There is a coupling in the standard model of the form $\theta F^{\mu\nu}\tilde{F}_{\mu\nu}$ where $F^{\mu\nu}$ is the QCD field strength. I've read that $\theta<10^{-8}$, which is speculated to be related to Peccei-Quinn symmetry.
What would be the consequences of this term if $\theta$ were not small? In particular, I know that this term is CP-violating. How will this CP-violation be manifested in experiments?
How do we know $\theta<10^{-8}$?
Answer
To answer point $2$, $\theta$ is derived from the neutron elecric dipole moment that is measured to be like $ <10^{-18}\,\text{e}\cdot\text{m}$. You can prove that the dipole moment is proportional to $\theta$ and from that you can get your upper limit for $\theta$. See Donoghue, Golowich, Holstein - "Dynamics of the Standard Model", pages 44-45 for more details.
As far as your first question is concerned, a value of $\theta$ different from $0$ would make $CP$ not a symmetry for QCD, but I don't really know what effect this could have in detail.
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