Abstract
We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called “spinning polynomials”. Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin cannot be “elementary”. The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can’t be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g − 2) as well as the beta function and rational terms in QCD. “Off-shell” observables like correlation functions and form-factors can be thought of as scattering amplitudes with external “probe” particles of general mass and spin, so all these objects — amplitudes, form factors and correlators, can be studied from a common on-shell perspective.
Highlights
Recent years have seen an explosion of progress in our understanding of scattering amplitudes in gauge theories and gravity
This has enabled the determination of scattering amplitudes of direct interest to collider physics experiments, while at the same time opening up novel directions of theoretical research into the foundations of quantum field theory, amongst other things revealing surprising and deep connections of this basic physics with areas of mathematics ranging from algebraic geometry to combinatorics to number theory
While it is clear that the conventional field-theoretic description of massless particles with spin, which involves the introduction of huge gauge redundancy, leaves ample room for improvement — provided by on-shell methods that directly describe particles, eliminating any reference to quantum fields and their attendant redundancies — the advantage of “on-shell physics” seems to disappear for the case of massive particles where no gauge redundancies are needed
Summary
Recent years have seen an explosion of progress in our understanding of scattering amplitudes in gauge theories and gravity. If the amazing structures unearthed in the study of gauge and gravity scattering amplitudes are an indication of a radical new way of thinking about quantum particle interactions in space-time, they must naturally extend beyond photons, gravitons and gluons to electrons, W, Z particles and top quarks as well Keeping this central motivation in mind, in this paper we initiate a systematic exploration of the physics of scattering amplitudes in four dimensions, for particles of general masses and spins. 8, to compute the correlation functions for (say) the stress tensor (in momentum-space), we need only imagine weakly coupling a continuum of massive spin 2 particle to the system with a universal (and arbitrarily weak) coupling; the leading scattering amplitudes for these massive particles is literally the correlation function for the stress tensor in momentum space This should allow us to explore both on- and off-shell physics in a uniform “on-shell” way
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