^{3}

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

Effective potentials improved by a renormalization group (RG) equation are widely applied in particle physics. In Refs. [

There has been a great deal of research into the RG improvement of the effective potential since a study by Coleman and Weinberg [

If multiple scalar fields are introduced, the analysis of the RG improved effective potential is complicated because the masses appearing in the logarithms depend on multiple classical background fields such as

In this paper, extending the method of Ref. [

This paper is organized as follows: In

In this section, we clarify the logarithmic structure of the effective potential based on Ref. [

We suppose that this model has ^{1}

Next, we shift the fields (

From the rewritten Lagrangian (

Moreover, since it is inconvenient for the mass matrix not to be diagonal, we rotate the mass matrix by introducing new states (

For later discussion, the coordinate

From now on, the mass eigenvalues and the effective potential are written with the polar coordinate

At this stage, we can replace the three mass parameters (

This information is so important that using these parameters we can write down the effective potential at the

Let us explain why the

As is well known, since the

Finally, by summing up

In this expression the power of

Next, we consider the choice of renormalization scale. As is well known, the effective potential satisfies the RG equation

These specific

However, because of

Since this choice leads to

Here, if we assume

If

We notice that the term of

We comment on the variables of the effective potential. Originally, the effective potential has three variables ^{2}

Since the above prescription is correct only in the case of

We specifically calculate the RG improved effective potential by the method constructed in

In order to obtain

So we can easily obtain

As mentioned above, now

The logarithm of

A contour plot of

A 3D plot of the RG improved effective potential at the leading log order divided by the initial renormalization scale

In this section we consider the massive theory in a two real scalar model. In particular, we treat the effective potential causing spontaneous symmetry breaking. The procedure for the construction of the RG improved effective potential is the same as the previous method. We solve Eq. (

Squaring both sides of

We can obtain the solution

Since we solve the quadratic equation, there are two solutions for

Although it is difficult to analytically prove whether either solution satisfies the condition or not, by using the initial values as inputs in the following subsections we confirm numerically the following results:

Therefore we adopt the solution

Since we get the solution

The expression is provided at leading log order as

In the following subsections, we consider two situations for inputting the initial value of the renormalization scale. First, taking

Since we set the initial condition on the vacuum in this subsection, we derive the stationary condition for the effective potential. Introducing a convenient notation for the mass parameter and quartic coupling constant:

We calculate the stationary conditions for the effective potential:

From

Combining this condition and

Substituting this

Using Eqs. (

Taking [

The logarithm of the ratio of

Left: The dot-dashed green, dotted orange, and solid blue lines correspond to the RG improved effective potential in

We give a more complete discussion for the logarithmic perturbative expansion. As explained above, there are regions in which the logarithm is beyond

In this subsection we impose the initial condition at a high-energy scale and gradually decrease the renormalization scale to a scale around

The logarithm of the ratio of

Left: The logarithm of the ratio of

In order to avoid a large logarithm, we should modify the RG improved effective potential for the low-energy scale. The way to modify the RG improvement is to utilize the decoupling theorem. In the present case, since

Additionally, we expand the

In this expression we see that

Note that because there is no contribution to the wavefunction renormalization in this model, the classical background fields do not change:

Since we use the parameters in the low-energy effective theory below

Hence we can get the

We notice that the effect of the heavy field disappears from the RG equation in Eqs. (

Let us consider a decoupling point at which the theory is separated into the full theory and the low-energy effective theory. From the left panel of

Left: The running of the quartic coupling constants is solved. The solid blue, dotted orange, and dot-dashed green lines denote the running of

Left: The 3D plot of the RG improved effective potential is evaluated as a function of

In this paper we have studied the RG improvement of the effective potential in a two real scalar system. In

There are three features in this method. First, we do not need to change the choice of the renormalization scale beyond the leading log order. This is because, since we analyze the logarithmic structure of the effective potential at any loop order, the choice

Our method can be applied to other multiple scalar models. If multiple scalar fields are introduced in a model, one represents the classical background fields in terms of polar coordinates such as

We thank T. Morozumi and Y. Shimizu for reading our manuscript and giving useful comments.

Open Access funding: SCOAP

In this appendix, we provide the

^{1}In this paper, we assume that all the quartic coupling constants are comparable to each other

^{2}Moreover, note that although the dimensionless parameters