{"id":"dc8d306d-9253-462b-923e-f29c4d30d4a5","arxiv_id":"2607.13769","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"At the Efimov-unatomic critical dimension, the three-body momentum tail acquires a quadratic-logarithmic term requiring a new contact parameter C3'' that changes sign with mass imbalance and vanishes for identical particles.","lead":"This paper derives the high-momentum tail of a mass-imbalanced three-body system exactly at the critical dimension where Efimov physics gives way to continuous scale invariance, finding a new logarithmic structure and a third three-body contact parameter. A generalist should read it because it closes a technical gap in the Efimov-to-unatomic transition framework and predicts a momentum scale where three-body contributions vanish, which is potentially measurable in cold-atom a","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim hinges on C3'' extracted from undocumented fits; the reduction from the integral representations (3.10)-(3.13) to the fitted form (3.14) is not shown.","rationale":"Reader's weakest_assumption is the non-integer dimension continuation/harmonic-mapping, a framework-level concern inherited from prior work. I agree that is unverified, but the single most load-bearing concern for the paper's new claim is narrower: the quantitative value and sign of C3'' are extracted by fits that are not documented, and the analytic step from (3.10)-(3.13) to (3.14) is omitted. If C3'' is a fit artifact, the paper's novelty collapses even if the dimensional continuation is accepted. The reader did flag the same derivation gap in the rationale, though not in the weakest_assumption, hence 'partial'. A concrete numerical recomputation and an independent asymptotic expansion would settle whether the log-squared term and its sign change are real. The verdict should remain CONDITIONAL (no change) because the concern is serious enough to block unconditional acceptance but is resolvable by the proposed test.","tokens_in":20453,"tokens_out":4846,"duration_ms":49375,"concrete_test":"For one mass-imbalanced system (e.g., 23Na2-40K at Dc=2.296) and the equal-mass limit, compute the four contributions (3.10)-(3.13) from the full spectator function (A16) on a qB/kappa0 grid spanning at least 10^2 to 10^7. Fit the sum to nB = C2/qB^4 + qB^{-(Dc+2)} [a + b L + c L^2], L = ln(qB/kappa0) - (1/4)ln(4 muA muB), using a documented nonlinear least-squares routine with several fit windows and weights. Record c (= C3'') with covariance estimates. If c does not stabilize, does not change sign between HHL and HLL mixtures, or is not consistent with zero for equal masses, the central claim fails. As a cross-check, derive the L^2 coefficient by asymptotic expansion of (3.10)-(3.13) and compare with the fitted c.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's new result is the quadratic-log coefficient C3'' and its sign change. Section III.C states C2, C3, C3', C3'' are 'obtained from fits to the asymptotic form, Eq. (3.14)', but it does not specify the fitting procedure, momentum range, number of fitted parameters, weighting, or statistical uncertainties. More fundamentally, Eqs. (3.10)-(3.13) each contain ln^2 terms with different momentum arguments and prefactors; the text jumps from these four integrals to the compact expansion (3.14) with the phrase 'collecting all four terms', without showing the algebra. No closed-form integrals or explicit definitions are given for C3, C3', or C3''. Consequently, the sign change in Fig. 8 and the claimed zero for equal masses are not independently verifiable from the manuscript. The vanishing for identical particles is asserted but does not follow trivially from the integrals, where terms like n1 and n4 have different coefficients; a numerical or analytic cancellation must be demonstrated. Since the abstract's central signature is precisely this sign-changing C3'', an undocumented fit artifact would invalidate the main claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a resonantly interacting mass-imbalanced three-body system in a continuous effective dimension D, focusing on the critical dimension D_c where the Efimov scaling parameter vanishes. It claims that at D_c the single-particle momentum distribution has a distinct asymptotic structure n_B(q_B) = C_2/q_B^4 + C_3'/q_B^{D_c+2} + C_3 q_B^{-(D_c+2)} ln[...] + C_3'' q_B^{-(D_c+2)} ln^2[...] (Eq. 3.14), requiring a new three-body contact parameter C_3'' associated with a quadratic logarithm. The paper further claims that C_3'' changes sign between heavy-heavy-light and heavy-light-light mass configurations, vanishes for equal masses, and can produce a momentum scale at which the three-body contribution cancels. It also analyzes the narrow intermediate region between D_c and a lower dimension D_*, which it calls the intermediate scaling regime, and relates its width to the mass ratio.","tokens_in":20702,"tokens_out":2532,"duration_ms":47010,"significance":"If the central claims are correct, the paper identifies a genuinely new scaling structure at the Efimov-unatomic transition: a marginal logarithmic regime distinct from both log-periodic Efimov behavior and power-law unatomic behavior. The proposal of an additional three-body contact parameter C_3'' and its mass-ratio dependence would be a valuable extension of contact theory, with experimentally testable consequences for high-momentum tails and possibly for three-body losses in confined ultracold mixtures. The paper also contributes explicit integral representations for the four contributions to the momentum distribution and compares them with full numerical solutions, which is a useful feature. However, the manuscript is not yet self-contained at the point where its central new quantity is defined: the reduction from the integrals (3.10)-(3.13) to the compact expansion (3.14) is not shown, and the coefficients C_3, C_3', C_3'' are extracted from unspecified fits. Because the sign change and equal-mass cancellation of C_3'' are the paper's principal physical message, this derivational gap currently prevents independent verification of the main result.","major_comments":[{"comment":"The central derivation is missing. The four integrals (3.10)-(3.13) contain logarithmic terms with different momentum arguments (q_B, q_A', p_B', and combinations such as q_B p_B' ±) and different prefactors. The text jumps from these integrals to the compact expansion (3.14) with 'collecting all four terms', without showing the algebra that produces the constant, linear, and quadratic logarithmic terms, and without giving closed-form definitions of C_3, C_3', and C_3''. Since Eq. (3.14) is the basis for the new contact parameter C_3'', the manuscript should either provide the analytic derivation or at minimum state the precise definitions of these coefficients and how they follow from (3.10)-(3.13). The present gap makes the sign change in Fig. 8 and the equal-mass limit unverifiable by the reader.","section":"§III.B, Eqs. (3.10)-(3.14)"},{"comment":"The paper states that C_2, C_3, C_3', and C_3'' are 'obtained from fits to the asymptotic form, Eq. (3.14)', but gives no details: no momentum fitting range, number of fitted parameters, weighting, or statistical uncertainties. Fig. 8 shows smooth curves over a wide range of m_B/m_A, which suggests these are fits to numerical data sets. If C_3'' is determined by fitting the same asymptotic form that it is then used to explain, the risk of a fitting artifact is real; the reader cannot assess whether the claimed sign change and the zero at equal masses are robust. The authors should provide the analytic expressions or, failing that, a complete description of the fitting procedure and the resulting uncertainties.","section":"§III.C, Fig. 8"},{"comment":"The text asserts that C_3'' vanishes for three identical particles. This does not follow trivially from the integral representations: in Eqs. (3.11)-(3.13) the pieces n_2, n_3, n_4 have different structures and different coefficients, and the equal-mass limit A=1 does not obviously make the quadratic-logarithmic terms cancel. A direct analytic or numerical demonstration of this cancellation should be given. This point is load-bearing because the paper repeatedly contrasts the C_3'' sign change against the identical-particle case as a limiting check.","section":"§III.C, equal-mass limit"},{"comment":"The experimental mapping between effective dimension and confinement aspect ratio is used to claim that the intermediate scaling regime is accessible in principle. The footnote reference to Ref. [41] may support the mapping, but the manuscript does not verify that the continuous-D Faddeev solution with Bethe-Peierls boundary conditions describes the actual quasi-two-dimensional trapped system without additional effective-range or confinement-induced corrections. Since the paper's experimental conclusions rely on this mapping, a brief quantitative check or a discussion of its regime of validity should be included. This is not a fatal issue for the central scaling result, but it is necessary to support the experimental framing.","section":"Eq. (2.1) and Sec. II"}],"minor_comments":[{"comment":"The paper alternates between 'Intermediate Scaling Regime (ISR)' and 'Scale Invariant Regime (SIR)' for the same interval D_* < D < D_c. The terminology should be unified to avoid confusion. Also, Fig. 1 uses open circles and green points whose meaning should be made explicit in the caption.","section":"§II and Fig. 1"},{"comment":"There are notational inconsistencies: some expressions show q'_A with prime and p'_B with prime, while others omit primes in the same formula; the arguments of the logarithms contain factors like '2q' that are not fully defined. Please normalize notation so that the momentum variables and the overall constants are unambiguous.","section":"§III.B, Eqs. (3.10)-(3.13)"},{"comment":"The restriction 2 < D < 4 is stated after Eq. (A12), but the paper also discusses D=3 and the limit D→2^+. The limiting cases should be handled explicitly, including how the formulas behave at D=3 where some gamma functions have special values.","section":"Appendix A, Eq. (A12)"},{"comment":"The figure caption and text refer to '6Li–133Cs2' while other parts use '133Cs2–6Li' or '23Na2–40K'. The ordering and naming of the heavy-heavy-light and heavy-light-light configurations should be consistent throughout, because the sign of C_3'' is configuration-dependent.","section":"Fig. 6"},{"comment":"The statement that the cancellation of the three-body contribution 'may also indicate reduced loss rates' is speculative. It is correctly labeled as future work, but the sentence should perhaps be more clearly separated from the established results.","section":"§IV, Summary"}],"recommendation":"major_revision","confidential_remarks":"The central idea is interesting and the manuscript has solid numerical support for the spectator function and the shape of the momentum distribution. The main problem is that the paper's central new object, C_3'', is introduced through an undocumented fit and an unshown algebraic reduction. This is fixable: the authors can supply the analytic derivation or, if the coefficients are numerical, a complete description of the fitting procedure and error estimates. I would not reject, but the current version does not meet the standard of verifiability required for the central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper identifies a genuinely new object — a quadratic-logarithmic contact parameter C3'' at the Efimov–unatomic critical dimension — and shows it changes sign with mass ratio. That is worth someone's time. But as written, the derivation stops short of proving C3'' is not a fit artifact. I'd send it to peer review and ask for the missing steps.\n\nWhat is actually good: the asymptotic spectator function at Dc is derived (Eq. 3.5) and has a clean logarithmic form; the comparison with the full numerical solution in Fig. 3 shows rapid convergence, which is a real check. The paper also gives the first systematic map of the intermediate scaling regime width versus mass ratio, and the confinement-aspect-ratio translation (Eq. 2.1) makes the regime concrete. The construction of the four momentum integrals in Appendix B is standard but careful.\n\nThe soft spot is exactly where the reader and stress-test point: Eqs. (3.10)–(3.13) each contain log-squared pieces with different momentum arguments, and then the text says 'collecting all four terms' and jumps to the compact expansion (3.14). That is a serious gap for the central result. The contact coefficients, including the new C3'', are obtained 'from fits' with no range, weighting, or uncertainty. I cannot verify the sign change or the equal-mass vanishing from the manuscript. The equal-mass case is especially bothersome because it is asserted but not demonstrated; the integrals do not obviously cancel term-by-term. This is a load-bearing gap, but it is a presentation gap, not obviously a wrong result. The underlying framework (non-integer dimension STM equation, Bethe–Peierls) is established in their prior work, and the new derivation of the log-tail is plausible.\n\nAlso, the harmonic-confinement mapping is imported and not cross-checked against a direct quasi-2D calculation; the experimental and nuclear remarks are speculative. Minor.\n\nBottom line: the paper deserves a serious referee. The referee should request either an explicit derivation of C3'' from the integrals or a detailed fitting protocol with diagnostics and error bars, and a demonstration of the equal-mass cancellation. If those are supplied, this would be a solid addition to the Tan-contact literature at the marginal dimension.","headline":"Plausible new claim that the critical dimension needs its own three-body contact, but the new coefficient comes from undocumented fits — referee it, with demands.","tokens_in":21231,"tokens_out":2863,"would_cite":false,"duration_ms":28690,"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":"At the Efimov-unatomic critical dimension, the three-body momentum tail gains a quadratic-logarithmic contact term that changes sign with mass imbalance, making Dc a regime of its own.","keywords":["Efimov effect","unatomic regime","critical dimension","momentum distribution","contact parameters","mass imbalance","scale invariance","three-body universality"],"falsifier":"Measure the single-particle momentum distribution of a resonantly interacting heavy-light-light mixture tuned to the critical dimension and test whether the subtracted tail qB^{Dc+2}[n(qB) - C2/qB^4] follows a quadratic logarithm with a positive C3'' and a zero crossing; alternatively, solve the three-body problem with a finite-range potential or an explicit harmonic trap and check whether the ln^2 term and its sign change survive beyond the zero-range, continuous-dimension model.","tokens_in":20285,"feed_emoji":"⚛️","tokens_out":3546,"duration_ms":35876,"temperature":0.7,"pith_summary":"This paper aims to establish that the critical dimension Dc, where the Efimov regime with discrete scale invariance gives way to the unatomic regime with continuous scale invariance, is not a smooth limit of either side but a distinct regime with its own scaling structure. At Dc, the high-momentum tail of the single-particle momentum distribution develops a quadratic-logarithmic contribution, requiring a fourth contact parameter, C3'', alongside the known two-body and three-body contacts. This new parameter vanishes for identical particles and changes sign as the mass configuration crosses from heavy-heavy-light to heavy-light-light, so in some systems the three-body contribution can vanish entirely at a characteristic momentum. If correct, the result rewrites the standard contact description at the transition and offers an experimentally accessible signature of the Efimov-unatomic crossover.","feed_headline":"Quadratic log term appears at the Efimov-unatomic critical point","feed_subtitle":"New contact parameter C3'' flips sign with mass ratio and can erase the three-body tail at one momentum.","key_machinery":"The argument is carried by the asymptotic spectator function of the Faddeev equations in a continuous effective dimension D, with Bethe-Peierls boundary conditions at unitarity. Near Dc the scaling exponent sn tends to zero, and the spectator function acquires a single logarithmic dependence on momentum, Eq. (3.5). Feeding this spectator function into the four Faddeev contributions to the single-particle momentum distribution and expanding at large qB produces Eq. (3.14), whose coefficients define the contact parameters; the quadratic-logarithmic coefficient C3'' is the newly identified three-body contact. The dimension Dc itself is located through the characteristic transcendental equation","core_discovery":"The central claim is that at the critical dimension Dc the asymptotic single-particle momentum distribution has the form nB(qB) = C2/qB^4 + C3'/qB^{Dc+2} + C3/qB^{Dc+2} ln[qB/(4 muA muB)^{1/4} kappa0] + C3''/qB^{Dc+2} [ln(...)]^2, with the coefficient C3'' of the squared logarithm nonzero only for mass-imbalanced systems, negative for heavy-heavy-light, positive for heavy-light-light, and zero for identical particles. The paper argues that this logarithmic hierarchy is intrinsic to Dc, not a degenerate limit of the log-periodic Efimov oscillations or the pure power law of the unatomic regime. Consequently, a consistent finite description of the momentum tail at the transition requires the ne","pith_inferences":["The sign change of C3'' suggests a practical calibration: a zero crossing in the subtracted momentum tail could serve as an experimental fiducial for tuning a trapped mixture exactly to Dc.","If the effective-dimension mapping holds for nuclear halo systems, analog logarithmic structure could appear in high-momentum knockout observables from heavy-light-light nuclei, offering a test beyond ultracold atoms.","The emergence of a marginal quadratic-logarithmic term at Dc resembles the logarithmic corrections familiar at upper critical dimensions in critical phenomena, hinting that the Efimov-unatomic transition may have an effective renormalization-group description with a marginally relevant operator.","A direct check of the mapping between harmonic confinement and effective dimension could be made by computing the same contacts with an explicit trap potential; the paper does not perform that check."],"forward_implications":["Any universal relation at Dc involving energy, momentum distribution, or response functions must include the new three-body contact C3''; omitting it leaves the leading sub-asymptotic behavior ill-defined.","In heavy-light-light systems, the three-body contribution vanishes at a finite momentum scale, making the asymptotic tail locally pure 1/q^4 and giving a direct experimental handle on the two-body contact.","The intermediate scaling regime between Dc and the lower dimension Dbar narrows strongly with mass imbalance, so for heavy-heavy-light systems the transition is sharply localized in effective dimension and requires fine confinement control to resolve.","The sign of C3'' distinguishes heavy-heavy-light from heavy-light-light configurations and vanishes for identical particles, providing a mass-ratio-dependent diagnostic of the transition.","The critical dimension should be treated as its own scaling regime in future few-body studies, not as a singular limit obtained by extrapolating Efimov or unatomic formulas."],"fun_headline_variants":["A quadratic log contact flips sign across mass configurations","Critical dimension yields squared-log term and vanishing three-body tail","C3'' sign change suppresses 3-body tail at one momentum","Efimov-unatomic critical point hosts new quadratic-log contact","Mass-imbalanced trio shows emergent scaling via C3''"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole prediction depends on treating spatial dimension as a continuous parameter and assuming that this noninteger-dimension problem faithfully represents a real quasi-two-dimensional confined gas; if the confinement-to-dimension mapping fails, the logarithmic-squared contact has no experimental realization.","fun_headline_variants_meta":{"raw":{"variants":["A quadratic log contact flips sign across mass configurations","Critical dimension yields squared-log term and vanishing three-body tail","C3'' sign change suppresses 3-body tail at one momentum","Efimov-unatomic critical point hosts new quadratic-log contact","Mass-imbalanced trio shows emergent scaling via C3''"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000763,"raw_usage":{"total_tokens":3253,"prompt_tokens":804,"completion_tokens":2449,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":2366}},"tokens_in":548,"tokens_out":2449,"duration_ms":15744,"temperature":1.0,"reasoning_tokens":2366,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T03:43:43.680735+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the single-particle momentum distribution of a resonantly interacting heavy-light-light mixture tuned to the critical dimension and test whether the subtracted tail qB^{Dc+2}[n(qB) - C2/qB^4] follows a quadratic logarithm with a positive C3'' and a zero crossing; alternatively, solve the three-body problem with a finite-range potential or an explicit harmonic trap and check whether the ln^2 term and its sign change survive beyond the zero-range, continuous-dimension model.","supporting_citations":[],"review_version":1}