Abstract
We use a discrete worldline representation in order to study the continuum limit of the one-loop expectation value of dimension two and four local operators in a background field. We illustrate this technique in the case of a scalar field coupled to a non-Abelian background gauge field. The first two coefficients of the expansion in powers of the lattice spacing can be expressed as sums over random walks on a d-dimensional cubic lattice. Using combinatorial identities for the distribution of the areas of closed random walks on a lattice, these coefficients can be turned into simple integrals. Our results are valid for an anisotropic lattice, with arbitrary lattice spacings in each direction.
Highlights
1.1 MotivationThe classical statistical approximation (CSA) is an approximate scheme to study in real time the dynamics of a system of fields, as an initial value problem
In the appendix A, we show how this formalism is modified on a finite lattice with periodic boundary conditions
We have applied a discrete version of the worldline formalism in order to obtain expressions for 1-loop expectation values in a lattice scalar field theory, in the presence of a non-Abelian gauge background
Summary
The classical statistical approximation (CSA) is an approximate scheme to study in real time the dynamics of a system of fields, as an initial value problem. We use this formalism in order to obtain useful expressions for 1-loop expectations values in a lattice field theory coupled to a (fixed) gauge background. The worldline formalism is well suited for this application because it enables one to have only gauge invariant objects at all stages of the calculation We use these expressions in order to study the limit of small lattice spacing. In order to keep things rather simple and focus on the main aspects of the worldline formalism, we consider a complex scalar field coupled to an external non-Abelian gauge field This background field is given once for all, and does not fluctuate.
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