Quantum Field Theory (QFT) provides the essential background for formulating the standard model of elementary particles and, moreover, practically all other theories attempting to explore the physical laws of nature at the sub-atomic level. One of the main observables in QFT are the scattering amplitudes, physical quantities which encode the information of the scattering process of particles. Accordingly, having authentic, well-defined and feasible prescriptions for the calculations of amplitudes is of huge importance to theoretical physicists. Actual calculations show that the text-book prescription, the Feynman method, besides in general being very cumbersome also hides some of the beautiful mathematical features of amplitudes. The last decade has seen tremendous efforts and achievements to improve such calculations, particularly in supersymmetric gauge theories, which have also led to better understanding of QFT itself. Among the known physically and mathematically interesting quantum field theories is perturbative gravity and its supersymmetric version, N=8 supergravity- much less understood than gauge theory. Following the developments in gauge theory, this dissertation mainly aims at exploring scattering amplitudes in gravity as a quantum field theory, using the modern approaches to QFT. The goal is not only to improve our understanding of gravity amplitudes by applying part of the known modern methods of calculations to it but also to introduce and develop new ones.

Amplitudes in theories with a massless gauge boson suffer from so-called infrared divergences where off-shell states become asymptotically close to the mass-shell in loop or phase-space momentum integrals. These singularities have been shown to cancel intricately order-by-order in the perturbative expansion. However, in order to obtain meaningful and precise predictions for physical observables, we must understand and compute such divergences to high orders. This can be accomplished by calculating webs: weighted sets of Feynman diagrams which, when exponentiated give the complete infrared singular component of the amplitude, known as the soft function. This quantity is formally equivalent to a vacuum expectation value of a product of Wilson lines. In this thesis we shall study webs correlating multiple Wilson lines, which differs from the two line case due to the possibility of non-trivial colour flows. This renders the soft function matrix valued in the space of colour flows, thus making its calculation and renormalisation non-trivial. At present, the infrared singularities of non-abelian, multiparton scattering amplitudes are known only to two loops in general kinematics, and to three loops in a simplifying kinematic limit. This thesis will thus form part of a program of work aimed at calculating and understanding the three-loop singularities in general kinematics and in doing so we aim to gain all-order insights into the pertubative structure of non-abelian gauge theories. We first specialise to a subset of webs which we have called Multiple Gluon Exchange Webs (MGEWs), which contain only those diagrams with direct exchanges of soft gauge bosons directly between Wilson lines with no intervening three- or four- boson vertices. Studing their properties allows us to construct a basis of functions which describes all examples of such webs, and we conjecture will continue to do so at any order. Furthermore, we find that the basis functions can be described by a simple, one-dimensional integral over only logarithms. We go on to compute several examples providing evidence for the validity of our basis and demonstrate the utility of the framework we have built by computing a four-loop web and providing some all-order results for particular classes of MGEW. We then consider a step beyond MGEWs, that is, webs which contain a single three-gluon vertex sub-diagram. In particular we study the simplest web in this class correlating four lines at three loops and attempt to calculate it through the numerical fitting of a physically motivated ansatz. We show that this web cannot carry kinematic dependence through conformal invariant cross ratios, which arise when connected subdiagrams correlate at least four lines. Hence, it is subject to the same constraints as MGEWs with regards to their symbol alphabet, from the physical considerations in their lightlike limit and spacelike/timelike analytic continuation. Like all other known webs satisfying such constraints, we therefore argue that it can be written in terms of sums of products of MGEW basis functions. Symmetries inherent to our parameterisation of the cusp angles, Bose symmetry and transcendental weight further constrain this ansatz, resulting in forty parameters for which we present preliminary results of a numerical fit.

The tree amplitudes of string theory low-energy effective actions admit a diagrammatic expansion in terms of higher-than-cubic contact terms. These tree amplitudes can be used to build loop amplitudes using unitary cuts or the forward limit. In this thesis we study the possibility of constructing four-point cubic representations for the resulting one-loop contact terms that obey the Bern-Carrasco-Johansson (BCJ) color-kinematics relation. From the string theory effective action we study the contact terms carrying ζ2, ζ3, ζ4, and ζ5. For the even ζ2 and ζ4 cases we find that the cubic representations are incompatible with the BCJ relations, as expected from their disappearance in the closed-string effective action. We find a unique, local set of numerators at ζ3 that obey the BCJ relations. For ζ5 we find two choices of representations: one obeys BCJ but requires non-trivial contributions for the tadpole; the other contains no tadpoles but breaks one of the BCJ relations.

When calculating scattering processes in theories involving massless gauge bosons, such as gluons in Quantum Chromodynamics (QCD), one encounters infrared (IR), or soft, divergences. To obtain precise predictions, it is important to have exact expressions for these IR divergences, which are present in any on-shell scattering amplitude. Due to their long wavelength, soft gluons factorise with respect to short-distance, or hard, interactions and can be captured by correlators of semi-infinite Wilson lines. The latter obey a renormalisation group equation, which gives rise to exponentiation. The exponent can be represented diagrammatically in terms of weighted sums of Feynman diagrams, called webs. A web with L external legs, each with ni gluon attachments, is denoted (n1; n2; : : : ; nL). In this way all soft gluon interactions can be described by a soft anomalous dimension. It is currently known at three loops with lightlike kinematics, and at two loops with general kinematics. Our work is a step towards a three-loop result in general kinematics. In recent years, much progress has been made in understanding the general physical properties of scattering amplitudes and in exploiting these properties to calculate specific amplitudes. At the same time, we have discovered a lot of structure underpinning the space of multiple polylogarithms, the functions in terms of which most known amplitudes can be written. General properties include analyticity, implying that scattering amplitudes are analytic functions except on certain branch cuts, and unitarity, or conservation of probability. These two properties are both exploited by unitarity cuts. Unitarity cuts provide a diagrammatic way of calculating the discontinuities of a Feynman diagram across its branch cuts, which is often simpler than calculating the diagram itself. From this discontinuity, the original function can be reconstructed by performing a dispersive integral. In this work, we extend the formalism of unitarity cuts to incorporate diagrams involving Wilson-line propagators, where the inverse propagator is linear in the loop momenta, rather than the quadratic case which has been studied before. To exploit this for the calculation of the soft anomalous dimension, we first found a suitable momentum-space IR regulator and corresponding prescription, and then derived the appropriate largest time equation (LTE). We find that, as in the case of the scalar diagrams, most terms contributing to the LTE turn out to be zero, albeit for different reasons. This simplifies calculations considerably. This formalism is then applied to the calculation of webs with non-lightlike Wilson lines. As a test, we first looked at webs that have been previously studied using other methods. It emerges that, when using the correct variables, the dispersive integrals one encounters here are trivial, illustrating why unitarity cuts are a particularly useful tool for the calculation of webs. We observe that our technique is especially efficient when looking at diagrams involving three-gluon vertices, such as the (1; 1; 1) web and the Y diagram between two lines. We then focus on three-loop diagrams connecting three or four external non-lightlike lines and involving a three-gluon vertex. We calculate the previously unknown three-loop three-leg (1; 1; 3) web in general kinematics. We obtain a result which agrees with the recently calculated lightlike limit. We also develop a technique to test our results numerically using the computer program SecDec, and we find agreement with our analytical result. The result for the (1; 1; 3) web can then be exploited to gain insight into the more complicated three-loop four-leg (1; 1; 1; 2) web. Indeed, the (1; 1; 1; 2) web reduces to the (1; 1; 3) web in a certain collinear limit. We propose an ansatz for the (1; 1; 1; 2) web in general kinematics, based on a conjectured basis of multiple polylogarithms. The result for the (1; 1; 3) web, together with the known result for the lightlike limit of the (1; 1; 1; 2) web, imposes strong constraints on the ansatz. Using these constraints, we manage to fix all but four coefficients in the ansatz. We fit the remaining coefficients numerically, but find that the quality of the fit is not good. We find possible explanations for this poor quality. This calculation is still a work in progress. Our results provide a major step towards the full calculation of the three-loop soft anomalous dimension for non-lightlike Wilson lines. We calculated new results for three-loop webs, and also deepened the understanding of webs in general. We confirm a conjecture about the functional dependence of the soft anomalous dimension on the cusp angles. We also confirm earlier findings about the symbol alphabet of the relevant functions. This confirms the remarkable simplicity found earlier in the expressions for the soft anomalous dimension.

This thesis is divided into two parts. The first part concerns the study of the ambitwistor string and the scattering equations, while the second concerns the interplay of the symmetries of the asymptotic null boundary of Minkowski space, called [scri], and scattering amplitudes. The first part begins with a review of the CHY formulas for scattering amplitudes, the scattering equations and the ambitwistor string including its pure spinor version. Next are the results of this thesis concerning these topics, they are: generalizing the ambitwistor model to higher genus surfaces; calculating the one-loop NS-NS scattering amplitudes and studying their modular and factorization properties; deriving the one-loop scattering equations and analyzing their factorization; showing that, in the case of the four graviton amplitude, the ambitwistor amplitude gives the expected kinematical prefactor; matching this amplitude to the field theory expectation in a particular kinematical regime; solving the one loop scattering equations in this kinematical regime; a conjecture for the IR behaviour of the one-loop ambitwistor integrand; computing the four graviton, two-loop amplitude using pure spinors; showing that this two-loop amplitude has the correct kinematical prefactor and factorizes as expected for a field theory amplitude; generalizing the ambitwistor string to curved backgrounds; obtaining the field equations for type II supergravity as anomaly cancellation on the worldsheet; generalizing the scattering equations for curved backgrounds. The second part begins with a review of the definition of the null asymptotic boundary of four dimensional Minkowski space, its symmetry algebra, and their relation to soft particles in the S-matrix. Next are the results of this thesis concerning these topics, they are: constructing two models consisting of maps from a worldsheet to [scri], one containing the spectrum of N=8 supergravity, and the other the spectrum of N=4 super Yang-Mills; showing how certain correlators in these theories calculate the tree-level S-matrix of N=8 sugra and N=4 sYM respectively; defining worldsheet charges which encode the action of the appropriate asymptotic symmetry algebra and showing that their Ward-identities recover the soft graviton, and soft gluon factors; defining worldsheet charges for proposed extensions of these symmetry algebras and showing that their Ward-identities give the subleading soft graviton and subleading soft gluon factors.

Nous étudions des fluctuations qui sont omniprésentes dans des collisions entre particules aux hautes énergies. Ces fluctuations peuvent être de nature classique ou quantique et nous allons considérer ces deux cas. D'abord, nous étudions les fluctuations quantiques qui sont présentes dans des collisions entre protons. Celles-ci sont calculables en théorie quantique des champs, et nous allons nous concentrer sur une certaine classe de diagrammes dans ce cadre. Dans un second temps nous allons étudier des fluctuations qui sont présentes dans des collisions entre particules plus lourdes que le proton. Celles-ci sont décrites par les lois quantiques de la nature qui donnent les positions des nucléons dans le noyau, ou bien des fluctuations classiques, d'origine thermique, qui affectent l'évolution hydrodynamique du milieu produit dans une collision. Les fluctuations dans des collisions entre protons peuvent être calculées analytiquement jusqu'à un certain ordre en théorie quantique des champs. Nous allons nous concentrer sur des diagrammes à une boucle, d'une topologie donnée. Ces diagrammes aux boucles donnent des intégrales, qui typiquement sont difficiles à calculer. Nous allons démontrer comment des outils des mathématiques modernes peuvent être utilisés pour faciliter leur évaluation. En particulier, nous allons étudier des relations entre des coupures d'un diagramme, la discontinuité à travers d'un branchement et le coproduit. Nous allons démontrer comment l'intégrale originale peut être reconstruit à partir de l'information contenue dans le coproduit. Nous nous attendons à ce que ces méthodes seront utiles pour le calcul des diagrammes avec des topologies plus difficiles et ainsi aident au calcul des nouvelles amplitudes de diffusion. A la fin, nous étudions les deux types de fluctuations qui ont lieu dans des collisions entre ions lourds. Celles-ci sont liées soit à l'état initial de la matière, soit à l'état intermédiaire produit dans une telle collision. Les fluctuations de l'état initial ont été mesurées expérimentalement, et on voit qu'elles donnent lieu à des non-Gaussianités dans le spectre final de particules. Nous allons démontrer comment ces non-Gaussianités peuvent être comprises comme des positions et des énergies d'interaction aléatoires des 'sources' dans les noyaux entrant en collision. En plus, nous étudions le bruit hydrodynamique dans le milieu produit juste après une collision. Le comportement de ce milieu est celui d'un fluide à basse viscosité. / We study fluctuations that are omnipresent in high-energy particle collisions. These fluctuations can be either of either classical or quantum origin and we will study both. Firstly, we consider the type of quantum fluctuations that arise in proton-proton collisions. These are computable perturbatively in quantum field theory and we will focus on a specific class of diagrams in this set-up. Secondly, we will consider the fluctuations that are present in collisions between nuclei that can be heavier than protons. These are the quantum laws of nature that describe the positions of nucleons within a nucleus, but also the hydrodynamic fluctuations of classical, thermal origin that affect the evolution of the medium produced in heavy-ion collisions. The fluctuations arising in proton-proton collisions can be computed analytically up to a certain order in perturbative quantum field theory. We will focus on one-loop diagrams of a fixed topology. Loop diagrams give rise to integrals that typically are hard to evaluate. We show how modern mathematical methods can be used to ease their computation. We will study the relations among unitarity cuts of a diagram, the discontinuity across the corresponding branch cut and the coproduct. We show how the original integral corresponding to a given diagram can be reconstructed from the information contained in the coproduct. We expect that these methods can be applied to solve more complicated topologies and help in the computation of new amplitudes in the future. Finally, we study the two types of fluctuations arising in heavy-ion collisions. These are related either to the initial state or the intermediate state of matter produced in such collisions. The initial state fluctuations are experimentally observed to give rise to non-Gaussianities in the final-state spectra. We show how these non-Gaussianities can be explained by the random position and interaction energy of `sources' in the colliding nuclei. Furthermore, we investigate the effect of hydrodynamical noise in the evolution of the medium produced just after a collision. This medium behaves like a fluid with a very low viscosity, and so the corresponding evolution is hydrodynamical.

In dieser Arbeit entwickeln wir Rechenwerkzeuge zur Berechnung von Baum– und Einschleifensuperstringamplituden. Insbesondere stellen wir eine rekursive Methode zur Konstruktion kinematischer Faktoren für die Amplitude offener Superstrings auf Baumniveau bereit und präsentieren systematischeWerkzeuge, um die supersymmetrischen Auslöschungen in n-Boson- Zwei-Fermion Amplituden auf Einschleifenniveau des RNS Superstrings zu manifestieren. Für offene Superstringamplituden auf Baumniveau stellen wir vereinfachte Rekursionen für Mehrteilchensuperfelder vor, mit denen kinematische Teile von offenen Superstringamplituden auf Baumniveau konstruiert werden können. Wir diskutieren auch die Eichtransformationen, die ihre Lie-Symmetrien erzwingen, wie dies durch die Bern-Carrasco-Johansson-Dualität zwischen Farbe und Kinematik nahegelegt wird. Eine weitere Eichtransformation aufgrund von Harnad und Shnider soll die Theta-Expansion von Mehrteilchensuperfeldern vereinfachen und die Notwendigkeit umgehen, ihre Rekursionsrelationen über die niedrigsten Komponenten hinaus zu verwenden. Die Ergebnisse dieser Arbeit vereinfachen die Komponentenextraktion aus kinematischen Faktoren im reinen Spinor Superspace erheblich. Wir untersuchen dann masselose n-Punkt-Ein-Schleifen-Amplituden des offenen RNS Superstrings mit zwei externen Fermionen und bestimmen ihre Weltflächen-Integranden. Die beitragenden Korrelationsfunktionen, an denen Spin-1/2– und Spin-3/2-Operatoren aus den Fermionen-Vertices beteiligt sind, werden zu einer beliebigen Multiplizität ausgewertet. Darüber hinaus führen wir Techniken ein, um diese Korrelatoren über die Spinstrukturen derWeltflächen- Fermionen zu summieren, um alle Auslöschungen aufgrund der Supersymmetrie der Raumzeit zu manifestieren. Diese spinsummierten Korrelatoren können in Form von doppeltperiodischen Funktionen ausgedrückt werden, die aus der mathematischen Literatur über elliptische Multiple-Zeta-Werte bekannt sind. Unsere spinsummierten Korrelatoren an der Grenze des Modulraums sind auf kompakte Darstellungen von fermionischen Ein-Schleifen-Integranden für ambitwistorische Strings spezialisiert. / In this thesis, we develop computational tools to calculate tree and one-loop superstring amplitudes. In particular, we provide a recursive method to construct kinematic factors of tree level open superstring amplitudes and present systematic tools to manifest the supersymmetric cancellations in n-boson-two-fermion amplitudes at the one-loop order of the RNS superstring. For tree level open superstring amplitudes, we present simplified recursions for multiparticle superfields, which can be applied to construct kinematic parts of open superstring amplitudes at tree level. We also discuss the gauge transformations which enforce their Lie symmetries as suggested by the Bern-Carrasco-Johansson duality between color and kinematics. Another gauge transformation due to Harnad and Shnider is shown to streamline the theta-expansion of multiparticle superfields, bypassing the need to use their recursion relations beyond the lowest components. The findings of this work greatly simplify the component extraction from kinematic factors in pure spinor superspace. We then investigate massless n-point one-loop amplitudes of the open RNS superstring with two external fermions and determine their world-sheet integrands. The contributing correlation functions involving spin-1/2 and spin-3/2 operators from the fermion vertices are evaluated to any multiplicity. Moreover, we introduce techniques to sum these correlators over the spin structures of the world-sheet fermions, such as to manifest all cancellations due to spacetime supersymmetry. These spin-summed correlators can be expressed in terms of doubly-periodic functions known from the mathematics literature on elliptic multiple zeta values. On the boundary of moduli space, our spin-summed correlators specialize to compact representations of fermionic one-loop integrands for ambitwistor strings.

We calculate tree-level scattering amplitudes in φ^3 theory and Yang-Mills theory by means of the perturbiner expansion. This involves solving the Euler-Lagrange equations of motion perturbatively via a multi-particle ansatz, and using Berends-Giele recursion relations to extract the solution from simple on-shell data. The results are Berends-Giele currents which are then used to calculate the scattering amplitudes. The theoretical calculations are implemented into a Mathematica script which effectively handles recursive calculations and allows us to calculate amplitudes for an arbitrary number of particles.

Cette thèse décrit quelques développements dans les techniques de calcul des amplitudes de diffusion en théorie supersymétrique de champ de jauge . L'accent est mis sur les relations de récurrence on-shell et sur l'utilisation de méthodes d'unitarité pour des calculs de boucle. En particulier, la récurrence on-shell est liée aux règles BCFW pour calculer les amplitudes de jauge au niveau des arbres. Les combinaisons de techniques de coupe d'unitarité et la récurrence sont utilisées pour calculer les amplitudes de boucle, et finalement, à partir des amplitudes, pour obtenir la fonction de corrélation énergie-énergie en théorie de super-Yang-Mills N = 4 à l'aide de la représentation de Mellin-Barnes. Dans le dernier chapitre, nous tentons de trouver un contour convergent pour les intégrales de Mellin Barnes en multi-dimension obtenu par une certaine approximation d'un contour de phase stationnaire.

Die Doppelkopie-Relation besagt, dass Observable in einer Gravitationstheorie durch "Quadrieren" entsprechender Größen in einer Eichtheorie abgeleitet werden können. Es ermöglicht die Verwendung moderner Techniken der Eichtheorien, um Probleme wie die Streuung von Schwarzen Löchern in der Gravitation anzugehen. Wir betrachten zunächst die massive skalare Quantenchromodynamik und führen die Doppelkopie für deren Streuamplituden durch. Aus den resultierenden Amplituden rekonstruieren wir die effektive Lagrange-Funktion. Diese besteht aus einer Graviationstheorie gekoppelt an massive Skalare, ein Axion und ein Dilaton. Der entstehende Lagrangian wird explizit bis zur sechsten Ordnung von Skalarfeldern konstruiert, und es wird eine Form aller Ordnungen postuliert. Es folgt die Erforschung der Doppelkopie massiver Punktteilchen. Die Quellen werden durch Weltlinien-Quantenfeldtheorien formuliert, die mit Yang-Mills, biadjungiertem Skalar und Zwei-Form-Dilaton-Gravitation gekoppelt sind. Wir schlagen eine Doppelkopievorschrift für die eikonalen Phase vor, und explizit bis zur nächstführenden Ordnung zu überprüfen. Wir untersuchen ferner die nicht-perturbative Doppelkopie klassischer Lösungen. Insbesondere erweitern wir die Kerr-Schild-Abbildung auf den Fall eines Probeteilchens, das sich im Kerr-Schild-Hintergrund bewegt. Wir finden darüberhinaus eine neue Doppelkopie zwischen den erhaltenen Ladungen auf der Eichtheorie und den Gravitationsseiten. Schließich untersuchen wir die Post-Minkowski'sche (PM) und Post-Newton'sche Entwicklungen des gravitativen effektiven Drei-Körper-Potentials. Wir liefern auf 2PM Ebene ein formelles nicht-lokales Ergebnis und entwickeln es in der Geschwindigkeit. / This thesis focuses on the double copy relation between gauge theories and gravity and its application in the classical scattering of massive compact objects. The double copy relation states that observables in a gravitational theory can be derived from “squaring” corresponding quantities in a gauge theory. It allows using modern techniques of gauge theories to tackle problems such as black hole scattering in gravity. We first consider massive scalar quantum chromodynamics and perform the double copy procedure for the scattering amplitudes. We reconstruct the effective Lagrangian from the resulting amplitudes. It yields a gravitational theory of massive scalars coupled to gravity, axion, and dilaton. The emerging Lagrangian is constructed explicitly up to the sixth order of scalar fields, and an all-order form is conjectured. It is followed by exploring the double copy of classical massive point particles. The source objects are formulated by worldline quantum field theories coupled to Yang-Mills, bi-adjoint scalar, and two-form-dilaton-gravity. We propose a double copy prescription for the eikonal phases, and check it explicitly up to next-to-leading order. We also investigate the non-perturbative double copy of classical solutions. Specifically, we extend the Kerr-Schild mapping, which allows obtaining solutions of the Einstein equation from that of gauge theory, to the case of a probe particle moving in the Kerr-Schild background. We find a new double copy between the conserved charges on the gauge theory and the gravity sides, which works naturally for both bound and unbound states. Additionally, we study the Post-Minkowskian (PM) and Post-Newtonian expansions of the gravitational three-body effective potential. We provide a formal non-local result at 2PM and expand it in the slow-motion limit.