Construction of a renormalization group improved effective potential in a two real scalar system

Abstract

We study the improvement of an effective potential by a 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 renormalization scale does not change even in a calculation of higher leading log order. Following our method one can derive the RG improved effective potential in a multiple scalar model.