  The Nonrelativistic Effective 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 bridges this gap by utilizing an innovative tensor
decomposition method, extending NRET to incorporate previously overlooked
tensor interactions. We find additional operators in the $\mu \to e$ conversion
that are not present in the scalar and vector couplings. 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. To support further research and experimental analyses,
comprehensive tables featuring tensor matrix elements and their corresponding
operators are provided.
