Seesaw mechanism

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 of the form

${\displaystyle A={\begin{pmatrix}0&M\\M&B\end{pmatrix}}{\text{.}}}$

It has two eigenvalues:

${\displaystyle \lambda _{\pm }={\frac {B\pm {\sqrt {B^{2}+4M^{2}}}}{2}}{\text{.}}}$

The geometric mean of λ₊ and −λ₋ equals |M|, since the determinant λ₊λ₋ = −M².

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, B is taken to be much larger than M. Then the larger eigenvalue, λ₊, is approximately equal to B, while the smaller eigenvalue is approximately equal to

${\displaystyle \lambda _{-}\approx -{\frac {M^{2}}{B}}.}$

This mechanism serves to explain why the neutrino masses are so small.[1][2][3][4][5][6] The matrix A is essentially the mass matrix for the neutrinos. The Majorana mass component B is comparable to the GUT scale and violates lepton number; while the components Dirac mass M, are of order of the much smaller electroweak scale – called the "VEV" below (where VEV stands for Vacuum Expectation Value). The smaller eigenvalue λ₋ then leads to a very small neutrino mass, comparable to 1 eV, which is in qualitative accord with experiments – sometimes regarded as supportive evidence for the framework of Grand Unified Theories.

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 l),

${\displaystyle L={\begin{pmatrix}\chi \\l\end{pmatrix}}~,}$

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

${\displaystyle {\frac {1}{2}}\,B'\,\chi ^{\alpha }\chi _{\alpha }\,{\text{,}}\quad {\frac {1}{2}}\,B\,\eta ^{\alpha }\eta _{\alpha }\,{\text{,}}\quad {\text{or}}\quad M\,\eta ^{\alpha }\chi _{\alpha }\,{\text{,}}}$

and their complex conjugates, which can be written as a quadratic form,

${\displaystyle {\frac {1}{2}}\,{\begin{pmatrix}\chi &\eta \end{pmatrix}}{\begin{pmatrix}B'&M\\M&B\end{pmatrix}}{\begin{pmatrix}\chi \\\eta \end{pmatrix}}.}$

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, like the Dirac masses of the charged leptons. In particular, since χ ∈ L has weak isospin ​¹⁄₂ like the Higgs field H, and η has weak isospin 0, the mass parameter M can be generated from Yukawa interactions with the Higgs field, in the conventional standard model fashion,

${\displaystyle {\mathcal {L}}_{yuk}=y\,\eta L\epsilon H^{*}+~...}$

This means that M is naturally of the order of the vacuum expectation value of the standard model Higgs field,

${\displaystyle {\text{VEV }}v\approx 246{\text{ GeV}},\qquad \qquad |\langle H\rangle |=v/{\sqrt {2}}}$

${\displaystyle M_{t}=O(v/{\sqrt {2}})\approx 174{\text{ GeV}}~,}$

if the dimensionless Yukawa coupling is of order 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 forbidden, since no renormalizable singlet under weak hypercharge and isospin can be formed using these doublet components – only a nonrenormalizable, dimension 5 term is allowed. This is the origin of the pattern and hierarchy of scales of the mass matrix A within the "Type 1" seesaw mechanism.

The large size of B can be motivated 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 large, non-vanishing value B ≈ M_GUT ≈ 10¹⁵ GeV, around the scale of their spontaneous symmetry breaking. So given a mass M ≈ 100 GeV , one has λ₋ ≈ 0.01 eV . A huge scale has thus induced a dramatically small neutrino mass for the eigenvector ν ≈ χ − (^M⁄_B) η .

• Majoron
• Spinor

• "Seesaw Mechanism details". quantumfieldtheory.info.