Reply to "Comment on "How (not) to renormalize integral equations with singular potentials in effective field theory"

TL;DR

This paper responds to criticism of a previous work on renormalizing integral equations with singular potentials in effective field theory, clarifying the authors' approach and addressing concerns raised.

Contribution

The authors provide a clarification and defense of their renormalization method in effective field theory against recent criticism.

Findings

Clarifies the authors' renormalization approach.

Addresses specific criticisms raised by Pavon Valderrama.

Reinforces validity of their method in effective field theory.

Abstract

We offer a brief response to the criticism put forward by Pavon Valderrama about our recent paper on "How (not) to renormalize integral equations with singular potentials in effective field theory".

Equations

$$T=T_{-1}{ϵ}^{-1}+T_1{ϵ}^1+T_3{ϵ}^3+\mathcal{O}({ϵ}^5),$$

$$T$$

Full text

The criticism raised by Pavon Valderrama in Ref. Valderrama:2019yiv concerns two issues summarized in the first paragraph of that paper. Below, we address both issues and show that the arguments of Ref. Valderrama:2019yiv are of no relevance for the conclusions reached in our paper Epelbaum:2018zli .

The first issue addressed by Pavon Valderrama concerns our statement about the inconsistency of taking the cutoff limit $\Lambda\to\infty$ in non-perturbative expressions for the scattering amplitude without having subtracted the relevant counterterms beforehand. The author does not point out any specific flaw in our arguments but simply declares that our “diagnosis is incorrect” because “redundant counterterms (RC)”, apparently discussed in Ref. Valderrama:2016koj 111We failed to find the definition of “redundant counterterms” in Ref. Valderrama:2016koj ., “can be ignored in practice when $\Lambda\to\infty$”. He then claims that “RCs are, however, regularly included in EFTs using the power divergence subtraction scheme (PDS) regularization Kaplan:1998we ”. Notice that Ref. Kaplan:1998we actually uses dimensional regularization222Pavon Valderrama apparently confuses the regularization and renormalization schemes. and therefore all loop integrals appearing in pionless EFT are finite in four space-time dimensions, i.e. the scattering amplitude considered in Ref. Kaplan:1998we has no residual cutoff dependence. This example is thus of no relevance for our paper. While these imprecise formulations and misleading statements make it difficult to follow the arguments of Ref. Valderrama:2019yiv , we have a feeling that the author has misinterpreted the statement in Epelbaum:2018zli he is objecting to. We do by no means claim any inconsistency of removing the cutoff in non-perturbative expressions for the scattering amplitude provided one follows the steps: (i) calculate the amplitude regularized with a cutoff $\Lambda$, (ii) subtract all ultraviolet divergences in loop integrals emerging from iterations of the integral equation and (iii) take the limit $\Lambda\to\infty$ afterwards. We do, however, claim that performing (iii) without having carried out step (ii) generally leads to results which cannot be regarded as renormalized and are incompatible with the principles of EFT, even if a finite limit $\Lambda\to\infty$ happens to exist for the amplitude. For pionless EFT, the algorithm specified above can indeed be easily implemented in the non-perturbative environment. In particular, Eq. (8) of our paper Epelbaum:2018zli contains all counterterms needed to remove the divergences from all terms in the expansion of the amplitude in powers of $\hbar$ in Eq. (6). We are, however, not aware of any calculations in spin-triplet nucleon-nucleon channels based on a non-perturbative treatment of the one-pion exchange potential, where the step (ii) could be carried out (except for the approach proposed in Ref. Epelbaum:2012ua ).

Regarding the second issue, the author of Ref. Valderrama:2019yiv has indeed succeeded to obtain a good description of the toy-model phase shifts based on a perturbative inclusion of contact interactions for several values of the cutoff parameter at the cost of fitting up to four adjustable parameters.333Still, no evidence of the convergence with respect to the coordinate-space cutoff $R_{c}$ and thus of the existence of the limit $R_{c}\to 0$ is provided for the phase shifts outside of the fitted region. However, as pointed out in our paper Epelbaum:2018zli , the large difference between the full phase shifts and the leading-order (LO) ones suggests that the results obtained from a perturbative inclusion of higher-order terms are strongly dependent on the employed unitarization procedure, thus being model-dependent. While repeating the numerical analysis of Ref. Valderrama:2019yiv goes beyond the scope of this comment, we can illustrate the origin of the problem using the following simple considerations. We start with assuming that the perturbative expansion of Ref. Valderrama:2019yiv is indeed convergent, i.e. the full scattering amplitude is well approximated by the first several terms in the perturbative expansion

[TABLE]

where we introduced a parameter $\epsilon$ to keep track of orders in small parameters (we set $\epsilon=1$ after the relative orders are established). As $T_{-1}$ is a solution to the LO integral equation, it is unitary by construction. Thus, $1/T-1/T_{-1}$ has to be real. Expanding this expression in powers of $\epsilon$ and demanding that each term is real, one can express the imaginary parts of $T_{1}$ and $T_{3}$ in terms of their real parts and the LO amplitude $T_{-1}$. Modulo higher-order corrections, the real parts of $T_{1}$ and $T_{3}$ can then be uniquely determined by demanding that the real and imaginary parts of the toy-model amplitude are reproduced. According to Fig. 1 of Ref. Valderrama:2019yiv , the phase shift at e.g. $k_{\rm cm}=0.22$ GeV is accurately described at order $\nu=3$ for the smallest considered cutoff $R_{\rm c}=0.3$ fm. Using the corresponding numerical values of the LO and the full phase shifts $\delta_{\rm LO}=-63.61^{\circ}$ and $\delta=-14.46^{\circ}$, we obtain two solutions for the real parts of $T_{1}$ and $T_{3}$ corresponding to the following expansion of the amplitude:

[TABLE]

Obviously, none of the expansions shows any sign of convergence. Moreover, while both expressions do exactly reproduce the full, unitary amplitude $-(4\pi/m)(1/T)=-3.90-i$ when truncated at order $\epsilon^{3}$, none of the expressions for the amplitude is approximately unitary when truncated at next-to-leading order $\epsilon^{1}$. Specifically, we obtain $-(4\pi/m)(1/T_{\rm NLO})=-1.11-0.26\,i$ for the first solution, while $-(4\pi/m)(1/T_{\rm NLO})=0.03-0.71\,i$ for the second one. Using different unitarization prescriptions one can, in fact, obtain a broad range of phase shifts at NLO including the ones shown in Fig. 1 of Ref. Valderrama:2019yiv . It is important to stress that the aim of an EFT is, however, not to describe the data at any price but rather to provide a systematic approach with controlled accuracy and reliable error estimations.

Acknowledgments

This work was supported in part by the Georgian Shota Rustaveli National Science Foundation (Grant No. FR17-354), by DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”) and the BMBF (Grant No. 05P18PCFP1). Further support was provided by the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (grant no. 2018DM0034) and by VolkswagenStiftung (grant no. 93562).