Sum rules for x-ray circular and linear dichroism based on complete magnetic multipole basis

X-ray magnetic circular dichroism (XMCD) and X-ray magnetic linear dichroism (XMLD) are powerful spectroscopic techniques for probing magnetic properties in solids. In this study, we revisit the XMCD and XMLD sum rules within a complete magnetic multipole basis that incorporates both spinless and spinful multipoles. We demonstrate that these multipoles can be clearly distinguished and individually detected through the sum-rule formalism. Within this framework, the anisotropic magnetic dipole term is naturally derived in XMCD, offering a microscopic origin for ferromagnetic-like behavior in antiferromagnets. Furthermore, we derive the sum rules for out-of-plane and in-plane XMLD regarding electric quadrupole contributions defined based on the complete multipole basis. Our theoretical approach provides a unified, symmetry-consistent framework for analyzing dichroic signals in various magnetic materials. These findings deepen the understanding of XMCD and XMLD and open pathways to exploring complex magnetic structures and spin-orbit coupling effects in emergent magnetic materials.