{"id":"76703f9f-bf9d-4d60-bc79-1e02a08bb505","arxiv_id":"2607.24648","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An on-shell decomposition of 2+J\to2 electroweak amplitudes isolates OPE poles, logs, and triangle singularities, leaving only smooth short-distance functions.","lead":"The paper derives an on-shell formula for two-hadron electroweak transition amplitudes that keeps left-hand cuts from light-particle exchange explicit. This is a needed step before lattice QCD can extract form factors of near-threshold states such as the Tcc or the deuteron.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Eq. (101) is a long, multi-step algebraic reassembly of four triangle classes and three endcap-dressed exchange terms with no independent cross-check; a toy-model verification — of the kind Refs. [40,41] provided for the OPE-free predecessor — is the missing load-bearing evidence.","rationale":"The reader identified the spinless/single-channel/leading-order-current restriction as the weakest assumption. That is a scoping limitation the authors state explicitly and flag as future work; it does not bear on whether the claim holds within its stated regime, so I rate it less load-bearing than the reader does. My concern is different: the internal verification status of the central algebraic result. The derivation's strategy (isolate on-shell poles, keep OPE kernels partially off-shell to avoid spurious left-hand cuts, regroup skeleton series into M_0, M_E, L, R, C) closely follows Refs. [19, 47], both of which have withstood scrutiny and seen application, which lends the pattern real credibility — this is why I do not recommend downgrading the verdict. The specific novelty here, however, is exactly where the risk concentrates: the four-way split of triangle diagrams by which kernel type (eB vs. E) flanks the loop, and the resulting dressed combinations in Eqs. (88)-(100) and the final Eq. (101). These steps have no analogue that has been independently checked, and the paper offers no toy-model or truncated-order validation, whereas the predecessor formalism was explicitly validated in Refs. [40, 41]. A single concrete computational check (described in concrete_test) would settle the matter and is entirely feasible with the authors' existing numerical toolbox (1D integral equations plus 2D triangle integrals after the azimuthal reduction of Sec. IV C). Secondary observations, noted but not load-bearing: (i) the reality of h^A_on below the 1+J→2 threshold (Eq. 47) relies on analytic continuation that is locally unambiguous but would deserve care if the subprocess itself has nearby left-hand structure; (ii) the endpoint singularities of E^A identified in Sec. IV D (√δ log δ, Eq. A79) live inside W^A_E and its dressings by construction, so they do not threaten the smoothness of A^A, but they do mean the reconstructed amplitude is nonanalytic on parts of the real (s_i, s_f) plane — consistent with, and worth confirming in, the proposed toy-model scan. Overall: no demonstrated error, a well-scoped and careful formal extension; the verdict should stand, with the toy-model cross-check as the single most valuable addition.","tokens_in":102571,"tokens_out":3868,"duration_ms":153672,"concrete_test":"Build a scalar toy model: contact short-range kernel eB plus one spinless exchange with constant Yukawa couplings, scalar current. Compute W^A_df two ways: (a) directly from the Bethe-Salpeter/skeleton series truncated at two loops, evaluating Feynman integrals numerically; (b) from Eq. (101) with the same inputs, solving the M_E integral equation (30) and evaluating the G and W functions as prescribed. Scan (s_i, s_f) across the two-particle threshold and across the left-hand branch point s = 4m² − m_e², at β = 0 and small β > 0. If (a) and (b) agree within integration error and the remainder A^A extracted from (a) is smooth at the branch point, the assembly is confirmed; disagreement localizes which of Eqs. (88)-(100) needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (101) isolates every singularity of W^A_df in the stated kinematic region, leaving A^A real and smooth. The derivation is a chain of skeleton-expansion reorganizations: splitting W^A_df into 1B/2B pieces (Eq. 51), expanding in eB and E insertions (Eqs. 54, 69, 79), regrouping into M_0/M_E and endcaps êL/êR (Eqs. 65-66, 74-75), and finally combining four distinct triangle classes into Ĝ^A_j, Ĝ^A_{R,j}, Ĝ^A_{L,j} (Eqs. 88-100). The structure is plausible and mirrors Refs. [19, 47], but the final assembly — in particular the relative signs and which dressing factors multiply which triangle class in Eqs. (88)-(100), and the combination [f·iĜ^A_R] − W^A_R vs. [f·iĜ^A_L] − W^A_L in Eq. (101) — is verified only by the internal logic of the derivation. Unlike the OPE-free 2+J→2 formalism, which was checked against solvable models in Refs. [40, 41], no cross-check is presented here: no toy model, no truncated-loop-order comparison, no numerical reproduction of even the E^A singularities against a Feynman-diagram evaluation (Figs. 8-9 check only the partial-wave projection of E^A itself, not its placement in Eq. 101). Given the number of moving parts, a single sign or endcap misassignment would leave the claimed smoothness of A^A intact in form but wrong in content, and nothing in the paper would detect it. This is a correctness risk on the central claim, not a scope limitation; it is the least-secured point in an otherwise careful argument.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper derives an on-shell representation of the fully connected 2+J→2 electroweak amplitude for spinless two-hadron states in the presence of one-particle t/u-channel exchange (e.g. OPE). Working at leading order in the current and to all orders in the strong interaction, the authors merge the OPE-free 2+J→2 formalism of Ref. [19] with the left-hand-cut 2→2 formalism of Ref. [47]. The main result, Eq. (101), decomposes W^A_df into (i) an exchange-current piece W^A_E built from a new kernel E^A containing single- and double-pole diagrams, (ii) four classes of triangle functions (G^A_0, G^A_R, G^A_L, G^A_E) with their endcap dressings, and (iii) a residual short-distance matrix A^A claimed to be real and smooth in the stated kinematic region. Appendix A analyzes the partial-wave-projected singularities of E^A via Landau conditions and cross-checks analytic S-wave projections against numerical integration (Figs. 8–9).","tokens_in":103096,"tokens_out":2871,"duration_ms":115950,"significance":"If correct, this closes a real gap: no existing formalism treats 2+J→2 amplitudes when left-hand cuts lie near or under the region of interest, which is precisely the situation for T_cc, the deuteron, and related near-threshold systems. Strengths deserving explicit note: the construction is parameter-free — every object in Eq. (101) except A^A is fixed by on-shell sub-processes; the only integral equation to solve is the 1D partial-wave-projected Eq. (30), with all remaining quantities reduced to 3D (and, via the azimuthal reduction of Sec. IV.C, 2D) integrals; and the partial-wave projection of the new E^A kernel is obtained analytically and verified numerically (Figs. 8–9). The paper also provides a concrete input list for a future T_cc application (Sec. II.1). These are the properties that make the result usable rather than merely formal.","major_comments":[{"comment":"The final assembly (Eq. 101, built from Eqs. 88–100) recombines four triangle classes and three endcap-dressed exchange terms with specific relative signs and dressing assignments (e.g. the minus sign on G^A_{EC,j} in Eq. 98, and the [f·iĜ^A_R]−W^A_R vs. [f·iĜ^A_L]−W^A_L pairing in Eq. 101). This assembly is verified only by the internal logic of the derivation. Unlike the OPE-free predecessor, which was checked against solvable models in Refs. [40, 41], no independent cross-check is given. A misassigned sign or endcap would leave A^A formally smooth but wrong. A tractable concrete test: expand both sides of Eq. (101) to first order in E and E^A and compare against a direct skeleton/Feynman-diagram evaluation of W^A_df at that order. I would like the authors to either supply such a check or state explicitly what has and has not been verified.","section":"Sec. IV.B, Eq. (101)"},{"comment":"Sec. IV.D and App. A identify endpoint singularities for γ>1 with leading behavior √δ log δ (Eq. A79) when one channel is above and the other below threshold, plus 'further regions of non-analytic behavior' deferred to the appendix (visible as yellow contours in Fig. 8). The central claim is that A^A is real and smooth in the whole stated kinematic region. The manuscript should state explicitly whether these non-analytic structures lie inside or outside the region of validity, and confirm that they are entirely carried by W^A_E and the triangle functions rather than leaking into A^A. As written, the smoothness claim is asserted for A^A but the singularity inventory in Sec. IV.D is presented without an explicit one-to-one assignment to terms in Eq. (101).","section":"Sec. IV.D / Appendix A"},{"comment":"The 3D triangle function G^A_{0,j} (Eqs. 82–83) differs from the 4D G^A_j of Ref. [19] (Eq. 42) by a smooth function, with the difference 'absorbed into the definition of A^A' (text below Eq. 83). This means A^A of the present paper is not the same function as A^A in the E→0 limit of Ref. [19], even though Eq. (72)/(87) recover the old structure. Since eventual EFT matching of one- and two-body currents (motivation given in the Introduction) depends on the precise definition of A^A, the manuscript should state this scheme dependence explicitly in Sec. II and Sec. IV.B, not only in passing below Eq. (83).","section":"Sec. IV.A.2, Eqs. (82)–(83)"}],"minor_comments":[{"comment":"The endcaps êL and êR are introduced twice: Eqs. (6)–(7) in Sec. II and Eqs. (65)–(66) in Sec. IV. A cross-reference at the first occurrence would help the reader.","section":"Sec. II / Sec. IV.A.1"},{"comment":"Typo below Eq. (26): 'as long as the cutoff is energy is much higher than all masses' — superfluous 'is'.","section":"Sec. III.A"},{"comment":"The notation switch between E^A_t (Eq. 48) and the figure labels E_t1a, E_t2 (Figs. 8–9, footnote 8) should be unified or a dictionary given; likewise Ĝ vs G hats/tildes in Eqs. (98)–(100) are dense and would benefit from a summary table of the four triangle classes and their dressings.","section":"Sec. IV.D / Figs. 8–9"},{"comment":"In Eq. (50), the third term on the right-hand side uses 'P−k' while the surrounding terms use P_i/P_f; please check for consistency of notation.","section":"Eq. (50)"},{"comment":"The claim in Sec. IV.C that the azimuthal integral in Eq. (112) can be done analytically is illustrated only for rank-1 K^A_j with ℓ≤1; a remark on the general rank/ℓ case (or a reference) would strengthen the reduction claim.","section":"Sec. IV.C"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a competent extension of the authors' (and collaborators') prior formalism line, and the self-citation density is high but the cited building blocks are themselves published derivations, so this is within field norms. The main judgment call for the editor is the absence of any independent check of the assembled Eq. (101): precedent in this subfield (Refs. [40, 41] checked Ref. [19] only after publication) supports publishing the formalism first, and the E^A components are numerically verified, so I do not think the missing toy-model check should block publication, but the authors should be pressed on major comment 1."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this paper finally writes the infinite-volume 2+J→2 amplitude with explicit one-pion-exchange left-hand cuts. That is the missing piece if you want lattice form factors of Tcc-like states or the deuteron once the finite-volume and spin extensions exist.\n\nWhat is new is concrete: the exchange-current kernel E^A (double-pole piece when the exchanged particle couples to the current, plus the single-pole h^A pieces), four classes of triangle functions dressed by M_E, and the final on-shell decomposition (Eq. 101) that isolates every singularity they claim to control and leaves a real smooth A^A. The skeleton expansion, 3D reduction, and geometric resummation into the familiar M_0/M_E endcaps follow the same controlled steps as Refs. [19] and [47]. Appendix A’s Landau analysis and the numerical heat-maps of the partial-wave projected E^A are careful and useful. Self-citation is heavy but legitimate; those earlier papers are the actual building blocks.\n\nThe soft spot is real but proportionate. The assembly of Eq. (101)—especially the relative signs and which endcap multiplies which triangle class—is verified only by internal logic. The OPE-free predecessor had solvable-model checks (Refs. [40,41]); this paper does not. Figs. 8–9 only test the projection of E^A itself. A single mis-assignment would leave A^A formally smooth but wrong in content, and nothing here would catch it. That is a correctness risk on the central claim, not a scope quibble. Spin and multi-channel are flagged as future work and do not undermine the scalar single-channel result as stated.\n\nThis is for people already working in the lattice on-shell amplitude program or nuclear EFTs who need the infinite-volume side before finite-volume matching. It deserves a serious referee. I would accept it for peer review, ask for at least a truncated-order or toy-model sanity check of Eq. (101), and then cite the infinite-volume formulae once they clear.","headline":"Solid formal merger of the authors’ 2+J→2 and OPE left-hand-cut frameworks; Eq. (101) is the real deliverable, but it still lacks the toy-model cross-check that secured the OPE-free predecessor.","tokens_in":104617,"tokens_out":547,"would_cite":true,"duration_ms":15304,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"On-shell 2+J\to2 amplitudes isolate every left-hand cut from light-particle exchange, leaving one smooth real short-distance function.","keywords":["2+J\to2 amplitudes","left-hand cuts","one-particle exchange","on-shell representation","triangle singularities","electroweak form factors","lattice QCD","hadron structure"],"falsifier":"Compute the partial-wave projection of the double-pole exchange kernel for a toy mass spectrum that places the left-hand cut near threshold and verify that the only remaining non-analyticities match the Landau singularities predicted in the appendix; any extra singularity would falsify the completeness of the decomposition.","tokens_in":104254,"feed_emoji":"⚛️","tokens_out":947,"duration_ms":16392,"temperature":0.7,"pith_summary":"This paper supplies a complete analytic decomposition of electroweak transition amplitudes in which two hadrons scatter both before and after a current insertion, when light-particle exchange (for example one-pion exchange) produces left-hand cuts near threshold. Every singularity that can appear—two-particle thresholds, triangle singularities, logarithmic branch points, and even one-particle poles when the exchanged particle itself couples to the current—is written in closed form from known subprocesses. What remains is a single real matrix of short-distance transition functions that stays smooth below the first unaccounted-for threshold. The construction therefore converts the problem of extracting form factors of near-threshold bound states or resonances into the practical task of determining that smooth matrix, either from lattice matrix elements or from data. The result is restricted to spinless particles in one elastic channel and treats the current at leading order, yet it already covers the kinematic region needed for systems such as the deuteron or the Tcc tetraquark once spin is restored.","feed_headline":"Every left-hand cut in 2+J\to2 amplitudes is isolated","feed_subtitle":"One smooth real function remains, ready for lattice or data constraints on near-threshold form factors","key_machinery":"The exchange-current kernel E^A (single- and double-pole diagrams in which the current couples to the exchanged particle or to the exchange vertices) together with the dressed end-caps built from the hadronic one-particle-exchange amplitude; these objects allow every singular loop to be written as a three-dimensional integral that can be reduced to two dimensions.","core_discovery":"The fully connected 2+J\to2 amplitude admits an on-shell representation in which the divergence-free piece is expressed solely through the purely hadronic amplitude (including its one-particle-exchange dressing), a new exchange-current kernel E^A, a finite set of triangle integrals, and one real smooth short-distance matrix A^A. All left-hand singularities generated by light-particle exchange are thereby isolated and removed from the unknown dynamical content.","pith_inferences":["Once the finite-volume counterpart is written, lattice calculations of Tcc or deuteron electromagnetic radii become feasible without uncontrolled left-hand-cut systematics.","The double-pole term in E^A is the relativistic analogue of the meson-exchange current familiar from nuclear physics; the decomposition therefore unifies lattice and effective-field-theory treatments of two-body currents.","The reduction of all triangle integrals to two-dimensional numerical integrals opens a practical path to global fits of multi-hadron form factors below inelastic thresholds."],"forward_implications":["Form factors of near-threshold states controlled by one-pion exchange can be extracted once the smooth matrix A^A is fixed by lattice or data.","The same skeleton expansion supplies the infinite-volume side of a future finite-volume matching formula for 2+J\to2 matrix elements.","Long-range contributions to processes such as deuteron photodisintegration or coherent neutrino-deuteron scattering become analytically controlled.","Coupled-channel and spin extensions reduce to matrix enlargements of the same objects once the angular-momentum algebra is enlarged."],"fun_headline_variants":["On-shell form isolates every left-hand cut in 2+J→2 amplitudes","Left-hand singularities fully isolated from electroweak 2+J→2 amplitudes","Exchange cuts removed, one smooth real short-distance function remains","All light-particle left-hand cuts peeled from connected 2+J→2 amplitudes","One-pion-exchange poles and logs isolated in two-hadron electroweak amplitudes"],"cache_read_input_tokens":98432,"weakest_assumption_plain":"Everything is derived for spinless particles in a single elastic two-body channel with the current kept strictly at leading order; if spin or multi-channel dynamics introduce qualitatively new singular structures, the isolation claim no longer holds as written.","fun_headline_variants_meta":{"raw":{"variants":["On-shell form isolates every left-hand cut in 2+J→2 amplitudes","Left-hand singularities fully isolated from electroweak 2+J→2 amplitudes","Exchange cuts removed, one smooth real short-distance function remains","All light-particle left-hand cuts peeled from connected 2+J→2 amplitudes","One-pion-exchange poles and logs isolated in two-hadron electroweak amplitudes"]},"model":"grok-4.5","effort":"low","cost_usd":0.005282,"raw_usage":{"total_tokens":1436,"prompt_tokens":779,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":52824000,"prompt_tokens_details":{"text_tokens":779,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":567,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":779,"tokens_out":90,"duration_ms":9105,"temperature":1.0,"reasoning_tokens":567,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T09:09:34.306577+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the partial-wave projection of the double-pole exchange kernel for a toy mass spectrum that places the left-hand cut near threshold and verify that the only remaining non-analyticities match the Landau singularities predicted in the appendix; any extra singularity would falsify the completeness of the decomposition.","supporting_citations":[],"review_version":1}