Left-hand cut and the HAL QCD method

Sinya Aoki; Takumi Doi; Yan Lyu

REVIEW 2 major objections 3 minor 1 cited by

Left-hand cut and the HAL QCD method

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.16804 v1 pith:6IW7CZEP submitted 2025-01-28 hep-lat hep-phnucl-th

Sinya Aoki , Takumi Doi , Yan Lyu This is my paper · ORCID

classification hep-lathep-phnucl-th

keywords left-handcutHALQCDmethodbindingenergyYukawapotentialinfraredcutoffS-matrixTcctetraquarklattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The 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.

Where 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.

Other 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.

Who 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[Section 3, item 4; Section 4] 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 2.3, Eqs. (9)-(10)] 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.

minor comments (3)

[Eq. (9)] 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 2.4 and Fig. 2] 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 3, item 4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the QM derivation is self-contained; the HAL QCD transfer is an unproven assumption, not a circular reduction.

full rationale

The central derivation is self-contained. Section 2.3 defines S(k,R) via the Volterra equation (7)-(8) and proves convergence/divergence of F(-k,R) by Eq. (9), using only the asymptotic form (10) of the regular solution; the pole k_b(R) is a zero of F(k,R), and its convergence to the exact binding momentum follows from F(k_b,∞)=0 rather than from any fitted parameter. The numerical example in Sec. 2.4 has parameters adjusted for illustration, but the comparison between S(k,R) and S_anal(k) is not used as input to the theorem. The application to HAL QCD in Sec. 3, item 4 ("the long distance behavior of the potential controls the position of the LHC, as discussed in the previous section") involves an unproven transfer from local, energy-independent potentials to the nonlocal, energy-dependent HAL QCD potentials; this is a gap in the argument, not a circular reduction, since no equation makes the HAL QCD binding energy equal to an input by construction. The T_cc validity statement in Sec. 4 relies on Ref. [6], a self-citation, but Ref. [6] is an independent lattice QCD calculation with data, so it is external evidence rather than a self-referential derivation. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in via citation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper's own derivation is a toy model with one explicitly varied parameter (the IR cutoff R) and an illustrative potential. The central new claims about HAL QCD rest on domain assumptions that the HAL QCD potential behaves like a local Yukawa-type potential and that the T_cc potential from Ref [6] is reliable.

free parameters (2)

Yukawa coupling g and Gaussian short-range parameters in Section 2.4 = not stated in the paper
The example potential is 'adjusted so that one bound state appears below the branch point of the LHC' (Section 2.4). No numerical values are given, so the demonstration is not fully specified.
Fit coefficients a_i, b_i of V_2pi_fit(r) in Eq. (12), taken from Ref [6] = not given in this paper
The discussion of T_cc in Section 3 depends on this fitted potential and its long-distance behavior.

assumptions (6)

standard math A compactly supported potential gives a meromorphic S(k,R) with no left-hand cut for finite R (Eq. 8).
Standard Volterra and Jost function theory; the paper relies on it without proof.
domain assumption The asymptotic approximation phi(k,r) ~ -F(k,∞) e^{-ikr}/(2ik) for large r and Im k ≥ 0 governs the convergence of F(-k,R) (Eqs. 9-10).
This approximation fails at k_b where F(k,∞)=0; the paper argues convergence anyway, but the proof is only sketched.
domain assumption A single-channel non-relativistic Yukawa potential captures the LHC structure relevant to the HAL QCD method (Section 2.2).
The entire transfer to lattice QCD rests on the analogy between the pion-exchange tail in D*D scattering and the Yukawa term.
domain assumption The LHC position for D*D scattering is determined by u-channel pion exchange; Eqs. (1)-(3) give the branch points.
The paper uses this to estimate k^2_-/m_pi^2 = -0.02 (1-pion) and -0.77 (2-pion) in Section 3.
domain assumption The HAL QCD potential is effectively local and energy-independent enough that the 1D quantum-mechanics conclusion transfers.
Section 3 item 4 asserts that long-distance behavior controls the LHC, so analytic continuation through the potential sees the LHC automatically. This is the least defended step.
domain assumption The D*D potential and virtual pole of Ref [6] are correct, so the T_cc HAL QCD result is above the LHC branch point.
The final validity claim for T_cc uses the authors' own previous lattice calculation as input.

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Cite this review

Pith. "Pith review of Left-hand cut and the HAL QCD method." pith.science (2026). https://pith.science/paper/6IW7CZEP

@misc{pith2026250116804,
  author       = {Pith},
  title        = {Pith review of: Left-hand cut and the HAL QCD method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6IW7CZEP}},
  note         = {Machine review of arXiv:2501.16804}
}

abstract

We investigate how the left-hand cut (LHC) problem is treated in the HAL QCD method. For this purpose, we first consider the effect of the LHC to the scattering problem in non-relativistic quantum mechanics with potentials. We show that the $S$-matrix or the scattering phase shift obtained from the potential including the Yukawa term ($e^{- m_\pi r}/r$) with the infra-red (IR) cutoff $R$ is well-defined even for the complex momentum $k$ as long as $R$ is finite, and they are compared with those obtained by the analytic continuation without the IR cutoff. In the $R\to\infty$ limit, the phase shift approaches the result from the analytic continuation at ${\rm Im}\, k < m_\pi/2$, while they differ at ${\rm Im}\, k > m_\pi/2$, except $k= k_b$, where $k_b$ is the binding momentum. We also observe that $k_b$ can be correctly obtained even at finite but large $R$. Using knowledge obtained in the non-relativistic quantum mechanics, we present how we should treat the LHC in the HAL QCD potential method.

Figures

Figures reproduced from arXiv: 2501.16804 by the authors.

**Figure 1.** (Left) The inverse scattering length 1/𝑎0 for the 𝑆-wave 𝐷 ∗𝐷 system with 𝐼 = 0 as a function of 𝑚 2 𝜋 , obtained from lattice QCD calculations by Refs. [3] (blue square), [4](green diamond), [5](yellow triangle), and [6](magenta circle), together with its chiral extrapolation of 1/𝑎0 linear in 𝑚 2 𝜋 (violet solid line). The black plus shows the value obtained from the potential extrapolated to 𝑚𝜋 = 135 MeV[6], whil… view at source ↗

**Figure 2.** (Left) A comparison of 𝑘 cot 𝛿(𝑘)/𝑚𝜋 at 𝑘 2 < 0 region between the analytic continuation (black line, the real part only) and results with the IR cutoff, 𝑅 = 5 fm(green), 10 fm(blue), 15 fm(red), 20 fm(violet) and 25 fm(magenta), together with the bound state condition − √︁ −(𝑘/𝑚𝜋) 2 (black dashed line). (Right) A difference between 𝑘 cot 𝛿𝑅(𝑘)/𝑚𝜋 and the real part of 𝑘 cot 𝛿(𝑘)/𝑚𝜋. behaviors observed in the left of… view at source ↗

**Figure 4.** FIG. 4. Fit results for the lattice data from Ref. [ Fi3(f) h ∗ il () [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Recent progress on charmed hadron interactions from lattice QCD
hep-lat 2025-09 conditional novelty 2.0 of 10

A review of lattice QCD results on charmed hadron interactions: T_cc near threshold, an attractive N-J/psi force, and an Omega_ccc-Omega_ccc state near unitarity.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Y. Lyu, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda and J. Meng, Phys. Rev. Lett.131(2023) no.16, 161901 doi:10.1103/PhysRevLett.131.161901 [arXiv:2302.04505 [hep-lat]]

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