A minimal approach to the scattering of physical massless bosons

Abstract

Abstract:Tree and loop level scattering amplitudes which involve physical massless bosons are derived directly from physical constraints such as locality, symmetry and unitarity, bypassing path integral constructions. Amplitudes can be projected onto a minimal basis of kinematic factors through linear algebra, by employing four dimensional spinor helicity methods or at its most general using projection techniques. The linear algebra analysis is closely related to amplitude relations, especially the Bern-Carrasco-Johansson relations for gluon amplitudes and the Kawai-Lewellen-Tye relations between gluons and graviton amplitudes. Projection techniques are known to reduce the computation of loop amplitudes with spinning particles to scalar integrals. Unitarity, locality and integrationby-parts identities can then be used to fix complete tree and loop amplitudes efficiently. The loop amplitudes follow algorithmically from the trees. A number of proof-of-concept examples are presented. These include the planar four point two-loop amplitude in pure Yang-Mills theory as well as a range of one loop amplitudes with internal and external scalars, gluons and gravitons. Several interesting features of the results are highlighted, such as the vanishing of certain basis coefficients for gluon and graviton amplitudes. Effective field theories are naturally and efficiently included into the framework. Dimensional regularisation is employed throughout; different regularisation schemes are worked out explicitly. The presented methods appear most powerful in non-supersymmetric theories in cases with relatively few legs, but with potentially many loops. For instance, in the introduced approach iterated unitarity cuts of four point amplitudes for non-supersymmetric gauge and gravity theories can be computed by matrix multiplication, generalising the socalled rung-rule of maximally supersymmetric theories. The philosophy of the approach to kinematics also leads to a technique to control colour quantum numbers of scattering amplitudes with matter, especially efficient in the adjoint and fundamental representations.

Two-, three-, and four-body decays in the constructive standard model

“…In order to conjugate any amplitude, we also describe the conjugation of complete spinor-chains with momenta in Eqs. (7) through (10) and (12) and (13).…”

“…The other two particle's spinors are then contracted with each other as either square or angle brackets. Therefore, the amplitude must be a linear combination of the terms [13].…”

“…Following this, [12] found the complete set of constructive SM 3-point vertices, found their high-energy limits and compared them with the massless version. Other authors have looked at how spin-spinors can be used outside of path integral constructions [13], how to use this formalism to study the spin structure of QCD [14], as well as studies involving gravity [15]. There still remains much to do to fully establish this massive constructive theory to determine what kind of simplifications are manifest in a massive theory when compared to Feynman diagrams.…”

Two-Loop QCD Corrections to the Higgs Plus Three-parton Amplitudes with Top Mass Correction

“…Since amplitude is gauge invariant, we can expand it using a set of gauge invariant basis B α (see e.g. [27] and also [59,78] for recent general discussion):…”

The analytic structure of amplitudes on backgrounds from gauge invariance and the infra-red

“…It has been shown that gauge invariance is enough to completely determine scattering amplitudes and their underlying analytical structure in certain theories [1][2][3][4][5][6][7], and it has been conjectured that locality and unitarity emerge as a consequence of imposing gauge invariance [2,8]. The investigation of which principles determine scattering amplitudes is not limited to gauge theories; it has been shown that soft theorems are enough to fix tree-level scattering amplitudes in the non-linear sigma model and Dirac-Born-Infeld [9,10], and to impose strong constraints on the Lagrangians of both scalar and vector effective field theories [11][12][13].…”