Chern-Simons class of EW theory and ghost

Question

Suppose we havetopological theta-term in the action:
$$S_{\theta} = \theta \int d^{4}x\text{tr}[F_{EW}\tilde{F}_{EW}] = \theta \int d^{4}x \partial_{\mu}K^{\mu},$$

where $K_{\mu}$ is called Chern-Simons class.

There is known that in the case of QCD theory with massive quarks there exist a massless ghost state which is coupled to $K_{\mu}$, called Veneziano ghost, so that the topological susceptibility $\kappa$ is nonzero

$$\kappa_{QCD}  \equiv  \lim_{p \to 0} p_{\mu}p_{\nu}\int d^{4}x e^{ipx}\langle |K^{\mu}(x)K^{\nu}(0) |\rangle \sim \langle |\bar{u}(0)u(0)|\rangle$$

Is there some ghost state which is coupled to EW $K_{\mu}$, so that EW topological term VEV is nonzero?