Editing Seesaw mechanism (section)

== Type 1 seesaw ==
This model produces a light neutrino, for each of the three known neutrino flavors, and a corresponding very heavy [[neutrino]] for each flavor, which has yet to be observed.

The simple mathematical principle behind the seesaw mechanism is the following property of any 2×2 [[Matrix (mathematics)|matrix]] of the form
: <math> A = \begin{pmatrix}  0  &  M   \\
                             M  &  B   \end{pmatrix} .</math>
It has two [[eigenvalue]]s:
: <math>\lambda_{(+)} = \frac{B + \sqrt{ B^2 + 4 M^2 }}{2} ,</math>
and
: <math>\lambda_{(-)} = \frac{B - \sqrt{ B^2 + 4 M^2 } }{2} .</math>
The [[geometric mean]] of <math>\lambda_{(+)}</math> and <math>\lambda_{(-)} </math> equals <math>\left| M \right|</math>, since the [[determinant]] <math> \lambda_{(+)} \; \lambda_{(-)} = -M^2 </math>.

Thus, if one of the eigenvalues goes up, the other goes down, and vice versa. This is the point of the name "[[seesaw]]" of the mechanism.

In applying this model to neutrinos, <math> B </math> is taken to be much larger than <math> M .</math>
Then the larger eigenvalue, <math>\lambda_{(+)},</math> is approximately equal to <math> B ,</math> while the smaller eigenvalue is approximately equal to
: <math> \lambda_- \approx -\frac{M^2}{B} .</math>

This mechanism serves to explain why the [[neutrino]] masses are so small.<ref name=Minkowski-1977-1biln-μ/><ref>[[Tsutomu Yanagida|Yanagida, T.]] (1979). "Horizontal gauge symmetry and masses of neutrinos", Proceedings: Workshop on the Unified Theories and the Baryon Number in the Universe: published in KEK Japan, February 13-14, 1979, Conf. Proc. C7902131, p.95- 99.</ref><ref>{{Cite journal |last=Yanagida |first=Tsutomu |date=1979-12-01 |title=Horizontal symmetry and mass of the $t$ quark |url=https://link.aps.org/doi/10.1103/PhysRevD.20.2986 |journal=Physical Review D |volume=20 |issue=11 |pages=2986–2988 |doi=10.1103/PhysRevD.20.2986|bibcode=1979PhRvD..20.2986Y |url-access=subscription }}</ref><ref name=GellMann-1979/><ref name=Yanagida-1980-HorizSym/><ref name=Glashow-1979-NATO/><ref name=Mohapatra-Senjanovic-1980/><ref name=Schechter-Valle-1980/>
The matrix {{mvar|A}} is essentially the [[mass matrix]] for the neutrinos. The [[Majorana spinor|Majorana]] mass component <math> B </math> is comparable to the [[GUT scale]] and violates lepton number conservation; while the [[Dirac spinor|Dirac]] mass components <math> M </math> are of order of the much smaller [[electroweak scale]], called the VEV or ''vacuum expectation value'' below. The smaller eigenvalue <math> \lambda_{(-)} </math> then leads to a very small neutrino mass, comparable to {{val|1|ul=eV}}, which is in qualitative accord with experiments—sometimes regarded as supportive evidence for the framework of Grand Unified Theories.