Scattering amplitudes in four- and six-dimensional gauge theories

We study scattering amplitudes in quantum chromodynamics (QCD), N = 4 super Yang-Mills (SYM) theory and the six-dimensional N = (1, 1) SYM theory, focusing on the symmetries of and relations between the tree-level scattering amplitudes in these three gauge theories. We derive the tree-level superconformal and dual superconformal symmetries of N = 4 SYM theory in non-chiral superspace using the non-chiral BCFW recursion to prove the latter. Similarly we give a complete derivation of the symmetries of the tree-level superamplitudes of N = (1, 1) SYM theory in six dimensions, again using the BCFW recursion to prove their dual conformal symmetry and thereby correcting some minor mistakes in the proof previously known in the literature. We provide the means necessary for efficient next to leading order QCD calculations by deriving the tree level and one-loop color decomposition of an arbitrary QCD amplitude into primitive amplitudes. This generalizes the known analytic results for amplitudes of arbitrary multiplicity with up to one quark anti-quark pair and provides an alternative to the Feynman diagram based algorithm for the determination of the decomposition into primitive amplitudes. Furthermore, we derive general fermion flip and reversion identities, spanning the null space among the primitive amplitudes. An accompanying, public Mathematica package QCDcolor contains implementations of the color decompositions as well as the flip and reversion identities. Based on the obtained color decomposition of QCD tree amplitudes and the fermion flip identities we prove that every color ordered tree amplitude of massless QCD can be written as a linear combination of gluon-gluino amplitudes of N = 4 SYM theory. The proof includes a general construction of these linear combinations. For all color ordered QCD amplitudes with up to four quark-antiquark pairs and an arbitrary number of gluons, we determine representations containing a minimal number of gluon-gluino amplitudes. Furthermore, we derive analytical formulae for all gluon-gluino amplitudes relevant for QCD by projecting the previously-found expressions for the superamplitudes of N = 4 SYM onto the relevant components. The obtained analytical formulae are implemented in the public Mathematica package GGT, and in turn yield all QCD tree amplitudes as well as the cut constructable part of all QCD loop amplitudes. We compare the numerical efficiency of evaluating these closed analytic formulae for color ordered QCD tree amplitudes to a numerically efficient implementation of the Berends-Giele recursion. We compare calculation times for tree amplitudes with parton numbers ranging from 4 to 25 with no, one, two and three external quark lines. It turns out that the analytical results are generally faster in the case of MHV and NMHV amplitudes. Starting with the NNMHV amplitudes the Berends-Giele recursion becomes more efficient. In addition to the runtime we also compare the numerical accuracy, finding that the analytical formulae are on average more accurate than the off-shell recursion relations though both are well suited for complicated phenomenological applications. In both cases we observe a reduction in the average accuracy when phase space configurations close to singular regions are evaluated. We believe that the above findings provide valuable information to select the right method for phenomenological applications. As a first step of trying to expand our results to massive QCD amplitudes, we investigate massive tree amplitudes on the coulomb branch of N = 4 SYM