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Variable-phase methods and data for long-range scattering

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Marcos M. Flores, Kalliopi Petraki, Thomas Pignard

High Energy Physics - Phenomenology · Submitted 21 Oct 2026 · 42 references

Pith's verdict: Solid, well-executed resource paper for Yukawa scattering in dark-matter phenomenology; the integration-domain concern is real but minor.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 23, 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 ID2610.30261 (version 1)
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CitationKalliopi Petraki, Marcos M. Flores, Thomas Pignard. "Variable-phase methods and data for long-range scattering." arXiv:2610.30261 (2026). https://pith.science/paper/XDVYDZVJ
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The paper

The paper claims

A variable-phase numerical framework for Yukawa scattering provides phase shifts, Sommerfeld factors, and irregular-solution inputs needed for self-consistent unitarization of dark-matter inelastic processes.

The reading

This paper presents a numerical implementation of the variable-phase method for the Yukawa potential, computing both the regular solution (phase shifts and Sommerfeld factors) and the irregular solution (the projected s-wave Green's function) from a single system of first-order differential equations. The method is validated against Born, WKB, and Coulomb limits, and covers a broad region of parameter space relevant for dark-matter phenomenology, spanning the Born, resonant, semiclassical, and Coulomb-like regimes. The authors release code and tabulated data for ℓ=0 to 100, including elastic, transfer, and viscosity cross sections, and the quantity T(k;k⋄) required for unitarization of contact-like inelastic processes. This enables self-consistent calculations of elastic and inelastic dark-matter scattering without repeated numerical scans.

What carries the argument

The variable-phase method recasts the radial Schrödinger equation for each partial wave into first-order differential equations for coefficient functions c(x) and s(x) that multiply the free Riccati–Bessel and Riccati–Neumann functions. The regular solution is obtained by outward integration from the origin with boundary conditions (3.10); the irregular solution by inward integration from large x with boundary conditions (3.27). The integration domain is determined dynamically by comparing the scales of the eigenvalue, centrifugal, and potential terms, Eqs. (4.6)–(4.7). Phase shifts and Sommerfeld factors follow from the asymptotic values of c and s; the logarithmic derivative of the irregular solution yields the finite part of the s-wave Green’s function.

Extended reading notes

Core claim

For the Yukawa potential, the variable-phase method yields the asymptotic coefficients of the regular solution, from which phase shifts and Sommerfeld factors are obtained via Eqs. (3.14) and (3.23). The irregular solution, integrated inward from large distances using the same equations (3.8) with boundary conditions (3.27), provides the finite part of the projected s-wave Green’s function, T(k;k⋄), via Eq. (3.30). This combination of quantities from a single framework, together with analytic estimates of integration boundaries and partial-wave truncation errors, makes nonperturbative long-range scattering calculations efficient and self-consistent. The tabulated data span ℓ∈[0,100] and a parameter grid covering the Born, resonant, semiclassical, and Coulomb-like regimes, with truncation errors below 1%.

Load-bearing premise

The specific choices of integration boundaries (x_min and x_max) with fixed constants epsilon=1e-7 and Lambda=1e4 in Eqs. (4.6) and (4.7) are assumed to guarantee the claimed numerical accuracy across the entire parameter grid, but this is not proven for all regimes.

Editorial extensions

If this is right

  • Dark-matter studies can directly use the tabulated data without performing computationally intensive scans of the Yukawa potential.
  • The provided T(k;k⋄) enables self-consistent unitarization of contact-like inelastic processes, preventing unitarity violations in combined elastic–inelastic calculations.
  • Partial-wave sums up to ℓ=100 keep truncation errors below 1% for elastic, transfer, and viscosity cross sections across the tabulated grid.
  • The method generalizes to any real, local, central potential satisfying the regularity conditions (2.7) and (2.22).
  • Comparisons with Born, WKB, and Coulomb limits validate the numerical implementation and define the regimes where each approximation applies.

Reading between the lines

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

  • The framework could be extended to coupled-channel problems, where multiple potentials mix partial waves, as occurs in dark matter co-annihilation scenarios.
  • The tabulated data facilitate rapid exploration of how resonant structure in Yukawa scattering impacts the thermal relic abundance and indirect detection signals.
  • The method's efficiency may allow its incorporation into Monte Carlo scanning codes for dark matter models, replacing slower direct integration.
  • The same variable-phase approach can be applied to other screened potentials (e.g., Hulthen, exponential) by modifying the potential form in the code, enabling cross-potential comparisons.
Original abstract

Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales. Reliable predictions often require nonperturbative solutions of the Schr\"odinger equation over broad parameter ranges and many partial waves. We present a numerical framework based on the variable-phase method, which recasts the radial Schr\"odinger equation as a system of first-order equations and allows both regular and irregular solutions to be computed efficiently. The former determine phase shifts and Sommerfeld factors, while the latter provide additional information relevant for the self-consistent unitarization of contact-like inelastic interactions. We discuss the numerical implementation, integration-domain selection, partial-wave convergence, and comparisons with analytic limits. We release the code together with tables for the Yukawa potential containing phase shifts, Sommerfeld factors, elastic and angularly weighted cross sections, and the irregular-solution information relevant for unitarization. These resources facilitate dark-matter studies without requiring repeated computationally intensive scans.

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, 2026-10-23: 1 request, 5,595 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.73 · mentions code or data availability 0.99.

The abstract, sentence by sentence

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

Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales. Reliable predictions often require nonperturbative solutions of the Schr\"odinger equation over broad parameter ranges and many partial waves. We present a numerical framework based on the variable-phase method, which recasts the radial Schr\"odinger equation as a system of first-order equations and allows both regular and irregular solutions to be computed efficiently. The former determine phase shifts and Sommerfeld factors, while the latter provide additional information relevant for the self-consistent unitarization of contact-like inelastic interactions. We discuss the numerical implementation, integration-domain selection, partial-wave convergence, and comparisons with analytic limits. We release the code together with tables for the Yukawa potential containing phase shifts, Sommerfeld factors, elastic and angularly weighted cross sections, and the irregular-solution information relevant for unitarization. These resources facilitate dark-matter studies without requiring repeated computationally intensive scans.

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 Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales. background 1.00 0.05 0.04 asserted 1.00
s1 Reliable predictions often require nonperturbative solutions of the Schr\"odinger equation over broad parameter ranges and many partial waves. background 0.90 0.06 0.03 asserted 0.81
s2 We present a numerical framework based on the variable-phase method, which recasts the radial Schr\"odinger equation as a system of first-order equations and allows both regular and irregular solutions to be computed efficiently. method 0.71 0.94 0.26 asserted 1.00
s3 The former determine phase shifts and Sommerfeld factors, while the latter provide additional information relevant for the self-consistent unitarization of contact-like inelastic interactions. method 0.45 0.49 0.10 asserted 1.00
s4 We discuss the numerical implementation, integration-domain selection, partial-wave convergence, and comparisons with analytic limits. method 0.73 0.85 0.05 asserted 0.80
s5 We release the code together with tables for the Yukawa potential containing phase shifts, Sommerfeld factors, elastic and angularly weighted cross sections, and the irregular-solution information relevant for unitarization. contribution 0.94 0.95 0.18 asserted 1.00
s6 These resources facilitate dark-matter studies without requiring repeated computationally intensive scans. interpretation 0.90 0.35 0.25 asserted 0.93

What kind of paper the classifier reports this is: method or tool 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. Illustration of the different length scales needed to compute the numerical
Figure 2. Regular coefficient functions c rgl k,ℓ(x), s rgl k,ℓ(x) for an attractive Yukawa potential with ζ = 103 , ξ = 102 and for several values of ℓ. 10-2 10-1 1 101 102 103 104 10-7 10-5 10-3 10-1 101 10-1 1 101 102 103 10-11 10-9 10-7 10-5 10-3 10-1 101
Figure 3. The s-wave elastic cross section normalized to its unitarity limit, for attractive and repulsive Yukawa potentials, as a function of |ζ| (left) and ξ (right)
Figure 4. The s-wave Sommerfeld factor for various combinations of parameters, as￾suming an attractive Yukawa potential. Here, ξ = ξ res n=5,ℓ=0 corresponds to the fifth resonance, labeled in the left panel. Both panels are compared to the analytic result for the Coulomb potential, Eq. (B.2b). The right panel instead shows the Sommerfeld factor as a function of ζ for fixed values of ξ. Away from resonances, the enhancement saturates at large ζ due to the finite range of the Yukawa interaction. At the bound-state thresholds, however, it grows as S0 ∝ ζ 2 , without saturation. This differs from
Figure 5. Trajectories of elastic-scattering extrema for attractive (solid) and repulsive
Figure 6. Numerical computation of the irregular solution. The left panel shows
Figure 7. (left) |T(k; k⋄)| as a function of k/k⋄ for an attractive Yukawa potential with ζ⋄ = 1. The numerical result follows the Coulomb limit of Eq. (B.2c) for k/k⋄ ≳ 1/ξ. (right) Sensitivity of T(k; k⋄) around bound-state thresholds. 24
Figure 8. T(k; k⋄)| as a function of ξ and ζ = k⋄/k for ζ⋄ = 1 and for an attractive po￾tential. The gray region, defined by 10 |ζ| < ξ, is excluded from our numerical evaluation because approaching the Coulomb limit increases the computational cost significantly. 25
Figure 9. The elastic (upper left), transfer (upper right), and viscosity (lower ) scattering cross sections as a function of ξ. The cross sections include partial waves up to ℓ = 100, 99, and 98 for elastic, transfer, and viscosity scattering, respectively. The numerical results are shown together with the Born and WKB approximations presented in the text. The transfer cross section uses the form valid for distinguishable particles (Cℓ = 1); for identical particles σelas and σT coincide. 30
Figure 10. (upper ) The elastic cross section for an attractive Yukawa potential as a function of ζ and ξ, including partial waves up to ℓ = 100. The gray region, 10 |ζ| < ξ, is excluded as approaching the Coulomb limit requires increasingly many partial waves. (lower ) Smallest ℓmax beyond which the geometric-tail estimate of the relative truncation error, Eq. (5.25), is at most 10−2 for all larger ℓ computed; the color bar shows log10(1 + ℓ elas max). We assumed Cℓ = 1 for all ℓ. 31
Figure 12. (left) A comparison of the elastic cross-section truncation error obtained nu￾merically and analytically using Eq. (5.18). The horizontal line indicates the asymptotic value of the elastic cross section, while the vertical line marks the point at which the WKB-approximated phase shift becomes of order unity. (right) The full relative trunca￾tion error for the elastic, transfer, and viscosity cross sections, computed directly (solid) and estimated via Eq. (5.25) (dashed). The difference between the cross sections reflects the additional suppression of the transfer and viscosity…
Figure 13. The normalized elastic cross section for

Showing 12 of the paper's 13 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.30261 by the authors.

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