Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to $n$-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.Comment: 115 pages, 29 figures; references added; additional examples added; matches published versio

Construcción de adenovectores y magnetoadenovectores para aplicaciones de terapia génica en el miocardio senil

Objetivos: 1) Construir un vector bidireccional que permita la coexpresión regulada del gen de IGF-I y GFP, ambos bajo el control de un promotor inducible por doxiciclina. Dicho vector constituirá la unidad funcional de expresión del adenovirus a construirse en pasos futuros. 2) Construir magnetoadenovectores a partir del complejamiento de MNP con nuestro adenovirus control que expresa GFP (RAd-GFP) y realizar experimentos de magnetofeccion in-vitro.Facultad de Ciencias Médica

Equivalenza fra modello circuitale ed adiabatico di computazione quantistica

La tesi ha lo scopo di approfondire il tema della computazione quantistica. Il candidato dovrà studiare un'introduzione al modello circuitale di computaizone quantistica e approfondire l'approccio di "computer adiabatico". Grazie al Teorema Adiabatico è possibile implementare una computazione partendo dal ground state di un'Hamiltoniana semplice e facendo variare l'Hamiltoniana in manera adiabatica verso una configurazione di cui non si consce il ground state. Lo studente dovrà studiare come questo approccio permette di realizzare un computer quantistico

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

Multiparticle Scattering Amplitudes at Two-Loop

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

N3LON^3LO calculations for 2→22 \to 2 processes using Simplified Differential Equations

We present the computation of the massless three-loop ladder-box family with one external off-shell leg using the Simplified Differential Equations (SDE) approach. We also discuss the methods we used for finding a canonical differential equation for the two tennis-court families with one off-shell leg, and the application of the SDE approach on these two families