20. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Short-distance matrix elements to 2-jet production at next-to- leading order: tree level + one loop + real emission 2

21. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Matrix element Integrate Real-Emission Singularities

22. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Physical quantities are finite Depend on resolution parameter Finiteness thanks to combination of Kinoshita–Lee–Nauenberg theorem and factorization

23. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Scattering Scattering matrix element Decompose it Invariant matrix element M Differential cross section

24. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Lorentz-invariant phase-space measure Compute invariant matrix element by crossing

25. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Lagrangian

26. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Feynman Rules Propagator (like QED) Three-gluon vertex (unlike QED) Four-gluon vertex (unlike QED)

27. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 From the Faddeev–Popov functional determinant anticommuting scalars or ghosts Propagator coupling to gauge bosons

28. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 So What’s Wrong with Feynman Diagrams? Huge number of diagrams in calculations of interest But answers often turn out to be very simple Vertices and propagators involve gauge-variant off-shell states Each diagram is not gauge invariant — huge cancellations of gauge-noninvariant, redundant, parts in the sum over diagrams Simple results should have a simple derivation — attr to Feynman Want approach in terms of physical states only

29. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Light-Cone Gauge Only physical (transverse) degrees of freedom propagate physical projector — two degrees of freedom

30. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Color Decomposition Standard Feynman rules  function of momenta, polarization vectors , and color indices Color structure is predictable. Use representation to represent each term as a product of traces, and the Fierz identity

31. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 To unwind traces Leads to tree-level representation in terms of single traces Color-ordered amplitude — function of momenta & polarizations alone; not not Bose symmetric

32. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Symmetry properties Cyclic symmetry Reflection identity Parity flips helicities Decoupling equation

33. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Color-Ordered Feynman Rules

34. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Amplitudes Functions of momenta k, polarization vectors  for gluons; momenta k, spinor wavefunctions u for fermions Gauge invariance implies this is a redundant representation:   k: A = 0

35. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Spinor Helicity Spinor wavefunctions Introduce spinor products Explicit representation where

36. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 We then obtain the explicit formulæ otherwise so that the identity always holds

37. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Introduce four-component representation corresponding to  matrices in order to define spinor strings

38. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Properties of the Spinor Product Antisymmetry Gordon identity Charge conjugation Fierz identity Projector representation Schouten identity

39. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Spinor-Helicity Representation for Gluons Gauge bosons also have only ± physical polarizations Elegant — and covariant — generalization of circular polarization Xu, Zhang, Chang (1984) reference momentum q Transverse Normalized

40. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 What is the significance of q ?

41. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Properties of the Spinor-Helicity Basis Physical-state projector Simplifications