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REVIEW 3 major objections 4 minor 39 references

Hermitizing the HAL QCD potential in the derivative expansion

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The non-Hermitian HAL QCD potential can be hermitized order by order to all orders in the derivative expansion, with the next-to-leading-order case exact.

desk verdict A genuine and useful technical advance for the HAL QCD program—hermitizing the derivative-expanded potential order by order—with explicit NLO formulas and a worked Xi-Xi application, but the central existence claim rests on regularity conditions that are stated but not verified. read the letter →

arxiv 1909.00656 v2 pith:EADDB2HF submitted 2019-09-02 hep-lat nucl-th

Sinya Aoki , Takumi Iritani , Koichi Yazaki This is my paper · ORCID
classification hep-latnucl-th
keywords HALQCDpotentialderivativeexpansionhermitizationNambu-Bethe-SalpeterwavefunctionXi-Xiscatteringlatticephaseshiftnon-local
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 aims to show that the non-Hermiticity of the HAL QCD potential is removable, not fundamental. It claims that, order by order in the derivative expansion, a multiplicative change of wavefunction produces a Hermitian Hamiltonian with exactly the same scattering phase shifts, and that the next-to-leading order (first- and second-derivative terms) can be hermitized exactly in closed form. The motive is practical: a Hermitian two-body potential can be compared with phenomenological interactions and used in many-body calculations, where non-Hermitian potentials are awkward. The paper applies the construction to $\Xi\Xi(^1S_0)$ scattering from lattice QCD, finding that the NLO terms shift the phase shift only slightly, so that the leading-order potential is a good approximation for that system.

What carries the argument

The machine of the paper is the multiplicative wavefunction redefinition $\psi = R\varphi$. $R$ is chosen order by order so that odd-derivative terms in the transformed Hamiltonian vanish, leaving a Hermitian operator built only from local terms and even-derivative terms. For the NLO block $U_1 = V_1 + V_2$, $R_1$ is fixed by a first-order differential equation whose rotationally symmetric solution is an exponential of an integral containing the denominator $1 - \frac{m}{2}V_2(r)$. At order $n$, the transformation factor is $R_{(n)} = R_{(n-1)}(1 + R_n)$ with $R_n$ containing only even derivatives, and the conditions that cancel the highest remaining odd-derivative terms are linear first-order PDEs solved sequentially down from $2n+1$ derivatives. The induction pairs $V_{2n-1}$ with $V_{2n}$ as the $n$-th order, so each order contributes one Hermitian even-derivative block and one transformation factor.

What would settle it

Evaluate $1 - m_\Xi V_2^{NLOA}(r)$ for the fitted Xi-Xi potential over $0 < r < 3.5$ fm. If it vanishes anywhere, $R_1$ diverges and the claimed exact hermitization fails in that channel; if it stays positive on the whole domain, the NLO construction is regular and reproduces the phase shifts.

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Extended reading notes

Core claim

The paper's central claim is that non-Hermiticity in the HAL QCD potential is a gauge artifact of the derivative expansion, not an intrinsic property: there exists an equivalent Hermitian Hamiltonian at every truncation. For next-to-leading order, meaning the first- and second-derivative terms $V_1$ and $V_2$, the hermitization is exact and explicit: one solves a first-order equation for the transformation factor $R_1$, and the resulting local potential $\widetilde V_0$ is given in closed form. For higher orders, the paper argues by induction that choosing $R_{n+1,2n}$, then $R_{n+1,2n-2}$, and so on down to $R_{n+1,0}$ removes all odd-derivative terms, so each truncated Hamiltonian has a Hermitian image containing only even derivatives. Applied to $\Xi\Xi(^1S_0)$, the NLO corrections to the phase shift are small, and the Hermitized local part $\widetilde V_0^{NLOA}$ tracks the full phase shift over a wider energy range than the bare $V_0^{NLOA}$.

Load-bearing premise

The construction works only if the derivative expansion of the non-local potential converges and the transformation factors $R$ stay finite, in particular if denominators like $1 - \frac{m}{2}V_2(r)$ never vanish where the potential is fitted.

Editorial extensions

If this is right

  • Hermitized NLO potentials can be fed into standard quantum many-body methods that require Hermitian interactions, without changing the infinite-volume phase shifts.
  • In systems where the denominator $1 - \frac{m}{2}V_2(r)$ stays positive, the NLO Hermitian potential is not a perturbative approximation; it is exactly equivalent to the original truncated non-Hermitian Hamiltonian.
  • At higher orders, the induction gives an algorithmic recipe: choose $R_{n+1,2n}$, then $R_{n+1,2n-2}$, and so on, to cancel odd-derivative terms, so each truncation has a Hermitian image with only even derivatives.
  • For $\Xi\Xi(^1S_0)$, the Hermitized local part $\widetilde V_0^{NLOA}$ tracks the exact phase shift over a wider energy range than the original local $V_0^{NLOA}$, so the Hermitized potential is a better local approximation.

Reading between the lines

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

  • An extension the paper leaves implicit: the same denominator condition $1 - \frac{m}{2}V_2(r) > 0$ could be checked on future $NN$ or $N\Lambda$ potentials, where NLO effects are larger, to see whether hermitization stays regular beyond $\Xi\Xi$.
  • Since the Hermitized potential is energy-independent and phase-shift equivalent by construction, it should plug directly into many-body methods that require Hermitian two-body interactions; the paper states the motivation but does not carry out any many-body test.
  • The paper's pairing of $V_{2n-1}$ with $V_{2n}$ suggests a bookkeeping rule for higher-order HAL QCD analyses: each order adds one Hermitian even-derivative block and one transformation factor, so truncation at order $n$ needs $(n+1)^2$ independent NBS wave functions. This counting is in the paper; the suggestion that it defines a practical convergence criterion is an extension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a formalism to make the HAL QCD potential Hermitian order by order in the derivative expansion. The Hamiltonian is written as the free part plus a nonlocal potential expanded in derivatives; the leading local term is Hermitian while higher-order terms are not. The authors introduce a local, invertible, energy-independent transformation R that maps the non-Hermitian Hamiltonian to a Hermitian one, and show explicitly that the NLO (first- and second-derivative) terms can be hermitized exactly. They then argue by induction that all higher orders can be hermitized as well, with each step requiring the solution of first-order linear partial differential equations. The formalism is applied to ΞΞ(^1S_0) scattering using lattice potentials from a previous HAL QCD calculation, comparing the phase shifts from the original non-Hermitian NLO potentials, their Hermitian counterparts, and the leading-order local potential. The paper concludes that the NLO corrections to the phase shift are small and that the LO analysis with the wall source is justified in this system.

Significance. If the central existence claim holds, the paper provides a systematic way to replace the non-Hermitian HAL QCD potential by an equivalent Hermitian potential at each order in the derivative expansion, which is valuable for many-body applications and for comparing lattice potentials with phenomenological interactions. The NLO construction is explicit and algebraically checkable, and the application to a realistic lattice system, using previously published fitted potentials, is a useful demonstration. The identity between phase shifts from the original and hermitized potentials is by construction, so the genuinely independent numerical content is the comparison between LO and NLO results; that comparison is presented for a concrete system and is a strength. However, the proof that the hermitizing transformation exists globally rests on regularity assumptions, and the numerical evidence for the smallness of NLO corrections is presented without statistical uncertainties, so the strength of the claims exceeds what is strictly demonstrated.

major comments (3)
  1. [Sec. II.A, Eq. (15); Sec. III.A, Eq. (70)] The existence of the hermitizing transformation requires the denominator 1 - (m/2)V2(r), or in the application 1 - m_Xi V2^{NLOA}(r), not to vanish in the integration region. The paper never checks the fitted V2 against this condition. If the denominator crosses zero, R1(r) in Eq. (15) acquires an exponential singularity and the hermitized potential (16) or (70) is not a well-defined local operator. The statement in Sec. II.B.2 that 'singularities of the integrand are all integrable' is an assumption, not a proof, and the all-orders induction inherits this gap. The authors should either prove that the relevant denominators are nonvanishing for the fitted potentials or demonstrate numerically that the singularity condition is not encountered in the region where the potentials are used.
  2. [Sec. II.C, Eqs. (62)-(64)] The induction argument asserts that each condition such as Eq. (62) or (64) 'fixes' the next coefficient Rn+1,2k through the term Xn+1,2k-1[k], but the paper does not establish that the resulting first-order linear partial differential equations admit global regular solutions. Local solvability of such equations does not guarantee global existence, and the explicit rotational example in Sec. II.B.2 is only one special case, relying on the same unproven integrability assumption. Since the central claim is that every truncation can be hermitized to all orders, this is a load-bearing gap. The authors should either provide a rigorous existence argument under explicit regularity conditions or state the theorem as conditional on those conditions, and verify those conditions for the lattice potentials used in Sec. III.
  3. [Sec. III, Figs. 2 and 4] The phase-shift comparisons that support the claim that the NLO correction to the phase shift is relatively small, and that the LO analysis is justified, show only central values. The text says 'For visibility, only central values are given here,' but the accompanying discussion treats differences among the curves as significant and even states that NLOA and NLOB agree 'within the uncertainties of the calculations.' Without error bars or numerical uncertainties on the phase shifts, this agreement cannot be assessed, and the claimed smallness of the NLO corrections is not quantitatively supported. The authors should provide the uncertainties, at least by propagating the errors of the fitted potentials, or weaken the phenomenological conclusions accordingly.
minor comments (4)
  1. [Sec. III, first paragraph] The sentence 'Since the V3(r) term does not contribute to the S wave scattering, we ignor this term in the present analysis' contains a typo: 'ignor' should be 'ignore'.
  2. [Sec. III.B, Fig. 4 caption and text] In the left panel of Fig. 4, the curve is labeled 'V LOB 0'; this should presumably read 'V NLOB 0' to be consistent with the notation introduced in Eq. (71) and used in the text.
  3. [Sec. II.A, Eqs. (13)-(16) and Sec. III.A, Eqs. (66)-(70)] The notational relationship between the tensor decomposition V2a(r) and V2b(r) in Sec. II.A and the single function V2^{NLOA}(r) used in Sec. III.A is not spelled out. The replacement rules in Eq. (67) define V2 as (1/2)(V2a+V2b), but Eq. (70) is written directly in terms of V2^{NLOA} and its derivatives; the authors should clarify how the terms involving V2a and V2b (for example, the -V2a/r term in the definition of \tilde V1 in Eq. (16)) reduce to the form used in Eq. (70).
  4. [Appendix A] The convergence argument in Appendix A applies to the application of the derivative expansion to a plane wave, but the hermitization procedure treats the potential coefficients as local functions of r. The assumption that the nonlocal potential is mild enough for the derivative expansion to converge is stated at the end of the appendix, but no criterion is given for the fitted lattice potentials used in the paper; a brief assessment for the ΞΞ system would strengthen the presentation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: hermitization derivation is self-contained; NLO/LO comparison uses independent lattice-fitted potentials.

full rationale

The paper's central claim is a mathematical construction: a local similarity transformation R that makes each truncation of the derivative-expanded HAL QCD potential hermitian. The derivation is carried out in the paper itself, starting from the Hamiltonian (1), with explicit formulas for R1 (Eq. 15), R2 (Eqs. 45-47, 49), and an induction argument in Sec. IIC. No fitted parameter is renamed as a prediction, and no input is defined in terms of the output. The statement that the hermitized and non-hermitian potentials give identical phase shifts is explicitly acknowledged as 'by construction' (Sec. IIIA), and this is an equivalence relation, not a predicted numerical result obtained from fitted data. The independent numerical content is the comparison of LO and NLO phase shifts for Xi-Xi(1S0), which uses lattice-fitted potentials from Ref. [12]; comparing different truncations of the same derivative expansion is a genuine model comparison rather than a circular prediction. Self-citation is heavy, especially the HAL QCD framework and the lattice data source, but the load-bearing derivation does not reduce to those citations: the hermitization theorem is proven in-paper, and the numerical inputs are external lattice results. The unproven regularity conditions (e.g., the denominators 1-(m/2)V2 and 1-m_Xi V2 in Eqs. 14-15 and 70, and the asserted integrability in Sec. IIB2) are potential correctness gaps, not circularity: failure of those conditions would invalidate the derivation, but would not make the derivation equivalent to its inputs. Appendix A likewise assumes convergence of the derivative expansion rather than proving it from fitted potentials, which is a stated assumption, not a circular step. Overall, no load-bearing circular step is identifiable; the paper's central result is self-contained, and the only minor concern is the heavy reliance on prior work by the same authors for the HAL QCD method, which is normal scientific context rather than circular evidence.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities; the transformation R is a mathematical device. The central numerical inputs are the lattice-fitted potentials from Ref. [12], which are free parameters in the application but not in the hermitization formalism itself. The main axioms are the convergence of the derivative expansion and the regularity of the transformation.

free parameters (5)
  • V0^{NLOA}(r) local potential = fitted to lattice NBS wave functions from Ref. [12]
    The hermitized potential in eq. (70) and the phase shifts depend directly on this fitted local function and its derivatives.
  • V2^{NLOA}(r) second-derivative coefficient = fitted to lattice NBS wave functions from Ref. [12]
    Appears in the NLOA potential and in the denominator 1 - m_Xi V2(r) of eq. (70), so its fitted values control the hermitization.
  • V0^{NLOB}(r) local potential = fitted to lattice NBS wave functions from Ref. [12]
    Used in eq. (72) to build the local hermitized NLOB potential.
  • V1^{NLOB}(r) first-derivative coefficient = fitted to lattice NBS wave functions from Ref. [12]
    Appears in eq. (72); its fitted values determine the deviation of the NLOB local potential.
  • Hadron mass m_Xi = 1.46 GeV
    Input from the lattice ensemble of Ref. [12], used as the particle mass in the Schrodinger equation and in the hermitization formulas.
assumptions (3)
  • domain assumption The derivative expansion of the non-local HAL QCD potential converges for the wave functions considered.
    Stated at the start of Sec. II; Appendix A gives sufficient conditions on moments of non-locality but does not verify them for the lattice potential.
  • domain assumption The NBS wave function formalism correctly relates the potential to scattering phase shifts.
    Standard HAL QCD methodology, cited to Refs. [5-7,9], used without re-derivation.
  • ad hoc to paper The transformation R is local, invertible, and regular; denominators such as 1 - (m/2)V2(r) do not vanish in the integration region.
    Required for eqs. (14)-(15) and eq. (70). The paper assumes singularities are integrable (Sec. II B 2) but does not check the fitted V2 against this condition.

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Pith. "Pith review of Hermitizing the HAL QCD potential in the derivative expansion." pith.science (2026). https://pith.science/paper/EADDB2HF

@misc{pith2026190900656,
  author       = {Pith},
  title        = {Pith review of: Hermitizing the HAL QCD potential in the derivative expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EADDB2HF}},
  note         = {Machine review of arXiv:1909.00656}
}
abstract

A formalism is given to hermitize the HAL QCD potential, which needs to be non-hermitian except the leading order (LO) local term in the derivative expansion as the Nambu-Bethe-Salpeter (NBS) wave functions for different energies are not orthogonal to each other. It is shown that the non-hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of $\Xi \Xi (^{1}S_{0}) $ scattering, for which the HAL QCD calculation is available to the NLO. The NLO term gives relatively small corrections to the scattering phase shift and the LO analysis seems justified in this case. We also observe that the local part of the hermitized NLO potential works better than that of the non-hermitian NLO potential. The hermitian version of the HAL QCD potential is desirable for comparing it with phenomenological interactions and also for using it as a two-body interaction in many body systems.

Figures

Figures reproduced from arXiv: 1909.00656 by the authors.

Figure 1
Figure 1. FIG. 1: NLO potentials without [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (Left) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. (Left) shows V NLOB 0 (r) (yellow open triangles), together with V LO(wall) 0 (r) (red open circles) and V NLOA 0 (r) (black open triangles), at 0.45 fm ≤ r ≤1.05 fm, while [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (Left) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Continue with ORCID to comment.

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