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In the theory of grand unification of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, and, in particular, in theories of
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
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 eV, compared to those of quarks and charged leptons, 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.


Type 1 seesaw

This model produces a light neutrino, for each of the three known neutrino flavors, and a corresponding very heavy
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
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 : A = \begin 0 & M \\ M & B \end . It has two
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s: :\lambda_ = \frac , and :\lambda_ = \frac . The geometric mean of \lambda_ and \lambda_ equals \left, M \, since the determinant \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 A seesaw (also known as a teeter-totter or teeterboard) is a long, narrow board supported by a single pivot point, most commonly located at the midpoint between both ends; as one end goes up, the other goes down. These are most commonly found a ...
" of the mechanism. In applying this model to neutrinos, B is taken to be much larger than M . Then the larger eigenvalue, \lambda_, is approximately equal to B , while the smaller eigenvalue is approximately equal to : \lambda_- \approx -\frac . This mechanism serves to explain why the
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
masses are so small. The matrix is essentially the mass matrix for the neutrinos. The Majorana mass component B is comparable to the
GUT scale The grand unification energy \Lambda_, or the GUT scale, is the energy level above which, it is believed, the electromagnetic force, weak force, and strong force become equal in strength and unify to one force governed by a simple Lie group. The exa ...
and violates lepton number; while the
Dirac Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
mass components M are of order of the much smaller electroweak scale, called the VEV or ''vacuum expectation value'' below. The smaller eigenvalue \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.


Background

The 2×2 matrix arises in a natural manner within the standard model by considering the most general mass matrix allowed by
gauge invariance In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
of the standard model
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
, and the corresponding charges of the lepton- and neutrino fields. Call the
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
part of a Weyl spinor \chi , a part of a
left-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subject ...
lepton
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...
doublet; the other part is the left-handed charged lepton \ell, : L = \begin \chi \\ \ell \end , as it is present in the minimal standard model with neutrino masses omitted, and let \eta be a postulated right-handed neutrino Weyl spinor which is a singlet under
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...
– i.e. a neutrino that fails to interact weakly, such as a
sterile neutrino Sterile neutrinos (or inert neutrinos) are hypothetical particles (neutral leptons – neutrinos) that are believed to interact only via gravity and not via any of the other fundamental interactions of the Standard Model. The term ''sterile neutri ...
. There are now three ways to form Lorentz covariant mass terms, giving either : \tfrac \, B' \, \chi^\alpha \chi_\alpha \, , \quad \frac \, B\, \eta^\alpha \eta_\alpha \, , \quad \mathrm \quad M \, \eta^\alpha \chi_\alpha \, , and their
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s, which can be written as a quadratic form, : \frac \, \begin \chi & \eta \end \begin B' & M \\ M & B \end \begin \chi \\ \eta \end . 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 spontaneous broken by a Higgs mechanism, like the Dirac masses of the charged leptons. In particular, since has
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...
like the Higgs field , and \eta has
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...
0, the mass parameter can be generated from
Yukawa interaction In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is a scalar field (or pseudoscalar field) and a Dirac field of the ...
s with the Higgs field, in the conventional standard model fashion, : \mathcal_=y \, \eta L \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)\quad v \; \approx \; \mathrm, \qquad \qquad , \langle H \rangle, \; = \; v / \sqrt : M_t = \mathcal \left( v / \sqrt \right) \; \approx \; \mathrm , if the dimensionless Yukawa coupling is of order y \approx 1 . It can be chosen smaller consistently, but extreme values y \gg 1 can make the model nonperturbative. The parameter B' on the other hand, is forbidden, since no
renormalizable Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
singlet under
weak hypercharge In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y_\mathsf and corresponds to the gauge ...
and
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
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 symmetries may be present, which initially force B = 0 in the unbroken phase, but generate a large, non-vanishing value B \approx M_\mathsf \approx \mathrm, around the scale of their
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
. So given a mass M \approx \mathrm one has \lambda_ \; \approx \; \mathrm. A huge scale has thus induced a dramatically small neutrino mass for the eigenvector \nu \approx \chi - \frac \eta .


See also

* Majoron *
Spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...


Footnotes


References


External links

* {{DEFAULTSORT:Seesaw Mechanism Neutrinos Physics beyond the Standard Model