Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force

Akaki Rusetsky; Bai-Long Hoid; Hans-Werner Hammer; Jia-Jun Wu; Jin-yi Pang; Rishabh Bubna

REVIEW 3 major objections 6 minor 1 cited by

Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Treating the long-range force in a plane-wave basis makes the modified Lüscher equation converge so fast that S-wave-only truncation reproduces exact energy levels, while the proposed renormalization keeps parameters natural.

desk verdict Solid implementation paper with a clean renormalization scheme and honest checks, but the headline claim about all partial waves is only demonstrated in a repulsive perturbative regime. read the letter →

arxiv 2507.18399 v1 pith:FCJXJBE6 submitted 2025-07-24 hep-lat

Rishabh Bubna , Hans-Werner Hammer , Bai-Long Hoid , Jin-Yi Pang , Akaki Rusetsky , Jia-Jun Wu This is my paper · ORCID

classification hep-lat PACS 11.15.Ha12.38.Gc13.75.Cs

keywords modifiedLüscherequationlong-rangeforceeffectiverangeexpansionleft-handcutdimensionalregularizationfinite-volumequantizationlatticeQCDYukawapotential

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

This paper gives a practical numerical recipe for the Lüscher quantization condition when the two particles also feel a known long-range force such as one-pion exchange. It argues that treating this long-range part in a plane-wave basis, and expanding only the short-range part in partial waves, makes the partial-wave expansion converge very fast: in the toy model the modified quantization condition with the S-wave alone already reproduces the exact finite-volume energy level, and adding the G-wave changes nothing. The paper further works out the ultraviolet renormalization of the modified zeta-function in dimensional regularization, subtracting a few terms of the Taylor expansion at threshold so that the modified effective-range parameters stay of natural size in all partial waves. If this carries over to real lattice data, scattering analyses on the lattice would need only a small number of partial waves even when a long-range pion exchange is present.

What carries the argument

The load-bearing object is the modified Lüscher zeta-function, the finite-volume analogue of the loop function $M_\ell(q_0)$ formed from the Green function of the long-range potential (a sum of one-pion-exchange ladder diagrams). Two decompositions do the work. The free propagator is split into a subtracted piece with no singular denominator in the physical region and a residual piece; this renders the finite-volume shift $\Delta H$ ultraviolet-finite and reduces the problem to numerically solving a Lippmann-Schwinger equation in a plane-wave basis, with exponentially small corrections dropped. Then, because each term of the Born series of $G_L$ has a lower divergence index than the previous one, $G_L$ is split into a divergent part containing only $2\ell+2$ terms and a finite part governed by an integral equation for $T_{\rm fin}$; the divergent part is handled in dimensional regularization and renormalized by subtracting a polynomial at threshold. That subtraction prescription is what keeps the modified effective-range parameters of natural size in all partial waves.

What would settle it

A direct falsifier is the attractive counterpart of the toy model: set $g$ to $-g$ in Eq. (3.10) and check whether the modified S-wave-only quantization condition still reproduces the exact Hamiltonian spectrum. The paper itself predicts it will not, because attractive Yukawa forces produce bound states and the Born series stops converging; observing the failure would confirm the stated limitation, while observing success would contradict it. On the repulsive side, rerunning the $M_S=10M$ comparison and checking that the S-wave-only modified solution agrees with the exact level to within the quoted exponentially small corrections would refute the decoupling claim if it departed visibly.

Watch

Extended reading notes

Core claim

The central claim is that the modified Lüscher equation—where the known long-range potential is kept in a plane-wave basis and only the unknown short-range part is expanded in partial waves—can be implemented numerically and converges far better than the standard Lüscher equation. In the toy model with a short-range scale $M_S = 10M$, the modified quantization condition with the S-wave only already reproduces the exact energy level, and adding the G-wave does not change anything; even in the borderline case $M_S = 2M$ its convergence is much better than the standard one. The paper also establishes a renormalization scheme: the modified zeta-function's ultraviolet divergences are isolated in a finite number of Born terms, evaluated in dimensional regularization, and removed by subtracting the first terms of the Taylor expansion at threshold. In this scheme the modified effective range expansion parameters are of natural size in all partial waves, and the modified effective range function is smooth and almost linear across the region of the left-hand cut, where the standard K-matrix becomes singular and complex.

Load-bearing premise

The method assumes the long-range force is weak enough to be handled as a small perturbation and does not by itself create bound states or near-threshold resonances; the authors flag this as their most restrictive assumption, noting it breaks down for attractive forces.

Editorial extensions

If this is right

For lattice analyses of systems with one-pion exchange, the number of partial waves one must keep is small; in the toy model S-wave-only truncation already suffices when the short-range force is not too light.
The left-hand-cut region, where the standard Lüscher method fails, becomes accessible: the modified effective range function stays smooth and real there, so energy levels in that region can be analyzed.
Once the modified zeta-function is tabulated for a known long-range potential, extracting scattering information from lattice data proceeds exactly as in the standard Lüscher approach.
Exponentially suppressed finite-volume corrections are much smaller in the modified equation (e.g., a momentum-scale ratio of 11 versus 1.1 for the ground state at $M_S=10M$), improving accuracy near the left-hand cut.
Dimensional regularization with threshold subtraction avoids the unnaturally large cutoff polynomials that arise in higher partial waves, so the effective-range parameters remain of natural size.

Reading between the lines

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

The successful Coulomb limit suggests the same subtraction scheme could be applied to other massless-exchange forces, where analytic results exist to benchmark the numerics; the paper only verifies the leading logarithm and a few coefficients.
The restriction to repulsive interactions is the main obstacle to pion-exchange channels with attractive partial waves; extending the method would require resumming or analytically continuing past the breakdown of the Born series, a problem the paper leaves open.
The group-theoretic shell analysis implies a practical selection rule: which excited states can be trusted with S-wave-only truncation depends on the lattice-momentum shell, not just on the energy, so users can choose states whose G-wave mixing coefficients are small.
Because real lattice calculations use moving frames and other irreps, the practical payoff for systems like $T_{cc}$ will depend on how well the decoupling survives those generalizations; the paper only demonstrates the center-of-mass $A_1^+$ case.

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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

This is the numerical follow-up to the authors' modified Lüscher formalism, and the new material is genuinely useful: an efficient algorithm for the modified zeta-function and a dimensional-regularization subtraction scheme that avoids the unpleasantly large cutoff-scale polynomials of cutoff regularization. The paper earns its keep. The checks are real: the Coulomb limit is reproduced, the result is μ-independent, and the modified quantization condition matches the exact finite-volume Hamiltonian spectrum in the toy model. The demonstration that S-wave-only truncation already works, while the standard Lüscher approach converges slowly, is convincing within the stated regime. The appendix on partial-wave contributions to energy shifts is a nice touch and explains the uneven convergence pattern quantitatively.

The soft spots are real but mostly acknowledged by the authors. The load-bearing assumption is that the long-range force is perturbative and does not create bound states or low-lying resonances by itself. They state this explicitly in Conclusions item (ii), and Section 4.2 restricts to repulsive interactions because the Born series can fail for attractive ones. That restriction matters for the stated physics motivation: one-pion exchange in Tcc and baryon-baryon systems has attractive channels, and the method's accuracy there is not established. The claim about natural-size modified effective range parameters "in all partial waves" is also extrapolated from ℓ=0 and ℓ=4; the ℓ=4 subtraction scheme is demonstrated for the repulsive Yukawa case, not for an attractive OPE-like channel. Minor issues: the numerical integrations carry no error bars, and no code or data are shipped, which would have made the algorithm reproducible without re-implementation. The self-citation of Ref. [1] is fine—it is the formalism paper, and this is the implementation.

Overall, the paper is honest, technically careful, and useful to lattice practitioners working on systems with long-range forces. The central derivation holds up; the limitations are explicit and addressable. I would send this to a serious referee. It is not ready as is—the attractive-channel question and the missing error bars should be addressed—but it deserves full review rather than desk rejection.

Referee Report

3 major / 6 minor

Summary. The paper presents a numerical implementation of the modified L\"uscher equation proposed in Ref. [1] for two-body systems with a known long-range potential plus a short-range interaction. It develops a subtraction procedure for the finite-volume Green function that separates exponentially suppressed corrections from a finite-volume remainder, and it introduces a dimensional-regularization scheme with threshold subtractions for the infinite-volume loop function M_\ell(q_0), giving explicit Feynman-parameter formulas for \ell=0 and \ell=4. In a two-Yukawa toy model restricted to the A_1^+ irrep, the authors claim that the modified quantization condition truncated to the S-wave already reproduces the exact finite-volume spectrum and that including the G-wave changes almost nothing, in contrast to the standard L\"uscher equation. They further claim that the proposed renormalization scheme yields modified effective range expansion parameters of natural size in all partial waves. The paper checks the M \to 0 limit against the known Coulomb result and compares the quantization condition with exact plane-wave Hamiltonian eigenvalues.

Significance. If the central claims hold, the paper would provide a practically useful tool for lattice analyses of systems with one-pion exchange and near-threshold left-hand cuts: partial-wave truncation in the modified L\"uscher equation would converge much faster than in the standard approach, and dimensional regularization would avoid the large cutoff subtraction terms that make higher-partial-wave effective range parameters unnatural. The paper's strengths are its explicit checks: the result is \mu-independent in the tested range, the Coulomb limit is reproduced, and the modified quantization condition matches the exact Hamiltonian spectrum in the toy model. The authors also honestly state in Conclusions item (ii) that the long-range force is assumed to be perturbative and not to create bound states or low-lying resonances by itself. However, the regime of validity is narrower than the abstract suggests, and the 'natural size' assertion is not backed by a quantitative extraction of ERE parameters.

major comments (3)

[§4.2 and Conclusions item (ii)] The central claim that the modified quantization condition can be truncated to the S-wave and yields natural-size ERE parameters in all partial waves is established only for repulsive, perturbative long-range forces. Section 4.2 explicitly says that only repulsive interactions are considered and that for attractive interactions bound states can emerge and the Born series is no longer convergent; Conclusions item (ii) flags the perturbativity assumption as 'more restrictive and might require additional scrutiny.' This matters because the decomposition in Eq. (4.1), where G_L is split into a finite number of divergent Born terms plus G_0 T_fin^L G_0, and the subsequent numerical solution of the Lippmann-Schwinger equation for T_fin^L, rely on T_fin^L being regular in the physical region. For an attractive OPE-like potential with a near-threshold bound or virtual state, this split can fail. Since the abstract and Section 3.2 advertise the method without this caveat, the paper should either restrict the headline claims to the perturbative repulsive regime or demonstrate, for example with a weakly attractive Yukawa benchmark, that the method remains accurate when the Born series is not convergent.
[§4.4 and §4.2] The abstract claims that the proposed renormalization scheme gives modified effective range expansion parameters of natural size in all partial waves, but the paper never actually extracts or tabulates those parameters. Figure 9 shows Re K_M^\ell(q_0^2) only for \ell=0 and in arbitrary units, while Figure 8 shows M_\ell(q_0) for \ell=4 but not the resulting ERE coefficients. 'Natural size' cannot be assessed from these plots without specifying the dimensionless ratios in which the coefficients are measured. Please provide a quantitative statement, e.g. the first few coefficients of K_M^\ell(q_0^2) for \ell=0 and \ell=4 in units of the available scales m and M, or revise the abstract so that it does not overstate what is demonstrated.
[§3.2 and §4.2] The headline result that the S-wave-only modified quantization condition 'already reproduces the exact energy level' and that adding the G-wave changes nothing is presented through D(q_0) curves superimposed on vertical lines rather than through quantitative residuals. The reader cannot judge the size of the residual or the actual effect of truncating at \ell=4. Moreover, the VEGAS integrations used for M_\ell(q_0) in Figures 7 and 8 are shown without statistical uncertainties, so the claim that higher loops are 'visually indistinguishable' from the full solution is not quantitatively supported. Please quote explicit numbers, such as |E(\ell_{\rm max}=0)-E_{\rm exact}| and |E(\ell_{\rm max}=4)-E_{\rm exact}| in units of q_0^2/M^2 or M, and report integration uncertainties for the numerical results in Section 4.2.

minor comments (6)

[Abstract] There is a typo in 'algoritm'; it should read 'algorithm'.
[§3.2] The sentence 'The potential of the toy model, which are used to produce the synthetic lattice data' has a subject-verb agreement error; it should be 'which is used'.
[§4.1.1] In the sentence 'One could use, for example, MS or MS renormalization scheme', the second 'MS' appears to be missing an overline; please clarify whether the \overline{MS} scheme is intended.
[Fig. 9 caption] The caption refers to the 'left-land threshold'; this should be 'left-hand threshold'.
[§3.1] The warning about subthreshold poles of G_\mu and G_f that cancel in the sum is important, but the statement that 'one can always adjust the free parameter \mu' is not supported by a concrete criterion. Please state how \mu should be chosen in practice or show a scan over \mu for a representative case.
[§4.2] The values plotted for \ell=4 in Figure 8 are very small (of order 10^{-8}); please specify the normalization and mass dimensions of M_\ell(q_0) so that the reader can interpret the 'natural size' of the resulting ERE parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central numerical claims are benchmarked against an independently diagonalized finite-volume Hamiltonian and the analytic Coulomb limit.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity test. The modified Luscher equation and the long-range/short-range splitting are taken from the authors' prior Ref. [1], but this is attribution of prior work rather than a load-bearing self-citation: the present implementation is independently checked against the exact finite-volume spectrum obtained by diagonalizing the Hamiltonian in the plane-wave basis (Section 3.2), and against the known analytic Coulomb limit (Section 4.3). The synthetic data are generated from a specified two-Yukawa potential, not fitted; the KM and H objects are computed from the same known potential, so the S-wave-only-versus-S+G-wave comparison is a controlled test of the quantization-condition reformulation, not a fitted parameter renamed as a prediction. The renormalization scheme (subtraction at threshold in dimensional regularization) is a prescription, and the claim of natural-size modified ERE parameters is supported by the explicit numerical smallness of the finite remainder (for ℓ=4, "M fin ℓ (q0) is so small that it cannot be distinguished with a bare eye"), not by defining naturalness into the subtraction. The stated assumption that the long-range force is perturbative and creates no bound states or low-lying resonances on its own (Conclusions, item ii) is an explicitly acknowledged limitation in scope, not a circular step. No step in the paper reduces a prediction to its own input by construction, and no uniqueness or validity conclusion is imported solely from a same-author citation.

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

The toy model parameters are chosen by hand to mimic a heavy particle exchanging a light pion; they are not fitted. The method assumes a known local long-range potential, perturbativity of that potential, and negligible exp(-mu L) corrections from the subtracted part of the propagator. No new physical entities are introduced.

free parameters (5)

Yukawa coupling g = 0.073 m
Input from one-pion exchange chiral EFT, used as the long-range coupling in the toy model; not fitted.
Short-range coupling ratio gS/g = -2.74
Chosen by hand for the toy model to mimic short-range repulsion.
Mass ratio m/M = 6.7
Chosen to mimic a heavy particle exchanging a light pion.
Short-range exchange mass MS/M = 2 and 10
Chosen as two extreme cases in the synthetic-data study.
Dimensionless box size ML = 3
Chosen box size for the finite-volume toy model.

assumptions (5)

domain assumption The long-range potential is exactly given by a Yukawa form
Section 2, Eq. (2.2); the method assumes the long-range part is known and local.
domain assumption The long-range force is perturbative and does not bind by itself
Conclusions item ii and Section 4.2; restricts to repulsive interactions, otherwise the Born-series decomposition in Eq. (4.1) may fail.
domain assumption Exponentially suppressed corrections G_mu - G_infinity are negligible
Section 3.1, Eq. (3.9) and following text; relies on exp(-mu L) suppression, with a warning about subthreshold poles.
ad hoc to paper Dimensional regularization with subtraction at threshold gives the physical modified ERE function
Section 4.1; the subtraction scheme is chosen to avoid large polynomials; it is a renormalization prescription, not derived from data.
domain assumption Analytic continuation below threshold is valid for q0^2 > -M^2
Section 4.4; the integration contour must be deformed below the left-hand cut, restricting the energy interval.

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Cite this review

Pith. "Pith review of Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force." pith.science (2026). https://pith.science/paper/FCJXJBE6

@misc{pith2026250718399,
  author       = {Pith},
  title        = {Pith review of: Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCJXJBE6}},
  note         = {Machine review of arXiv:2507.18399}
}

read the original abstract

An efficient numerical algoritm is proposed for the calculation of the modified L\"uscher zeta-function in the presence of a long-range force. Using the formalism developed in Ref.~\cite{Bubna:2024izx} for the analysis of synthetic data on the finite-volume energy levels in a toy model, it is demonstrated that, in contrast to the standard L\"uscher approach, the truncation of the higher partial waves has very little effect on the final result. Furthermore, the regularization and renormalization of the modified L\"uscher zeta-function is discussed in detail, as well as the problems arising within the cutoff regularization. It is shown that, using the renormalization scheme proposed in the present paper, one obtains modified effective range expansion parameters of natural size in all partial waves.

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Forward citations

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Works this paper leans on

4 extracted references · cited by 1 Pith paper

[1]

L¨ uscher equation with long-range forces.JHEP, 05:168, 2024

Rishabh Bubna, Hans-Werner Hammer, Fabian M¨ uller, Jin-Yi Pang, Akaki Rusetsky, and Jia-Jun Wu. L¨ uscher equation with long-range forces.JHEP, 05:168, 2024

2024
[2]

Epelbaum

Lu Meng and E. Epelbaum. Two-particle scattering from finite-volume quantization conditions using the plane wave basis. JHEP, 10:051, 2021

2021
[3]

Filin, and Ashot M

Lu Meng, Vadim Baru, Evgeny Epelbaum, Arseniy A. Filin, and Ashot M. Gasparyan. Solving the left-hand cut problem in lattice QCD: Tcc(3875)+ from finite volume energy levels. Phys. Rev. D , 109(7):L071506, 2024

2024
[4]

Andr´ e Bai˜ ao Raposo and Maxwell T. Hansen. Finite-volume scattering on the left-hand cut. JHEP, 08:075, 2024

2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

L¨ uscher equation with long-range forces.JHEP, 05:168, 2024

Rishabh Bubna, Hans-Werner Hammer, Fabian M¨ uller, Jin-Yi Pang, Akaki Rusetsky, and Jia-Jun Wu. L¨ uscher equation with long-range forces.JHEP, 05:168, 2024

2024

[2] [2]

Epelbaum

Lu Meng and E. Epelbaum. Two-particle scattering from finite-volume quantization conditions using the plane wave basis. JHEP, 10:051, 2021

2021

[3] [3]

Filin, and Ashot M

Lu Meng, Vadim Baru, Evgeny Epelbaum, Arseniy A. Filin, and Ashot M. Gasparyan. Solving the left-hand cut problem in lattice QCD: Tcc(3875)+ from finite volume energy levels. Phys. Rev. D , 109(7):L071506, 2024

2024

[4] [4]

Andr´ e Bai˜ ao Raposo and Maxwell T. Hansen. Finite-volume scattering on the left-hand cut. JHEP, 08:075, 2024

2024