Variable-phase methods and data for long-range scattering

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.