Cutting rules for non-relativistic dark matter in solids based on Kohn-Sham orbitals

TL;DR

This work develops a diagram-by-diagram cutting-rule framework for non-relativistic dark matter interacting with electrons in solids, grounded in Kohn–Sham orbitals from density functional theory. It adapts Cutkosky rules to Coulomb- and instantaneous DM–electron interactions by introducing NR-specific underlining rules and demonstrates the approach with vertex corrections and RPA screening, connecting the imaginary part of the DM self-energy to electron–hole excitations and the dielectric response. The main contribution is a practical, first-principles–compatible method to compute DM decay rates in materials, enabling unitarity checks and systematic inclusion of medium effects such as screening and plasmons. The framework balances accuracy and efficiency by using KS orbitals as a starting point, while acknowledging KS band-gap limitations and pointing toward GW refinements for improved quasi-particle energies.

Abstract

The Cutkosky cutting rules establish a direct connection between the imaginary parts of loop amplitudes and physical observables such as decay rates and cross sections, providing heuristic insights into the underlying processes. This work lays a robust theoretical foundation for the application of cutting rules in solid-state systems involving instantaneous dark matter (DM)-electron Yukawa interaction as well as the Coulomb potential. The cutting rules are formulated using the single-electron wavefunctions and corresponding energy eigenvalues obtained from the Kohn-Sham equations within density functional theory (DFT). This framework is not only of considerable theoretical interest but also holds significant practical relevance for studying DM phenomenology in condensed matter systems.

Figures (1)

• Figure 1: Two specific ways of underlining vertices in a diagram for the scattering process $p_{1}p_{2}\rightarrow k_{1}k_{2}$.