A Brief Introduction to Modern Amplitude Methods

Lance J. Dixon
2015 Journeys Through the Precision Frontier: Amplitudes for Colliders  
I provide a basic introduction to modern helicity amplitude methods, including color organization, the spinor helicity formalism, and factorization properties. I also describe the BCFW (on-shell) recursion relation at tree level, and explain how similar ideas -unitarity and on-shell methods -work at the loop level. These notes are based on lectures delivered at the 2012 CERN Summer School and at TASI 2013. Introduction Scattering amplitudes are at the heart of high energy physics. They lie at
more » ... e intersection between quantum field theory and collider experiments. Currently we are in the hadron collider era, which began at the Tevatron and has now moved to the Large Hadron Collider (LHC). Hadron colliders are broadband machines capable of great discoveries, such as the Higgs boson [1], but there are also huge Standard Model backgrounds to many potential signals. If we are to discover new physics (besides the Higgs boson) at the LHC, we will need to understand the old physics of the Standard Model at an exquisitely precise level. QCD dominates collisions at the LHC, and the largest theoretical uncertainties for most processes are due to our limited knowledge of higher order terms in perturbative QCD. Many theorists have been working to improve this situation. Some have been computing the nextto-leading order (NLO) QCD corrections to complex collider processes that were previously only known at leading order (LO). LO uncertainties are often of order one, while NLO uncertainties can be in the 10-20% range, depending on the process. Others have been computing the next-to-next-to-leading order (NNLO) corrections to benchmark processes that are only known at NLO; most NNLO predictions have uncertainties in the range of 1-5%, allowing precise experimental measurements to be interpreted with similar theoretical precision. Higher-order computations have a number of technical ingredients, but they all require loop amplitudes, one-loop for NLO, and both one-and two-loop for NNLO, as well as tree amplitudes of higher multiplicity. The usual textbook methods for computing an unpolarized cross section involve squaring the scattering amplitude at the beginning, then summing analytically over the spins of external states, and transforming the result into an expression that only involves momentum invariants (Mandelstam variables) and masses. For complex processes, this approach is usually infeasible. If there are N Feynman diagrams for an amplitude, then there are N 2 terms in the square of the amplitude. It is much better to calculate the N terms in the amplitude, as a complex number, and then compute the cross section by squaring that number. This approach of directly computing the amplitude benefits greatly from the fact that many amplitudes are much simpler than one might expect from the number of Feynman diagrams contributing to them. In order to compute the amplitude directly, one has to pick a basis for the polarization states of the external particles. At collider energies, most of these particles are effectively massless: the light quarks and gluons, photons, and the charged leptons and neutrinos (decay products of W and Z bosons). Massless fermions have the property that their chirality and helicity coincide, and their chirality is preserved by the gauge interactions. Therefore the helicity basis is clearly an optimal one for massless fermions, because many matrix elements (the helicity-flip ones) will always vanish. Around three decades ago, it was realized that the helicity basis was extremely useful for massless vector bosons as well [2] . Many tree-level amplitudes were found to vanish in this basis as well (which Published by CERN in the
doi:10.1142/9789814678766_0002 fatcat:ka33vobjizagrprrxqpudtf2uq