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There is a possible CP-violating term in the strong sector of the standard model proportional to $\theta_\text{QCD}$. In the absence of this term, the strong interactions are CP-invariant. In the weak interactions, there is CP-violation that comes from the lone phase $\delta$ appearing in the CKM quark-mixing matrix.

When the strong interactions are coupled to the weak interactions, and in the absence of new physics, do the weak interactions renormalize $\theta_\text{QCD}$?

My best guess by how this would show up diagrammatically is when the self energies of quarks pick up phases through loops, which via the U(1) anomaly could be interpreted as $\theta_\text{QCD}$. Is this guess correct?

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The CP phase in the strong sector comes from a topological term (a total derivative) $$\require{cancel} \mathcal L_{\cancel{CP}} \propto \theta_\mathrm{QCD} \epsilon^{\mu \nu \rho \sigma} G_{\mu \nu} G_{\rho \sigma}, $$ where $G_{\mu \nu}$ is the gluon field-strength.

Such an operator can never be produced as an effective operator from weak interactions, therefore $\theta_\mathrm{QCD}$ will not be renormalized.

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