Effective Field Theories

TL;DR

This paper discusses methods for deriving effective field theories at low energies, including analytical, numerical, and neural network approaches, with applications to gauge theories and phase structure analysis.

Contribution

It introduces a neural network strategy for optimizing kernels in effective field theory computations and discusses special features of blockspins in nonabelian gauge theories.

Findings

Neural network approach for kernel optimization

Methods for deriving effective actions from fundamental theories

Analysis of phase structure via the constraint effective potential

Abstract

Effective field theories encode the predictions of a quantum field theory at low energy. The effective theory has a fairly low ultraviolet cutoff. As a result, loop corrections are small, at least if the effective action contains a term which is quadratic in the fields, and physical predictions can be read straight from the effective Lagrangean. Methods will be discussed how to compute an effective low energy action from a given fundamental action, either analytically or numerically, or by a combination of both methods. Basically,the idea is to integrate out the high frequency components of fields. This requires the choice of a "blockspin",i.e. the specification of a low frequency field as a function of the fundamental fields. These blockspins will be the fields of the effective field theory. The blockspin need not be a field of the same type as one of the fundamental fields, and it may be composite. Special features of blockspins in nonabelian gauge theories will be discussed in some detail. In analytical work and in multigrid updating schemes one needs interpolation kernels $\A$ from coarse to fine grid in addition to the averaging kernels $C$ which determines the blockspin. A neural net strategy for finding optimal kernels is presented. Numerical methods are applicable to obtain actions of effective theories on lattices of finite volume. The constraint effective potential) is of particular interest. In a Higgs model it yields the free energy, considered as a function of a gauge covariant magnetization. Its shape determines the phase structure of the theory. Its loop expansion with and without gauge fields can be used to determine finite size corrections to numerical data.