# Maximal mixing as a ‘sum’ of small mixings

Joydeep Chakrabortty<EMAIL_ADDRESS>Harish-Chandra Research Institute,
Allahabad 211 019, India  Anjan S. Joshipura<EMAIL_ADDRESS>Theory Group,
Physical Research Laboratory, Ahmedabad 380 009, India  Poonam Mehta
<EMAIL_ADDRESS>Theoretical Physics Group, Raman Research Institute,
Bangalore 560 080, India  Sudhir K. Vempati<EMAIL_ADDRESS>Centre
for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India

###### Abstract

In models with two sources of neutrino masses, we look at the possibility of
generating maximal/large mixing angles in the total mass matrix, where both
the sources have only small mixing angles. We show that in the two generation
case, maximal mixing can naturally arise only when the total neutrino mass
matrix has a quasi-degenerate pattern. The best way to demonstrate this is by
decomposing the quasi-degenerate spectrum in to hierarchial and inverse-
hierarchial mass matrices, both with small mixing. Such a decomposition of the
quasi-degenerate spectra is in fact very general and can be done irrespective
of the mixing present in the mass matrices. With three generations, and two
sources, we show that only one or all the three small mixing angles in the
total neutrino mass matrix can be converted to maximal/large mixing angles.
The decomposition of the degenerate pattern in this case is best realised in
to sub-matrices whose dominant eigenvalues have an alternating pattern. On the
other hand, it is possible to generate two large and one small mixing angle if
either one or both of the sub-matrices contain maximal mixing. We present
example textures of this. With three sources of neutrino masses, the results
remain almost the same as long as all the sub-matrices contribute equally. The
Left-Right Symmetric model where Type I and Type II seesaw mechanisms are
related provides a framework where small mixings can be converted to large
mixing angles, for degenerate neutrinos.

###### pacs:

14.60.Pq, 14.60.St, 11.30.Hv

††preprint: RECAPP-HRI-2009-016

##  I Introduction

While neutrino masses have been thoroughly established experimentally
conchamaltoni , the question of how they attain their masses still needs to be
understood. Perhaps, the most elegant mechanism of generating neutrino masses
is through the seesaw mechanism seesaw . Here one trades the tininess of the
neutrino masses with high scale Majorana masses for right-handed neutrinos
introduced for this purpose. While the original seesaw mechanism dealt only
with right-handed heavy neutrino states, in recent years, it has been realized
that there could be other heavy triplet scalars seesaw2  or even triplet
fermions seesaw3 ; goran  which could play the same role as right-handed
neutrinos in the original seesaw mechanism. These mechnaisms are named as Type
I, Type II and Type III seesaw mechanisms respectively (for recent reviews
please see strumiareview ; nirdavidsonreview ). While one of the three seesaw
mechanisms suffices to generate non-zero neutrino masses, it is interesting to
note that in most Grand Unified Theory (GUT) models, there is more than one
seesaw mechanism at work. For example, in SO(10) models both Type I and Type
II seesaw mechanisms are simultaneously present as soon as one considers
representations of the type $\overline{126}$ babumohapatra . In Left-Right
Symmetric (LRS) models, the Type I and Type II seesaw mechanisms are not just
present, but they are also related to each other akhmedov . Similarly, Type I
and Type III mechanisms co-exist in SU(5) model with an adjoint fermion
representation goran . In most of these investigations, typically one
considers one of them to be dominant while the other to be subdominant.

One of the crucial features of seesaw mechanism was its ability to generate
large or maximal mixing even though the mixing present in the Dirac neutrino
Yukawa couplings is small like in the hadronic sector. In fact this is what
typically happens in a SO(10) GUT smirnovaf , where neutrino Dirac Yukawa
couplings have the same structure as the top Yukawa couplings; even in such
cases large mixing in the neutrino sector is possible. However this would
require large hierarchies in the masses of the right-handed neutrinos which is
in conflict with thermal leptogenesis in these models thermallepto  111This is
true when the mixing angles in neutrino Dirac Yukawa are exactly like CKM
angles.. In the present work, we look for an alternative method to generate
large/maximal mixings instead of using the ‘seesaw-effect’. We will use the
fact that most models GUT models like SO(10) have more than one seesaw
mechanism at work. However, instead of restricting ourselves to any particular
GUT model or the seesaw mechanism, we analyze the general situation where
there are two sources for neutrino masses and both of these contain small
neutrino mixing.

Our analysis shows that the total neutrino mass matrix which is given by the
sum of the two neutrino sources can have large or maximal mixing only if the
resulting pattern of the neutrino masses is of the quasi-degenerate form. A
crucial condition which needs to be satisfied to reach this conclusion is that
the large eigenvalues of the sub-matrices do not cancel in the total mass
matrix. This results in the sub-matrices taking the form of hierarchial and
inverse-hierarchial matrices whose sum leads to the quasi-degenerate form.
Given that the decomposition of the degenerate spectrum in to hierarchial and
inverse-hierarchial mass matrices is quite generic, as we will demonstrate
here, one can enumerate the possible forms the individual sub-matrices can
take. It should be noted that the decomposition itself is independent of the
actual mechanism responsible for generating neutrino masses i.e, doesn’t
depend on whether there is a seesaw mechanism at work or not. It is well known
that the quasi-degenerate pattern for neutrino masses can be achieved both
with joshipura  and without marajasekaran  seesaw mechanism. However, as will
demonstrate later, the model dependence enters, if one wants to realise the
decomposition in terms of independent Lagrangian parameters which for example
is possible in Type I seesaw mechanism.

The simple example where our scheme of things can be realised is the LRS model
where both Type I and Type II seesaw mechanisms are simultaneously present. We
will explicitly present the conditions on the LRS parameters required in order
to realize the mechanism. The paper is organised as follows. In Sec. II, we
analyse the two generation case and show how only when the quasi-degeneracy is
satisfied in the final matrix, one can have large or maximal mixing. We also
describe all the possible decompositions of the degenerate spectra. We further
discuss how this scheme can be incorporated within the LRS models. In Sec.
III, we consider two cases (a) with two seesaw mechanisms or two sources, and
(b) with three sources. We then demonstrate the decomposition of the quasi-
degenerate spectrum and discuss the subtleties which arise in this case. We
also determine the required parameter values within the LRS model for both the
cases. We close with summary and outlook in Sec. IV. Generalisation of our
result to the case of $n$ sources of neutrino masses is given in Appendix A.

##  II Large mixing as sum of small mixing angles

Consider a model for neutrino masses in which the total neutrino mass matrix
is given by

${\mathbb{M}}_{\nu}={\mathbb{M}}_{\nu}^{(1)}+{\mathbb{M}}_{\nu}^{(2)},$ (1)

where ${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$ can be thought
of as two individual sources of neutrino mass. For example,
${\mathbb{M}}_{\nu}^{(1)}$ could have its origin in Type I seesaw whereas
${\mathbb{M}}_{\nu}^{(2)}$ could have its origin in Type II seesaw mechanism
in a model like SO(10) where both these mechanisms are simultaneously present
222In fact, in most models of neutrino masses, one of them, say
${\mathbb{M}}_{\nu}^{(1)}$ could correspond to zeroth order mass while the
other ${\mathbb{M}}_{\nu}^{(2)}$ could correspond to perturbations required to
make contact with the experimental results.. Irrespective of their origin, let
us assume that both ${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$
contain only small mixing angles. We now ask the question whether it is
possible to have in the total mass matrix ${\mathbb{M}}_{\nu}$ (a) maximal or
large mixing, and (b) a reasonable $\Delta\mbox{m}^{2}$ without fine-tuning.
By this we mean, that the $\Delta m^{2}$ is determined in terms of the
dominant eigenvalues of ${\mathbb{M}}_{\nu}^{(i)}$ (where $i=1,2$). To make
the discussion concrete, we will stick to two generation case in the present
section. Denoting

${\mathbb{M}}_{\nu}^{(1)}=\left(\begin{array}[]{cc}m_{ee}^{(1)}&m_{e\mu}^{(1)}\\\
m_{e\mu}^{(1)}&m_{\mu\mu}^{(1)}\end{array}\right),\;\;\;\;\;\;{\mathbb{M}}_{\nu}^{(2)}=\left(\begin{array}[]{cc}m_{ee}^{(2)}&m_{e\mu}^{(2)}\\\
m_{e\mu}^{(2)}&m_{\mu\mu}^{(2)}\end{array}\right),$ (2)

we can easily derive the following relations :

$\displaystyle\tan 2\theta$ $\displaystyle=$
$\displaystyle{2m_{e\mu}^{(1)}+2m_{e\mu}^{(2)}\over
m_{\mu\mu}^{(2)}+m_{\mu\mu}^{(1)}-m_{ee}^{(2)}-m_{ee}^{(1)}}$ (3)
$\displaystyle=$ $\displaystyle\tan 2\theta^{(1)}{1\over(1+d)}+\tan
2\theta^{(2)}{d\over(1+d)}~{},$ (4)

where $d=(m_{\mu\mu}^{(2)}-m_{ee}^{(2)})/(m_{\mu\mu}^{(1)}-m_{ee}^{(1)})$ and
$\theta^{(1)}$ and $\theta^{(2)}$ are the mixing angles of
${\mathbb{M}}_{\nu}^{(1)}$ and ${\mathbb{M}}_{\nu}^{(2)}$ respectively. From
this expression, it is obvious that when both the mixing angles,
$\theta^{(1)}$ and $\theta^{(2)}$ are small, the only region where $\theta$
would be maximal is when $d=-1$. Notice that the small mixing in
${\mathbb{M}}_{\nu}^{(i)}$ would mean (a) 2
$m_{e\mu}^{(i)}\ll|m_{\mu\mu}^{(i)}-m^{(i)}_{ee}|$, and (b)
$m^{(i)}_{\mu\mu}\neq m^{(i)}_{ee}$ for $i=(1,2)$ i.e, the splitting in the
diagonal entries is much larger than the off-diagonal entry such that the
mixing remains small. Assuming at least one of the diagonal entries in each of
the matrix ${\mathbb{M}}_{\nu}^{(i)}$ is large, we have the following three
solutions for $d=-1$

  1. (A) $m^{(2)}_{\mu\mu}=-m^{(1)}_{\mu\mu}$ ,

  2. (B) $m^{(2)}_{ee}=-m^{(1)}_{ee}$ , and

  3. (C) $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or $m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$ .

The solution of the type (A) would represent the case in which both the
matrices ${\mathbb{M}}_{\nu}^{(i)}$ are of the hierarchial form with one
dominant diagonal element (the $\mu\mu$ entry). However in the total mass
matrix ${\mathbb{M}}_{\nu}$ this entry gets cancelled. To illustrate this,
consider the following textures for ${\mathbb{M}}_{\nu}^{(i)}$

${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{cc}z&x\\\
x&1+z^{\prime}\end{array}\right)+m_{2}\left(\begin{array}[]{cc}0&y\\\
y&-1\end{array}\right)~{},$ (5)

where $x,y,z$ are the small entries compared to
$m_{\mu\mu}^{(1)}/m_{1}\equiv(1+z^{\prime})$ and
$m_{\mu\mu}^{(2)}/m_{2}\equiv-1$. Notice that the dominant eigenvalues of
${\mathbb{M}}_{\nu}^{(i)}$ have opposite CP parities as the maximal mixing
requirement condition is now given as $m_{1}\approx m_{2}\approx m$. In this
limit, the total mass matrix has the form

${\mathbb{M}}_{\nu}=m\left(\begin{array}[]{cc}z&x+y\\\
x+y&z^{\prime}\end{array}\right)~{}.$ (6)

The total mass matrix here has no trace of the dominant element of the
$\mathcal{O}(m)$ which was present in the sub-matrices. It has been cancelled
in such a way that the condition
$m^{(2)}_{\mu\mu}+m^{(1)}_{\mu\mu}=m^{(1)}_{ee}+m^{(2)}_{ee}$ is satisfied,
which is the same as the full condition of $d=-1$ which would mean
$z^{\prime}=z$, rather than the sub-condition, (A)
$m^{(2)}_{\mu\mu}=-m^{(1)}_{\mu\mu}$, which would instead mean $z^{\prime}=0$.
The role of the large element of the $\mathcal{O}(m)$ has only been to
generate the small mixing in the respective ${\mathbb{M}}_{\nu}^{(i)}$. Thus,
at the level of total mass matrix ${\mathbb{M}}_{\nu}$, the properties are
determined by the small entries of the original sub-matrices. The mass-squared
splitting $\Delta\mbox{m}^{2}$ of the total mass matrix in terms of the
elements of ${\mathbb{M}}_{\nu}^{(i)}$ is given by

$\Delta\mbox{m}^{2}=(m^{(1)}_{ee}+m^{(1)}_{\mu\mu}+m^{(2)}_{ee}+m^{(2)}_{\mu\mu})\sqrt{(m^{(1)}_{\mu\mu}+m^{(2)}_{\mu\mu}-m^{(1)}_{ee}-m^{(2)}_{ee})^{2}+4(m^{(1)}_{e\mu}+m^{(2)}_{e\mu})^{2}}~{},$
(7)

which reduces in the present case to

$\Delta\mbox{m}^{2}=m^{2}(z+z^{\prime})\sqrt{4(x+y)^{2}+(z-z^{\prime})^{2}}~{}.$
(8)

In the limit $z^{\prime}\rightarrow 0$, the mixing $\tan 2\theta=2(x+y)/z$
would depend on the relative magnitudes of $x,y$ and $z$ with large mixing
being possible as long as $x+y\gg z$. Similarly, in the limit
$z^{\prime}\approx z$, maximal/large mixing is possible, and a hierarchial
pattern for the neutrinos can arise if $x,y\sim z,z^{\prime}$. Solutions of
the class (B) also lead to similar results with the dominant entries of the
sub-matrices ${\mathbb{M}}_{\nu}^{(i)}$ being cancelled in the total mass
matrix. We do not find these solutions attractive as large mixing can only
come when the dominant elements (‘$ee$’ elements in this case) cancel
precisely to such an extent to be equal to the sum of the other diagonal
elements (‘$\mu\mu$’ elements). We now go on to discuss the solutions (C)
which we find more natural.

The solutions of the type (C) are given by $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or
$m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$. The condition now requires that the opposite
diagonal elements of the sub-matrices are equal. This naturally sets the
${\mathbb{M}}_{\nu}^{(i)}$ to have an opposite ordering of their eigenvalues
i.e, one with normal hierarchy and the other has inverse hierarchy. For
illustration, let us consider the (sub)-case with
$m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$. This can be represented as

$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{M}}^{\rm{(1)}}_{\nu}+{\mathbb{M}}^{\rm{(2)}}_{\nu}$
$\displaystyle=$ $\displaystyle m_{1}\begin{pmatrix}0&x\\\
x&1\end{pmatrix}+m_{2}\begin{pmatrix}1&-y\\\ -y&0\end{pmatrix}$ (9)
$\displaystyle=$ $\displaystyle\begin{pmatrix}m_{2}&m_{1}x-m_{2}y\\\
m_{1}x-m_{2}y&m_{1}\end{pmatrix}~{},$

where $x,y$ are small entries with $m_{\mu\mu}^{(1)}\equiv m_{1}$ and
$m_{ee}^{(2)}\equiv m_{2}$. Note that here too as in the earlier case the
mixing angles in the individual sub-matrices are small, $\theta\simeq$ $x$ or
$y$, where as the total mixing matrix is given by

$\tan 2\theta={2(m_{1}x-m_{2}y)\over m_{1}-m_{2}}~{}.$ (10)

In the limit of exact degeneracy between $m_{1}$ and $m_{2}$, the mixing is
maximal as is evident. However, an important assumption is that both the
$m_{1}$ and $m_{2}$ carry the same sign or equivalently have the same CP
parity 333If the CP parities are opposite the mixing will remain small.. In a
more general situation, say when the zeros of the matrices on the RHS are of
Eq. (9) are filled with small entries (‘$ee$’ element in
${\mathbb{M}}_{\nu}^{(1)}$ and ‘$\mu\mu$’ element in
${\mathbb{M}}_{\nu}^{(2)}$), the condition for the large mixing is given by
$|m_{1}-m_{2}|<2(m_{1}x-m_{2}y)$. Thus, the splitting in the diagonal entries
should be much smaller than the off-diagonal elements. The spectrum of the
total mass matrix points towards a quasi-degenerate pattern. The eigenvalues
are given by :

$\lambda_{1,2}={\displaystyle\frac{1}{2}}\left[m_{1}+m_{2}\mp\sqrt{(m_{1}-m_{2})^{2}+4(m_{1}x-m_{2}y)^{2}}\right],$
(11)

which in the limit $m_{1}\approx m_{2}\approx m$ take the form
$m-\epsilon,m+\epsilon$, with $\epsilon=m(x-y)$ being the order of the off-
diagonal entry. The $\Delta\mbox{m}^{2}=4m\epsilon$. The other solution of
(C), $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$, corresponds to an interchange of $m_{1}$
and $m_{2}$ and would lead to similar conclusions.

Finally, let us consider a class of solutions with two large diagonal entries
in each of the ${\mathbb{M}}_{\nu}^{(i)}$. However given that the mixing in
each of them is small, as per the discussion above, the splitting between the
diagonal elements should be larger than the off-diagonal entry. This can be
parameterised by the following set of matrices :

$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{M}}^{\rm{(1)}}_{\nu}+{\mathbb{M}}^{\rm{(2)}}_{\nu}$
$\displaystyle=$ $\displaystyle m_{1}\begin{pmatrix}1+\rho&x\\\
x&1\end{pmatrix}+m_{2}\begin{pmatrix}1&x^{\prime}\\\
x^{\prime}&1+\rho^{\prime}\end{pmatrix}$ (12) $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{1}(1+\rho)+m_{2}&m_{1}x+m_{2}x^{\prime}\\\
m_{1}x^{\prime}+m_{2}y&m_{1}+m_{2}(1+\rho^{\prime})\end{pmatrix},$

where $x,x^{\prime}$ are small entries compared to one as before and
$\rho,\rho^{\prime}$ are chosen such that $2|x/\rho|\ll 1$ and
$2|x^{\prime}/\rho^{\prime}|\ll 1$ to keep the mixing small in
${\mathbb{M}}_{\rm{i}}^{\nu}$. This would mean a relative hierarchy of the
elements in the individual matrices, $m_{ee}^{(1)}\gg m_{\mu\mu}^{(1)}\gg
m_{e\mu}^{(1)}$ in ${\mathbb{M}}_{\rm{1}}^{\nu}$ and $m_{\mu\mu}^{(2)}\gg
m_{ee}^{(2)}\gg m_{e\mu}^{(2)}$ in ${\mathbb{M}}_{\rm{2}}^{\nu}$, which is
very similar to the case of solutions (C) with a large $m_{ee}$ ($m_{\mu\mu}$)
in ${\mathbb{M}}^{\rm{(1)}}_{\nu}$ (${\mathbb{M}}^{\rm{(2)}}_{\nu}$).
Qualitatively, these could form a different class of solutions compared to
type (C) with each of the sub-matrices here forming a quasi-degenerate pair
with small mixing. However, notice that the total mixing is now given by $\tan
2\theta\approx(x+x^{\prime})/(\rho^{\prime}-\rho)$ would remain small as
$x,x^{\prime}\ll\rho,\rho^{\prime}$ unless $\rho=\rho^{\prime}$. With this
additional condition, this class of solutions again falls in to the class (C)
i.e, $m^{(2)}_{ee}=m^{(1)}_{\mu\mu}$ or $m^{(2)}_{\mu\mu}=m^{(1)}_{ee}$.
However, to distinguish from the solutions in Eq. (9), we will call the class
of solutions represented by Eq. (12) as type (C1) 444Solutions with three
large entries in each sub-matrix violate the small mixing assumption..

In summary, the sum of two mass matrices with small mixing angles would
naturally lead to a degenerate spectrum with maximal/large mixing provided we
insist there are no cancellations of the large eigenvalues of the individual
sub-matrices. The individual sub-matrices could be (a) ordered as hierarchial
+ inverse-hierarchial with small mixing (solutions of type (C)) or (b) be
quasi-degenerate themselves but with small mixing (C1). However as we have
seen, solutions of the type (C1) require further precise cancellation in the
differences of their large diagonal elements. For this reason, we consider
solutions of the type (C) i.e, matrices as parameterised in Eq. (9) to be the
most natural. Thus to convert one small mixing angle in two matrices to one
maximal mixing in the total matrix, we would require a pair of
(quasi)-degenerate eigenvalues with the same CP parities, ordered oppositely
in the sub-matrices. This count would be useful when we extend this degeneracy
induced large mixing to three generations.

###  II.1 Decomposition of the Degenerate Spectrum

In the previous section we have seen that a quasi-degenerate pattern naturally
emerges if two mass matrices of small mixing are added and we demand large
mixing in the total mass matrix. One can instead reverse the argument and
might say that the quasi-degenerate spectrum with large mixing can be
decomposed in to two matrices with small mixing. In fact, the decomposition of
the quasi-degenerate spectrum in to two matrices is more generic and is
independent of the mixing present in them. This can be easily be demonstrated
by considering zeroth order neutrino mass matrices in the flavour basis. Let
us denote the neutrino mass matrix in the flavour basis by

$\displaystyle{\mathbb{M}}_{\nu}={\mathbb{U}}_{PMNS}{\mathbb{M}}_{diag}{\mathbb{U}}_{PMNS}^{\dagger}~{},$
(13)

where ${\mathbb{U}}_{PMNS}={\mathbb{U}}_{l}^{\dagger}{\mathbb{U}}_{\nu}$ is
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary leptonic mixing matrix.
${\mathbb{U}}_{PMNS}={\mathbb{U}}_{\nu}$ in a basis in which charged lepton
mass matrix is diagonal, i.e, ${\mathbb{U}}_{l}={\mathbb{I}}_{n\times n}$. In
Table 1, we have listed the zeroth order mass matrices for hierarchal,
inverse-hierarchal and degenerate spectra for the case of small mixing and
maximal mixing. In writing down these textures, we have followed Altarelli and
Feruglio afreview  method, where each of these (zeroth order) mass matrices
has to be multiplied by a mass scale $m$ representing the heaviest eigenvalue
of the mass matrix. From Table 1 we can see that, as we go along each column,
the degenerate mass matrices $\mathbb{C}_{i}$ can be expressed as a sum of
hierarchal, $\mathbb{A}$ and inverse hierarchal, $\mathbb{B}$ matrices. For
example, $\mathbb{C}_{0}=\mathbb{A}+\mathbb{B}$,
$\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$,
$\mathbb{C}_{2}=\mathbb{B}-\mathbb{A}$. Note that the mass scale $m$
multiplying $\mathbb{C}_{i}$ now multiplies both $\mathbb{A}$ and
$\mathbb{B}$. These equations hold irrespective of the mixing being small or
maximal. Thus every degenerate mass matrix can be expressed a sum (or
difference) of a hierarchial and inverse-hierarchial mass matrices, but with
common mass scale given by the degenerate mass $m$, which is an obvious
observation if one just sees the diagonal eigenvalues of each mass matrix in
the first column.

Mixing $\Rightarrow$ |  Small |  Maximal
---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{M}$
Hierarchial |  |
${\mathbb{A}}$: Diag[0,1] | $\begin{pmatrix}0&\epsilon\\\ \epsilon&1\end{pmatrix}$ | $\begin{pmatrix}1/2&1/2\\\ 1/2&1/2\end{pmatrix}$
Inverse hierarchial |  |
${\mathbb{B}}$: Diag[1,0] | $\begin{pmatrix}1&-\epsilon\\\ -\epsilon&0\\\ \end{pmatrix}$ | $\begin{pmatrix}1/2&-1/2\\\ -1/2&1/2\end{pmatrix}$
Degenerate |  |
${\mathbb{C}}_{0}$: Diag[1,1] | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
${\mathbb{C}}_{1}$: Diag[-1,1] | $\begin{pmatrix}-1&2\epsilon\\\ 2\epsilon&1\\\ \end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$
${\mathbb{C}}_{2}$: Diag[1,-1] | $\begin{pmatrix}1&-2\epsilon\\\ -2\epsilon&-1\\\ \end{pmatrix}$ | $\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}$
Table 1: Zeroth order textures for small and maximal mixing (setting $m_{1}$
and $m_{2}$ as dimensionless quantities which are either zero or one depending
on the different cases listed) for the two-generation case.

Let us now turn to the question of mixing for the degenerate cases mentioned
above. The mixing in $\mathbb{C}_{0}=\mathbb{A}+\mathbb{B}$ in undetermined as
it is proportional to the identity matrix. This is also the exact degeneracy
limit. This situation arises if the mixing angles of $\mathbb{A}$ and
$\mathbb{B}$ are not only small, but are also equal. On the other hand, the
mixing in $\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$ can be maximal again as we
explained above in the previous section. The mixing in $\mathbb{C}_{1}$ and
$\mathbb{C}_{2}$ will remain small as they have opposite CP parities. An
important exception to generate large mixing in terms of small mixing angles
through quasi-degeneracy is the pseudo-Dirac pattern. The pseudo-Dirac pair
can come as a sum (difference) of two sub-matrices both with maximal mixing,
one hierarchal and the other inverse-hierarchal. This is clearly evident from
the last column of Table 1. We see the pseudo-Dirac pairs
$\mathbb{C}_{1}=\mathbb{A}-\mathbb{B}$ and
$\mathbb{C}_{2}=\mathbb{B}-\mathbb{A}$ with both $\mathbb{A}$ and $\mathbb{B}$
containing maximal mixing.

The decomposition of quasi-degenerate spectra can easily be incorporated
within models of neutrino masses. For example, in the Type I seesaw mechanism
(with two generations) the mass matrix is given by

$\displaystyle-{\mathbb{M}}^{\rm{I}}_{\nu}$ $\displaystyle=$ $\displaystyle
v^{2}\begin{pmatrix}h_{ee}^{D}&h_{\mu e}^{D}\\\
h_{e\mu}^{D}&h_{\mu\mu}^{D}\end{pmatrix}\begin{pmatrix}1/M_{R1}&0\\\
0&1/M_{R2}\end{pmatrix}\begin{pmatrix}h_{ee}^{D}&h_{e\mu}^{D}\\\ h_{\mu
e}^{D}&h_{\mu\mu}^{D}\end{pmatrix}$ (18) $\displaystyle=$ $\displaystyle
m_{1}\left(\begin{array}[]{cc}(h_{ee}^{D})^{2}&h_{ee}^{D}h_{e\mu}^{D}\\\
h_{ee}^{D}h_{e\mu}^{D}&(h_{e\mu}^{D})^{2}\end{array}\right)+m_{2}\left(\begin{array}[]{cc}(h_{\mu
e}^{D})^{2}&h_{\mu e}^{D}h_{\mu\mu}^{D}\\\ h_{\mu
e}^{D}h_{\mu\mu}^{D}&(h_{\mu\mu}^{D})^{2}\end{array}\right),$

where $m_{1}$ and $m_{2}$ are given as $v^{2}/(M_{R_{1}})$ and
$v^{2}/(M_{R_{2}})$ respectively. Each of these sub-matrices is result of a
seesaw mechanism with one right-handed neutrino. Comparing the above with Eq.
(9), we can determine the parameter regions required for quasi-degeneracy and
large mixing. For, $M_{R_{1}}=M_{R_{2}}$, we see that for the Yukawa
parameters, there are two choices where the mixing in the sub-matrices is
small

$\displaystyle
h_{e\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{ee}^{D}~{}\sim~{}x~{},\;\;h^{D}_{\mu\mu}~{}\sim~{}-y~{},\;\;h_{\mu
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},~{}~{}~{}\mbox{or}$ $\displaystyle
h_{ee}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\mu}^{D}~{}\sim~{}-x~{},\;\;h^{D}_{\mu
e}~{}\sim~{}y~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{}.$ (19)

Thus each right-handed neutrino couples with the Standard Model neutrinos with
only small mixing angles, but total mass matrix ensures maximal mixing angles
for the above choice of parameters. These conditions are already known in the
literature for some time afreview . So this is an alternative approach of
arriving at these conditions. The interesting aspect of Type I seesaw
mechanism is that the decomposition at the neutrino mass matrix level can be
realised at the Lagrangian level in terms of independent parameters with
‘independent mass’ scales for the the individual sub-matrices, for instance
the sub-matrices have mass scales $v^{2}/M_{R_{1}}$ and $v^{2}/M_{R_{2}}$.
Such a realisation might not be possible in other models for degenerate
neutrinos like in Type II seesaw mechanism. A further interesting possibility
would be to consider the case when there are two independent seesaw mechanisms
at work.

###  II.2 Left-Right Symmetric Model

The simplest model where the above mechanism can be realised is the LRS model.
In the recent years, this model has been thoroughly analyzed for its duality
properties akhmedov . The LRS model naturally contains both Type I and Type II
seesaw contributions, which can be thought of as two sub-matrices discussed
above. Further more, these models are characterized by a common Yukawa
coupling to both the left-handed and right-handed Majorana mass matrices

$\mathcal{L}_{M}=-{f\over
2}\left(\overline{\nu_{L}^{c}}\nu_{L}\Delta^{0}_{L}+\overline{\nu_{R}^{c}}\nu_{R}\Delta^{0}_{R}\right)+h.c.~{},$
(20)

where $\Delta_{L(R)}$ is the triplet Higgs field whose neutral component
attains a vacuum expectation value (vev) giving rise to the Majorana mass to
the left (right) handed neutrino fields. In addition the Dirac neutrino Yukawa
coupling is also present

$\mathcal{L}_{D}=-Y\overline{\nu}_{L}\nu_{R}\phi^{0}+h.c.$ (21)

In the limit where $v_{R}\gg v$, the Type I seesaw mechanism becomes operative
and the total neutrino mass matrix is now given as

${\mathbb{M}}_{\nu}=fv_{L}-{v^{2}\over v_{R}}Yf^{-1}Y^{T}~{}.$ (22)

Along the lines of the discussion we had for the two-generation case, Eq. (9),
we can assume the contribution (first term on the RHS of Eq. (22)), due to
Type II to be hierarchial with small mixing and second part due to the Type I
contribution inverse hierarchial with small mixing. The appropriate choice of
the Yukawa textures in this case are as follows

$f=\left(\begin{array}[]{cc}0&x\\\ x&1\\\
\end{array}\right)\;\;,\;\;Y=\left(\begin{array}[]{cc}1&y\\\ y&0\\\
\end{array}\right)~{}.$ (23)

With this choice the total mass matrix takes the form

$\displaystyle{\mathbb{M}}_{\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&m_{1}x\\\ m_{1}x&m_{1}\\\
\end{array}\right)+{m_{2}\over x^{2}}\left(\begin{array}[]{cc}1-2xy&y(1-xy)\\\
y(1-xy)&y^{2}\\\ \end{array}\right)$ (28) $\displaystyle=$
$\displaystyle{1\over
x^{2}}\left(\begin{array}[]{cc}m_{2}(1-2xy)&m_{1}x^{3}+m_{2}y(1-xy)\\\
m_{1}x^{3}+m_{2}y(1-xy)&m_{1}x^{2}+m_{2}y^{2}\\\ \end{array}\right)~{},$ (31)

where $m_{1}=v_{L}$ and $m_{2}=v^{2}/v_{R}$. The mixing angle in the above
mass matrix is given by

$\tan
2\theta={2[m_{1}x^{3}+m_{2}y(1-xy)]\over[m_{1}x^{2}+m_{2}y^{2}]-m_{2}(1-2xy)}~{}.$
(32)

From the above it is clear that the degeneracy requirement $m_{1}x^{2}\approx
m_{2}$ automatically leads to large mixing, $\tan 2\theta\sim{\cal
O}(\frac{1}{2y})$. A rough idea of how stable this mixing would be under
radiative corrections can be obtained by considering the modification of the
neutrino mass matrix below the seesaw scale. The modification is set by the
matrix ${\mathbb{P}}=\mbox{Diag}\\{1,1+\delta_{\mu}\\}$ and is given as
${\mathbb{M}}_{\nu}={\mathbb{P}}{\mathbb{M}}_{\nu}{\mathbb{P}}$. In this case,
the mixing angle now takes the form

$\tan
2\theta={2[m_{1}x^{3}+m_{2}y(1-xy)](1+\delta_{\mu})\over(1+\delta_{\mu})^{2}[m_{1}x^{2}+m_{2}y^{2}]-m_{2}(1-2xy)}~{},$
(33)

where $\delta_{\mu}=c~{}h_{\mu}^{2}/(16\pi^{2})\log(M_{X}/M_{W})$ specifies
the size of the radiative corrections induced by the Yukawa coupling of the
$\mu$, $h_{\mu}$. Here $c$ is a constant depending on whether the theory is
supersymmetric or not and $M_{X}$ is the high scale just below the seesaw
scale antusch . The condition for large mixing case now gets modified as
$[m_{1}x^{2}+m_{2}y^{2}](1+\delta_{\mu})^{2}\approx m_{2}$. Note that this
same condition is also required to keep the degeneracy stable even after
radiative corrections. Of course, the splitting of the degeneracy can come
from the radiative effects. A more detailed analysis of radiative corrections
will be presented elsewhere.

##  III Extension to Three Generations

Let us extend the analysis of the previous section to the case of three
generations. Here we will consider two cases - (a) Case I: two seesaw
mechanisms or two sources of neutrino masses (Sec. III.1), and (b) Case II:
three seesaw mechanisms or three sources of neutrino masses (Sec. III.2).

###  III.1 Case I: Two Seesaw Mechanisms

As before, let us consider two $3\times 3$ mass matrices each with a small
mixing angle and one large eigenvalue,
${\mathbb{M}}_{\nu}={\mathbb{M}}_{\nu}^{(1)}+{\mathbb{M}}_{\nu}^{(2)}$.
Instead of representing them as general mass matrices as we have done for the
case of two generations, we will represent them by using

$\mathbb{M}_{\nu}^{(i)}=[{\mathbb{U}}_{\mbox{mix}}^{(i)}]^{T}\cdot\mbox{Diag}[\mathbb{M}_{\nu}^{(i)}]\cdot{\mathbb{U}}_{\mbox{mix}}^{(i)}~{},$
(34)

where
$\mbox{Diag}[\mathbb{M}_{\nu}^{(i)}]=\mbox{Diag}[\\{m_{1}^{(1)},m_{2}^{(2)},m_{3}^{(3)}\\}]$,
the eigenvalues of the mass matrices and ${\mathbb{U}}_{\mbox{mix}}^{(i)}$
represents the mixing present in each of the mass matrices, with $i=1,2$.
Given that the mixing angles in $\mathbb{M}_{\nu}^{(i)}$ are small, we can
expand ${\mathbb{U}}_{\mbox{mix}}^{(i)}$ in terms of small parameters
$\cos\theta_{m}^{(i)}\approx 1$,
$\sin\theta_{m}^{(i)}\approx\epsilon^{(i)}_{m}$, where $m=\\{12,23,13\\}$
labels the three angles. The total mass matrix now takes the form

$\mathbb{M}_{\nu}=\left(\begin{array}[]{ccc}m_{1}^{(1)}+m_{1}^{(2)}&(m_{2}^{(1)}-m_{1}^{(1)})\epsilon^{(1)}_{12}+(m_{2}^{(2)}-m_{1}^{(2)})\epsilon^{(2)}_{12}&(m_{3}^{(1)}-m_{1}^{(1)})\epsilon^{(1)}_{13}+(m_{2}^{(2)}-m_{1}^{(2)})\epsilon^{(2)}_{13}\\\
&m_{2}^{(1)}+m_{2}^{(2)}&(m_{3}^{(1)}-m_{2}^{(1)})\epsilon^{(1)}_{23}+(m_{3}^{(2)}-m_{2}^{(2)})\epsilon^{(2)}_{23}\\\
&*&m_{3}^{(1)}+m_{3}^{(2)}\end{array}\right),$ (35)

where the symmetric elements of the matrix have been represented by $*$. We
can determine the mixing present in the total mass matrix by diagonalising the
above matrix. We have

$\mathbb{M}^{\prime}_{\nu}={\mathbb{U}}_{23}^{T}\mathbb{M}_{\nu}{\mathbb{U}}_{23}~{},$
(36)

where

${\mathbb{U}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&\sin\theta_{23}\\\
0&-\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{},$

with

$\tan
2\theta_{23}=2~{}{(m_{3}^{(1)}-m_{2}^{(1)})\epsilon^{(1)}_{23}+(m_{3}^{(2)}-m_{2}^{(2)})\epsilon^{(2)}_{23}\over
m_{3}^{(1)}+m_{3}^{(2)}-m_{2}^{(1)}-m_{2}^{(2)}}~{}.$ (37)

For this mixing to be maximal the condition would be
$(m_{3}^{(1)}-m_{2}^{(1)})=-(m_{3}^{(2)}-m_{2}^{(2)})$. This condition is
similar to the one we have seen earlier for the two generation case and as
argued in that case, the only natural solution is to have
$m_{3}^{(1)}=m_{2}^{(2)}$ with $m_{2}^{(1)},m_{3}^{(2)}$ negligible or
$m_{2}^{(1)}=m_{3}^{(2)}$ with 555 More precisely, we should have
$m_{3}^{(1)}-m_{2}^{(2)}\approx{\cal
O}(m_{3}^{(1)}(\epsilon_{23}^{1}-\epsilon_{23}^{2}))$and
$m_{2}^{(1)},m_{2}^{(2)}$ much smaller compared to them.
$m_{3}^{(1)},m_{2}^{(2)}$ negligible. We now proceed to show that if we accept
either of these two solutions, it would not be possible to have one another
large mixing angle in $\mathbb{M}_{\nu}$, if they have to satisfy the
naturalness criteria that the large eigenvalues of the individual matrices
should not cancel in the total mass matrix. Defining

${\mathbb{U}}_{13}=\left(\begin{array}[]{ccc}\cos\theta_{13}&0&\sin\theta_{13}\\\
0&1&0\\\ -\sin\theta_{13}&0&\cos\theta_{13}\end{array}\right)~{},$ (38)

we have

$\mathbb{M}^{{}^{\prime\prime}}_{\nu}={\mathbb{U}}_{13}^{T}\mathbb{M}^{\prime}_{\nu}{\mathbb{U}}_{13}~{}.$
(39)

$\tan 2\theta_{23}$ in the limit where the solution for maximal mixing of the
$23$ angle, $m_{3}^{(1)}=m_{2}^{(2)}=\bar{m}$ with
$m_{2}^{(1)},m_{3}^{(2)}\sim 0$ is taken is given by

$\tan
2\theta_{13}\approx{m_{1}^{(1)}(\epsilon^{(1)}_{12}+\epsilon^{(1)}_{13})-\bar{m}(\epsilon^{(1)}_{13}+\epsilon^{(2)}_{12})+m_{1}^{(2)}(\epsilon^{(2)}_{12}+\epsilon^{(2)}_{13})\over\sqrt{2}(m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}(1+\epsilon^{(1)}_{23}-\epsilon^{(2)}_{23}))}~{}.$
(40)

From the above we realize the following conditions for (a) small mixing :
$|m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}|\gg 0$, and (b) maximal mixing :
$|m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}|=0$. Finally, defining

${\mathbb{U}}_{12}=\left(\begin{array}[]{ccc}\cos\theta_{12}&\sin\theta_{12}&0\\\
-\sin\theta_{12}&\cos\theta_{12}&0\\\ 0&0&1\end{array}\right)~{},$ (41)

we have

$\mathbb{M}^{{}^{\prime\prime\prime}}_{\nu}={\mathbb{U}}_{12}^{T}\mathbb{M}^{{}^{\prime\prime}}_{\nu}{\mathbb{U}}_{12}~{}.$
(42)

$\tan 2\theta_{12}$ has the following form in the limiting case when
$\theta_{13}$ is very small

$\tan
2\theta_{12}\approx{m_{1}^{(1)}(\epsilon_{12}^{(1)}-\epsilon_{13}^{(1)})+\bar{m}(\epsilon_{13}^{(1)}-\epsilon_{12}^{(2)})+m_{1}^{(1)}(\epsilon^{(2)}_{12}-\epsilon^{(2)}_{13})\over\sqrt{2}(m_{1}^{(1)}+m_{1}^{(2)}-\bar{m}(1-\epsilon_{23}^{(1)}+\epsilon_{23}^{(2)}))}+\mathcal{O}(\theta_{13})~{}.$
(43)

From the above we see that the conditions for the mixing are the same for both
$\theta_{12}$ and $\theta_{13}$ in this limit. Thus either both become
maximal/large or both remain small. Finally, in the limit of maximal
$\theta_{13}$ mixing, the expression for $\tan 2\theta_{12}$ becomes

${(m_{1}^{(1)}(\epsilon_{12}^{(1)}-\epsilon_{13}^{(1)})+\bar{m}(\epsilon_{13}^{(1)}-\epsilon_{12}^{(2)})+m_{1}^{(2)}(\epsilon_{12}^{(2)}-\epsilon_{13}^{(2)}))\over
m_{1}^{(1)}+m_{1}^{(2)}+\bar{m}(-1+3\epsilon_{23}^{(1)}-3\epsilon^{(2)}_{23})+\sqrt{2}(m_{1}^{(1)}(\epsilon_{12}^{(1)}+\epsilon_{13}^{(1)})-\bar{m}(\epsilon_{13}^{(1)}+\epsilon_{12}^{(2)})+m_{1}^{(2)}(\epsilon^{(2)}_{12}+\epsilon_{13}^{(2)}))}~{},$
(44)

which is also automatically maximal/large within the small
$\epsilon_{ij}^{(k)}$ limit. Before we proceed, a few comments are in order
regarding the ordering of the eigenvalues. In the case where there is only one
maximal/large mixing, the sub matrices can have hierarchal and inverse-
hierarchal patterns, with the hierarchal sub-matrix containing one large
eigenvalue and the inverse-hierarchal containing two large eigenvalues. The
only condition is on their CP parties; the eigenvalues taking part in the
enhancement of the mixing should have the same CP parities. The list of
possible forms the sub-matrices can take is discussed in the subsection
III.1.2 where decomposition of the degenerate spectrum is considered in three
generation case.

On the other hand, for the case with all the three large/maximal mixing case,
as per our arguments earlier, i.e, the large eigenvalues of the individual
matrices should not cancel in the total matrix, the present solution
necessarily favours an alternating pattern for the eigenvalues for the
individual mass matrices 666The zeroth order textures for alternating pattern
of neutrino mass matrices are given in Table 4.

$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(1)}]=\mbox{Diag}[\\{m_{1}^{(1)},0,m_{3}^{(1)}\\}]$
,
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(2)}=\mbox{Diag}[\\{0,m_{2}^{(2)},0\\}]$
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(1)}]=\mbox{Diag}[\\{0,m_{2}^{(1)},0\\}]$
,
$\displaystyle\mbox{Diag}[\mathbb{M}_{\nu}^{(2)}]=\mbox{Diag}[\\{m_{1}^{(2)},0,m_{3}^{(2)}\\}]~{}.$
(45)

In this case, the mixing pattern corresponds to the truly maximal mixing
matrix of Cabibbo and Wolfenstein cabwolf  along with the degeneracy condition
$m_{1}\approx m_{2}\approx m_{3}$. In the recent years, the truly maximal
mixing matrix has been achieved from $A_{4}$ symmetry by Ma and Rajasekaran
also for degenerate case marajasekaran . As it stands this matrix is not
phenomenologically viable as all the three mixing angles it predicts are
large. However there could be other corrections to this mass matrix depending
on the model which would rectify this situation vallemababu  and make the mass
matrix phenomenologically viable.

####  III.1.1 Two Equivalent Textures

From our arguments above, it appears that we can generate only one large
mixing angle in the case when there are only two sub-matrices, because of the
important constraint that the third mixing angle ($\theta_{13}$) must not be
large 777This would be case in models where there are no large radiative
corrections effecting the mixing angles strongly.. Given that we can only
generate one large mixing from the small mixing using the degenerate
conditions, we will have to assume that at least one of the sub-matrices has
intrinsically one maximal/large mixing angle. However, the presence of this
mixing should not disturb the smallness of $\theta_{13}$ angle in the total
mass matrix. In the following, we will consider one of the sub-matrices to
have pseudo-Dirac structure and other one to have one large eigenvalue and all
the three mixing angles small. This is because the pseudo-Dirac structure not
only gives maximal mixing but also has the eigenvalues with opposite CP
parities.

${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&0&1\\\
y^{2}&1&0\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}1&z&t^{3}\\\
z&z^{3}&t^{3}\\\ t^{3}&t^{3}&z^{3}\end{array}\right)~{},$ (46)

where $x,y,z,t$ are small entries compared to $m_{1},m_{2}$. We will
diagonalise this matrix in the following manner. Rotating by
${\mathbb{O}}_{23}$ on both sides, we have

${\mathbb{O}}_{23}^{T}{\mathbb{M}}_{\nu}{\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}m_{2}+m_{1}x^{2}&m_{1}x\cos\theta_{23}+\widetilde{m}_{12}&-m_{1}x\sin\theta_{23}+\widetilde{m}_{13}\\\
m_{1}x\cos\theta_{23}+\widetilde{m}_{12}&m_{1}\sin
2\theta_{23}+\widetilde{m}_{22}&0\\\
-m_{1}x\sin\theta_{23}+\widetilde{m}_{13}&0&-m_{1}\sin
2\theta_{23}+\widetilde{m}_{33}\end{array}\right)~{},$ (47)

where ${\mathbb{O}}_{23}$ is defined as

${\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&-\sin\theta_{23}\\\
0&\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{},$ (48)

with the angle $\theta_{23}$ given by

$\theta_{23}={1\over
2}\tan^{-1}\left[{2(m_{1}+m_{2}t^{3})\over(m_{2}z^{3}-m_{2}z^{3})}\right]={\pi\over
4}~{}.$ (49)

The explicit forms for $\widetilde{m}_{ij}$ can be easily deduced. A crucial
point to note is that the diagonal elements of the matrix in Eq. (47) carry
opposite sign for the dominant element ($m_{1}$). This would have the
consequence of keeping the $13$ mixing small, while making $23$ mixing large,
when the degeneracy condition $m_{1}\approx m_{2}\approx m$ is imposed. The
total mixing matrix is given by
${\mathbb{O}}={\mathbb{O}}_{12}{\mathbb{O}}_{13}{\mathbb{O}}_{23}$ with the
angles $\theta_{13}$ and $\theta_{12}$ defined as

$\displaystyle\theta_{13}$ $\displaystyle=$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(-m_{1}x\sin\theta_{23}+\widetilde{m}_{13})\over-m_{1}\sin
2\theta_{23}+\widetilde{m}_{33}-m_{1}x^{2}-{m}_{2}}\right]~{},$
$\displaystyle\theta_{12}$ $\displaystyle=$ $\displaystyle{1\over
2}\tan^{-1}\left[{2\widetilde{m}^{\prime}_{12}\over\widetilde{m}^{\prime}_{22}-\widetilde{m}^{\prime}_{11}}\right]~{},$
(50)

where the explicit form of $\widetilde{m}^{\prime}_{ij}$ can easily be
deduced. From the above, we can see that the degeneracy induced large mixing
mechanism works for the $12$ mixing, while it does not generate large
(maximal) mixing for the $13$ mixing angle. This is due to the choice of
having $\tau\tau$ element with opposite sign (loosely speaking CP parity)
compared to the $\mu\mu$ element.

The above Yukawa matrices can be easily incorporated in the LRS model by
choosing $f$ and $Y$ of Eq. (22) (at the leading order) as

$f=\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&0&1\\\
y^{2}&1&0\end{array}\right)\;\;\;Y=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)~{}.$ (51)

Notice that it reproduces the Eq. (46) at the zeroth order. From the
discussion in the previous section, we also know that the pseudo-Dirac mass
matrix can be decomposed in to maximally mixing sub-matrices. Thus another
texture which could equally give the same results is given by

${\mathbb{M}}_{\nu}=m_{1}\left(\begin{array}[]{ccc}x^{2}&x&y^{2}\\\ x&{1\over
2}&{1\over 2}\\\ y^{2}&{1\over 2}&{1\over
2}\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}1&z&t^{3}\\\ z&{1\over
2}&-{1\over 2}\\\ t^{3}&-{1\over 2}&{1\over 2}\end{array}\right)~{},$ (52)

The first of the matrices has only one large eigenvalue in a hierarchial
pattern with maximal mixing, whereas the second one has two large eigenvalues
with one maximal mixing and two small mixings with inverted hierarchy. Lets
emphasize once more that one needs opposite eigenvalues $m_{1}\approx-m_{2}$
to obtain the large atmospheric mixing in this case.

####  III.1.2 Decomposition of the Degenerate Spectrum

For three generations the decomposition of the degenerate spectrum in to
hierarchal and inverse-hierarchal mass patterns is straight forward. In Table
9, we present the zeroth order mass matrices for the three generation case.

Mixing $\Rightarrow$ |  Small |  Single maximal |  Bimaximal |  Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
Hierarchial |  |  |  |
${\mathbb{A}}$: Diag[0,0,1] | $\begin{pmatrix}0&0&\epsilon_{13}\\\ 0&0&\epsilon_{23}\\\ \epsilon_{13}&\epsilon_{23}&1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$
Inverse hierarchial |  |  |  |
${\mathbb{B}}_{1}$: Diag[1,-1,0] | $\begin{pmatrix}1&-2\epsilon_{12}&-\epsilon_{13}\\\ -2\epsilon_{12}&-1&\epsilon_{23}\\\ -\epsilon_{13}&\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&-\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&-\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ -\frac{1}{\sqrt{2}}&0&0\\\ \frac{1}{\sqrt{2}}&0&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}\\\ -\frac{2}{3}&-\frac{1}{6}&\frac{1}{6}\\\ \frac{2}{3}&\frac{1}{6}&-\frac{1}{6}\end{pmatrix}$
${\mathbb{B}}_{2}$: Diag[1,1,0] | $\begin{pmatrix}1&0&-\epsilon_{13}\\\ 0&1&-\epsilon_{23}\\\ -\epsilon_{13}&-\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ |  $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$
Degenerate |  |  |  |
${\mathbb{C}}_{0}$: Diag[1,1,1] | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ |  $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$
${\mathbb{C}}_{1}$: Diag[-1,1,1] | $\begin{pmatrix}-1&2\epsilon_{12}&2\epsilon_{13}\\\ 2\epsilon_{12}&1&0\\\ 2\epsilon_{13}&0&1\end{pmatrix}$ |  $\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\ \frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\\ -\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}-\frac{1}{3}&\frac{2}{3}&-\frac{2}{3}\\\ \frac{2}{3}&\frac{2}{3}&\frac{1}{3}\\\ -\frac{2}{3}&\frac{1}{3}&\frac{2}{3}\end{pmatrix}$
${\mathbb{C}}_{2}$: Diag[1,-1,1] | $\begin{pmatrix}1&-2\epsilon_{12}&0\\\ -2\epsilon_{12}&-1&2\epsilon_{23}\\\ 0&2\epsilon_{23}&1\end{pmatrix}$ |  $\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$ | $\begin{pmatrix}0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ -\frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\\\ \frac{1}{\sqrt{2}}&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}\\\ -\frac{2}{3}&\frac{1}{3}&\frac{2}{3}\\\ \frac{2}{3}&\frac{2}{3}&\frac{1}{3}\end{pmatrix}$
${\mathbb{C}}_{3}$: Diag[1,1,-1] | $\begin{pmatrix}1&0&-2\epsilon_{13}\\\ 0&1&-2\epsilon_{23}\\\ -2\epsilon_{13}&-2\epsilon_{23}&-1\end{pmatrix}$ |  $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ |  $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$
Table 2: Different standard textures (zeroth order) for different combinations
of mixings (setting $m_{1}$, $m_{2}$ and $m_{3}$ as dimensionless quantities
which are either zero or one depending on the different cases listed)
consistent with data 999In Ref. afreview, , only two cases (single and
bimaximal mixing) were considered and they used
${\mathbb{M}}_{\nu}={\mathbb{U}}_{PMNS}^{\dagger}{\mathbb{M}}_{diag}{\mathbb{U}}_{PMNS}$,
which is different from our definition (Eq. (13)). .

Note that the present notation has been previously used in the literature
afreview  and all the matrices present in this table have been used previously
to describe the neutrino mass matrix at the zeroth order. After adding small
perturbations to these matrices they can explain the neutrino data. However,
as before we are interested in only decomposing the degenerate mass matrix in
terms of the hierarchal and inverse-hierarchal mass matrices. As before, from
each of the columns, we can see that each degenerate case can be constructed
as a sum of hierarchal and inverse hierarchal textures. For example,
$\mathbb{C}_{0}$ can be considered as $\mathbb{A}~{}+~{}\mathbb{B}_{2}$.
Similarly, $\mathbb{C}_{1}$ can be considered as
$-\mathbb{B}_{1}~{}+~{}\mathbb{A}$ and so on. And this is true as we go along
each of the columns, i.e for all kinds of mixing angles. This simple
observation can be restated as every degenerate neutrino mass matrix can be
thought of a sum of hierarchal and inverse hierarchal sub-mass matrices while
the converse is not generally true.

Mixing $\Rightarrow$ |  Small |  Single maximal |  Bimaximal |  Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
$\widetilde{\mathbb{A}}_{1}$: Diag[0,1,1] | $\begin{pmatrix}0&\epsilon_{12}&\epsilon_{13}\\\ \epsilon_{12}&1&0\\\ \epsilon_{13}&0&1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&\frac{3}{4}&\frac{1}{4}\\\ -\frac{1}{2\sqrt{2}}&\frac{1}{4}&\frac{3}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&\frac{5}{6}&\frac{1}{6}\\\ -\frac{1}{3}&\frac{1}{6}&\frac{5}{6}\end{pmatrix}$
$\widetilde{\mathbb{A}}_{2}$: Diag[0,1,-1] | $\begin{pmatrix}0&\epsilon_{12}&-\epsilon_{13}\\\ \epsilon_{12}&1&-2\epsilon_{23}\\\ -\epsilon_{13}&-2\epsilon_{23}&-1\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&0&-1\\\ 0&-1&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&-\frac{1}{4}&-\frac{3}{4}\\\ -\frac{1}{2\sqrt{2}}&-\frac{3}{4}&-\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&-\frac{1}{6}&-\frac{5}{6}\\\ -\frac{1}{3}&-\frac{5}{6}&-\frac{1}{6}\end{pmatrix}$
$\widetilde{\mathbb{B}}$: Diag[1,0,0] | $\begin{pmatrix}1&-\epsilon_{12}&-\epsilon_{13}\\\ -\epsilon_{12}&0&0\\\ -\epsilon_{13}&0&0\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&-\frac{1}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}\\\ -\frac{1}{2\sqrt{2}}&\frac{1}{4}&-\frac{1}{4}\\\ \frac{1}{2\sqrt{2}}&-\frac{1}{4}&\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\ -\frac{1}{3}&\frac{1}{6}&-\frac{1}{6}\\\ \frac{1}{3}&-\frac{1}{6}&\frac{1}{6}\end{pmatrix}$
Table 3: Novel textures (leading order) for different mixing scenarios which
by themselves need not be consistent with data. These cases are useful when we
consider adding two different textures to obtain the degenerate cases. The
labels with tilde sign are new textures by taking into account the fact that
hierarchy or inverse hierarchy can appear in either 1-2 sector or the 2-3
sector respectively. The standard textures considered degeneracy in 1-2 sector
and hierarchy or inverse hierarchy only in the 2-3 sector.

In three generations, the above set of decomposition which is based on
neutrino data is not exhaustive. This is essentially because the constraints
of the neutrino data are not on the individual sub-matrices but on the total
mass matrix. In such a case, the normal and inverse hierarchial sub matrices
can take other possible forms ${\mathbb{A}}$ and ${\mathbb{B}}_{i}$ than those
listed in Table 9. From Table 3, it is easy to see that the combinations of
$\widetilde{\mathbb{A}}_{i}$ and $\widetilde{\mathbb{B}}$ would produce one of
the degenerate textures ${\mathbb{C}}_{i}$ of the original Table 9. However,
even this list is not exhaustive for the degenerate case. We could have
textures which are not traditionally ordered as either hierarchial or inverse
hierarchial in the three generation case. These cases are listed in Table 4
and we call them as alternating textures (see Eq. (III.1)). Thus in summary,
we have covered all possible ways of ordering the three degenerate eigenvalues
in to two sub-matrices, which are not degenerate themselves.

Mixing $\Rightarrow$ |  Small |  Single maximal |  Bimaximal |  Tribimaximal
---|---|---|---|---
$\mathbb{M}_{diag}$ | ${\mathbb{X}}_{\epsilon}$ | ${\mathbb{X}}_{SM}$ | ${\mathbb{X}}_{BM}$ | ${\mathbb{X}}_{TBM}$
${\mathbb{T}}_{1}$: Diag[0,1,0] | $\begin{pmatrix}0&\epsilon_{12}&0\\\ \epsilon_{12}&1&-\epsilon_{23}\\\ \epsilon_{13}&-\epsilon_{23}&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}&-\frac{1}{2}\\\ 0&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&\frac{1}{2\sqrt{2}}&-\frac{1}{2\sqrt{2}}\\\ \frac{1}{2\sqrt{2}}&\frac{1}{4}&-\frac{1}{4}\\\ -\frac{1}{2\sqrt{2}}&-\frac{1}{4}&\frac{1}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ \frac{1}{3}&\frac{1}{3}&-\frac{1}{3}\\\ -\frac{1}{3}&-\frac{1}{3}&\frac{1}{3}\end{pmatrix}$
${\mathbb{T}}_{2}$: Diag[1,0,1] | $\begin{pmatrix}1&-\epsilon_{12}&0\\\ -\epsilon_{12}&0&\epsilon_{23}\\\ 0&\epsilon_{23}&1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&\frac{1}{2}&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&-\frac{1}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}\\\ -\frac{1}{2\sqrt{2}}&\frac{3}{4}&\frac{1}{4}\\\ \frac{1}{2\sqrt{2}}&\frac{1}{4}&\frac{3}{4}\end{pmatrix}$ | $\begin{pmatrix}\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\\ -\frac{1}{3}&\frac{2}{3}&\frac{1}{3}\\\ \frac{1}{3}&\frac{1}{3}&\frac{2}{3}\end{pmatrix}$
Table 4: Alternating textures (leading order) for different mixing scenarios.

###  III.2 Case II: Three Sources

For more than two seesaw mechanisms at work, the generalisation is straight
forward. Lets consider the case where there are three sources of neutrino
masses. The total mass matrix in this case is given by

$\mathbb{M}_{\nu}=\mathbb{M}_{\nu}^{(1)}+\mathbb{M}_{\nu}^{(2)}+\mathbb{M}_{\nu}^{(3)}~{},$
(53)

where each of the sub matrices can be thought of having independent origin
through a seesaw mechanism or any other scheme to generate non-zero neutrino
masses. As with the two-generation case, we will now consider the case where
all the mixings present in each of the sub matrices are taken to be small and
each sub-matrix is assumed to have only one large eigenvalue. The second
assumption is a direct consequence of assuming that all the three sources
contribute equally and there are no cancellations between the dominant
eigenvalues of the sub-matrices. With these assumptions, the total mass matrix
can now be written in terms of the individual mass matrices as

$\mathbb{M}_{\nu}=m_{1}\left(\begin{array}[]{ccc}\epsilon_{13}^{2}&\epsilon_{13}\epsilon_{23}&\epsilon_{13}\\\
\epsilon_{13}\epsilon_{23}&\epsilon_{23}^{2}&\epsilon_{23}\\\
\epsilon_{13}&\epsilon_{23}&1\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}\epsilon_{12}^{{}^{\prime}2}&\epsilon^{\prime}_{12}&\epsilon^{\prime}_{12}\epsilon^{\prime}_{23}\\\
\epsilon^{\prime}_{12}&1&-\epsilon^{\prime}_{23}\\\
\epsilon^{\prime}_{12}\epsilon^{\prime}_{23}&-\epsilon^{\prime}_{23}&\epsilon_{23}^{{}^{\prime}2}\end{array}\right)+m_{3}\left(\begin{array}[]{ccc}1&-\epsilon^{\prime\prime}_{12}&-\epsilon^{\prime\prime}_{13}\\\
-\epsilon^{\prime\prime}_{12}&\epsilon_{12}^{{}^{\prime\prime}2}&\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13}\\\
-\epsilon^{\prime\prime}_{13}&\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13}&\epsilon_{13}^{{}^{\prime\prime}2}\end{array}\right)~{},$
(54)

where $\epsilon_{ij},\epsilon^{\prime}_{ij},\epsilon^{\prime\prime}_{ij}$
($i,j=1,2,3$) are small entries corresponding to small mixing angles in
$U^{(i)}_{mix}$. This total matrix can be diagonalised by an orthogonal matrix
${\mathbb{O}}\equiv{\mathbb{O}}_{23}{\mathbb{O}}_{13}{\mathbb{O}}_{12}$, such
that
${\mathbb{O}}^{T}{\mathbb{M}}_{\nu}{\mathbb{O}=\mbox{Diag}[\mathbb{M}}_{\nu}]$.
${\mathbb{O}}_{ij}$ represents a rotation in the ${ij}^{th}$ plane. For
example

${\mathbb{O}}_{23}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}&\sin\theta_{23}\\\
0&-\sin\theta_{23}&\cos\theta_{23}\end{array}\right)~{}.$ (55)
$\displaystyle\theta_{23}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})\over
m_{1}(1-\epsilon_{23}^{2})-m_{2}(1-\epsilon_{23}^{{}^{\prime}2})+m_{3}(\epsilon_{13}^{{}^{\prime\prime}2}-\epsilon_{12}^{"2})}\right]~{},$
$\displaystyle\theta_{13}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(\widetilde{m}_{13})\over\widetilde{m}_{33}-\widetilde{m}_{11}}\right]~{},$
$\displaystyle\theta_{12}$ $\displaystyle\approx$ $\displaystyle{1\over
2}\tan^{-1}\left[{2(\widetilde{m}^{\prime}_{12})\over\widetilde{m}^{\prime}_{22}-\widetilde{m}^{\prime}_{11}}\right]~{},$
(56)

where

$\displaystyle\widetilde{m}_{13}$ $\displaystyle=$ $\displaystyle
s_{23}(m_{1}\epsilon_{13}\epsilon_{23}+m_{2}\epsilon^{\prime}_{12}-m_{3}\epsilon^{\prime\prime}_{12})+c_{23}(m_{1}\epsilon_{13}+m_{2}\epsilon^{\prime}_{12}\epsilon^{\prime}_{13}-m_{3}\epsilon^{\prime\prime}_{13})~{},$
$\displaystyle\widetilde{m}_{33}$ $\displaystyle=$ $\displaystyle
s_{23}[s_{23}(m_{2}+m_{1}\epsilon_{23}^{2}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})+c_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})]$
$\displaystyle+$ $\displaystyle
c_{23}[s_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})+c_{23}(m_{1}+m_{2}\epsilon^{{}^{\prime}2}_{23}+m_{3}\epsilon^{{}^{\prime\prime}2}_{13})]~{},$
$\displaystyle\widetilde{m}_{11}$ $\displaystyle=$ $\displaystyle
m_{3}+m_{1}\epsilon_{13}^{2}+m_{2}\epsilon_{12}^{{}^{\prime}2}~{},$
$\displaystyle\widetilde{m}^{\prime}_{12}$ $\displaystyle=$ $\displaystyle
c_{13}\widetilde{m}_{12}=c_{13}[c_{23}(m_{1}\epsilon_{13}\epsilon_{23}+m_{2}\epsilon^{\prime}_{12}-m_{3}\epsilon^{\prime\prime}_{12})-s_{23}(m_{1}\epsilon_{13}+m_{2}\epsilon^{\prime}_{12}\epsilon^{\prime}_{13}-m_{3}\epsilon^{\prime\prime}_{13})]~{},$
$\displaystyle\widetilde{m}^{\prime}_{22}$ $\displaystyle=$
$\displaystyle\widetilde{m}_{22}=c_{23}[c_{23}(m_{2}+m_{1}\epsilon_{23}^{2}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})-s_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})]$
$\displaystyle-$ $\displaystyle
s_{23}[c_{23}(m_{1}\epsilon_{23}-m_{2}\epsilon^{\prime}_{23}+m_{3}\epsilon^{\prime\prime}_{12}\epsilon^{\prime\prime}_{13})-s_{23}(m_{1}+m_{2}\epsilon_{23}^{{}^{\prime}2}+m_{3}\epsilon_{13}^{{}^{\prime\prime}2})]~{},$
$\displaystyle\widetilde{m}^{\prime}_{11}$ $\displaystyle=$ $\displaystyle
c_{13}(\widetilde{m}_{11}c_{13}-\widetilde{m}_{13}s_{13})-s_{13}(\widetilde{m}_{13}c_{13}-\widetilde{m}_{33}s_{13})~{}.$
(57)

Notice that all the three mass eigenvalues are of the same CP parity in the
above and the degeneracy induced mixing thus works for the all the three
mixing angles. Thus all the three mixing angles are large. One can then ask
the question whether choosing one of the mass eigenvalues with a negative CP
parity would help in keeping one of the mixing angles small. The answer is
negative, choosing one of the eigenvalues to have CP parity negative leads to
at least two of the mixing angles to remain small as the degeneracy induced
large mixing mechanism is no longer operative for two of the mixing angles.
Thus we are back to the case of two seesaw mechanisms which we have seen in
the previous subsection.

While it is possible to visualise GUT models where there are three seesaw
mechanisms at work, it much easier to suitably split a single Type I seesaw
mass matrix into three sub-matrices. In this case, we can extend Eq. (18) to
three generations as

$\displaystyle-{\mathbb{M}}^{\rm{I}}_{\nu}$ $\displaystyle=$ $\displaystyle
m_{1}\left(\begin{array}[]{ccc}(h_{ee}^{D})^{2}&h_{ee}^{D}h_{e\mu}^{D}&h_{ee}^{D}h^{D}_{e\tau}\\\
h_{ee}^{D}h_{e\mu}^{D}&(h_{e\mu}^{D})^{2}&h^{D}_{e\mu}h^{D}_{e\tau}\\\
h_{ee}^{D}h^{D}_{e\tau}&h^{D}_{e\mu}h^{D}_{e\tau}&(h^{D}_{e\tau})^{2}\end{array}\right)+m_{2}\left(\begin{array}[]{ccc}(h_{\mu
e})^{2}&h_{\mu e}^{D}h_{\mu\mu}^{D}&h_{\mu e}^{D}h^{D}_{\mu\tau}\\\ h_{\mu
e}^{D}h_{\mu\mu}^{D}&(h_{\mu\mu}^{D})^{2}&h^{D}_{\mu\mu}h^{D}_{\mu\tau}\\\
h_{\mu
e}^{D}h^{D}_{\mu\tau}&h^{D}_{\mu\mu}h^{D}_{\mu\tau}&(h^{D}_{\mu\tau})^{2}\end{array}\right)$
(64) $\displaystyle+$ $\displaystyle m_{3}\left(\begin{array}[]{ccc}(h_{\tau
e})^{2}&h_{\tau e}^{D}h_{\tau\mu}^{D}&h_{\tau e}^{D}h^{D}_{\tau\tau}\\\
h_{\tau
e}^{D}h_{\tau\mu}^{D}&(h_{\tau\mu}^{D})^{2}&h^{D}_{\tau\mu}h^{D}_{\tau\tau}\\\
h_{\tau
e}^{D}h^{D}_{\tau\tau}&h^{D}_{\tau\mu}h^{D}_{\tau\tau}&(h^{D}_{\tau\tau})^{2}\end{array}\right)~{}.$
(68)

Comparing this with Eq. (54), we see that we will have three possible
solutions for the Yukawa couplings in this case. The first solution is

$\displaystyle
h_{ee}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{e\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{e\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\mu\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{\tau\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{}.$
(69)

There are two more possibilities given by

$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{\mu\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle
h_{ee}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{e\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{\tau\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{},$
(70)

or

$\displaystyle h_{\tau
e}^{D}~{}\sim~{}\epsilon_{13}~{},\;\;h_{\tau\mu}^{D}~{}\sim~{}\epsilon_{23}~{},\;\;h^{D}_{\tau\tau}~{}\sim~{}\mathcal{O}(1)~{},\;\;$
$\displaystyle h_{\mu
e}^{D}~{}\sim~{}\epsilon^{\prime}_{12}~{},\;\;h_{\mu\mu}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{\mu\tau}^{D}~{}\sim~{}-\epsilon^{\prime}_{23}~{},\;\;$
$\displaystyle
h_{ee}^{D}~{}\sim~{}\mathcal{O}(1)~{},\;\;h_{e\mu}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{12}~{},\;\;h_{e\tau}^{D}~{}\sim~{}-\epsilon^{\prime\prime}_{13}~{}.$
(71)

From the above we see that even if each of the leptonic generation couples
minimally with each of the right-handed neutrino, the total mixing can be
maximal, purely due to the degeneracy requirement. The comments at the end of
subsection III.1 regarding maximally symmetric leptonic mixing matrix hold in
this case too. Finally, note that each set of these solutions is related by
$S_{3}$ symmetry to the other set.

##  IV Summary

In the present work, we have concentrated on the case with two seesaw
mechanisms at work which occurs naturally in many examples like LRS models,
SO(10) based GUT models etc. We have shown that if both these seesaw
mechanisms result in mass matrices which only have small mixing in them, then
the only pattern of mass eigenvalues which is naturally consistent with
maximal/large mixing is the quasi-degenerate pattern for the total mass
matrix.

All the arguments presented in the present work are independent of the details
of the sources of neutrino masses. However, depending on the specifics of the
model, there could be radiative corrections which could significantly modify
the mixing angles. For example, if one has Type I + Type II seesaw mechanism
operating at the high scale, radiative corrections could significantly modify
the mixing angles at the weak scale. These effects should be taken in to
account when applying the results of the present work to any particular model.
The impact of radiative corrections, models and implications for leptogenesis
within this class of hybrid degenerate models are being studied for a future
publication ourupcoming .

##  Appendix A Generalization of the result for $n$ sources

If there are $n$ sources of neutrino masses in a particular model such that
the total mass matrix is given by

${\mathbb{M}}_{\nu}={\mathbb{M}}^{{(1)}}_{\nu}+{\mathbb{M}}^{{(2)}}_{\nu}+\ldots+{\mathbb{M}}^{{(n)}}_{\nu}~{}.$
(72)

And further each of the ${\mathbb{M}}^{{(i)}}_{\nu}$ have one dominant
diagonal element proportional to its largest eigenvalue $m_{i}$, and rest of
the entries to be tiny (all the mixing angles in all the
${\mathbb{M}}^{{(i)}}_{\nu}$ are small); ${\mathbb{M}}^{{(i)}}_{\nu}$ are
ordered in such a way that the ${ii}^{th}$ element is dominant. There are $n$
possible orderings of ${\mathbb{M}}^{{(i)}}_{\nu}$. Then the total mass matrix
would naturally have a quasi-degenerate pattern with maximal/large mixing
depending on the number of pairs of eigenvalues which have the same CP parity,
if $m_{1}\approx m_{2}\approx\ldots\approx m_{n}$. If there are $l$
eigenvalues with the same CP parity 101010And if the splitting between
relevant $m_{i}$ is smaller than the tiny off-diagonal entries., then
${}^{n}{\cal{C}}_{2}+^{n-l}{\cal{C}}_{2}$ (if $(n-l)>2$) angles will be large
or maximal and the remaining will be small. An important exception to the
above is the pseudo-Dirac pattern of degenerate masses, which can only result
from a ‘sum’ of two mass matrices both containing maximal mixing and equal
eigenvalues with opposite ordering in hierarchy. Conversely, at the zeroth
order a $n\times n$ quasi-degenerate matrix with eigenvalues
$m_{1},m_{2},\ldots m_{i}\ldots m_{n}$ (by definition $m_{1}\approx
m_{2}\approx\ldots\approx m_{n}$) can be decomposed in to $n$ sub-matrices
${\mathbb{M}}^{{(n)}}_{\nu}$, with eigenvalues distributed as

$\displaystyle{\mathbb{M}}_{\nu}$ $\displaystyle=$
$\displaystyle{\mathbb{M}}^{{(1)}}_{\nu}+{\mathbb{M}}^{{(2)}}_{\nu}+\ldots+{\mathbb{M}}^{{(n)}}_{\nu}$
$\displaystyle\begin{pmatrix}m_{1}&&&&\\\ &m_{2}&&&\\\ &&\ddots&&\\\
&&&&m_{n}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{1}&&&&\\\ &0&&&\\\ &&\ddots&&\\\
&&&&0\end{pmatrix}+\begin{pmatrix}0&&&&\\\ &m_{2}&&&\\\ &&\ddots&&\\\
&&&&0\end{pmatrix}+\ldots+\begin{pmatrix}0&&&&\\\ &0&&&\\\ &&\ddots&&\\\
&&&&m_{n}\end{pmatrix}~{}.$ (73)

This holds true irrespective of the mixing present in the total mass matrix
${\mathbb{M}}_{\nu}$.

###### Acknowledgements.

J.C. thanks A. Raychaudhuri for encouragement and discussions. He also
acknowledges support from the Neutrino Project and RECAPP under the XIth plan
of Harish-Chandra Research Institute. J.C. further acknowledges the
hospitality and support from CHEP, IISc., Bangalore where part of the work was
carried out. P.M. acknowledges the kind hospitality received from the Institut
für Theoretische Physik und Astrophysik, Universität Würzburg; Institut für
Theoretische Physik E, RWTH Aachen; CFTP, Instituto Superior Técnico -
Universidade Técnica de Lisboa as well as the organisers of “Workshop towards
neutrino technologies” at ICTP, Italy and “Lepton Photon 2009” in Hamburg
during the final stages of this work.

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