  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.
