Abstract
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 \( \overline{\theta} \) = −arg det \( {\mathcal{M}}_q^{\textrm{loop}} \) 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 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.
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Acknowledgments
This work is supported by the JSPS Grant-in-Aid for Scientific Research Grant No. 20H01895 (J.H.) and No. 21K03572 (J.H.). The work of J.H. is also supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. This work is also supported by JSPS Core-to-Core Program Grant No. JPJSCCA20200002. This work was financially supported by JST SPRING, Grant Number JPMJSP2125. The author (N.O.) would like to take this opportunity to thank the “Interdisciplinary Frontier Next-Generation Researcher Program of the Tokai Higher Education and Research System.”
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Banno, T., Hisano, J., Kitahara, T. et al. Closer look at the matching condition for radiative QCD θ parameter. J. High Energ. Phys. 2024, 195 (2024). https://doi.org/10.1007/JHEP02(2024)195
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DOI: https://doi.org/10.1007/JHEP02(2024)195