  In a class of beyond-standard-model theories, CP-odd observables, such as the
neutron electric dipole moment, receive significant contributions from
flavor-neutral P-odd and CP-odd four-quark operators. However, considerable
uncertainties exist in the hadronic matrix elements of these operators strongly
affecting the experimental constraints on CP-violating parameters in the
theories. Here we study their hadronic matrix elements in combined chiral
perturbation theory and nucleon models. We first classify the operators in
chiral representations and present the leading-order QCD evolutions. We then
match the four-quark operators to the corresponding ones in chiral hadronic
theory, finding symmetry relations among the matrix elements. Although this
makes lattice QCD calculations feasible, we choose to estimate the
non-perturbative matching coefficients in simple quark models. We finally
compare the results for the neutron electric dipole moment and P-odd and CP-odd
pion-nucleon couplings with the previous studies using naive factorization and
QCD sum rules. Our study shall provide valuable insights on the present
hadronic physics uncertainties in these observables.
