# How do we know $\theta_\mathrm{QCD} \ne \pi$?

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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.

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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 $\pi$).

answered Sep 16, 2014 by (5,120 points)
edited Sep 16, 2014 by 40227

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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