# Renormalization group analysis of the hyperbolic sine-Gordon model -- Asymptotic freedom from cosh interaction --

## TL;DR

This paper performs a renormalization group analysis of the two-dimensional hyperbolic sine-Gordon model, demonstrating its asymptotic freedom and a non-renormalization property of the parameter t.

## Contribution

It derives RG equations for the sinh-Gordon model using two methods and shows the model's asymptotic freedom and a vanishing beta function for t.

## Key findings

- The interaction parameter α flows to zero at high energies.
- The beta function for t is zero in two dimensions.
- The RG equations are consistent across different regularization methods.

## Abstract

We present a renormalization group analysis for the hyperbolic sine-Gordon (sinh-Gordon) model in two dimensions. We derive the renormalization group equations based on the dimensional regularization method and the Wilson method. The same equations are obtained using both these methods. We have two parameters $\alpha$ and $\beta\equiv \sqrt{t}$ where $\alpha$ indicates the strength of interaction of a real salar field and $t=\beta^2$ is related with the normalization of the action. We show that $\alpha$ is renormalized to zero in the high-energy region, that is, the sinh-Gordon theory is an asymptotically free theory. We also show a non-renormalization property that the beta function of $t$ vanishes in two dimensions.