Renormalization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:math> -violating nuclear forces

Electric dipole moments of nuclei, diamagnetic atoms, and certain molecules are induced by $CP$-violating nuclear forces. Naive dimensional analysis predicts these forces to be dominated long-range one-pion-exchange processes with short-range entering only at next-to-next-to-leading order in the chiral expansion. Based on renormalization arguments we argue that a consistent picture requires leading-order short-distance operator contributing $^{1}S_{0}\text{\ensuremath{-}}^{3}P_{0}$ transitions due attractive singular nature strong tensor force $^{3}P_{0}$ channel. The leads $O(1)$ corrections static oscillating, relevant for axion searches, electric moments. We discuss strategies how finite part associated low-energy constant can determined case $CP$ violation from QCD $\overline{\ensuremath{\theta}}$ term connection charge-symmetry systems.