Scattering amplitude calculation and intersection theory

In this thesis, we present new developments for the analytic calculation of multi-loop level amplitudes. Similarly, we study the underlying mathematical structure of such key objects for modern high energy physics research. In this thesis we elaborate on the new and powerful tools provided by intersection theory. This mathematical tool sheds new light on the algebraic structure of Feynman integrals, paving a new way to performing multi-loop precision computation. Specifically, multi-loop scattering amplitudes for state of the art calculations are built upon a large number of scalar multi-loop integrals, whose reduction in terms of a smaller set of Master Integrals (MIs) can be a bottleneck in amplitudes computation. Such reduction is possible thanks to the Integration By Parts Identities (IBPs), which consist in linear relations among Feynman integrals generated by the vanishing of a total derivative under the integral sign. The reduction is usually achieved thanks to the Laporta algorithm by solving a huge system of such relations which, depending on the number of scales involved, can require very demanding algebraic manipulations. In a different approach, intersection theory allows us to embed Feynman integrals in a vector space, defining a scalar product between them: the intersection number. In this way, obtaining the coefficients that multiplies a MI in the reduction of a Feynman integral is equivalent to finding the decomposition of a vector in terms of its basis vector in a vector space. It consists of using simple linear algebra methods to project the multi-loop integrals directly on the MIs basis, bypassing the system-solving procedure otherwise required in the standard approach to multi-loop calculations. In the first part of the thesis, we describe the main features of the multi-loop calculations. We briefly overview the adaptive integrand decomposition (AID), a variant of the standard integrand reduction algorithm. AID exploits the decomposition of the space-time dimension in parallel and orthogonal subspaces. We then proceed to introduce IBPs and the Differential Equation method for the computation of master integrals, finally outlining the key steps that allowed the computation of the two-loop four-fermion scattering amplitude in qed, with one massive fermion. We then elaborate on the properties of intersection theory and how to apply it in relation with Feynman Integrals. After showing the successful application of it to a wide variety of Feynman integrals admitting a univariate integral representation, we present the implementation of a recursive algorithm for multivariate intersection number to extend this method to generic Feynman integrals. We also present alternative algorithm for the application of multivariate intersection number to Feynman integrals decomposition, showing the flexibility of this powerful tool, combining the advantages of the decomposition by intersection numbers with the subtraction algorithm traditionally used in methods of integrand decomposition. Aside from the reduction to MIs, we apply intersection theory to the derivation of contiguity relations and of differential equations for MIs, as first steps towards potential applications to generic multi-loop integrals.In this thesis, we present new developments for the analytic calculation of multi-loop level amplitudes. Similarly, we study the underlying mathematical structure of such key objects for modern high energy physics research. In this thesis we elaborate on the new and powerful tools provided by intersection theory. This mathematical tool sheds new light on the algebraic structure of Feynman integrals, paving a new way to performing multi-loop precision computation. Specifically, multi-loop scattering amplitudes for state of the art calculations are built upon a large number of scalar multi-loop integrals, whose reduction in terms of a smaller set of Master Integrals (MIs) can be a bottleneck in amplitudes computation. Such reduction is possible thanks to the Integration By Parts Identities (IBPs), which consist in linear relations among Feynman integrals generated by the vanishing of a total derivative under the integral sign. The reduction is usually achieved thanks to the Laporta algorithm by solving a huge system of such relations which, depending on the number of scales involved, can require very demanding algebraic manipulations. In a different approach, intersection theory allows us to embed Feynman integrals in a vector space, defining a scalar product between them: the intersection number. In this way, obtaining the coefficients that multiplies a MI in the reduction of a Feynman integral is equivalent to finding the decomposition of a vector in terms of its basis vector in a vector space. It consists of using simple linear algebra methods to project the multi-loop integrals directly on the MIs basis, bypassing the system-solving procedure otherwise required in the standard approach to multi-loop calculations. In the first part of the thesis, we describe the main features of the multi-loop calculations. We briefly overview the adaptive integrand decomposition (AID), a variant of the standard integrand reduction algorithm. AID exploits the decomposition of the space-time dimension in parallel and orthogonal subspaces. We then proceed to introduce IBPs and the Differential Equation method for the computation of master integrals, finally outlining the key steps that allowed the computation of the two-loop four-fermion scattering amplitude in qed, with one massive fermion. We then elaborate on the properties of intersection theory and how to apply it in relation with Feynman Integrals. After showing the successful application of it to a wide variety of Feynman integrals admitting a univariate integral representation, we present the implementation of a recursive algorithm for multivariate intersection number to extend this method to generic Feynman integrals. We also present alternative algorithm for the application of multivariate intersection number to Feynman integrals decomposition, showing the flexibility of this powerful tool, combining the advantages of the decomposition by intersection numbers with the subtraction algorithm traditionally used in methods of integrand decomposition. Aside from the reduction to MIs, we apply intersection theory to the derivation of contiguity relations and of differential equations for MIs, as first steps towards potential applications to generic multi-loop integrals