Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

Question

From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like

\begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation}

where $ \tilde{G}^{a,\mu \nu} = \frac{1}{2} \epsilon^{\mu \nu \rho \sigma} G^a_{ \rho \sigma} $ is the dual field strength tensor. This term is said to violate CP, giving rise to the strong CP problem.

I understand the CP violation comes from the epsilon tensor in the dual field strength but I am looking for a simple straightforward demonstration of the CP violating nature of a term like $G \tilde{G}$.

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user user42865

Comments

I have answered this question here: physics.stackexchange.com/a/111178/6178

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user Frederic Brünner

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possible duplicate of Yang-Mills CP violation

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user Robin Ekman

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Ok thanks. Is there another way to answer without using electric and magnetic fields, nor differential forms ?

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user user42865

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Well the simplest answer then is what you wrote, namely that it's because the Levi-Civita is a pseudo-tensor, so this term changes sign under parity.

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user Robin Ekman