OnShell Methods in Field Theory David A Kosower

On-Shell Methods in Field Theory David A. Kosower International School of Theoretical Physics, Parma, September 10 -15, 2006 Lecture I

Tools for Computing Amplitudes • Focus on gauge theories …but they are useful for gravity too • Motivations and connections – Particle physics – N =4 supersymmetric gauge theories and Ad. S/CFT – Witten’s twistor string On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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? Now 450 to 600 days away… On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

CDF event On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

CMS Higgs event simulation On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

D 0 event On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 Field Theory 2006 , Parma, September 10– 15,

Guenther Dissertori (Jan ’ 04) On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 Field Theory 2006 , Parma, September 10– 15,

• Complexity is due to QCD • Perturbative QCD: Gluons & quarks → gluons & quarks • Real world: Hadrons → hadrons with hard physics described by p. QCD • Hadrons → jets On-Shell Methods in Field Theory 2006 narrow nearly collimated streams of hadrons , Parma, September 10– 15,

Jets • Defined by an experimental resolution parameter invariant mass in e+e− – cone algorithm in hadron colliders: cone size in and minimum ET – k. T algorithm: essentially by a relative transverse momentum – CDF (Lefevre 2004) 1374 Ge. V On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

In theory, theory and practice are the same. In practice, they are different — Yogi Berra On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

QCD-Improved Parton Model On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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: p. T(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) • On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Precision Perturbative QCD • • • Predictions of signals, signals+jets Predictions of backgrounds Everything at ahadron Measurement of luminosity collider involves QCD Measurement of fundamental parameters ( s, m t) Measurement of electroweak parameters Extraction of parton distributions — ingredients in any theoretical prediction On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Leading-Order, Next-to-Leading Order • LO: Basic shapes of distributions but: no quantitative prediction — large scale dependence Anastasiou, Dixon, Melnikov, & Petriello 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 On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

What Contributions Do We Need? • Short-distance matrix elements to 2 -jet production at leading order: tree level On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

• Short-distance matrix elements to 2 -jet production at next-toleading order: tree level + one loop + real emission 2 On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Real-Emission Singularities Matrix element Integrate On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

• Physical quantities are finite • Depend on resolution parameter • Finiteness thanks to combination of Kinoshita–Lee–Nauenberg theorem and factorization On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Scattering matrix element Decompose it Invariant matrix element M Differential cross section On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Lorentz-invariant phase-space measure Compute invariant matrix element by crossing On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Lagrangian On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Feynman Rules Propagator (like QED) Three-gluon vertex (unlike QED) Four-gluon vertex (unlike QED) On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

From the Faddeev–Popov functional determinant anticommuting scalars or ghosts Propagator coupling to gauge bosons On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 — • Want approach in terms of physical states only On-Shell Methods in Field Theory 2006 , Parma, September 10– 15, attr to Feynman

Light-Cone Gauge Only physical (transverse) degrees of freedom propagate physical projector — two degrees of freedom On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 Field Theory 2006 , Parma, September 10– 15,

To unwind traces Leads to tree-level representation in terms of single traces Color-ordered amplitude — function of momenta & polarizations alone; not Bose symmetric On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Symmetry properties • Cyclic symmetry • Reflection identity • Parity flips helicities • Decoupling equation On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Color-Ordered Feynman Rules On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 Field Theory 2006 , Parma, September 10– 15,

Spinor Helicity Spinor wavefunctions Introduce spinor products Explicit representation where On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

We then obtain the explicit formulæ otherwise so that the identity On-Shell Methods in Field Theory 2006 , Parma, September 10– 15, always holds

Introduce four-component representation corresponding to matrices in order to define spinor strings On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Properties of the Spinor Product • Antisymmetry • Gordon identity • Charge conjugation • Fierz identity • Projector representation • Schouten identity On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

What is the significance of q? On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Properties of the Spinor-Helicity Basis Physical-state projector Simplifications On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Examples By explicit calculation (or other arguments), every term in the gluon tree-level amplitude has at least one factor of Look at four-point amplitude Recall three-point color-ordered vertex On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

Calculate choose identical reference momenta for all legs all amplitude vanishes Calculate choose reference momenta 4, 1, 1, 1 all amplitude vanishes Calculate choose reference momenta 3, 3, 2, 2 only nonvanishing is only s 12 channel contributes On-Shell Methods in Field Theory 2006 , Parma, September 10– 15, vanish

On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

No diagrammatic calculation required for the last helicity amplitude, Obtain it from the decoupling identity On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,

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 On-Shell Methods in Field Theory 2006 , Parma, September 10– 15,