Nonrelativistic nuclear reduction for tensor couplings in dark matter direct detection and $\mu \to e$ conversion

The nonrelativistic effective field theory (NRET) is widely used in dark matter direct detection and charged-lepton flavor violation studies through $\mu \to e$ conversion. However, existing literature has not fully considered tensor couplings. This study fills this gap by utilizing an innovative tensor decomposition method, extending NRET to incorporate previously overlooked tensor interactions. This development is expected to have a significant impact on ongoing experiments seeking physics beyond the Standard Model and on our understanding of the new-physics interactions. Notably, we identify additional operators in $\mu \to e$ conversion that are absent in scalar and vector couplings. To support further research and experimental analyses, comprehensive tables featuring tensor matrix elements and their corresponding operators are provided.