Academic literature on the topic 'Multi-loop Feynman integrals'

The Loop-Tree Duality (LTD) theorem is an innovative technique to deal with multi-loop scattering amplitudes, leading to integrand-level representations over a Euclidean space. In this article, we review the last developments concerning this framework, focusing on the manifestly causal representation of multi-loop Feynman integrals and scattering amplitudes, and the definition of dual local counter-terms to cancel infrared singularities.

We present Star–Triangle Relations (STRs), a Mathematica® package designed to solve Feynman diagrams by means of the method of uniqueness in any Euclidean space-time dimension. The method of uniqueness is a powerful technique to solve multi-loop Feynman integrals in theories with conformal symmetry imposing some relations between the powers of propagators and the space-time dimension. In our algorithm, we include both identities for scalar and Yukawa type integrals. The package provides a graphical environment in which it is possible to draw the desired diagram with the mouse input and a set of tools to modify and compute it. Throughout the use of a graphic interface, the package should be easily accessible to users with little or no previous experience on diagrams computation. This manual includes some pedagogical examples of computation of Feynman graphs as the scalar two-loop kite master integral and a fermionic diagram appearing in the computation of the spectrum of the [Formula: see text]-deformed [Formula: see text] SYM in the double scaling limit.

Die Auswertung von Mehrschleifen-Feynman-Integralen ist eine der größten Herausforderungen bei der Berechnung präziser theoretischer Vorhersagen für die am LHC gemessenen Wirkungsquerschnitte. In den vergangenen Jahren hat sich die Nutzung von Differentialgleichungen bei der Berechnung von Feynman-Integralen als sehr erfolgreich erwiesen. Es wurde dabei beobachtet, dass die von den Feynman-Integralen erfüllte Differentialgleichung oftmals in eine sogenannte kanonische Form transformiert werden kann, welche die Integration der Differentialgleichung mittels iterierter Integrale wesentlich vereinfacht. Das zentrale Ergebnis der vorliegenden Arbeit ist ein Algorithmus zur Berechnung rationaler Transformationen von Differentialgleichungen von Feynman-Integralen in eine kanonische Form. Neben der Existenz einer solchen rationalen Transformation stellt der Algorithmus keinerlei weitere Bedingungen an die Differentialgleichung. Insbesondere ist der Algorithmus auf Mehrskalenprobleme anwendbar und erlaubt eine rationale Abhängigkeit der Differentialgleichung vom dimensionalen Regulator. Bei der Anwendung des Algorithmus wird zunächst das Transformationsgesetz im dimensionalen Regulator entwickelt, um Differentialgleichungen für die Koeffizienten in der Entwicklung der Transformation herzuleiten. Diese Differentialgleichungen werden dann mit einem rationalen Ansatz für die gesuchte Transformation gelöst. Es wird zudem eine Implementation des Algorithmus in dem Mathematica Paket CANONICA vorgestellt, welches das erste veröffentlichte Programm dieser Art ist, das auf Mehrskalenprobleme anwendbar ist. CANONICAs Potential für moderne Mehrschleifenrechnungen wird anhand mehrerer nicht trivialer Mehrschleifen-Integraltopologien demonstriert. Die gezeigten Topologien hängen von bis zu drei Variablen ab und umfassen auch vormals ungelöste Topologien, die zu Korrekturen höherer Ordnung zum Wirkungsquerschnitt der Produktion einzelner Top-Quarks am LHC beitragen.<br>The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transformations of differential equations of Feynman integrals into a canonical form. Apart from requiring the existence of such a rational transformation, the algorithm needs no further assumptions about the differential equation. In particular, it is applicable to problems depending on multiple kinematic variables and also allows for a rational dependence on the dimensional regulator. First, the transformation law is expanded in the dimensional regulator to derive differential equations for the coefficients of the transformation. Using an ansatz in terms of rational functions, these differential equations are then solved to determine the transformation. This thesis also presents an implementation of the algorithm in the Mathematica package CANONICA, which is the first publicly available program to compute transformations to a canonical form for differential equations depending on multiple variables. The main functionality and its usage are illustrated with some simple examples. Furthermore, the package is applied to state-of-the-art integral topologies appearing in recent multi-loop calculations. These topologies depend on up to three variables and include previously unknown topologies contributing to higher-order corrections to the cross section of single top-quark production at the LHC.

In this thesis we discuss, within the framework of the Standard Model (SM) of particle physics, advanced methods for the computation of scattering amplitudes at higher-order in perturbation theory. We offer a new insight into the role played by the unitarity of scattering amplitudes in the theoretical understanding and in the computational simplification of multi-loop calculations, at both the algebraic and the analytical level. On the algebraic side, generalized unitarity can be used, within the integrand reduction method, to express the integrand associated to a multi-loop amplitude as a sum of fundamental, irreducible contributions, yielding to a decomposition of the amplitude as a linear combination of master integrals. In this framework, we propose an adaptive formulation of the integrand decomposition algorithm, which systematically adjusts to the kinematics of the individual integrands the dimensionality of the momentum space, where unitarity cuts are performed. This new formulation makes the integrand decomposition method, which in the past played a key role in streamlining one-loop computations, an efficient tool also at multi-loop level. We provide evidence of the generality of the proposed method by determining a universal parametrization of the integrand basis for two-loop amplitudes in arbitrary kinematics and we illustrate its technical feasibility in the first automated implementation of the analytic integrand decomposition at one- and two-loop level. On the analytic side, we discuss the role of maximal-unitarity for the solution of differential equations for dimensionally regulated Feynman integrals. The determination of the analytic expression of the master integrals as a Laurent expansion in the dimensional regulating parameter requires the knowledge of the solutions of the homogeneous part of their differential equations at d=4. In all cases where Feynman integrals fulfil genuine first-order differential equations with a linear dependence on d, the corresponding homogeneous solutions can be determined through the Magnus exponential expansion. In this work we apply the latter to two-loop corrections to several SM processes such as the Higgs decay to weak vector bosons, H → WW, triple gauge couplings ZWW and γ∗WW and to the elastic scattering μe → μe in quantum electrodynamics. In some cases, the inadequacy of the Magnus method hints at the presence of master integrals that obey higher-order differential equations, for which no general theory exists. In this thesis we show that maximal-cuts of Feynman integrals solve, by construction, such homogeneous equations regardless of their order and complexity. Hence, whenever a Feynman integral obeys an irreducible higher-order differential equation, the computation of its maximal-cut along independent contours provides a closed integral representation of the full set of independent homogeneous solutions. We apply this strategy to the two-loop elliptic integrals that appear in heavy-quark mediated corrections to gg → gg and gg → gH as well as to the three-loop massive banana graph, which constitute the first example of Feynman integral that obeys a third-order differential equation. In the light of the results presented in this thesis, generalized unitarity emerges as a powerful tool not only for handling the algebraic complexity of perturbative calculations but also for investigating the nature of new classes of mathematical functions encountered in particle physics.<br>In questa tesi si discutono metodi avanzi per il calcolo dei contributi perturbativi alle ampiezze di scattering nel contesto del Modello Standard delle interazioni fondamentali. In particolare, si avanzano nuove interpretazioni del ruolo dell’unitarietà delle ampiezze di scattering, sia nella comprensione teorica che nella semplificazione computazionale dei calcoli a multi-loop. Dal punto di vista prettamente algebrico della decomposizione a livello integrando, l’unitarietà generalizzata consente di esprimere l’integrando associato ad una qualunque ampiezza a multi-loop in termini di una combinazione lineare di un numero minimo di integrandi irriducibili, a loro volta associati ad una base di integrali indipendenti, generalmente chiamati master integral. In questo contesto, viene avanzata una formulazione adattiva dell’algoritmo di decomposizione integranda, che parametrizza in maniera sistematica lo spazio dei momenti di ciascun integrando a seconda della relativa configurazione cinematica. Questa riformulazione rende la decomposizione a livello integrando, che in passato ha svolto un ruolo fondamentale nella semplificazione ad automazione dei calcoli a un loop, uno strumento versatile ed efficiente anche a multi-loop. A riprova della generalità del metodo proposto, in questa tesi vengono determinate le basi integrande universali per ampiezze a due loop con cinematica arbitraria e si illustra la sua fattibilità tecnica attraverso la prima implementazione automatica della decomposizione integranda analitica a uno e due loop. Sul piano analitico, invece, si discute il ruolo dell’unitarietà generalizzata nella soluzione delle equazioni differenziali per integrali di Feynman in regolarizzazione dimensionale. La determinazione dell’espressione analitica dei master integral in termini di un’espansione di Laurent nel parametro regolarizzatore richiede la conoscenza delle soluzioni omogenee del sistema di equazioni differenziali a d=4. Nei casi in cui gli integrali di Feynman soddisfino equazioni differenziali del primo ordine con una dipendenza lineare in d, tali soluzioni omogenee posso essere determinate attraverso la soluzione esponenziale di Magnus. In questa tesi, quest’ultima viene applicata al calcolo dei master integrals a due loop per diversi processi di scattering nel Modello Standard, quali il decadimento del bosone di Higgs in the bosoni elettro-deboli W, gli accoppiamenti di triplo gauge ZWW e γ∗WW, nonché lo scattering elastico tra elettrone e muone in elettrodinamica quantistica. In un certo numero di casi, l’inadeguatezza del metodo di Magnus è indice della presenza di master integral che soddisfano equazioni differenziali di ordine più elevato, per le quali non esiste una sistematica trattazione matematica. In questa tesi mostriamo che i maximal-cut degli integrali di Feynman soddisfano, per costruzione, la parte omogenea delle relative equazioni differenziali, indipendentemente dal loro ordine e complessità. Di conseguenza, ogniqualvolta un integrale di Feynman soddisfa un’equazione di ordine elevato, il calcolo del maximal-cut su domini di integrazione indipendenti fornisce una rappresentazione integrale chiusa di un insieme completo di soluzioni omogenee. Questa strategia viene applicata agli integrali ellittici a due-loop che compaiono nelle correzioni ai processi gg → gg e gg → gH mediate da quark pesanti e al diagramma a banana a tre loop, che costituisce il primo esempio di integrali di Feynman associato ad un’equazione differenziale del terzo ordine. I risultati presentati in questa tesi illustrano l’efficacia dei metodi di unitarietà, sia nella gestione della complessità algebrica dei calcoli a multi-loop, sia nell’indagine matematica delle nuove classi di funzioni incontrate nella fisica delle interazioni fondamentali.

In this thesis, we present new developments for the analytic calculation of tree- and multi-loop level amplitudes. Similarly, we study and extend their analytic properties. We propose a Four-dimensional formulation (FDF) equivalent to the four-dimensional helicity scheme (FDH). In our formulation, particles propagating inside the loop are represented by four dimensional massive internal states regulating the divergences. We provide explicit four-dimensional representations of the polarisation and helicity states of the particles propagating in the loop. Within FDF, we use integrand reduction and four dimensional unitarity to perform analytic computations of one-loop scattering amplitudes. The calculation of tree level scattering amplitude, in this framework, allows for a simultaneous computation of cut-constructible and rational parts of one-loop scattering amplitudes. We present a set of non-trivial examples, showing that FDF scheme is suitable for computing important $2\to2,3,4$ partonic amplitudes at one-loop level. We start by considering two gluons production by quark anti-quark annihilation. Then, the (up to four) gluon production, $gg\to ng$ with $n=2,3,4$. And finally, the Higgs and (up to three) gluons production via gluon fusion, $gg\to ng\,H$ with $n=1,2,3$, in the heavy top mass limit. We also investigate, by following a diagrammatic approach, the role of colour-kinematics (C/K) duality of off-shell diagrams in gauge theories coupled to matter. We study the behaviour of C/K-duality for theories in four- and in $d$-dimensions. The latter follows the prescriptions given by FDF. We show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a C/K-violating term due to the contributions of sub-graphs only. We discuss the role of the off-shell decomposition in the direct construction of higher-multiplicity numerators satisfying C/K-duality. We present the QCD process $gg\to q\bar{q}g$. An analogous study, within FDF, is carried out for $d$-dimensionally regulated amplitudes. The computation of dual numerators generates, as byproduct, relations between tree-level amplitudes with different orderings. These relations turn to be the Bern-Carrasco-Johansson (BCJ) identities for four- and $d$-dimensionally regulated amplitudes. We combine BCJ identities and integrand reduction methods to establish relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. We also elaborate on the radiative behaviour of tree-level scattering amplitudes in the soft regime. We show that the subleading soft term in single-gluon emission of quark-gluon amplitudes in QCD is controlled by differential operators, whose universal form can be derived from both Britto-Cachazo-Feng-Witten recursive relations and gauge invariance, as it was shown to hold for graviton and gluon scattering. In the last part of the thesis, we describe the main features of the multi-loop calculations. We briefly describe 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 focus, in particular, on the calculation of $2\to2,3$ partonic amplitudes at two loop-level<br>In questa tesi discutiamo le proprietà di analiticità delle ampiezze di scattering e presentiamo nuovi metodi per il loro calcolo analitico, sia a tree-level e agli ordini perturbativi successivi. Proponiamo un nuovo schema di regolarizzazione dimensionale, la Four-dimensional formulation (FDF), che mostriamo equivalente al Four-dimensional helicity scheme (FDH). Nella nostra formulazione, consideriamo le particelle che si propagano all'interno dei loop in quattro dimensioni, fornendo una rappresentazione esplicitamente quadridimensionale dei loro stati di polarizzazione ed elicità. La massa di tali particelle virtuali agisce da regolatore delle divergenze. Lavorando in FDF, utilizziamo le tecniche di unitarietà e il metodo dell'integrand reduction per calcolare analiticamente ampiezze di scattering a un loop, mostrando che la conoscenza delle ampiezze a tree-level consente, in questo formalismo, di ottenere sia la cosiddetta parte cut-constructibile dell'ampiezza di loop sia i suoi termini razionali. Presentiamo una serie di esempi non banali e illustriamo come FDF consenta di calcolare ampiezze partoniche per processi $2\to 2,3,4$ di notevole rilevanza fenomenologica. In particolare, iniziamo considerando la produzione di due gluoni a partire da una coppia di quark-antiquark per poi analizzare ampiezze puramente gluoniche del tipo $gg\to ng$, con $n=2,3,4$. Infine, lavorando nel limite di massa infinita del quark top, presentiamo i risultati per la produzione via gluon-fusion di un bosone di Higgs in associazione con jet gluonici, $gg\to ngH$, $n=1,2,3$. Seguendo un approccio diagrammatico, investighiamo il ruolo della colour-kinematics duality (C/K) in teorie di gauge accoppiate alla materia, sia in quattro che in $d$ dimensioni, adottando, nel secondo caso, le prescrizioni di FDF. Mostriamo che le identità di Jacobi tra i numeratori cinematici dei diagrammi di Feynman off-shell (per i quali utilizziamo il gauge assiale) producono violazioni della C/K dualità riconducibili all'esclusivo contributo di sottodiagrammi. Discutiamo il ruolo di tale decomposizione off-shell nella costruzione diretta di numeratori esplicitamente duali. In particolare, analizziamo il processo $gg\to q\bar{q}g$ in quattro dimensioni per poi estendere tale studio, mediante l'utilizzo di FDF, al caso $d$-dimensionale. Nel seguito, studiamo il comportamento delle ampiezze a tree-level di QCD nel limite di emissione di radiazione soffice. Nel caso dell'emissione di un singolo gluone, mostriamo che il termine sottodominante nell'approssimazione soffice dell'ampiezza è descritto da operatori differenziali la cui espressione universale può essere derivata sia delle relazioni di ricorrenza di Britto-Cachazo-Feng-Witten sia dalle proprietà di invarianza di gauge dell'ampiezza. Tali proprietà si rivelano valide, oltre che per processi gluonici, per lo scattering tra gravitoni. Nell'ultima parte di questa tesi, discutiamo le caratteristiche principali del calcolo di ampiezze di scattering oltre un loop. Descriviamo brevemente il metodo dell'adaptive integrand decomposition (AID), una formulazione alternativa della tecnica di integrand decomposition tradizionale, che sfrutta la scomposizione dello spazio-tempo nei sottospazi parallelo ed ortogonale alla cinematica esterna. In particolare, ci concentriamo su calcolo di ampiezze partoniche $2\to2,3$ a due loop.

It is very challenging to solve multi-scale, multi-loop Feynman diagrams analytically. The presence of different kinematic scales makes the computation of Feynman diagrams very difficult, sometimes impossible to get the analytic results. One way to tackle this problem is to consider systematic approximations based on the hierarchies of the scales. The basic idea is to simplify the integral before the integration. The Method of Regions (MoR) is one of the powerful methods for handling the evaluation of multi-scale, multi-loop Feynman diagrams asymptotically. The whole loop momentum domain is divided into several regions and the integrand of the given Feynman diagram is expanded, in each of the regions, in a Taylor series based on a small expansion parameter, which is the ratio of low scale and the high scale. After the expansion, the sum of the contributions which are obtained from the integration of the expanded terms over the whole range of momentum, gives the result for the original Feynman diagram in an expanded form. It is a non-trivial task to identify the correct set of regions required for the asymptotic analysis of the Feynman integrals. In one of the projects reported in this thesis, we have designed an algorithm for unveiling the regions associated with the multi-scale multi-loop Feynman integrals in given limits. We show that the regions can be unveiled from the neighborhood of the singular surfaces of the Feynman diagrams. The associated singularities are known as the Landau singularities. The Feynman diagrams are characterized by two homogeneous polynomials, called the Symanzik polynomials. The location of the singularities of the Feynman diagrams are determined from the Landau equations, which are obtained by equating the Symanzik polynomial of second kind and all of its partial derivatives with respect to the Feynman parameters to zero. In our framework, we consider the set of the Landau equations for a given multi-loop, multi-scale Feynman diagram and express them via the Gröbner basis elements. By equating the Gröbner basis elements of the Landau equations to zero, we derive a set of linear transformations which map the singular surfaces to the origin, co-ordinate axes or co-ordinate planes in the parametric space. The so obtained linear transformations are then applied to the sum of the Symanzik polynomials of first and second kind. The obtained set of polynomials are then analyzed within the framework of Power Geometry. The analysis consists of several steps. The first one is to find the support of the obtained polynomials which basically are the vector exponents for each of the terms of the considered polynomials. The second step is to find the convex hulls of the obtained supports, which are called the Newton polytopes. We then find the normal vector for each of the facets of the Newton polytopes based on certain rules. The set of unique normal vectors then give the set of required regions for the given Feynman diagram. We call our algorithm ASPIRE. Within our approach, we show that all the regions including the potential and Glauber can be unveiled. The algorithm ASPIRE thus provides a useful method for identifying the regions required for the asymptotic expansion of Feynman diagrams based on rigorous mathematical tools. In another project of this thesis, we consider the notion of top facets of the ASPIRE program. The top facet scalings with equal components under the consideration of the Landau equations and the analysis of Power Geometry gives rise to a criterion which allows us to correlate the top facet scalings with equal components to the maximally cut Feynman diagrams. We use the method of residue calculus for one loop cases for finding out the maximally cut Feynman diagrams. The integrals for the top facets with equal components have been evaluated considering the parametric representation. The top facets with equal components have the property that the external momenta can be set to zero (neglected with respect the internal masses) for a given diagram and thus corresponds to the case of the large expansion parameter. We call this property to be the top facet condition. When the top facet condition is applied to the case of maximal cut of the given Feynman diagram, we show that the top facet with equal components is exactly proportional to the maximal cut. We also provide the results for the other top facets with non-equal components. In this work, we have connected two independent approaches via the consideration of Landau equations and the Power Geometry. In another work, we analyze a two loop planar three point Feynman diagram asymptotically using the method of regions in the high energy limit. We consider the Schwinger parametric representation for Feynman diagrams with the consideration of the analytic regulators. The standard method of dimensional regularization is not sufficient to regularize the contributions of some of the regions. For the most generic consideration, we can take the help of extra analytic regulators for the regularization of the contributions of those regions. For the considered diagram, we obtain six regions. Out of six regions, we have solved the contribution from three regions. The contribution of two regions are expressed via Mellin-Barnes integrations and the contribution from the hard region remain to be evaluated.

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.