A novel functional renormalization group framework for gauge theories and gravity

In this thesis we develop further the functional renormalization group (RG) approach to quantum field theory (QFT) based on the effective average action (EAA) and on the exact flow equation that it satisfies. The EAA is a generalization of the standard effective action that interpolates smoothly between the bare action for krightarrowinfty and the standard effective action rnfor krightarrow0. In this way, the problem of performing the functional integral is converted into the problem of integrating the exact flow of the EAA from the UV to the IR. The EAA formalism deals naturally with several different aspects of a QFT. One aspect is related to the discovery of non-Gaussian fixed points of the RG flow that can be used to construct continuum limits. In particular, the EAA framework is a useful setting to search for Asymptotically Safe theories, i.e. theories valid up to arbitrarily high energies. A second aspect in which the EAA reveals its usefulness are non-perturbative calculations. In fact, the exact flow that it satisfies is a valuable starting point for devising new approximation schemes. In the first part of this thesis we review and extend the formalism, in particular we derive the exact RG flow equation for the EAA and the related hierarchy of coupled flow equations for the proper-vertices. We show how standard perturbation theory emerges as a particular way to iteratively solve the flow equation, if the starting point is the bare action. Next, we explore both technical and conceptual issues by means of three different applications of the formalism, to QED, to general non-linear sigma models (NLsigmaM) and to matter fields on curved spacetimes. In the main part of this thesis we construct the EAA for non-abelian gauge theories and for quantum Einstein gravity (QEG), using the background field method to implement the coarse-graining procedure in a gauge invariant way. We propose a new truncation scheme where the EAA is expanded in powers of the curvature or field strength. Crucial to the practical use of this expansion is the development of new techniques to manage functional traces such as the algorithm proposed in this thesis. This allows to project the flow of all terms in the EAA which are analytic in the fields. As an application we show how the low energy effective action for quantum gravity emerges as the result of integrating the RG flow. In any treatment of theories with local symmetries that introduces a reference scale, the question of preserving gauge invariance along the flow emerges as predominant. In the EAA framework this problem is dealt with the use of the background field formalism. This comes at the cost of enlarging the theory space where the EAA lives to the space of functionals of both fluctuation and background fields. In this thesis, we study how the identities dictated by the symmetries are modified by the introduction of the cutoff and we study so called bimetric truncations of the EAA that contain both fluctuation and background couplings. In particular, we confirm the existence of a non-Gaussian fixed point for QEG, that is at the heart of the Asymptotic Safety scenario in quantum gravity; in the enlarged bimetric theory space where the running of the cosmological constant and of Newton's constant is influenced by fluctuation couplings.

Quantum Einstein gravity - advancements of heat kernel-based renormalization group studies

The asymptotic safety scenario allows to define a consistent theory of quantized gravity within the framework of quantum field theory. The central conjecture of this scenario is the existence of a non-Gaussian fixed point of the theory's renormalization group flow, that allows to formulate renormalization conditions that render the theory fully predictive. Investigations of this possibility use an exact functional renormalization group equation as a primary non-perturbative tool. This equation implements Wilsonian renormalization group transformations, and is demonstrated to represent a reformulation of the functional integral approach to quantum field theory.rnAs its main result, this thesis develops an algebraic algorithm which allows to systematically construct the renormalization group flow of gauge theories as well as gravity in arbitrary expansion schemes. In particular, it uses off-diagonal heat kernel techniques to efficiently handle the non-minimal differential operators which appear due to gauge symmetries. The central virtue of the algorithm is that no additional simplifications need to be employed, opening the possibility for more systematic investigations of the emergence of non-perturbative phenomena. As a by-product several novel results on the heat kernel expansion of the Laplace operator acting on general gauge bundles are obtained.rnThe constructed algorithm is used to re-derive the renormalization group flow of gravity in the Einstein-Hilbert truncation, showing the manifest background independence of the results. The well-studied Einstein-Hilbert case is further advanced by taking the effect of a running ghost field renormalization on the gravitational coupling constants into account. A detailed numerical analysis reveals a further stabilization of the found non-Gaussian fixed point.rnFinally, the proposed algorithm is applied to the case of higher derivative gravity including all curvature squared interactions. This establishes an improvement of existing computations, taking the independent running of the Euler topological term into account. Known perturbative results are reproduced in this case from the renormalization group equation, identifying however a unique non-Gaussian fixed point.rn

On the flavor problem in strongly coupled theories

rnThis thesis is on the flavor problem of Randall Sundrum modelsrnand their strongly coupled dual theories. These models are particularly wellrnmotivated extensions of the Standard Model, because they simultaneously address rntherngauge hierarchy problem and the hierarchies in the quarkrnmasses and mixings. In order to put this into context, special attention is given to concepts underlying therntheories which can explain the hierarchy problem and the flavor structure of the Standard Model (SM). ThernAdS/CFTrnduality is introduced and its implications for the Randall Sundrum model withrnfermions in the bulk andrngeneral bulk gauge groups is investigated. It will be shown that the differentrnterms in the general 5D propagator of a bulk gauge field can be related tornthe corresponding diagrams of the strongly coupled dual, which allows for arndeeperrnunderstanding of the origin of flavor changing neutral currents generated by thernexchange of the Kaluza Klein excitations of these bulk fields.rnIn the numerical analysis, different observables which are sensitive torncorrections from therntree-levelrnexchange of these resonances will be presented on the basis of updatedrnexperimental data from the Tevatron and LHC experiments. This includesrnelectroweak precision observables, namely corrections to the S and Trnparameters followed by corrections to the Zbb vertex, flavor changingrnobservables with flavor changes at one vertex, viz. BR (Bd -> mu+mu-) and BR (Bs -> mu+mu-), and two vertices,rn viz. S_psiphi and |eps_K|, as well as bounds from direct detectionrnexperiments. rnThe analysis will show that all of these bounds can be brought in agreement withrna new physics scale Lambda_NP in the TeV range, except for the CPrnviolating quantity |eps_K|, which requires Lambda_NP= Ord(10) TeVrnin the absencernof fine-tuning. The numerous modifications of the Randall Sundrum modelrnin the literature, which try to attenuate this bound are reviewed andrncategorized.rnrnSubsequently, a novel solution to this flavor problem, based on an extendedrncolor gauge group in the bulk and its thorough implementation inrnthe RS model, will be presented, as well as an analysis of the observablesrnmentioned above in the extended model. This solution is especially motivatedrnfromrnthe point of view of the strongly coupled dual theory and the implications forrnstrongly coupled models of new physics, which do not possess a holographic dual,rnare examined.rnFinally, the top quark plays a special role in models with a geometric explanation ofrnflavor hierarchies and the predictions in the Randall-Sundrum model with andrnwithout the proposed extension for the forward-backward asymmetryrnA_FB^trnin top pair production are computed.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Hadronic correlation functions with quark-disconnected contributions in lattice QCD

One of the fundamental interactions in the Standard Model of particle physicsrnis the strong force, which can be formulated as a non-abelian gauge theoryrncalled Quantum Chromodynamics (QCD). rnIn the low-energy regime, where the QCD coupling becomes strong and quarksrnand gluons are confined to hadrons, a perturbativernexpansion in the coupling constant is not possible.rnHowever, the introduction of a four-dimensional Euclidean space-timernlattice allows for an textit{ab initio} treatment of QCD and provides arnpowerful tool to study the low-energy dynamics of hadrons.rnSome hadronic matrix elements of interest receive contributionsrnfrom diagrams including quark-disconnected loops, i.e. disconnected quarkrnlines from one lattice point back to the same point. The calculation of suchrnquark loops is computationally very demanding, because it requires knowledge ofrnthe all-to-all propagator. In this thesis we use stochastic sources and arnhopping parameter expansion to estimate such propagators.rnWe apply this technique to study two problems which relay crucially on therncalculation of quark-disconnected diagrams, namely the scalar form factor ofrnthe pion and the hadronic vacuum polarization contribution to the anomalousrnmagnet moment of the muon.rnThe scalar form factor of the pion describes the coupling of a charged pion torna scalar particle. We calculate the connected and the disconnected contributionrnto the scalar form factor for three different momentum transfers. The scalarrnradius of the pion is extracted from the momentum dependence of the form factor.rnThe use ofrnseveral different pion masses and lattice spacings allows for an extrapolationrnto the physical point. The chiral extrapolation is done using chiralrnperturbation theory ($chi$PT). We find that our pion mass dependence of thernscalar radius is consistent with $chi$PT at next-to-leading order.rnAdditionally, we are able to extract the low energy constant $ell_4$ from thernextrapolation, and ourrnresult is in agreement with results from other lattice determinations.rnFurthermore, our result for the scalar pion radius at the physical point isrnconsistent with a value that was extracted from $pipi$-scattering data. rnThe hadronic vacuum polarization (HVP) is the leading-order hadronicrncontribution to the anomalous magnetic moment $a_mu$ of the muon. The HVP canrnbe estimated from the correlation of two vector currents in the time-momentumrnrepresentation. We explicitly calculate the corresponding disconnectedrncontribution to the vector correlator. We find that the disconnectedrncontribution is consistent with zero within its statistical errors. This resultrncan be converted into an upper limit for the maximum contribution of therndisconnected diagram to $a_mu$ by using the expected time-dependence of therncorrelator and comparing it to the corresponding connected contribution. Wernfind the disconnected contribution to be smaller than $approx5%$ of thernconnected one. This value can be used as an estimate for a systematic errorrnthat arises from neglecting the disconnected contribution.rn

Ein, zwei Punkte zu Nichtkommutativer Geometrie und Quantenfeldtheorie auf diskreten Räumen

Über viele Jahre hinweg wurden wieder und wieder Argumente angeführt, die diskreten Räumen gegenüber kontinuierlichen Räumen eine fundamentalere Rolle zusprechen. Unser Zugangzur diskreten Welt wird durch neuere Überlegungen der Nichtkommutativen Geometrie (NKG) bestimmt. Seit ca. 15Jahren gibt es Anstrengungen und auch Fortschritte, Physikmit Hilfe von Nichtkommutativer Geometrie besser zuverstehen. Nur eine von vielen Möglichkeiten ist dieReformulierung des Standardmodells derElementarteilchenphysik. Unter anderem gelingt es, auch denHiggs-Mechanismus geometrisch zu beschreiben. Das Higgs-Feld wird in der NKG in Form eines Zusammenhangs auf einer zweielementigen Menge beschrieben. In der Arbeit werden verschiedene Ziele erreicht:Quantisierung einer nulldimensionalen ,,Raum-Zeit'', konsistente Diskretisierungf'ur Modelle im nichtkommutativen Rahmen.Yang-Mills-Theorien auf einem Punkt mit deformiertemHiggs-Potenzial. Erweiterung auf eine ,,echte''Zwei-Punkte-Raum-Zeit, Abzählen von Feynman-Graphen in einer nulldimensionalen Theorie, Feynman-Regeln. Eine besondere Rolle werden Termini, die in derQuantenfeldtheorie ihren Ursprung haben, gewidmet. In diesemRahmen werden Begriffe frei von Komplikationen diskutiert,die durch etwaige Divergenzen oder Schwierigkeitentechnischer Natur verursacht werden könnten.Eichfixierungen, Geistbeiträge, Slavnov-Taylor-Identität undRenormierung. Iteratives Lösungsverfahren derDyson-Schwinger-Gleichung mit Computeralgebra-Unterstützung,die Renormierungsprozedur berücksichtigt.

Renormalization of fermion mixing

Precision measurements of phenomena related to fermion mixing require the inclusion of higher order corrections in the calculation of corresponding theoretical predictions. For this, a complete renormalization scheme for models that allow for fermion mixing is highly required. The correct treatment of unstable particles makes this task difficult and yet, no satisfactory and general solution can be found in the literature. In the present work, we study the renormalization of the fermion Lagrange density with Dirac and Majorana particles in models that involve mixing. The first part of the thesis provides a general renormalization prescription for the Lagrangian, while the second one is an application to specific models. In a general framework, using the on-shell renormalization scheme, we identify the physical mass and the decay width of a fermion from its full propagator. The so-called wave function renormalization constants are determined such that the subtracted propagator is diagonal on-shell. As a consequence of absorptive parts in the self-energy, the constants that are supposed to renormalize the incoming fermion and the outgoing antifermion are different from the ones that should renormalize the outgoing fermion and the incoming antifermion and not related by hermiticity, as desired. Instead of defining field renormalization constants identical to the wave function renormalization ones, we differentiate the two by a set of finite constants. Using the additional freedom offered by this finite difference, we investigate the possibility of defining field renormalization constants related by hermiticity. We show that for Dirac fermions, unless the model has very special features, the hermiticity condition leads to ill-defined matrix elements due to self-energy corrections of external legs. In the case of Majorana fermions, the constraints for the model are less restrictive. Here one might have a better chance to define field renormalization constants related by hermiticity. After analysing the complete renormalized Lagrangian in a general theory including vector and scalar bosons with arbitrary renormalizable interactions, we consider two specific models: quark mixing in the electroweak Standard Model and mixing of Majorana neutrinos in the seesaw mechanism. A counter term for fermion mixing matrices can not be fixed by only taking into account self-energy corrections or fermion field renormalization constants. The presence of unstable particles in the theory can lead to a non-unitary renormalized mixing matrix or to a gauge parameter dependence in its counter term. Therefore, we propose to determine the mixing matrix counter term by fixing the complete correction terms for a physical process to experimental measurements. As an example, we calculate the decay rate of a top quark and of a heavy neutrino. We provide in each of the chosen models sample calculations that can be easily extended to other theories.

Lattice preferred orientation in omphacite: Examples from the Tauern Window, Austria and the Western Gneiss Region, Norway

It is lively debated how eclogites find their way from deep to mid-crustal levels during exhumation. Different exhumation models for high-pressure and ultrahigh-pressure rocks were suggested in previous studies, based mainly on field observations and less on microstructural studies on the exhumed rocks. The development and improvement of electron microscopy techniques allows it, to focus interest on direct investigations of microstructures and crystallographic properties in eclogites. In this case, it is of importance to study the applicability of crystallographic measurements on eclogites for exhumation processes and to unravel which processes affect eclogite textures. Previous studies suggested a strong relationship between deformation and lattice preferred orientation (LPO) in omphacite but it is still unclear if the deformation is related to the exhumation of eclogites. This study is focused on the questions which processes affect omphacite LPO and if textural investigations of omphacite are applicable for studying eclogite exhumation. Therefore, eclogites from two examples in the Alps and in the Caledonides were collected systematically and investigated with respect to omphacite LPO by using the electron backscattered diffraction (EBSD) technique. Omphacite textures of the Tauern Window (Austria) and the Western Gneiss Region (Norway) were studied to compare lattice preferred orientation with field observations and suggested exhumation models from previous studies. The interpretation of omphacite textures, regarding the deformation regime is mainly based on numerical simulations in previous studies. Omphacite LPO patterns of the Eclogite Zone are clearly independent from any kind of exhumation process. The textures were generated during omphacite growth on the prograde path of eclogite development until metamorphic peak conditions. Field observations in the Eclogite Zone show that kinematics in garnet mica schist, surrounding the eclogites, strongly indicate an extrusion wedge geometry. Stretching lineations show top-N thrusting at the base and a top-S normal faulting with a sinistral shear component at the top of the Eclogite Zone. The different shear sense on both sides of the unit does not affect the omphacite textures in any way. The omphacite lattice preferred orientation patterns of the Western Gneiss Region can not be connected with any exhumation model. The textures were probably generated during the metamorphic peak and reflect the change from subduction to exhumation. Eclogite Zone and Western Gneiss Region differ significantly in size and especially in metamorphic conditions. While the Eclogite Zone is characterized by constant P-T conditions (600-650°C, 20-25 kbar), the Western Gneiss Region contains a wide P-T range from high- to ultrahigh pressure conditions (400-800°C, 20-35 kbar). In contrast to this, the omphacite textures of both units are very similar. This means that omphacite LPO is independent from P-T conditions and therefore from burial depth. Further, in both units, omphacite LPO is independent from grain and subgrain size as well as from any shape preferred orientation (SPO) on grain and subgrain scale. Overall, omphacite lattice preferred orientation are generated on the prograde part of omphacite development. Therefore, textural investigations on omphacite LPO are not applicable to study eclogite exhumation.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Deriving Deligne-Mumford stacks with perfect obstruction theories

Intersection theory on moduli spaces has lead to immense progress in certain areas of enumerative geometry. For some important areas, most notably counting stable maps and counting stable sheaves, it is important to work with a virtual fundamental class instead of the usual fundamental class of the moduli space. The crucial prerequisite for the existence of such a class is a two-term complex controlling deformations of the moduli space. Kontsevich conjectured in 1994 that there should exist derived version of spaces with this specific property. Another hint at the existence of these spaces comes from derived algebraic geometry. It is expected that for every pair of a space and a complex controlling deformations of the space their exists, under some additional hypothesis, a derived version of the space having the chosen complex as cotangent complex. In this thesis one version of these additional hypothesis is identified. We then show that every space admitting a two-term complex controlling deformations satisfies these hypothesis, and we finally construct the derived spaces.

Interacting bosons and fermions in three-dimensional optical lattice potentials

This thesis reports on the realization, characterization and analysis of ultracold bosonic and fermionic atoms in three-dimensional optical lattice potentials. Ultracold quantum gases in optical lattices can be regarded as ideal model systems to investigate quantum many-body physics. In this work interacting ensembles of bosonic 87Rb and fermionic 40K atoms are employed to study equilibrium phases and nonequilibrium dynamics. The investigations are enabled by a versatile experimental setup, whose core feature is a blue-detuned optical lattice that is combined with Feshbach resonances and a red-detuned dipole trap to allow for independent control of tunneling, interactions and external confinement. The Fermi-Hubbard model, which plays a central role in the theoretical description of strongly correlated electrons, is experimentally realized by loading interacting fermionic spin mixtures into the optical lattice. Using phase-contrast imaging the in-situ size of the atomic density distribution is measured, which allows to extract the global compressibility of the many-body state as a function of interaction and external confinement. Thereby, metallic and insulating phases are clearly identified. At strongly repulsive interaction, a vanishing compressibility and suppression of doubly occupied lattice sites signal the emergence of a fermionic Mott insulator. In a second series of experiments interaction effects in bosonic lattice quantum gases are analyzed. Typically, interactions between microscopic particles are described as two-body interactions. As such they are also contained in the single-band Bose-Hubbard model. However, our measurements demonstrate the presence of multi-body interactions that effectively emerge via virtual transitions of atoms to higher lattice bands. These findings are enabled by the development of a novel atom optical measurement technique: In quantum phase revival spectroscopy periodic collapse and revival dynamics of the bosonic matter wave field are induced. The frequencies of the dynamics are directly related to the on-site interaction energies of atomic Fock states and can be read out with high precision. The third part of this work deals with mixtures of bosons and fermions in optical lattices, in which the interspecies interactions are accurately controlled by means of a Feshbach resonance. Studies of the equilibrium phases show that the bosonic superfluid to Mott insulator transition is shifted towards lower lattice depths when bosons and fermions interact attractively. This observation is further analyzed by applying quantum phase revival spectroscopy to few-body systems consisting of a single fermion and a coherent bosonic field on individual lattice sites. In addition to the direct measurement of Bose-Fermi interaction energies, Bose-Bose interactions are proven to be modified by the presence of a fermion. This renormalization of bosonic interaction energies can explain the shift of the Mott insulator transition. The experiments of this thesis lay important foundations for future studies of quantum magnetism with fermionic spin mixtures as well as for the realization of complex quantum phases with Bose-Fermi mixtures. They furthermore point towards physics that reaches beyond the single-band Hubbard model.

Baryons in the chiral regime

Quantum Chromodynamics (QCD) is the theory of strong interactions, one of the four fundamental forces in our Universe. It describes the interaction of gluons and quarks which build up hadrons like protons and neutrons. Most of the visible matter in our universe is made of protons and neutrons. Hence, we are interested in their fundamental properties like their masses, their distribution of charge and their shape. \\rnThe only known theoretical, non-perturbative and {\it ab initio} method to investigate hadron properties at low energies is lattice Quantum Chromodynamics (lattice QCD). However, up-to-date simulations (especially for baryonic quantities) do not achieve the accuracy of experiments. In fact, current simulations do not even reproduce the experimental values for the form factors. The question arises wether these deviations can be explained by systematic effects in lattice QCD simulations.rnrnThis thesis is about the computation of nucleon form factors and other hadronic quantities from lattice QCD. So called Wilson fermions are used and the u- and d-quarks are treated fully dynamically. The simulations were performed using gauge ensembles with a range of lattice spacings, volumes and pion masses.\\rnFirst of all, the lattice spacing was set to be able to make contact between the lattice results and their experimental complement and to be able to perform a continuum extrapolation. The light quark mass has been computed and found to be $m_{ud}^{\overline{\text{MS}}}(2\text{ GeV}) = 3.03(17)(38)\text{ MeV}$. This value is in good agreement with values from experiments and other lattice determinations.\\rnElectro-magnetic and axial form factors of the nucleon have been calculated. From these form factors the nucleon radii and the coupling constants were computed. The different ensembles enabled us to investigate systematically the dependence of these quantities on the volume, the lattice spacing and the pion mass.\newpage Finally we perform a continuum extrapolation and chiral extrapolations to the physical point.\\rnIn addition, we investigated so called excited state contributions to these observables. A technique was used, the summation method, which reduces these effects significantly and a much better agreement with experimental data was achieved. On the lattice, the Dirac radius and the axial charge are usually found to be much smaller than the experimental values. However, due to the carefully investigation of all the afore-mentioned systematic effects we get $\langle r_1^2\rangle_{u-d}=0.627(54)\text{ fm}^2$ and $g_A=1.218(92)$, which is in agreement with the experimental values within the errors.rnrnThe first three chapters introduce the theoretical background of form factors of the nucleon and lattice QCD in general. In chapter four the lattice spacing is determined. The computation of nucleon form factors is described in chapter five where systematic effects are investigated. All results are presented in chapter six. The thesis ends with a summary of the results and identifies options to complement and extend the calculations presented. rn

Investigations on asymptotic safety of metric, tetrad and Einstein-Cartan gravity

Among the different approaches for a construction of a fundamental quantum theory of gravity the Asymptotic Safety scenario conjectures that quantum gravity can be defined within the framework of conventional quantum field theory, but only non-perturbatively. In this case its high energy behavior is controlled by a non-Gaussian fixed point of the renormalization group flow, such that its infinite cutoff limit can be taken in a well defined way. A theory of this kind is referred to as non-perturbatively renormalizable. In the last decade a considerable amount of evidence has been collected that in four dimensional metric gravity such a fixed point, suitable for the Asymptotic Safety construction, indeed exists. This thesis extends the Asymptotic Safety program of quantum gravity by three independent studies that differ in the fundamental field variables the investigated quantum theory is based on, but all exhibit a gauge group of equivalent semi-direct product structure. It allows for the first time for a direct comparison of three asymptotically safe theories of gravity constructed from different field variables. The first study investigates metric gravity coupled to SU(N) Yang-Mills theory. In particular the gravitational effects to the running of the gauge coupling are analyzed and its implications for QED and the Standard Model are discussed. The second analysis amounts to the first investigation on an asymptotically safe theory of gravity in a pure tetrad formulation. Its renormalization group flow is compared to the corresponding approximation of the metric theory and the influence of its enlarged gauge group on the UV behavior of the theory is analyzed. The third study explores Asymptotic Safety of gravity in the Einstein-Cartan setting. Here, besides the tetrad, the spin connection is considered a second fundamental field. The larger number of independent field components and the enlarged gauge group render any RG analysis of this system much more difficult than the analog metric analysis. In order to reduce the complexity of this task a novel functional renormalization group equation is proposed, that allows for an evaluation of the flow in a purely algebraic manner. As a first example of its suitability it is applied to a three dimensional truncation of the form of the Holst action, with the Newton constant, the cosmological constant and the Immirzi parameter as its running couplings. A detailed comparison of the resulting renormalization group flow to a previous study of the same system demonstrates the reliability of the new equation and suggests its use for future studies of extended truncations in this framework.

Supersymmetry searches in dilepton final states with the ATLAS experiment

One of the main goals of the ATLAS experiment at the Large Hadron Collider (LHC) at CERN in Geneva is the search for new physics beyond the Standard Model. In 2011, proton-proton collisions were performed at the LHC at a center of mass energy of 7 TeV and an integrated luminosity of 4.7 fb^{-1} was recorded. This dataset can be tested for one of the most promising theories beyond limits achieved thus far: supersymmetry. Final states in supersymmetry events at the LHC contain highly energetic jets and sizeable missing transverse energy. The additional requirement of events with highly energetic leptons simplifies the control of the backgrounds. This work presents results of a search for supersymmetry in the inclusive dilepton channel. Special emphasis is put on the search within the Gauge-Mediated Symmetry Breaking (GMSB) scenario in which the supersymmetry breaking is mediated via gauge fields. Statistically independent Control Regionsrnfor the dominant Standard Model backgrounds as well as Signal Regions for a discovery of a possible supersymmetry signal are defined and optimized. A simultaneous fit of the background normalizations in the Control Regions via the profile likelihood method allows for a precise prediction of the backgrounds in the Signal Regions and thus increases the sensitivity to several supersymmetry models. Systematic uncertainties on the background prediction are constrained via the jet multiplicity distribution in the Control Regions driven by data. The observed data are consistent with the Standard Model expectation. New limits within the GMSB and the minimal Supergravity (mSUGRA) scenario as well as for several simplified supersymmetry models are set or extended.

Hadronic matrix elements in lattice QCD

The lattice formulation of Quantum ChromoDynamics (QCD) has become a reliable tool providing an ab initio calculation of low-energy quantities. Despite numerous successes, systematic uncertainties, such as discretisation effects, finite-size effects, and contaminations from excited states, are inherent in any lattice calculation. Simulations with controlled systematic uncertainties and close to the physical pion mass have become state-of-the-art. We present such a calculation for various hadronic matrix elements using non-perturbatively O(a)-improved Wilson fermions with two dynamical light quark flavours. The main topics covered in this thesis are the axial charge of the nucleon, the electro-magnetic form factors of the nucleon, and the leading hadronic contributions to the anomalous magnetic moment of the muon. Lattice simulations typically tend to underestimate the axial charge of the nucleon by 5 − 10%. We show that including excited state contaminations using the summed operator insertion method leads to agreement with the experimentally determined value. Further studies of systematic uncertainties reveal only small discretisation effects. For the electro-magnetic form factors of the nucleon, we see a similar contamination from excited states as for the axial charge. The electro-magnetic radii, extracted from a dipole fit to the momentum dependence of the form factors, show no indication of finite-size or cutoff effects. If we include excited states using the summed operator insertion method, we achieve better agreement with the radii from phenomenology. The anomalous magnetic moment of the muon can be measured and predicted to very high precision. The theoretical prediction of the anomalous magnetic moment receives contribution from strong, weak, and electro-magnetic interactions, where the hadronic contributions dominate the uncertainties. A persistent 3σ tension between the experimental determination and the theoretical calculation is found, which is considered to be an indication for physics beyond the Standard Model. We present a calculation of the connected part of the hadronic vacuum polarisation using lattice QCD. Partially twisted boundary conditions lead to a significant improvement of the vacuum polarisation in the region of small momentum transfer, which is crucial in the extraction of the hadronic vacuum polarisation.