{"id":"d11b8108-5d50-4ba6-90cf-92b886f201b9","arxiv_id":"2501.16804","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A quantum-mechanics model shows that the left-hand cut does not change the binding momentum extracted from IR-regulated potentials, validating the HAL QCD method's treatment of near-threshold states like T_cc.","lead":"This paper studies whether the 'left-hand cut', a mathematical obstruction in scattering calculations, affects binding energies extracted with the HAL QCD method used in lattice quantum chromodynamics. Using a simple quantum mechanical model with a Yukawa potential, it shows that the binding momentum survives the infinite-range limit even where analytic continuation fails, and gives practical guidance for HAL QCD analyses.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of LHC-insensitive binding energies is given only for local, energy-independent potentials; the transfer to nonlocal, energy-dependent HAL QCD potentials is asserted, not derived.","rationale":"The reader's weakest assumption correctly identifies the bridge from the 1D local Yukawa model to the nonlocal, energy-dependent HAL QCD potential. This is the single most load-bearing gap because every broader conclusion in §3 and §4 depends on it: the proposed fitting strategy, the treatment of virtual states, and the reassurance about T_cc all rely on the claim that the long-distance tail of the HAL QCD potential controls the LHC in the same way as in the local case. The local-potential part of the paper is coherent: the Volterra representation with a finite IR cutoff is standard, the convergence/divergence analysis in §2.3 is plausible, and the numerical example in §2.4 illustrates the expected behavior. The problem is specifically that the paper does not extend this analysis to the nonlocal setting. That gap warrants a conditional verdict rather than rejection, because the conclusion may well be true for HAL QCD potentials, but it is currently unproven. The reader's verdict of CONDITIONAL is therefore appropriate, and this stress-test does not change it. The suggested concrete test would settle the issue by directly comparing pole extraction and LHC position for a nonlocal toy potential against the local-tail prediction.","tokens_in":10343,"tokens_out":5239,"duration_ms":50398,"concrete_test":"Construct a toy two-body system with a genuinely nonlocal potential, e.g., a separable potential V(k,k') = lambda g(k) g(k') with g(k) = 1/(k^2 + m_pi^2), tuned so that a bound state exists with Im k_b > m_pi/2. Generate the corresponding HAL QCD potential from the NBS wave functions using the standard time-dependent formula, introduce the IR cutoff R, and compute the pole of the resulting finite-R amplitude. If the pole converges to the exact k_b only when the local tail is separately imposed, or if the LHC position differs from the prediction of the local-tail argument, then the assertion in §3.4 fails for nonlocal HAL QCD potentials.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central mathematical result in §2.3 — that the IR-cutoff S-matrix S(k,R) reproduces the binding momentum even when Im k_b > m_pi/2 — is proven for a local, single-channel, energy-independent potential U(r) (Eqs. 4–5). The paper's headline conclusion in §4, 'the binding energy in the HAL QCD method is not affected by the LHC even if it exists,' requires this result to hold for actual HAL QCD potentials, which are nonlocal and energy-dependent. Section 3, item 4 asserts that 'the long distance behavior of the potential controls the position of the LHC, as discussed in the previous section,' but the previous section contains no treatment of nonlocal kernels or energy-dependent potentials. For a nonlocal potential, the left-hand cut is not simply determined by the Fourier transform of the coordinate-space tail; singularities of the momentum-space potential V(k,k') can generate cuts at different locations, and energy dependence can shift poles relative to cuts. This is not an objection to the local-potential derivation, which appears internally consistent; it is a missing step in the argument that the local result carries over to the HAL QCD setting. Unless this bridge is supplied, the broad claim about HAL QCD binding energies, including the T_cc statement, is not proven.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript investigates whether the left-hand cut (LHC) of a scattering amplitude affects binding energies extracted by the HAL QCD method. For a local, single-channel, energy-independent Schrödinger potential with a Yukawa tail, the authors show that the S-matrix S(k,R) with an infrared cutoff R is well defined for finite R, that S(k,R) approaches the analytic continuation of the cutoff-free S-matrix for Im k < m_pi/2, and that for Im k > m_pi/2 the two differ except at the bound-state pole k_b. A numerical Yukawa-plus-Gaussian example illustrates these behaviors and suggests that k_b(R) is nearly R independent at large R. The paper then proposes a practical protocol for HAL QCD analyses: determine the long-distance tail of the potential, estimate the LHC position, solve the Schrödinger equation directly if the bound state lies below the LHC, and include expected one- or two-pion tails as alternative fits. The concluding claim is that the HAL QCD binding energy is not affected by the LHC even if it exists, and that the T_cc virtual pole from Ref. [6] lies above the 1-pion LHC and remains valid.","tokens_in":10665,"tokens_out":8788,"duration_ms":83010,"significance":"The question addressed is timely and relevant: the LHC has been invoked as a possible obstacle to the reliability of lattice extractions of the T_cc pole, and the HAL QCD method is one of the main tools in that discussion. The finite-R construction in Section 2 is elementary but useful: Eq. (8) makes manifest that the IR-truncated S-matrix is meromorphic in k and that the LHC is an artifact of taking R to infinity. The mathematical derivation is self-contained and does not reduce to fitted quantities; the numerical example is illustrative. If the missing bridge to actual HAL QCD potentials is supplied, this work would provide a clear protocol (long-distance fit, LHC position estimate, direct Schrödinger solution) and would directly address an active controversy. As it stands, the strongest conclusion in Section 4 outruns the proof, because the theorem is proven only for local potentials while HAL QCD potentials are in general nonlocal and, in truncated derivative expansions, energy dependent.","major_comments":[{"comment":"The central conclusion that the binding energy in the HAL QCD method is unaffected by the LHC is derived only for a local, single-channel, energy-independent potential U(r) in Section 2. HAL QCD potentials are nonlocal objects extracted from NBS wave functions, and once a derivative expansion is truncated they are also energy dependent. The statement in Section 3, item 4 that 'the long distance behavior of the potential controls the position of the LHC' is asserted, not derived, for such kernels. For a nonlocal kernel V(r,r'), the singularities of the momentum-space kernel V(k,k') need not be fixed by the coordinate-space tail of the diagonal part V(r), and energy dependence can shift poles relative to cuts. Without a concrete argument or a nonlocal/energy-dependent model showing that the local result carries over, the claims in Section 4, including the T_cc statement, are not established by the paper's mathematics. At minimum, the conclusion should be restricted to the local-potential setting or explicitly supported by a derived bridge for HAL QCD potentials.","section":"Section 3, item 4; Section 4"},{"comment":"The proof that S(k,R) converges for Im k < m_pi/2, diverges for Im k > m_pi/2 except at k=k_b, and that k_b(R) converges to k_b is incomplete at the bound-state pole. Equation (10), phi(k,r) approximately -F(k,infty) e^{-ikr}/(2ik), is the leading large-r behavior only when F(k,infty) is nonzero. At k=k_b the coefficient F(k_b,infty) vanishes, and the bound-state tail is governed by the first term in Eq. (7), so the vanishing of the divergent integral in Eq. (9) does not by itself prove convergence of F(-k_b,R) or give the rate of approach. To establish lim_{R->infty} k_b(R)=k_b one needs a uniform estimate of the zeros of F(k,R) in a neighborhood of k_b, for example F(k_b,R)->0 together with a lower bound on F'(k_b,R); the manuscript does not provide this. The numerical example in Section 2.4 is consistent with the claim but is only a single illustrative potential.","section":"Section 2.3, Eqs. (9)-(10)"}],"minor_comments":[{"comment":"The displayed integration limits appear reversed: the second term should run from the fixed cutoff \\bar R to the large radius R, not from R to \\bar R.","section":"Eq. (9)"},{"comment":"The figure caption refers to 'black symbols' while the text describes a 'black line' for the analytic continuation; these should be made consistent. In addition, the statement that the R dependence of k_b(R) is 'very very small' should be replaced by a quantitative statement, for example the shift |k_b(R)-k_b(25 fm)| in units of m_pi.","section":"Section 2.4 and Fig. 2"},{"comment":"The statement that 'the analytic continuation knows the existence of the LHC' is vague; the authors should specify which object is being continued, through which Riemann sheet, and how the complex nature of k cot delta emerges from the HAL QCD potential in practice.","section":"Section 3, item 4"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings-style contribution whose local-potential section is sound as far as it goes, but the headline claim in Section 4 is broader than the proof. The use of Ref. [6], the authors' own previous lattice result, to support the T_cc statement is acceptable, but the revision should either supply the missing bridge to nonlocal HAL QCD potentials or substantially narrow the claim. I would ask the editor to require one of these before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper's genuinely new bit is in Section 2: for a local potential with a Yukawa tail, putting an IR cutoff R at large r gives an S-matrix S(k,R) that, as R goes to infinity, converges to the analytic continuation for Im k < m_pi/2, diverges for Im k > m_pi/2, and does so in a way that the bound-state pole k_b(R) still converges to the true binding momentum. That explicit statement — including the exceptional pole point — I haven't seen written out before, and it clears up a piece of folk wisdom. The practical recipe in Section 3 (look at the long-range tail, locate the LHC, solve the Schroedinger equation directly rather than using ERE crossing below the LHC) is sensible and actionable. I also credit the paper for being honest that the finite-R path is well-defined even when the analytic continuation is not.\n\nWhere I part company with the paper is at the transfer step. The theorem is for a local, energy-independent, single-channel potential. HAL QCD potentials are nonlocal and energy-dependent. The left-hand cut of a nonlocal kernel is not simply fixed by the coordinate-space tail; singularities of V(k,k') land at their own locations. Section 3, item 4 asserts that 'the long distance behavior of the potential controls the position of the LHC, as discussed in the previous section,' but the previous section contains no nonlocal or energy-dependent analysis. So the headline claim in Section 4 — that binding energies in HAL QCD are not affected by the LHC — is not proven for the actual HAL QCD setting. That is a missing step, not a contradiction; the local derivation looks internally consistent.\n\nOther soft spots are minor by comparison. The proof that k_b(R) converges when Im k_b > m_pi/2 is sketched via a vanishing asymptotic coefficient; I'd like the argument made explicit. The numerical example does not ship code or data, so the R-independence of k_b(R) is illustrative rather than reproducible. The T_cc defense in the final paragraph is tied to Ref. [6], the authors' own lattice result, but that is not circular — they are just checking where their previous pole sits relative to the 1-pion LHC.\n\nWho should read this: lattice practitioners using HAL QCD, and anyone worried that the LHC invalidates potential-derived binding energies. It is a proceedings-style paper, short, and the central claim outruns the math, but the math it does contain is worth having. I would send it to a referee. The referee should ask for either a derivation of the nonlocal case or a modest rewording that limits the claim to local potentials and flags the HAL QCD transfer as an assumption.","headline":"Clean local-potential result about IR-cutoff S-matrices and the left-hand cut, but the headline claim for HAL QCD outruns the proof because nonlocal, energy-dependent potentials are never treated.","tokens_in":11188,"tokens_out":2938,"would_cite":true,"duration_ms":24465,"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":"This paper shows that binding energies obtained with the HAL QCD method are unaffected by the left-hand cut, even when a bound state sits on it, because the infrared-regulated S-matrix has a pole that converges to the exact binding…","keywords":["left-hand cut","HAL QCD method","binding energy","Yukawa potential","infrared cutoff","S-matrix","Tcc tetraquark","lattice QCD"],"falsifier":"Take a non-local or energy-dependent potential with a known bound state and a Yukawa tail, compute the pole of the regulated S-matrix as $R$ grows, and check whether the pole converges to the exact binding momentum when $\\operatorname{Im} k_b > m_{\\pi}/2$; a failure would falsify the claim that binding energies in HAL QCD are generically LHC-independent. A simpler version: in the same model as the paper, increase the coupling until the bound state lies well above the LHC branch point and test whether $k_b(R)$ is still stable at $R = 25$ fm.","tokens_in":10169,"feed_emoji":"⚛️","tokens_out":10172,"duration_ms":76302,"temperature":0.7,"pith_summary":"The paper asks whether the left-hand cut (LHC) caused by pion exchange invalidates binding energies extracted with the HAL QCD method. Working first in non-relativistic quantum mechanics with a Yukawa potential, it shows that the S-matrix with a finite infrared cutoff $R$ is well defined for every complex momentum. As $R$ goes to infinity, this regulated S-matrix converges to the analytic continuation of the exact S-matrix for $\\operatorname{Im} k < m_{\\pi}/2$, diverges for $\\operatorname{Im} k > m_{\\pi}/2$, but always keeps a pole at the binding momentum $k_b$, which converges to the exact $k_b$. From this the authors conclude that the binding energy in the HAL QCD method is not affected by the LHC, and they give a step-by-step recipe for treating the LHC in practice. This matters because the doubly charmed tetraquark $T_{cc}$ is a shallow state that sits near such cuts.","feed_headline":"HAL QCD binding energies survive the left-hand cut","feed_subtitle":"Even when a bound state sits on the pion-exchange cut, the HAL QCD pole converges to the exact binding momentum.","key_machinery":"The load-bearing object is the infrared-regulated S-matrix $S(k,R)=F(-k,R)/F(k,R)$, built from the Volterra integrals $F(k,R)=1+\\int_0^R dr' e^{ikr'} U(r') \\varphi(k,r')$ for a potential $U(r)$, with the Yukawa tail $e^{-m_{\\pi} r}/r$ as the model LHC source. The key estimate is that the second term in $F(-k,R)$ contains a factor $e^{(2\\operatorname{Im} k - m_{\\pi}) r'}$, so it converges as $R \\to \\infty$ only for $\\operatorname{Im} k < m_{\\pi}/2$, diverges for $\\operatorname{Im} k > m_{\\pi}/2$, yet remains convergent at the bound-state momentum $k_b$ because the bound-state condition $F(k_b,\\infty)=0$ kills the dangerous term. This single mechanism simultaneously explains the convergence below the LHC, the divergence above it, and the stability of the binding pole.","core_discovery":"The central claim is that the binding momentum (hence binding energy) obtained by the HAL QCD method is free of left-hand-cut contamination. The mechanism is that any realistic potential has an infrared cutoff: the potential vanishes beyond some large $R$, as a lattice simulation effectively enforces. With that cutoff, the S-matrix $S(k,R)$ is analytic and well defined everywhere in the complex $k$ plane; in the $R \\to \\infty$ limit, $S(k,R)$ agrees with the analytic continuation of the exact S-matrix for $\\operatorname{Im} k < m_{\\pi}/2$, while for $\\operatorname{Im} k > m_{\\pi}/2$ it diverges except at the binding pole $k = k_b$. Because the pole $k_b(R)$ of $S(k,R)$ converges to the exact binding momentum $k_b$ even when the pole lies on or beyond the LHC, the HAL QCD binding energy remains correct. For virtual states below the LHC branch point, the paper recommends analytic continuation through the HAL QCD potential and, if needed, including the offending one-pion exchange tail in the fit to estimate systematic errors.","pith_inferences":["The proof is carried out for local, single-channel, energy-independent potentials; real HAL QCD potentials are non-local and energy-dependent, so the claim that binding energies are LHC-free should be re-checked numerically in a non-local toy model before being taken as universal.","The same infrared-cutoff logic suggests that any finite-volume lattice calculation implicitly regulates the LHC, so exact finite-volume energies are not polluted by the cut; the danger is only in the analytic continuation or effective-range fits used to interpret them.","A concrete testable extension: in the Yukawa-plus-Gaussian model, scan the coupling so the bound state moves from $\\operatorname{Im} k < m_{\\pi}/2$ to $\\operatorname{Im} k > m_{\\pi}/2$ and verify that $k_b(R)$ remains within numerical error for a fixed large $R$; the paper shows this for one parameter set.","The procedure's emphasis on fitting the long-range tail suggests that systematic uncertainties of future HAL QCD studies of near-threshold states should be dominated by the uncertainty in the one- and two-pion exchange tails, not by the LHC itself."],"forward_implications":["The previous HAL QCD determination of $T_{cc}$ as a virtual state above the one-pion left-hand cut remains valid; the LHC does not move its pole.","When a bound state lies below the LHC branch point, the correct procedure is to solve the Schrödinger equation directly rather than find an intersection of $k \\cot \\delta$ with the bound-state condition.","For virtual states below the LHC branch point, analytic continuation through the HAL QCD potential automatically makes $k \\cot \\delta$ complex on the cut, so apparent virtual poles from simple effective-range fits can disappear.","Including the one-pion exchange tail in the potential fit, even when the data do not show it, gives a systematic error estimate for LHC effects.","The long-distance part of the potential controls the LHC position, so the HAL QCD method can explicitly control LHC effects better than finite-volume methods."],"supporting_citations":[{"why":"Reports the experimental observation of the $T_{cc}$ tetraquark, the physical state whose binding the paper shields from LHC effects.","marker":"[1]"},{"why":"Provides the experimental $T_{cc}$ parameters and pole position compared with lattice results in the introduction.","marker":"[2]"},{"why":"Supplies the finite-volume effective-range expansion results whose virtual pole is reanalyzed and shown to be affected by the LHC.","marker":"[5]"},{"why":"Gives the HAL QCD $D^*D$ potential and the $T_{cc}$ virtual-state result that is the main application defended in the paper.","marker":"[6]"},{"why":"Raises the left-hand-cut objection to finite-volume analyses, the challenge the paper answers with the HAL QCD method.","marker":"[7]"},{"why":"Provides a modified finite-volume formula including the LHC, a competing approach the paper contrasts with the HAL QCD potential method.","marker":"[10]"}],"fun_headline_variants":["HAL QCD binding poles immune to left-hand cut","Lattice binding energies beat the left-hand cut","Infrared cutoff saves HAL QCD binding energies","Left-hand cut no match for HAL QCD poles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the HAL QCD potential can be treated as an effectively local, single-channel, energy-independent potential whose long-distance tail sets the left-hand cut; if the actual potential violates this, the convergence argument for the binding pole need not apply.","fun_headline_variants_meta":{"raw":{"variants":["HAL QCD binding poles immune to left-hand cut","Lattice binding energies beat the left-hand cut","Infrared cutoff saves HAL QCD binding energies","Left-hand cut no match for HAL QCD poles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1375,"prompt_tokens":992,"completion_tokens":383,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":321}},"tokens_in":608,"tokens_out":383,"duration_ms":3911,"temperature":1.0,"reasoning_tokens":321,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T10:36:15.041604+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-local or energy-dependent potential with a known bound state and a Yukawa tail, compute the pole of the regulated S-matrix as $R$ grows, and check whether the pole converges to the exact binding momentum when $\\operatorname{Im} k_b > m_{\\pi}/2$; a failure would falsify the claim that binding energies in HAL QCD are generically LHC-independent. A simpler version: in the same model as the paper, increase the coupling until the bound state lies well above the LHC branch point and test whether $k_b(R)$ is still stable at $R = 25$ fm.","supporting_citations":[],"review_version":1}