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In the [[Grand Unified Theory|theory of grand unification]] of [[particle physics]], and, in particular, in theories of [[neutrino]] masses and [[neutrino oscillation]], the '''seesaw mechanism''' is a generic model used to understand the relative sizes of observed neutrino masses, of the order of [[electronvolt|eV]],  compared to those of [[quark]]s and charged [[lepton]]s, which are millions of times heavier.

There are several types of models, each extending the [[Standard Model]].  The simplest version, type 1, extends the Standard Model by assuming two or more additional right-handed neutrino fields
inert under the electroweak interactions,<ref>It is possible to generate two light but massive neutrinos with only one right-handed neutrino, but the resulting spectra are generally not viable.</ref> and the existence of a very large mass scale.  This allows the mass scale to be identifiable with the postulated scale of grand unification.

== 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]]
:<math>A = \begin{pmatrix}0&M\\M&B\end{pmatrix} \text{,}</math>
where ''B'' is taken to be much larger than ''M''.

It has two very disproportionate [[eigenvalue]]s:
:<math>\lambda_\pm = \frac{B\pm \sqrt{B^2+4M^2}}{2} \text{.}</math>
The larger eigenvalue, ''λ''<sub>+</sub>, is approximately equal to {{mvar|B}}, while the smaller eigenvalue is approximately equal to
:<math>\lambda_- \approx -\frac{M^2}B.</math>
Thus, |''M'' | is the [[geometric mean]] of ''λ''<sub>+</sub> and {{math| −''λ''<sub>−</sub>}}, since the [[determinant]] equals {{nowrap|''λ''<sub>+</sub>''λ''<sub>−</sub> {{=}} −''M'' <sup>2</sup>}}.

''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.

This mechanism serves to explain why the [[neutrino]] masses are so small.<ref name="Minkowski1977">{{cite journal|author=P. Minkowski|authorlink=Peter Minkowski|doi=10.1016/0370-2693(77)90435-X|title=μ --> e γ at a Rate of One Out of 1-Billion Muon Decays?|bibcode = 1977PhLB...67..421M|year=1977|journal=Physics Letters B|volume=67|issue=4|pages=421 }}</ref><ref name="Gell-Mann1979">[[Murray Gell-Mann|M. Gell-Mann]], [[Pierre Ramond|P. Ramond]] and [[Richard Slansky|R. Slansky]], in ''Supergravity'', ed. by D. Freedman and P. Van Nieuwenhuizen, North Holland, Amsterdam (1979), pp. 315-321. ISBN 044485438X</ref><ref name="Yanagida1980" >{{cite journal|journal=Progress of Theoretical Physics|volume=64|issue=3|pages=1103–1105|year=1980|author=T. Yanagida|doi=10.1143/PTP.64.1103|title=Horizontal Symmetry and Masses of Neutrinos}}</ref><ref name="MohapatraSenjanovic1980" >{{cite journal|journal=Phys. Rev. Lett.|volume=44|issue=14|pages=912–915|year=1980|author=[[Rabindra Mohapatra|R. N. Mohapatra]], G. Senjanovic|doi=10.1103/PhysRevLett.44.912|title=Neutrino Mass and Spontaneous Parity Nonconservation|bibcode = 1980PhRvL..44..912M }}</ref><ref name="SchechterValle1980" >{{cite journal|journal=Phys. Rev.|volume=22|issue=9|pages=2227–2235|year=1980|author=J. Schechter, [[José W. F. Valle]] |doi=10.1103/PhysRevD.22.2227|title=Neutrino masses in SU(2) ⊗ U(1) theories|bibcode = 1980PhRvD..22.2227S|last2=Valle|first2=J. }}</ref> The matrix ''A'' is essentially the [[mass matrix]] for the neutrinos. The [[Majorana spinor|Majorana]] mass component ''B'' is comparable to the [[GUT scale]] and violates lepton number; while the components [[Dirac spinor|Dirac]] mass ''M'', are of order of the much smaller [[electroweak scale]], the ''VEV'' below. The smaller eigenvalue  λ<sub>−</sub> 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.

== 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 theory|gauge invariance]] of the standard model [[Action (physics)|action]], and the corresponding charges of the lepton- and neutrino fields.

Let the  [[Weyl spinor]] ''χ'' be the [[neutrino]] part of a [[Chirality (physics)|left-handed]] [[lepton]] [[Weak isospin|isospin]] [[Doublet state|doublet]] (the other part being the left-handed charged lepton),
:<math> L=\begin{pmatrix}\chi \\ \eta \end{pmatrix}   ~,</math>
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 state|singlet]] under [[weak isospin]] (i.e. does not interact weakly, such as a [[sterile neutrino]]).

There are now three ways to form [[Lorentz covariance|Lorentz covariant]] mass terms, giving either
:<math>\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{,}</math>
and their [[complex conjugate]]s, which can be written as a [[quadratic form]],
:<math>\frac{1}{2}\,\begin{pmatrix}\chi& \eta\end{pmatrix} \begin{pmatrix}B' & M \\ M& B\end{pmatrix}\begin{pmatrix}\chi\\ \eta\end{pmatrix}.</math>
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 interaction|electroweak gauge symmetry]], and can only appear after its [[spontaneous symmetry breaking|spontaneous breakdown]] through a [[Higgs mechanism]], like the Dirac masses of the charged leptons. In particular, since {{nowrap|''χ'' ∈ ''L''}} has [[weak isospin]] ½ like the [[Higgs field]] ''H'', and ''η'' has [[weak isospin]] 0, the mass parameter ''M'' can be generated from [[Yukawa interaction]]s with the [[Higgs field]], in the conventional standard model fashion,
:<math>\mathcal{L}_{yuk}=y \, \eta L \epsilon H^*+...</math>
This means that ''M'' is [[Naturalness (physics)|naturally]] of the order of the [[vacuum expectation value]] of the standard model [[Higgs field]], 
:<math>\text{VEV }v\approx 246 \text{ GeV}, \qquad\qquad |\langle H \rangle|= v/\sqrt{2}
 </math>
:<math>M_t= O( v/\sqrt{2})\approx 174 \text{ GeV}~,</math>
if the dimensionless [[Yukawa interaction|Yukawa coupling]] is of order {{nowrap|''y'' ≈ 1}}. It can be chosen smaller consistently, but extreme values {{nowrap|''y'' ≫ 1}} can make the model [[Perturbation theory (quantum mechanics)|nonperturbative]].

The parameter ''B''',  on the other hand, is forbidden, since no [[renormalizable]] singlet under [[weak hypercharge]] and [[weak isospin|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 theory|grand unification]]. In such models, enlarged [[Gauge theory|gauge symmetries]] may be present, which initially force {{nowrap|''B'' {{=}} 0}} in the unbroken phase, but generate a non vanishing large value {{nowrap|''B'' ≈ ''M''<sub>GUT</sub>}} ≈  10<sup>15</sup> GeV, around the scale of their [[spontaneous symmetry breaking]], so, given an ''M'' ≈ 100 GeV,  one has   λ<sub>−</sub> ≈ 0.01 eV.    A  huge scale has thus induced a dramatically small neutrino mass for the eigenvector  ''ν'' ≈ ''χ'' − ''(M/B) η''  .

== See also ==
* [[Majoron]]
* [[Spinor]]

==References==
<references/>

==External links==
* [http://www.quantumfieldtheory.info/TheSeesawMechanism.htm Seesaw Mechanism details]

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[[Category:Particle physics]]
[[Category:Neutrinos]]
[[Category:Quantum field theory]]