Tunneling with physics-informed RG flows in the anharmonic oscillator

1) Explores the use of field redefinitions and the non-perturbative RG to capture non-perturbative physics.

2) Introduces a new observable that seems more accessible numerically.

3) The results suggest that the current approach and approximation can reveal the key aspects of the weak coupling regime.

4) Numerical advances in solving the flow equations for the anharmonic oscillator.

1) The relation between the mean composite field and the mean fundamental field is not clearly defined.

2) The important step to arrive at the expression for the new observable is obscure.

3) It is not clear at this stage whether reconstructing the map between the fields is essential to obtain the desired observables.

This work tests non-perturbative renormalization group techniques with an application to the anharmonic oscillator. In particular, scale-dependent field redefinitions are used to simplify the form of the flowing action. Results are then obtained in the first order of the derivative expansion.

At present, the paper is not clear in some aspects, and therefore these points need to be addressed by the authors. I list my specific comments below.

1. Regarding field redefinitions, the composite field $\hat{\phi}[\hat{\varphi}]$ is coupled to a source to arrive at the effective action $\Gamma_\phi[\phi]$. However, in equation (10) a relation $\phi[\varphi]$ is referred to, and it is not clear to me how this relation is defined. In particular, any expectation $\phi = \langle \hat{\phi} \rangle$ value depends not only on the operator $\hat{\phi}$, but also on which field is coupled to the source. Is $\phi[\varphi] = \langle \hat{\phi} \rangle $ when the source is coupled to the fundamental field $\hat{\varphi}$ or to the composite field $\hat{\phi}[\hat{\varphi}]$? My confusion is made worse when I try to understand equation (14); I'm not sure where this comes from given that $\dot{\phi}$ is not the derivative of the mean field but the expectation value of the derivative of the fundamental field. Can one derive (14) from some other definition of $\phi[\varphi]$, or is (14) in fact the definition?

2. Because the definition of the map between mean fields is obscure, it is hard to access the approximation (16). Is there some limiting case where (16) could hold with an equality sign?

3. I'm unsure where equation (35) comes from. It is key to defining the alternative observable evaluated by the authors and hence obtaining the main result of the work. I understand that, according to the authors, this does not rely on the approximation (16). This seems promising if it does not entail reconstructing the field transformation to compute the observables $a_{\rho_\phi}$. The authors refer to (35) as a ``leading order map''. The authors need to better guide the readers to arrive at (35), to understand its significance, and to elaborate on how higher orders could be obtained.

4. In relation to the last point, can the authors make it clear whether it is possible to calculate $a_{\rm inst}$ {\it without} knowing the details of the field transformation? Is this perhaps an open question?

5. Minor point. In equation (B7) there appears to be a typo: there should be a factor $Z_{\varphi}$ in the denominator rather than $Z_{\varphi}^{1/2}$, I believe.

To conclude, the paper touches on some important questions and suggests the power of field transformations in addressing truly non-perturbative problems. However, in its current form the paper is rather opaque in parts. I think this is somewhat understandable given the exploratory nature of the work. It seems to me like many steps are based on educated guesses.

My recommendation is that the paper could be published after addressing the points I have raised.

1. The relation between the mean fields appearing in equation (10) should be defined clearly.

2. The nature of the approximation (16) should be clarified.

3. The origin of equation (35) needs to be elaborated on and made clear to the reader.

4. The authors should make it clear whether it is possible to calculate a_inst without knowing the details of the field transformation.

5. A typo in (B7) should be fixed.

Ask for major revision

1- The paper exploits a recently developed method and a particular expansion (the ground state expansion) to simplify the study of strongly non-perturbative regimes, by means of Renormalization group flow techniques.

2- The suggestion of an original observable, in principle simpler to compute with large accuracy, in order to point out the non-perturbative features of the analysed problem.

3- The use of a precise numerical approach, crucial in the resolution of the flow equations.

1- The method does not seem much simpler than the standard approach. Again a couple of differential equations are involved.

2- While the results on the new observable look more promising, those on the energy gap, the usually studied observable, only slightly improve on the existing results.

3- The full improvement on the existing results is unclear, as the new observable was never determined within the older approach, and the error on these observable, associated with spurious regulator dependence is not analysed.

The purpose of the paper is certainly interesting and it contains original ingredients, introduced to reach the prefixed goal.

However, I find the comparison with the existing similar tests quite incomplete, so that the improvement achieved along this new route is less evident.

Therefore, in my opinion, the paper deserves to be reconsidered for publication, after some changes and improvements are implemented by the authors.

Below, a detail report is given.

The paper deals with the application of the recently developed physics-informed renormalization group (PIRG) flow to the quartically anharmonic double well quantum mechanical oscillator as a test to verify the capability of this approach to describe the instanton dominated regime, corresponding to small anharmonicity. In particular, the authors not only employ a novel flow but they also focus on the determination of a new parameter, indicated as $a_{inst}$, instead of the commonly studied energy gap $\Delta E$.

The purpose of the paper is certainly interesting and it contains original ingredients, introduced to reach the prefixed goal. However, I find the comparison with the existing similar tests quite incomplete, so that the improvement achieved along this new route is less evident.

Here is the list of the issues that, in my opinion, the authors should reconsider:

1) The novel technique, the PIRG, was developed in advance and the explanation of many details is addressed to a group of papers quoted in the references (many of which co-authored by some of the authors of the present paper). This makes the understanding of the paper rather heavy and leaves few points obscure.

    1.a) For instance the inequalities in Eq. (15) and the approximation in Eq. (16) should be supported by a direct measurement of $\Delta E$, which   is actually

reported in Figure 7 and shows only a marginal improvement with respect to the existing literature. Then, just after Eq. (16), the authors say that they prefer to devise the new observable $a_{inst}$, therefore leaving the reader still questioning about the accuracy of the approximation in Eq. (16).

  1.b)  Another example is given by Eq. (35), that I find very difficult to derive from Appendix B2, as suggested by the authors, whereas it is

crucial in determining Eqs. (37) and (40), which are at the heart of the main numerical test. In order to help the readers, I would suggest to expand on this point in the manuscript.

2) Again on the computation of $\Delta E$ in Appendix B1. We notice that the authors probe the instanton dominated region down to $\lambda_\phi =0.22$ (according to their Eq. (33)) and claim that such highly non-perturbative region, dominated by exponential decay, starts below $\lambda_\phi =0.4$. They also claim in the Introduction that none of the previous computations - available in the literature - could confirm the exponential decay and that, in addition, already at larger couplings, outside the exponential regime, sizeable deviations from the correct scaling with the couplings have been reported. Moreover, at the end of Section IV they state that, up until now, solving the Wetterich equation deep in the instanton regime ... was unsuccessful.

Now, according to the results shown in Figure 7, three numerical points taken from reference [7], lie below $\lambda_\phi =0.4$, therefore within the instanton regime. Moreover, in reference [7] one finds two determinations of $\Delta E$ below $\lambda_\phi =0.4$, obtained with the Wetterich equation, that, although with large errors, clearly show an exponential trend with the coupling, especially if the first order approximation in the derivative expansion gets compared with the LPA of the Wetterich equation.

Therefore, one must conclude that the statements of the authors quoted above are too strict with respect to the past results and too generous toward their own achievements. In fact, by looking at Figure 7, both the extension into the instanton regime and the actual determination of $\Delta E$ do not seem very much improved by their analysis which, in my opinion, would also require a more accurate analysis of the uncertainties related to the use of different regulators. In fact, according to [7], regulators different from the one reported in Figure 7, do further reduce the distance from the actual value of $\Delta E$. Additionally, unlike stated by the authors, results reported in Figure 7 from [7] for larger $\lambda_\phi$, seem in equal agreement with the actual $\Delta E$ as those from PIRG.

3) I understand that my criticism in 2) concerns a point on which the authors intend to further improve. However, even for the main result of the paper, i.e. Eq. (45), it is evident from figure 4 that their analysis involves values of the couplings down to $\lambda_\phi\simeq 0.3 $, which is not very different from the results collected in [7], so that the instanton dominated explored region is essentially the same. Moreover it would be extremely instructive to compare the result in Eq. (45) with the analogous number obtained by the standard (not PIRG) flow, possibly including the uncertainty coming from different regulators, to get a quantitative estimate of accuracy of this approach and substantial comparison with other analyses.

4) Some typos or mistakes to be corrected: (A) In the term proportional to $\lambda_\phi$ in the central term of Eq. (5.b), $1/\sqrt{2}$ is missing.
(B) In the list of references of papers that analysed this issue in the past, the paper by A.S. Kapoyannis and N. Tetradis Phys.Lett.A 276 (2000) 225-232
e-Print: hep-th/0010180 [hep-th] should be included. (C) The quotation - PT RG from [6] - in Figure 7, left and right, is wrong and it should be changed into [7], coherently with the caption. Also in the Appendix B1 text, when the authors mention - the data taken from [7], Table II -, as far as I understand, they intend to refer to Table 4 of [7].

In conclusion I think that the paper in the present form is not suitable for publication, but it should be reconsidered after some amendments and suggested improvements are properly taken into account.

As explained in detail in the full report , the major requested changes are:

1- A better explanation of the validity of Eq. (16) in the light of the results in Fig. 7

2- An explanation on the derivation of Eq. (35), and consequently of (37) , (40)

3- A more accurate comparison of the results in Fig. 7 which includes (at least indicatively) the error associated with the use of different regulators.

4- If possible, the determination of $a_{inst}$ with the standard technique of FRG (not PIRG), again including the uncertainty associated with different regulators.

5- The revision of the statements concerning the great progress of this approach with respect to the old one, as it is not so far, sufficiently corroborated.

6- the correction of the typos (A), (B), (C), illustrated in point 4) of the above report.

Ask for major revision