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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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