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arxiv: 2605.16977 · v1 · pith:Y2GDXQ6Cnew · submitted 2026-05-16 · ✦ hep-lat · hep-ph· nucl-th

Two-nucleon systems at m_(π)approx292 MeV from lattice QCD

Kuan Zhang , Kang Yu , Yiqi Geng , Chuan Liu , Liuming Liu , Peng Sun , Jia-Jun Wu , Ruilin Zhu This is my paper

Pith reviewed 2026-05-19 18:54 UTC · model grok-4.3

classification ✦ hep-lat hep-phnucl-th
keywords lattice QCDnucleon-nucleon scatteringvirtual statedeuterondi-neutronfinite volumeLuscher methodpion mass
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0 comments X

The pith

Lattice QCD at 292 MeV pion mass finds virtual state poles in deuteron and di-neutron channels

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

This paper computes finite-volume energies of two-nucleon systems in lattice QCD at a pion mass of roughly 292 MeV using three different spatial volumes. It converts those energies into scattering amplitudes in the ^3S1 and ^1S0 channels with Lüscher's method and locates a virtual-state pole in each channel. The poles lie on the unphysical sheet with binding energies of 6^{+5}_{-3} MeV and 11^{+6}_{-5} MeV. A reader cares because the result supplies a concrete datum on how nucleon-nucleon binding changes with light-quark mass, which is required to understand the physical deuteron and to guide extrapolations toward the real world.

Core claim

Finite-volume energies are computed for nucleon-nucleon systems in the ^3S1 and ^1S0 channels on three volumes at m_π ≈ 292 MeV. Lüscher's method is used to extract the scattering amplitudes, revealing a virtual state pole in each channel with binding energies 6^{+5}_{-3} MeV for the deuteron channel and 11^{+6}_{-5} MeV for the di-neutron channel. An alternative Non-Perturbative Hamiltonian framework yields consistent results, indicating that left-hand cut effects do not alter the conclusion.

What carries the argument

Lüscher's finite-volume quantization condition that connects the discrete spectrum in a finite spatial volume to the infinite-volume scattering phase shift or amplitude.

If this is right

  • The ^3S1 channel has a virtual state rather than a bound deuteron at this pion mass.
  • The ^1S0 channel has a virtual state for the di-neutron.
  • The position of these poles can be used to constrain effective field theories at this quark mass.
  • Consistent results from two different analysis methods strengthen the reliability of the virtual state identification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the poles move to the physical sheet at lighter pion masses, this would explain the emergence of the physical deuteron bound state.
  • Similar calculations at other pion masses could map the trajectory of the pole as a function of m_π.
  • These results may help test whether the di-neutron becomes bound or remains virtual in the chiral limit or at physical masses.

Load-bearing premise

Higher partial waves and residual finite-volume distortions do not affect the extracted low-energy scattering parameters at the volumes and momenta studied.

What would settle it

A direct computation of the infinite-volume scattering length or effective range in these channels at the same pion mass, or a calculation at the physical pion mass to see if the poles become bound states.

Figures

Figures reproduced from arXiv: 2605.16977 by Chuan Liu, Jia-Jun Wu, Kang Yu, Kuan Zhang, Liuming Liu, Peng Sun, Ruilin Zhu, Yiqi Geng.

Figure 1
Figure 1. Figure 1: FIG. 1. The energy levels in the center-of-momentum frame [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ω( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. log [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The results for the parameters [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The distributions of binding energy in the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Finite volume energies from the FVH method com [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Fitting details on for the ensemble C24P29, in the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Details of the fitting for the ensemble C32P29, in the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Details of the fitting process for the ensemble C48P29, in the [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

Nucleon-nucleon systems in the $^3S_1$ and the $^1S_0$ channels are studied in lattice quantum chromodynamics at a pion mass of approximately $m_{\pi}\approx292$ MeV, employing three $N_f = 2+1$ ensembles with the same pion mass and lattice spacing $a=0.10530(18)$ fm but different spatial volumes. Finite-volume energies of the nucleon-nucleon systems are determined in both the rest frame and a moving frame. The distillation quark smearing method is applied to improve the precision and to ensure the symmetric correlators by using the same interpolating operators at sink and source. The scattering amplitudes are extracted from the finite-volume spectra using the L\"uscher's finite-volume method. At the studied pion mass, both the $^3S_1$ (deuteron) and $^1S_0$(di-neutron) channels exhibit a virtual state pole, with binding energies of $6^{+5}_{-3}$ MeV and $11^{+6}_{-5}$ MeV, respectively. To investigate the effects of the left-hand cut, an alternative method -- the Non-Perturbative Hamiltonian framework (NPHF) -- is used for the scattering analysis and yields consistent results with those from the L\"uscher method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports lattice QCD results for two-nucleon systems in the ^3S1 and ^1S0 channels at m_π ≈ 292 MeV on three Nf=2+1 ensembles with fixed lattice spacing but varying spatial volumes. Finite-volume energies are extracted in both rest and moving frames using distillation-improved operators. Scattering amplitudes and infinite-volume poles are then obtained via Lüscher's quantization condition, with a cross-check using the Non-Perturbative Hamiltonian framework (NPHF). The central result is the identification of virtual-state poles in both channels, with binding energies 6^{+5}_{-3} MeV (^3S1) and 11^{+6}_{-5} MeV (^1S0).

Significance. If the quoted pole positions hold after systematic checks, the work supplies useful lattice data on the pion-mass dependence of NN interactions near the physical point, aiding chiral EFT calibrations and extrapolations. The agreement between independent Lüscher and NPHF extractions, together with the use of multiple volumes and frames, strengthens the analysis and provides a concrete benchmark for future studies at lighter pion masses.

major comments (2)
  1. [Lüscher analysis (methods and results sections)] Lüscher analysis (methods and results sections): The extraction of the virtual-state poles assumes S-wave dominance with negligible higher partial-wave mixing, particularly in the moving-frame data, and that residual finite-volume corrections beyond the leading Lüscher term lie inside the quoted uncertainties. No explicit tests—such as volume-by-volume consistency of extracted phase shifts, fits that float P-wave parameters, or direct per-volume comparison of Lüscher versus NPHF amplitudes—are presented. Because the reported poles lie close to threshold, even modest contamination would shift the binding energies outside the stated errors.
  2. [Finite-volume spectra extraction] Finite-volume spectra extraction: The manuscript states that higher partial-wave contamination and discretization effects are under control at the quoted precision, yet no quantitative assessment (e.g., comparison of spectra across the three volumes after subtracting leading Lüscher contributions or inclusion of lattice-spacing artifacts in the quantization condition) is shown. This assumption is load-bearing for the central claim that both channels exhibit virtual states at the reported locations.
minor comments (2)
  1. [Results] Notation: The definition of the binding energy relative to the two-nucleon threshold should be stated explicitly in the text when the pole positions are first quoted, to avoid ambiguity with the sign convention for virtual states.
  2. [Figures] Figures: The error bands on the phase-shift plots would benefit from an additional panel or inset showing the sensitivity to the choice of fitting range or to the inclusion of a small P-wave term.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important aspects of the analysis that warrant additional clarification and explicit checks. We address each major comment below and will revise the manuscript to incorporate the suggested improvements where feasible.

read point-by-point responses

  1. Referee: [Lüscher analysis (methods and results sections)] Lüscher analysis (methods and results sections): The extraction of the virtual-state poles assumes S-wave dominance with negligible higher partial-wave mixing, particularly in the moving-frame data, and that residual finite-volume corrections beyond the leading Lüscher term lie inside the quoted uncertainties. No explicit tests—such as volume-by-volume consistency of extracted phase shifts, fits that float P-wave parameters, or direct per-volume comparison of Lüscher versus NPHF amplitudes—are presented. Because the reported poles lie close to threshold, even modest contamination would shift the binding energies outside the stated errors.

    Authors: We agree that more explicit tests would strengthen the presentation. The manuscript already demonstrates consistency of the extracted poles across three volumes and both rest and moving frames, together with agreement between the independent Lüscher and NPHF determinations. To address the referee's concern directly, the revised version will include: (i) volume-by-volume phase-shift extractions showing stability, (ii) an extended fit in which P-wave parameters are floated (yielding values consistent with zero within uncertainties), and (iii) a per-volume comparison of the scattering amplitudes obtained from the two methods. These additions confirm that higher partial-wave mixing remains negligible at the quoted precision and that residual finite-volume effects are captured within the reported uncertainties. revision: yes

  2. Referee: [Finite-volume spectra extraction] Finite-volume spectra extraction: The manuscript states that higher partial-wave contamination and discretization effects are under control at the quoted precision, yet no quantitative assessment (e.g., comparison of spectra across the three volumes after subtracting leading Lüscher contributions or inclusion of lattice-spacing artifacts in the quantization condition) is shown. This assumption is load-bearing for the central claim that both channels exhibit virtual states at the reported locations.

    Authors: We have performed internal checks that compare the finite-volume spectra across the three volumes after subtracting the leading Lüscher contributions; these show good consistency within statistical uncertainties. For discretization effects, all ensembles share the same lattice spacing, precluding a direct multi-spacing comparison. In the revision we will add a quantitative discussion estimating the expected size of residual discretization and higher-wave effects based on the observed volume dependence and on existing literature at comparable pion masses and lattice spacings. This discussion will be placed in the methods and results sections to make the control of systematics explicit. revision: partial

standing simulated objections not resolved
  • A fully quantitative inclusion of lattice-spacing artifacts directly into the quantization condition or a multi-spacing comparison of spectra cannot be performed with the present set of ensembles at fixed lattice spacing; such an assessment would require additional simulations at different a that lie beyond the scope of the current work.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper computes finite-volume two-nucleon energies on three lattice ensembles using the distillation method, then maps those energies to infinite-volume scattering amplitudes via Lüscher's quantization condition (and the independent NPHF cross-check) to locate virtual-state poles. This mapping is a standard, externally defined relation between discrete FV spectra and continuum phase shifts; the reported binding energies (6^{+5}_{-3} MeV and 11^{+6}_{-5} MeV) are derived outputs, not inputs that are fitted or redefined by construction. No step invokes a self-citation whose validity depends on the present result, renames a known pattern as a new derivation, or smuggles an ansatz that makes the final pole positions tautological with the raw spectra. The analysis therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the validity of Lüscher's quantization condition for two-nucleon systems, the assumption that the chosen interpolating operators have good overlap with the ground states, and the neglect of higher partial waves at the energies considered. No new free parameters are introduced beyond the standard lattice QCD action parameters already fixed by the ensembles.

axioms (2)
  • domain assumption Lüscher's finite-volume quantization condition accurately relates the discrete spectrum to the infinite-volume scattering amplitude for S-wave NN systems at these volumes and energies.
    Invoked in the scattering analysis section when converting finite-volume energies to phase shifts.
  • domain assumption The distillation smearing and symmetric correlators sufficiently suppress excited-state contamination and ensure reliable ground-state energies.
    Stated in the methods when describing the operator construction.

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discussion (0)

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    work page internal anchor Pith review Pith/arXiv arXiv 2014