Editing Seesaw mechanism

{{Short description|Model in particle physics}}
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. The name of the seesaw mechanism was given by [[Tsutomu Yanagida]] in a Tokyo conference in 1981.

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 interaction,{{efn|
It is possible to generate two low-mass neutrinos with only one right-handed neutrino, but the resulting mass spectra are generally not viable.
}}
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]] 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.

== Background ==
The 2×2 matrix {{mvar|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.

Call the [[neutrino]] part of a [[Weyl spinor]] <math> \chi ,</math> a part of a [[Chirality (physics)|left-handed]] [[lepton]] [[weak isospin]] [[Doublet state|doublet]]; the other part is the left-handed charged lepton <math>\ell,</math>
:<math> L = \begin{pmatrix} \chi \\ \ell \end{pmatrix} ,</math>
as it is present in the minimal [[standard model]] with neutrino masses omitted, and let <math>\eta</math> be a postulated right-handed neutrino Weyl spinor which is a [[Singlet state|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 covariance|Lorentz covariant]] mass terms, giving either
: <math> \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 \, ,</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, {{mvar|B}} is a free parameter which can in principle take any arbitrary value.

The parameter {{mvar|M}} is forbidden by [[Electroweak interaction|electroweak gauge symmetry]], and can only appear after the symmetry has been [[spontaneous symmetry breaking|spontaneously broken]] by a [[Higgs mechanism]], like the Dirac masses of the charged leptons. In particular, since {{nowrap|{{mvar|χ}} ∈ {{mvar|L}}}} has [[weak isospin]] {{sfrac|1|2}} like the [[Higgs field]] {{mvar|H}}, and <math> \eta </math> has [[weak isospin]] 0, the mass parameter {{mvar|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 {{mvar|M}} is naturally of the order of the [[vacuum expectation value]] of the standard model [[Higgs field]], 
: the [[vacuum expectation value]] (VEV)<math>\quad v \; \approx \; \mathrm{ 246 \; GeV }, \qquad \qquad | \langle H \rangle| \; = \; v / \sqrt{2} </math>
: <math> M_t = \mathcal{O} \left( v / \sqrt{2} \right) \; \approx \; \mathrm{ 174 \; GeV } ,</math>
if the dimensionless [[Yukawa interaction|Yukawa coupling]] is of order <math> y \approx 1 </math>. It can be chosen smaller consistently, but extreme values <math> y \gg 1 </math> can make the model [[Perturbation theory (quantum mechanics)|nonperturbative]].

The parameter <math> B' </math> on the other hand, is forbidden, since no [[renormalizable]] singlet under [[weak hypercharge]] and [[weak isospin|isospin]] can be formed using these doublet components &ndash; only a nonrenormalizable, dimension&nbsp;5 term is allowed. This is the origin of the pattern and hierarchy of scales of the mass matrix <math> A </math> within the "Type&nbsp;1" seesaw mechanism.

The large size of {{mvar|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 <math> B = 0 </math> in the unbroken phase, but generate a large, non-vanishing value <math> B \approx  M_\mathsf{GUT} \approx \mathrm{10^{15}~GeV},</math> around the scale of their [[spontaneous symmetry breaking]]. So given a mass <math>M \approx \mathrm{ 100 \; GeV }</math> one has <math>\lambda_{(-)} \; \approx \; \mathrm{ 0.01 \; eV }.</math> A huge scale has thus induced a dramatically small neutrino mass for the eigenvector <math> \nu \approx \chi -  \frac{\; M \;}{B} \eta .</math>

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

== Footnotes ==
{{notelist|1}}

== References ==
{{reflist|25em|refs=

<ref name=GellMann-1979>
{{cite book
 |first1=M.  |last1=Gell-Mann |author1-link=Murray Gell-Mann
 |first2=P.  |last2=Ramond    |author2-link=Pierre Ramond
 |first3=R.  |last3=Slansky   |author3-link=Richard Slansky
 |year=1979
 |title=Supergravity
 |editor1-first=D.  |editor1-last=Freedman
 |editor2-first=P.  |editor2-last=van Nieuwenhuizen
 |publisher=North Holland
 |place=Amsterdam, NL
 |pages=315–321
 |isbn=044485438X
}}
</ref>

<ref name=Glashow-1979-NATO>
{{cite journal
 |first=S.L.  |last=Glashow   |author-link=Sheldon Glashow
 |year=1980
 |title=The future of elementary particle physics
 |editor1-first=Maurice    |editor1-last=Lévy
 |editor2-first=Jean-Louis |editor2-last=Basdevant
 |editor3-first=David      |editor3-last=Speiser
 |editor4-first=Jacques    |editor4-last=Weyers
 |editor5-first=Raymond    |editor5-last=Gastmans
 |editor6-first=Maurice    |editor6-last=Jacob
 |journal=NATO Sci. Ser. B
 |volume=61  |pages=687
 |doi=10.1007/978-1-4684-7197-7
 |isbn=978-1-4684-7199-1
}}
</ref>

<ref name=Minkowski-1977-1biln-μ>
{{cite journal
 |first=P. |last=Minkowski |author-link=Peter Minkowski
 |year=1977
 |title={{nobr|μ → e ''γ''}} at a rate of one out of 1&nbsp;billion muon decays?
 |bibcode=1977PhLB...67..421M
 |journal=Physics Letters B
 |volume=67  |issue=4  |page=421
 |doi=10.1016/0370-2693(77)90435-X
}}
</ref>

<ref name=Mohapatra-Senjanovic-1980>
{{cite journal
 |first1=R.N.  |last1=Mohapatra    |author1-link=Rabindra Mohapatra
 |first2=G.    |last2=Senjanovic   |author2-link=Goran Senjanović
 |year=1980
 |title=Neutrino mass and spontaneous parity non-conservation
 |journal=Phys. Rev. Lett.
 |volume=44  |issue=14  |pages=912–915
 |doi=10.1103/PhysRevLett.44.912
 |bibcode=1980PhRvL..44..912M
}}
</ref>

<ref name=Schechter-Valle-1980>
{{cite journal
 |first1=J. |last1=Schechter
 |first2=J. |last2=Valle     |author2-link=José W. F. Valle
 |year=1980
 |journal=Phys. Rev.
 |volume=22  |issue=9  |pages=2227–2235
 |doi=10.1103/PhysRevD.22.2227
 |title=Neutrino masses in SU(2) ⊗ U(1) theories
 |bibcode=1980PhRvD..22.2227S
}}
</ref>

<ref name=Yanagida-1980-HorizSym>
{{cite journal
 |first=T. |last=Yanagida  |author-link=Tsutomu Yanagida
 |year=1980
 |title=Horizontal symmetry and masses of neutrinos
 |journal=Progress of Theoretical Physics
 |volume=64  |issue=3  |pages=1103–1105
 |doi=10.1143/PTP.64.1103   |doi-access=free
 |bibcode=1980PThPh..64.1103Y
}}
</ref>

}}


{{DEFAULTSORT:Seesaw Mechanism}}
[[Category:Neutrinos]]
[[Category:Physics beyond the Standard Model]]