  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
integration-by-parts identities can then be used to fix complete tree and loop
amplitudes efficiently. The loop amplitudes follow algorithmically from the
trees. A range of proof-of-concept examples is 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. The presented methods appear most powerful in non-supersymmetric
theories in cases with relatively few legs, but with potentially many loops.
For instance, iterated unitarity cuts of four point amplitudes for
non-supersymmetric gauge and gravity theories can be computed by matrix
multiplication, generalising the so-called rung-rule of maximally
supersymmetric theories. The philosophy of the approach to kinematics also
leads to a technique to control color quantum numbers of scattering amplitudes
with matter.
