Auswahl der wissenschaftlichen Literatur zum Thema „Ultraviolet regularisation“

Abstract We derive CP-violating transport equations for fermions for electroweak baryogenesis from the CTP-formalism including thermal corrections at the one-loop level. We consider both the VEV-insertion approximation (VIA) and the semiclassical (SC) formalism. We show that the VIA-method is based on an assumption that leads to an ill-defined source term containing a pinch singularity, whose regularisation by thermal effects leads to ambiguities including spurious ultraviolet and infrared divergences. We then carefully review the derivation of the semiclassical formalism and extend it to include thermal corrections. We present the semiclassical Boltzmann equations for thermal WKB-quasiparticles with source terms up to the second order in gradients that contain both dispersive and finite width corrections. We also show that the SC-method reproduces the current divergence equations and that a correct implementation of the Fick's law captures the semiclassical source term even with conserved total current ∂μ j μ = 0. Our results show that the VIA-source term is not just ambiguous, but that it does not exist. Finally, we show that the collisional source terms reported earlier in the semiclassical literature are also spurious, and vanish in a consistent calculation.

Within the Local Unitarity formalism, any physical cross-section is re-written in such a way that cancellations of infrared singularities between real and virtual contributions are realised locally. Consequently, phase-space and loop integrals are regulated locally without the need of any counter-terms or dimensional regularisation aside from those strictly needed to achieve ultraviolet regularisation. This Local Unitarity representation is especially suitable to direct Monte Carlo integration. As such, it offers a clear and direct path to automate the computation of fixed-order differential observables in Quantum Field Theories.

We present a new on-shell method for the matching of ultraviolet models featuring massive states onto their massless effective field theory. We employ a dispersion relation in the space of complex momentum dilations to capture, in a single variable, the relevant analytic structure of scattering amplitudes at any multiplicity. Multivariate complex analysis and crossing considerations are therefore avoided. Remarkably, no knowledge about the infrared effective field theory is required in dimensional regularisation. All matching information is extracted from the residues and iscontinuities of the ultraviolet scattering amplitudes, which unitarity expresses in terms of lower-point and lower-loop results, respectively. This decomposition into simpler building blocks could deliver new insight in the structure of the effective field theories obtained from classes of ultraviolet scenarios and facilitate computations at higher loop orders.

AbstractA geometric formulation of Wilson’s exact renormalisation group is presented based on a gauge invariant ultraviolet regularisation scheme without the introduction of a background field. This allows for a manifestly background independent approach to quantum gravity and gauge theories in the continuum. The regularisation is a geometric variant of Slavnov’s scheme consisting of a modified action, which suppresses high momentum modes, supplemented by Pauli–Villars determinants in the path integral measure. An exact renormalisation group flow equation for the Wilsonian effective action is derived by requiring that the path integral is invariant under a change in the cutoff scale while preserving quasi-locality. The renormalisation group flow is defined directly on the space of gauge invariant actions without the need to fix the gauge. We show that the one-loop beta function in Yang–Mills and the one-loop divergencies of General Relativity can be calculated without fixing the gauge. As a first non-perturbative application we find the form of the Yang–Mills beta function within a simple truncation of the Wilsonian effective action.

Abstract Numerical approaches to computations typically reconstruct the numerators of Feynman diagrams in four dimensions. In doing so, certain rational terms arising from the (D − 4)-dimensional part of the numerator multiplying ultraviolet (UV) poles in dimensional regularisation are not captured and need to be obtained by other means. At one-loop these rational terms of UV origin can be computed from a set of process-independent Feynman rules. Recently, it was shown that this approach can be extended to two loops. In this paper, we show that to all loop orders it is possible to compute rational terms of UV origin through process-independent vertices that are polynomial in masses and momenta.

Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this quantum Hamiltonian captures the near horizon dynamics of the scattering matrix of the Schwarzschild black hole, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix.

Abstract The Local Unitarity (LU) representation of differential cross-sections locally realises the cancellations of infrared singularities predicted by the Kinoshita-Lee-Nauenberg theorem. In this work we solve the two remaining challenges to enable practical higher-loop computations within the LU formalism. The first concerns the generalisation of the LU representation to graphs with raised propagators. The solution to this problem results in a generalisation of distributional Cutkosky rules. The second concerns the regularisation of ultraviolet and spurious soft singularities, solved using a fully automated and local renormalisation procedure based on Bogoliubov’s R-operation. We detail an all-order construction for the hybrid $$ \overline{\textrm{MS}} $$ MS ¯ and On-Shell scheme whose only analytic input is single-scale vacuum diagrams. We validate this novel technology by providing (semi-)inclusive results for two multi-leg processes at NLO, study limits of individual supergraphs up to N3LO and present the first physical NNLO cross-sections computed fully numerically in momentum-space, namely for the processes γ∗→ jj and $$ {\gamma}^{\ast}\to t\overline{t} $$ γ ∗ → t t ¯ .

Analysing the large-scale structure (LSS) in the Universe with galaxy surveys demands accurate structure formation models. Such models should ideally be fast and have a clear theoretical framework in order to rapidly scan a variety of cosmological parameter spaces without requiring large training data sets. This study aims to extend Lagrangian perturbation theory (LPT), including viscosity and vorticity, to reproduce the cosmic evolution from dark matter $N$-body calculations at the field level. We extend LPT to a Eulerian framework, which we dub eALPT. An ultraviolet regularisation through the spherical collapse model provided by Augmented LPT turns out to be crucial at low redshifts. This iterative method enables modelling of the stress tensor and introduces vorticity. The eALPT model has two free parameters apart from the choice of cosmology, redshift snapshots, cosmic volume, and the number of particles. Mpc black and $z=0$ from $ with the Zel'dovich approximation ($ with ALPT), to sim 95<!PCT!> with the three-timestep eALPT, and the power spectra show percentage accuracy up to $k Mpc

Cette thèse de doctorat est principalement consacrée à l'étude mathématique du vide de Dirac, potentiellement en présence d'un champ électromagnétique et d'électrons. Ce système est décrit à l'aide de modèles non linéaires issus d'approximations de champ moyen de l'électrodynamique quantique. Dans le Chapitre 2, on étudie la dynamique d'un nombre fini d'électrons relativistes interagissant avec un nombre fini de noyaux classiques dans la mer de Dirac, en prouvant que le problème de Cauchy correspondant est globalement bien posé. Dans le Chapitre 3, on fournit la première dérivation rigoureuse du lagrangien magnétique effectif à une boucle à température positive, une fonctionnelle non linéaire décrivant l'énergie libre du vide de Dirac dans un champ magnétique classique. Finalement, dans le Chapitre 4, on présente quelques résultats numériques concernant un modèle unidimensionnel décrivant un atome hydrogénoïde relativiste. En particulier, on propose une approche pour calculer la polarisation du vide ainsi que le décalage de Lamb dans une approximation de base finie
This doctoral thesis is mainly devoted to the mathematical study of the Dirac vacuum, potentially in presence of an electromagnetic field and a finite number of electrons. This system is described by means of non-linear models originating from mean-field approximations of quantum electrodynamics. In Chapter 2, we study the dynamics of a finite number of relativistic electrons interacting with finitely many classical nuclei in the Dirac sea, by proving a global well-posedness result for the corresponding Cauchy problem. In Chapter 3, we provide the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of the Dirac vacuum in a classical magnetic field. Finally, in Chapter 4, we present some numerical results concerning a one-dimensional model describing a relativistic hydrogen-like atom. In particular, we present an approach to compute the vacuum polarisation as well as the Lamb shift in a finite basis set approximation

AbstractIn this chapter we introduce methods for evaluating Feynman loop integrals. We introduce basic methods such as Feynman and Mellin parametrisations, and present a number of one-loop examples. Working in dimensional regularisation, we discuss ultraviolet and infrared divergences. We then introduce special functions encountered in loop calculations and discuss their properties. Focusing on their defining differential equations, we show how the symbol method is a useful tool for keeping track of functional identities. We then connect back to Feynman integrals by showing how differential equations can be effectively used to read off the special functions appearing in them. In particular, we discuss residue-based methods that streamline such computations.

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.