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

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

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

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 $G \tilde{G}$.

• I have answered this question here: physics.stackexchange.com/a/111178/6178 – Frederic Brünner May 26 '14 at 9:08
• possible duplicate of Yang-Mills CP violation – Robin Ekman May 26 '14 at 9:27
• Ok thanks. Is there another way to answer without using electric and magnetic fields, nor differential forms ? – user42865 May 26 '14 at 9:30
• 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. – Robin Ekman May 26 '14 at 9:32