Novel loop-diagrammatic approach to QCD $\theta$ parameter and application to the left-right model

TL;DR

This paper introduces a new loop-diagrammatic method for calculating radiative corrections to the QCD $ heta$ parameter, especially in models with extra heavy quarks, and applies it to the minimal left-right symmetric model.

Contribution

It develops a robust, direct loop-diagrammatic approach for radiative $ar heta$ calculations, improving upon the traditional method that struggles with extra heavy quarks.

Findings

Two-loop radiative $ar heta$ vanishes in the minimal left-right model.

Three-loop corrections can induce a neutron EDM near current experimental bounds.

The size of the induced neutron EDM is constrained by perturbative unitarity.

Abstract

When the QCD axion is absent in full theory, the strong $CP$ problem has to be explained by an additional mechanism, e.g., the left-right symmetry. Even though tree-level QCD $\bar\theta$ parameter is restricted by the mechanism, radiative corrections to $\bar\theta$ are mostly generated, which leads to a dangerous neutron electric dipole moment (EDM). The ordinary method for calculating the radiative $\bar\theta$ utilizes an equation $\bar \theta= - \text{arg}\, \text{det}\, m_q^{\rm loop}$ based on the chiral rotations of complex quark masses. In this paper, we point out that when full theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark contributions and does not contain its full radiative corrections. We formulate a novel method to calculate the radiative corrections to $\bar\theta$ through a direct loop-diagrammatic approach, which should be more robust than the ordinary one. As an application, we investigate the radiative $\bar\theta$ in the minimal left-right symmetric model. We first confirm a seminal result that two-loop level radiative $\bar\theta$ completely vanishes (corresponding to one-loop corrections to the quark mass matrices). Furthermore, we estimate the size of a non-vanishing radiative $\bar\theta$ at three-loop level. It is found that the resultant induced neutron EDM is comparable to the current experimental bound, and the expected size is restricted by the perturbative unitarity bound in the minimal left-right model.