Relativistic Atomic Effects of Dark Matter Electron Scattering

The dark matter scattering with atomic bound electrons is a crucial avenue for exploring the sub-GeV mass range. The commonly used factorization, where atomic effects are encoded in an overall form factor multiplying the free-electron scattering matrix element, is not necessarily true. Especially, the free-electron kinematics and phase space cannot consistently apply for off-shell bound electrons. Starting from the first principles of quantum field theory, we establish a theoretically consistent formalism to account for the atomic effects. By taking the scalar-type interaction as an example, we investigate the difference between the non-relativistic and relativistic calculations to show that the relativistic effects can lead to a $30\% \sim 50\%$ reduction in the scattering phase space and differential cross section. In other words, not just a theoretically consistent formalism for the atomic effects but also relativistic calculation with Dirac equation are necessary.