A systematic investigation on vector dark matter-nucleus scattering in effective field theories

In this paper, we systematically investigate the general spin-one dark matter-nucleus interactions within the framework of effective field theories (EFT). We consider both the nonrelativistic (NR) and the relativistic EFT descriptions of the DM interactions with nucleons and quarks. In the NREFT framework, we present a complete list of NR operators for spin-one DM coupling to nucleons and compute their contributions to the DM response functions. Next, we consider all possible leading-order relativistic EFT operators between DM and light quarks and the photon, and perform NR reductions to match them onto the NREFT. We then derive the nuclear scattering rate from these interactions, and employ recent DM direct detection data (from both the nuclear recoil and the Migdal effect) to constrain all these EFT operators and DM electromagnetic properties. We find the elastic nuclear recoil data (from PandaX-4T, XENONnT, LZ, and DarkSide-50) set stringent bounds on the EFT coefficients for a DM mass above a few GeV while the Migdal effect datasets (from PandaX-4T, XENONnT, and DarkSide-50) can probe the DM mass region as small as 20 MeV. Lastly, we construct a UV complete model that can provide a complex spin-one DM candidate, and at the same time generate DM-quark/photon operators discussed in this work.

💡 Research Summary

This paper presents a comprehensive effective‑field‑theory (EFT) study of spin‑1 (vector) dark matter (DM) scattering off atomic nuclei. The authors work in parallel with two complementary frameworks: a non‑relativistic EFT (NREFT) that captures the low‑energy interactions relevant for direct‑detection experiments, and a relativistic EFT that describes the most general couplings of a vector DM particle to light quarks and photons.

First, the NREFT sector is built from the basic spin operators of the nucleon (1_N, S_N) and of the vector DM field (1_X, S_X, ˜S_X). Using the momentum transfer q and the transverse relative velocity v⊥ N as kinematic building blocks, the authors enumerate a complete and independent set of 26 Hermitian Galilean‑invariant operators up to second order in q and linear order in v⊥ N. These operators are classified as spin‑independent (SI) or spin‑dependent (SD) and are labeled O₁–O₂₆. Operators O₁, O₃, O₇, O₁₀ contribute for any DM spin, while O₁₄–O₂₆ are exclusive to vector DM. Each operator carries a Wilson coefficient c_i^N that encodes the underlying physics.

Next, the relativistic side lists all leading‑order (dimension‑5, 6, 7) operators that couple a complex vector DM field X_μ to quark bilinears (scalar (\bar q q), pseudoscalar (\bar q iγ₅ q), vector (\bar q γ^μ q), axial‑vector (\bar q γ^μγ₅ q), tensor (\bar q σ^{μν} q), etc.) and to the electromagnetic field strength F^{μν} (magnetic dipole κ, electric dipole (\tilde κ), and higher‑dimensional X‑γ interactions). The paper provides explicit matching formulas that reduce each relativistic operator to a linear combination of the NREFT basis, with the matching coefficients displayed in Table 2. For a complex vector DM particle, operators marked with a “×” survive; for a real vector field they vanish, and surviving terms acquire an extra factor of two.

The scattering rate is then derived by folding the NREFT operators with nuclear response functions. The authors adopt state‑of‑the‑art nuclear structure calculations (shell‑model and ab‑initio results) to evaluate the six standard DM response functions (M, Σ′, Σ″, Δ, Φ′′, Φ) for each operator. This formalism yields the differential recoil spectrum dR/dE_R for elastic nuclear recoils and, after incorporating the Migdal ionization probability, the electron‑recoil spectrum associated with the Migdal effect.

Armed with this machinery, the authors confront the theory with the latest direct‑detection data. Elastic‑recoil limits from PandaX‑4T, XENONnT, LZ, and DarkSide‑50 are used to bound the Wilson coefficients for DM masses above a few GeV. Two isospin scenarios are considered: an isospin‑universal case (single coefficient for protons and neutrons) and an isospin‑specific case (separate coefficients). For sub‑GeV masses, the Migdal‑effect searches from the same experiments are employed, extending sensitivity down to ≈ 20 MeV. The Migdal analysis is especially powerful for operators that couple DM to the electromagnetic current (magnetic/electric dipoles) or to scalar/pseudoscalar quark currents, because the ionized electron carries away a sizable fraction of the transferred energy. The resulting 90 % confidence limits are presented for each operator, showing improvements of up to one or two orders of magnitude relative to previous studies.

Finally, the paper constructs a UV‑complete model that realizes a complex vector DM candidate. A new dark gauge group U(1)_X is introduced, spontaneously broken by a complex scalar Φ. The gauge boson X_μ acquires mass and is stabilized by a residual Z₂ symmetry. Kinetic mixing between the dark photon and the SM hypercharge, together with higher‑dimensional operators involving Φ, generate the relativistic DM‑quark and DM‑photon interactions listed earlier. The model reproduces the EFT operator set, respects all experimental bounds, and provides a concrete playground for further phenomenological investigations (e.g., collider signatures, indirect detection).

In summary, the work delivers the most exhaustive EFT treatment of vector DM–nucleus scattering to date. It supplies a complete non‑relativistic operator basis, a systematic matching from relativistic quark‑level operators, a robust nuclear‑response formalism, and up‑to‑date experimental constraints—including the Migdal effect for ultra‑light DM. The accompanying UV model demonstrates that such an EFT can arise from a well‑motivated high‑energy theory. This study therefore sets a solid benchmark for future theoretical model building and for interpreting forthcoming direct‑detection data in the context of vector dark matter.