There are a confusing variety of models of neutrino mass. Here, I give a brief survey of the principle classes. For more detail, see [4] and [5].
Mass terms describe transitions between right (R)
and left (L)-handed
states.
A Dirac mass term, which conserves lepton number, involves transitions
between two different Weyl neutrinos,
and 
.
That is, the right-handed state
is different from 
, the CPT partner of the
. The form is
where the Dirac field is defined as 
. Thus a
Dirac neutrino has four components 
(the CPT partner of 
),
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
Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix), which
can lead to neutrino oscillations between the light neutrinos.
For an ordinary Dirac neutrino
the 
is active ( i.e., is in an 
doublet)
and the 
is
sterile
( i.e., is an 
singlet, with no weak
interactions except those due to mixing).
The transition is 
,
where I is the weak isospin. The mass requires 
breaking and
is generated by a Yukawa coupling
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 would require
in order to have 
eV.
A Majorana mass, which violates lepton number by two units 
, makes use of the right-handed 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 self-conjugate two-component state
satisfying 
, where C is the
charge conjugation matrix. If 
is active then
and m must be generated by either an elementary Higgs
triplet or by an effective operator involving two Higgs doublets
arranged to transform as a triplet.
For an elementary triplet 
, where 
is a Yukawa
coupling and 
is the triplet VEV.
The simplest implementation
is the Gelmini-Roncadelli (GR) model [6],
in which lepton number is spontaneously broken by 
. The
original GR model is now excluded by the LEP data on the Z
width.
Variant models involving explicit lepton number violation or in which
the Majoron (the Goldstone boson associated with lepton number
violation) is mainly a weak singlet ( invisible Majoron
models) are still possible.
For an effective operator one
expects 
, where C is a dimensionless constant and
M is the scale of the new physics which generates the operator.
The most familiar example is the seesaw model, to be discussed below.
It is also possible to consider mixed models in which both Majorana and Dirac mass terms are present. For two Weyl neutrinos one has a mass term
where 
and 
are the two Weyl states. 
and 
are Majorana masses which
transform as weak triplets and singlets, respectively (assuming that
the states are respectively active and sterile), while 
is a
Dirac mass term. Diagonalizing this 
matrix one finds that
the physical particle content is given by two Majorana mass
eigenstates
.
An especialy interesting case is the seesaw limit
[7], 
, in which there are
two Majorana neutrinos
with masses
Thus, there is one heavy neutrino and one neutrino much lighter than the typical Dirac scale. Such models are a popular and natural way of generating neutrino masses much smaller than the other fermion masses.
There are literally hundreds of versions of the
seesaw and related models [5]. The heavy scale 
can
range anywhere from the TeV scale to the Planck scale.
The TeV scale models are motivated, for example, by left-right
symmetric models [8]. Typically, the Dirac masses 
are of the order of magnitude of the corresponding charged
lepton masses, so that one expects masses of order
eV, 10 keV, and 1 MeV for the 
,
and 
, respectively. (The latter two violate
cosmological bounds unless they decay rapidly and invisibly.)
Intermediate scales, such as 
GeV, are motivated
by grand unification and typically yield masses in the range
relevant to hot dark matter, and solar and atmospheric
neutrino oscillations. The grand unified theories often
imply Dirac masses 
, where 
is the mass of
the up-type quark of the corresponding family. Depending on whether
there is also a family hierarchy of heavy masses 
, the
light masses
of the
family may vary approximately
quadratically with 
(the quadratic seesaw) or
linearly (the linear seesaw) [9].
in (7) is a
radiative correction.
Typical light neutrino masses in the quadratic seesaw are
(
eV, 
eV, 10 eV) for 
GeV (the intermediate seesaw, expected in
some superstring models or in grand unified theories with
multiple breaking stages). Such masses would correspond to
in the Sun, and 
a dark matter candidate (or, for a somewhat smaller 
mass, 
atmospheric neutrino
oscillations). Similarly,
for 
GeV (the grand unified seesaw, expected in old-fashioned
grand unified theories with large Higgs representations) one typically
finds smaller masses around
(
eV, 
, 
eV), suggesting
in the Sun.
In such models one often (but not always)
finds that the lepton and quark mixing matrices are similar.
A very different class of models are those in which the neutrino
masses are zero at the tree level (typically because no Weyl singlets
or elementary Higgs triplets are introduced), but only generated by
loops [10], i.e.,
radiative generation. Such models
are very attractive in principle and explain the smallness of 
.
However, the actual implementation generally requires
the ad hoc introduction of new Higgs particles with nonstandard
electroweak quantum numbers and lepton number-violating couplings.
