As far as we are aware, no one has yet suggested that these operators play a role in baryogenesis.

Refs. [7] and [8] differ in that the former proposes a search based on observables that are genuinely odd under $CP$, while the latter’s observables are odd only under the naïve time-reversal transformation (which reverses momenta and spins [7]), meaning that effects could be generated by $CP$-conserving physics in the presence of, e.g., final state interactions.

That ${\mathcal{O}}_B$ is not $SU(2{)}_L$ invariant is best seen by writing ${\mathcal{O}}_B={W^{+}}_{\nu }^{\mu }{W^{-}}_{\lambda }^{\nu }{\tilde{B}}_{\mu }^{\lambda }=i{W^1}_{\nu }^{\mu }{W^2}_{\lambda }^{\nu }{\tilde{B}}_{\mu }^{\lambda }$, whereas the only two-field-strength invariant is $W^a{W}^b{\delta }^{ab}=W^a{W}^a$. A different operator involving $W^a{W}^a$ vanishes identically: ${W^a}_{\nu }^{\mu }{W^a}_{\lambda }^{\nu }{\tilde{B}}_{\mu }^{\lambda }=(-{W^a}_{\mu }^{\nu })(-{W^a}_{\nu }^{\lambda })(-{\tilde{B}}_{\lambda }^{\mu })=-{W^a}_{\nu }^{\lambda }{W^a}_{\mu }^{\nu }{\tilde{B}}_{\lambda }^{\mu }=0$.

$CP$-odd operators such as ${\mathcal{O}}_W$ and ${\mathcal{O}}_B$ also contribute to $CP$-even electroweak precision observables, via diagrams containing $\ge 2$ insertions. For example, ${\mathcal{O}}_B$ gives contributions to $g^2\widehat{U}$ and $g^{-2}{m}_W^2\widehat{T}$ (defined as in [12]) of order ${\alpha }_B^2\frac{m_W^4}{16{\pi }^2}$ and ${\alpha }_B^2{g}^2\frac{m_W^6}{(16{\pi }^2{)}^2}$, respectively, implying weak bounds of ${\alpha }_B\lesssim (100\mathrm{GeV}{)}^{-2}$ and ${\alpha }_B\lesssim (20\mathrm{GeV}{)}^{-2}$, respectively. Note that the bound from $\widehat{T}$ is unusually poorly constraining as ${\mathcal{O}}_B$’s contribution must be 2-loop (any 1-loop diagram of two ${\mathcal{O}}_B$s has derivatives on the external legs). ${\mathcal{O}}_W$’s contribution to $2g^{-2}{m}_W^{-2}W$ of $\sim {\alpha }_W^2\frac{m_W^2}{16{\pi }^2}$ gives a similarly loose limit of ${\alpha }_W\lesssim (90\mathrm{GeV}{)}^{-2}$.

For a discussion of the RG corrections to these bounds, see [18].

Note that the power counting given in (5) only applies for terms where the symmetry is nonlinearly realized. It does not apply to the gauge kinetic terms or ${\mathcal{O}}_W$, for example.

Ref. [26] obtains a similar result for both ${\alpha }_{\gamma }$ and ${\alpha }_Z$ “within the nonlinear realization scenario”; in fact, it is only correct for ${\alpha }_B$. Moreover, our discussion shows that there is no alternative scenario.

Though, as we see, the cutoff needed to generate observable effects from ${\mathcal{O}}_B$ at the LHC is so low that we are almost justified in using a description in which a 125 GeV Higgs is integrated out.

DELPHI has a bound of ${\alpha }_Z{m}_W^2=-0.08\pm 0.07$ [29].