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So, as far as I know, the Strong CP Problem in QCD results from the theta angle term in the action: $i\theta\int_X F_\nabla\wedge F_\nabla$ where $\nabla$ is the gauge connection and $X$ is a manifold on which the theory is defined. This term obviously breaks CP symmetry with non-zero choice of theta angle. Correct me if I am wrong.

At any rate, experimental evidence has shown CP symmetry to be a consistent aspect of QCD, and the Strong CP Problem is essentially to discover CP violation or to prove CP symmetry in the lagrangian. Now, I am wondering about the particulars of various solutions to the problem. In particular, is it necessary to fabricate a new particle such as the axion, or are there (hypothetically) simpler and more easily verifiable ways of solving the problem?

Also, how important is the problem? In other words, would making experiment and theory consistent warrant a Nobel Prize? Or is it simply an irritating discrepancy that is not fundamentally important to our understanding of the Standard Model?

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"Would making experiment and theory consistent warrant a Nobel Prize? Or is it simply an irritating discrepancy that is not fundamentally important to our understanding of the Standard Model?"

As far as I can tell:

It depends on what the experimental results will be :) If strong CP is found to be broken it is quite important, since CP violation is considered a cornerstone of understanding the matter-antimatter asymmetry.

If it is preserved and no further explanation is found to work, then it's an intriguing feature since there's a term in the SM Lagrangian which simply doesn't manifest.

If axions are found, it's obviously the most interesting case.

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