BRST view of spontaneous symmetry breaking

After the phase transition of a grand unified theory (GUT), its single coupling splits into the couplings of the basic interactions. On the other side, the measurement of the Weinberg mixing angle in the electroweak theory determines that the couplings associated to the SU(2) and U(1) symmetries were already distinct in its symmetric phase, before the electroweak breaking. This picture is consistent as we understand that such couplings evolve independently with the energy scale, meeting a common point together with the strong interaction coupling at the GUT scale. From a BRST symmetry point of view, this implies that each independent coupling should be associated to independent cocycles defined on a cohomological basis. The problem is that, before the GUT symmetry breaking, the BRST operator associated to this symmetric phase predicts a single Yang-Mills coupling. This sets the question of what should be the BRST operator of the asymmetric phase, that would allow for the splitting of this coupling but at the same time, remain compatible with the BRST operator of the symmetric phase. In this work, we proceed to answer this question.