Ronca, Jonathan (2017) Unitarity-based methods and Integration-by-parts identities for Feynman Integrals. [Magistrali biennali]
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In this thesis, we present a novel idea to address the evaluation of multi-loop Feynman integrals, inspired by unitarity of S-matrix. In the first part of this work, we present the Feynman integral formalism. Within the dimensional regularization scheme, Feynman integrals are known to obey integration-by-parts identities (IBPs), yielding the identification of an independent integral basis, dubbed master integrals. We describe the currently adopted strategy for the amplitudes decomposition, known as reduction algorithm, which is based on IBPs for integrands with denominators that depend quadratically on the loop momenta (quadratic denominators). In the second part of this thesis, we present a novel strategy to decompose Feynman integrals based on the use of partial fractions decomposition of its integrand. Within this approach any multi-loop integrand is first decomposed into a combination of integrands that contain just a minimal, irreducible number of quadratic denominators and several other denominators that carry a linear dependence of the loop momenta (linear denominators). After partial fractioning, IBPs are applied to integrals with linear denominators, in order to identify an alternative set of master integrals. Finally, the obtained relations are combined back to restore the IBPs for the original integrals containing quadratic denominators only. We examine the underlying algebraic structure of dimensionally regulated integrals with linear denominators, classifying all spurious, vanishing classes of integrals that may emerge after partial fractioning. In the last part of the thesis, we present the implementation of the novel algorithm within a Mathematica code called Parsival (Partial fractions-baSed method for feynman Integral eVALuation), which has been interfaced to the public package Reduze, for the IBPs decomposition. Preliminary results for the application of Parsival+Reduze framework to 1-loop integrals for 2 -- n (n = 1; 2; 3) scattering processes, and to 2-loop integrals, corresponding to planar and non-planar diagrams for 2 -- n (n = 1; 2) scattering amplitudes are given in the final chapter. The proposed algorithm is suitable for parallelization, and the preliminary results show that its effectiveness can be improved by exploiting the symmetries of the integrand under redefinition of loop-momenta, not accounted for in the present version of the code. The proposed strategy is very general and it can be applied to any scattering reduction.
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