Closer look at the matching condition for radiative QCD $\theta$ parameter

In this paper, we scrutinize a radiatively generated QCD $\theta$ parameter at the two-loop level based on both full analytical loop functions with the Fock-Schwinger gauge method and the effective field theory approach, using simplified models. We observe that the radiatively generated $\theta$ parameters at the low energy scale precisely match between them. It provides validity to perturbative loop calculations of the QCD $\theta$ parameter with the Fock-Schwinger gauge method. Furthermore, it is also shown that the ordinary Fujikawa method for the radiative $\theta$ parameter by using $\bar\theta = - arg det M_q^{loop}$ does not cover all contributions in the simplified models. But, we also find that when there is a scale hierarchy in $CP$-violating sector, evaluation of the Fujikawa method is numerically sufficient. As an application, we calculate the radiative $\theta$ parameter at the two-loop level in a slightly extended Nelson-Barr model, where the spontaneous $CP$ violation occurs to solve the strong $CP$ problem. It is found a part of the radiative $\theta$ parameters cannot be described by the Fujikawa method.