# Construction of RG improved effective potential in a two real scalar system

## TL;DR

This paper extends the method of constructing RG improved effective potentials to a two real scalar system, clarifying the logarithmic structure and addressing decoupling of heavy particles, thus enabling more accurate potential calculations in multi-scalar models.

## Contribution

It develops a method to construct RG improved effective potentials in a two scalar system, including decoupling effects, building on previous single scalar approaches.

## Key findings

- RG improved potential up to L-th-to-leading log order can be derived from L-loop potential and (L+1)-loop beta and gamma functions.
- Decoupling of heavy particles is incorporated when large logarithms appear at the renormalization scale.
- The method maintains a consistent renormalization scale choice even at higher log orders.

## Abstract

We study the improvement of effective potential by renormalization group (RG) equation in a two real scalar system. We clarify the logarithmic structure of the effective potential in this model. Based on the analysis of the logarithmic structure of it, we find that the RG improved effective potential up to $L$-th-to-leading log order can be calculated by the $L$-loop effective potential and $(L+1)$-loop $\beta$ and $\gamma$ functions. To obtain the RG improved effective potential, we choose the mass eigenvalue as a renormalization scale. If another logarithm at the renormalization scale is large, we decouple the heavy particle from the RG equation and we must modify the RG improved effective potential. In this paper we treat such a situation and evaluate the RG improved effective potential. Although this method was previously developed in a single scalar case, we implement the method in a two real scalar system. The feature of this method is that the choice of the renormalization scale does't change even in a calculation of higher leading log order. Following the our method one can derive the RG improved effective potential in a multiple scalar model.