Final Report Summary - TOPPHYSICS (Precision Physics and Discovery at Hadron Colliders with Heavy Quarks) In the last two years, we witnessed the triumph of the Standard Model of fundamental interactions (SM). Thanks to the great performance of the run I of the Large Hadron Collider of CERN, it was possible to pose the last tile and complete the picture of the Physics in the microscopic regime up to energies in the TeV scale. The last missing particle of the SM, the Brout-Engler-Higgs boson, responsible for the generation of particle masses via the mechanism of spontaneous symmetry breaking, was discovered in 2012. In the following year, F. Engler and P. Higgs were awarded the Nobel Prize in Physics for the underlying theoretical formulation. Since that date, the Community of particle physicists was captured by the study of the properties of the new boson, that now seams to correspond exactly to the particle formulated by Brout, Engler and Higgs. For a large part of the Community, another result was widely expected: the striking manifestation, already at the energies of the LHC, of New Physics beyond the just completed SM. This expectation was not satisfied so far and, at the moment, the search for New Physics consists in the study of possible small deviations of the measurements from the theoretical SM predictions. This is the scenario of New Physics search that was considered when TOPPhysics was proposed.The key ingredients, in this scenario, are high-quality data from the experimental side and an extremely good control on the SM predictions from the theoretical side, at a level of sophistication that was never reached in the past. After the "Next-to-leading Order (NLO) revolution", that took place in the last ten years, it is now common to think about theoretical predictions for the LHC that include at least the first quantum perturbative corrections to the basic process. As it was shown in many occasions, the NLO perturbative corrections to observables at hadron colliders can have a big theoretical impact. They can change dramatically a prediction, revealing that an analysis based on the leading order is largely unreliable. With the run I and even more with the forthcoming run II at the LHC, we are in a situation in which the "Next-to-next-to-leading Order" (NNLO) perturbative corrections have to be definitely included in the theoretical analysis of many physical observables. TOPPhysics was proposed to contribute to the improvement of the theoretical predictions in a very important sector of the high-energy physics search at the LHC: the production and decay of heavy quarks. Heavy quarks constitute a very important instrument both for testing the SM and for the search of New Physics beyond it. The top quark (t) in particular, being the heaviest particle so far produced at colliders, represents a unique probe of the dynamics that breaks the electroweak symmetry. Differently from the other quarks, it does not hadronize: it decays in a b quark and a W boson in a time shorter than the usual hadronization time. This feature of the top quark gives us the opportunity to study a quark as a single particle, thus providing us with the access to all its properties that are not diluted in the hadronization process, such as its spin, interaction vertices and mass. Moreover, the top quark is important in many models of New Physics, in which it plays a direct role in the electroweak symmetry breaking or it has enhanced interactions with new gauge bosons coming, for instance, from the breaking of grand unified symmetry groups.Heavy quarks are profusely produced at the LHC. The statistics is so high, that the observables related to heavy-quark production (and decay) are measured with an extremely high accuracy. To compare these data with the SM and possibly reveal deviations from it and evidences of New Physics, we need an equally accurate theoretical prediction.In our project, we want to address this issue and provide precise theoretical predictions for a class of observables related to the production (and decay) of heavy quarks at hadron colliders. TOPPhysics is structured in two main parts, with some individual (self consistent) investigation subjects. The first part regards the theory predictions within the SM of fundamental interactions. The second part aims at studying top production in a wide category of New Physics models. In both parts, the goal is to provide precise reliable predictions and tools that are demanded by the experimental analysis.The SM part is focused on the calculation of the NNLO QCD corrections to the production of top-antitop pairs at hadron colliders. The ingredients needed to achieve the objective are essentially two: matrix elements and proper subtraction terms. First, we need to calculate the two-loop matrix elements, both for the production and the decay processes. This is a very challenging calculation, that we chose to afford by using analytic techniques. The contribution to the cross section is expressed in terms of many (thousands of) dimensionally regularized scalar integrals. These integrals are not all independent, but they are related by a class of identities called ``integration-by-parts identities". Using these relations, which are worked out following a well defined algorithm called "Laporta algorithm", we were able to reduce all the scalar integrals to a (smaller) set of independent integrals, called the "master integrals" of the problem under consideration. This first step was carried out for both the production and the decay processes. Then, the calculation of the master integrals was done using the differential equations method. The masters fulfill a system of first-order linear differential equations in the kinematic invariants. The solution of the system is achieved expanding in Laurent series of (d-4) (d is the dimension of the space-time) the masters and solving order-by-order in (d-4) the system. The results are expressed through a well defined class of functions. The analytic calculation has of course advantages and drawbacks. It is characterized by a marked flexibility and a fast numerical evaluation of the final result, suitable for a future implementation in a Monte Carlo computer program. However, the solution of the system of differential equations is not always an easy task and to cast the solution in a suitable form for numerical calculations can be of great difficulty. In the last years we succeeded to calculate a large part of the matrix elements in both relevant partonic channels, quark-antiquark and gluon-gluon. In particular, the last completed set regarded the corrections which arise from diagrams involving a closed light-quark loop in the gluon-gluon channel. Due to technical difficulties that delayed the completion of the calculation, a certain number of matrix elements are still under investigation. Once the matrix elements are calculated, in order to be able to predict observables in a way that can be used for the direct comparison with experimental data, we need to define and calculate infrared subtraction terms, at the same perturbative order. The counter-terms regularize locally the behaviour of the observable in the singular regions and allow for a reliable and stable numerical evaluation. We defined the counter-terms in the framework of the Qt subtraction scheme, proposed by the scientist in charge and M. Grazzini, some years ago. This was possible thanks to the new result by S. Catani and collaborators that extends the Qt formalism to processes with heavy-quark final states. The NLO counter-terms are known, and their calculation at the NNLO is set up and at an advanced stage.The part devoted to New Physics is focused on the calculation and implementation in Monte Carlo event generators of the NLO QCD corrections to the production of top-antitop pairs and single top in presence of new resonant states, such as Z' and W' bosons, that can originate from Grand Unification scenarios for the Physics beyond the SM. We calculated the corresponding matrix elements and we implemented the corrections in the NLO Monte Carlo program POWHEG, in a model-independent way. We outlined a strategy to discriminate between different models of New Physics and we applied it to a class of models called G221. The release of the program, that will be public soon, goes exactly in the direction of the objectives of the project: to provide tools for a precise and reliable analysis of new physics effects in the run II of the LHC.To conclude, TOPPhysics was conceived to provide precise theoretical predictions in the sector of heavy quarks, as demanded by the experimental analysis at the LHC. Many objectives were achieved during the two years of fellowship. In particular: many two-loop matrix elements were calculated, both for the production and the decay of top quarks, addressing and solving technical difficulties that, nevertheless, delayed the completion of the calculation; the infrared counter-terms were identified in the context of the Qt subtraction formalism and their calculation at the NNLO is at an advanced stage; finally, a model independent NLO analysis of new physics effects in top-antitop and single top production was performed and a Monte Carlo program that can be used by the experimental collaborations in the search of new physics at the LHC was released. The parts of the project that have to be completed will be the object of a devoted study in the next months.