# Exact Renormalization Group and Sine Gordon Theory

## TL;DR

This paper applies the exact renormalization group method to analyze the flow of quantities in field theories, simplifying calculations of beta functions and c-functions, and confirming results in Sine-Gordon and phi^4 theories, including entanglement entropy.

## Contribution

It introduces a perturbative evolution operator approach to the exact renormalization group, simplifying calculations and reproducing known results in two and four dimensions.

## Key findings

- Simplified calculation of beta functions in field theories.
- Reproduced known c-function results in Sine-Gordon theory.
- Confirmed entanglement entropy results consistent with holographic calculations.

## Abstract

The exact renormalization group is used to study the RG flow of quantities in field theories. The basic idea is to write an evolution operator for the flow and evaluate it in perturbation theory. This is easier than directly solving the differential equation. This is illustrated by reproducing known results in four dimensional $\phi^4$ field theory and the two dimensional Sine-Gordon theory. It is shown that the calculation of beta function is somewhat simplified. The technique is also used to calculate the c-function in two dimensional Sine-Gordon theory. This agrees with other prescriptions for calculating c-functions in the literature. If one extrapolates the connection between central charge of a CFT and entanglement entropy in two dimensions, to the c-function of the perturbed CFT, then one gets a value for the entanglement entropy in Sine-Gordon theory that is in exact agreement with earlier calculations (including one using holography) in arXiv:1610.04233.