Effective models of strong and electroweak interactions are extensively applied in particle physics phenomenology, and in many instances can compete with large-scale numerical simulations of Standard Model physics. These contexts include but are not limited to providing indications for phase transitions and the nature of elementary excitations of strong and electroweak matter. A precondition for obtaining high-precision predictions is the application of some advanced functional techniques to the effective models, where the sensitivity of the results to the accurate choice of the input parameters is under control and the insensitivity to the actual choice of ultraviolet regulators is ensured. The credibility of such attempts ultimately requires a clean renormalization procedure and an error estimation due to a necessary truncation in the resummation procedure.

In this concise primer we discuss systematically and in sufficient technical depth the features of a number of approximate methods, as applied to various effective models of chiral symmetry breaking in strong interactions and the BEH-mechanism of symmetry breaking in the electroweak theory. After introducing the basics of the functional integral formulation of quantum field theories and the derivation of different variants of the equations which determine the n-point functions, the text elaborates on the formulation of the optimized perturbation theory and the large-N expansion, as applied to the solution of these underlying equations in vacuum. The optimisation aspects of the 2PI approximation is discussed. Each of them is presented as a specific reorganisation of the weak coupling perturbation theory. The dimensional reduction of high temperature field theories is discussed from the same viewpoint. The renormalization program is described for each approach in detail and particular attention is paid to the appropriate interpretation of the notion of renormalization in the presence of the Landau singularity. Finally, results which emerge from the application of these techniques to the thermodynamics of strong and electroweak interactions are reviewed in detail.

First self-contained primer on this topic Authored by leading specialists in the field Contains detailed derivations of results

Autorentext

Tamás Sándor Biró born at March 22, 1956 in Budapest I was half a year old at the time of the Hungarian Revolution. It prevented my parents from trying to make their living in the West, unlike many other fellow Hungarians did. I studied physics at Eotvos University from 1975 til 1980, and then received a diploma (equivalent to MSc) in physics and biophysics. I achieved my PhD degree at the same university (dr.rer.nat.) in 1982 based upon a research in theoretical heavy ion physics, about strange quark production in quark-gluon plasma, supervised and guided by Prof. Zimanyi. I continued my carreier as researcher at the Central Research Institute for Physics, Budapest, at the Niels Bohr Institute, Copenhagen, at the GSI, Darmstadt until 1984. Then I took an opportunity in 1985 to join to the Institute fur Theoretische Physik at the University of Giessen, Germany, where I stayed until my return to Budapest in 1994. Habilitation in theoretical physics in 1991, and an academic DSc degree in Hungary in 1994 signed some academic qualication of my eorts. I suggested the color rope model for describing the early phase of ultrarelativistic heavy ion collisions in 1983, took part in studies of chaotic dynamics in strongly interacting non-abelian gauge elds (1991-1999), then developed more and more a feeble for statistical physics of complex systems. Since 2005 I was active on studying and improving non-extensive thermodynamics. Since 2013 I am vice director of the Institute for Particle and Nuclear Physics, the half of theWigner Research Centre for Physics, in Budapest. I had the pleasant opportunity to collaboratewith colleagues spread over the world: Duke University in North Carolina, US, Bergen Universityin Norway, University of Cape Town and of Johannesburg, South Africa, Central China NormalUniversity in Wuhan, China, Yukawa Institute in Kyoto, Japan and some other places. I am alsotrying to help the research communities with editorial work: Acta Physica Hungarica, followed bythe EPJ A (Hadrons and Nuclei), where my term as Editor-in-Chief just ends soon (September30, 2018).By now we are witnessing a restructuring of the Hungarian research support landscape, byestablishing a new Ministry for Innovation and Techology and attaching the research institutenetwork to this ministry (formerly maintained by the Hungarian Academy of Science). Lookingforward to new challenges with new exciting possibilities for the research activity here.

Inhalt

Effective Theories From Nuclear to Particle Physics.- Finite Temperature Field Theories: Review.- Divergences in the Perturbation Theory.- Optimized Perturbation Theory.- The Large-N Expansion.- Dimensional Reduction and Infrared Improved Treatment of Finite Temperature Transitions.- Thermodynamics of Strong Matter.- Finite Temperature Restoration of the Brout-Englert-Higgs Effect.- The Spectral Function.- Computation of the Basic Diagrams.- Integrals Relevant for Dimensional Reduction.