{"id":"ff1eedeb-f410-4554-821d-1a83f70f8120","arxiv_id":"2411.15730","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A finite-volume quantization condition based on the N/D representation is derived and claimed to remain valid when the energy coincides with left-hand cuts, where the original Luescher condition fails.","lead":"This paper derives a new formula connecting the energy levels of two particles in a finite lattice box to the infinite-volume scattering amplitude, based on the N/D decomposition of the amplitude. If correct, it extends the standard Luescher method into energy regions where crossed-channel particle exchanges create left-hand cuts, a regime relevant to resonance studies such as the T_cc tetraquark.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The below-threshold validity of Eq. (20) is the load-bearing step; Supplement C only proves N_L=N for p*>0, and no analytic-continuation or numerical check is supplied for s<slhc.","rationale":"The reader's weakest assumption identifies the right location: Supplement C explicitly restricts the N_L=N proof to p*>0, and the new quantization condition is then used where p* is imaginary. I partially agree. The actual mathematical hole may be smaller than the reader states, because Eq. (17) only needs N_L at physical internal momenta k*, and if that equality is established for all k* in the sum, Eq. (19) follows for every s by the same pole-sum identity. But the paper does not make this argument, and it also does not prove that the exact D_L has no extra singularities in the below-threshold region or that the replacement is uniform in the infinite sum. The advertised extension of Lüscher's condition to left-hand-cut energies is therefore not fully supported as written. Since the gap is addressable by an explicit analytic-continuation argument or a small numerical experiment, the conditional verdict is appropriate. I would not reject the paper: the N/D construction and the elastic-region equivalence to Lüscher in Supplement D are nontrivial and internally consistent, and the proposed test is straightforward.","tokens_in":29136,"tokens_out":28452,"duration_ms":288659,"concrete_test":"Implement a toy model with a known left-hand singularity (e.g., the one-pole model of Supplement E or the OPE Q_l model) and compute the finite-volume spectrum exactly by solving the Lippmann-Schwinger equation in a box for L/a = 4, 6, 8, 10, or by direct diagonalization if the interaction is finite-rank. Then evaluate Eq. (20) using the IV numerator N of the same model and locate its zeros for at least one energy below the threshold and, if possible, below the nearest left-hand branch point. If the two sets of energies agree up to O(e^{-m L}) with m the lightest physical scale, the below-threshold continuation is confirmed; if they disagree beyond that, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (20) with D_L from Eq. (19) quantizes the finite-volume spectrum even for s below the nearest left-hand cut. The exact FV identity (17) expresses D_L in terms of N_L(k*,P) evaluated at the discrete internal momenta k*. Supplement C proves N_L(p*)=N(p*) only under the assumption p*>0 ('physical kinematics'), and the derivation of Eq. (32) uses T(s) for physical s. Since every k* in the sum is physical, a charitable reading is that this is enough for the term-by-term replacement N_L->N in Eq. (17); however, the paper never states that justification. The load-bearing gap is that Eq. (19) is then asserted to be the correct quantization condition for imaginary external p* (s<slhc), where N_L is not controlled. In particular, the paper does not show that the exact FV denominator D_L admits the pole-sum representation (17) with N in place of N_L throughout this region, nor that finite-volume images of the left-hand singularities (which Supplement A places in N_L) cannot feed power-like corrections into D_L. If N_L(k*) differs from N(k*) by non-exponential terms, or if the pole-sum representation ceases to be valid when s is continued below slhc, the zeros of Eq. (20) below slhc need not coincide with the FV spectrum. The self-flagged restriction in Supplement C is exactly this missing step, and no numerical demonstration is provided.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a finite-volume (FV) quantization condition derived from the N/D representation of a two-body partial-wave amplitude. After defining an FV analog of the N/D decomposition, the authors derive a pole-sum expression for the FV denominator D_L, Eq. (17), replace the FV numerator N_L by the infinite-volume numerator N using an argument in Supplement C, and obtain the quantization condition det D_L = 0 with D_L given by Eq. (19). The central claim is that this condition remains valid for energies below the nearest left-hand cut, where the standard Lüscher condition is inapplicable. Supplement D derives the reduction to the Lüscher condition in the elastic region, and Supplement E proposes pole and one-particle-exchange parametrizations of N.","tokens_in":29467,"tokens_out":29867,"duration_ms":261021,"significance":"If the derivation is correct, the result is a useful and conceptually clean extension of Lüscher's framework: it avoids evaluating singular crossed-channel contributions at finite volume and provides a direct route from lattice spectra to the N/D numerator. The paper is strong in its explicit setup of the FV unitarity equations, its formal supplement with definitions of FV partial waves, and its concrete parametrization suggestions; it also honestly flags the need for subtractions and for a truncation study of the angular-momentum basis. The main weakness is that the proof of the key replacement N_L→N contains an apparent error in the written kernel, and the analytic continuation to s<slhc is not explicitly stated, so the advertised extension is not yet fully supported as printed. No numerical demonstration is included, which would materially increase confidence in the new condition.","major_comments":[{"comment":"The kernel in Eq. (31) is written as ρℓ(s'') Im[T(s'')-T(s)]/(s''-s-iϵ) N(s''). For s and s'' above sthr, T(s) is real because it contains only left-hand singularities, so this imaginary part vanishes identically and Eq. (31) reduces to N(s)=T(s). That result is inconsistent with the pole-model solution Nℓ(s)=g Dℓ(slhc)/(s-slhc) in Supplement E.1 and with the standard N/D integral equation, whose correct kernel is [T(s'')-T(s)]/(s''-s-iϵ) without the operation Im. The same spurious Im appears in Eq. (32). As written, the comparison between Eqs. (31) and (32) does not prove N_L=N+O(e^{-ΔL}); please correct both equations and re-derive the comparison.","section":"Supplement C, Eqs. (31)-(32)"},{"comment":"Supplement C establishes the equality of FV and IV numerator functions only for physical kinematics p*>0. Since the sum in Eq. (17) runs over physical discrete momenta, the termwise replacement N_L(k*,P)→N(k*) is justified at each k*, and Eq. (19) can then be read as an identity of meromorphic functions in the external variable s, valid also for s<slhc. This analytic-continuation step is not stated in the paper. Please add an explicit sentence or short argument explaining that the pole-sum representation defines D_L for all s away from the unitarity cut and that the replacement is made inside the sum; without this clarification, the advertised extension to left-hand-cut energies is not directly supported by the proof as presented.","section":"Main text, Eqs. (17)-(19); Supplement C"},{"comment":"No numerical demonstration of the new condition is provided. For a proposal whose advertised advantage is validity below the left-hand cut, a simple test—for example, using the one-pole model of Supplement E.1 to generate synthetic FV energies and showing that the zeros of Eq. (20) match them—would directly exercise the continued Eq. (19) in the region where Lüscher's condition is not available. Without such a check, the reader cannot distinguish the claimed extension from an uncontrolled analytic continuation.","section":"Conclusions and overall validation"}],"minor_comments":[{"comment":"Please verify the sign convention in Eq. (51). With the standard 1/(s'-s-iϵ) prescription, Im Iℓ(s)=+ρℓ(s)/(s-slhc) for s above sthr, whereas the displayed minus sign gives the opposite sign. Equation (52) appears to use the opposite sign convention and also appears to drop a term proportional to (s-slhc)Iℓ(slhc) that follows from the consistency condition Nℓ(s)=gDℓ(slhc)/(s-slhc); please reconcile these expressions.","section":"Supplement E.1, Eqs. (51)-(52)"},{"comment":"The statement 'at no point were we required to assume s > slhc' should be qualified, because the proof of N_L=N in Supplement C does assume physical kinematics for the momenta appearing in the sum; the stronger statement is that Eq. (19) is an identity in the external variable s after the termwise replacement, not that no physical-kinematics assumption enters the derivation.","section":"Main text, paragraph after Eq. (20)"},{"comment":"The caption does not clearly identify which shaded band corresponds to the 'validity of Lüscher' region and which to 'validity of this work' in panels (a) and (b); please clarify the legend and the meaning of the blue region.","section":"Figure 1 and caption"},{"comment":"The angular-momentum truncation of the determinant in Eq. (20) is mentioned only in the conclusions. A brief statement in the main text about the expected convergence of the ℓmℓ basis for the parametrizations in Supplement E would help practitioners assess the practical cost of the new condition.","section":"Main text, Eq. (20) and Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The apparent spurious Im in Supplement C looks like a typographical error rather than a deep conceptual flaw; the intended kernel without Im is standard and consistent with the pole-model solution. If that correction is made and the analytic-continuation step is stated explicitly, the central idea is likely sound. No concerns about citation practice or scope. The manuscript fits hep-lat and would be of interest to the lattice-resonance community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely new construction. The N/D-based finite-volume quantization condition, det D_L=0 with D_L from Eq. (19), is not in the prior literature, and the equivalence with Luescher in the elastic region is a sensible consistency check. The derivation from unitarity to Eqs. (16)-(17) is clean, and the supplement is honest about its assumptions.\n\nThe reader's report pins the below-threshold concern on Supplement C proving N_L=N only for physical kinematics p*>0. That is right, but it may not be the fatal gap it appears. In Eq. (19) the sum runs over internal momenta k* that are physical--their invariant masses sit above threshold--so the proven equality applies term by term. For external s below threshold and below slhc, D_L(s) continues meromorphically; the only obstruction would be non-exponential terms in N_L(k*) that are not controlled. The paper never explicitly states this per-term justification, and it should. That is a presentation gap more than a derivation gap, but a careful reader cannot fully close it without more detail.\n\nThe larger weakness is the lack of a numerical test. A simple toy model with a known t-channel pole would check that zeros of Eq. (20) below slhc actually track the finite-volume spectrum. Without that, confidence rests on the formal arguments and the elastic-region equivalence, which is good but not complete.\n\nWho this is for: lattice practitioners working on T_cc or NN, and formalists in finite-volume methods. It deserves serious refereeing. I would send it to review, asking the authors to clarify the analytic continuation and to add a numerical demonstration--at least in an appendix.","headline":"A genuinely new N/D quantization condition; the below-threshold claim is plausible but needs an explicit continuity argument and a numerical check.","tokens_in":630,"tokens_out":1441,"would_cite":true,"duration_ms":80177,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that finite-volume lattice spectra determine the numerator of the N/D representation at energies below the left-hand cut, a region the standard quantization condition cannot reach.","keywords":["finite-volume quantization condition","N/D representation","lattice QCD","left-hand cuts","hadronic resonances","partial-wave amplitude","unitarity","analyticity"],"falsifier":"Take a non-relativistic or relativistic model with a known left-hand cut, such as one-pion exchange, put it in a finite volume, and compute the exact spectrum by diagonalizing the Hamiltonian at several box sizes. Then compare those energies to the zeros of $\\det D_L$ from Eq. (20), with $\\mathcal{N}$ obtained by analytically continuing the infinite-volume numerator below threshold; a discrepancy that grows as a power of $L$, rather than shrinking exponentially, would falsify the central claim.","tokens_in":28927,"feed_emoji":"⚛️","tokens_out":6686,"duration_ms":59345,"temperature":0.7,"pith_summary":"This paper proposes a new model-independent way to extract two-body scattering amplitudes from spectra computed in a finite cubic volume. The idea is to write the partial-wave amplitude in the N/D form, in which the numerator $\\mathcal{N}$ carries all left-hand singularities from crossed-channel exchanges and the denominator $D$ carries the right-hand unitarity cut. The paper derives a quantization condition $\\det D_L = 0$ that relates the finite-volume energies to the infinite-volume numerator $\\mathcal{N}$, and argues that, unlike the original quantization condition, it is valid at energies below the nearest left-hand cut. If correct, it would let lattice QCD constrain resonances whose physics is dominated by particle exchange in regions that standard methods cannot reach.","feed_headline":"Quantization condition now works below left-hand cuts","feed_subtitle":"A determinant condition built from the N/D numerator ties box energies to scattering data where standard methods fail.","key_machinery":"The load-bearing object is the finite-volume denominator matrix $D_L$ arising from the $N/D$ decomposition of the finite-volume partial-wave amplitude, $\\widetilde{M}_L = \\mathcal{N}_L D_L^{-1}$. Unitarity in the box fixes $\\operatorname{Im} D_L$ as a sum over discrete momenta, and the solution $D_L$ has poles at free energies; zeros of $\\det D_L$ are the interacting energies. The argument then replaces $\\mathcal{N}_L$ by the infinite-volume numerator $\\mathcal{N}$ using a Poisson summation (sum-integral difference) step that leaves only exponentially suppressed corrections, so that the quantization condition involves only $\\mathcal{N}$ and known finite-volume kinematic factors.","core_discovery":"The central claim is that, for a single-channel two-body reaction of spinless particles, the finite-volume spectrum is determined by $\\det D_L = 0$, with $D_L$ built from the infinite-volume numerator $\\mathcal{N}$ of the $N/D$ representation via a sum over finite-volume momenta, $D_L = 1 + (\\xi/L^3)\\sum_k L(k,P) \\mathcal{Y}^*(k^*) \\mathcal{Y}(k^*)^T \\mathcal{N}(k^*)/(s-E^*(k)^2)$. The derivation never assumes $s > s_{\\mathrm{lhc}}$, so the condition is claimed to hold arbitrarily far below the left-hand branch point, where the standard condition breaks down. The finite-volume numerator is shown to equal the infinite-volume one up to $O(e^{-\\Delta L})$ corrections in the physical region, and the new condition reduces to the standard quantization condition in the elastic region above the left-hand cut.","pith_inferences":["A natural next test would be a benchmark in a solvable model with a known exact finite-volume spectrum and a left-hand cut, comparing the energies predicted by $\\det D_L = 0$ to exact diagonalization; the paper does not include such a numerical demonstration.","The same replacement logic might extend to coupled-channel and three-body finite-volume equations, where left-hand cuts also complicate standard quantization conditions, though the paper only sketches these as future work.","If the below-threshold analytic continuation holds, the method effectively turns lattice spectra into direct constraints on the couplings of exchanged particles, since those couplings enter $\\mathcal{N}$ through the known positions and residues of left-hand singularities."],"forward_implications":["Lattice spectra in energy regions below left-hand cuts, such as near-threshold states with important $t$-channel exchanges, become usable for amplitude constraints without modeling the singular part of the amplitude.","Parameters of $\\mathcal{N}$ fit from lattice energies feed directly into the $N/D$ integral equations, reconstructing the full partial-wave amplitude and its resonance poles.","In the elastic region above the left-hand cut, the new condition is equivalent to the original quantization condition, so existing extractions remain valid.","The formalism opens a route to systems like $DD^*$ scattering relevant to tetraquark candidates, where one- and multi-particle left-hand cuts lie inside the energy region of interest.","The construction generalizes the standard condition to arbitrary one- and multi-particle exchanges encoded in $\\mathcal{N}$, extending the reach of finite-volume methods."],"supporting_citations":[{"why":"Defines the original quantization condition relating finite-volume spectra to infinite-volume scattering, the baseline this work extends.","marker":"[25]"},{"why":"Provides the standard finite-volume formalism for moving frames used in later lattice applications.","marker":"[26]"},{"why":"Introduces the dispersive ratio representation of the amplitude used throughout the derivation.","marker":"[77]"},{"why":"Establishes the N/D form with the left-hand and right-hand cut separation.","marker":"[78]"},{"why":"Provides the coupled integral equations that reconstruct the full amplitude from the numerator.","marker":"[79]"},{"why":"Gives the matrix manipulations used to turn finite-volume unitarity into the equation for Im D_L.","marker":"[86]"},{"why":"Supplies the explicit form of the F-function matrix used in the reduction to the standard quantization condition.","marker":"[55]"}],"fun_headline_variants":["N/D determinant extends Luscher below left-hand cuts","Finite-volume spectra from N/D, no cut barrier","Quantization condition now reaches below cuts","Unit-analytic condition beats Luscher below cuts","New lattice condition works under left-hand cuts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the finite-volume numerator stays exponentially close to the infinite-volume one even when the momentum is imaginary, but the proof in the paper is written for real physical momenta only.","fun_headline_variants_meta":{"raw":{"variants":["N/D determinant extends Luscher below left-hand cuts","Finite-volume spectra from N/D, no cut barrier","Quantization condition now reaches below cuts","Unit-analytic condition beats Luscher below cuts","New lattice condition works under left-hand cuts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1570,"prompt_tokens":815,"completion_tokens":755,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":684}},"tokens_in":431,"tokens_out":755,"duration_ms":7450,"temperature":1.0,"reasoning_tokens":684,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:58:10.302281+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-relativistic or relativistic model with a known left-hand cut, such as one-pion exchange, put it in a finite volume, and compute the exact spectrum by diagonalizing the Hamiltonian at several box sizes. Then compare those energies to the zeros of $\\det D_L$ from Eq. (20), with $\\mathcal{N}$ obtained by analytically continuing the infinite-volume numerator below threshold; a discrepancy that grows as a power of $L$, rather than shrinking exponentially, would falsify the central claim.","supporting_citations":[],"review_version":1}