Renormalization group and effective potential: a simple non-perturbative approach

1 - Well written.
2 - Thorough comparison with relevant literature.
3 - Explicit computations.

1 - Lacking clear motivation for the main object under study.
2 - Manipulations leading to the main equation under study require additional substantiation.
3 - The utility of the method to future applications is unclear.

The author considers an effective potential $U_{DS}$ introduced by Shepard et al, which is an approximate solution to the exact RG and satisfies a second order differential equation. They propose a two-step integration procedure to approximate the solution to this equation, and demonstrate that implementing this procedure reduces error. They then proceed to analyze the resulting effective potential.

The author convincingly makes the point that their two-step integration procedure is an improvement on the one-step procedure, providing concrete calculations to support this claim.

However, this paper would greatly benefit by establishing a firmer initial motivation for analyzing $U_{DS}$. This object was introduced by Shepard et al as a “crude integration of the ERG equation” (see page 5 of the paper under review), which itself is an approximation of the effective potential to lowest order in the derivative expansion. In this reviewer’s opinion, the author should more convincingly address why this particular approximation is interesting and worth detailed study.

Relatedly, it would be helpful to have a clearer discussion of the sense in which this$~U_{DS}$ is approximate, since this point is so central to the paper. The author discusses the manner of deviation of equation (2) from the sharp cutoff ERG equation (equations (3)-(5) in various representations), but the effect of the deviation is unclear. Is there (for instance) an estimate for the error accrued in making the assumption that one can replace$~U’’(\phi,k)$ in equation (5) with its value at the lower integration limit?

Furthermore, this reviewer does not understand the author’s claim that the “requirement of convexity of the effective potential prohibits the study of symmetry broken phases”. The study of spontaneous symmetry breaking in the context of (e.g.) the derivative expansion of the effective potential has a long and famous history, going back to the work of Coleman and Weinberg. Even if spontaneous symmetry breaking is not studied in the current paper, the discussion should address how one might do so in the future.

1 - Clarify the motivations for a detailed study of $U_{DS}$.
2 - Explain the error in going from equation (5) to equation (2).
3 - Comment on the application to spontaneous symmetry breaking.

1- Incremental study of an arbitrary equation for the effective potential
2- No clear goal and motivation
2- Reference to the numerous studies of LPA missing

This manuscript is focused on the approximate solution of an Exact Renormalization Group (ERG) equation at the Local Potential Approximation (LPA), see Eq. (3). The latter is a PDE for the effective potential, which depends on the "classical field" phi and an RG scale Lambda.
Twenty-five years ago, Shepard et al. gave an approximate solution to this ERG equation to obtain a self-consistent equation for the renormalized (Lambda=0) effective potential.
Here, the author gives another approximate solution by splitting the integration range in two, and replacing in an ad hoc way the flowing potential by either its classical value (at the UV scale Lambda_0) or its renormalized value. After some more approximations, Eq. (11) is obtained and then studied at length in the rest of the paper.

My main issue with the manuscript is that there is no clear motivation as to why this should be an interesting equation to study, and what new information we can gain from it. Indeed, in the end, and as stated by the author himself in the conclusion, Eq. (11) is just an approximate solution to the LPA flow equation. Then, why not just study that equation? One issue of course is that the LPA has been studied at length in the last thirty years, with a variety of regulator functions, and therefore it is hard to see what new insight we would get from yet another study of the LPA.

Furthermore, the "difficulties" listed in the conclusion, such as "the requirement of convexity of the effective potential prohibits the study of symmetry broken phase" and "The massless limit of the effective potential [...] is troublesome [...] because the massless (or critical) theory should have a non-analytic effective potential. " are very well understood in the ERG context.
However, to correctly capture the behavior at criticality (non-analyticity) and in the broken-symmetry phase, it is well-known that one needs to: 1) be functional (i.e. not expand the effective potential); 2) solve the RG flow (i.e. keep a flowing effective potential in the RHS of the flow equation). Among the many possible references on the second aspect: Nuclear Phys. B 855 (2012) 854, Phys. Rev. E 94, 042136 (2016).

To summarize, this manuscript studies in detail an equation that is not able to capture some phenomena that are very well understood in the context of ERG (and leave that for future work). As such, I do not see how it could be published without major revisions and a thorough study of the literature.

1- Clear language;
2- Detailed computations;
3- Improvement on previous results from the literature;
4- Comparison between the results obtained in the paper with previous results from literature.

1- It is not presented a deeper study of the tricritical behavior of the theory;
2- The author doesn't adress theories with spontaneously bronken symmetries;
3- It is not discussed possible implications in adding more fields to the model.

The paper adress a interesting topic and presents an improvement in computing the nonperturbative effective potential. This is of interest to the field in question. Yet there are some questions that need to be clarified before publication.

The first is regard to the substitution of the integration variable by the lower integration limit in equation (5). The author argues that if one uses this limit, it will obtain the desired result while if one uses the upper integration limit it will obtain the one-loop aproximation. This is crucial to the paper since it consists of a improvement of this method.

Secondly, the author discusses the massless limit of this model and states that it is problematic using one method and not using the one developed in the paper. However, one expects that the computation of the effective potential for a classical massless theory dinamically breaks the symmetry, see (Sidney Coleman and Erick Weinberg, Phys. Rev. D 7, 1888 (1973)), (Steven Weinberg
Phys. Rev. D 13, 974 (1976)), (D. G. C. Mckeon
International Journal of Theoretical Physics volume 37, pages817–826 (1998)), (Huan Souza, L. Ibiapina Bevilaqua, and A. C. Lehum
Phys. Rev. D 102, 045004 (2020)). It would be interesting put into perspective the results from dynamical symmetry breaking and the ones developed in the paper.

Moreover, the paper is well written and gives a positive contribution to ongoing discussions of the field.

1- Needs clarifications about the procedure to identify equation (5) as equation (2).

2- It would be nice to discuss if it is possible to have dynamical symmetry break using this method and put the results from this method into perspective with the ones obtained via perturbation theory.