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We calculate the screening charge density distribution due to a point charge,such as that of a positive muon ($\mu^+$), placed between the planes of ahighly anisotropic layered metal. In underdoped hole cuprates the screeningcharge converts the charge density in the metallic-plane unit cells in thevicinity of the $\mu^+$ to nearly its value in the insulating state. Thecurrent-loop ordered state observed by polarized neutron diffraction thenvanishes in such cells, and also in nearby cells over a distance of order theintrinsic correlation length of the loop-ordered state. This in turn stronglysuppresses the loop-current field at the $\mu^+$ site. We estimate thissuppressed field in underdoped YBa$_2$Cu$_3$O$_{6+x}$ andLa$_{2-x}$Sr$_x$CuO$_4$, and find consistency with the observed 0.2--0.3 Gfield in the former case and the observed upper bound of $\sim$0.2 G in thelatter case. This resolves the controversy between the neutron diffraction and$\mu$SR experiments. The screening calculation also has relevance for theeffect of other charge impurities in the cuprates, such as the dopantsthemselves.

Screening of point charge impurities in highly anisotropic metals: application to $\mu^+$ spin relaxation in underdoped cuprates