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

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.

💡 Research Summary

The paper develops a rigorous theoretical framework that extends the Cutkosky cutting rules—originally formulated for relativistic free particles—to the non‑relativistic regime of dark‑matter (DM) scattering in solid‑state detectors. The authors recognize that conventional cutting rules rely on plane‑wave electron states, which are inadequate for describing electrons bound in a crystal lattice where the electronic wavefunctions are highly inhomogeneous and subject to band structure effects. To overcome this limitation, they employ Kohn‑Sham (KS) orbitals obtained from density‑functional theory (DFT) as the fundamental building blocks for the electron‑hole propagator.

The DM model considered is a Dirac fermion χ that interacts with electrons through a massive vector mediator A′. In the non‑relativistic limit, the interaction reduces to an instantaneous Yukawa potential (V_{\chi e}(\mathbf{r})\sim g_\chi g_e e^{-m_{A’}r}/r). The authors further simplify the electron‑electron and electron‑ion interactions to the pure Coulomb potential, arguing that transverse photon exchange is suppressed for low‑velocity electrons in solids.

A central result is the derivation of a self‑energy expression for the DM particle in the medium, (\Sigma_\chi), whose imaginary part directly yields the decay (or absorption) rate (\Gamma_\chi = 2,\Im\mathcal{M}_{\chi\to\chi}). The self‑energy diagram contains a “blob” that represents the electron‑hole sector. By substituting the KS‑based electron‑hole propagator
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