Variable-phase methods and data for long-range scattering

arXiv:2610.30261 · hep-ph, astro-ph.CO, astro-ph.HE · Submitted 2026-10-21 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Variable-phase methods and data for long-range scattering".

Jocelyn: Here's the gist of the paper. The gist Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales Variable-phase approach The paper presents a numerical framework based on the variable-phase method to recast the radial Schrödinger equation as a system of first-order equations This method allows for the efficient computation of both regular and irregular solutions, which are crucial for studying nonperturbative solutions over broad parameter ranges The formulation replaces the second-order radial equation by a system of first-order differential equations and is well suited to scans over broad regions of parameter space Regular solution and phase shift The regular solution yields the phase shifts and Sommerfeld factors, which are determined by the asymptotic behavior of the coefficient functions as r → ∞ The phase shift is defined through the relation tan θl(k) ≡ limx→∞ tk,l(x) = limx→∞ s rglk,l(x)/c rglk,l(x) The Sommerfeld factor is defined as the ratio Sl(k) ≡ limr→0 Fk,l(r)/Ffree k,l (r) These quantities are determined by the integration of Eqs.

Vera: (3.8), which are derived from differentiating the coefficient functions Irregular solution and projected s-wave Green’s function The irregular solution is obtained by integrating inward from large distances to construct it, and this allows for the extraction of information needed for self-consistent unitarization of contact-like inelastic interactions For the s-wave, the quantity T(k; k⋄) is defined as 1/k limr→0 "G'k,0(r) Gk,0(r) - G'k⋄,0(r) Gk⋄,0(r)" This quantity is related to Wfin 0 (k; k⋄) defined in Ref.

17: as Wfin 0 (k; k⋄) = -4k kf Cfm2 T(k; k⋄) Application to the Yukawa potential The method is applied to the Yukawa potential, V Y(r) = −α/r e−mmedr Dimensionless parameters ζ ≡ µα k = α vrel and ξ ≡ µα mmed are defined in terms of which the rescaled potential is V Y˜ (x) = −2ζ/x e−x ζ/ξ The numerical implementation requires choosing a finite integration domain x ∈

xmin, xmax: based on characteristic length scales derived from pairwise balances Scattering cross sections and numerical convergence The paper computes elastic, transfer, and viscosity cross sections using the partial-wave phase shifts derived from the regular solution The truncation error for all three cross sections is exponentially suppressed with lmax, controlled by the parameter η = 2 arcsinh

ζ/(2ξ): The tabulated data for partial waves up to l = 100, 99, and 98 keep the truncation error below 1% for the elastic, transfer, and viscosity cross sections, respectively Coulomb limit comparison The analysis compares results with analytic limits such as the Born approximation and WKB approximation For the strict Coulomb potential, individual Yukawa partial-wave elastic cross sections develop increasingly rapid oscillations rather than approaching Eq.

Vera: (B.3) pointwise However, the Sommerfeld factor and T(k; k⋄) possess well-defined Coulomb limits given above The common divergent contribution cancels in differences of partial-wave phase shifts, while the Sommerfeld factor and T(k; k⋄) possess the well-defined Coulomb limits given above The variable-phase formulation enables reliable nonperturbative calculations of long-range scattering across a broad parameter space relevant for dark-matter phenomenology, while providing the inputs needed for the self-consistent unitarization of contact-like inelastic processes The accompanying code and tabulated data provide phase shifts and Sommerfeld factors for l ∈

0, 100: , the total elastic, transfer, and viscosity cross sections, and T(k; k⋄) The tabulated parameter space spans the Born and resonant regimes and extends to the onset of the semiclassical and Coulomb-like regimes The framework thus enables reliable nonperturbative calculations of long-range scattering across a broad parameter space relevant for dark-matter phenomenology, while providing the inputs needed for the self-consistent unitarization of contact-like inelastic processes.

Title and authors: Vera: Let's talk about the title and the folks who wrote this paper, "Variable-phase methods and data for long-range scattering". It sounds like it’s offering a new toolset for handling these tricky interactions.

Jocelyn: The authors are Marcos M. Flores, Petraki Kalliopi, and Thomas Pignard. I haven't heard of them before, so they seem to be the ones putting this numerical framework together for us to look at.

Subrahmanyan: I’ve read their work on cosmology a bit; this paper is connecting those fundamental scattering calculations directly to how dark matter interacts with the environment on galactic scales.

Vera: It’s about taking that complex scattering problem and solving it in a way that lets you scan through lots of different parameters, which is what we need when testing these theories.

Jocelyn: They are essentially providing the machinery—the variable-phase approach—to get the necessary data points for things like annihilation rates or those velocity-dependent cross sections in SIDM scenarios.

Subrahmanyan: If this method works robustly, it means we can move beyond just simplified models and actually calculate how these long-range effects modify structure formation on galactic scales.

The paper's summary: Vera: So what’s the actual summary of "Variable-phase methods and data for long-range scattering"? It boils down to using this method to solve the radial Schrödinger equation by breaking it into first-order equations.

Jocelyn: They use this setup so they can find both the regular solution, which gives us phase shifts and Sommerfeld factors, and the irregular solution, which helps with unitarization of those inelastic interactions.

Subrahmanyan: The paper mentions that in the multi-TeV dark matter models, these long-range interactions can substantially modify the thermal relic abundance through a Sommerfeld effect and radiative bound-state formation.

Vera: Right, so we’re talking about how these long-range forces change how much dark matter is left over from the early universe or how it annihilates in indirect searches.

Jocelyn: They apply this framework to the Yukawa potential, which is a very common choice for modeling these kinds of interactions, and they define some dimensionless parameters like zeta and xi based on that potential.

Subrahmanyan: It’s important that they are connecting these general scattering solutions to a specific physical setup—the Yukawa potential—so we can actually test the results against something concrete.

The paper's improvements: Vera: Now, what are the actual improvements this paper suggests? They focus on how they use the regular solution to get phase shifts and Sommerfeld factors, which then feed into calculating elastic, transfer, and viscosity cross sections.

Jocelyn: They also discuss using the irregular solution to extract something called T(k; k⋄), which is related to the finite part of the projected s-wave Green’s function, useful for self-consistent unitarization of those contact-like inelastic interactions.

Subrahmanyan: A major improvement they point out is that they can select their integration domain, xmin and xmax, based on characteristic length scales derived from pairwise balances, which helps ensure numerical accuracy across different regimes.

Vera: That makes sense because choosing the right boundaries for integration is where you control the errors when you're scanning over broad regions of parameter space.

Jocelyn: And they show that by truncating the partial waves up to, say, one hundred or ninety-eight they keep the truncation error below one percent for all three types of cross sections—elastic, transfer, and viscosity.

Conclusion: Vera: So to wrap up "Variable-phase methods and data for long-range scattering", this paper gives us a reliable nonperturbative calculation tool for long-range scattering across a broad parameter space relevant for dark matter phenomenology.

Jocelyn: It provides the phase shifts and Sommerfeld factors, along with the total elastic, transfer, and viscosity cross sections, which are all tabulated for partial waves up to one hundred <ref:2610.30261#pg3>.

Subrahmanyan: This framework is powerful because it covers regimes from the Born approximation right up to resonant regimes and even extends into the semiclassical and Coulomb-like regimes.

Vera: The implication is that we now have a solid way to get those inputs needed for unitarizing contact-like inelastic processes, which is a necessary step for more complete dark matter models.

Jocelyn: It gives us the data we need to test those models against what we observe in galactic structure formation and indirect detection signals.

Subrahmanyan: For the big picture, this method allows us to see how these long-range forces sculpt the universe on galactic scales, which is exactly what we need to connect theory with observation.

Marcos M. Flores, Kalliopi Petraki, Thomas Pignard

Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, University of Paris-Saclay, CNRS, Sorbonne University, University of Paris Citພ · Department of Physics, University of Oslo

hep-ph, astro-ph.CO, astro-ph.HE

Submitted: 2026-10-21

Updated: 2026-10-21

Comments: 48 pages, 13 figures; for the associated code and instructions for accessing the data, see https://github.com/mmfloresnm/YukawaVPS

Code: https://github.com/mmfloresnm/YukawaVPS

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales

Key concepts

Variable-phase approach: This numerical technique transforms the radial Schrödinger equation into a system of first-order differential equations. This formulation is highly effective for computing both regular and irregular solutions across wide parameter ranges, making it ideal for studying complex scattering phenomena in dark matter models.
Regular solution and phase shift: The regular solution determines physical quantities like phase shifts ($ an heta_ ext{l}(k)$) and Sommerfeld factors ($S_ ext{l}(k)$). These are calculated by analyzing the asymptotic behavior of coefficient functions as the radial distance $r$ goes to infinity, providing essential information about long-range scattering.
Irregular solution and T(k; k⋄): The irregular solution is constructed by integrating inward from large distances. This allows for extracting information needed to self-consistently unitarize contact-like inelastic interactions. A key quantity derived here is $T(k; k ext{⋄})$, which relates to the finite part of the projected s-wave Green's function.
Yukawa potential application: The method is applied to the Yukawa potential, a common model for dark matter interactions. Dimensionless parameters are defined based on $ ilde{V}_Y(x) = -2 ilde{ ho}/x e^{-x | ilde{ ho}|/ ilde{eta}}$, allowing the numerical implementation to be performed over specific finite integration domains determined by physical balances.

Summary

The gist Long-range interactions play an important role in dark-matter phenomenology, affecting the relic abundance, indirect-detection signals, and structure formation on galactic scales

Variable-phase approach

The paper presents a numerical framework based on the variable-phase method to recast the radial Schrödinger equation as a system of first-order equations This method allows for the efficient computation of both regular and irregular solutions, which are crucial for studying nonperturbative solutions over broad parameter ranges The formulation replaces the second-order radial equation by a system of first-order differential equations and is well suited to scans over broad regions of parameter space

Regular solution and phase shift

The regular solution yields the phase shifts and Sommerfeld factors, which are determined by the asymptotic behavior of the coefficient functions as r → ∞ The phase shift is defined through the relation tan θl(k) ≡ limx→∞ tk,l(x) = limx→∞ s rglk,l(x)/c rglk,l(x) The Sommerfeld factor is defined as the ratio Sl(k) ≡ limr→0 Fk,l(r)/Ffree k,l (r) These quantities are determined by the integration of Eqs. (3.8), which are derived from differentiating the coefficient functions

Irregular solution and projected s-wave Green’s function

The irregular solution is obtained by integrating inward from large distances to construct it, and this allows for the extraction of information needed for self-consistent unitarization of contact-like inelastic interactions For the s-wave, the quantity T(k; k⋄) is defined as 1/k limr→0 G'k,0(r) Gk,0(r) - G'k⋄,0(r) Gk⋄,0(r) This quantity is related to Wfin 0 (k; k⋄) defined in Ref. [17] as Wfin 0 (k; k⋄) = -4k kf Cfm2 T(k; k⋄)

Application to the Yukawa potential

The method is applied to the Yukawa potential, V Y(r) = −α/r e−mmedr Dimensionless parameters ζ ≡ µα k = α vrel and ξ ≡ µα mmed are defined in terms of which the rescaled potential is V Y˜ (x) = −2ζ/x e−x ζ/ξ The numerical implementation requires choosing a finite integration domain x ∈ [xmin, xmax] based on characteristic length scales derived from pairwise balances

Scattering cross sections and numerical convergence

The paper computes elastic, transfer, and viscosity cross sections using the partial-wave phase shifts derived from the regular solution The truncation error for all three cross sections is exponentially suppressed with lmax, controlled by the parameter η = 2 arcsinh[ζ/(2ξ)] The tabulated data for partial waves up to l = 100, 99, and 98 keep the truncation error below 1% for the elastic, transfer, and viscosity cross sections, respectively

Coulomb limit comparison

The analysis compares results with analytic limits such as the Born approximation and WKB approximation For the strict Coulomb potential, individual Yukawa partial-wave elastic cross sections develop increasingly rapid oscillations rather than approaching Eq. (B.3) pointwise However, the Sommerfeld factor and T(k; k⋄) possess well-defined Coulomb limits given above The common divergent contribution cancels in differences of partial-wave phase shifts, while the Sommerfeld factor and T(k; k⋄) possess the well-defined Coulomb limits given above

Conclusion

The variable-phase formulation enables reliable nonperturbative calculations of long-range scattering across a broad parameter space relevant for dark-matter phenomenology, while providing the inputs needed for the self-consistent unitarization of contact-like inelastic processes The accompanying code and tabulated data provide phase shifts and Sommerfeld factors for l ∈ [0, 100], the total elastic, transfer, and viscosity cross sections, and T(k; k⋄) The tabulated parameter space spans the Born and resonant regimes and extends to the onset of the semiclassical and Coulomb-like regimes The framework thus enables reliable nonperturbative calculations of long-range scattering across a broad parameter space relevant for dark-matter phenomenology, while providing the inputs needed for the self-consistent unitarization of contact-like inelastic processes.

Improvements for AI systems

System can perform nonperturbative calculations of long-range scattering across a broad parameter space relevant for dark matter phenomenology by solving Schrodinger equation using first-order equations based on the variable phase method.
The improved system can accurately determine phase shifts and Sommerfeld factors from regular solution coefficient functions, which are then used to calculate elastic, transfer, and viscosity cross sections via Eq. (5.3a) through (5.3c).
The system can compute the finite part of the projected s-wave Green's function T(k; k⋄) by integrating the irregular solution inward and extracting it from Eqs. (3.8) to facilitate unitarization of contact-like inelastic interactions.
The AI can select appropriate integration boundaries xmin and xmax based on physical scales derived from pairwise balances, as prescribed by Eqs. (4.6a) and (4.7), ensuring numerical accuracy across different regimes like Born, resonant, and Coulomb limits for the Yukawa potential.
The system can analyze partial-wave truncation errors for cross sections using analytic estimates from Eq. (5.18) and evaluate them against direct numerical results to determine the required number of partial waves for a given accuracy tolerance in different parameter spaces.

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 Schrödinger equation over broad parameter ranges and many partial waves. We present a numerical framework based on the variable-phase method, which recasts the radial Schrödinger 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.