Relating spontaneous and explicit symmetry breaking in the presence of the Higgs mechanism

One common way to define spontaneous symmetry breaking involves explicit symmetry breaking. This definition can be used in any approach to Effective Field Theory, from perturbation theory to lattice simulations. It allows us to study the spontaneous breakdown of global symmetries without assuming that the local gauge symmetry is spontaneously broken. This is important since perturbation theory is insufficient to study extended Higgs sectors: it is insufficient to predict the physical spectrum of the SU(5) Grand Unified Theory (Georgi-Glashow) or to predict the spontaneous breakdown of global symmetries. We also study background symmetries: these are symmetries that despite they are already explicitly broken, can be still spontaneously broken. We analyse examples where a background CP (charge-parity) symmetry is not spontaneously broken: in the Standard Model, in rephasing symmetries and in geometrical CP-violation. We show that all fields are real representations of the group of symmetries, since CP is a unitary transformation. There are consequences: to study accidental symmetries (e.g. custodial symmetry, pseudo-golstone bosons) we must consider real representations; CP is a symmetry of order 4 if the neutrinos are Majorana particles and the notion of CP-violating phases is inconsistent in some Lagrangians; a recent claim that a toy model exhibits physical CP-violation while the CP symmetry is conserved by the Lagrangian and the vacuum is false.