Renormalizability and nonrenormalizability of nonlocal potentials

A. M. Gasparyan; E. Epelbaum; N. Jacobi

Renormalizability and nonrenormalizability of nonlocal potentials

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read When long-range forces are nonlocal, renormalization may require nonlocal counterterms or fail.

desk verdict A clean separable-toy-model proof that nonlocal long-range potentials can force nonlocal counterterms or fail renormalization at NLO, though the leap to realistic chiral EFT potentials is not established. read the letter →

arxiv 2509.08512 v1 pith:ADB2TPT6 submitted 2025-09-10 nucl-th

N. Jacobi , A. M. Gasparyan , E. Epelbaum This is my paper · ORCID

classification nucl-th

keywords chiraleffectivefieldtheoryrenormalizationnonlocalpotentialsseparabletoymodelsnucleon-nucleoninteractionpowercountingLippmann-Schwingerequationlong-rangeforces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether renormalizability in nuclear chiral effective field theory is a generic property of quantum-mechanical potentials or a specific feature of interactions derived from pion exchange. Working with separable toy models of the two-nucleon S-wave scattering, the authors show that a singular long-range piece that depends on the initial and final momenta separately, rather than on their difference, generates power-counting-breaking terms that local counterterms cannot absorb. In the model motivated by two-pion exchange at next-to-leading order, the only route to a finite, cutoff-independent amplitude is to put the same long-range structure already into the leading-order potential, which defeats the purpose of an EFT expansion. The upshot is that the locality of the long-range force is a load-bearing assumption behind the renormalizability of chiral nuclear EFT.

What carries the argument

The argument runs on separable toy potentials built from a small basis of form factors: the nonlocal regulator $F_\Lambda(p)=\Lambda^2/(\Lambda^2+p^2)$, the one-pion-exchange-like $F_{1\pi}(p)=M_\pi^2/(M_\pi^2+4p^2)$, and the two-pion-exchange-like $F_{2\pi}(p)=M_\pi^2/(M_\pi^2+p^2)$. The leading-order Lippmann-Schwinger equation is solved nonperturbatively; the NLO amplitude is then computed through $T_2=\bar R V_2 R$, and power-counting breaking is identified in the loop function $\Sigma_{2,21}(p_{\rm on})$, which scales as $\Lambda^3/\Lambda_b^3$ instead of $Q^2$. The counterterm $\delta V_0$ must carry the same separable structure as the term that produced the divergence, and because the $F_{2\pi}$ structure is absent from the LO basis, the corresponding power-counting-breaking piece cannot be absorbed without enlarging the leading-order potential.

What would settle it

Take the NLO two-pion-exchange-like model with $g_{2\pi}\neq 0$ and allow the counterterm $\delta V_0$ to be any separable kernel, not just one restricted to the leading-order operator content. If some choice of counterterm constants makes the renormalized NLO amplitude cutoff-independent to $O(Q^2)$, the claim that renormalization is impossible in this model is refuted; if no such choice exists, the claim is confirmed.

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Extended reading notes

Core claim

The central discovery is that nonlocality of the long-range part of a separable two-nucleon potential can introduce power-counting-breaking contributions proportional to positive powers of the cutoff $\Lambda$ that cannot be absorbed by counterterms made of the operator content already present at leading order. For the singular one-pion-exchange-like term, renormalization is still possible, but only with a long-range counterterm. For the two-pion-exchange-like term at NLO, renormalization is impossible unless a term of the same long-range structure is added to the leading-order potential; the paper reads this as evidence that renormalizability is not a universal property of arbitrary model potentials but a specific consequence of the local character of pion-exchange forces in chiral EFT.

Load-bearing premise

The whole conclusion rests on the assumption that these simple separable potentials capture the singular behavior of real pion exchange, and that a counterterm may only use the structures that already appear in the leading-order potential.

Editorial extensions

If this is right

A chiral-EFT calculation that regulates the long-range part nonlocally must verify that its NLO power-counting-breaking terms are absorbable by local counterterms; the paper shows this is not automatic.
The sufficient criterion of local long-range interactions used in the renormalizability proofs behaves like a necessary condition in this toy class: violating it changes the counterterm structure or destroys renormalizability.
A regular, non-singular nonlocal long-range interaction can stay renormalizable, so nonlocality alone is not the problem; singularity and nonlocality together are.
Phenomenological nonlocal potentials cannot be assumed to inherit renormalizability from chiral EFT; that property is structural, not generic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If realistic chiral potentials turn out to contain nonlocal long-range pieces beyond the toy-model basis, the same mechanism could generate power-counting-breaking terms at higher orders, so the published renormalizability proof may need to be extended rather than applied unchanged.
Repeating the analysis with a Gaussian regulator would test whether the conclusion is tied to power-law cutoffs or holds for the whole regulator class admitted by the bounds in the paper.
The failure mode suggests a possible connection to EFTs that require an infinite tower of counterterms; one could look for a limit-cycle-like pattern in the renormalization group flow of the separable model.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper does what it says, and the toy-model math is solid. The new result is the separable NLO two-pion-exchange-like model, where the PCB term cannot be absorbed without putting a TPE-like structure into the LO potential. That is a legitimate and interesting observation.

The paper works through Lippmann-Schwinger solutions for separable S-wave potentials, gives closed-form loop functions, and checks the threshold subtraction. The three-model separation (short-range only, singular OPE-like, TPE-like) cleanly shows when renormalization works, when it needs a nonlocal counterterm, and when it fails. I checked Eq. (46); the subtraction is correct. The authors do not oversell the calculation itself.

The soft spot is the scope of the conclusion. The failure in the TPE-like model depends on the separable ansatz: the PCB loop Sigma_{2,21} factorizes from the external F2pi(p), which is what forces the counterterm to be nonlocal. In a realistic chiral potential the long-range part is local in q, and only the regulator is nonlocal; the loop integral does not factorize that way. So the statement ‘nonlocality of long-range forces makes renormalization impossible’ is not established for realistic nonlocal regulators. It is established for separable nonlocal long-range potentials. The paper’s cautions in the text (‘might’, ‘toy models’) are fine, but the abstract can be read as broader.

The citations to the authors’ earlier renormalizability proofs are used as background, not to derive the toy-model results; that’s fine. No fitted parameters, no invented entities.

Who should read it: anyone working on renormalization of nonlocal potentials in chiral EFT or in phenomenological NN models. It’s a good lecture-style illustration of a subtle point. I would not treat it as a no-go theorem for general nonlocal chiral potentials.

Recommendation: send to peer review. It’s a focused, correct analytic note. With a modest revision to scope the abstract and final paragraph to separable models, it is publishable. I would not desk-reject.

Referee Report

0 major / 6 minor

Summary. This paper analyzes a set of separable S-wave toy potentials modeled on the chiral EFT NN interaction. After reviewing the locality bounds that underlie the renormalizability proofs of Refs. [8,9], the authors introduce LO potentials built from a nonlocal regulator F_Λ(p), a one-pion-exchange-like form factor F_{1π}(p) with a pole at p^2 = -M_π^2/4, and an NLO two-pion-exchange-like form factor F_{2π}(p) with a pole at p^2 = -M_π^2. The LO Lippmann-Schwinger equation is solved in closed form, and the NLO amplitude is computed as \bar R V_2 R. The authors identify the power-counting-breaking contributions proportional to Σ_{2,21}, show that in the purely short-range and regular-long-range cases a local contact counterterm suffices, find that the singular OPE-like LO term requires a nonlocal long-range counterterm, and argue that the TPE-like NLO term cannot be renormalized without promoting the F_{2π} structure to LO. The central conclusion is that in these separable toy models, nonlocality of the long-range interaction either forces nonlocal counterterms or destroys renormalizability at NLO.

Significance. Within its stated scope, the paper delivers a clean and fully explicit demonstration that the locality criterion for the singular long-range part of the chiral EFT potential is not just a technical sufficient condition but can become necessary in a controlled class of models. The closed-form loop functions in Eqs. (27) and (37), the explicit PCB decompositions in Eqs. (42)-(43), and the threshold subtraction in Eq. (46) make every step machine-checkable, and the derivation is self-contained with no fitted parameters. The result sharpens the message of Refs. [8-10] and gives a concrete warning against nonlocal phenomenological potentials. I do not see the skeptic's concern as an internal flaw: the paper consistently says 'separable toy models' and hedges the general conclusion with 'might'; the nonrenormalizability statement is proven for the specific separable TPE-like model, not for all nonlocal potentials, and the text does not claim otherwise.

minor comments (6)

[Abstract and Sec. IV] The abstract's statement that nonlocality 'causes the need for nonlocal counter terms, or even makes the whole approach nonrenormalizable' can be overread as a theorem about all nonlocal long-range potentials; since the demonstration is restricted to the separable toy models, I suggest adding 'in such separable models' or 'can' to the abstract and to the final paragraph of Sec. IV.
[Sec. III.A, Eq. (14)] The labels 'singular' and 'regular' for the two terms in V_{0,long} are not defined; please clarify that 'singular' refers to the spin-triplet-like term carrying the pole at p^2 = -M_π^2/4 and 'regular' to the product F_{1π}(p')F_{1π}(p).
[Sec. IV] The sentence 'the same counter term renormalizes the theory with \tilde g_{1π}\neq0' is ambiguous, because the counterterm matrix in Eq. (45) is diagonal in the (1,1) entry; please state explicitly that the same constant is used while ψ_0(p_on) is modified by the new LO term.
[Sec. IV, Eqs. (42)-(43)] The labels 'PCB-I' and 'PCB-II' are never defined; a one-sentence explanation of the two classes would help the reader follow the classification.
[Sec. II, Eq. (9)] The domain conditions 'if p' > p' and 'if p > p'' appear only in the preceding sentence; making them explicit inside Eq. (9) would remove a possible source of confusion.
[References] Reference [10] is cited as a conference contribution without a preprint identifier; if an arXiv number is available, it should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the toy-model renormalization calculations are self-contained and the self-citations are motivational only.

full rationale

The paper's central results are obtained by explicit solution of the Lippmann-Schwinger equation for the stated separable potentials, not by importing a conclusion. The PCB contributions in Eqs. (42)-(43) are computed from the NLO amplitude via Eqs. (34)-(37); the stated need for an F2π structure in the LO counterterm follows from the matrix identity that ψ2,3(pon) does not reduce to ψ0(pon) (Sec. IV), i.e., from the algebraic structure of the separable ansatz, not from an assumed output. No parameter is fitted and then renamed a prediction. The self-citations to Refs. [8-10] supply the locality criterion and the bounds on chiral potentials, but those are used only to motivate the toy models and to provide contrast; the toy-model derivations do not depend on the validity of Refs. [8-10]. The paper explicitly describes the locality condition as a sufficient criterion ('one of the sufficient renormalizability criteria') and limits its own conclusion with 'might' and 'separable toy models', so any overgeneralization to all nonlocal chiral potentials would be a scope question, not a circular step. There is therefore no circular step to report.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical free parameters are fitted: all coupling constants are symbolic, and the cutoff is fixed by the EFT assumption Lambda ~ Lambda_b. The central claim rests on four structural assumptions about the toy models, the EFT operator content, and the renormalizability criteria.

assumptions (4)

standard math Lippmann-Schwinger resolvent algebra: the solutions T0 = V0 R and T2 = Rbar V2 R with R=(1-GV0)^{-1} are valid for the separable potentials.
Sec. III.B uses these identities without derivation; standard operator resolvent formulas.
domain assumption Sufficient renormalizability criteria of Refs [8,9]: local long-range potentials with the stated bounds are renormalizable in the two-nucleon sector.
Sec. II reviews the bounds; the paper uses them as the baseline against which nonlocal toy models are compared, but the toy-model amplitudes are computed independently.
domain assumption The separable long-range structures F1pi(p) and F2pi(p) represent the essential analytic features of OPE and TPE partial-wave potentials.
Sec. III.A constructs the potentials as 'motivated by' chiral EFT; the transferability of the results to realistic potentials depends on this assumption.
domain assumption A renormalizable theory must be able to absorb all PCB terms by redefining operators already present in the LO potential; counterterms cannot introduce new long-range structures without promoting those structures to LO.
Sec. IV uses this to conclude the TPE-like PCB term is unrenormalizable; it is the standard EFT power-counting requirement, but the paper does not prove a no-go theorem for all possible counterterm bases.

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Cite this review

Pith. "Pith review of Renormalizability and nonrenormalizability of nonlocal potentials." pith.science (2026). https://pith.science/paper/ADB2TPT6

@misc{pith2026250908512,
  author       = {Pith},
  title        = {Pith review of: Renormalizability and nonrenormalizability of nonlocal potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADB2TPT6}},
  note         = {Machine review of arXiv:2509.08512}
}

read the original abstract

We consider separable toy models of the nucleon-nucleon interaction inspired by chiral effective field theory. We show that nonlocality of the long-range forces causes the need for nonlocal counter terms, or even makes the whole approach nonrenormalizable.

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Works this paper leans on

16 extracted references · 4 canonical work pages

[1]

Leading order To obtain the LO scattering amplitude, we solve the partial-wave projected Lippmann-Schwinger equation,T 0 = V0 +V 0GT0, or, explicitly: T0(p′,p;p on) =V 0(p′,p) + Z p′′2dp′′ (2π)3 V0(p′,p′′)G(p′′;p on)T0(p′′,p;p on), G(p′′;p on) = mN p2on−p′′2 +iϵ .(21) Its general solution is T0 =V 0R= ¯RV0,(22) whereR( ¯R) is the resolvent of the Lippmann...

work page
[2]

The unrenormalized NLO amplitude is given by T2 = ¯RV2R.(33) For the separable potential defined in Eq

Next-to-leading order The NLO amplitude is calculated perturbatively to explicitly control the contributions of different EFT orders. The unrenormalized NLO amplitude is given by T2 = ¯RV2R.(33) For the separable potential defined in Eq. (19), we obtain T2(p′,p;p on) =VNNψ⊺ 2 (pon)v2ψ2(pon),(34) whereψ 2 is given by ψ2(pon) =f 2(pon) + Σ2(pon)t0(pon)f0(po...

work page 2020
[3]

Weinberg, Phys

S. Weinberg, Phys. Lett.B251, 288 (1990)

1990
[4]

Weinberg, Nucl

S. Weinberg, Nucl. Phys.B363, 3 (1991)

1991
[5]

P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci.52, 339 (2002), nucl-th/0203055

arXiv 2002
[6]

Epelbaum, H.-W

E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.Mod.Phys.81, 1773 (2009), 0811.1338

arXiv 2009
[7]

Machleidt and D

R. Machleidt and D. Entem, Phys.Rept.503, 1 (2011), 1105.2919

arXiv 2011
[8]

Epelbaum, H

E. Epelbaum, H. Krebs, and P. Reinert, Front. in Phys.8, 98 (2020), 1911.11875

arXiv 2020

Show all 16 references

[9]

H. W. Hammer, S. K¨ onig, and U. van Kolck, Rev. Mod. Phys.92, 025004 (2020), 1906.12122

2020 arXiv
[10]

A. M. Gasparyan and E. Epelbaum, Phys. Rev. C105, 024001 (2022), 2110.15302

2022 arXiv
[11]

A. M. Gasparyan and E. Epelbaum, Phys. Rev. C107, 044002 (2023), 2301.13032

2023 arXiv
[12]

A. M. Gasparyan and E. Epelbaum, in11th International Workshop on Chiral Dynamics(2025), 2504.16512

2025 arXiv
[13]

Epelbaum and J

E. Epelbaum and J. Gegelia, Eur. Phys. J.A41, 341 (2009), 0906.3822

2009 arXiv
[14]

Epelbaum, J

E. Epelbaum, J. Gegelia, and U.-G. Meißner, Nucl. Phys. B925, 161 (2017), 1705.02524

2017 arXiv
[15]

R. Peng, S. Lyu, S. K¨ onig, and B. Long, Phys. Rev. C105, 054002 (2022), 2112.00947

2022 arXiv
[16]

H. F. Arellano and N. A. Adriazola, Eur. Phys. J. A60, 158 (2024), 2406.12827

2024 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Leading order To obtain the LO scattering amplitude, we solve the partial-wave projected Lippmann-Schwinger equation,T 0 = V0 +V 0GT0, or, explicitly: T0(p′,p;p on) =V 0(p′,p) + Z p′′2dp′′ (2π)3 V0(p′,p′′)G(p′′;p on)T0(p′′,p;p on), G(p′′;p on) = mN p2on−p′′2 +iϵ .(21) Its general solution is T0 =V 0R= ¯RV0,(22) whereR( ¯R) is the resolvent of the Lippmann...

work page

[2] [2]

The unrenormalized NLO amplitude is given by T2 = ¯RV2R.(33) For the separable potential defined in Eq

Next-to-leading order The NLO amplitude is calculated perturbatively to explicitly control the contributions of different EFT orders. The unrenormalized NLO amplitude is given by T2 = ¯RV2R.(33) For the separable potential defined in Eq. (19), we obtain T2(p′,p;p on) =VNNψ⊺ 2 (pon)v2ψ2(pon),(34) whereψ 2 is given by ψ2(pon) =f 2(pon) + Σ2(pon)t0(pon)f0(po...

work page 2020

[3] [3]

Weinberg, Phys

S. Weinberg, Phys. Lett.B251, 288 (1990)

1990

[4] [4]

Weinberg, Nucl

S. Weinberg, Nucl. Phys.B363, 3 (1991)

1991

[5] [5]

P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci.52, 339 (2002), nucl-th/0203055

arXiv 2002

[6] [6]

Epelbaum, H.-W

E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.Mod.Phys.81, 1773 (2009), 0811.1338

arXiv 2009

[7] [7]

Machleidt and D

R. Machleidt and D. Entem, Phys.Rept.503, 1 (2011), 1105.2919

arXiv 2011

[8] [8]

Epelbaum, H

E. Epelbaum, H. Krebs, and P. Reinert, Front. in Phys.8, 98 (2020), 1911.11875

arXiv 2020