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

Manuel Pavon Valderrama

arXiv:1901.10398·nucl-th·May 22, 2019

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

Manuel Pavon Valderrama

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TL;DR

This paper critically examines recent claims of inconsistencies in renormalizing singular potentials in effective field theory, concluding that these supposed issues do not actually occur.

Contribution

It clarifies and refutes two claimed inconsistencies in the renormalization process for singular potentials, supporting the validity of current methods.

Findings

01

No mismatch between $$ expansion and renormalization.

02

Non-perturbative renormalization is feasible for repulsive singular interactions.

03

The claimed inconsistencies are resolved or do not exist.

Abstract

I critically discuss two of the potential inconsistencies pointed out in the recent manuscript by Epelbaum, Gasparyan, Gegelia and Meissner, published in Eur. Phys.J. A54, 186 (2018). The potential inconsistencies are: (i) a possible mismatch between the $ℏ$ expansion of the scattering amplitude and renormalization, (ii) an impossibility of "non-perturbative renormalization" to deal with repulsive singular interactions. The conclusion is that these inconsistencies do not happen.

Equations4

V_{L} (r) \to V_{L} (r; R_{c}) = V_{L} (r) (1 - e^{- (r / R_{c})^{2}}),

V_{L} (r) \to V_{L} (r; R_{c}) = V_{L} (r) (1 - e^{- (r / R_{c})^{2}}),

δ (r) \to δ (r; r_{c}) = \frac{1}{π ^{3/2} R _{c}^{3}} e^{- (r / R_{c})^{2}},

δ (r) \to δ (r; r_{c}) = \frac{1}{π ^{3/2} R _{c}^{3}} e^{- (r / R_{c})^{2}},

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Full text

11institutetext: School of Physics and Nuclear Energy Engineering,

International Research Center for Nuclei and Particles in the Cosmos and

Beijing Key Laboratory of Advanced Nuclear Materials and Physics,

Beihang University, Beijing 100191, China, 11email: [email protected]

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

Manuel Pavon Valderrama 11

(Received: date / Revised version: date)

[ I critically discuss two of the potential inconsistencies pointed out in the recent manuscript by Epelbaum, Gasparyan, Gegelia and Meißner, published in Eur. Phys.J. A54, 186 (2018); the conclusion is that these inconsistencies do not happen.]

In a recent manuscript Epelbaum et al. Epelbaum et al. (2018) discuss non-perturbative renormalization within effective field theory (EFT). The term refers mostly to renormalization as applied in a purely non-perturbative context, but tangentially includes the mixture of perturbative and non-perturbative renormalization advocated, for instance, by Nogga et al. Nogga et al. (2005). The idea is that for really hard cut-offs this type of renormalization will lead to a series of (apparent) inconsistencies. Two possible inconsistencies discussed in detail are: (i) a possible mismatch between the $\hbar$ expansion of the scattering amplitude and renormalization, (ii) an impossibility of non-perturbative renormalization to deal with repulsive singular interactions. Here I review these inconsistencies.

First, I will discuss the $\hbar$ mismatch. It can be explained as follows: we consider a scattering amplitude $T$ in a theory with a cutoff $\Lambda$ . The amplitude $T$ is generated from the iteration of the leading order potential $V=C_{0}+C_{2}(p^{2}+p^{\prime 2})$ . $T$ has a well-defined $\Lambda\to\infty$ limit, from which we are tempted to conclude that it has been renormalized in agreement with EFT principles. But when we consider its $\hbar$ expansion, i.e. $T=T^{[0]}+\hbar\,T^{[1]}+\mathcal{O}(\hbar^{2})$ , a closer inspection reveals that $T^{[1]}$ contains a linear divergence. That is, according to Ref. Epelbaum et al. (2018) the amplitude $T$ has not been renormalized in agreement with the EFT principles (notice that this is controversial because $T^{[1]}$ is not observable, but let us concede that there is an inconsistency in what follows). The diagnosis of Ref. Epelbaum et al. (2018) is that the problem lies in a misguided insistence in taking the $\Lambda\to\infty$ limit, which is not necessary within the EFT framework and can potentially lead to more harm than good. Though the inconsistency might indeed be there, this diagnosis is incorrect owing to a subtlety in the renormalization process that is regularly ignored in cutoff regularization (because, except for formal settings as this one, it is generally of no consequence). This subtlety is the existence of two contributions to the EFT counterterms, one that contains physical information and one that doesn’t. I refer to the second type of contribution as redundant counterterms (RC), which are discussed in Ref. Valderrama (2016). The RCs are there to cancel the residual cutoff dependence, but can be ignored in practice when $\Lambda\to\infty$ . RCs are however regularly included in EFTs using power divergence subtraction (PDS) regularization Kaplan et al. (1998), for instance. In cutoff EFT a trivial way to remove the divergences in the $\hbar$ expansion is to explicitly include the RCs, which for the $T$ matrix discussed here take the form $V^{R}=\sum_{n\geq 2}C^{R}_{2n}(p^{2n}+p^{\prime 2n})$ , with $C^{R}_{2n}=\sum_{k}\hbar^{k}\,{C^{R}_{2n}}^{[k]}$ . The number of RCs is infinite, but this is inconsequential because they are not observable and carry no physical information; instead, they are an analysis tool. Concrete calculations for this redundant potential are easy to perform with PDS, where explicit solutions exist for the $C_{2n}^{R}$ and where it can be explicitly shown that no positive power of $\Lambda_{\rm PDS}$ (i.e. the PDS cutoff) appear in the $\hbar$ expansion of $T$ . With a sharp cutoff in momentum space, calculations are more involved and cannot be solved in closed form, yet it is trivial to check that ${C^{R}_{4}}^{[1]}$ removes the $\hbar$ inconsistency.

Second, I will discuss the non-perturbative renormalization of the interesting toy model of Epelbaum et al. Epelbaum et al. (2018), which contains a repulsive singular interaction at leading order. The potential in this toy model contains a long- and short-range piece, $V(r)=V_{L}+V_{S}$ , with $V_{L}$ singular and repulsive and $V_{S}$ attractive. Despite the singular nature of $V_{L}$ , the full potential $V$ is regular, where the details of the toy model can be consulted in Ref. Epelbaum et al. (2018). As shown by explicit calculations $V_{L}$ is not renormalizable if treated purely non-perturbatively in combination with a contact-range interaction Epelbaum et al. (2018) (in agreement with Ref. Valderrama and Arriola (2006)). Here I stress that the inconsistency is not in the renormalization process, but in the power counting employed: EFT requires to iterate according to power counting, but repulsive singular interaction likely entail a demotion of the contact interactions, not a promotion, see the discussion below (or see Ref. Birse (2006) for a different opinion). The authors of Ref. Epelbaum et al. (2018) put into question that a mixture of non-perturbative and perturbative methods could reproduce the fundamental theory.

This viewpoint, though sensible, is premature: here I present calculations for this toy model within a mixture of perturbative and non-perturbative renormalization. For that I simply include the long range potential $V_{L}$ as the leading order of the calculation and include contact interactions according to their power counting, which I determine by adapting the ideas of Ref. Valderrama and Phillips (2015). The outcome is that $V_{L}$ will be counted as $Q^{-1}$ (to justify its iteration), while the contact-range couplings will be demoted by one order with respect to naive dimensional analysis, i.e. $C_{2n}(p^{2n}+p^{\prime 2n})$ enters at order $Q^{2n+1}$ . Concrete calculations of the phase shifts are shown in Fig. 1, where we can see that the perturbative expansion indeed converges well. We regularize the long-range potential with a Gaussian regulator in coordinate space

[TABLE]

while for the contact-range potential we regularize the Dirac-delta as

[TABLE]

plus similar expressions for its derivatives, where a local contact-range potential is used: $C_{2n}q^{2n}$ . The regularization is slightly different than in the original manuscript, but it is certainly simpler and nonetheless equivalent. The details of the calculation are analogous to those of Ref. Pavon Valderrama (2011), but extended to higher orders. The $C_{2n}$ couplings are determined by fitting to the toy model phase shifts in the $20-80\,{\rm MeV}$ ( $80-200\,{\rm MeV}$ ) range for $\nu=1,3$ ( $\nu=5,7$ ). Calculations are shown for the cutoffs $R_{c}=0.3,0.6,1.2$ and $1.8\,{\rm fm}$ up to order $Q^{7}$ ( $\rm N^{8}LO$ ) in the EFT expansion. The conclusion is that the standard EFT approach of Ref. Nogga et al. (2005) is perfectly able to describe the physics of the toy model of Epelbaum et al. Epelbaum et al. (2018). In addition it improves over the proposal of Ref. Epelbaum et al. (2018) (namely, a purely non-perturbative approach with a judiciously chosen cutoff), in the sense that there are no strong restrictions on the cutoff (besides the numerical ones, $R_{c}\geq 0.3\,{\rm fm}$ in this case), which can be taken harder than the breakdown scale if one wishes to. Notice that even though the existence of the $R_{c}\to 0$ limit has not been proven, this is not a necessary condition for the present approach to be useful.

Acknowledgements.

This work is partly supported by the National Natural Science Foundation of China under Grants No.11522539, No.11735003, the Fundamental Research Funds for the Central Universities and the Thousand Talents Plan for Young Professionals.

Bibliography8

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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