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.
