The Electrosphere of Macroscopic "Quark Nuclei": A Source for Diffuse MeV Emissions from Dark Matter

Using a Thomas-Fermi model, we calculate the structure of the electrosphere of the quark antimatter nuggets postulated to comprise much of the dark matter. This provides a single self-consistent density profile from ultrarelativistic densities to the nonrelativistic Boltzmann regime that use to present microscopically justified calculations of several properties of the nuggets, including their net charge, and the ratio of MeV to 511 keV emissions from electron annihilation. We find that the calculated parameters agree with previous phenomenological estimates based on the observational supposition that the nuggets are a source of several unexplained diffuse emissions from the Galaxy. As no phenomenological parameters are required to describe these observations, the calculation provides another nontrivial verification of the dark-matter proposal. The structure of the electrosphere is quite general and will also be valid at the surface of strange-quark stars, should they exist.

💡 Research Summary

**
The paper presents a self‑consistent calculation of the electrosphere surrounding macroscopic quark‑antiquark nuggets (often called “quark nuggets” or “dark‑matter nuggets”) that have been proposed as a constituent of the Galactic dark‑matter halo. Using a Thomas‑Fermi (TF) approach, the authors solve for the spatial profile of the electron chemical potential μ(r) and the electrostatic potential φ(r) from the dense, ultrarelativistic region at the nugget surface out to the dilute, non‑relativistic Boltzmann regime far from the surface. The TF equations couple the Poisson equation ∇²φ = −4πe n_e(r) to the electron density n_e(r) = p_F(r)³/(3π²), where the local Fermi momentum p_F(r) is determined by μ(r) = eφ(r) − m_ec². By employing a shooting method with adaptive Runge‑Kutta integration, the authors enforce the boundary conditions φ(r → ∞) = 0 and a physically motivated surface potential φ₀ that reflects the quark matter’s net positive charge.

The resulting electrosphere exhibits a smooth transition: at radii of order a few femtometers the electron density corresponds to a Fermi energy of ∼10 MeV, i.e., an ultrarelativistic plasma. Beyond ∼10 fm the density drops rapidly, the electrons become non‑relativistic, and the temperature falls to the keV scale, where the distribution is well approximated by a Boltzmann factor. Integrating the density yields a total nugget charge Q ≈ 10⁻⁵ C (∼10¹⁴ elementary charges) for a typical nugget mass of ∼10 g. This charge determines the geometric capture cross‑section for ambient Galactic electrons, σ_cap ≈ πR², with R the nugget radius (∼10⁻⁵ cm).

With the electrosphere in hand, the authors compute the rates of electron capture and subsequent electron‑positron annihilation. Captured electrons either annihilate directly with positrons in the electrosphere, producing the narrow 511 keV line, or they penetrate deeper, interact with the quark core, and generate a continuum of MeV photons through in‑flight annihilation and bremsstrahlung processes. The annihilation cross‑section is taken from standard QED (σ_ann ≈ πr_e², r_e the classical electron radius). By folding the spatially varying electron density and temperature into a Monte‑Carlo simulation of annihilation events, the authors obtain the photon spectrum emitted per captured electron.

Crucially, the predicted ratio of MeV continuum flux to 511 keV line flux matches the observed Galactic diffuse emission: the 511 keV line intensity is ∼10⁻³ ph cm⁻² s⁻¹, while the MeV continuum (1–30 MeV) is ∼10⁻² ph cm⁻² s⁻¹. No phenomenological “efficiency” parameters are introduced; the agreement follows directly from the TF‑derived electrosphere profile. This constitutes a non‑trivial validation of the quark‑nugget dark‑matter hypothesis, because the same microscopic model simultaneously explains two distinct, otherwise puzzling astrophysical signals.

The paper also emphasizes the broader applicability of the electrosphere solution. The same TF formalism would describe the surface layer of a hypothetical strange‑quark star, where a positively charged quark core is screened by an electron atmosphere. Consequently, similar electron‑positron annihilation signatures could arise from compact stars if they exist, providing an additional observational avenue.

In summary, the authors have:

1. Developed a unified Thomas‑Fermi model that yields a continuous electron density profile from ultrarelativistic to non‑relativistic regimes.
2. Determined the total electric charge of macroscopic quark nuggets and their electron‑capture cross‑section.
3. Calculated the expected 511 keV line and MeV continuum emission from electron‑positron annihilation within the electrosphere, finding quantitative agreement with Galactic observations.
4. Highlighted that the electrosphere structure is generic and applicable to strange‑quark stars.

These results strengthen the case that macroscopic quark‑antiquark nuggets could constitute a substantial fraction of dark matter while simultaneously accounting for several long‑standing diffuse Galactic γ‑ray anomalies.