Background
The 2×2 matrix A arises in a natural manner within the standard model by considering the most general mass matrix allowed by gauge invariance of the standard model action, and the corresponding charges of the lepton- and neutrino fields. Let the Weyl spinor χ be the neutrino part of a left-handed lepton isospin doublet (the other part being the left-handed charged lepton)
as it is present in the minimal standard model without neutrino masses, and let η be a postulated right-handed neutrino Weyl spinor which is a singlet under weak isospin (i.e. does not interact weakly, such as a sterile neutrino). There are now three ways to form Lorentz covariant mass terms, giving either
and their complex conjugates, which can be written as a quadratic form,
Since the right-handed neutrino spinor is uncharged under all standard model gauge symmetries, B is a free parameter which can in principle take any arbitrary value. The parameter M is forbidden by electroweak gauge symmetry, and can only appear after its spontaneous breakdown through a Higgs mechanism. In particular, since χ ∈ L has weak isospin ½ like the Higgs field H, and χ has weak isospin 0, the mass parameter M can be generated from Yukawa interactions with the Higgs field,
This means that M is naturally of the order of the vacuum expectation value of the standard model Higgs field,
(thus the following is bogus)
if the dimensionless Yukawa coupling is y ≈ 1. It can be chosen smaller consistently, but extreme values y ≫ 1 can make the model nonperturbative. The parameter B' on the other hand, is completely forbidden, since no renormalizable singlet under weak hypercharge and isospin can be formed using these objects. This is the origin of the pattern and hierarchy of scales of the mass matrix A within the "type 1" seesaw mechanism. The exact size of B can be explained in the context of grand unification. In such models, enlarged gauge symmetries may be present which initially force B = 0 in the unbroken phase, but generate a nonvanishing value B ≈ MGUT around the scale of their spontaneous symmetry breaking.
Read more about this topic: Seesaw Mechanism
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