2.2. The Dirac and Majorana Mass Matrices
In absence of
, the only possibility to give mass to the active neutrinos is by a Dirac mass matrix. Since that involves products of left and right handed fields, this presupposes the existence of
sterile neutrinos, that must be right handed and represented by quantized fermionic fields
,
where the Dirac mass matrix
is a complex
matrix. The sterile fields do not enter the weak interactions; they are singlets under the U(1)
SU(2)
SU(3)
gauge groups of the SM and affect neither gauge invariance, anomalies nor renormalization. Hence they preserve its full functioning while accounting for neutrino masses. Moreover, the sterile fields may have a mutual mass term like Equation (
1),
where the right handed Majorana mass matrix
is symmetric and complex valued, and where
is the charge conjugate of
,
While
is a right handed field,
is left handed (see
Appendix A for properties of
and
matrices).
The kinetic term has a common form for all species [
12],
where the slash denotes contraction with
matrices, and the partial derivatives acting as
2.3. The General Mass Matrix for Three Sterile Neutrinos
Though the number of right handed neutrinos is not fixed in principle, the case
has, if not a practical value [
8,
9,
10,
11], at least an esthetic one: for each left handed neutrino there is a right handed one, in the way it occurs for charged leptons and quarks. The three families of active left and sterile right handed neutrinos have the flavor three vectors (in our case
one may be tempted to denote
as
)
With the combined left handed flavor vector
the above mass Lagrangians combine into
In general, the mass matrix consists of four
blocks,
As stated, we take
. In the (“standard”, “pure” or “trivial”) Dirac limit also
. For pseudo-Dirac neutrinos
will be small with respect to
, or, more precisely, small with respect to the variation in the eigenvalues of
. Though we consider general
, we are inspired by the case of galaxy cluster lensing where it has nearly equal eigenvalues with central value 1.5–1.9 eV [
8,
9,
10,
11]; in
Section 3.1 we shall show that the entries of
then typically lie well below 1 meV.
2.4. Intermezzo: One Neutrino Family
In case of one family, the flavor vector is
. The entries of Equation (
10) are scalars, so
The physical masses are their absolute values [
12]. For
nonnegative, this leads to
The corresponding eigenvectors are
For small
these are 45
rotations, i.e., maximal mixing of the active and sterile basis vectors. Formally we may undo the rotations over 45
, by considering
where the approximations are to first order in
, the pseudo Dirac regime. The first vector,
, has its main weight on the first component, so it is mainly active, which we indicate by the tilde on
a. The second one,
, is mainly sterile. But unless
, the masses
are different, so that
and
are not eigenvectors and have no physical meaning. In fact, the mass squares have the difference
An initially active state,
with momentum
p will at time
t have oscillated into
where
. The occurrence probability is
where for
In practice there will not be a pure initial state but some wave packet [
12]. For
the cosine in Equation (
19) will average out, so that the fraction of observable neutrinos is approximately
. In plain terms: for
t large enough, half of the neutrinos are sterile and thus unobservable. For the solar neutrino problem the one-family approximation happens to work quite well [
15] and the detection rates are well established. Hence for the pseudo Dirac model it would mean that twice as many neutrinos should be emitted as in the standard solar model. The corresponding doubling of heat generated by nuclear reactions is ruled out by the measurements of the solar luminosity, so the case is rarely discussed.
Only in the pure Dirac case, i.e., with Majorana mass , the oscillations will not take place, since and . When starting from an initial active state , it now equals , and this can be taken as eigenstate. The sterile state will merely be a spectator, “just sitting there and wasting its time”. This can be generalized to three families. If one would follow the Franciscan William of Ockham (Occam’s razor), it would be preferable for active neutrinos to be Majorana rather than Dirac with unobservable right handed partners.
The SM differs from the neutrino sector in the SM by accounting for finite masses of its three Majorana neutrinos. Below we discuss a “Diracian” setup in which the sterile fields become physical, namely partly active, and the active fields partly sterile, even though the mass eigenstates have Dirac signature in vacuum.
2.5. Diagonalization of the Dirac Mass Matrix
We return to the three family case and its total mass matrix Equation (
10) with
. We notice that any
unitary matrix
U can be decomposed as a product of five standard ones,
The diagonal matrix
is called the Majorana phase matrix. Likewise we denote the diagonal phase matrix
by
) (for
U in Equation (
21) only five of the
and
are needed; this can be seen by factoring out
from
and
from
and setting
. Both sides of Equation (
21) thus involve nine free parameters). The matrix
is the product of
where
,
where the angles
are termed in standard notation
,
and
. The Dirac phase
is also called weak CP violation phase.
The complex valued Dirac mass matrix
can be diagonalized by two unitary matrices of the form (
21), viz.
and
. The result reads
with the real positive
(we denote Dirac mass eigenvalues by
to distinguish them from the physical masses
, the eigenvalues in absolute value of the total mass matrix. Notice also that while the left hand side of Equation (
23) has nine complex or 18 real parameters, the right hand side has
; but since
is diagonal, the diagonal matrices
and
only act as a product. Hence it is allowed to fix
before solving
, see below Equation (
28). The number of parameters available for the diagonalization is then still 18). We identify
with the PMNS mixing matrix
and
with the Majorana matrix
employed in literature.
To connect the transformation (
23) to
, we introduce the
unitary matrix
and define, using that
since it is diagonal,
New active and sterile fields
,
, merged as
With these steps the right handed Majorana mass matrix transforms into
Like
, it is complex symmetric, but since
was needed to diagonalize
, it will in general not result in a diagonal
. With the decomposition
as in Equation (
21), one can, however, use the phases in
to make the off-diagonal elements of
real and nonnegative (While the left hand side of Equation (
23) has nine complex or 18 real parameters, the right hand side has
; but since
is diagonal, the diagonal matrices
and
only act as a product. Hence it is allowed to fix
before solving
, see below Equation (
28). The number of parameters available for the diagonalization is then still 18. Moreover, for
n lepton families there are
independent complex valued off-diagonal elements and
n Majorana phases, so making all off-diagonal elements real and nonnegative is possible for
or 2).
We denote the diagonal elements of the Majorana matrix
by
, that may still be complex, and the real positive off-diagonal elements by
. The right handed Majorana mass matrix
then takes the form
so that the total mass matrix
reads
Except in the pure Dirac limit where , the are not rotations of mass eigenstates.
2.6. Diracian Limit
For reasons explained above, we wish to achieve pairwise degeneracies in the masses. The standard Dirac limit, just taking and , is a trivial way to achieve this; we shall, however, need finite values for them and design the more subtle “Diracian” limit.
To start, we notice that the eigenvalues of the mass matrix (
29) follow from
, where
The criterion to get pairwise degeneracies in the eigenvalues (up to signs), is simply that the odd powers in
vanish. Let us denote
and express the
in a common dimensionless parameter
through
The relations
make the coefficients of
and
of Equation (
31) vanish, respectively. To condense further notation, we express the
into dimensionless non-negative parameters
,
For normal ordering of the
(notice that these are Dirac masses, not the physical masses),
implies
,
,
, hence
; this is also the case for the inverted ordering
whence
,
,
. It thus holds that
Equating the
coefficient of Equation (
31) to zero requires
This cubic equation has the solutions for
, and positive or negative
,
We restrict ourselves to real solutions; there is always one. Then the matrix
is real-valued. All solutions are real when
, which occurs in particular when the
are small, i.e., in the pseudo-Dirac case. Then there exist the large solutions
with
, which in both cases leads to the eigenvalues
for
. For small
the
solution has a small
and
, viz.
The Diracian limit, defined by Equations (
33), (
34) and (
37), reduces Equation (
31) to a cubic polynomial in
,
Its analytical roots are intricate, but they are easily calculated numerically. Denoting them as
, the squares of the physical masses, the eigenvalues of
are
and
for
. From det
it holds that
. The eigenvectors are set by
and they are real and orthonormal. They can be expressed as
with orthonormal
and
for
. For small
and
the
and
read to first order
Notice that the
i and
components of
and
stem with the one-family case Equation (
15).
With the first three components of these vectors relating to active neutrinos and the last three to sterile ones, it is seen that for small
and
the
are mainly active and the
mainly sterile, which we indicate by the tildes. The
and
are
rotations of the
and
, which is maximal mixing. In the standard Dirac limit, it is customary to work with Dirac states and not with Majorana states. Likewise, in our Diracian limit the mass degeneracies allow the rotations to be circumvented by working with the
and
themselves. Indeed, there holds the exact decomposition
These steps allow us to retrieve the standard Dirac expressions in the limit where the Majorana masses , vanish, whence the and become purely active and purely sterile states, respectively.
For neutrino oscillation probabilities in vacuum (see
Section 2.4 and
Section 3.2) one needs the eigenvalues
and eigenvectors
and
of
,
In terms of
defined above Equation (
27) and related there to the flavor states (
8), the fields for the mass eigenstates are
Here label active fields and sterile ones. Hence the fields annihilate chiral left handed, mainly active neutrinos and create similar right handed antineutrinos, while the annihilate chiral right handed, mainly sterile neutrinos and create similar left handed antineutrinos.
The mass term of the 6 Majorana fields now takes the form of 3 Dirac terms,
because fermion fields anticommute and left and right handed fields are orthogonal. The here introduced Dirac fields,
combine left and right handed chiral fields, as usual. They are the mass eigenstates. In this basis the Dirac–Majorana neutrino Lagrangian is a sum of Dirac terms,
2.7. Charged and Neutral Current
Neutrinos also enter the currents coupled to the
W and
Z gauge bosons, which are part of the covariant derivatives in the Lagrangian, see Equation (
A11) below. The
W boson couples to the charged weak current. On the flavor basis it reads
with
g the weak coupling constant. The neutral weak current reads on the flavor basis
with
the weak or Weinberg angle.
To express these in the mass eigenstates, we define
and
as matrices consisting of the active components of the 6-component eigenvectors
and
, respectively,
and, likewise,
and
for the sterile components
From (
42) we read off that for small
and
From the orthonormality of the eigenvectors it follows that the real valued
matrices
and
satisfy the unitarity relation
while .
From Equations (
7), (
8), (
21), (
22), (
24) and (
27), and denoting
, we have
. As shown below Equation (
50), the diagonal phase matrix
can be absorbed in the fields. Inverting Equation (
45) leaves Equation (
48) invariant and expresses the flavor eigenstates as superpositions of mass eigenstates
. In vector notation, and using
, one has
Here
, with
is the standard PMNS matrix, see (
21), while
is the Majorana matrix of the three-neutrino problem; its
are Dirac phases now (a word on nomenclature: the Majorana phases in the matrix
stem from the 3 + 0 S
M, without sterile neutrinos. While they become physical Dirac phases in the 3 + 3 Dirac–Majorana neutrino standard model (DM
SM), there appear no true 3 + 3 Majorana phases, so we propose to keep this name for them. Hence the DM
SM has three physical phases: one Dirac and two “Majorana” phases. They all appear in the CP-invariance breaking part of the neutrino oscillation probabilities, see Equation (81). We also introduced
The sterile field
) similar to (
56) reads
The only current knowledge of the involved matrix elements lies in (54).
The flavor eigenstates can also be written as single sums over mass eigenstates,
with the
PMNS matrix
U having elements
the latter deriving from (
55), while
, because
U represents the three active rows of a unitary
matrix which also involves
and
. Hence the GIM theorem that
has the same form on flavor and mass basis, does not hold [
12].
Inserted in the currents the relations (
56) yield
2.8. Lepton Number for Sterile Neutrinos
There is an ambiguity in defining the lepton number of the sterile neutrinos. The lepton number of neutrinos is investigated by making the transformation
This leaves the kinetic terms invariant and for the standard choice
also the Dirac mass terms (
2). Only the Majorana mass terms (
1) and (
3) will vary by factors
: they violate lepton number conservation by
. This approach connects the lepton number
of active neutrinos also to sterile neutrinos, hence
for sterile antineutrinos (charge conjugated sterile ones). This assigns lepton number
to the components
of
of Equation (
8) and
of Equation (
26), but
to the components
. Then the mixing (
45), or its reverse (
56), (
58), enforced by the nonvanishing right handed Majorana mass matrix, makes it impossible to consistently connect a lepton number to the particles connected to the mass eigenstates
and
.
The opposite choice
,
circumvents this problem for general models with active and sterile neutrinos. According to (
8), (
45) and (
56) the lepton number
of
particles is consistent with
of a
particle and
of
particles. This choice is henceforward consistent with (
58). The benefit of this convention is that in pion and neutron decay both channels
,
, and
,
, respectively, satisfy lepton number conservation.
The minor price to pay is that now both the Dirac mass term and the Majorana terms violate lepton number conservation by two units, so that the unsolvable problem of lepton number violation remains unsolved. Indeed, as we shall discuss below, the Majorana mass terms still allow for neutrinoless double decay, where a nucleus decays by emitting two electrons (or two positrons) but no (anti)neutrinos.