  Measurements of a permanent neutron electric dipole moment (EDM) potentially
probe Beyond-the-Standard Model (BSM) sources of CP-violation. At low energy
the CP-violating BSM interactions are parametrized by flavor-conserving
CP-violating operators of dimension higher than four. QCD calculations of the
nucleon matrix elements of these operators are required to fully reconstruct
the sources and magnitudes of the different CP-violating contributions to the
nucleon EDM. Herein we study the quark-chromo electric dipole moment (qCEDM)
operator and the three-gluon Weinberg operator. The non-perturbative
determination, using lattice QCD, of the nucleon matrix elements of these
CP-violating operators is hampered by their short-distance behavior. Under
renormalization these operators mix with lower dimensional operators, which
induces power divergences in the lattice spacing, as the continuum limit is
approached. We study the short-distance behavior of the qCEDM and the Weinberg
operators using the gradient flow. We perform a short flow time expansion and
determine, in perturbation theory, the expansion coefficients of the
linearly-divergent terms stemming from the mixing with the pseudoscalar density
and the topological charge, confirming the expectations of the operator product
expansion. We introduce a new method to perform calculations at non-zero
flow-time for arbitrary values of the external momenta. This method allows us
to work in four dimensions for most of the calculations described in this
paper, avoiding the complications associated with defining $\gamma_5$ in
generic d dimensions. We show that leading contributions in the external
momenta can be reproduced by defining $\gamma_5$ using the 't
Hooft-Veltman-Breitenlohner-Maison scheme.
