# Implications of the Zero 1-3 Flavour Mixing
Hypothesis: Predictions for $\theta_{23}^{\mathrm{PMNS}}$ and
$\delta^{\mathrm{PMNS}}$

Stefan Antusch<EMAIL_ADDRESS>Kevin Hinze<EMAIL_ADDRESS>Shaikh Saad<EMAIL_ADDRESS>Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland

###### Abstract

We revisit mixing sum rule relations in the lepton and quark sectors under the
assumption that the 1-3 elements of the flavour mixing matrices
($V^{u}_{L},V^{d}_{L},V^{e}_{L},V^{\nu}_{L}$) are zero in the flavour basis.
We consider the exact relations resulting from the validity of this “zero 1-3
flavour mixing hypothesis” and analyse their implications based on the current
experimental data, including effects from RG running. In particular, we
analyse how the existing precise measurement of $\theta_{13}^{\mathrm{PMNS}}$
allows to derive predictions for $\theta_{23}^{\mathrm{PMNS}}$ in models with
constrained $\theta_{12}^{\mathrm{e}}$. As examples, we calculate the
predictions for $\theta_{23}^{\mathrm{PMNS}}$ which arise in classes of Pati-
Salam models and SU(5) GUTs that relate $\theta_{12}^{\mathrm{e}}$ to
$\theta_{12}^{\mathrm{d}}$. We also derive a novel “lepton phase sum rule”,
valid under the additional assumption of small charged lepton mixing
contributions. We furthermore point out that, in the context of GUT flavour
models, the quark and lepton CP violating phases $\delta^{\mathrm{CKM}}$ and
$\delta^{\mathrm{PMNS}}$ can both be predicted from a single imaginary element
in the mass matrices.

##  I Introduction

Mixing sum rule relations are a useful tool for understanding how classes of
flavour models can give rise to predictions for the observable parameters of
the PMNS and CKM mixing matrices in the lepton and quark sectors. By providing
such understanding, they can be a valuable guidance for model building. In the
lepton sector, it has been shown that under the assumption that the charged
lepton left mixing angles $\theta_{ij}^{e}$ are small and
$\theta_{13}^{e},\theta_{13}^{\nu}\approx 0$, the approximate lepton mixing
relations King:2005bj ; Masina:2005hf ; Antusch:2005kw ; Antusch:2007rk ;
Antusch:2012fb

$\displaystyle\theta_{12}^{\mathrm{PMNS}}-\theta_{13}^{\mathrm{PMNS}}\cot{\theta_{23}^{\mathrm{PMNS}}}\cos{\delta^{\mathrm{PMNS}}}\approx\theta_{12}^{\nu}\>,$
(1)
$\displaystyle\theta_{13}^{\mathrm{PMNS}}\approx\sin(\theta_{23}^{\mathrm{PMNS}})\theta_{12}^{e}$
(2)

can be derived. The relations show that flavour models which satisfy the
assumptions and predict a value of $\theta_{12}^{\nu}$ or $\theta_{12}^{e}$,
inevitably predict a correlation among the parameters of the PMNS matrix, i.e.
here among the mixing angles $\theta_{ij}^{\mathrm{PMNS}}$ and the Dirac CP
phase $\delta^{\mathrm{PMNS}}$. Crucially, apart from the one model dependent
quantity ($\theta_{12}^{\nu}$ or $\theta_{12}^{e}$), the relations involve
only quantities measurable in neutrino oscillation experiments.

In the quark sector, analogous relations, derived under the condition of
hierarchical up and down quark mass matrices (i.e. small left mixing angles
$\theta_{ij}^{u}$ and $\theta_{ij}^{d}$) and
$\theta_{13}^{u},\theta_{13}^{d}\approx 0$ read Antusch:2009hq :

$\displaystyle\left|\theta^{\textrm{CKM}}_{12}-\frac{\theta^{\textrm{CKM}}_{13}}{\theta^{\textrm{CKM}}_{23}}e^{-i\delta^{\textrm{CKM}}}\right|\approx\theta^{d}_{12}\>,$
(3)
$\displaystyle\frac{\theta_{13}^{\mathrm{CKM}}}{\theta_{23}^{\mathrm{CKM}}}\approx\theta_{12}^{u}\>.$
(4)

Furthermore, a “quark phase sum rule” has been derived, which relates the two
model-dependent phases of the 1-2 mixings in the up- and down-quark sectors to
the observable CP violation via the relation Antusch:2009hq

$\displaystyle\delta_{12}^{d}-\delta^{u}_{12}\approx\alpha\approx\frac{\pi}{2}\>,$
(5)

where $\alpha$ is the unitarity triangle angle, with current measured value
$\alpha=\left(91.98^{+0.82}_{-1.40}\right)^{\circ}$ at $2\sigma$ CL CKMFIT ,
defined as

$\displaystyle\alpha$
$\displaystyle=\arg\left(-\frac{V_{td}V_{tb}^{\ast}}{V_{ud}V_{ub}^{\ast}}\right)$
(6)
$\displaystyle\approx\arg\left(1-\frac{\theta^{\mathrm{CKM}}_{12}\theta^{\mathrm{CKM}}_{23}}{\theta^{\mathrm{CKM}}_{13}}e^{-i\delta^{\mathrm{CKM}}}\right).$
(7)

The goal of this paper is to revisit these sum rule relations and to analyse
their predictive power in the light of the current experimental data. We focus
on the following aspects:

  * •

To start with, we note that a minimal assumption, under which relations can be
derived that do not depend on unphysical phases, is that of zero 1-3 elements
of the flavour mixing matrices in the flavour basis, or, in terms of the most
common parametrization, that the individual 1-3 flavour mixing angles
$\theta_{13}^{i}$, with $i\in\\{u,d,e,\nu\\}$ vanish (approximately). We will
apply this zero 1-3 flavour mixing hypothesis to obtain exact versions of the
mixing sum rule relations (cf. also Ballett:2014dua ; Marzocca:2013cr ;
Petcov:2014laa ).

  * •

We then address the question how one can use these relations, given the
improved experimental data on the leptonic mixing parameters, to understand
how predictions for the PMNS parameters can arise in classes of flavour models
that satisfy the zero 1-3 flavour mixing hypothesis. In particular, we will
show that the existing precise measurement of $\theta_{13}^{\mathrm{PMNS}}$
leads to predictions for $\theta_{23}^{\mathrm{PMNS}}$ (which currently has a
rather wide experimentally allowed range), from relations between
$\theta_{12}^{e}$ and $\theta_{12}^{d}$ that can emerge in Pati-Salam models
and SU(5) GUTs. We also discuss the predictions for $\delta^{\mathrm{PMNS}}$
in models that feature a fixed value of $\theta_{12}^{\nu}$.

  * •

Furthermore, extending the concept of phase sum rules from the quark sector to
the lepton sector, we derive a novel “lepton phase sum rule”. We find,
however, that in contrast to the mixing sum rule relations, the phase sum
rules in lepton (quark) sectors only hold free of unphysical phases in a small
angle approximation for the charged lepton (quark) mixing contributions.

  * •

Finally, applying the phase sum rules in both sectors, we point out that in
the context of GUT flavour models, the quark and lepton CP violating phases
can both be predicted from a single imaginary element in the mass matrices.

The paper is organized as follows: In Sec. II, we fix the notation for the
parametrizations of unitary matrices used in this work. In Sec. III, we
motivate and discuss the zero 1-3 flavour mixing hypothesis. Exact mixing sum
rules in the lepton and quark sectors are discussed in Secs. IV and V,
respectively. Predictions for $\theta_{23}^{\mathrm{PMNS}}$ in quark-lepton
unified models are exemplified in Sec. VI. Furthermore, we derive a phase sum
rule in the lepton sector in Sec. VII and examine the possible unified origin
of the CP-violating Dirac phases in the quark and leptons sectors. Finally, we
conclude in Sec. IX.

##  II Parametrization of PMNS and CKM matrices

Parametrization of PMNS matrix:– The part of the Lagrangian that contains
lepton masses is given by

$\displaystyle-\mathcal{L}_{Y}\supset\overline{\ell}_{L}M_{e}\ell_{R}+\frac{1}{2}\overline{\nu^{c}_{L}}M_{\nu}\nu_{L}+h.c.,$
(8)

which are diagonalized as follows:

$\displaystyle M_{e}=V^{e}_{L}\textrm{ diag($m_{e},m_{\mu},m_{\tau}$)
}V^{e\dagger}_{R},$ (9) $\displaystyle M_{\nu}=V^{\nu\ast}_{L}\textrm{
diag($m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}}$) }V^{\nu\dagger}_{L}.$ (10)

Then the mixing matrix in the lepton sector $U$ arises in the leptonic charged
current interactions in the mass basis,

$\displaystyle-\mathcal{L}_{cc}=\frac{g}{\sqrt{2}}\overline{\ell}_{L}\gamma^{\mu}U\nu_{L}W^{-}_{\mu}+h.c.,$
(11)

where,

$\displaystyle U=V^{e\dagger}_{L}V^{\nu}_{L}.$ (12)

The leptonic mixing matrix $U$ can be written as,

$\displaystyle U=P\;U_{\textrm{PMNS}},$ (13)

where $P$ is a diagonal phase matrix containing three unphysical phases
$P=\textrm{diag}\\{e^{i\phi_{1}},e^{i\phi_{2}},e^{i\phi_{3}}\\}$ and
$U_{\textrm{PMNS}}$ is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix for
which we use the PDG parametrization ParticleDataGroup:2020ssz :

$\displaystyle U_{\textrm{PMNS}}=R_{23}U_{13}R_{12}Q,$ (14) $\displaystyle
R_{23}=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix},\;U_{13}=\scalebox{0.9}{\mbox{$\displaystyle\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\ -s_{13}e^{i\delta_{CP}}&0&c_{13}\end{pmatrix}$}},$ (15)
$\displaystyle R_{12}=\begin{pmatrix}c_{12}&s_{12}&0\\\ -s_{12}&c_{12}&0\\\
0&0&1\end{pmatrix},\;Q=\begin{pmatrix}e^{i\beta_{1}}&0&0\\\
0&e^{i\beta_{2}}&0\\\ 0&0&1\end{pmatrix},$ (16)

where we have used $c_{ij}=\cos\theta_{ij}^{\textrm{PMNS}}$,
$s_{ij}=\sin\theta_{ij}^{\textrm{PMNS}}$. $U_{\textrm{PMNS}}$ contains three
physical phases: one Dirac CP-violating phase $\delta^{\textrm{PMNS}}$ and two
Majorana phases $\beta_{1,2}$ (these latter two phases become unphysical if
neutrinos are Dirac particles). In this work, we assume neutrinos to be
Majorana in nature.

Parametrization of CKM matrix:– Similarly, the part of the Lagrangian that
contains quark masses is given by

$\displaystyle-\mathcal{L}_{Y}\supset\overline{u}_{L}M_{u}u_{R}+\overline{d}_{L}M_{d}d_{R}+h.c.,$
(17)

which are diagonalized by,

$\displaystyle M_{u}=V^{u}_{L}\textrm{ diag($m_{u},m_{c},m_{t}$)
}V^{u\dagger}_{R},$ (18) $\displaystyle M_{d}=V^{d}_{L}\textrm{
diag($m_{d},m_{s},m_{b}$) }V^{d\dagger}_{R}.$ (19)

The corresponding mixing matrix in the quark sector $V$ arises from charged
quark current interactions in the mass basis,

$\displaystyle-\mathcal{L}_{cc}=\frac{g}{\sqrt{2}}\overline{u}_{L}\gamma^{\mu}Vd_{L}W^{+}_{\mu}+h.c.,$
(20)

where

$\displaystyle V=V^{u\dagger}_{L}V^{d}_{L}.$ (21)

The quark mixing matrix $V$ can be parametrised as

$\displaystyle V=P^{\prime}\;V_{\textrm{CKM}}Q^{\prime},$ (22) $\displaystyle
V_{\textrm{CKM}}=R^{\textrm{CKM}}_{23}U^{\textrm{CKM}}_{13}R^{\textrm{CKM}}_{12},$
(23)

where $V_{\textrm{CKM}}$ is called the Cabibbo-Kobayashi-Maskawa (CKM) mixing
matrix that contains a single (Dirac) physical phase in
$U^{\textrm{CKM}}_{13}$, which we denote by $\delta^{\textrm{CKM}}$. Then two
diagonal phase matrices $P^{\prime}$ and $Q^{\prime}$ contain five unphysical
phases.

Alternative parametrization of unitary matrices:– It is often convenient to
use an alternative parametrization for the unitary matrices that diagonalize
the mass matrices of the individual flavour sectors, making use of the fact
that a general $3\times 3$ unitary matrix $W$, containing nine parameters
(three angles and six phases), can be parametrized as

$\displaystyle W=W_{23}W_{13}W_{12}S,$ (24)

where three unitary matrices $W_{ij}$ are defined as

$\displaystyle W_{23}=\begin{pmatrix}1&0&0\\\
0&c_{23}&s_{23}e^{-i\delta_{23}}\\\
0&-s_{23}e^{i\delta_{23}}&c_{23}\end{pmatrix},$ (25)

and analogously for the other two matrices, and where $S$ is a diagonal phase
matrix $S=\textrm{diag}\\{e^{i\chi_{1}},e^{i\chi_{2}},e^{i\chi_{3}}\\}$.
Throughout this work, we use this general parametrization Eq. (24) for the two
unitary matrices $V^{e}_{L},V^{\nu}_{L}$ in the leptonic sector (Eq. (12)) as
well as for $V^{u}_{L},V^{d}_{L}$ in the quark sector (Eq. (21)).111For a
similar parametrization, cf. Ref. King:2002nf .

##  III The Zero 1-3 Flavour Mixing Hypothesis

The fact that in both sectors, quarks and leptons, the 1-3 elements of the CKM
and PMNS mixing matrices are much smaller than the other elements motivates
the hypothesis that the 1-3 elements of all flavour mixing matrices
$V^{u}_{L},V^{d}_{L},V^{e}_{L},V^{\nu}_{L}$ in the up-quark, down-quark,
charged lepton and neutrino sectors, or, in terms of the most common
parametrization, that the individual 1-3 flavour mixing angles, vanish
(approximately). With left mixing angles $\theta^{u,d,e,\nu}_{13}=0$ in the
flavour basis (equivalent to $W^{u,d,e,\nu}_{13}=\mathcal{I}_{3\times 3}$),
the 1-3 mixings in the observable CKM and PMNS mixing matrices are then
induced by the non-commutativity of the individual flavour rotations. In the
following, we will revisit the mixing sum rule relations under this
assumption.

We note that although we will work here with the exact hypothesis, our results
remain valid as approximate statements when the $\theta^{x}_{13}$ are
sufficiently small. One example is the case when $\theta^{u,d,e}_{13}\approx
0$ is realised with hierarchical mass matrices via “texture zeros” in the 1-3
positions of the mass matrices (with LR convention as used in this paper). In
the neutrino sector, $\theta^{\nu}_{13}\approx 0$ can e.g. be realised in the
framework of “Sequential Dominance (SD)” King:1999mb ; King:2002nf ;
Antusch:2004gf ; Antusch:2010tf  for neutrino masses via the type I seesaw
mechanism, when the dominant right-handed neutrino coupling vector to lepton
doublets has a zero in the $e$-position and is orthogonal to the subdominant
one, in the limit of the third right-handed neutrino being approximately
decoupled. This is the case e.g. in Constrained Sequential Dominance
King:2005bj . In the following, we will study the implications of the zero 1-3
flavour mixing hypothesis.

Figure 1: Predictions for the 1$\sigma$ and 2$\sigma$ HPD intervals of the low
scale PMNS mixing angle $\theta_{23}^{\textrm{PMNS}}$ as a function of the GUT
scale parameter $\theta_{12}^{e}$, including RG effects. The white region
indicates the current experimental 1$\sigma$ range, whereas the light gray
region represents the current experimental 3$\sigma$ range. Left: SM. Right:
MSSM. Top: NH. Bottom: IH.

##  IV Exact Mixing Sum Rule Relations in the Lepton Sector

Exact relation between $\theta_{13}^{\mathrm{PMNS}}$,
$\theta_{23}^{\mathrm{PMNS}}$ and $\theta^{e}_{12}$:– In this section, we
consider an exact form of the relation between
$\theta_{23}^{\mathrm{PMNS}},\theta_{13}^{\mathrm{PMNS}}$ and
$\theta_{12}^{e}$ King:2005bj ; Masina:2005hf ; Antusch:2005kw ;
Antusch:2007rk ; Antusch:2012fb , which arises under the zero 1-3 flavour
mixing hypothesis when we divide in Eq. (12) the (13)-entry by the (23)-entry,
leading to (cf. also Ballett:2014dua )

$\displaystyle
e^{i(-\delta+\phi_{1}-\phi_{2})}\frac{t_{13}}{s_{23}}=-t^{e}_{12}e^{-i(\delta^{e}_{12}+\chi^{e}_{1}-\chi^{e}_{2})}.$
(26)

Here we have dropped the PMNS labels for brevity, and abbreviated
$t_{ij}:=\tan(\theta_{ij})$. By taking the modulus of the above equation we
obtain the relation

$\displaystyle
s^{\mathrm{PMNS}}_{23}=\frac{t^{\mathrm{PMNS}}_{13}}{t^{e}_{12}}.$ (27)

Currently, $\theta^{\textrm{PMNS}}_{13}$ is measured with great accuracy, its
$1\sigma$ allowed range corresponds to
$\left(8.57^{+0.13}_{-0.12}\right)^{\circ}$ NUFIT ; Esteban:2020cvm . On the
contrary, $\theta^{\textrm{PMNS}}_{23}$ has large uncertainty associated with
its measurements. While in the past the relation of Eq. (2) has been used to
derive the value of $\theta^{\textrm{PMNS}}_{13}$ generated by the charged
lepton 1-2 mixing contribution $\theta_{12}^{e}$ (often assuming maximal
$\theta_{23}^{\mathrm{PMNS}}$), we point out that it is now more valuable to
use it with the precise measurement of $\theta_{13}^{\mathrm{PMNS}}$ to
predict the less well measured PMNS mixing angle
$\theta_{23}^{\mathrm{PMNS}}$.

Numerical analysis:– We can now quantitatively analyse, using the precisely
measured value of $\theta^{\textrm{PMNS}}_{13}$ (cf. NUFIT ; Esteban:2020cvm
), how Eq. (27) leads to a prediction for $\theta^{\textrm{PMNS}}_{23}$ in
terms of $\theta_{12}^{e}$. However, since the relation Eq. (27) is valid at
the flavour scale, which we here set equal to the GUT scale, RG running
effects (cf. Antusch:2003kp ; Antusch:2005gp  and references therein) must be
properly taken into account to find the accurate prediction for
$\theta^{\textrm{PMNS}}_{23}$ at the low scale.222We note that we assume for
our RG analysis a realisation of the neutrino mass matrix by a type I seesaw
mechanism where, however, the neutrino Yukawa couplings are significantly
smaller than unity such that they do not notably affect the RG evolution. This
essentially leaves us with the running of the dimension 5 operator. For this
purpose, we have implemented our setup in REAP Antusch:2005gp  with zero 1-3
mixings for the charged fermion Yukawa coupling matrices $Y^{u,d,e}$ as well
as for the neutrino Majorana mass matrix $M_{\nu}$. For a fixed value of
$\theta^{e}_{12}$ within the range $(10-14)^{\circ}$, all other parameters (at
the high scale) both in the quark and the lepton sectors are freely varied.
After running down to the low scale, all observables are then fitted to their
experimental values Antusch:2013jca ; NUFIT , except
$\theta^{\textrm{PMNS}}_{23}$ for which we obtain a theory prediction. We have
performed fits and Markov Chain Monte Carlo (MCMC) analyses both in the SM as
well as in the MSSM framework for both normal (NH) and inverted (IH) neutrino
mass hierarchies, assuming strong hierarchies with the lightest neutrino mass
set to zero. We consider the running from $M_{\rm{GUT}}=2\times 10^{16}$ GeV
to $M_{\rm{Z}}=91.18$ GeV. For the case of the MSSM, in our numerical
analysis, the SUSY scale is chosen to be 3 TeV, and $\tan\beta$ is taken as a
free parameter (which we vary from 10 to 50). Outcomes (i.e. predictions for
$\theta^{\textrm{PMNS}}_{23}$) of our fitting procedure as a function of
$\theta^{e}_{12}$ are presented in Fig. 1. One can use the plots to read off
the predicted range for $\theta^{\textrm{PMNS}}_{23}$ for models that feature
a fixed value of $\theta^{e}_{12}$. As can be seen from these plots, due to
$\tan\beta$ effects in the MSSM that can enhance the running, the allowed
range of $\theta^{e}_{12}$ consistent with experimentally measured values of
$\theta^{\textrm{PMNS}}_{23}$ is slightly wider in comparison to the SM
scenario, in the case of NH. In the case of IH, the two plots for the SM and
MSSM look identical. The reason for this is that for IH the running of the
ratio
${\tan(\theta_{13}^{\mathrm{PMNS}})}/{\sin(\theta_{23}^{\mathrm{PMNS}})}$ is
practically zero, and thus both plots look as if we had just evaluated the
formula of Eq. (27) at low scale. This stability against RG running is
illustrated in Fig. 2 (cf. also Antusch:2012fb ).

Furthermore, we can also state the currently preferred values of
$\theta^{e}_{12}$, given the experimentally allowed region for
$\theta^{\textrm{PMNS}}_{23}$. To this end we have varied the fitting
procedure described above by including $\theta^{e}_{12}$ as a fit parameter
and by adding $\theta^{\textrm{PMNS}}_{23}$ to the list of observables to be
fitted. Performing MCMC analyses we now obtain the highest posterior density
(HPD) intervals for $\theta^{e}_{12}$ consistent with the experimental data
for $\theta^{\textrm{PMNS}}_{23}$, which are listed in Table 1. These allowed
ranges for $\theta^{e}_{12}$ can be used as a guideline for building flavour
models consistent with the current results from $\theta^{\textrm{PMNS}}_{23}$
measurements.

Figure 2: Running of the quotient ${\tan(\theta_{13}^{\mathrm{PMNS}})}/{\sin(\theta_{23}^{\mathrm{PMNS}})}$ for both NH and IH in the MSSM with $\tan\beta=50$. As it can be seen, for the case of IH the RG effects for $\theta_{13}^{\mathrm{PMNS}}$ and $\theta_{23}^{\mathrm{PMNS}}$ almost cancel each other such that the quotient remains nearly invariant under RG running. |  $\theta_{12}^{e}$ (1$\sigma$) |  $\theta_{12}^{e}$ (2$\sigma$)
---|---|---
SM: NH | $10.99^{\circ}-11.68^{\circ}$ | $10.84^{\circ}-13.08^{\circ}$
| $12.25^{\circ}-12.86^{\circ}$ |
SM: IH | $10.90^{\circ}-11.80^{\circ}$ | $10.81^{\circ}-13.02^{\circ}$
MSSM: NH | $10.98^{\circ}-11.86^{\circ}$ | $10.75^{\circ}-13.18^{\circ}$
| $12.17^{\circ}-12.72^{\circ}$ |
MSSM: IH | $10.88^{\circ}-11.81^{\circ}$ | $10.81^{\circ}-12.99^{\circ}$
Table 1: Viable 1$\sigma$ and 2$\sigma$ HPD intervals for $\theta^{e}_{12}$
which predict the correct experimental value for
$\theta^{\textrm{PMNS}}_{23}$. The RG running effects have been considered in
the case of the SM as well as of the MSSM for both NH as well as IH, as
discussed in the main text.

Exact lepton mixing sum rule involving $\theta_{12}^{\nu}$ and
$\delta_{\textrm{PMNS}}$:– A second consequence of the zero 1-3 mixing
hypothesis is that the value of $\theta^{\nu}_{12}$, which is a model
parameter, can be related to the leptonic Dirac CP phase
$\delta_{\textrm{PMNS}}$ King:2005bj ; Masina:2005hf ; Antusch:2005kw ;
Antusch:2007rk ; Antusch:2012fb . In the following we consider the exact
relation which arises when we divide in Eq. (12) the (31)-entry by the
(32)-entry. This yields (cf. also Ballett:2014dua )

$\displaystyle
e^{i(\beta_{1}-\beta_{2})}\frac{-t_{12}t_{23}+e^{i\delta}s_{13}}{t_{23}+e^{i\delta}t_{12}s_{13}}=-t^{\nu}_{12}e^{i(\delta^{\nu}_{12}+\chi^{\nu}_{1}-\chi^{\nu}_{2})},$
(28)

where we again dropped the PMNS labels for brevity. Taking the modulus leads
to

$\displaystyle
t^{\nu}_{12}=\left|\frac{t^{\textrm{PMNS}}_{12}t^{\textrm{PMNS}}_{23}-e^{i\delta^{\textrm{PMNS}}}s^{\textrm{PMNS}}_{13}}{t^{\textrm{PMNS}}_{23}+e^{i\delta^{\textrm{PMNS}}}t^{\textrm{PMNS}}_{12}s^{\textrm{PMNS}}_{13}}\right|.$
(29)

Using the experimentally measured values of neutrino mixing angles as well as
the Dirac CP phase, one can calculate the preferred value of
$\theta^{\nu}_{12}$, which subsequently can be used as a guideline for flavour
model building. Alternatively, since the neutrino mixing angles are much more
precisely measured than the Dirac CP phase $\delta^{\textrm{PMNS}}$, each
flavour model that features a fixed value of $\theta^{\nu}_{12}$ and that
satisfies the zero 1-3 mixing hypothesis predicts (utilizing the experimental
values of the neutrino mixing angles) a specific Dirac CP phase by which it
can be experimentally tested.

Numerical analysis:– As discussed above, since the sum rule relations such as
Eq. (29) hold at the flavour scale (here taken as the GUT scale), RG effects
have to be considered in order to give the low scale prediction for
$\delta^{\textrm{PMNS}}$ in terms of the (high scale) value of
$\theta^{\nu}_{12}$. To obtain the predicted range for
$\delta^{\textrm{PMNS}}$ in terms of $\theta^{\nu}_{12}$, we have again
implemented the zero 1-3 flavour mixing setup in REAP. Keeping
$\theta_{12}^{\nu}$ fixed, we now vary the remaining GUT scale input
parameters, compute the RG evolution from the GUT scale $M_{\rm{GUT}}=2\times
10^{16}$ GeV down to the $Z$ scale $M_{Z}=91.18$ GeV at which we fit, apart
from $\delta^{\textrm{PMNS}}$, all the observables to the experimental data.
We have performed fits and MCMC analyses with NH and IH for both the SM and
the MSSM. In the latter case, we have again chosen a SUSY scale of 3 TeV and
included $\tan\beta$ in the parameter list (varied from 10 to 50). Fig. 3
visualizes our numerical results. The results can be used to read off the
predicted range for $\delta^{\textrm{PMNS}}$ for models with fixed
$\theta^{\nu}_{12}$. One can see that the allowed parameter region is largely
extended in the MSSM for IH. This is due to RG running effects which can
particularly strongly affect $\theta^{\textrm{PMNS}}_{12}$ when the Majorana
phase difference $\beta_{2}-\beta_{1}$ is zero and when $\tan\beta$ is large,
as can be seen e.g. from the analytical formulae for the running derived in
Antusch:2003kp . On the other hand, in many flavour models with strong inverse
neutrino mass hierarchy the Majorana phases satisfy
$\beta_{2}-\beta_{1}\approx\pi$, since this can explain the almost degenerate
neutrino masses $m_{1}$ and $m_{2}$ from a pseudo-Dirac structure of the
neutrino mass matrix. We therefore show in Fig. 4 another plot with the
results for the MSSM with IH and the Majorana phase difference fixed to
$\beta_{2}-\beta_{1}=\pi$, which can be applied to this class of models.

Moreover, in order to obtain values of $\theta_{12}^{\nu}$ preferred by the
current experimental data, we have performed additional fits and MCMC analyses
for which we added $\theta_{12}^{\nu}$ to the list of input parameters and
also included $\delta^{\textrm{PMNS}}$ in the fit to the experimental data.
These results are presented in Table 2 and can provide useful guidance for
building models which agree with the current preferred values of the Dirac CP
phase $\delta^{\textrm{PMNS}}$. Note that the preferred value for
$\theta_{12}^{\nu}$ is different for NH compared to IH also in the SM. The
reason for this is the different experimentally preferred range for
$\delta^{\textrm{PMNS}}$ and not an effect coming from enhanced RG running
with $\beta_{2}-\beta_{1}=0$, which only has a large effect in the MSSM when
also $\tan\beta$ is large.

|  $\theta_{12}^{\nu}$ (1$\sigma$) |  $\theta_{12}^{\nu}$ (2$\sigma$)
---|---|---
SM: NH | $36.37^{\circ}-43.43^{\circ}$ | $26.75^{\circ}-44.23^{\circ}$
SM: IH | $28.11^{\circ}-35.64^{\circ}$ | $25.80^{\circ}-39.62^{\circ}$
MSSM: NH | $36.24^{\circ}-43.29^{\circ}$ | $26.64^{\circ}-44.11^{\circ}$
MSSM: IH | $19.06^{\circ}-34.22^{\circ}$ | $8.68^{\circ}-37.41^{\circ}$
Table 2: Viable 1$\sigma$ and 2$\sigma$ HPD intervals for $\theta^{\nu}_{12}$
which predict the correct experimental value for $\delta^{\textrm{PMNS}}$. We
have considered the RG evolution for the SM and the MSSM for both NH as well
as for IH), as discussed in the main text.

Figure 3: Viable 1$\sigma$ and 2$\sigma$ HPD intervals for $\theta_{12}^{\nu}$
wich predict the correct experimental value for the PMNS Dirac CP phase
$\delta^{\textrm{PMNS}}$ as a function of $\theta_{12}^{\nu}$ including
running effects. The experimental 1$\sigma$ and 3$\sigma$ ranges are
represented by white and light gray regions, respectively. Left: SM. Right:
MSSM. Top: NH. Bottom: IH.

Figure 4: Viable 1$\sigma$ and 2$\sigma$ HPD intervals for $\theta_{12}^{\nu}$
which predict the correct experimental value for the PMNS Dirac CP phase
$\delta^{\textrm{PMNS}}$ as a function of $\theta_{12}^{\nu}$ including
running effects for the MSSM with IH and a Majorana phase difference
$\beta_{2}-\beta_{1}=\pi$. The experimental 1$\sigma$ and 3$\sigma$ ranges are
represented by white and light gray regions, respectively.

##  V Exact Mixing Sum Rule Relations in the Quark Sector

Exact versions of the quark mixing sum rules Antusch:2009hq  valid under the
assumption of zero 1-3 flavour mixing hypothesis can be obtained similarly to
the lepton sector. First, we divide the (13)-entry by the (23)-entry in Eq.
(21) and then take the absolute values, which yields,

$\displaystyle
t^{u}_{12}=\frac{t^{\mathrm{CKM}}_{13}}{s^{\mathrm{CKM}}_{23}}.$ (30)

Furthermore, we divide the (31)-entry by (32)-entry in Eq. (21) and take the
absolute value, which provides,

$\displaystyle
t^{d}_{12}=\left|\frac{t^{\textrm{CKM}}_{12}t^{\textrm{CKM}}_{23}-e^{i\delta^{\textrm{CKM}}}s^{\textrm{CKM}}_{13}}{t^{\textrm{CKM}}_{23}+e^{i\delta^{\textrm{CKM}}}t^{\textrm{CKM}}_{12}s^{\textrm{CKM}}_{13}}\right|.$
(31)

Since CKM parameters are very precisely measured in the experiments, these
relations essentially fix the 1-2 mixings. The results for both, the SM and
MSSM cases, are $\theta_{12}^{u}\in(4.75^{\circ},\,5.14^{\circ})$ and
$\theta_{12}^{d}\in(11.85^{\circ},\,12.40^{\circ})$ at $1\sigma$.

##  VI Example Predictions for $\theta_{23}^{\mathrm{PMNS}}$ in Quark-Lepton
Unified Models

Predictions for $\theta^{e}_{12}$ can arise e.g. from quark-lepton unification
in Pati-Salam Pati:1973rp ; Pati:1974yy  or GUT models Georgi:1974sy ;
Georgi:1974yf ; Georgi:1974my ; Fritzsch:1974nn , which can dictate strong
correlations between down-type ($Y_{d}$) and charged-lepton ($Y_{e}$) Yukawa
matrices. To establish such predictive mass relations from quark-lepton
unification, we assume that the entries of $Y_{d}$ and $Y_{e}$ are each
generated by one single joint GUT operator (also referred to as “single
operator dominance”, cf. Antusch:2011qg ; Antusch:2012fb ; Antusch:2019avd ).
Consequently, the matrix entries are linked only via group theoretical Clebsch
factors (cf. Antusch:2009gu ; Antusch:2013rxa ).

To illustrate how this assumption can lead to predictions for
$\theta^{e}_{12}$, and thus for $\theta_{23}^{\mathrm{PMNS}}$, we focus on the
1-2 sector of the down-type and charged lepton mass matrices, ignoring
possible 2-3 rotations (which are here taken to be small). Writing the 1-2
block of the $Y_{d}$ matrix as

$\displaystyle Y_{d}=\begin{pmatrix}y_{11}&y_{12}\\\
y_{21}&y_{22}\end{pmatrix},$ (32)

$Y_{e}$ then, at the unification scale, has the form

$\displaystyle Y_{e}^{SU(5)}=\begin{pmatrix}c_{11}y_{11}&c_{21}y_{21}\\\
c_{12}y_{12}&c_{22}y_{22}\end{pmatrix},\;\;\;Y_{e}^{PS}=\begin{pmatrix}c_{11}y_{11}&c_{12}y_{12}\\\
c_{21}y_{21}&c_{22}y_{22}\end{pmatrix},$ (33)

for SU(5) GUT and Pati-Salam (PS) partial unification, respectively. Here,
$c_{ij}$ are the Clebsch factors that relate the $Y_{d}$ and $Y_{e}$ entries.
For example, possible options for the Clebsch factors in SU(5) GUTs are
$|c_{ij}|\in\\{1/6,\,4/9,\,1/2,\,2/3,\,1,\,3/2,\,2,\,3,\,9/2,\,6,\,9,\,18\\}$
and in PS models $|c_{ij}|\in\\{1/3,\,3/4,\,1,\,3/2,\,2,\,3,\,9\\}$
Antusch:2009gu ; Antusch:2013rxa . These coefficients are constrained by the
requirement that the down-quark and charged-lepton masses of the first two
generations have to be consistent with the measured values at low energy,
taking RG running and possible SUSY threshold corrections into account. From
the above Yukawa matrices, one can infer a relation between the left 1-2
mixing angles $\theta^{d}_{12}$ (fixed from the zero 1-3 flavour mixing
hypothesis to be $\theta_{12}^{d}\in(11.85^{\circ},\,12.40^{\circ})$ and
$\theta^{e}_{12}$ (cf. also Antusch:2011qg ).

For the PS case, in the limit of hierarchical Yukawa matrices with $y_{e}\ll
y_{\mu}$ and $y_{d}\ll y_{s}$, one finds

$\displaystyle\tan(\theta^{e,PS}_{12})\approx\left|\frac{c_{12}y_{12}}{c_{22}y_{22}}\right|\approx\left|\frac{c_{12}}{c_{22}}\right|\tan(\theta^{d}_{12})\>,$
(34)

which directly implies a prediction for $\theta^{e}_{12}$ in terms of
$|{c_{12}}/{c_{22}}|$ and the allowed range for $\theta_{12}^{d}$.

In SU(5) GUTs, without any further constraint, the analogous relation would
additionally depend on the ratio $|{y_{21}}/{y_{12}}|$, since $Y^{SU(5)}_{e}$
is related to $Y_{d}^{T}$ and not as in PS models directly to $Y_{d}$.
Predictivity can however be restored by, e.g., requiring a zero 1-1 element of
$Y_{d}$ (and thus also of $Y_{e}$). Then, again in the limit of hierarchical
Yukawa matrices where $y_{\mu}\approx c_{22}y_{22}$,
$y_{e}=c_{21}y_{21}c_{12}y_{12}/(c_{22}y_{22})$,
$\tan(\theta^{d}_{12})=y_{12}/y_{22}$ and
$\tan(\theta^{e,SU(5)}_{12})=c_{21}y_{21}/(c_{22}y_{22})$, we can write

$\displaystyle\frac{y_{e}}{y_{\mu}}\approx\left|\frac{c_{21}y_{21}c_{12}y_{12}}{(c_{22}y_{22})^{2}}\right|\approx\left|\frac{c_{12}}{c_{22}}\right|\tan(\theta^{e,SU(5)}_{12})\tan(\theta^{d}_{12})\>.$
(35)

We further note that the Yukawa coupling ratio $y_{e}/y_{\mu}$ is very stable
under RG running and also under possible SUSY threshold corrections (cf.
discussion in Antusch:2013jca ), such that we can evaluate it using the
running low scale masses (or, alternatively, the running Yukawa couplings at
$M_{Z}$ provided in Antusch:2013jca ). This leads to the relation

$\displaystyle\tan(\theta^{e,SU(5)}_{12})\approx\left|\frac{c_{22}}{c_{12}}\right|\frac{m_{e}}{m_{\mu}}\frac{1}{\tan(\theta^{d}_{12})}\>.$
(36)

which implies a prediction for $\theta^{e}_{12}$, depending only on the
Clebsch factor ratio $|{c_{22}}/{c_{12}}|$, the precisely known ratio
$m_{e}/m_{\mu}$, and the allowed range for $\theta_{12}^{d}$.

Among the Clebsch factors mentioned above and discussed in Antusch:2009gu ;
Antusch:2013rxa ), only a few lead (utilizing Eq. (27)) to predictions for
$\theta_{23}^{\mathrm{PMNS}}$ compatible with the current experimental data.
Two promising combinations of Clebsches in SU(5) are $c_{22}=9/2,\,c_{12}=1/2$
and $c_{22}=6,\,c_{12}=2/3$, which yield the same ratio $|{c_{22}}/{c_{12}}|$
and thus identical predictions for $\theta_{23}^{\mathrm{PMNS}}$, and
$c_{22}=9/2,\,c_{12}=4/9$. We like to remark that the Clebsch factor $4/9$
stems from a higher order operator, as discussed in Antusch:2013rxa , whereas
the other Clebsch factors appear at dimension 5. Furthermore, in PS, Clebsch
combinations with $|c_{12}|=|c_{22}|$ lead to a (currently) viable prediction
for $\theta_{23}^{\mathrm{PMNS}}$.

The three choices of Clebsch factors mentioned above, combined with the
resulting values for $\theta_{12}^{e}$ as well as their respective predicted
range for $\theta_{23}^{\text{PMNS}}$ for the example case of the MSSM with
NH, are shown in Table 3. The predictions for $\theta_{23}^{\text{PMNS}}$ are
obtained from an MCMC analysis considering the RG running effects in the MSSM
with NH and utilizing the HPD results for $\theta_{12}^{d}$.

SU(5) | ($c_{22},c_{12}$) |  $\theta_{12}^{e}$ (1$\sigma$) |  |  |  $\theta_{23}^{\text{PMNS}}$ (1$\sigma$) |
---|---|---|---|---|---|---
| ($\frac{9}{2},\frac{1}{2}$) | $10.97^{\circ}-11.49^{\circ}$ |  |  | $47.04^{\circ}-51.65^{\circ}$ |
| ($\frac{9}{2},\frac{4}{9}$) | $12.31^{\circ}-12.88^{\circ}$ |  |  | $40.67^{\circ}-44.16^{\circ}$ |
PS | ($c_{22},c_{12}$) |  $\theta_{12}^{e}$ (1$\sigma$) |  |  |  $\theta_{23}^{\text{PMNS}}$ (1$\sigma$) |
| (3,3) | $11.85^{\circ}-12.40^{\circ}$ |  |  | $43.05^{\circ}-47.07^{\circ}$ |
Table 3: SU(5) and PS Clebsch factor combinations with predicted ranges of
$\theta_{12}^{e}$, together with their respective predictions for
$\theta_{23}^{\text{PMNS}}$. As mentioned in the main text, the Clebsch
combination ($9/2,1/2)$ can be replaced by ($6,2/3$) giving the same
prediction for $\theta_{23}^{\text{PMNS}}$.

We like to remark that, at this level of model independence, there are
comparatively large uncertainties in the predictions. They are, to some
extend, caused by the uncertainty in the amount of RG running, which could be
significantly reduced if $\tan\beta$ was known. Additionally, a further
reduction of the uncertainties in the predictions for
$\theta_{23}^{\text{PMNS}}$ could be achieved by even more precise
measurements of $\theta_{13}^{\text{PMNS}}$ and of the CKM parameters
(reducing the allowed range of $\theta_{12}^{d}$).

Furthermore, the predictions are also expected to get more precise when the
Clebsch factor combinations are extended to specify the other relevant Yukawa
matrix entries. For instance in the SU(5) case with zero 1-1 elements of
$Y_{d}$ (and $Y_{e}$), fixing the Clebsch factor $c_{21}$ leads to an
additional prediction for the ratio of the running masses $m_{s}/m_{d}$ from
the relation $m_{s}/m_{d}\approx
y_{22}^{2}/(y_{12}y_{21})\approx(m_{\mu}/m_{e})(c_{12}c_{21}/c_{22}^{2})$,
which has to be taken into account when fitting a specific scenario, with all
Clebsch factors fixed and model-specific restrictions implemented, to the
experimental data.

Such fits have been performed in Antusch:2018gnu , and one can see from the
tables provided there that the best-fit values for $\theta^{e}_{12}$ are
sensitive to the choice of the additional Clebsch factor $c_{21}$, but remain
within the ranges given in Table 3. Cases considered in Antusch:2018gnu
include the possibility to complement the choice $c_{22}=9/2,\,c_{12}=1/2$ by
$c_{21}=9/2$ or $3$, or $c_{22}=6,\,c_{12}=2/3$ by $c_{21}=9/2$ or $6$. We
note a possible completion of the Clebsch combination
$c_{22}=9/2,\,c_{12}=4/9$ could be $c_{21}=9/2$. Finally, we remark that a
possible way to judge the viability of Clebsch combinations is provided by the
double ratio $d=(y_{\mu}/y_{e})(y_{d}/y_{s})$ Antusch:2013jca , which is very
stable under RG running also also under possible SUSY threshold corrections.
It can here be evaluated as $c_{22}^{2}/(c_{12}c_{21})$ and is experimentally
constrained as $d=10.7^{+1.8}_{-0.8}$ at $1\sigma$.

##  VII A Phase Sum Rule in the Lepton Sector

Leptonic phase sum rule:– In this section we derive a phase sum rule in the
lepton sector, which has not been discussed in the literature before. In
contrast to the mixing sum rules discussed above, we find a relation that does
not contain unphysical phases only in the limit of small angle approximation
for the charged lepton mixings (i.e. requiring small $\theta^{e}_{ij}$), which
is thus an extra assumption. Then, to the leading order, the (11)-, (22)-, and
(33)-entries in Eq. (12) lead to the following relations,

$\displaystyle e^{i(\beta_{1}+\phi_{1})}c_{12}c_{13}$ $\displaystyle\approx
e^{i(\chi^{\nu}_{1}-\chi^{e}_{1})}c^{\nu}_{12}+...\;,$ (37) $\displaystyle
e^{i(\beta_{2}+\phi_{2})}c_{12}c_{23}$ $\displaystyle\approx
e^{i(\chi^{\nu}_{2}-\chi^{e}_{2})}c^{\nu}_{12}c^{\nu}_{23}+...\;,$ (38)
$\displaystyle e^{i\phi_{3}}c_{13}c_{23}$ $\displaystyle\approx
e^{i(\chi^{\nu}_{3}-\chi^{e}_{3})}c^{\nu}_{23}+...\;,$ (39)

where the ellipses represent next-to-leading order terms. These relations can
be used to relate the unphysical phases,

$\displaystyle\phi_{1}=\chi^{\nu}_{1}-\chi^{e}_{1}-\beta_{1},$ (40)
$\displaystyle\phi_{2}=\chi^{\nu}_{2}-\chi^{e}_{2}-\beta_{2},$ (41)
$\displaystyle\phi_{3}=\chi^{\nu}_{3}-\chi^{e}_{3}.$ (42)

Using these relations and combining Eqs. (26) and (28), we obtain (dropping
the PMNS labels for brevity)

$\displaystyle
e^{i(\delta_{12}^{\nu}-\delta^{e}_{12})}t^{e}_{12}t^{\nu}_{12}=\frac{t_{13}}{s_{23}}\frac{s_{13}-t_{12}t_{23}e^{-i\delta}}{t_{23}+t_{12}s_{13}e^{i\delta}},$
(43)

which yields

$\displaystyle\delta_{12}^{\nu}-\delta^{e}_{12}=\arg\left(\frac{s^{\textrm{PMNS}}_{13}-t^{\textrm{PMNS}}_{12}t^{\textrm{PMNS}}_{23}e^{-i\delta^{\textrm{PMNS}}}}{t^{\textrm{PMNS}}_{23}+t^{\textrm{PMNS}}_{12}s^{\textrm{PMNS}}_{13}e^{i\delta^{\textrm{PMNS}}}}\right).$
(44)

This new phase sum rule in the lepton sector determines the difference
$\delta_{12}^{\nu}-\delta^{e}_{12}$ between two model-dependent phases. In
models satisfying the zero 1-3 flavour mixing hypothesis, this can provide
insight into the possibilities for predicting $\delta^{\textrm{PMNS}}$ (the
least well measured PMNS parameter) in flavour models. Preferred ranges of
$\delta_{12}^{\nu}-\delta^{e}_{12}$ from MCMC analyses are summarized in Table
4.

|  $\delta_{12}^{\nu}-\delta_{12}^{e}$ (1$\sigma$) |  $\delta_{12}^{\nu}-\delta_{12}^{e}$ (2$\sigma$)
---|---|---
SM: NH | $297.9^{\circ}-14.6^{\circ}$ | $217.3^{\circ}-37.6^{\circ}$
SM: IH | $235.9^{\circ}-296.6^{\circ}$ | $208.8^{\circ}-327.0^{\circ}$
MSSM: NH | $299.0^{\circ}-15.0^{\circ}$ | $218.7^{\circ}-37.7^{\circ}$
MSSM: IH | $234.1^{\circ}-308.9^{\circ}$ | $196.6^{\circ}-342.6^{\circ}$
Table 4:  Viable 1$\sigma$ and 2$\sigma$ HPD intervals for
$\delta_{12}^{\nu}-\delta^{e}_{12}$ consistent with current experimental
values of neutrino observables. RG evolution for the SM and the MSSM are
considered for both NH and IH, as discussed in the main text. We note that in
all cases the 2$\sigma$ intervals contain $270^{\circ}=-\frac{\pi}{2}$.

Quark phase sum rule:– For completeness, let us state that in the quark
sector, the analogous calculation (assuming small $\theta^{u}_{ij}$) yields

$\displaystyle\delta_{12}^{d}-\delta^{u}_{12}=\arg\left(\frac{s^{\textrm{CKM}}_{13}-t^{\textrm{CKM}}_{12}t^{\textrm{CKM}}_{23}e^{-i\delta^{\textrm{CKM}}}}{t^{\textrm{CKM}}_{23}+t^{\textrm{CKM}}_{12}s^{\textrm{CKM}}_{13}e^{i\delta^{\textrm{CKM}}}}\right).$
(45)

Expanding in leading order in the small CKM mixing angles reproduced the known
result from Antusch:2009hq  (cf. Eqs. (5) - (7)). From our MCMC analyses, the
results obtained for $\delta_{12}^{d}-\delta_{12}^{u}$ are listed in Table 5.
Interestingly, in both sectors the experimental data points to a difference
between the 1-2 phases of about $\pm 90^{\circ}$, which might hint towards a
common origin, as will be discussed in the next section.

|  $\delta_{12}^{d}-\delta_{12}^{u}$ (1$\sigma$)
---|---
SM | $84.38^{\circ}-91.02^{\circ}$
MSSM | $84.38^{\circ}-91.05^{\circ}$
Table 5: The 1$\sigma$ HPD range for $\delta_{12}^{d}-\delta_{12}^{u}$
including running effects within the SM and the MSSM scenarios.

##  VIII Unified origin of $\delta^{\textrm{PMNS}}$ and
$\delta^{\textrm{CKM}}$

With the experimental data pointing to a difference between the 1-2 phases
$\delta_{12}^{\nu}-\delta^{e}_{12}$ and $\delta_{12}^{d}-\delta^{u}_{12}$ of
about $\pm 90^{\circ}$, we might think of textures of GUT Yukawa matrices
capable of explaining this situation in a simple way. In the following, we
like to point out a possibility which predicts both CP phases from a single
imaginary Yukawa matrix element. We consider an SU(5) GUT, where $Y_{d}$ and
$Y_{e}$ have the following form:333We remark that alternatively, one can also
choose a texture with $y_{32}=0$ but instead $y_{23}\not=0$ (and a
corresponding term $c_{23}y_{23}$ in $Y_{e}^{T}$), leading to the same
predictions for the Dirac CP phases.

$\displaystyle Y_{d}=\begin{pmatrix}0&y_{12}&0\\\ y_{21}&i\;y_{22}&0\\\
0&y_{32}&y_{33}\end{pmatrix},\;Y_{e}^{T}=\begin{pmatrix}0&c_{12}y_{12}&0\\\
c_{21}y_{21}&i\;c_{22}y_{22}&0\\\ 0&c_{32}y_{32}&c_{33}y_{33}\end{pmatrix},$
(46)

with all parameters $y_{ij}$ being real (as well as the Clebsch factors
$c_{ij}$). Only the 2-2 elements are purely imaginary, all other Yukawa matrix
elements are real. We note that in supersymmetric flavour models, with the
flavour structure generated via the breaking of family symmetries by “flavon
fields”, such specific values of the CP phases of Yukawa matrix entries can
readily be generated with spontaneous CP breaking and the method of “discrete
vacuum alignment” Antusch:2011sx . Interestingly, textures of this type can
also solve the strong CP problem, as has been pointed out in Antusch:2013rla .

Now, diagonalizing the mass matrices following Eq. (18), and collecting (12)-
and (22)-entries, we get,

$\displaystyle(M_{e})_{12}\approx
e^{i(-\delta^{e}_{12}+\chi^{e}_{2}-\chi^{e\prime}_{2})}\theta^{e}_{12}m_{\mu}+...\;,$
(47) $\displaystyle(M_{e})_{22}\approx
e^{i(\chi^{e}_{2}-\chi^{e\prime}_{2})}m_{\mu}+...\;,$ (48)

here small mixing angle approximation is adopted (both in the charged-lepton
and the down-quark sectors), and we ignore terms proportional to the first
generation fermion mass. Ellipses contain next-to-leading order contributions
that are small. Phases $\chi^{e}_{i}$ and $\chi^{e\prime}_{i}$ belong to the
left and the right rotation matrices. Finally, by dividing (12)- and
(22)-entries in Eq. (46) and comparing with Eqs. (47) and (48), we obtain444We
remark that variations of the above texture could also result in somewhat
modified predictions. Ultimately, the full flavour model will have to be
analysed in order to obtain the specific precise predictions.

$\displaystyle\delta^{d,e}_{12}=\frac{\pi}{2},$ (49)

and thus

$\displaystyle\arg\left(\frac{s^{\textrm{CKM}}_{13}-t^{\textrm{CKM}}_{12}t^{\textrm{CKM}}_{23}e^{-i\delta^{\textrm{CKM}}}}{t^{\textrm{CKM}}_{23}+e^{i\delta^{\textrm{CKM}}}t^{\textrm{CKM}}_{12}s^{\textrm{CKM}}_{13}}\right)=\frac{\pi}{2},$
(50)
$\displaystyle\arg\left(\frac{s^{\textrm{PMNS}}_{13}-t^{\textrm{PMNS}}_{12}t^{\textrm{PMNS}}_{23}e^{-i\delta^{\textrm{PMNS}}}}{t^{\textrm{PMNS}}_{23}+e^{i\delta^{\textrm{PMNS}}}t^{\textrm{PMNS}}_{12}s^{\textrm{PMNS}}_{13}}\right)=-\frac{\pi}{2},$
(51)

from Eqs. (45) and (44), respectively. As discussed above, these relations can
be used to calculate the predicted ranges for $\delta^{\textrm{CKM}}$ and
$\delta^{\textrm{PMNS}}$. The first relation corresponds to
$\alpha\approx\pi/2$, compatible with the current experimental data, whereas
the second relation corresponds (using central values of mixing angles and
without running) to the predictions $\delta^{\textrm{PMNS}}\approx
286^{\circ}$ ($\delta^{\textrm{PMNS}}\approx 290^{\circ}$) for
$\theta_{23}>45^{\circ}$ and ($\theta_{23}<45^{\circ}$). It is remarkable to
see that a single imaginary vacuum expectation value can be behind the origin
of two CP-violating phases, one in the quark sector and another in the lepton
sector. Future more precise measurements of $\delta^{\textrm{PMNS}}$ (and also
of $\delta^{\textrm{CKM}}$) will tell if this possibility could be realised in
nature.

##  IX Discussion and conclusions

The observation that in the quark sector as well as in the lepton sector, the
1-3 elements of the CKM and PMNS mixing matrices are much smaller than the
other elements, motivates the hypothesis that the 1-3 elements of all flavour
mixing matrices in the up-quark, down-quark, charged lepton and neutrino
sectors in fact all vanish (approximately) in the flavour basis, i.e. that
$\theta^{u,d,e,\nu}_{13}=0$ in terms of the most common parametrization. The
1-3 mixing in the observable CKM and PMNS mixing matrices are then induced by
the non-commutativity of the individual flavour rotations.

In this paper, we have revisited mixing sum rule relations in the lepton and
quark sectors under the assumption of this “zero 1-3 flavour mixing
hypothesis”, and noted that it provides a minimal assumption under which sum
rule relations can be derived that do not depend on unphysical phases.
Assuming that the zero 1-3 flavour mixing hypothesis holds, we have obtained
exact versions of the sum rule mixing relations and analysed their
implications based on the current experimental data, including effects from RG
running. In particular, we have analysed how the existing precise measurement
of $\theta_{13}^{\mathrm{PMNS}}$ allows to derive predictions for
$\theta_{23}^{\mathrm{PMNS}}$ in models with constrained 1-2 mixing
$\theta_{12}^{\mathrm{e}}$ in the charged lepton sector, from the exact
relation between $\theta_{13}^{\mathrm{PMNS}}$, $\theta_{23}^{\mathrm{PMNS}}$
and $\theta_{12}^{e}$ (cf. Eq. (27)). Our numerical results are visualized in
Fig. 1. As specific examples, we have calculated the predictions for
$\theta_{23}^{\mathrm{PMNS}}$ which arise in classes of Pati-Salam models and
SU(5) GUTs that relate $\theta_{12}^{\mathrm{e}}$ to
$\theta_{12}^{\mathrm{d}}$ (cf. Table 3). Furthermore, under the zero 1-3
flavour mixing hypothesis, we have also considered the exact lepton mixing
relation involving $\theta_{12}^{\nu}$ and $\delta_{\textrm{PMNS}}$ (cf. Eq.
(29)), and visualized our numerical results, which allow to read off the
predictions for $\delta_{\textrm{PMNS}}$ for given 1-2 mixing
$\theta_{12}^{\nu}$ in the neutrino sector, in Figs. 3 and 4. We have also
discussed the analogous exact relations that hold in the quark sector.

Let us now look at a few example applications to illustrate the usefulness of
our results. For instance, one can immediately judge if some of the the
existing GUT flavour models still provide a good fit to the present
experimental data on the PMNS parameters. For example, the two SU(5) GUT
flavour models presented in Antusch:2013kna  and Antusch:2013tta  satisfy the
zero 1-3 flavour mixing hypothesis to a good approximation. While
Antusch:2013kna  features a NH, it is inverted in Antusch:2013tta . Both
models use the Clebsch factor combination ($6,6,\frac{1}{2}$) and thus,
following the discussion in section VI and using the Table provided in
Antusch:2018gnu , predict $\theta_{12}^{e}\in(14.50^{\circ},15.16^{\circ})$.
From Fig. 1 one can clearly see that the resulting predictions for
$\theta_{23}^{\mathrm{PMNS}}$ are outside the current experimental $3\sigma$
range, so both models - which provided good fits to the data at the time they
were constructed - are now disfavoured by the current more precise
experimental results. From Figs. 3 and 4 one can also evaluate if models with,
e.g. tri-bimaxial or bimaximal neutrino mixing can be viable hypothesis for
model building (assuming the charged fermion sectors also satisfy the zero 1-3
flavour mixing hypothesis): one can see that while tri-bimaximal mixing,
leading to $\theta_{12}^{e}\approx 35.3^{\circ}$, predicts
$\delta_{\textrm{PMNS}}$ inside the current $1\sigma-2\sigma$ range, models
with bimaximal mixing can not be realised in consistency with the data.

Furthermore, we have also derived a novel “lepton phase sum rule” which,
however, in contrast to the mixing sum rule relations, only holds (free of
unphysical phases) in a small angle approximation for the charged lepton
mixing contributions. Confronting both, the quark and lepton phase sum rules
with the present data for the PMNS and CKM parameters, we point out that in
the context of GUT flavour models, the quark and lepton CP violating phases
can both be predicted from a single imaginary element in the mass matrices.

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