  We study local CP-violation on the lattice by measuring the local correlation
between the topological charge density and the electric dipole moment of
quarks, induced by a constant external magnetic field. This correlator is found
to increase linearly with the external field, with the coefficient of
proportionality depending only weakly on temperature. Results are obtained on
lattices with various spacings, and are extrapolated to the continuum limit
after the renormalization of the observables is carried out. This
renormalization utilizes the gradient flow for the quark and gluon fields. Our
findings suggest that the strength of local CP-violation in QCD with physical
quark masses is about an order of magnitude smaller than a model prediction
based on nearly massless quarks in domains of constant gluon backgrounds with
topological charge. We also show numerical evidence that the observed local
CP-violation correlates with spatially extended electric dipole structures in
the QCD vacuum.
