Seesaw mechanism

 ( Neutrinos ) 

In the 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, compared to those of and charged , 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, and the existence of a very large mass scale. This allows the mass scale to be identifiable with the postulated scale of grand unification.

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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 of the form

                            M  &  B   \end{pmatrix} .
     

It has two :

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

and

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

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

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, $B$ is taken to be much larger than $M\; .$ Then the larger eigenvalue, $\backslash lambda\_\{(+)\},$ is approximately equal to $B\; ,$ while the smaller eigenvalue is approximately equal to

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

This mechanism serves to explain why the neutrino masses are so small.Tsutomu Yanagida (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. The matrix is essentially the mass matrix for the neutrinos. The Majorana spinor mass component $B$ is comparable to the GUT scale and violates lepton number conservation; while the Dirac spinor mass components $M$ are of order of the much smaller electroweak scale, called the VEV or vacuum expectation value below. The smaller eigenvalue $\backslash lambda\_\{(-)\}$ then leads to a very small neutrino mass, comparable to , which is in qualitative accord with experiments—sometimes regarded as supportive evidence for the framework of Grand Unified Theories.

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Background The 2×2 matrix arises in a natural manner within the standard model by considering the most general mass matrix allowed by Gauge theory of the standard model action, and the corresponding charges of the lepton- and neutrino fields.

Call the neutrino part of a Weyl spinor $\backslash chi\; ,$ a part of a left-handed lepton weak isospin Doublet state; the other part is the left-handed charged lepton $\backslash ell,$

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

as it is present in the minimal standard model with neutrino masses omitted, and let $\backslash eta$ be a postulated right-handed neutrino Weyl spinor which is a Singlet state 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

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

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

$$

\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, is a free parameter which can in principle take any arbitrary value.

The parameter 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 has weak isospin like the Higgs field , and $\backslash eta$ has weak isospin 0, the mass parameter can be generated from Yukawa interactions with the Higgs field, in the conventional standard model fashion,

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

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

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

$M\_t\; =\; \backslash mathcal\{O\}\; \backslash left(\; v\; /\; \backslash sqrt\{2\}\; \backslash right)\; \backslash ;\; \backslash approx\; \backslash ;\; \backslash mathrm\{\; 174\; \backslash ;\; GeV\; \}\; ,$

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

The parameter $B\text{'}$ on the other hand, is forbidden, since no renormalizable singlet under weak hypercharge and weak 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 can be motivated in the context of grand unification. In such models, enlarged Gauge theory may be present, which initially force $B\; =\; 0$ in the unbroken phase, but generate a large, non-vanishing value $B\; \backslash approx\; M\_\backslash mathsf\{GUT\}\; \backslash approx\; \backslash mathrm\{10^\{15\}~GeV\},$ around the scale of their spontaneous symmetry breaking. So given a mass $M\; \backslash approx\; \backslash mathrm\{\; 100\; \backslash ;\; GeV\; \}$ one has $\backslash lambda\_\{(-)\}\; \backslash ;\; \backslash approx\; \backslash ;\; \backslash mathrm\{\; 0.01\; \backslash ;\; eV\; \}.$ A huge scale has thus induced a dramatically small neutrino mass for the eigenvector $\backslash nu\; \backslash approx\; \backslash chi\; -\; \backslash frac\{\backslash ;\; M\; \backslash ;\}\{B\}\; \backslash eta\; .$

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See also

• Majoron
• Spinor

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Footnotes

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