compat=1.1.0

On quark-lepton mixing and the leptonic CP violation
Alessio Giarnetti111e-mail address<EMAIL_ADDRESS>, Simone
Marciano222e-mail address<EMAIL_ADDRESS>, Davide
Meloni333e-mail address<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica, Università di Roma Tre
INFN Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy

Abstract

> In the absence of a Grand Unified Theory framework, connecting the values of
> the mixing parameters in the quark and lepton sector is a difficult task,
> unless one introduces ad-hoc relations among the matrices that diagonalize
> such different kinds of fermions. In this paper, we discuss in detail the
> possibility that the PMNS matrix is given by the product
> $U_{PMNS}=V_{CKM}^{\star}\,T^{\star}$, where $T$ comes from the
> diagonalization of a see-saw like mass matrix that can be of a Bimaximal
> (BM), Tri-Bimaximal (TBM) and Golden Ratio (GR) form, and identify the
> leading corrections to such patterns that allow a good fit to the leptonic
> mixing matrix as well as to the CP phase. We also show that the modified
> versions of BM, TBM and GR can easily accommodate the solar and atmospheric
> mass differences.

###### Contents

  1. 1 Introduction
  2. 2 Corrections to BM, TBM and GR
  3. 3 On the neutrino masses
  4. 4 Conclusions

##  1 Introduction

In the last years, neutrino experiments confirmed that neutrinos oscillate and
measured with a great precision the values of the mixing angles. Some neutrino
oscillation properties are still unknown/not really clear (as, for example,
whether CP violation exists in the lepton sector or whether the mass hierarchy
is of normal or inverted type) but the emerging picture is quite intriguing:
differently from the mixing angles in the quark sector, described by an almost
diagonal matrix $V_{CKM}$, neutrino mixing is dictated by two large and one
small angle, thus making the $U_{PMNS}$ a matrix with large entries, except
for the $(13)$ element. In spite of this huge discrepancy (that has been
dubbed as the flavor problem), the current numerical values of fermion mixings
seem to be inextricably wedged into well-defined relations [1] which, using
the standard parametrization of mixing matrices, are summarized as follows:

$\theta_{12}^{PMNS}+\theta_{12}^{CKM}\sim\pi/4\,,\qquad\theta_{23}^{PMNS}+\theta_{23}^{CKM}\sim\pi/4\,.$
(1.1)

The previous structure, which is presumed to exist behind such empirical
relations, is known as quark-lepton complementarity (QLC) and, while being
appealing from a theoretical and phenomenological point of view, does not give
any clue on which kind of symmetry could be responsible for them.

The usual answer to this problem is grand unification (GUT) in which quarks
and leptons are unified into the same multiplets [2, 3, 4, 5, 6, 7, 8]; on the
other hand, in non-GUT scenarios, one is somehow forced to input the CKM
(PMNS) matrix into the relations that define the PMNS (CKM). Several authors
have explored such a possibility [1, 9, 10, 11, 12, 13, 14, 15] and discussed
the observable consequences of scenarios leading to QLC [16, 17, 18, 19, 20],
including the effect of the RGE running on the stability of eq.(1.1) [21, 22,
23, 24]. An extension of eq.(1.1) to the $(13)$ sector results in a complete
failure, as the sum $\theta_{13}^{PMNS}+\theta_{13}^{CKM}\sim 10^{\circ}$;
thus, it is necessary to find a new connection between neutrinos and quarks
that involves the reactor angle. The most promising suggestion is, once again,
GUT-inspired and reads:

$\theta_{13}^{PMNS}=\alpha\,\theta_{12}^{CKM}\,,$ (1.2)

where $\alpha$ can be any ${\cal O}(1)$ number [25, 10, 26, 27, 28]. One
possibility to recover eq.(1.2) is to assume that the mixing matrices are
related through:

$\displaystyle U_{PMNS}$ $\displaystyle\sim$ $\displaystyle V_{CKM}\,T\,,$
(1.3)

where $T$ is an appropriate unitary matrix that we parametrize as the product
of three sub-roations:

$T\equiv U_{23}U_{13}U_{12}\,.$ (1.4)

In this paper we want to elaborate more on eq.(1.3), finding the exact
theoretical relation among $U_{PMNS}$ and $V_{CKM}$ allowed by specific ansatz
on the diagonalization procedure of the fermion mass matrices. This involves
the determination of the matrix $T$; by assuming an initial form for $T$ of
Bimaximal (BM), Tri-Bimaximal (TBM) [29] and Golden Ratio (GR) type [30], we
compute in a systematic way all relevant corrections that allow to reproduce
the neutrino mixing angles as well as the Jarlskog invariant [31]. Instead of
performing an overall fit involving general perturbations of BM, TBM and GR
mixings, we preferred to introduce three different corrections, one for each
$U_{ij}$ quoted in eq.(1.4), and study the prediction of mixing parameters
determined by each of them. In this way, we are able to keep track of the
relevant source of deviations from the initial form for $T$ that allows a good
fit to the experimental data. We find that a complex parameter $u$ is needed
in the $U_{13}$ rotation to increase the amount of leptonic CP violation up to
the current experimental values while a simple real correction in the $(12)$
plane is mandatory to account for the solar angle. Finally, deviation to
maximality for the atmospheric angle can be accounted by a real shift $\omega$
in the (23) sector. In addition to mixing parameters, the newly found
corrections are also compatible with the solar and atmospheric mass
differences to a high degree of precision.

The paper is organized as follows: in Sect.(2) we discuss all the above-
mentioned corrections in detail, showing how to include them in a perturbative
approach to the determination of the mixing parameters; in Sect.(3) we show
how to reproduce the experimental mass differences within our framework for
all perturbed mixing patterns; finally, Sect.(4) is devoted to our
conclusions. We close the paper with the Appendix where we report the
expressions of the mixing parameters up to ${\cal O}(\lambda^{3})$.

##  2 Corrections to BM, TBM and GR

###  2.1 Notation

Let us first fix our notation; we are working in the left-right (LR) basis
and, with no loss of generality, we assume diagonal heavy right-handed
neutrinos $M_{R}=M_{R}^{diag}$ and diagonal charged leptons
$M_{\ell}=M_{\ell}^{diag}$. The diagonalization of the Dirac neutrino mass is
achieved through $W_{L}^{\dagger}m_{\nu_{D}}U_{R}=m^{diag}_{\nu_{D}}$, so that
the hermitean matrix $m_{\nu_{D}}m_{\nu_{D}}^{\dagger}$ is such that
$W_{L}^{\dagger}m_{\nu_{D}}m_{\nu_{D}}^{\dagger}W_{L}=(m^{diag}_{\nu_{D}})^{2}$,
where the eigenvalues of $(m^{diag}_{\nu_{D}})^{2}$ are real and non-negative,
and the columns of $W_{L}$ are the eigenvectors of the
$m_{\nu_{D}}m_{\nu_{D}}^{\dagger}$ matrix. Applying the see-saw formula in the
LR basis, we get:

$\displaystyle m_{\nu}$ $\displaystyle=$ $\displaystyle-
m_{\nu_{D}}\,(M_{R}^{diag})^{-1}\,m_{\nu_{D}}^{T}$ (2.1) $\displaystyle=$
$\displaystyle
W_{L}\,m^{diag}_{\nu_{D}}\,U_{R}^{\dagger}\,(M_{R}^{diag})^{-1}\,U_{R}^{\star}\,m^{diag}_{\nu_{D}}\,W_{L}^{T}\,.$

At this point, the matrix
$m_{0}=m^{diag}_{\nu_{D}}\,U_{R}^{\dagger}\,(M_{R}^{diag})^{-1}\,U_{R}^{\star}\,m^{diag}_{\nu_{D}}$
is a complex symmetric matrix and, thus, it can be diagonalized by an unitary
matrix $T$ such that:

$\displaystyle m_{0}$ $\displaystyle=$ $\displaystyle T\,S\,T^{T}\,,$ (2.2)

where $S$ is a diagonal matrix with, in principle, complex entries. Thus, for
the light neutrino mass we have the following decomposition:

$\displaystyle m_{\nu}$ $\displaystyle=$ $\displaystyle-
W_{L}\,T\,S\,T^{T}\,W_{L}^{T}\,.$ (2.3)

To get the proper structure of $U_{PMNS}$, we assume a neutrino change of
basis of the following type:

$\displaystyle\nu^{\prime}$ $\displaystyle=$ $\displaystyle U_{PMNS}\,\nu\,,$
(2.4)

where the mass eigenstate are those indicated with $\nu$. At the Lagrangian
level, the symmetric mass term, in the basis of interaction eigenstates, is as
follows:

$\displaystyle(\nu^{T})^{\prime}\,m_{\nu}\nu^{\prime}$ $\displaystyle=$
$\displaystyle\nu^{T}\,U_{PMNS}^{T}\,m_{\nu}\,U_{PMNS}\,\nu\equiv\nu^{T}\,m_{\nu}^{diag}\,\nu$
(2.5)

so that:

$\displaystyle m_{\nu}$ $\displaystyle=$ $\displaystyle
U_{PMNS}^{\star}\,m_{\nu}^{diag}\,U_{PMNS}^{\dagger}\,,$ (2.6)

and we can identify:

$\displaystyle U_{PMNS}$ $\displaystyle=$ $\displaystyle
W_{L}^{\star}\,T^{\star}\,$ (2.7)

and

$\displaystyle m_{\nu}^{diag}$ $\displaystyle=$ $\displaystyle S\,.$ (2.8)

In the following we will assume $W_{L}\equiv V_{CKM}$, whose structure in the
Wolfenstein paramterization is reported below:

$V_{CKM}=\begin{pmatrix}1-\lambda^{2}/2&\lambda&A\lambda^{3}(-i\eta+\rho)\\\
-\lambda&1-\lambda^{2}/2&A\lambda^{2}\\\
A\lambda^{3}(1-i\eta-\rho)&-A\lambda^{2}&1\end{pmatrix}\,.$ (2.9)

The values of the $V_{CKM}$ parameters used in our simulations are provided in
Tab.(1).

Parameter | Best-fit value and $1\sigma$ range
---|---
$\lambda$ | $0.2251\pm 0.0008$
$A$ | $0.828\pm 0.01$
$\eta$ | $0.355\pm 0.009$
$\rho$ | $0.164\pm 0.009$
Table 1: Best-fit value and $1\sigma$ range of the $V_{CKM}$ parameters, from
[32].

For the $T$ matrix, instead, one can in principle assume an exact Tri-
Bimaximal mixing (TBM), Bimaximal mixing (BM) or Golden Ratio (GR) forms:

$U_{BM}=\left(\begin{array}[]{ccc}\displaystyle\frac{1}{\sqrt{2}}&\displaystyle-\frac{1}{\sqrt{2}}&0\\\
\displaystyle\frac{1}{2}&\displaystyle\frac{1}{2}&\displaystyle\frac{1}{\sqrt{2}}\\\
-\displaystyle\frac{1}{2}&-\displaystyle\frac{1}{2}&\displaystyle\frac{1}{\sqrt{2}}\end{array}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
U_{TBM}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ U_{GR}=\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\ \
\displaystyle\frac{s_{12}}{\sqrt{2}}&-\displaystyle\frac{c_{12}}{\sqrt{2}}&\displaystyle\frac{1}{\sqrt{2}}\\\
\displaystyle\frac{s_{12}}{\sqrt{2}}&-\displaystyle\frac{c_{12}}{\sqrt{2}}&-\displaystyle\frac{1}{\sqrt{2}}\end{array}\right)$
(2.10)

where $c_{12}=\cos\theta_{12},s_{12}=\sin\theta_{12}$ and
$\tan\theta_{12}=1/\phi$, with $\phi=(1+\sqrt{5})/2$. However, it turns out
that the $U_{PMNS}$ implied by them is unsatisfactory in the predicted values
of the mixing angles and Jarlskog invariant $J_{\text{CP}}$, for which we use
the following expression:

$\displaystyle
J_{\text{C}P}=\text{Im}\,\left[(U_{PMNS})_{11}(U_{PMNS})_{12}^{*}(U_{PMNS})_{21}^{*}(U_{PMNS})_{22}\right]\,.$
(2.11)

We have summarized the situation in Tab.(3) where, for each mixing pattern, we
have reported the perturbative prediction on
$\sin(\theta_{13}),\tan(\theta_{12}),\tan(\theta_{23})$ (up to ${\cal
O}(\lambda$)) and $J_{\text{CP}}$ (up to ${\cal O}(\lambda^{3}$)). In the last
column we have computed the distance $\Delta$ between such predictions and the
current experimental values for a Normal Ordering (NO) of the neutrino
masses111For our purposes, it is enough to consider the normal hierarchy only,
as the only significant difference with respect to the inverted ordering case
is a slight preference for the opposite $\theta_{23}$ octant., reported in
Tab.(2). Such a distance is computed according the following formula:

$\displaystyle\Delta=\Sigma_{i=1}^{3}\left[\frac{P_{i}-B_{i}}{\sigma_{i}}\right]^{2}\,,$
(2.12)

where $\vec{P}$ is a vector of parameters
$\vec{P}=\left[\tan(\theta_{12}),\tan(\theta_{13}),\tan(\theta_{23}),J_{\text{CP}}\right]$
as predicted by TBM, BM and GR (see Tab.(3)), $\vec{\sigma}$ are the related
$1\sigma$ errors and $\vec{B}$ contains the best-fit values of Tab.(2),
$\vec{B}=\left[\tan^{bf}(\theta_{12}),\tan^{bf}(\theta_{13}),\tan^{bf}(\theta_{23}),J^{bf}_{\text{CP}}\right]$.
$\Delta$ allows us to estimate how far a given texture is from the current
values of the mixing parameters.

Parameter | Best-fit value and $1\sigma$ range
---|---
$r\equiv\Delta m^{2}_{\text{sol}}/|\Delta m^{2}_{\text{atm}}|$ | $0.0295\pm 0.0008$
$\tan(\theta_{12})$ | $0.666\pm 0.019$
$\sin(\theta_{13})$ | $0.149\pm 0.002$
$\tan(\theta_{23})$ | $0.912\pm 0.035$
$J_{\text{CP}}$ | $-0.027\pm 0.010$
Table 2: Neutrino observables and their 1$\sigma$ ranges as derived from NuFIT 5.3 [33, 34], using the dataset with SK atmospheric data [35]. For the extraction of the best-fit value and $1\sigma$ uncertainty of the Jarlskog invariant, we refer to its one-dimensional $\chi^{2}$ projection from NuFIT 5.3.  $T$ | $\sin(\theta_{13})$ | $\tan(\theta_{12})$ | $\tan(\theta_{23})$ | $J_{\text{CP}}$ | $\Delta$
---|---|---|---|---|---
$U_{TBM}$ | $\frac{\lambda}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}+\frac{3\lambda}{2\sqrt{2}}$ | 1 | $-\frac{1}{6}A\eta\lambda^{3}$ | 2715
$U_{BM}$ | $\frac{\lambda}{\sqrt{2}}$ | $1-\sqrt{2}\lambda$ | 1 | $\frac{1}{4\sqrt{2}}A\eta\lambda^{3}$ | 2500
$U_{GR}$ | $\frac{\lambda}{\sqrt{2}}$ | $\frac{2\sqrt{5}}{5+\sqrt{5}}+\frac{5\sqrt{2}}{5+\sqrt{5}}\lambda$ | 1 | $-\frac{1}{2\sqrt{10}}A\eta\lambda^{3}$ | 2580
Table 3: Perturbative predictions on
$\sin(\theta_{13}),\tan(\theta_{12}),\tan(\theta_{23})$ (up to ${\cal
O}(\lambda))$ and $J_{\text{CP}}$ (up to ${\cal O}(\lambda^{3}))$ as obtained
from the ansatz $U_{PMNS}=V_{CKM}^{\star}T^{\star}$, where $T$ can be TBM, BM
and GR mixing patterns. In the last column we report the values of the
variable $\Delta$ defined in eq.(2.12).

While all patterns predict maximal $(23)$ mixing and the same
$\sin(\theta_{13})$, the differences come from $J_{\text{CP}}$ (strongly
suppressed for all patterns) and from the solar sector; in particular, for the
latter the BM mixing results in a better agreement with the current
experimental value than TBM and GR, as evident by the smaller $\Delta$. The
predictions in Tab.(3) are also reported in Fig.(1), together with their
1$\sigma$ experimental spread (red rectangles)222We do not report the spread
of $J_{\text{CP}}$ as, for any patters, its absolute value is around two
orders of magnitude smaller than the experimental best-fit..

Figure 1: Current 1$\sigma$ experimental spread on
$\tan(\theta_{12}),\tan(\theta_{23})$ and $\sin(\theta_{13})$ (red rectangles)
and the predictions derived from $U_{PMNS}=V_{CKM}^{\star}T^{\star}$, where
$T$ can be TBM, BM and GR mixing patterns.

From this we learn that, after the shifts of ${\cal O}(\lambda)$ provided by
$V_{CKM}$, negative corrections are needed for all patterns to jump into the
1$\sigma$ allowed range for all mixing angles. It is worth to mention that, if
3$\sigma$ allowed ranges for the atmospheric mixing angle and the Jarlskog
invariant are taken into account, the BM scenario is compatible with
experimental data. Indeed, both $\sin\theta_{23}\sim 1$ and $J_{\text{CP}}\sim
0$ are not yet excluded by neutrino experiments [34].

In the next section we will analyze, in a systematic way, which corrections of
$U_{ij}$ in eq.(1.4) are the most appropriate to better fit the neutrino
mixing parameters.

###  2.2 Corrections from the $(13)$-sector to BM, TBM and GR

We start our analysis by studying in detail the correction to the standard
patterns from the $(13)$-sector. The main idea is that, given the absence of
any CP phase in (2.10), eq.(2.7) implies a very low CP violation in the lepton
sector [36], of the order of ${\cal O}(\lambda^{3})$ and proportional to
$\eta$ as shown from the expressions of $J_{\text{CP}}$ in Tab.(2). Thus, to
allow for a larger CP violation, which seems to be preferred by recent
oscillation results, new sources of symmetry violation are needed. Assuming
for $T$ the decomposition as in eq.(1.4), larger CP violation can be generated
by slightly shifting the $(13)$-rotation from the identity; to this aim, we
introduce a complex parameter $u$ [37] such that $|u|\ll 1$ and we rescale it
by one power of the Cabibbo angle $\lambda$. This also implies that the
rescaled $|u|\sim{\cal O}(1)$. Thus, the $(13)$-rotation has the following
structure:

$U_{13}=\begin{pmatrix}1-\frac{\lambda^{2}}{2}|u|^{2}&0&u\lambda\\\ 0&1&0\\\
-u^{*}\lambda&0&1-\frac{\lambda^{2}}{2}|u|^{2}\end{pmatrix}\,.$ (2.13)

To construct completely the matrix $T$, we need to specify the rotations in
the other two sectors, the $(12)$ and $(23)$-rotations. In order to
contemplate the BM, TBM and GR mixings simultaneously, we leave unspecified
the rotation in the $(12)$-sector and, since the sign of such a rotation is
not fixed a priori, we leave it as free, encoding this uncertainty into the
parameter $\sigma$, that can assume values $\pm 1$. At this stage, the
rotation in the $(23)$-sector is maximal (so, from our ansatz, we expect all
deviations to $\theta_{23}$ coming from $V_{CKM}$, see below). Thus, we have:

$U_{23}=\begin{pmatrix}1&0&0\\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}\,,\qquad
U_{12}=\begin{pmatrix}\tilde{c}_{12}&\sigma\tilde{s}_{12}&0\\\
-\sigma\tilde{s}_{12}&\tilde{c}_{12}&0\\\ 0&0&1\end{pmatrix}\,,$ (2.14)

where
$\tilde{c}_{12}\equiv\cos(\tilde{\theta}_{12}),\tilde{s}_{12}\equiv\sin(\tilde{\theta}_{12})$
are the cosinus and sinus functions of a rotation in the (12)-sector (not to
be confused with the usual solar angle). This, in turn, implies the following
structure of the $T$ matrix:

$T\equiv
U_{23}U_{13}U_{12}=\begin{pmatrix}\tilde{c}_{12}\left(1-\frac{\lambda^{2}}{2}|u|^{2}\right)&\sigma\tilde{s}_{12}\left(1-\frac{\lambda^{2}}{2}|u|^{2}\right)&u\lambda\\\
-(\sigma\tilde{s}_{12}+\tilde{c}_{12}u^{*}\lambda)/\sqrt{2}&(\tilde{c}_{12}-\sigma\tilde{s}_{12}u^{*}\lambda)/\sqrt{2}&\left(1-\frac{\lambda^{2}}{2}|u|^{2}\right)/\sqrt{2}\\\
(\sigma\tilde{s}_{12}-\tilde{c}_{12}u^{*}\lambda)/\sqrt{2}&-(\tilde{c}_{12}+\sigma\tilde{s}_{12}u^{*}\lambda)/\sqrt{2}&\left(1-\frac{\lambda^{2}}{2}|u|^{2}\right)/\sqrt{2}\end{pmatrix}\,.$
(2.15)

Notice that unitarity is fully respected up to ${\cal O}(\lambda^{3})$. With
our parametrization, the relevant patterns are recovered once we fix $u=0$
(for all of them) and $\tilde{s}_{12}=1/\sqrt{3},\tilde{s}_{12}=1/\sqrt{2}$
and $\tilde{s}_{12}^{2}=2/(5+\sqrt{5})$ for TBM, BM and GR, respectively (at
this stage, the value of $\sigma$ is irrelevant). For the Jarlskog invariant
$J_{\text{C}P}$, up to ${\cal O}(\lambda^{3})$, we get the expression as
below:

$\displaystyle
J_{\text{C}P}=\frac{\lambda}{4}\,\sigma\,\text{Im}\,{(u)}\,\sin(2\tilde{\theta}_{12})+\frac{\lambda^{2}}{2\sqrt{2}}\,\text{Im}\,{(u)}\cos(2\tilde{\theta}_{12})+$
$\displaystyle-\frac{\lambda^{3}}{8}\sigma\sin(2\tilde{\theta}_{12})\left[\sqrt{2}A\eta+2\text{Im}\,(u)\left(2+|u|^{2}+\sqrt{2}\text{Re}\,(u)\right)\right]\,.$
(2.16)

Some comments are in order:

  * •

in the limit of exact TBM, BM and GR, the invariant $J$ reduces to:

$\displaystyle
J^{\text{TBM}}_{\text{CP}}=-\frac{A\eta\lambda^{3}\sigma}{6}\,,\qquad
J^{\text{BM}}_{\text{CP}}=-\frac{A\eta\lambda^{3}\sigma}{4\sqrt{2}}\,,\qquad\qquad
J^{\text{GR}}_{\text{CP}}=-\frac{A\eta\lambda^{3}\sigma}{2\sqrt{10}}\,,$
(2.17)

which all lead to a suppressed CP violation in the lepton sector, in agreement
with Tab.(3) for an appropriate choice of $\sigma$;

  * •

retaining terms proportional to $\text{Re}\,(u)$ (and setting
$\text{Im}\,(u)=0$) does not cure the previous problem since they appear only
to ${\cal O}(\lambda^{3})$;

  * •

to reconcile our prediction with the experimental value, we need to allow a
deviation from exact TBM, BM and GR forms provided by $\text{Im}\,(u)$. The
${\cal O}(\lambda)$ degeneracy between $\sigma$ and $\text{Im}\,(u)$ will
allow the latter to assume both positive and negative values.

To find the set of values of $\text{Re}\,(u),\text{Im}\,(u)$ that allows to
reproduce the best fit point of $J_{\text{CP}}$ ($J^{bf}_{\text{CP}}$), in
Fig.(2) we plot the ensemble of $u$ values which makes the modified versions
of TBM (black solid line), BM (red dashed line) and GR (blue dot-dashed line)
compatible with $J^{\text{bf}}_{\text{CP}}$ at 1$\sigma$, subject to the
constraint $|u|<1$. Given the similarities in the analytical structure of TBM,
BM and GR, we see that overlapping complex $u$-region are covered; in
addition, as commented above, we also expect a less relevant dependence on
$\text{Re}\,(u)$ compared to $\text{Im}\,(u)$.

Figure 2: Ensemble of $u$ values which make the modified versions of TBM
(black solid line), BM (red dashed line) and GR (blue dot-dashed line)
compatible with $J^{bf}_{\text{CP}}$ at 1$\sigma$. The upper (lower) plots
show the solutions obtained with $\sigma=-1$ ($\sigma=+1$).

The main conclusion is that $|\text{Im}\,(u)|\gtrsim 0.3$ (and almost any
$\text{Re}\,(u)$ in the $[-1,1]$ range) is enough to get the correct amount of
leptonic CP violation, for any choice of the starting matrix. The mild
$\text{Re}\,(u),\text{Im}\,(u)$ correlation is mainly dictated by the
constraint $|u|<1$.

Now we go for the expressions of the mixing angles. For the reactor angle we
get a formula which is independent on the $\tilde{\theta}_{12}$ parameter (and
thus on the sign of $\sigma$) up to ${\cal O}(\lambda^{3})$ terms (notice that
${\cal O}(\lambda^{2})$ terms vanish):

$\displaystyle\sin(\theta_{13})$ $\displaystyle=$
$\displaystyle\sqrt{1/2+|u|^{2}+\sqrt{2}\text{Re}\,(u)}\,\lambda+$ (2.18)
$\displaystyle\frac{\left[2\sqrt{2}A\rho-4A\eta\text{Im}\,{u}+(-2+4A\rho)\text{Re}\,{u}-|u|^{2}(3\sqrt{2}+2\text{Re}\,(u))\right]}{4\sqrt{1+2|u|^{2}+2\sqrt{2}\text{Re}\,(u)}}\,\lambda^{3}\,.$

In the limit of exact TBM, BM and GR mixing ($u=0$), we recover the well-known
relation $\sin(\theta_{13})=\frac{\lambda}{\sqrt{2}}+{\cal O}(\lambda^{3})$ ,
which is still a good approximation, see also Tab.(3). Moreover, eq.(2.18)
shows that, barring accidental cancellations, negative $\text{Re}\,(u)$ values
are needed to compensate for positive shifts driven by $|u|$ (unless
$\text{Im}\,(u)$ is also small, in that case small positive values of
$\text{Re}\,(u)$ are also allowed).

Not too much must be said for the atmospheric angle; up to ${\cal
O}(\lambda^{3})$ we get:

$\displaystyle\tan(\theta_{23})$ $\displaystyle=$ $\displaystyle
1+\frac{\lambda^{2}}{2}\left[-1+4A-2\sqrt{2}\text{Re}\,(u)\right]\,.$ (2.19)

The most interesting feature is the absence of any dependence on
$\text{Im}\,(u)$; thus, the small deviations from maximality are governed,
beside the Cabibbo angle, by $\text{Re}\,(u)$ only. We also have to mention
that the current best fit point is away from maximal mixing at the level of
3$\sigma$, see Tab.(2). Thus, relatively large positive $\text{Re}\,(u)$ are
needed to shift $\tan(\theta_{23})$ towards its 1$\sigma$ preferred value
which lies around $\tan(\theta_{23})^{bf}\sim 0.9$. As in the previous case,
no dependence on $\tilde{\theta}_{12}$ appears so that exact TBM, BM and GR
hypothesis give the same expression in eq.(2.19) with $\text{Re}\,(u)=0$.

Finally, for the solar angle we get:

$\displaystyle\tan(\theta_{12})$ $\displaystyle=$
$\displaystyle\tan(\tilde{\theta}_{12})+\frac{\lambda}{\sqrt{2}\tilde{c}^{2}_{12}}\sigma+\frac{\lambda^{2}}{2\tilde{c}^{3}_{12}}\tilde{s}_{12}+$
(2.20) $\displaystyle+$
$\displaystyle\frac{\lambda^{3}}{4\tilde{c}^{4}_{12}}\sigma\,\left[\sqrt{2}(1-2A\tilde{c}_{12}^{2}\rho)+\tilde{c}_{12}^{2}(\sqrt{2}|u|^{2}+2\text{Re}\,(u))\right]\,.$

The most considerable feature is that the corrections implied by $U_{13}$ of
eq.(2.13) are too small to be significant; thus, the expressions of
$\theta_{12}$ are very similar to those quoted in Tab.(3). In addition, once
we specify the values of $\tilde{\theta}_{12}$ for the relevant patterns,
there are no free parameters up to ${\cal O}(\lambda^{2})$; we can then derive
the following sum-rules among physical angles (that, for the sake of
simplicity, we report here up to first order in $\sin(\theta_{13})$):

$\tan(\theta_{12})=\begin{cases}\frac{1}{\sqrt{2}}+3\sigma\,\sin(\theta_{13})/2\qquad{\rm
for\;TBM}\\\ 1+2\sigma\sin(\theta_{13})\;\;\qquad{\rm for\;BM}\\\
2\sqrt{5}/(5+\sqrt{5})+10\sigma\sin(\theta_{13})/(5+\sqrt{5})\qquad{\rm
for\;GR}\,.\end{cases}$ (2.21)

The only possibility to (marginally) reconcile the previous sum rules with the
experimental value happens for BM mixing with $\sigma=-1$, which shows a
deviation from $\tan(\theta_{12})^{bf}$ at around $\sim 3\%$ (compare with
Fig.(1)); for the other mixing patterns, this difference amounts to values as
large as $\sim$20% for GR and $\sim$30% for TBM. To better quantify the
(dis-)agreements of the obtained $U_{\text{PMNS}}$ with the experimental data
after including the corrections in eq.(2.13), we perform a simple $\chi^{2}$
test, with the function:

$\chi^{2}=\frac{\left[J_{\text{CP}}-J_{\text{CP}}^{bf}\right]^{2}}{\sigma_{J_{\text{CP}}}^{2}}+\frac{\left[\sin(\theta_{13})-\sin(\theta_{13})^{bf}\right]^{2}}{\sigma_{\sin(\theta_{13})}^{2}}+\frac{\left[\tan(\theta_{23})-\tan(\theta_{23})^{bf}\right]^{2}}{\sigma_{\tan(\theta_{23})}^{2}}+\frac{\left[\tan(\theta_{12})-\tan(\theta_{12})^{bf}\right]^{2}}{\sigma_{\tan(\theta_{12})}^{2}}$

(2.22)

For all patterns, the minimum of the $\chi^{2}$ is very large, in the range
$(10-10^{3})$ and it is dominated by the $\tan(\theta_{12})$ term; in fact, if
we exclude $\theta_{12}$ from the $\chi^{2}$ function, the fit improves
considerably for all patterns, with $\chi^{2}_{min}\sim{\cal O}(20)$ (the best
performance being obtained by BM mixing with $\sigma=-1$). The problem related
to the deviation from maximality of $\theta_{23}$ is, instead, less relevant
because of a larger relative 1$\sigma$ error compared to $\theta_{12}$.
Finally, the corrections analyzed here help in improving the values of
$\Delta$, $\Delta=(2708,2492,2573)$ for TBM, BM and GR mixings, respectively.
Obviously, assuming for the variable $u$ a smaller value, that is shifting
$u\to\lambda^{N}\,u$, does not solve the problem for any integer $N$.

###  2.3 Perturbation on the (23)- sector

One possibility to alleviate the problem in the (23)-sector is to slightly
modify $U_{23}$ of eq.(1.4) by inserting a new real parameter $\omega$
according to333Since the complex variable $u$ was already enough to guarantee
the correct amount of leptonic CP violation, we prefer to reduce the number of
free parameters choosing a real correction $\omega$.:

$U_{23}=\begin{pmatrix}1&0&0\\\
0&\frac{1}{\sqrt{2}}-\lambda\omega-\sqrt{2}\lambda^{2}\omega^{2}-2\lambda^{3}\omega^{3}&\frac{1}{\sqrt{2}}+\omega\lambda\\\
0&-\frac{1}{\sqrt{2}}-\omega\lambda&\frac{1}{\sqrt{2}}-\lambda\omega-\sqrt{2}\lambda^{2}\omega^{2}-2\lambda^{3}\omega^{3}\end{pmatrix}\,.$
(2.23)

Notice that, to maintain the unitarity of $U_{23}$, we displayed up to ${\cal
O}(\lambda^{3})$ terms. We repeat the same calculations as before and indicate
with a prime the new expressions of the mixing parameters while leaving
unprimed the results of the previous section. The relevant corrections driven
by $\omega$ are as follows:

$\displaystyle J_{\text{CP}}^{\prime}$ $\displaystyle=$ $\displaystyle
J_{\text{CP}}-\frac{\lambda^{3}}{2}\,\text{Im}\,(u)\,\left[\omega\cos\left(2\tilde{\theta}_{12}\right)+2\sigma\omega^{2}\sin\left(2\tilde{\theta}_{12}\right)\right]$
$\displaystyle\sin^{\prime}(\theta_{13})$ $\displaystyle=$
$\displaystyle\sin(\theta_{13})+\frac{\lambda^{2}\omega}{\sqrt{2}}\,\frac{\left[\sqrt{2}+2\text{Re}\,(u)\right]}{\sqrt{1+2|u|^{2}+2\sqrt{2}\text{Re}\,(u)}}$
$\displaystyle\tan^{\prime}(\theta_{23})$ $\displaystyle=$
$\displaystyle\tan(\theta_{23})+2\sqrt{2}\lambda\omega$ (2.24)
$\displaystyle\tan^{\prime}(\theta_{12})$ $\displaystyle=$
$\displaystyle\tan(\theta_{12})-\frac{\lambda^{2}\omega\sigma}{\cos^{2}\left(\tilde{\theta}_{12}\right)}\,.$

We see that $J_{\text{CP}},\theta_{12}$ and $\theta_{13}$ acquire small ${\cal
O}(\lambda^{2-3})$ corrections that do not improve the fit compared to the
previous section. For the atmospheric angle, instead, an ${\cal O}(\lambda)$
is relevant, especially for negative values of $\omega$ as, starting from
maximality, we need a negative correction to jump into the experimental
value444The three main neutrino global fits [33, 38, 39] do not agree on the
preferred $\theta_{23}$ octant, even though the $3\sigma$ ranges are all
compatible. In our analysis, an higher octant value for $\theta_{23}$ can be
easily obtained with a positive $\omega$ value.. Notice that this is true for
any value of $\sigma$. However, even though the atmospheric angle turns out to
be in the correct range, the fits to the expressions in eq.(2.24) are only
slightly improved but still remain $\gtrsim{\cal O}(100)$ because of the poor
foreseen solar angle; as before, only the modified BM mixing case presents a
good minimum of the $\chi^{2}$ at $\chi^{2}_{min}=3.47$. For the sake of
illustration, the behaviour of the $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{min}$ as
a function of $\omega$ is presented in Fig.(3). For every $\omega$, we have
marginalized over $\text{Re}\,(u)$ and $\text{Im}\,(u)$ in the fit.

Figure 3: $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{min}$ behaviour as a function of
$\omega$ for the modified BM mixing. In the fit procedure, we have
marginalized over the $(\text{Re}\,(u),\text{Im}\,(u)))$ pair.

###  2.4 The full glory: perturbation on the (12)- sector

The results of the previous sections have shown that the predictions for
$J_{\text{CP}}$ and $\sin(\theta_{13})$ are good for all mixing once the
$u$-corrections are included. The $\omega$ corrections are needed to reconcile
the $\theta_{23}$ deviations from maximal mixing (common to all patterns)
while the solar angle remains sensitively away from its experimental value for
TBM and GR mixing but sufficiently close to it for BM. Thus, in order to
complete our program to match the data of Tab.(2), we need to add a (real)
correction of ${\cal O}(\lambda)$ to the (12)-sector, that we dub with $z$. We
parameterize it in the following way:

$U_{12}=\begin{pmatrix}K&\tilde{s}_{12}\sigma+z\lambda&0\\\
-\tilde{s}_{12}\sigma-z\lambda&K&0\\\ 0&0&1\end{pmatrix}\,,$ (2.25)

where
$K=\tilde{c}_{12}-\tilde{s}_{12}z\sigma/\tilde{c}_{12}\lambda-z^{2}/(2\tilde{c}_{12}^{3})\lambda^{2}-\tilde{s}_{12}z^{3}\sigma/(2\tilde{c}_{12}^{5})\lambda^{3}$.
The expression of the mixing parameters are modified accordingly; in
particular, $\theta_{13}$ and $\theta_{23}$ are unaffected by $z$, so their
expressions of eq.(2.24) are valid even in this case. The Jarlskog invariant
gets an ${\cal O}(\lambda^{2})$ correction of the form:

$J_{\text{CP}}^{\prime\prime}=J_{\text{CP}}^{\prime}+\frac{\lambda^{2}}{2}\,\frac{\cos(2\tilde{\theta}_{12})\text{Im}\,(u)}{\cos(\tilde{\theta}_{12})}\,z\,.$
(2.26)

By construction, the most interesting case is related to $\theta_{12}$; here,
corrections of ${\cal O}(\lambda)$ driven by $z$ compete with that shown in
eq.(2.20):

$\displaystyle\tan^{\prime\prime}(\theta_{12})$ $\displaystyle=$
$\displaystyle\tan(\theta_{12})^{\prime}+\lambda\,\frac{\sigma\,z}{\cos^{3}\left(\tilde{\theta}_{12}\right)}\,.$
(2.27)

Thus, we expect that a cancellation among the $\lambda$ coefficients could
bring the TBM and GR mixing in agreement with the data (for any $\sigma$)
while for BM the contribution from $z$ (and $\sigma=-1$) must be small in
order not to destroy the agreement found above; conversely, we expect that
$\sigma=1$ will be acceptable for non-vanishing $z$ corrections. To check
whether this is the case, we minimized the $\chi^{2}$ function of eq.(2.22)
over the four independent parameters $\text{Re}\,(u),\text{Im}\,(u),\omega$
and $z$ and reported their best fit values in Tab.(4).

Pattern | $\text{Re}\,(u)$ | $\text{Im}\,(u)$ | $\omega$ | $z$
---|---|---|---|---
TBM | -0.27(-0.27) | 0.57(-0.55) | -0.27(-0.27) | -0.50(-0.77)
BM | -0.27(-0.29) | 0.57(-0.56) | -0.27(-0.27) | 0.08(-1.17)
GR | -0.27(-0.27) | 0.57(-0.54) | -0.27(-0.27) | -0.73(-0.55)
Table 4: Values of the parameters $\text{Re}\,(u),\text{Im}\,(u),\omega$ and
$z$ that minimize the $\chi^{2}$ function of eq.(2.22), computed for
$\sigma=-1$ and, in parenthesis, for $\sigma=1$. For all patterns,
$\chi^{2}_{min}\sim 0$.

For all patterns, the minimum of the $\chi^{2}$ is very close to zero, so we
did not report it on the table. As expected, the magnitude and signs of the
needed $z^{\prime}s$ reflects our considerations below eq.(2.27). In addition,
the very similar values for $\text{Re}\,(u)$ and $\text{Im}\,(u)$ can be
understood from Fig.(2), where the acceptable regions for such parameters are
almost equivalent for each pattern. Finally, compared to the previous section,
the value of $\omega$ is compatible with the BM case previously analyzed and,
as expected, tends to assume a very similar strength for all other patterns
and signs of $\sigma$ (${\cal O}(\lambda)$ corrections are universal). The 90%
and 99% confidence levels of the $\chi^{2}$ function in the $(\omega,z)$-plane
for TBM (left panel), BM (middle panel) and GR (right panel) are reported in
Fig.(4); in each plots we included both $\pm 1$ possibilities for $\sigma$ and
marginalized over the $(\text{Re}\,(u),\text{Im}\,(u)))$ pair.

Figure 4: The 90% and 99% $\chi^{2}$ confidence levels in the
$(\omega,z)$-plane for TBM (left panel), BM (middle panel) and GR (right
panel). In each plots, we have reported both $\pm 1$ possibilities for
$\sigma$ and marginalized over the $(\text{Re}\,(u),\text{Im}\,(u)))$ pair.

##  3 On the neutrino masses

The next step is to ensure that our procedure is able to reproduce the solar
and atmospheric mass differences. Eq.(2.1) offers the structure of the
neutrino mixing matrix in terms of a right rotation $U_{R}$ (four real
parameters), three right-handed neutrino masses and three Dirac neutrino
masses, for a total of ten unknown parameters; of those, four have been used
to constrain the matrix $T$ in eq.(2.2), and the remaining six parameters are
left to describe neutrino masses. To determine them, one can try to figure out
the structure of the diagonal matrix $S$ by inverting eq.(2.2), so that:

$\displaystyle
S=T^{\dagger}m^{diag}_{\nu_{D}}\,U_{R}^{\dagger}\,(M_{R}^{diag})^{-1}\,U_{R}^{\star}m^{diag}_{\nu_{D}}T^{\star}\,.$
(3.1)

Notice that the matrix $S\equiv m_{\nu}^{diag}$ does not depend on the quark
mixing. One possibility to determine the unknown parameters is to rephrase
eq.(3.1) to the more useful form:

$\displaystyle S-T^{\dagger}m_{0}T^{\star}=0\,.$ (3.2)

Its left-hand side is a symmetric matrix made complex by the entries of
$m^{diag}_{\nu}$ and by the $T$ matrix, needed to successfully reproduce the
leptonic CP violation. Thus, eq.(3.2) is equivalent to 12 conditions, which
have to be simultaneously valid. However, we can easily verify that the
imaginary parts of the elements of $T$ are always smaller than the real part
(at the level of 20% or smaller) with a notable exception of the element (13),
for which the imaginary part is either larger (in the only case when $T$ is
the corrected BM mixing with $\sigma=-1$) or just half of the real part. With
the aim of catching the relevant physics, not obfuscated by useless details
(phases are of the uttermost importance for CP violation, not for neutrino
masses), we prefer to deal with real $S$ and $T$ matrices; this allows us to
reduce the number of constraints to six only555If, instead, we prefer to deal
with complex matrices, thus phases must be added to $U_{R}$ that helps in
making vanishing all imaginary parts of eq.(3.2).. Even in this case, the
large number of free parameters makes the expressions of neutrino masses quite
cumbersome. Thus, we only give a numerical solution to eq.(3.2). For the $S$
matrix we take the following expression, valid for the Normal Ordering (NO)
case:

$S=\text{diag}(m_{1},\sqrt{m_{1}^{2}+\Delta
m^{2}_{\text{s}ol}},\sqrt{m_{1}^{2}+\Delta m^{2}_{\text{a}tm}})\,,$ (3.3)

where $m_{1}$ is the absolute neutrino mass scale that, for the sake of
simplicity, we assume vanishing. We then construct the adimensional function:

$\displaystyle
F(\vec{m}_{\nu_{D}},\vec{M}_{R},\vec{\theta}_{R})=\frac{\sum_{j<i=1}^{3}\left[S_{ij}-(T^{T}m_{0}T)_{ij}\right]^{2}}{\Delta
m^{2}_{\text{sol}}}\,,$ (3.4)

and look for minima as close as possible to zero. Here the vectors have the
following entries, with obvious meaning:

$\displaystyle\vec{m}_{\nu_{D}}=\left(m_{\nu_{D_{1}}},m_{\nu_{D_{2}}},m_{\nu_{D_{3}}}\right)\qquad\vec{M}_{R}=\left(M_{R_{1}},M_{R_{2}},M_{R_{3}}\right)\qquad\vec{\theta}_{R}=\left(\theta_{R_{12}},\theta_{R_{13}},\theta_{R_{23}}\right)\,.$
(3.5)

We consider ourselves satisfied when
$F(\vec{m}_{\nu_{D}},\vec{M}_{R},\vec{\theta}_{R})<1$, meaning that all the
differences between the corresponding matrix elements of $S$ and $T^{T}m_{0}T$
are smaller than the smallest measured mass scale $\Delta m^{2}_{\text{sol}}$.
The minimization procedure has been carried out by means of the software
MultiNest, which is based on nested sampling normally used for calculation of
the Bayesian evidence [40, 41, 42]. The choice of priors in this context is
relevant. To prove that a solution to the system (3.2) exists, we set:

$\displaystyle 1\leq m_{\nu_{D_{1}}}/{\text{G}eV}$ $\displaystyle<10\,,\qquad$
$\displaystyle 10\leq m_{\nu_{D_{2}}}/{\text{G}eV}<100\,,\qquad 100\leq
m_{\nu_{D_{3}}}/{\text{G}eV}$ $\displaystyle<500\,,$ $\displaystyle
10^{13}\leq M_{R_{1}}/{\text{G}eV}$ $\displaystyle<10^{14}\,,\qquad$
$\displaystyle 10^{14}\leq M_{R_{2}}/{\text{G}eV}<10^{15}\,,\qquad 10^{15}\leq
M_{R_{3}}/{\text{G}eV}$ $\displaystyle<10^{16}\,,$ (3.6)
$\displaystyle\vec{\theta}_{R}\in[0,2\pi)\,.\phantom{\qquad 100\leq
m_{\nu_{D_{3}}}/{\text{G}eV}}$

Notice that, being the neutrino masses given by complicated expressions of
parameters, the position $m_{\nu_{D_{1}}}<m_{\nu_{D_{2}}}<m_{\nu_{D_{3}}}$ and
$M_{R_{1}}<M_{R_{2}}<M_{R_{3}}$ does not correspond a priori to a definite
mass hierarchy, as it would be the case for a standard see-saw mechanisms
where, for example, $m_{i}\sim m^{2}_{\nu_{D_{i}}}/M_{R_{i}}$ for NO. We have
analyzed the 6 different cases corresponding to modified BM, TBM and GR and
the two values of $\sigma=\pm 1$; for each texture, we reported in Tab.(5) the
minimum of $F(\vec{m}_{\nu_{D}},\vec{M}_{R},\vec{\theta}_{R})$ and the values
of the vectors $\vec{m}_{\nu_{D}},\vec{M}_{R}$ and $\vec{\theta}_{R}$ in which
the minimum is assumed. We also report in Fig.(5) an example of posterior
distributions for the BM case, $\sigma=-1$ (all cases are very similar to each
other).

|  | $F^{\text{m}in}$ |  $\vec{m}_{\nu_{D}}$ (GeV) |  $\vec{M}_{R}$ ($10^{13}$ GeV) |  $\vec{\theta}_{R}$ (∘)
---|---|---|---|---|---
| $\sigma=+1$ | 0.42 | (9.35, 55.35, 117.70) | (4.0, 70.10, 297.51) | (145.80, 195.44, 162.01)
BM | $\sigma=-1$ | 0.44 | (9.98, 36.84, 111.52) | (3.59, 67.78, 708.16) | (122.88, 12.88, 112.59)
| $\sigma=+1$ | 0.31 | (9.38, 58.54, 130.56) | (3.41, 93.21, 794.35) | (331.00, 347.73, 317.70)
TBM | $\sigma=-1$ | 0.34 | (9.66, 34.52, 159.29) | (2.09, 55.69, 485.30) | (321.58, 354.03, 221.83)
| $\sigma=+1$ | 0.19 | (9.83, 80.88, 201.77) | (3.55, 86.014, 728.73) | (341.65, 171.70, 188.76)
GR | $\sigma=-1$ | 0.45 | (9.26, 42.48, 156.88) | (2.97, 38.95, 268.45) | (145.00, 8.15, 341.63)
Table 5: Results of the minimization procedure of the function
$F(\vec{m}_{\nu_{D}},\vec{M}_{R},\vec{\theta}_{R})$ in eq.(3.4). $F^{min}$
stands for the minimum value of such a function; the meaning of the three
vectors $\vec{m}_{\nu_{D}}$, $\vec{M}_{R}$ and $\vec{\theta}_{R}$ has been
given in eq.(3.5).

Let us analyze more in detail the results of our minimizing procedure. First
of all, none of the analyzed patterns can be tagged as a preferred one, as the
minima of the $F$ function are very close to each other. This is in agreement
with what we found for the mixing angles where, after including all relevant
corrections, no preferred choice emerged. The vector $\vec{m}_{\nu_{D}}$ is
characterized by the fact that the first and third element prefer values at
their upper and lower limits, respectively while $m_{\nu_{D_{2}}}$ is
generally confined in the central region (with an exception for the case TBM,
$\sigma=+1$ which, instead, prefers larger values). As for the Majorana
masses, we observe similarities in all elements among the different patterns:
$M_{R_{1}}$ and $M_{R_{3}}$ tend to stay close to their allowed lower and
upper bounds, respectively, while $M_{R_{2}}$ is mostly concentrated in the
middle region around $[40-90]\cdot 10^{13}$ GeV. It is interesting to observe
that the posterior distributions (middle panels of Fig.(5)) are almost flat
for $M_{R_{3}}$ but peaked at large allowed values for $M_{R_{1}}$ and
$M_{R_{2}}$; while for the latter case this seems consistent with the values
at the minimum of $F$, for the former this behavior does not completely match
what reported in Tab.(5). We interpret this as that $M_{R_{1}}$ gives a
smaller contribution to $F$ as the other Majorana masses. This happens also
for the first Dirac neutrino mass $m_{\nu_{D_{1}}}$, whose best fit value is
close to its upper limit while the posterior distribution is essentially flat.
Finally, a look at Fig.(5) reveals that the posterior distributions for the
mixing angles are multi-modal; in particular, a clear bi-modal distribution is
seen for $\theta_{R_{12}}$, around $|\sin(\theta_{R_{12}})|\sim 1/2$, and for
$\theta_{R_{13}}$ around $|\sin(\theta_{R_{13}})|\sim 0$; this is also visible
in Tab.(5). A less clear bi-modal behaviour is also present for
$\theta_{R_{23}}$ but the spreads around the maximum posterior probability are
not negligible. Assuming the fixed values $\sin(\theta_{R_{12}})=1/2$ and
$\sin(\theta_{R_{13}})=0$, the right-handed rotation implied by our fit is as
follows:

$U_{R}=\begin{pmatrix}\sqrt{3}/2&1/2&0\\\ -c_{23}/2&\sqrt{3}c_{23}/2&s_{23}\\\
s_{23}/2&-\sqrt{3}s_{23}/2&c_{23}\end{pmatrix}\,.$ (3.7)

Figure 5: Posterior distributions for the elements of the vectors
$\vec{m}_{\nu_{D}}$ (first line) $\vec{M}_{R}$ (middle line) and
$\vec{\theta}_{R}$ (lower line) for the BM case, $\sigma=-1$. Darker and
lighter blue refer to 68% and 95% credible intervals, respectively.

##  4 Conclusions

In this paper we have investigated in detail the hypothesis that the PMNS
mixing matrix is given by the relation $U_{PMNS}=V_{CKM}^{\star}\,T^{\star}$,
where $T$ is a unitary matrix. By considering the decomposition $T\equiv
U_{23}U_{13}U_{12}$, we have shown that a $T$ matrix coinciding with TBM, BM
and GR mixing fails, among others, to reproduce the experimental preferred
value of the Jarlskog invariant, which is related to the third power of the
Cabibbo angle. To solve these issues, we have analyzed ${\cal O}(\lambda)$
corrections to the $U_{ij}$ matrices, showing that a complex parameter $u$ is
needed in the (13) rotation to reconcile our ansatz with the experimental
amount of leptonic CP violation. While a correction $\omega$ in the (23)
sector is needed for a substantial deviation of the atmospheric angle from
maximality, it only marginally improves the global fit to the experimental
values of the mixing angles, because of a wrong estimate of $\theta_{12}$ in
all cases but BM. Thus, a shift in the (12) plane is mandatory to account for
the solar angle and, consequently, to get an excellent fit for all mixing
parameters and for any initial choice of $T$. The ansatz illustrated here is
also appropriate to reproduce the value of solar and atmospheric mass
differences. Indeed, equipped with the best fit values of the
$\text{Re}\,(u),\text{Im}\,(u),\omega$ and $z$ parameters, we have shown that
a description of neutrino masses via the see-saw mechanism is possible.
Because of the cumbersome analytical expressions of $\Delta m^{2}_{sol,atm}$,
we relied on numerical scan of the vector components of $\vec{m}_{\nu_{D}}$,
$\vec{M}_{R}$ and $\vec{\theta}_{R}$ of eq.(3.5) and found that, with our
choice of priors, a complete description of neutrino masses and mixing under
the assumption $U_{PMNS}=V_{CKM}^{\star}\,T^{\star}$ is possible.

## Acknowledgments

We thank João Penedo and Matteo Parriciatu for useful comments and suggestions
on our manuscript.

## Appendix: Full ${\cal O}(\lambda^{3})$ formulae

For the sake of completeness, we report here the full ${\cal O}(\lambda^{3})$
expressions of the mixing parameters obtained from our ansatz
$U_{PMNS}=V_{CKM}^{\star}\,T^{\star}$.

$\displaystyle
J=\frac{\lambda}{4}\,\sigma\,\text{Im}\,{(u)}\,\sin(2\tilde{\theta}_{12})+\lambda^{2}\frac{\text{Im}\,(u)\cos^{2}(\tilde{\theta}_{12})}{4\cos(\tilde{\theta}_{12})}\left(\sqrt{2}\cos(\tilde{\theta}_{12})+2z\right)+$
$\displaystyle-\frac{\lambda^{3}}{8}\sigma\sin(2\tilde{\theta}_{12})\left[\sqrt{2}A\eta+2\text{Im}\,(u)\left(2+|u|^{2}+\sqrt{2}\text{Re}\,(u)\right)\right]+$
(4.1)
$\displaystyle-\frac{\lambda^{3}}{2}\,\text{Im}\,(u)\,\left[\omega\cos\left(2\tilde{\theta}_{12}\right)+2\sigma\omega^{2}\sin\left(2\tilde{\theta}_{12}\right)\right]+$
$\displaystyle-\lambda^{3}z\sigma\left\\{-8\sqrt{2}\sin(\tilde{\theta}_{12})\text{Im}\,(u)+\frac{2z\text{Im}\,(u)\sin(\tilde{\theta}_{12})}{\cos(\tilde{\theta}_{12})^{3}}\left[3\cos^{2}(\tilde{\theta}_{12})+\sin^{2}(\tilde{\theta}_{12})\right]\right\\}$

$\displaystyle\sin(\theta_{13})$ $\displaystyle=$
$\displaystyle\sqrt{1/2+|u|^{2}+\sqrt{2}\text{Re}\,(u)}\,\lambda+$
$\displaystyle+\frac{\lambda^{2}\omega}{\sqrt{2}}\,\frac{\left[\sqrt{2}+2\text{Re}\,(u)\right]}{\sqrt{1+2|u|^{2}+2\sqrt{2}\text{Re}\,(u)}}+$
$\displaystyle+\lambda^{3}\,\frac{\left[2\sqrt{2}A\rho-4A\eta\text{Im}\,{u}+(-2+4A\rho)\text{Re}\,{u}-|u|^{2}(3\sqrt{2}+2\text{Re}\,(u))\right]}{4\sqrt{1+2|u|^{2}+2\sqrt{2}\text{Re}\,(u)}}+\,$
$\displaystyle+\lambda^{3}\,\omega^{2}\,\frac{\sqrt{2}\text{Im}\,^{2}(u)}{\left[2\text{Re}\,(u)\left(\text{Re}\,(u)+\sqrt{2}\right)+2\text{Im}\,^{2}(u)+1\right]^{3/2}}$

$\displaystyle\tan(\theta_{23})$ $\displaystyle=$ $\displaystyle
1+2\sqrt{2}\lambda\omega+\frac{\lambda^{2}}{2}\left[-1+4A-2\sqrt{2}\text{Re}\,(u)\right]+\lambda^{3}\omega\left(4\sqrt{2}A-2\text{Re}\,(u)+12\sqrt{2}\omega^{2}-\sqrt{2}\right)$
$\displaystyle\,.$ (4.3)

$\displaystyle\tan(\theta_{12})$ $\displaystyle=$
$\displaystyle\tan(\tilde{\theta}_{12})+\frac{\lambda\,\sigma\left(\sqrt{2}\tilde{c}_{12}+2z\right)}{2\tilde{c}_{12}^{3}}+\frac{\lambda^{2}}{2\tilde{c}^{3}_{12}}\tilde{s}_{12}-\frac{\lambda^{2}\omega\sigma}{\tilde{c}^{2}_{12}}+\frac{\lambda^{2}\,z\sqrt{2}\tilde{s}_{12}}{\tilde{c}^{4}_{12}}+\frac{3\lambda^{2}z^{2}\tilde{s}_{12}}{2\tilde{c}^{5}_{12}}+$
$\displaystyle+$
$\displaystyle\frac{\lambda^{3}}{4\tilde{c}^{4}_{12}}\sigma\,\left[\sqrt{2}(1-2A\tilde{c}_{12}^{2}\rho)+\tilde{c}_{12}^{2}(\sqrt{2}|u|^{2}+2\text{Re}\,(u))\right]+$
$\displaystyle+$
$\displaystyle\frac{\lambda^{3}}{2\tilde{c}^{7}_{12}}\left\\{-2\sqrt{2}\tilde{c}^{4}_{12}\omega(\tilde{c}_{12}\sigma\omega+\tilde{s}_{12})+\tilde{c}^{2}_{12}z\left[\tilde{c}^{2}_{12}\sigma-2\omega\sin(2\tilde{\theta}_{12})+3\sigma\tilde{s}^{2}_{12}\right]+\right.$
$\displaystyle\left.\frac{\tilde{c}_{12}\sigma
z^{2}(5-3\cos(2\tilde{\theta}_{12}))}{\sqrt{2}}+\sigma
z^{3}(3-2\cos(2\tilde{\theta}_{12}))\right\\}\,$

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