Closer look at the matching condition for radiative QCD θ parameter
Tatsuya Banno, Junji Hisano, Teppei Kitahara, Naohiro Osamura
Abstract
A bstract In this paper, we scrutinize a radiatively generated QCD θ 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 θ parameters at the low energy scale precisely match between them. It provides validity to perturbative loop calculations of the QCD θ parameter with the Fock-Schwinger gauge method. Furthermore, it is also shown that the ordinary Fujikawa method for the radiative θ parameter by using $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> = −arg det $$ {\mathcal{M}}_q^{\textrm{loop}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>q</mml:mi> <mml:mtext>loop</mml:mtext> </mml:msubsup> </mml:math> 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 θ 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 θ parameters cannot be described by the Fujikawa method.