Mattiazzi, Luca (2018) Multiparticle Scattering Amplitudes at Two-Loop. [Magistrali biennali]
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In this thesis we present modern techniques needed for the evaluation of one and multi loop amplitudes, and apply some of them in a complete chain that allows the evaluation of a Feynman amplitude. In particular the automated evaluation of a 5 point 2 loop Feynman diagram contributing to the process e^+ e^-→μ^+ μ^- γ here is presented for the first time. Furthermore we investigate the properties of the integration domain of Feynman integrals in Baikov representation, presenting a new and general formula for their calculation, highlighting an interesting iterative structure beneath the Feynman Integrals. Given this key information in such representation, we found a new parameterization for the Feynman integrals, which needs further studies in order to be better understood. In this thesis, we firstly review the Unitarity based methods, which stems from the Unitarity of the S matrix. Such methods uses cuts (i.e. put internal lines on shell) in order to project the amplitude on to its component. For example, in the Cutkosky rule the amplitude is projected in to its imaginary part by means of cuts. Another techniques that relies on cut is the Feynman tree theorem, which by means of complex analysis connect loop level amplitude to tree level one. The most successful approach in such field was the Generalized Unitarity one. Applying the same idea as in the Cutkosky rule, it lead to major automation of one loop calculation. Afterwards we present the issues and the tools that one faces when tackling the calculation of a multiloop Feynman integral, arriving to the analyze the generalized cut and the IBP reduction on the Baikov representation. Lastly, present the Adaptive integrand decomposition and an algorithm for the complete automated evaluation of an amplitude. A complete software chain needed to complete such task is then presented, highlighting our contribution to such software.
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