We revisit the issue of the vacuum angle theta dependence in weakly coupled (Higgsed) Yang-Mills theories. Two most popular mechanisms for eliminating physical theta dependence are massless quarks and axions. Anselm and Johansen noted that the vacuum angle theta(EW), associated with the electroweak SU(2) in the Glashow-Weinberg-Salam model, is unobservable although all fermion fields obtain masses through Higgsing and there is no axion. We generalize this idea to a broad class of Higgsed Yang-Mills theories. In the second part we consider consequences of Grand Unification. We start from a unifying group, e.g. SU(5), at a high ultraviolet scale and evolve the theory down within the Wilson procedure. If on the way to infrared the unifying group is broken down into a few factors, all factor groups inherit one and the same theta angle -- that of the unifying group. We show that embedding the SM in SU(5) drastically changes the Anselm-Johansen conclusion: the electroweak vacuum angle theta(EW), equal to theta(QCD) becomes in principle observable in \Delta B=\Delta L =\pm 1 processes. We also note in passing that if the axion mechanism is set up above the unification scale, we have one and the same axion in the electroweak theory and QCD, and their impacts are interdependent.
Shifman, M., & Vainshtein, A. (2017). (In)dependence of Theta in the Higgs Regime without Axions. https://doi.org/10.1142/S0217732317500845