{"id":"51d9aa17-062a-47d2-9a10-a27874306d3e","arxiv_id":"2509.08512","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonlocal long-range separable potentials in a two-nucleon toy model require nonlocal counterterms at NLO, and a two-pion-exchange-like nonlocal term cannot be renormalized without moving that structure into the leading-order potential.","lead":"This paper uses simple separable models to show that when the long-range part of a nucleon-nucleon potential is nonlocal, renormalizing the next-to-leading-order amplitude may force you to add nonlocal counterterms, or may fail entirely. It matters because it sharpens the conditions under which chiral effective field theory potentials can be renormalized order by order.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nonrenormalizability is demonstrated for separable nonlocal long-range potentials, but the inference to general nonlocal chiral potentials is not established because the PCB term's external F2π factor rides on the separable ansatz.","rationale":"The paper is scoped as a study of separable toy models, and the derivation of the PCB terms is explicit and correct. The reader's weakest assumption is essentially transfer from toy models to realistic chiral EFT; this review sharpens that concern: the nonrenormalizable model relies on the factorization of the PCB loop from the external F2π factor, a property of the separable ansatz. A local chiral potential multiplied by a nonlocal regulator is the more common nonlocal case, and the Sec. II bounds cover it, making it plausible that no such irreducible nonlocal counterterm appears. Because the paper carefully says 'may' and 'separable toy models', this scope limitation does not invalidate the paper's claims; it only blocks the broader reading that nonlocality of long-range forces generically destroys renormalizability. The proposed test would settle whether the separable mechanism is representative or an artifact of the toy construction.","tokens_in":7785,"tokens_out":20889,"duration_ms":520536,"concrete_test":"Replace the separable NLO term of Eq. (18) by the S-wave projection of a local TPE-like potential with the same left-hand singularity, e.g. V2,long(p',p)=V_NN g2π (p'^2+p^2)/Λb^2 FΛ(p')FΛ(p) times the S-wave projection of ∫_{2Mπ}∞ ρ(μ)dμ/((p'-p)^2+μ^2), and recompute the NLO PCB contributions to T2 in Eq. (33). If the PCB external-momentum dependence is again an irreducible F2π-style factor, the nonrenormalizability survives the non-separable test; if it is polynomial and absorbable by the LO contacts and OPE-like counterterms already present in v0, the toy-model nonrenormalizability is specific to the separable ansatz and the universal claim should be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's explicit NLO calculation is internally consistent, and the toy models do show that separable nonlocal long-range potentials can require nonlocal counterterms (Eq. (49)) or fail renormalization without a TPE-like LO term. The load-bearing issue is the scope of the inference from these models. The nonrenormalizable case is produced by the separable form of Eq. (18): the PCB loop Sigma2,21 factorizes from the external F2π(p) factor in T_PCB,II (Eq. (43)), so the counterterm must contain F2π. In a realistic chiral potential the long-range part is a local function of q=p'-p; even after multiplying by a nonlocal regulator FΛ(p')FΛ(p), the PCB loop integral is not separable, and the large-momentum contribution need not carry an irreducible nonlocal external factor. The Sec. II bounds used in the authors' renormalizability proofs for chiral EFT explicitly allow nonlocal regulators FΛ(p^2), FΛ(p'^2) while requiring the long-range singular part to be local; the toy model violates this by replacing V(q) with products of functions of p and p'. Therefore the conclusion 'any EFT with sufficiently singular nonlocal long-range parts fails' (the reader's strongest reading) is not proven; the result is established for separable nonlocal long-range potentials. The paper's own hedging ('might', 'separable toy models') partially acknowledges this, but the abstract's causal phrasing can be overread.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7986,"tokens_out":12284,"duration_ms":105465,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Abstract and Sec. IV"},{"comment":"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).","section":"Sec. III.A, Eq. (14)"},{"comment":"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.","section":"Sec. IV"},{"comment":"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.","section":"Sec. IV, Eqs. (42)-(43)"},{"comment":"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.","section":"Sec. II, Eq. (9)"},{"comment":"Reference [10] is cited as a conference contribution without a preprint identifier; if an arXiv number is available, it should be added.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a good fit for the journal. The self-citations to Refs. [8-10] are appropriate because the toy models are explicitly designed to probe the locality criterion from those papers. The only thing to guard against in the published version is a reader overgeneralizing the 'nonrenormalizable' conclusion beyond separable potentials; the authors' hedged wording is largely sufficient, but tightening the abstract would help."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":8599,"tokens_out":2877,"would_cite":true,"duration_ms":25882,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"When long-range forces are nonlocal, renormalization may require nonlocal counterterms or fail.","keywords":["chiral effective field theory","renormalization","nonlocal potentials","separable toy models","nucleon-nucleon interaction","power counting","Lippmann-Schwinger equation","long-range forces"],"falsifier":"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.","tokens_in":7507,"feed_emoji":"⚛️","tokens_out":9861,"duration_ms":82077,"temperature":0.7,"pith_summary":"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.","feed_headline":"Nonlocal long-range forces can block renormalization","feed_subtitle":"Separable toy models show nonlocal long-range parts need counterterms that chiral EFTs lack.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Derives the sufficient renormalizability bounds for local long-range interactions in chiral EFT that the toy models are constructed to violate.","marker":"[8]"},{"why":"Extends the renormalizability demonstration to NLO in the two-nucleon sector, giving the standard against which the toy models are checked.","marker":"[9]"},{"why":"Defines the renormalizability criterion used to classify the toy models: power-counting-breaking contributions must be removable by redefining lower-order low-energy constants.","marker":"[10]"}],"fun_headline_variants":["Nonlocal potentials break renormalization in toy nucleon models","Nonlocality forces new counterterms or kills renormalizability","Chiral EFT toy models show nonlocal forces ruin renormalization","Nonlocal long-range terms block renormalization without extra ops","Renormalization fails for nonlocal two-pion-exchange-like terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal potentials break renormalization in toy nucleon models","Nonlocality forces new counterterms or kills renormalizability","Chiral EFT toy models show nonlocal forces ruin renormalization","Nonlocal long-range terms block renormalization without extra ops","Renormalization fails for nonlocal two-pion-exchange-like terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":1918,"prompt_tokens":709,"completion_tokens":1209,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":325,"completion_tokens_details":{"reasoning_tokens":1120}},"tokens_in":325,"tokens_out":1209,"duration_ms":7985,"temperature":1.0,"reasoning_tokens":1120,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:01:18.988763+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}