# Is a constant electric field CP violating?

Consider, for instance, a fundamental massless three-form field $$C_{\alpha\beta\gamma}$$ in the Coulomb phase:

$$\mathcal L = E_{\mu\alpha\beta\gamma}E^{\mu\alpha\beta\gamma} + C_{\alpha\beta\gamma}J^{\alpha\beta\gamma}\,,$$

where $$E_{\mu\alpha\beta\gamma} = \partial_{[\mu}C_{\alpha\beta\gamma]}$$ is the field strength and $$J^{\alpha\beta\gamma}$$ is a conserved external current.

In the absence of sources, the four-form electric field can take an arbitrary constant value, $$E_{\mu\alpha\beta\gamma} = E_0\epsilon_{\mu\alpha\beta\gamma}$$ with $$E_0$$ a constant number.

In a paper by Gia Dvali, it is claimed that

Any theory in which a three-form field is in the Coulomb phase ‘suffers’ from a generalized strong CP problem.

I would like to know why a constant electric field violates CP. Thanks.

This is because $$E$$ is a psuedo-scalar.
$$E$$ is a 4-form, which in 4-d is a volume form. Hence under a transformation $$T^\mu_\nu$$ it goes to $$\det(T) E$$. This makes it Lorenz invariant since Lorenz transformations have determinant 1, but not $$P$$ invariant since $$\det(P)=-1$$. Since we also expect $$C$$ to act trivially on $$E$$, we have $$CP(E)=-E$$.