On-Shell Methods in Gauge Theory

1. On-Shell Methods in Gauge Theory David A. Kosower IPhT, CEA–Saclay Taiwan Summer Institute, Chi-Tou (溪頭) August 10–17, 2008 Lecture I

2. Tools for Computing Amplitudes • New tools for computing in gauge theories — the core of the Standard Model(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. On-Shell Methods • Physical states • Use of properties of amplitudes as calculational tools • Kinematics: Spinor Helicity Basis  Twistor space • Tree Amplitudes: On-shell Recursion Relations  Factorization • Loop Amplitudes: Unitarity (SUSY) Unitarity + On-shell Recursion QCD

4. Outline • Review: motivations; jets; QCD and parton model; radiative corrections; • Color decomposition and color ordering; spinor product and spinor helicity • Factorization, collinear and soft limits • On-shell and off-shell recursion relations • Unitarity method and one-loop amplitudes • Loop-level on-shell recursion relations and the bootstrap • Numerical approaches • Higher loops and applications to N = 4 SUSY

5. Particle Physics The LHC is coming, the LHC is coming! • Why do we compute in field theory? • Why do we do hard computations? • What quantities should we compute in field theory? Within one month!

8. D0 event

9. 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

10. Guenther Dissertori (Jan ’04)

11. 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

12. Complexity is due to QCD • Perturbative QCD: Gluons & quarks → gluons & quarks • Real world: Hadrons → hadrons with hard physics described by pQCD • Hadrons → jets narrow nearly collimated streams of hadrons

13. Jets • Defined by an experimental resolution parameter • invariant mass in e+e− • cone algorithm in hadron colliders: cone size in and minimum ET • kT algorithm: essentially by a relative transverse momentum CDF (Lefevre 2004)1374 GeV

14. In theory, theory and practice are the same. In practice, they are different — Yogi Berra

15. QCD-Improved Parton Model

16. The Challenge • Everything at a hadron collider (signals, backgrounds, luminosity measurement) involves QCD • Strong coupling is not small: s(MZ)  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: pT(W) & MW • 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)

17. CDF, PRD 77:011108 Leading-Order, Next-to-Leading Order • LO: Basic shapes of distributionsbut: 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

18. What Contributions Do We Need? • Short-distance matrix elements to 2-jet production at leading order: tree level

19. Short-distance matrix elements to 2-jet production at next-to-leading order: tree level + one loop + real emission 2

20. Real-Emission Singularities Matrix element Integrate

21. Physical quantities are finite • Depend on resolution parameter • Finiteness thanks to combination of Kinoshita–Lee–Nauenberg theorem and factorization • Singularities in virtual corrections canceled by those in real emission

22. Tree Amplitudes • First step in any physics process — leading-order contribution • But also — the key ingredient in loop calculations

23. Traditional Tool: Feynman Diagrams • Write down all Feynman diagrams for the desired process • Write out all vertex factors, kinematic and color, and contract indices with propagators • Square amplitude, contracting polarization vectors or fermion wavefunctions by summing over helicities • Yields expressions in terms of momentum invariants • But unmanageably large ones

24. So What’s Wrong with Feynman Diagrams? • Huge number of diagrams in calculations of interest — factorial growth • 2 → 6 jets: 34300 tree diagrams, ~ 2.5 ∙ 107 terms ~2.9 ∙ 106 1-loop diagrams, ~ 1.9 ∙ 1010 terms • 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

25. Light-Cone Gauge Only physical (transverse) degrees of freedom propagate physical projector — two degrees of freedom

26. 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

27. To unwind traces Leads to tree-level representation in terms of single traces Color-ordered amplitude — function of momenta & polarizations alone; not Bose symmetric

28. Symmetry properties • Cyclic symmetry • Reflection identity • Parity flips helicities • Decoupling equation

29. Color-Ordered Feynman Rules

30. Amplitudes Functions of momenta k, polarization vectors  for gluons; momenta k, spinor wavefunctions ufor fermions Gauge invariance implies this is a redundant representation:   k: A = 0

31. Spinor Variables From Lorentz vectors to bi-spinors 2×2 complex matrices with det = 1 ‘Chinese Magic’ Xu, Zhang, Chang (1984) Spinor-helicity basis

32. We have explicit formulæ otherwise so that the identity always holds Properties Transverse Normalized

33. Properties of the Spinor Product • Antisymmetry • Gordon identity • Charge conjugation • Fierz identity • Projector representation • Schouten identity

34. Color decomposition & stripping Gauge-theory amplitude  Color-ordered amplitude: function of kiand i  Helicity amplitude: function of spinor variables and helicities ±1 Spinor-helicity basis

35. Fierz identity

36. Calculate choose identical reference momenta for all legs  all vanish amplitude vanishes Calculate choose reference momenta 4,1,1,1  all vanish amplitude vanishes Calculate choose reference momenta 3,3,2,2  only nonvanishing is  (*,2,3) vertex vanishes  only s12 channel contributes 

38. No diagrammatic calculation required for the last helicity amplitude, Obtain it from the decoupling identity

39. These forms hold more generally, for larger numbers of external legs: Parke-Taylor equations Mangano, Xu, Parke (1986) Maximally helicity-violating or ‘MHV’ Proven using the Berends–Giele recurrence relations Berends & Giele (1988)

40. Berends–Giele Recursion Relations

41. J5 appearing inside J10 is identical to J5 appearing inside J17 • Imagine computing all subcurrents ‘bottom up’: first compute J3, then J4 , and so on. • O(n) different color-ordered currents Jk for each k appearing in n-point amplitude, n different ks • Compute once numerically maximal reuse • Computing each takes O(n2) steps (because of the four-point vertex) •  Polynomial complexity per helicity: O(n4)