The study of scattering amplitudes beyond one loop is necessary for precision phenomenology for the Large Hadron Collider and may also provide deeper insights into the theoretical foundations of quantum field theory. In this thesis we develop a new method for computing two-loop amplitudes, based on unitarity rather than Feynman diagrams. In this approach, the two-loop amplitude is first expanded in a linearly independent basis of integrals. The process dependence thereby resides in the coefficients of the integrals. These expansion coefficients are then the object of calculation. Our main results include explicit formulas for a subset of the integral coefficients, expressing them as products of tree-level amplitudes integrated over specific contours in the complex plane. We give a general selection principle for determining these contours. This principle is then applied to obtain the coefficients of integrals with the topology of a double box. We show that, for four-particle scattering, each double-box integral in the two-loop basis is associated with a uniquely defined complex contour, referred to as its master contour. We provide a classification of the solutions to setting all propagators of the general double-box integral on-shell. Depending on the number of external momenta at the vertices of the graph, these solutions are given as a chain of pointwise intersecting Riemann spheres, or a torus. This classification is needed to define master contours for amplitudes with arbitrary multiplicities. We point out that a basis of two-loop integrals with as many infrared finite elements as possible allows substantial technical simplications, in terms of obtaining the coefficients of the integrals, as well as for the analytic evaluation of the integrals themselves. We compute two such integrals at four points, obtaining remarkably compact expressions. Finally, we provide a check on a recently developed recursion relation for the all-loop integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory, examining the two-loop six-gluon MHV amplitude and finding agreement. The validity of the approach to two-loop amplitudes developed in this thesis extends to all four-dimensional gauge theories, in particular QCD. The approach is suited for obtaining compact analytical expressions as well as for numerical implementations.

La these porte sur les corrections mixtes QCD-EW au niveau NNLO a la productionDrell-Yan de bosons Z et W. Le processus Drell-Yan est un processus fondamentalpermettant de tester avec precision le Modele Standard (MS) de physique des partic-ules au sein de collisionneurs hadroniques, car ce dernier presente une section ecaceimportante, une signature experimentale tres propre, ainsi qu'une tres haute sensi-bilite aux proprietes des bosons de jauge. Pour toutes ces raisons, une prediction theorique precise et able, siginant ici que l'on garde sous contr^ole lestermes provenant des corrections perturbatives d'ordre superieur de la section ecaceet des distributions du mecanisme de production de Drell-Yan, est exigee pour menera bien des etudes de physique au niveau de collisionneurs hadroniques.Dans cette thèse , nous étudions les corrections QCD mixtes - EW à Drell - Yan traite à la NNLO . D'un point de vue technique , le calcul d'un tel ensemble de corrections impliquerait le cal-tion de diagrammes de Feynman très compliquées , La plus grande contribution provient des diagrammes dans lesquels la particule de décomposition ( Z ou boson W ) est presque sur - coquille.En utilisant les règles Cutkosky , nous pouvons ré-écrire l'intégration sur l'espace de phase de latermes d'interférence ( une boucle 2 à 2 diagrammes interféré avec le niveau arbre 2 à 2 etarbre 2 ou 3 diagrammes carré ) en termes de combinaison des intégrales de propagationteurs ayant la prescription et propagateurs de causalité droite avec une face .Ces intégrales peuvent être traités de la même manière que les corrections virtuelles . Cette réduction se fait en utilisant l' algorithme Laporta \ " , sur la base del'intégration par parties identités . Le calcul de l' IM est réalisée en utilisant la méthode de la différenceéquations. En conséquence , nous obtenons l' IM exprimée en série de Laurent ,où D est la dimension de l'espace - temps , la multiplication d'un facteur qui prend entenir compte de la limite souple de l'intégrale en D dimensions . / The thesis concerns the NNLO mixed QCD-EW corrections to the Drell-Yan (DY)production of Z andW bosons, via the following reactions: pp(p) Z+X to l + Xand pp to W + X to l + X. This is a fundamental process for an accurate testof the Standard Model (SM) at hadron colliders, since it has a large cross section, aclean experimental signature. In particular, the Drell-Yan production of Ws is important for an accuratedetermination (via transverse mass and pT distributions) of the W mass, mW, aninput parameter of the model. Because of all these reasons, an accurate and reliable theoretical prediction forthe cross section and the distributions of the Drell-Yan production mechanism, thatmeans control on the higher-order perturbative corrections, is demanded for physicsstudies at hadron colliders. In this thesis, we study the mixed QCD-EW corrections to Drell-Yan processes at the NNLO. From a technical point of view, the calculation of such a set of corrections would involve the calcu-lation of very complicated Feynman diagrams, The biggest contribution comes from the diagrams in which the decaying particle(Z or W boson) is nearly on-shell. Using the Cutkosky rules, we can re-write the integration over the phase-space of theinterference terms (one-loop 2 to 2 diagrams interfered with the tree-level 2 to 2 andtree 2 to 3 diagrams squared) in terms of a combination of integrals with propaga-tors having the right causality prescription and propagators with the opposite one.These integrals can be treated in the same way as the virtual corrections. This reduction is done using the \Laporta Algorithm", based onthe Integration-by-Parts Identities. The calculation of the MIs is performed using the method of differentialequations. As a result, we get the MIs expressed as a Laurent series ,where D is the dimension of the space-time, multiplying a factor which takes intoaccount the soft limit of the integral in D dimensions.

In this thesis we consider the phenomenology of the theory of strong interactions, Quantum Chromodynamics (QCD), with particular reference to the ongoing experimental program at the Large Hadron Collider in CERN. The current progress in precision measurement of Standard Model processes at the LHC experiments must be matched with corresponding precision in theoretical predictions, and to this end we present calculations at next-to-next-to-leading order in perturbation theory of observable quantities involving quarks and gluons, the strongly interacting particles of the SM. Such calculations form the most important class of corrections to observables and are vital if we are to untangle signals of New Physics from LHC data. We consider in particular the amplitudes for five parton interactions at 1- and 2-loop order and present full (in the 1-loop case) and partial (in the 2-loop case) analytic results in terms of rational functions of kinematic invariants multiplying a basis of master integrals. We address the problem of the solution of a system of integration-by-parts identities for Feynman integrals and demonstrate how some current difficulties may be overcome. We consider also the properties of the top quark, and present the NNLO, real-virtual contributions to the calculation of its decay rate. The results are presented as helicity amplitudes so that the full behaviour of the top spin is retained. These amplitudes constitute a necessary ingredient in the complete calculation of top quark pair production and decay at NNLO which will be an important theoretical input to many experimental analyses. Turning to a more phenomenological study, we consider the extraction of two important SM parameters, the top mass and the strong coupling constant, from measurements of top pair production at the ATLAS and CMS experiments. We compare with NNLO theory predictions and use a least-squares method to extract the values of the parameters simultaneously. We find best fit values of the parameters which are compatible with previous extractions performed using top data with the current world averages published by the Particle Data Group. We consider the issue of PDF choice and the circumstances in which a heavy quark can be considered a constituent of the proton. In particular, we look at the production of a Higgs boson in association with bottom quarks in four and five flavour schemes, in which the b may or may not be included in the initial state. We show that theoretical predictions in both schemes are well-motivated and appropriate in different scenarios, and moreover that results in the schemes are consistent provided a judicious choice of the renormalisation and factorisation scales is made. We suggest a typical scale choice motivated by considerations of consistency and find it to be somewhat lower than the typical hard scale of the process.