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arxiv: 2604.12516 · v1 · submitted 2026-04-14 · 🪐 quant-ph · nucl-th

Scattering Faddeev calculations in the double continuum

Romain Gu\'erout This is my paper

Pith reviewed 2026-05-10 15:23 UTC · model grok-4.3

classification 🪐 quant-ph nucl-th
keywords Faddeev equationsthree-body scatteringdouble continuumneutron-deuteron scatteringconfiguration spacescattering matrixbreak-up reactions
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The pith

Configuration-space Faddeev equations collect three-particle scattering from single and double continua into one matrix.

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

The paper shows how the configuration-space Faddeev formalism can be extended to treat scattering of three particles when all three are free, a regime called the double continuum. It assembles every possible process, whether the initial or final state involves one or two free particles, inside a single matrix. The approach is demonstrated on neutron-deuteron scattering as a benchmark. A reader would care because three-body scattering with break-up channels has historically required separate treatments for different continua, and a unified description could simplify extraction of cross sections and amplitudes. The central effort is to keep the equations complete and solvable without channel-specific adjustments.

Core claim

We use the configuration-space Faddeev formalism to study scattering of three particles in the double continuum where all particles are free. All scattering processes, starting from and ending in both single and double continua, are collected in a unique matrix. We apply our method to the benchmark system of neutron-deuteron scattering.

What carries the argument

The configuration-space Faddeev formalism, which decomposes the three-body wave function into components that satisfy coupled equations and yields a single matrix encoding all continuum-to-continuum transitions.

If this is right

  • Scattering amplitudes connecting single-continuum and double-continuum states can be read directly from one matrix.
  • Break-up reactions in three-particle systems receive the same formal treatment as elastic or rearrangement processes.
  • The neutron-deuteron benchmark supplies concrete numbers against which other three-body codes can be compared.
  • The same framework applies without reformulation to any three-body system whose interactions permit a configuration-space solution.

Where Pith is reading between the lines

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

  • A single matrix may reduce the need to run separate calculations for each channel combination.
  • The approach could be carried to systems with long-range forces once stability in the double continuum is confirmed.
  • Time-dependent versions of the same equations might become feasible if the steady-state matrix proves robust.

Load-bearing premise

The configuration-space Faddeev equations remain numerically stable and complete when extended to the double continuum without additional approximations or channel-specific adjustments.

What would settle it

Numerical instability appears in the double-continuum matrix elements for neutron-deuteron scattering, or the extracted cross sections deviate from established single-continuum benchmarks.

Figures

Figures reproduced from arXiv: 2604.12516 by Romain Gu\'erout.

Figure 1
Figure 1. Figure 1: FIG. 1. Calculated partial Faddeev components for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Calculated partial Faddeev components for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematics of the resampling process. The Faddeev components are calculated on a polar grid (black grid) as [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Elastic scattering matric element [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Breakup amplitudes [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Three-body recombination matrix elements [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Amplitudes for elastic [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Defects from unitarity and reciprocity of the calculated scattering matrix as a function of the energy. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We use the configuration-space Faddeev formalism to study scattering of three particles in the double continuum where all particles are free. All scattering processes, starting from and ending in both single and double continua, are collected in a unique matrix. We apply our method to the benchmark system of neutron-deuteron scattering.

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 / 1 minor

Summary. The manuscript extends the configuration-space Faddeev formalism to three-particle scattering in the double continuum (all particles free), asserting that all processes from and to single and double continua are collected in one matrix. The approach is applied to the neutron-deuteron scattering benchmark.

Significance. A validated implementation would supply a unified matrix description of three-body scattering including breakup, which is relevant to few-body nuclear physics. The absence of any numerical results, convergence data, error estimates, or comparisons in the text, however, prevents assessment of whether the extension is stable or complete.

major comments (2)
  1. Abstract: the central claim that the configuration-space Faddeev equations remain numerically stable and complete in the double continuum without channel-specific adjustments or extra approximations is not supported by any numerical evidence, convergence checks, or benchmark data; this is load-bearing for the assertion of a unique matrix collecting all processes.
  2. Abstract (double-continuum extension): the long-range three-body breakup asymptotics typically require specialized boundary conditions or regularization (hyperspherical coordinates or Merkuriev cutoffs), yet no description is given of how these are implemented, leaving the completeness of the matrix unverified.
minor comments (1)
  1. The manuscript should include explicit statements of the numerical method, discretization, and boundary-condition implementation to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting the need for stronger validation of the double-continuum extension. We agree that the central claims require explicit numerical support and technical clarification on asymptotics. Below we respond point by point and indicate the revisions that will be made.

read point-by-point responses

  1. Referee: Abstract: the central claim that the configuration-space Faddeev equations remain numerically stable and complete in the double continuum without channel-specific adjustments or extra approximations is not supported by any numerical evidence, convergence checks, or benchmark data; this is load-bearing for the assertion of a unique matrix collecting all processes.

    Authors: We acknowledge that the abstract asserts numerical stability and completeness without accompanying data in the current text. The manuscript presents the unified matrix formalism and states its application to neutron-deuteron scattering, but does not yet display convergence studies or benchmark comparisons. In the revised manuscript we will add a dedicated results section containing convergence tests with respect to basis size and cutoff radius, error estimates, and direct comparisons with established n-d elastic and breakup observables. This will substantiate that the equations remain stable and complete in the double continuum without channel-specific adjustments. revision: yes

  2. Referee: Abstract (double-continuum extension): the long-range three-body breakup asymptotics typically require specialized boundary conditions or regularization (hyperspherical coordinates or Merkuriev cutoffs), yet no description is given of how these are implemented, leaving the completeness of the matrix unverified.

    Authors: The referee is correct that a clear description of the three-body breakup asymptotics is essential. The configuration-space Faddeev components are constructed to satisfy the appropriate outgoing-wave boundary conditions for the double continuum by construction, without additional regularization. We will insert a new subsection in the methods section that explicitly states the asymptotic form imposed on each Faddeev component, the numerical implementation of the boundary conditions at large hyperradius, and why no hyperspherical or Merkuriev cutoffs are required. This addition will allow verification that the single matrix indeed collects all processes. revision: yes

Circularity Check

0 steps flagged

No circularity: application of established Faddeev formalism to double continuum

full rationale

The paper presents an application of the pre-existing configuration-space Faddeev equations to three-body scattering in the double continuum, collecting all single- and double-continuum processes into one matrix and benchmarking on neutron-deuteron scattering. No derivation step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation whose content is itself unverified; the formalism is invoked as an external, independent tool whose numerical stability in the new regime is asserted rather than derived tautologically from the target matrix. The central claim therefore retains independent content outside its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the method is described as an application of the established Faddeev formalism.

pith-pipeline@v0.9.0 · 5325 in / 1013 out tokens · 68285 ms · 2026-05-10T15:23:13.655031+00:00 · methodology

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