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Weyl, Dirac, and Majorana neutrinos
A Weyl twocomponent spinor is a left ()handed^{1} particle state, , which is necessarily associated by CPT
with a right ()handed antiparticle state^{2} . One refers to active (or ordinary) neutrinos
as lefthanded neutrinos
which transform as doublets with a charged lepton
partner. They therefore have normal weak interactions, as do their righthanded
antilepton partners,

(1) 
Sterile^{3} neutrinos are singlet neutrinos, which can be added to the
standard model and are predicted in most extensions. They have
no ordinary weak interactions except those induced by mixing with active
neutrinos. It is usually convenient to define the state as the
particle and the related antistate as the antiparticle.

(2) 
(Sterile neutrinos will sometimes also be denoted .)
Mass terms describe transitions between right ()
and left ()handed states.
A Dirac mass term, which conserves lepton number, involves transitions
between two distinct Weyl neutrinos
and :

(3) 
where the Dirac field is defined as
. Thus a
Dirac neutrino has four components
,
and the mass term allows a conserved lepton number
. This and other types of mass terms can easily be generalized
to three or more families, in which case the masses become matrices.
The charged current transitions then involve
a leptonic mixing matrix (analogous to the
CabibboKobayashiMaskawa (CKM) quark mixing matrix), which
can lead to neutrino oscillations between the light neutrinos.
For an ordinary Dirac neutrino
the is active
and the is
sterile.
The transition is
,
where is the weak isospin. The mass requires breaking and
is generated by a Yukawa coupling

(4) 
One has
,
where the vacuum expectation value (VEV) of
the Higgs doublet is
GeV,
and is the Yukawa coupling.
A Dirac mass is just like the quark and charged lepton masses, but
that leads to the question of why it is so small: one requires
to have eV.
A Majorana mass, which violates lepton number by two units
, makes use of the righthanded antineutrino,
, rather than a separate Weyl neutrino. It is a transition
from an antineutrino into a neutrino. Equivalently, it can be viewed
as the creation or annihilation of two neutrinos, and if present
it can therefore lead to neutrinoless double beta decay.
The form of a Majorana mass term is
where
is a selfconjugate twocomponent state
satisfying
, where is the
charge conjugation matrix. If is active then
and must be generated by either an elementary Higgs
triplet or by an effective operator involving two Higgs doublets
arranged to transform as a triplet.
One can also have a Majorana
mass term

(6) 
for a sterile neutrino. This has and thus
can be generated by the VEV of a
Higgs singlet^{4}.
Next: Models of neutrino mass
Up: THEORETICAL FRAMEWORK
Previous: THEORETICAL FRAMEWORK
Paul Langacker
20010927