All-electron dark matter-electron scattering with random-phase approximation dielectric screening and local field effects

This paper presents an all-electron framework implemented in the open-source code QCDark2 that incorporates random-phase approximation dielectric screening with local field effects to accurately predict dark matter-electron scattering rates across various target materials, demonstrating that these effects significantly modify recoil spectra and detection sensitivities for both non-relativistic and boosted dark matter.

The Gist

Imagine you are trying to catch a ghost. In the world of physics, this "ghost" is Dark Matter, a mysterious substance that makes up most of the universe but refuses to interact with light or ordinary matter in any obvious way.

For decades, scientists have built giant, ultra-sensitive traps (detectors) made of solid materials like silicon to catch these ghosts. The theory is that if a dark matter particle bumps into an electron in the silicon, it will give the electron a tiny "kick," creating a spark of electricity that the detector can see.

However, predicting exactly how often this happens is incredibly difficult. It's like trying to predict how a pebble will bounce off a trampoline, but the trampoline is made of billions of tiny, interconnected springs, and the pebble is moving at different speeds.

This paper, written by a team of physicists, introduces a new, much more accurate way to calculate these "bounces." Here is the breakdown using simple analogies:

1. The Problem: The "Crowded Room" Effect

When a dark matter particle hits an electron in a solid material (like a silicon chip), it doesn't just hit that one electron in isolation. The electron is surrounded by a sea of other electrons.

• The Old Way: Previous calculations treated the electrons like people standing alone in a field. They assumed the "kick" from the dark matter would travel in a straight line to the electron.
• The Reality: The electrons are actually in a crowded room. When one person (the electron) gets pushed, they bump into their neighbors, who bump into others, creating a ripple effect. This is called dielectric screening. The crowd absorbs and redirects the energy.

Furthermore, the crowd isn't perfectly uniform. There are pockets of density and empty spaces. These tiny, microscopic irregularities are called Local Field Effects (LFEs).

2. The New Tool: A High-Definition Map

The authors developed a new computer code called QCDark2. Think of previous codes as using a low-resolution, pixelated map to navigate the crowded room. They missed the tiny details (the local field effects) and had to guess how the crowd would react.

QCDark2 is like a high-definition, 3D simulation of the entire room. It accounts for:

• All the electrons: It doesn't ignore the "inner" electrons deep inside the atoms (an "all-electron" treatment).
• The crowd's reaction: It calculates exactly how the electrons push and pull on each other (dielectric screening).
• The tiny bumps: It includes the "Local Field Effects," the microscopic irregularities that change how the energy spreads.

3. Two Types of "Ghosts" (Dark Matter)

The paper studies two different types of dark matter, which behave differently in the detector:

• The Slow Ghosts (Halo Dark Matter): These are the standard dark matter particles floating around our galaxy. They move relatively slowly. When they hit the silicon, they transfer a lot of momentum (a hard shove).

  • The Finding: The new code shows that because of the "crowded room" effects (LFEs), these slow ghosts are actually less likely to be detected than we thought. The crowd absorbs more of the energy than the old maps predicted. This means we might need to build bigger detectors or wait longer to catch them.

• The Fast Ghosts (Boosted Dark Matter): These are dark matter particles that have been "boosted" to high speeds, perhaps by bouncing off the Sun or other cosmic events. They move very fast.

  • The Finding: When these fast ghosts hit the silicon, they create a specific type of ripple called a plasmon (imagine a collective wave in the electron crowd). The new code shows that the "crowded room" effects broaden this wave. Instead of a sharp, distinct splash, the energy spreads out. This changes the shape of the signal, making it look different from background noise.

4. Why This Matters

Imagine you are trying to hear a whisper in a noisy stadium.

• Old Method: You assumed the noise was a steady hum. You might think you heard a whisper, but it was just a fluctuation in the hum.
• New Method: You realize the noise is actually a complex mix of thousands of individual voices shouting, whispering, and clapping (the Local Field Effects).

By understanding the exact nature of the "noise" (the electron response), the scientists can:

1. Stop wasting time: They know exactly what signal to look for, so they don't get fooled by false alarms.
2. Know what they missed: They realized that for slow dark matter, the old methods were overestimating how easy it is to find them. This saves the scientific community from chasing a ghost that might be harder to catch than we thought.
3. Find new signals: For fast dark matter, the new method reveals a unique "signature" (the broadened plasmon wave) that could be easier to spot if we know exactly what to look for.

The Bottom Line

This paper is a major upgrade to the "instruction manual" for dark matter hunters. By using a super-accurate simulation of how electrons behave in solid materials, the authors have refined our predictions. They found that the "crowded room" of electrons is more complex than we thought, changing both how easy it is to catch slow dark matter and what the signal looks like for fast dark matter.

This work is now open-source, meaning other scientists can use this new "high-definition map" to design better experiments and potentially solve one of the biggest mysteries in the universe.

Technical Summary

1. Problem Statement

Direct detection experiments for light dark matter (DM) in the mass range of $\mathcal{O}(\text{keV})$ to $\mathcal{O}(\text{GeV})$ rely on detecting electron recoils in solid-state materials. Accurate prediction of scattering rates requires a precise calculation of the material's response to momentum and energy transfer. Existing computational frameworks suffer from a trade-off between two critical physical aspects:

1. All-electron treatment: Necessary for high-momentum transfers (relevant for standard halo DM), which pseudopotential-based methods often miss.
2. Accurate dielectric screening: Necessary for low-to-intermediate momentum transfers, particularly to capture Local Field Effects (LFEs) arising from microscopic inhomogeneities in the electronic response.

Previous codes (e.g., QEDark, DarkELF, EXCEED-DM, QCDark1) typically excelled in one area but failed in the other. For instance, some neglected LFEs, while others used pseudopotentials that could not handle the high momentum transfers required for non-relativistic halo DM or failed to capture the plasmon resonance accurately for boosted DM. This paper addresses the need for a unified framework that combines all-electron Density Functional Theory (DFT) with Random-Phase Approximation (RPA) dielectric screening including LFEs across the full momentum spectrum.

2. Methodology

The authors developed QCDark2, an open-source code that implements a comprehensive all-electron framework.

• Theoretical Framework:

  • Scattering Rate: The differential scattering rate is derived for both non-relativistic halo DM and boosted DM (e.g., Solar-Reflected Dark Matter, SRDM). The rate depends on the dynamic structure factor $S(\omega, \vec{q})$, which is directly related to the imaginary part of the inverse dielectric function (loss function).
  • Dielectric Function: The longitudinal dielectric function $\epsilon(\omega, \vec{q})$ is calculated using the RPA based on Kohn-Sham wavefunctions from DFT.
  • Local Field Effects (LFEs): The full dielectric matrix $\epsilon_{\vec{G}\vec{G}'}$ is constructed in reciprocal space. The macroscopic dielectric function is obtained by inverting this matrix. Neglecting off-diagonal elements ($\vec{G} \neq \vec{G}'$) corresponds to neglecting LFEs. The authors explicitly calculate and invert the full matrix to include LFEs.
  • Composite Approach: To manage computational cost, a composite dielectric function is used: $\epsilon_{\text{comp}}$ includes full LFEs up to a cutoff momentum $q_{\text{LFE}}$ (typically $7\text{--}8 \, \alpha m_e$) and switches to a no-LFE approximation ($\epsilon_{\text{noLFE}}$) at higher momenta.

• Computational Implementation:

  • Codebase: Built on PySCF using localized Gaussian-type orbital basis sets (all-electron), avoiding the limitations of plane-wave pseudopotentials.
  • Materials: Calculations were performed for Silicon (Si), Germanium (Ge), Gallium Arsenide (GaAs), Silicon Carbide (SiC), and Diamond (C).
  • Parameters: The study uses the PBE exchange-correlation functional with a "scissor correction" to match experimental bandgaps. It employs dense Monkhorst-Pack k-grids and includes a large number of conduction bands (up to 64) to ensure convergence up to high energy transfers (50 eV for most, 150 eV for diamond).
  • Optical Limit: Special handling is applied for the $\vec{q} \to 0$ limit to avoid divergences in the dielectric matrix elements.

3. Key Contributions

1. Unified All-Electron RPA Framework: QCDark2 is the first code to simultaneously provide an all-electron treatment and full RPA dielectric screening including LFEs for DM-electron scattering.
2. Quantification of LFEs: The paper provides the first systematic quantification of how LFEs modify scattering rates across different momentum regimes and materials.
3. Benchmarking and Comparison: The authors rigorously compare QCDark2 results against previous codes (DarkELF, EXCEED-DM, QCDark1) and experimental data (EELS, refractive index), validating the accuracy of the new approach.
4. Public Resources: Precomputed dielectric functions and band structures for five target materials are provided, and the code is made open-source.

4. Key Results

A. Impact of Local Field Effects (LFEs)

• Low Momentum (Plasmon Region): LFEs significantly broaden the plasmon peak in the loss function. This broadening improves agreement with experimental Electron Energy Loss Spectroscopy (EELS) data. For boosted DM (SRDM), which is sensitive to this region, LFEs redistribute power from the peak to lower and higher energies, altering the electron recoil spectrum shape.
• High Momentum: LFEs cause a significant suppression of the dynamic structure factor at high momentum transfers ($q \gtrsim 3 \, \alpha m_e$).
  • For Halo DM (sensitive to high momentum), including LFEs reduces the projected sensitivity (reach) by 20–50% for DM masses above a few MeV across all materials.
  • For Boosted DM, the impact is less dramatic on the total rate but crucial for the spectral shape.

B. Comparison of Screening Approximations

The authors compared full RPA screening against:

• No Screening: Overestimates rates significantly.
• Modified Thomas-Fermi (MTF): Accurate for high momentum but fails at low momentum (singularity issues) and overestimates rates for boosted DM.
• Lindhard Approximation: Better than MTF for the plasmon region but fails to capture material-specific features (like interband transitions and LFEs) at higher momenta.
• Conclusion: Full RPA with LFEs is required for precision across the entire $(\omega, q)$ phase space.

C. Material-Specific Findings

• Germanium (Ge): The inclusion of all-electron effects is critical for Ge. The 3d orbitals (which are frozen in pseudopotential cores in some codes like DarkELF) contribute significantly to scattering at high energies ($\omega \approx 26$ eV). QCDark2 captures these transitions, enhancing sensitivity for heavy mediators compared to codes that miss them.
• Diamond: Due to its large bandgap, diamond requires calculations up to 150 eV, which QCDark2 successfully handles.

D. Sensitivity Projections

• Halo DM: The exclusion curves shift to higher cross-sections (lower sensitivity) when LFEs are included, particularly for heavy mediators where high-momentum transfers dominate.
• Solar-Reflected DM (SRDM): Sensitivity is less affected by the LFE cutoff magnitude but depends heavily on the accurate modeling of the plasmon peak (low momentum).
• Code Comparison: QCDark2 agrees well with DarkELF at low masses (low momentum) but diverges at higher masses due to DarkELF's momentum cutoff ($q=6 \, \alpha m_e$). It agrees qualitatively with EXCEED-DM but provides a more rigorous all-electron treatment.

5. Significance

This work resolves a long-standing gap in dark matter direct detection theory. By providing a tool that is accurate for both the low-momentum plasmon resonance (crucial for boosted DM and low-mass halo DM) and high-momentum transfers (crucial for standard halo DM), it enables:

1. More accurate exclusion limits: Preventing overestimation of detector sensitivity by 20–50% for certain mass ranges.
2. Optimized experimental design: Helping experiments distinguish between halo and boosted DM signals based on the shape of the electron recoil spectrum.
3. Reliable material selection: Providing a systematic comparison of Si, Ge, GaAs, SiC, and diamond, highlighting that material choice (e.g., Ge's 3d orbitals) significantly impacts discovery potential.

The release of QCDark2 and the associated precomputed data establishes a new standard for ab initio calculations in the field of dark matter-electron scattering.