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

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

It has two eigenvalues:

${\displaystyle \lambda _{(+)}={\frac {B+{\sqrt {B^{2}+4M^{2}}}}{2}},}$ 

and

${\displaystyle \lambda _{(-)}={\frac {B-{\sqrt {B^{2}+4M^{2}}}}{2}}.}$ 

The geometric mean of ${\displaystyle \lambda _{(+)}}$  and ${\displaystyle \lambda _{(-)}}$  equals ${\displaystyle \left|M\right|}$ , since the determinant ${\displaystyle \lambda _{(+)}\;\lambda _{(-)}=-M^{2}}$ .

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, ${\displaystyle B}$  is taken to be much larger than ${\displaystyle M.}$  Then the larger eigenvalue, ${\displaystyle \lambda _{(+)},}$  is approximately equal to ${\displaystyle B,}$  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][7][8]} The matrix A is essentially the mass matrix for the neutrinos. The Majorana mass component ${\displaystyle B}$  is comparable to the GUT scale and violates lepton number conservation; while the Dirac mass components ${\displaystyle M}$  are of order of the much smaller electroweak scale, called the VEV or vacuum expectation value below. The smaller eigenvalue ${\displaystyle \lambda _{(-)}}$  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.

Call the neutrino part of a Weyl spinor ${\displaystyle \chi ,}$  a part of a left-handed lepton weak isospin doublet; the other part is the left-handed charged lepton ${\displaystyle \ell ,}$ 

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

as it is present in the minimal standard model with neutrino masses omitted, and let ${\displaystyle \eta }$  be a postulated right-handed neutrino Weyl spinor which is a singlet under weak isospin – i.e. a neutrino that fails to interact weakly, such as a sterile neutrino.

There are now three ways to form Lorentz covariant mass terms, giving either

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

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 the symmetry has been spontaneously broken by 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 ${\displaystyle \eta }$  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,

the vacuum expectation value (VEV)${\displaystyle \quad v\;\approx \;\mathrm {246\;GeV} ,\qquad \qquad |\langle H\rangle |\;=\;v/{\sqrt {2}}}$ 

${\displaystyle M_{t}={\mathcal {O}}\left(v/{\sqrt {2}}\right)\;\approx \;\mathrm {174\;GeV} ,}$ 

if the dimensionless Yukawa coupling is of order ${\displaystyle y\approx 1}$ . It can be chosen smaller consistently, but extreme values ${\displaystyle y\gg 1}$  can make the model nonperturbative.

The parameter ${\displaystyle B'}$  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 ${\displaystyle A}$  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 ${\displaystyle B=0}$  in the unbroken phase, but generate a large, non-vanishing value ${\displaystyle B\approx M_{\mathsf {GUT}}\approx \mathrm {10^{15}~GeV} ,}$  around the scale of their spontaneous symmetry breaking. So given a mass ${\displaystyle M\approx \mathrm {100\;GeV} }$  one has ${\displaystyle \lambda _{(-)}\;\approx \;\mathrm {0.01\;eV} .}$  A huge scale has thus induced a dramatically small neutrino mass for the eigenvector ${\displaystyle \nu \approx \chi -{\frac {\;M\;}{B}}\eta .}$ 

• Majoron
• Spinor

• "Seesaw Mechanism details". quantumfieldtheory.info.