A real-time thermal field theoretical analysis of Kubo-type shear viscosity: Numerical understanding with simple examples

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

All one-loop scalar vertices in the effective potential approach

Using the one-loop Coleman-Weinberg effective potential, we derive a general analytic expression for all the derivatives of the effective potential with respect to any number of classical scalar fields. The result is valid for a renormalisable theory in four dimensions with any number of scalars, fermions or gauge bosons. This result corresponds to the zero-external momentum contribution to a general one-loop diagram with N scalar external legs. We illustrate the use of the general result in two simple scalar singlet extensions of the Standard Model, to obtain the dominant contributions to the triple couplings of light scalar particles under the zero external momentum approximation.

Light-cone gauge integrals: prescriptionlessness at two loops

The only calculations performed beyond one-loop level in the light-cone gauge make use of the Mandelstam-Leibbrandt (ML) prescription in order to circumvent the notorious gauge dependent poles. Recently we have shown that in the context of negative dimensional integration method (NDIM) such prescription can be altogether abandoned, at least in one-loop order calculations. We extend our approach, now studying two-loop integrals pertaining to two-point functions. While previous works on the subject present only divergent parts for the integrals, we show that our prescriptionless method gives the same results for them, besides finite parts for arbitrary exponents of propagators. (C) 2000 Elsevier B.V. B.V. All rights reserved.

Negative dimensional integration: Lab testing at two loops

Negative dimensional integration method (NDIM) is a technique to deal with D-dimensional Feynman loop integrals. Since most of the physical quantities in perturbative Quantum Field Theory (pQFT) require the ability of solving them, the quicker and easier the method to evaluate them the better. The NDIM is a novel and promising technique, ipso facto requiring that we put it to test in different contexts and situations and compare the results it yields with those that we already know by other well-established methods. It is in this perspective that we consider here the calculation of an on-shell two-loop three point function in a massless theory. Surprisingly this approach provides twelve non-trivial results in terms of double power series. More astonishing than this is the fact that we can show these twelve solutions to be different representations for the same well-known single result obtained via other methods. It really comes to us as a surprise that the solution for the particular integral we are dealing with is twelvefold degenerate.

Relativistic quark spin coupling effects in the nucleon electromagnetic form factors

We investigate the effect of different forms of relativistic spin coupling of constituent quarks in the nucleon electromagnetic form factors. The four-dimensional integrations in the two-loop Feynman diagram are reduced to the null-plane, such that the light-front wave function is introduced in the computation of the form factors. The neutron charge form factor is very sensitive to different choices of spin coupling schemes, once its magnetic moment is fitted to the experimental value. The scalar coupling between two quarks is preferred by the neutron data, when a reasonable fit of the proton magnetic momentum is found. (C) 2000 Elsevier Science B.V.

Loop Integrals in Three Outstanding Gauges: Feynman, Light-Cone, and Coulomb

We apply the negative dimensional integration method (NDIM) to three outstanding gauges: Feynman, light-cone, and Coulomb gauges. Our aim is to show that NDIM is a very suitable technique to deal with loop integrals, regardless of which gauge choice that originated them. In the Feynman gauge we perform scalar two-loop four-point massless integrals; in the light-cone gauge we calculate scalar two-loop integrals contributing to two-point functions without any kind of prescriptions, since NDIM can abandon such devices - this calculation is the first test of our prescriptionless method beyond one-loop order; and finally, for the Coulomb gauge we consider a four-propagator massless loop integral, in the split-dimensional regularization context. © 2001 Academic Press.

Cálculos de integrais de Feynman em teorias de campos

Pós-graduação em Física - IFT

Use of computer algebra in the calculation of Feynman diagrams

The increasing precision of current and future experiments in high-energy physics requires a likewise increase in the accuracy of the calculation of theoretical predictions, in order to find evidence for possible deviations of the generally accepted Standard Model of elementary particles and interactions. Calculating the experimentally measurable cross sections of scattering and decay processes to a higher accuracy directly translates into including higher order radiative corrections in the calculation. The large number of particles and interactions in the full Standard Model results in an exponentially growing number of Feynman diagrams contributing to any given process in higher orders. Additionally, the appearance of multiple independent mass scales makes even the calculation of single diagrams non-trivial. For over two decades now, the only way to cope with these issues has been to rely on the assistance of computers. The aim of the xloops project is to provide the necessary tools to automate the calculation procedures as far as possible, including the generation of the contributing diagrams and the evaluation of the resulting Feynman integrals. The latter is based on the techniques developed in Mainz for solving one- and two-loop diagrams in a general and systematic way using parallel/orthogonal space methods. These techniques involve a considerable amount of symbolic computations. During the development of xloops it was found that conventional computer algebra systems were not a suitable implementation environment. For this reason, a new system called GiNaC has been created, which allows the development of large-scale symbolic applications in an object-oriented fashion within the C++ programming language. This system, which is now also in use for other projects besides xloops, is the main focus of this thesis. The implementation of GiNaC as a C++ library sets it apart from other algebraic systems. Our results prove that a highly efficient symbolic manipulator can be designed in an object-oriented way, and that having a very fine granularity of objects is also feasible. The xloops-related parts of this work consist of a new implementation, based on GiNaC, of functions for calculating one-loop Feynman integrals that already existed in the original xloops program, as well as the addition of supplementary modules belonging to the interface between the library of integral functions and the diagram generator.

Higher-order calculations in manifestly Lorentz-invariant baryon chiral perturbation theory

This thesis is concerned with calculations in manifestly Lorentz-invariant baryon chiral perturbation theory beyond order D=4. We investigate two different methods. The first approach consists of the inclusion of additional particles besides pions and nucleons as explicit degrees of freedom. This results in the resummation of an infinite number of higher-order terms which contribute to higher-order low-energy constants in the standard formulation. In this thesis the nucleon axial, induced pseudoscalar, and pion-nucleon form factors are investigated. They are first calculated in the standard approach up to order D=4. Next, the inclusion of the axial-vector meson a_1(1260) is considered. We find three diagrams with an axial-vector meson which are relevant to the form factors. Due to the applied renormalization scheme, however, the contributions of the two loop diagrams vanish and only a tree diagram contributes explicitly. The appearing coupling constant is fitted to experimental data of the axial form factor. The inclusion of the axial-vector meson results in an improved description of the axial form factor for higher values of momentum transfer. The contributions to the induced pseudoscalar form factor, however, are negligible for the considered momentum transfer, and the axial-vector meson does not contribute to the pion-nucleon form factor. The second method consists in the explicit calculation of higher-order diagrams. This thesis describes the applied renormalization scheme and shows that all symmetries and the power counting are preserved. As an application we determine the nucleon mass up to order D=6 which includes the evaluation of two-loop diagrams. This is the first complete calculation in manifestly Lorentz-invariant baryon chiral perturbation theory at the two-loop level. The numerical contributions of the terms of order D=5 and D=6 are estimated, and we investigate their pion-mass dependence. Furthermore, the higher-order terms of the nucleon sigma term are determined with the help of the Feynman-Hellmann theorem.

Studies on Wilson loops, correlators and localization in supersymmetric quantum field theories

In this thesis we study at perturbative level correlation functions of Wilson loops (and local operators) and their relations to localization, integrability and other quantities of interest as the cusp anomalous dimension and the Bremsstrahlung function. First of all we consider a general class of 1/8 BPS Wilson loops and chiral primaries in N=4 Super Yang-Mills theory. We perform explicit two-loop computations, for some particular but still rather general configuration, that confirm the elegant results expected from localization procedure. We find notably full consistency with the multi-matrix model averages, obtained from 2D Yang-Mills theory on the sphere, when interacting diagrams do not cancel and contribute non-trivially to the final answer. We also discuss the near BPS expansion of the generalized cusp anomalous dimension with L units of R-charge. Integrability provides an exact solution, obtained by solving a general TBA equation in the appropriate limit: we propose here an alternative method based on supersymmetric localization. The basic idea is to relate the computation to the vacuum expectation value of certain 1/8 BPS Wilson loops with local operator insertions along the contour. Also these observables localize on a two-dimensional gauge theory on S^2, opening the possibility of exact calculations. As a test of our proposal, we reproduce the leading Luscher correction at weak coupling to the generalized cusp anomalous dimension. This result is also checked against a genuine Feynman diagram approach in N=4 super Yang-Mills theory. Finally we study the cusp anomalous dimension in N=6 ABJ(M) theory, identifying a scaling limit in which the ladder diagrams dominate. The resummation is encoded into a Bethe-Salpeter equation that is mapped to a Schroedinger problem, exactly solvable due to the surprising supersymmetry of the effective Hamiltonian. In the ABJ case the solution implies the diagonalization of the U(N) and U(M) building blocks, suggesting the existence of two independent cusp anomalous dimensions and an unexpected exponentation structure for the related Wilson loops.

Automated Equation Formulation for Causal Loop Diagrams

The annotation of Business Dynamics models with parameters and equations, to simulate the system under study and further evaluate its simulation output, typically involves a lot of manual work. In this paper we present an approach for automated equation formulation of a given Causal Loop Diagram (CLD) and a set of associated time series with the help of neural network evolution (NEvo). NEvo enables the automated retrieval of surrogate equations for each quantity in the given CLD, hence it produces a fully annotated CLD that can be used for later simulations to predict future KPI development. In the end of the paper, we provide a detailed evaluation of NEvo on a business use-case to demonstrate its single step prediction capabilities.

Supergraph techniques for D=3, N=1 broken supersymmetric theories

We enlarge the usual D = 3 N = 1 supergraph techniques to include the case of (explicitly or spontaneously) broken supersymmetric gauge theories. To illustrate the utility of these techniques, we calculate the two-loop effective potential of the SQED(3) by using the tadpole and the vacuum bubble methods. In these methods, to investigate the possibility of supersymmetry breaking, the superfields must be shifted by theta(alpha) dependent classical superfields (vacuum expectation values), what implies in the explicit breakdown of supersymmetry in the intermediate steps of the calculation. Nevertheless, after studying the minimum of the resulting effective potential, we find that supersymmetry is conserved, while gauge symmetry is dynamically broken, with a mass generated for the gauge superfield.