Abstract: The duality relation between one-loop integrals and phase-space integrals, developed in a previous work, is extended to higher-order loops. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators, which compensates for the absence of the multiple-cut contributions that appear in the Feynman tree theorem. We rederive the duality theorem at one-loop order in a form that is more suitable for its iterative extension to higher-loop orders. We explicitly show its application to two-and three-loop scalar master integrals, and we discuss the structure of the occurring cuts and the ensuing results in detail.

Abstract: We consider the singular behaviour of QCD scattering amplitudes in kinematical configurations where two or more momenta of the external partons become collinear. At the tree level, this behaviour is known to be controlled by factorization formulae in which the singular collinear factor is universal (process independent). We show that this strict (process-independent) factorization is not valid at one-loop and higher-loop orders in the case of the collinear limit in space-like regions (e. g., collinear radiation from initial-state partons). We introduce a generalized version of all-order collinear factorization, in which the space-like singular factors retain some dependence on the momentum and colour charge of the non-collinear partons. We present explicit results on one-loop and two-loop amplitudes for both the two-parton and multiparton collinear limits. At the level of squared amplitudes and, more generally, cross sections in hadron-hadron collisions, the violation of strict collinear factorization has implications on the non-abelian structure of logarithmically-enhanced terms in perturbative calculations (starting from the next-to-next-to-leading order) and on various factorization issues of mass singularities (starting from the next-to-next-to-next-to-leading order).

Abstract: Calculation of amplitudes in perturbative quantum field theory involve large loop integrals. The complexity of those integrals, in combination with the large number of Feynman diagrams, make the calculations very difficult. Reduction methods proved to be very helpful, lowering the number of integrals that need to be actually calculated. Especially reduction at the integrand level improves the speed and set-up of these calculations. In this article we demonstrate, by counting the numbers of tensor structures and independent coefficients, how to write such relations at the integrand level for one-and two-loop amplitudes. We clarify their connection to the so-called spurious terms at one loop and discuss their structure in the two-loop case. This method is also applicable to higher loops, and the results obtained apply to both planar and non-planar diagrams.

Abstract: We analyze oriented event-shapes in the context of Soft-Collinear Effective Theory (SCET) and in fixed-order perturbation theory. Oriented event-shapes are distributions of event-shape variables which are differential on the angle theta(T) that the thrust axis forms with the electron-positron beam. We show that at any order in perturbation theory and for any event shape, only two angular structures can appear: F-0 = 3/8 (1+cos(2) theta(T)) and F-1 = (1 – 3 cos(2) theta(T)). When integrating over theta(T) to recover the more familiar event-shape distributions, only F-0 survives. The validity of our proof goes beyond perturbation theory, and hence only these two structures are present at the hadron level. The proof also carries over massive particles. Using SCET techniques we show that singular terms can only arise in the F-0 term. Since only the hard function is sensitive to the orientation of the thrust axis, this statement applies also for recoil-sensitive variables such as Jet Broadening. We show how to carry out resummation of the singular terms at (NLL)-L-3 for Thrust, Heavy-Jet Mass, the sum of the Hemisphere Masses and C-parameter by using existing computations in SCET. We also compute the fixed-order distributions for these event-shapes at O(alpha(S)) analytically and at O(alpha(2)(S)) with the program Event2.

Abstract: Landau parameters determined from phenomenological finite-range interactions are used to get an estimation of next-to-next-to-next-to-leading order ((NLO)-L-3) pseudo-potentials parameters. The parameter sets obtained in this way are shown to lead to consistent results concerning saturation properties. The uniqueness of this procedure is discussed, and an estimate of the error induced by the truncation at (NLO)-L-3 is given.