Simplified Spin Dependence in Dark Matter Direct Detection

TL;DR

This work extends the DM–electron scattering framework to anisotropic detector media, providing a complete, symmetry-aware decomposition into seven SM response functions and five DM response coefficients. By separating scalar and vector form factors and exploiting parity, isotropy, and electronic adiabaticity, the authors derive substantial reductions in the independent terms needed to describe spin-dependent DM interactions, enabling efficient ab initio rate calculations. The analysis connects UV DM models to NR EFT coefficients and demonstrates concrete simplifications for common detectors, while also detailing when these simplifications fail (e.g., non-adiabatic or inelastic regimes). The results offer a practical toolkit for predicting electron recoil signals across diverse materials and guiding the design of spin-sensitive, anisotropic direct-detection experiments.

Abstract

The interactions of dark matter with Standard Model particles can be systematically studied in the language of effective field theories. We investigate dark matter interactions with Standard Model particles, including spin-dependent interactions, for direct detection experiments and demonstrate that, although the scattering rate generally depends on multiple types of material response functions, certain linear combinations of these material response functions vanish if the initial and final electronic states share the same Hamiltonian. We also find that several other response functions vanish in parity-symmetric materials, making these systems as simple as isotropic detectors in some respects. Finally, we present the scattering rate for an anisotropic, possibly chiral detector, for generic dark matter-electron spin interactions. These relations reduce the number of independent response functions needed, thereby simplifying the computational complexity for a broad class of dark matter models. Our results provide a complete and efficient toolkit for analyzing electron recoil signals in diverse detector materials.

Figures (2)

• Figure 1: The mapping of example UV DM models ( ovals) to the $c_i$ Wilson coefficients, the material response functions $f_S$ and $\vec{f}_V$ ( gray rectangles), and the NR DM couplings $a_{0,\ldots,4}$ (blue diamonds). The colored lines join the UV models (scalar real/complex DM or fermionic Dirac/Majorana DM) to the Wilson coefficients that they generate at tree level for dimension-4 operators, with the color of the line distinguishing between scalar ( dashed black) and vector ( solid) mediators. The SM photon (red) is a special case of vector mediator as it only couples to the electron through the vector current. Any $c_i$ that can be generated via the SM photon can also be generated by a generic BSM vector mediator. All lines belonging to "Majorana" (or "real") also belong to "Dirac" (or "complex"). Note that the coefficients in orange$c_{12},\ldots,c_{15}$ can be generated at one-loop as shown in Figure \ref{['fig:loops']}. Shown in green are the coefficients $c_{17},\ldots,c_{26}$ that can only be generated by a spin-1 DM particle (see Table 1 of Liang:2024ecw for the operators they correspond to). The coefficients $c_i$ are classified by their contribution to the scalar form factor $f_S$ ( left gray rectangle) or to the vector form factor $\vec{f}_V$ ( right gray rectangle). In the central gray box, we show all the five possible interference terms for scalar and fermionic DM. We denote in green the interference terms that are generated by spin-1 DM. In the lower part of the diagram, we show the relation between the $c_i$ coefficients and the $a_{0,\ldots,4}$ coefficients.
• Figure 2: From left to right, the first two loop diagrams generate $\mathcal{O}_{15}$; the second and third diagrams generate $\mathcal{O}_{14}$ and $\mathcal{O}_{13}$, respectively; and the last diagram generates $\mathcal{O}_{12}$. The $\mathcal{O}_{13,14}$ contributions from these loops vanish as $q \rightarrow 0$.