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Strong CP violation happens when the theta term gives a complex phase, but there are two values where the effect is real, $\theta=0$ which gives no phase for instantons, or $\theta=\pi$, which gives a -1 phase for each instanton. Do we know we are at 0, not at $\pi$?

For instance, do we have lattice simulations or other theoretical estimates for the $\pi$ version of QCD showing disagreement with the hadron spectrum? This might not be easy, as there is a sign problem.

This question is considered at the end of the paper by Crewther, Di Vecchia, Veneziano and Witten :

http://cds.cern.ch/record/133382/files/197909176.pdf

They argue that to go to $\theta_{QCD} =0$ to $\theta_{QCD} = \pi$ is equivalent to change the sign of one of the masses of quarks, for example $m_u$ (this step uses the usual relation between chiral rotation and $\theta$-term). Then they look for quantities sensible to the sign of $m_u$. A natural choice is $m_d - m_u$, that you can expect to show up in the difference of masses between the neutral kaon anf the positively charged kaon. Indeed, they use a formula obtained by current algebra techniques at $\theta_{QCD} =0$ relating this difference of masses to $m_d - m_u$ (formula (23) of the paper). For $\theta_{QCD}= \pi$, the same formula should be true with $m_u$ replaced by $-m_u$ (formula (24) of the paper). Comparison with experimental masses of kaons excludes the case $\theta_{QCD} = \pi$ (roughly, the smallness of the difference of masses between the neutral and the positively charged kaons suggests that $\theta_{QCD}$ is around 0 rather than around $ \pi$).

Aren't masses always supposed to be positive?

Is that link stable and permanent? I'd like to import the paper for review, and Dashen's paper linked as 6 in the references. Thanks, this is interesting. I figured some sort of SVZ/Veneziano-Witten thing would estimate it, but I didn't know it was considered.

Maybe a better link is http://cds.cern.ch/record/133382 which is a page of the CERN Document server, so I guess stable and permanent, where one can find a link to the pdf.

Dirac fermions in 3+1d can have complex masses and still be unitary.

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