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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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! Within one month!

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008

D0 event

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Guenther Dissertori (Jan ’04)

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 In theory, theory and practice are the same. In practice, they are different — Yogi Berra

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 QCD-Improved Parton Model

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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)

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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 CDF, PRD 77:011108

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 What Contributions Do We Need? Short-distance matrix elements to 2-jet production at leading order: tree level

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Short-distance matrix elements to 2-jet production at next-to- leading order: tree level + one loop + real emission 2

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Matrix element Integrate Real-Emission Singularities

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Tree Amplitudes First step in any physics process — leading-order contribution But also — the key ingredient in loop calculations

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 So What’s Wrong with Feynman Diagrams? Huge number of diagrams in calculations of interest — factorial growth 2 → 6 jets: tree diagrams, ~ 2.5 ∙ 10 7 terms ~2.9 ∙ loop diagrams, ~ 1.9 ∙ 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Light-Cone Gauge Only physical (transverse) degrees of freedom propagate physical projector — two degrees of freedom

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Symmetry properties Cyclic symmetry Reflection identity Parity flips helicities Decoupling equation

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Color-Ordered Feynman Rules

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Spinor Variables From Lorentz vectors to bi-spinors 2×2 complex matrices with det = 1 ‘Chinese Magic’ Xu, Zhang, Chang (1984) Spinor-helicity basis

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 We have explicit formulæ otherwise so that the identity always holds Properties Transverse Normalized

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Properties of the Spinor Product Antisymmetry Gordon identity Charge conjugation Fierz identity Projector representation Schouten identity

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Gauge-theory amplitude  Color-ordered amplitude: function of k i and  i  Helicity amplitude: function of spinor variables and helicities ±1 Color decomposition & stripping Spinor-helicity basis

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Fierz identity

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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 s 12 channel contributes 

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008

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

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 These forms hold more generally, for larger numbers of external legs: Parke-Taylor equations Mangano, Xu, Parke (1986) Proven using the Berends–Giele recurrence relations Berends & Giele (1988) Maximally helicity-violating or ‘MHV’

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Berends–Giele Recursion Relations

On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 J 5 appearing inside J 10 is identical to J 5 appearing inside J 17 Imagine computing all subcurrents ‘bottom up’: first compute J 3, then J 4, and so on. O ( n ) different color-ordered currents J k for each k appearing in n -point amplitude, n different k s Compute once numerically  maximal reuse Computing each takes O ( n 2 ) steps (because of the four-point vertex)  Polynomial complexity per helicity: O ( n 4 )