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Two-nucleon scattering amplitudes at light quark masses from lattice QCD

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Jorge Baeza-Ballesteros, André Baião-Raposo, Davide Laudicina, Malcolm Lazarow, Miguel Salg, John Bulava, Jeremy R. Green, Andrew Hanlon
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Renwick J. Hudspith, Daniel Mohler, Colin Morningstar, Amy Nicholson, Fernando Romero-López, Thomas R. Richardson, André Walker-Loud, Hartmut Wittig, for the Baryon Scattering (BaSc) Collaboration

High Energy Physics - Lattice · Submitted 16 Oct 2026 · 121 references

Pith's verdict: State-of-the-art lattice QCD for NN scattering at near-physical masses, with honest accounting of limitations; definitely deserves peer review.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 19, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2610.23332 (version 1)
Pith IDDX53I7GE, Pith's permanent record of this paper and its review
CitationAmy Nicholson, Andr\'e Bai\~ao-Raposo, Andr\'e Walker-Loud, Andrew Hanlon, Colin Morningstar, Daniel Mohler, Davide Laudicina, Fernando Romero-L\'opez, for the Baryon Scattering (BaSc) Collaboration, Hartmut Wittig, Jeremy R. Green, John Bulava, Jorge Baeza-Ballesteros, Malcolm Lazarow, Miguel Salg, Renwick J. Hudspith, Thomas R. Richardson. "Two-nucleon scattering amplitudes at light quark masses from lattice QCD." arXiv:2610.23332 (2026). https://pith.science/paper/DX53I7GE
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The paper

The paper claims

Lattice QCD finds virtual states in nucleon-nucleon scattering at near-physical quark masses.

The reading

This paper presents lattice-QCD calculations of nucleon-nucleon (NN) scattering at pion masses of approximately 200 and 280 MeV, the lightest such study to date. By employing two recently developed finite-volume formalisms that explicitly account for the long-range one-pion-exchange left-hand cut, the authors reliably extract scattering amplitudes from energy spectra obtained with the distillation framework. The central claim is that at both pion masses the amplitudes in both isospin channels (I=0 deuteron and I=1 dineutron) prefer a virtual-state pole on the unphysical sheet by more than 1σ. Furthermore, comparing with a previous calculation at Mπ≈710 MeV, the inverse scattering lengths and binding momenta evolve monotonically toward their physical values, suggesting a virtual-to-bound-state transition in the deuteron channel near Mπ≈200 MeV. This work provides a crucial step toward grounding nuclear forces directly in QCD.

What carries the argument

Two recently developed finite-volume quantization conditions that explicitly incorporate the one-pion-exchange left-hand cut: the plane-wave quantization condition (based on the Lippmann-Schwinger equation in a finite-volume plane-wave basis) and the relativistic left-hand-cut formalism (which introduces a modified K-matrix and solves integral equations to construct the full amplitude). These are combined with a large, Hermitian correlation matrix built from distillation, solved via a generalized eigenvalue problem and Bayesian multi-exponential conspiracy fits to extract finite-volume energy levels with controlled excited-state contamination.

Extended reading notes

Core claim

The paper establishes that at Mπ≈200 MeV and Mπ≈280 MeV, the NN scattering amplitudes in both the I=0 (deuteron) and I=1 (dineutron) channels contain virtual-state poles on the second Riemann sheet, with statistical significance exceeding 1σ. In the deuteron channel, the inverse scattering length approaches zero, consistent with proximity to the unitary limit. The analysis uses two independent spectrum determination strategies and two complementary finite-volume formalisms (the plane-wave quantization condition and the relativistic left-hand-cut formalism) that incorporate one-pion-exchange effects. The combined results show that as the pion mass decreases, the scattering lengths and pole positions approach the physical-point values monotonically, with the I=0 channel nearing a bound-state transition.

Load-bearing premise

The analysis is performed at a single lattice spacing (a≈0.064 fm) with no continuum extrapolation, so unquantified discretization effects could shift the scattering parameters, especially for the lightest pion mass where the left-hand cut lies close to threshold.

Editorial extensions

If this is right

  • If the virtual-state interpretation is correct, the deuteron bound state at the physical point can be understood as a crossover from a virtual state as the pion mass is lowered, providing a QCD prediction for the deuteron binding energy.
  • The extracted scattering lengths, effective ranges, and pole positions can directly constrain the low-energy constants of chiral effective field theory, reducing model dependence in nuclear force calculations.
  • The successful application of left-hand-cut formalisms to NN scattering paves the way for analogous studies of other systems with long-range pion exchange, such as hyperon-nucleon or charmed meson scattering.
  • The methodology demonstrated here—using multiple finite-volume formalisms, multiple spectrum analysis strategies, and model averaging—sets a new standard for controlling systematic uncertainties in lattice-QCD studies of hadron-hadron scattering.

Reading between the lines

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

  • The monotonic approach of the inverse scattering length toward zero in I=0 suggests that at the physical pion mass, the deuteron emerges from a fine-tuned cancellation, consistent with the observed unnaturally large scattering length; lattice QCD may be able to quantify how close to criticality the physical system lies.
  • The fact that two independent left-hand-cut formalisms yield consistent results, even though they treat off-shell effects differently, indicates that the essential long-range physics is well captured, and future calculations can focus on reducing statistical errors rather than addressing formalism discrepancies.
  • If the virtual state persists at still lower pion masses (e.g., Mπ≈170 MeV), it would challenge the simple monotonic extrapolation and suggest the need for additional dynamics (e.g., two-pion exchange or inelasticities) becoming relevant before the physical point is reached.
Original abstract

We present a lattice-QCD determination of nucleon-nucleon scattering amplitudes at pion masses of $M_\pi \approx 200$ MeV and $M_\pi \approx 280$ MeV at a single lattice spacing $a \approx 0.064$ fm on two CLS ensembles. The finite-volume energy spectra are extracted from large Hermitian correlation matrices constructed within the distillation framework using an extensive basis of two-nucleon interpolating operators. To reliably relate the finite-volume spectra to the infinite-volume scattering amplitudes, we employ two recently developed formalisms that explicitly account for the long-range interaction generated by one-pion exchange: the plane-wave quantization condition and the relativistic left-hand-cut formalism. We discuss the quark-mass dependence of the nucleon-nucleon scattering lengths, effective ranges, shape parameters and pole positions. At all pion masses considered, our analysis prefers virtual states in both isospin channels by more than $1\sigma$. Comparing also results from our previous calculation at $M_\pi \approx 710$ MeV, we observe that the inverse scattering lengths and binding momenta approach their physical-point phenomenological values monotonically as the pion mass is varied.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 5,202 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present a lattice-QCD determination of nucleon-nucleon scattering amplitudes at pion masses of $M_\pi \approx 200$ MeV and $M_\pi \approx 280$ MeV at a single lattice spacing $a \approx 0.064$ fm on two CLS ensembles. The finite-volume energy spectra are extracted from large Hermitian correlation matrices constructed within the distillation framework using an extensive basis of two-nucleon interpolating operators. To reliably relate the finite-volume spectra to the infinite-volume scattering amplitudes, we employ two recently developed formalisms that explicitly account for the long-range interaction generated by one-pion exchange: the plane-wave quantization condition and the relativistic left-hand-cut formalism. We discuss the quark-mass dependence of the nucleon-nucleon scattering lengths, effective ranges, shape parameters and pole positions. At all pion masses considered, our analysis prefers virtual states in both isospin channels by more than $1\sigma$. Comparing also results from our previous calculation at $M_\pi \approx 710$ MeV, we observe that the inverse scattering lengths and binding momenta approach their physical-point phenomenological values monotonically as the pion mass is varied.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a lattice-QCD determination of nucleon-nucleon scattering amplitudes at pion masses of $M_\pi \approx 200$ MeV and $M_\pi \approx 280$ MeV at a single lattice spacing $a \approx 0.064$ fm on two CLS ensembles. contribution 0.98 0.97 0.05 asserted 1.00
s1 The finite-volume energy spectra are extracted from large Hermitian correlation matrices constructed within the distillation framework using an extensive basis of two-nucleon interpolating operators. method 1.00 0.95 0.02 asserted 1.00
s2 To reliably relate the finite-volume spectra to the infinite-volume scattering amplitudes, we employ two recently developed formalisms that explicitly account for the long-range interaction generated by one-pion exchange: the plane-wave quantization condition and the relativistic left-hand-cut formalism. method 0.99 0.80 0.03 asserted 1.00
s3 We discuss the quark-mass dependence of the nucleon-nucleon scattering lengths, effective ranges, shape parameters and pole positions. contribution 0.87 0.84 0.05 unclear 0.51
s4 At all pion masses considered, our analysis prefers virtual states in both isospin channels by more than $1\sigma$. result 0.99 0.96 0.34 asserted 0.53
s5 Comparing also results from our previous calculation at $M_\pi \approx 710$ MeV, we observe that the inverse scattering lengths and binding momenta approach their physical-point phenomenological values monotonically as the pion mass is varied. result 0.84 0.93 0.21 asserted 0.97

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Overlap factors for the three lowest momentum operators in the isosinglet
Figure 2. FIG. 2: Determination of the D251
Figure 3. FIG. 3: Stability for the J250
Figure 4. FIG. 4: Full spectra on the D251 ensemble (top) and J250 ensemble (bottom). Two versions of the spectrum analysis
Figure 5. FIG. 5: Best-fit predictions to the finite-volume energies of D251 (Set 1) obtained with different fit models with a fixed
Figure 6. FIG. 6: Averaged results for the scattering phase shift of the lowest partial wave in the D251 (
Figure 7. FIG. 7: Results, in physical units, for the phase shift (left) and real part of the corresponding
Figure 8. FIG. 8: Results for the inverse scattering length, effective range, shape parameter and binding momentum for the
Figure 9. FIG. 9: Results, in physical units, for the
Figure 10. FIG. 10: Results, in physical units, for the bar phase shifts and mixing angle in the deuteron channel, as defined
Figure 11. FIG. 11: Comparison of the scattering phase shift of the lowest partial wave for all the methods used to analyze the
Figure 12. FIG. 12: Subset of results for scattering observables for the D251 and J250 ensembles from multiple combinations of
Figure 13. FIG. 13: Dispersion relation for the D251 (blue) and J250 (orange) single-nucleon spectrum. The data are obtained
Figure 14. FIG. 14: Left: stability of the D251
Figure 15. FIG. 15: Same as in Figure
Figure 16. FIG. 16: Model averaging for the
Figure 17. FIG. 17: Effect of varying
Figure 18. FIG. 18: Comparison of the results of the lowest partial wave in the C103 ensemble (
Figure 19. FIG. 19: Same as Figure
Figure 20. FIG. 20: Same as Figure
Figure 21. FIG. 21: Same as Figure

Figures reproduced from arXiv:2610.23332 by the authors.

References

Discussion

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