aainstitutetext: Institute for Theoretical Physics, Georg-August University
Göttingen,
Friedrich-Hund-Platz 1, Göttingen, D-37077 Germanybbinstitutetext: School of
Physics and Astronomy, University of Southampton,
Southampton SO17 1BJ, United Kingdom

# Neutrino mixing sum rules and the Littlest Seesaw

Francesco Costa b Stephen F. King<EMAIL_ADDRESS>goettingen.de<EMAIL_ADDRESS>
###### Abstract

In this work, we revisit neutrino mixing sum rules arising from discrete
symmetries, and the class of Littlest Seesaw neutrino models. These
theoretical models offer predictions for the leptonic CP phase and mixing
angles, and correlations among them, which can be tested in forthcoming
neutrino experiments. In particular we study the solar neutrino mixing sum
rules, arising from charged lepton corrections to Tri-bimaximal (TB), Bi-
maximal (BM), Golden Ratios (GRs) and Hexagonal (HEX) neutrino mixing, and
atmospheric neutrino mixing sum rules, arising from preserving one of the
columns of these types of mixing, for example the first or second column of
the TB mixing matrix (TM1 or TM2), and confront them with an up-to-date global
fit of the neutrino oscillation data. We show that some mixing sum rules, for
example an atmospheric neutrino mixing sum rule arising from a version of
neutrino Golden Ratio mixing (GRa1), are already excluded at 3$\sigma$, and
determine the remaining models allowed by the data. We also consider the
highly predictive Littlest Seesaw models, which are a special case of tri-
maximal mixing models (TM1), and discuss their prospects.

##  1 Introduction

Neutrino mass and mixing represents the first and so far only new physics
beyond the Standard Model (SM) of particle physics. We know it must be new
physics because its origin is unknown and it is not predicted by the SM.
Independently of the whatever the new (or nu) SM is, we do know that the
minimal paradigm involves three active neutrinos, the weak eigenstates
$\nu_{e},\nu_{\mu},\nu_{\tau}$ (the $SU(2)_{L}$ partners to the left-handed
charged lepton mass eigenstates) which are related to the three mass
eigenstates $m_{1,2,3}$ by a unitary PMNS mixing matrix Workman:2022ynf .

The PMNS matrix is similar to the CKM matrix which describes quark mixing, but
involves three independent leptonic mixing angles
$\theta_{23},\theta_{13},\theta_{12}$ (or $s_{23}=\sin\theta_{23}$,
$s_{13}=\sin\theta_{13}$, $s_{12}=\sin\theta_{12}$), one leptonic CP violating
Dirac phase $\delta$ which affects neutrino oscillations, and possibly two
Majorana phases which do not enter into neutrino oscillation formulas.
Furthermore neutrino oscillations only depend on the two mass squared
differences $\Delta m^{2}_{21}=m_{2}^{2}-m_{1}^{2}$, which is constrained by
data to be positive, and $\Delta m^{2}_{31}=m_{3}^{2}-m_{1}^{2}$, which
current data allows to take a positive (normal) or negative (inverted) value.
In 1998, the angle $\theta_{23}$ was first measured to be roughly $45^{o}$
(consistent with equal bi-maximal $\nu_{\mu}-\nu_{\tau}$ mixing) by
atmospheric neutrino oscillations, while $\theta_{12}$ was determined to be
roughly $35^{o}$ (consistent with equal tri-maximal
$\nu_{e}-\nu_{\mu}-\nu_{\tau}$ mixing) in 2002 by solar neutrino oscillation
experiments, while $\theta_{13}$ was first accurately found to be $8.5^{o}$ in
2012 by reactor oscillation experiments.

Various simple ansatzes for the PMNS matrix were proposed, the most simple
ones involving a zero reactor angle and bimaximal atmospheric mixing,
$s_{13}=0$ and $s_{23}=c_{23}=1/\sqrt{2}$, leading to a PMNS matrix of the
form,

$\displaystyle U_{0}=\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-\frac{s_{12}}{\sqrt{2}}&\frac{c_{12}}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
\frac{s_{12}}{\sqrt{2}}&-\frac{c_{12}}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right),$
(4)

where the zero subscript reminds us that this form has $\theta_{13}=0$ (and
$\theta_{23}=45^{\circ}$).

For golden ratio (GRa) mixing Datta:2003qg , the solar angle is given by
$\tan\theta_{12}=1/\phi$, where $\phi=(1+\sqrt{5})/2$ is the golden ratio
which implies $\theta_{12}=31.7^{\circ}$. There are two alternative versions
where $\cos\theta_{12}=\phi/2$ and $\theta_{12}=36^{\circ}$ Rodejohann:2008ir
which we refer to as GRb mixing, and GRc where $\cos\theta_{12}=\phi/\sqrt{3}$
and $\theta_{12}\approx 20.9^{\circ}$.

For bimaximal (BM) mixing (see e.g. Davidson:1998bi ; Altarelli:2009gn ;
Meloni:2011fx  and references therein), we insert $s_{12}=c_{12}=1/\sqrt{2}$
($\theta_{12}=45^{\circ}$) into Eq. (4),

$\displaystyle
U_{\mathrm{BM}}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\
-\frac{1}{2}&\frac{1}{2}&\frac{1}{\sqrt{2}}\\\
\frac{1}{2}&-\frac{1}{2}&\frac{1}{\sqrt{2}}\end{array}\right).$ (8)

For tri-bimaximal (TB) mixing Harrison:2002er , alternatively we use
$s_{12}=1/\sqrt{3}$, $c_{12}=\sqrt{2/3}$ ($\theta_{12}=35.26^{\circ}$) in Eq.
(4),

$\displaystyle
U_{\mathrm{TB}}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right).$
(12)

Finally another pattern studied in the literature with $\theta_{13}=0$ (and
$\theta_{23}=45^{\circ}$) is the hexagonal mixing (HEX) where
$\theta_{12}=\pi/6$.

These proposals are typically by finite discrete symmetries such as
$A_{4},S_{4},S_{5}$ (for a review see e.g. King:2013eh ). After the reactor
angle was measured, which excluded all these ansatze, there were various
proposals to rescue them and hence to maintain the notion of predictivity of
the leptonic mixing parameters, in particular the Dirac CP phase $\delta$,
which is not directly measured so far and remains poorly determined even
indirectly. Two approaches have been developed, in which some finite symmetry
(typically a subgroup of $A_{4},S_{4},S_{5}$) can enforce a particular
structure of the PMNS matrix consistent with a non-zero reactor angle, leading
to solar and atmospheric sum rules, as we now discuss.

The first approach, which leads to solar sum rules, is to assume that the
above patterns of mixing still apply to the neutrino sector, but receive
charged lepton mixing corrections due to the PMNS matrix being the product of
two unitary matrices, which in our convention is written as
$V_{eL}V^{\dagger}_{\nu_{L}}$, where $V^{\dagger}_{\nu_{L}}$ is assumed to
take the BM, TB or GR form, while $V_{eL}$ differs from the unit matrix. If
$V_{eL}$ involves negligible 13 charged lepton mixing, then it is possible to
generate a non-zero 13 PMNS mixing angle, while leading to correlations
amongst the physical PMNS parameters, known as solar mixing sum rules
King:2005bj ; Masina:2005hf ; Antusch:2005kw ; Antusch:2007rk . This scenario
may be enforced by a subgroup of $A_{4},S_{4},S_{5}$ which enforces the
$V_{\nu}$ structure King:2013eh  while allowing charged lepton corrections.

In the second approach, which leads to atmospheric sum rules, it is assumed
that the physical PMNS mixing matrix takes the BM, TB or GR form but only in
its first or second column, while the third column necessarily departs from
these structures due to the non-zero 13 angle. Such patterns again lead to
correlations amongst the physical PMNS parameters, known as atmospheric mixing
sum rules. This scenario may be enforced by a subgroup of $A_{4},S_{4},S_{5}$
which enforces the one column $V_{\nu}$ structure King:2013eh  while
forbidding charged lepton corrections.

Apart from the large lepton mixing angles, another puzzle is the extreme
lightness of neutrino masses. Although the type I seesaw mechanism can
qualitatively explain the smallness of neutrino masses through the heavy
right-handed neutrinos (RHNs), if one doesn’t make other assumptions, it
contains too many parameters to make any particular predictions for neutrino
mass and mixing. The sequential dominance (SD) King:1998jw ; King:1999cm  of
right-handed neutrinos proposes that the mass spectrum of heavy Majorana
neutrinos is strongly hierarchical, i.e. $M_{\text{atm}}\ll M_{\text{sol}}\ll
M_{\text{dec}}$, where the lightest RHN with mass $M_{\text{atm}}$ is
responsible for the atmospheric neutrino mass, that with mass $M_{\text{sol}}$
gives the solar neutrino mass, and a third largely decoupled RHN gives a
suppressed lightest neutrino mass. It leads to an effective two right-handed
neutrino (2RHN) model King:1999mb ; Frampton:2002qc  with a natural
explanation for the physical neutrino mass hierarchy, with normal ordering and
the lightest neutrino being approximately massless, $m_{1}=0$.

A very predictive minimal seesaw model with two right-handed neutrinos and one
texture zero is the so-called constrained sequential dominance (CSD) model
King:2005bj ; Antusch:2011ic ; King:2013iva ; King:2015dvf ; King:2016yvg ;
Ballett:2016yod ; King:2018fqh ; King:2013xba ; King:2013hoa ;
Bjorkeroth:2014vha . The CSD($n$) scheme, also known as the Littlest Seesaw,
assumes that the two columns of the Dirac neutrino mass matrix are
proportional to $(0,1,-1)$ and $(1,n,2-n)$ or $(1,2-n,n)$ respectively in the
RHN diagonal basis (or equivalently $(0,1,1)$ and $(1,n,n-2)$ or $(1,n-2,n)$)
where the parameter $n$ was initially assumed to be a positive integer, but in
general may be a real number. For example the CSD($3$) (also called Littlest
Seesaw model) King:2013iva ; King:2015dvf ; King:2016yvg ; Ballett:2016yod ;
King:2018fqh , CSD($4$) models King:2013xba ; King:2013hoa  and CSD($2.5$)
Chen:2019oey  can give rise to phenomenologically viable predictions for
lepton mixing parameters and the two neutrino mass squared differences $\Delta
m^{2}_{21}$ and $\Delta m^{2}_{31}$, corresponding to special constrained
cases of lepton mixing which preserve the first column of the TB mixing
matrix, namely TM1 and hence satisfy atmospheric mixing sum rules. As was
observed, modular symmetry remarkably suggests CSD($1+\sqrt{6}$) $\approx$
CSD($3.45$) Ding:2019gof ; Ding:2021zbg ; deMedeirosVarzielas:2022fbw ;
deAnda:2023udh .

In this paper we shall revisit neutrino solar and atmospheric mixing sum rules
arising from discrete symmetries, and also discuss the class of Littlest
Seesaw models. These theoretical models offer predictions for the leptonic CP
phase and mixing angles, and correlations among them, which can be tested in
forthcoming neutrino experiments. In particular we study the solar neutrino
mixing sum rules, arising from charged lepton corrections to TB, BM and GR
neutrino mixing, and atmospheric neutrino mixing sum rules, arising from
preserving one of the columns of these types of mixing, for example the first
or second column of the TB mixing matrix (TM1 or TM2), and confront them with
an up-to-date global fit of the neutrino oscillation data. We show that some
mixing sum rules, for example all the atmospheric neutrino mixing sum rule
arising from a Golden Ratio mixings are already excluded at 3$\sigma$ a part
from GRa2, and determine the remaining models allowed by the data. We also
consider the highly predictive Littlest Seesaw models, which are a special
case of tri-maximal mixing models (TM1), and discuss their prospects.

The layout of the remainder of the paper is as follows. In Chapter 2 we
introduce the notation for the PMNS matrix and discuss the symmetries of the
leptonic Lagrangian. In Chapter 3 and 4 we introduce the atmospheric and solar
sum rules for the different models we are studying and confront them with the
up-to-date neutrino data global fit. We proceed in Chapter 5 discussing the
CDS and the Littlest Seesaw model, showing its high predictivity and the
viable parameter space given the experimental data and its fit. Finally we
conclude in Chapter 6.

##  2 Lepton mixing and symmetries

The mixing matrix in the lepton sector, the PMNS matrix $U_{\mathrm{PMNS}}$,
is defined as the matrix which appears in the electroweak coupling to the $W$
bosons expressed in terms of lepton mass eigenstates. With the mass matrices
of charged leptons $M_{e}$ and neutrinos $m^{\nu}_{LL}$ written as111Although
we have chosen to write a Majorana mass matrix, all relations in the following
are independent of the Dirac or Majorana nature of neutrino masses.

$\displaystyle
L=-\overline{e_{L}}M^{e}e_{R}-\frac{1}{2}\overline{\nu_{L}}M^{\nu}\nu_{L}^{c}+{H.c.}\;,$
(13)

and performing the transformation from flavour to mass basis by

$\displaystyle
V_{e_{L}}\,M^{e}\,V^{\dagger}_{e_{R}}=\mbox{diag}(m_{e},m_{\mu},m_{\tau}),\quad~{}V_{\nu_{L}}\,M^{\nu}\,V^{T}_{\nu_{L}}=\mbox{diag}(m_{1},m_{2},m_{3}),$
(14)

the PMNS matrix is given by

$\displaystyle U_{\mathrm{PMNS}}=V_{e_{L}}V^{\dagger}_{\nu_{L}}\,.$ (15)

Here it is assumed implicitly that unphysical phases are removed by field
redefinitions, and $U_{\mathrm{PMNS}}$ contains one Dirac phase and two
Majorana phases. The latter are physical only in the case of Majorana
neutrinos, for Dirac neutrinos the two Majorana phases can be absorbed as
well.

According to the above discussion, the neutrino mass and flavour bases are
misaligned by the PMNS matrix as follows,

$\displaystyle\begin{aligned} \left(\begin{array}[]{c}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{array}\right)=&\left(\begin{array}[]{ccc}U_{e1}&U_{e2}&U_{e3}\\\
U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau
3}\end{array}\right)\left(\begin{array}[]{c}\nu_{1}\\\ \nu_{2}\\\
\nu_{3}\end{array}\right)\equiv U_{\text{PMNS
}}\left(\begin{array}[]{c}\nu_{1}\\\ \nu_{2}\\\
\nu_{3}\end{array}\right),\end{aligned}$ (16)

where $\nu_{e},\nu_{\mu},\nu_{\tau}$ are the $SU(2)_{L}$ partners to the left-
handed charged lepton mass eigenstates and $\nu_{1,2,3}$ are the neutrinos in
their mass basis. Following the standard convention we can describe
$U_{\text{PMNS }}$ in terms of three angles, one CP violation phase and two
Majorana phases

$\displaystyle\begin{aligned} U_{\text{PMNS
}}=&\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-\mathrm{i}\delta}\\\
0&1&0\\\
-s_{13}e^{\mathrm{i}\delta}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right)P,\end{aligned}$ (17)

$\displaystyle=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{array}\right)P,$
(21)

where $P$ contains the Majorana phases

$\displaystyle
P=\operatorname{diag}\left(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2}\right),$
(22)

The current $3\sigma$ parameters intervals coming from the global fit of the
neutrino oscillation data by the nuFIT collaboration Esteban:2020cvm  are

$\displaystyle\theta_{12}=[31.31^{\circ},35.74^{\circ}],\quad\theta_{23}=[39.6^{\circ},51.9^{\circ}],\quad\theta_{13}=[8.19^{\circ},8.89^{\circ}],\quad$
(23)
$\displaystyle\delta=[0^{\circ},44^{\circ}]\quad\&\quad[108^{\circ},360^{\circ}],\quad\frac{\Delta_{21}^{2}}{10^{-5}\mathrm{eV}^{2}}=[6.82,8.03],\quad\frac{\Delta_{3l}^{2}}{10^{-3}\mathrm{eV}^{2}}=[2.428,2.597].\quad\quad$
(24)

The PMNS matrix reads

$\displaystyle|U|_{3\sigma}^{\text{w/o SK-atm
}}=\left(\begin{array}[]{lll}0.803\rightarrow 0.845&0.514\rightarrow
0.578&0.142\rightarrow 0.155\\\ 0.233\rightarrow 0.505&0.460\rightarrow
0.693&0.630\rightarrow 0.779\\\ 0.262\rightarrow 0.525&0.473\rightarrow
0.702&0.610\rightarrow 0.762\end{array}\right).$ (28)

These results are obtained considering normal ordering, which is the current
best fit, and without including the Super-Kamiokande (SK) data.

Figure 1: Subgroups of $SU(3)$ with triplet representations. The smaller of
two groups connected in the graph is a subset of the other. Figure from
King:2013eh .

Simple mixing patter such TB, BM or GR could explain the first neutrino
oscillation data. These patterns can be enforced via symmetries of the mass
matrices. Let us take a basis where the charged lepton $M_{e}$ mass matrix is
diagonal and we notice that for 3 generations we have that $Z^{T}_{3}$ is a
symmetry of the Lagrangian

$\displaystyle
T^{\dagger}\left(M_{e}^{\dagger}M_{e}\right)T=M_{e}^{\dagger}M_{e},$ (29)

where $T=\operatorname{diag}\left(1,\omega^{2},\omega\right)$ and
$\omega=e^{i2\pi/3}$. The light Majorana neutrino mass matrix is invariant
under the Klein symmetry: $Z_{2}^{U}\times Z_{2}^{S}$. This can be seen taking
the diagonal neutrino mass matrix and performing the transformations

$\displaystyle M^{\nu}=S^{T}M^{\nu}S,\quad M^{\nu}=U^{T}M^{\nu}U,$ (30)

and $M^{\nu}$ is left invariant with

$\displaystyle\begin{gathered}S=U_{\mathrm{PMNS}}^{*}\operatorname{diag}(+1,-1,-1)U_{\mathrm{PMNS}}^{T}\\\
U=U_{\mathrm{PMNS}}^{*}\operatorname{diag}(-1,+1,-1)U_{\mathrm{PMNS}}^{T},\end{gathered}$
(33)

where this result follows from the fact that, in the charged lepton mass
eigenstate basis, the neutrino mass matrix is diagonalised by
$U_{\mathrm{PMNS}}$ as in Eq. (14), where any two diagonal matrices commute.
Then Eq. (33) shows that the matrices $S,U$ are both diagonalised by the same
matrix $U_{\mathrm{PMNS}}$ that also diagonalises the neutrino mass matrix.
Given this result, we can always find the two matrices $S,U$ for any PMNS
mixing matrix, and hence the Klein symmetry is present for any choice of the
PMNS mixing. However not all Klein symmetries may be identified with finite
groups of low order.

This description is meaningful if the charged leptons are diagonal (T is
conserved) or approximately diagonal (T is softly broken). We are therefore
interested in finite groups that are superset of $Z_{2}^{U}\times Z_{2}^{S}$
and $Z_{3}^{T}$ and have a triplet representation. Groups of low order that
satisfy these constraints are given in Figure 1.

Figure 2: A schematic diagram that illustrate the way that the two subgroups
$Z_{2}^{U}\times Z_{2}^{S}$ and $Z_{3}^{T}$ of a finite group work in the
charged lepton and neutrino sectors in order to enforce a particular pattern
of PMNS mixing. In this example, the group $S_{4}$ leads to TB mixing.

One simple example is the group $G=S_{4}$, of order 24, which is the group of
permutation of 4 objects. The generators follow the presentation rules
King:2013eh

$\displaystyle
S^{2}=T^{3}=(ST)^{3}=U^{2}=(TU)^{2}=(SU)^{2}=(STU)^{4}=\mathbf{1},$ (34)

The two possible $S_{4}$ triplet irreducible representations with a standard
choice of basis King:2009ap , gives the generators explicit expression

$\displaystyle S=\frac{1}{3}\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right),\quad T=\left(\begin{array}[]{ccc}1&0&0\\\
0&\omega^{2}&0\\\ 0&0&\omega\end{array}\right),\quad
U=\mp\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\end{array}\right),$
(44)

where again $\omega=e^{i2\pi/3}$ and the sign of the $U$ matrix corresponds to
the two different triplet representation. The group $S_{4}$ predicts a TB
mixing Harrison:2002er , see Figure LABEL:fig:tbm. This can be checked by the
fact that $S$ and $U$ are diagonalised by $U_{\mathrm{TB}}$, see Equations
(33). Another commonly used group is $A_{4}$, which has two generators $S$ and
$U$ that follow the same presentation rules as in Equation (34) and in a
standard basis Altarelli:2005yx , the generators have the same form as in
Equation (44).

Figure 3: In order to generate a non-zero (13) PMNS element, one or more of
the generators $S,T,U$ must be broken. In the left panel we depict $T$
breaking leading to charged lepton mixing corrections and possible solar sum
rules. In the right panel, $U$ is broken, while either $S$ or $SU$ is
preserved leading to neutrino mixing corrections and atmospheric sum rules.

In order to explain the experimental results $G$ needs to be broken and
generate a non-zero $(13)$ PMNS element. This will lead to corrections to the
leading order PMNS predictions from the discrete group $G$. In Figure 3 we
illustrate two possible direction we can proceed to do that. The first one is
to break the T generator while the Klein symmetry in the neutrino sector is
exact (left hand side). This means that the charged lepton matrix is
approximately diagonal. In the mass basis we will have then a correction to
the neutrino mixing matrix by a unitary matrix $V_{e}$ and the PMNS is now
$U_{\mathrm{PMNS}}=V_{e}^{\dagger}V_{\nu}$. Applying this to a group $G$ will
lead to solar sum rules. The second direction is to preserve $Z_{3}^{T}$ but
breaking $Z_{2}^{U}$ while keeping either $Z_{2}^{SU}$ or $Z_{2}^{S}$ unbroken
(right hand side). This leads to corrections to the prediction of $G$ within
the neutrino mixing and to atmospheric sum rules. It is convenient to
introduce small parameters that can simplify the sum rules expressions and
help us understand their physical behaviour since both in solar and
atmospheric sum rules we implement a small deviation from the prediction of
the exact finite discrete symmetries. We can consider the deviation parameters
$s,r,a$ King:2007pr

$\displaystyle\sin\theta_{12}\equiv\frac{1+s}{\sqrt{3}},\quad\sin\theta_{13}\equiv\frac{r}{\sqrt{2}},\quad\sin\theta_{23}\equiv\frac{1+a}{\sqrt{2}},$
(45)

that highlight the differences from TB mixing. Given the latest fit the
$3\sigma$ allowed range for the solar, reactor and atmospheric deviation are
respectively

$\displaystyle\begin{gathered}-0.0999<\;s<0.0117,\\\ 0.20146<\;r<0.21855,\\\
-0.0985<\;a<0.1129.\end{gathered}$ (49)

This shows that the reactor angle differs from zero significantly ($r\neq 0$),
but the solar and atmospheric angles remain consistent with TB mixing
($s=a=0$) at $3\sigma$. From a theoretical point of view, one of the goals of
the neutrino experiments would be to exclude the TB prediction $s=a=0$
King:2012vj , which is so far still allowed at $3\sigma$.

##  3 Solar sum rules

The first possibility to generate a non-zero reactor angle, whilst maintaining
some of the predictivity of the original mixing patterns, is to allow the the
charged lepton sector to give a mixing correction to the leading order mixing
matrix $U_{\nu}$. This will lead to the so-called solar sum rules, that are
relations between the parameters that can be tested. This operation is
equivalent to considering the $T$ generator of the $S_{4}$ symmetry which
enforces the charged lepton mass matrix to be diagonal (in our basis) to be
broken.

When the $T$ generator is broken, the charged lepton matrix is not exactly
diagonal and it will give a correction to the PMNS matrix predicted by the
symmetry group $G$. For example for the $S_{4}$, $U_{\mathrm{PMNS}}$ is not
exactly $U_{\mathrm{TB}}$ but it receives a correction that we will compute.
The fact that $S$ and $U$ are preserved leads to a set of correlations among
the physical parameters, the solar sum rules which are the prediction of the
model. For the solar sum rules we can obtain a prediction for $\cos\delta$ as
we shall now show.

For example consider the case of TB neutrino mixing with the charged lepton
mixing corrections involving only (1,2) mixing, so that the PMNS matrix in Eq.
15 is given by,

$\\!U_{\mathrm{PMNS}}=\left(\begin{array}[]{ccc}\\!c^{e}_{12}&s^{e}_{12}e^{-i\delta^{e}_{12}}&0\\!\\\
\\!-s^{e}_{12}e^{i\delta^{e}_{12}}&c^{e}_{12}&0\\!\\\
\\!0&0&1\\!\end{array}\right)\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right)=\left(\begin{array}[]{ccc}\\!\cdots&\
\ \\!\cdots&\\!\frac{s^{e}_{12}}{\sqrt{2}}e^{-i\delta^{e}_{12}}\\\ \\!\cdots&\
\ \\!\cdots&\\!\frac{c^{e}_{12}}{\sqrt{2}}\\!\\\
\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right)$
(50)

The elements of the PMNS matrix are clearly related by Antusch:2007rk ;
Ballett:2014dua

$\displaystyle\frac{|U_{\tau 1}|}{|U_{\tau
2}|}=\frac{s_{12}^{\nu}}{c_{12}^{\nu}}=t_{12}^{\nu}=\frac{1}{\sqrt{2}}.$ (51)

This relation is easy to understand if we consider only one charged lepton
angle to be non-zero, $\theta^{e}_{12}$ then the third row of the PMS matrix
in Eq. 50 is unchanged, so the elements $U_{\tau i}$ may be identified with
the corresponding elements in the uncorrected mixing matrix in Eq.4.
Interestingly, the above relation still holds even if both $\theta^{e}_{12}$
and $\theta^{e}_{23}$ are non-zero. However it fails if $\theta^{e}_{13}\neq
0$ Antusch:2022ufb .

The above relation in Eq.51 can be translated into a prediction for
$\cos\delta$ as Ballett:2014dua 222See also Marzocca:2013cr .

$\displaystyle\cos\delta=\frac{\tan\theta_{23}\sin\theta_{12}^{2}+\sin\theta_{13}^{2}\cos\theta_{12}^{2}/\tan\theta_{23}-(\sin\theta_{12}^{\nu})^{2}\left(\tan\theta_{23}+\sin\theta_{13}^{2}/\tan\theta_{23}\right)}{\sin
2\theta_{12}\sin\theta_{13}},$ (52)

where only the parameter $\sin\theta_{12}^{\nu}$ is model dependent and we
have respectively $\sin\theta_{12}^{\nu}=1/\sqrt{3}$,
$\sin\theta_{12}^{\nu}=1/\sqrt{2}$, $\tan\theta_{12}^{\nu}=1/\varphi$ and
$\theta_{12}^{\nu}=\pi/5$, $\cos\theta_{12}^{\nu}=\varphi/\sqrt{3}$ and
$\theta_{12}^{\nu}=\pi/6$ for mixing based on TB, BM, GRa, GRb, GRc and HEX
where $\varphi=(1+\sqrt{5})/2$.

Let us discuss an approximation of the sum rules for the TB mixing as an
example, where $\sin\theta_{12}^{\nu}=1/\sqrt{3}$. We can re-write Equation
(52) using the parameters $s$, $a$ and $r$ defined in Equation (45) and then
expand in them. The linearised sum rule reads King:2007pr

$\displaystyle\cos\delta=\frac{s}{r},$ (53)

but it does not describe adequately the exact sum rules as shown in the left
panel of Figure 4.

Figure 4: Solar mixing sum rule predictions for TB neutrino mixing. In the
both panels the red band is the allowed region of the exact TB solar sum rules
using the 3$\sigma$ range of r (i.e. the deviation of $\sin\theta_{13}$) and
it is plotted in the 3$\sigma$ range of s (i.e. the deviation of
$\sin\theta_{23}$ from the TB value) and using the best fit value for
$a=0.071$. The blue band is given by the linearised sum rule. In the right
panel the blue band is the second order expansion sum rule prediction, it
matches the exact sum rule.

Figure 5: Solar mixing sum rule predictions for BM neutrino mixing. In the
both panels the red band is the allowed region of the exact BM solar sum rules
using the 3$\sigma$ range of r (i.e. the deviation of $\sin\theta_{13}$) and
it is plotted in the 3$\sigma$ range of s (i.e. the deviation of
$\sin\theta_{23}$ from the TB value) and using the value $a=-0.1$. The blue
band is given by the linearised sum rule in the left panel. In the right panel
the blue band is the second order expansion sum rule prediction, it matches
the exact sum rule.

Therefore we can go to the second order expansion, which is

$\displaystyle\cos\delta=\frac{s}{r}+\frac{r^{2}+8as}{4r},$ (54)

and it matches the exact sum rule behaviour as seen on the right panel in
Figure 4. Similarly we can obtain higher order expansion for the other cases
and check them against the data, like for the BM case showed in Figure 5. In
this case we did not choose the best fit value for $a$ because otherwise it
would fall out of the physical range of $\cos\delta$ since BM is almost
excluded by the data. The approximated expression for the sum rules can help
us understand its behaviour and the dependence of $\cos\delta$ on the other
parameters that are in general non-linear and assess the deviation from the
non-corrected PMNS mixing. We then expect for the exact sum rules a first
order linear dependence on $s$.

Figure 6: Summary of exact solar sum rule predictions for different types of
neutrino mixing. In the top left hand panel we present with the different
colored band the sum rule prediction for TB for $\cos\delta$ letting
$\sin\theta_{12}$ vary in its $3\sigma$ range, the different color denoted
different choice of $\sin\theta_{23}$ given in the legend, in its $3\sigma$
range and the width of the band is given by the $3\sigma$ range in
$\sin\theta_{13}$. The green and yellow band are the $1\sigma$ range for
respectively $\cos\delta$ and $\sin\theta_{23}$. Similar plots for BM, GRa,
GRb, GRc and HEX are presented respectively on the top right, center right,
center left, bottom left, bottom right panels.

In Figure 6 we present the exact sum rules prediction from Equation (52) for
TB, BM, GRa, GRb, GRc and HEX and the constraints from the fit of the neutrino
oscillation data Esteban:2020cvm . We require $\cos\delta$ to fall in the
physical range $-1<\cos\delta<1$ and we present it in the y-axis. In all panel
the x-axis is $\sin^{2}\theta_{12}$ and the different colour bands are sampled
in the allowed $\sin\theta_{23}$ region. The width of the band is given by
allowing $\sin\theta_{13}$ to vary in its $3\sigma$ range. We notice that the
$\theta_{12}^{\nu}=45^{\circ}$ BM mixing (top-right panel) is closed to be
excluded at $3\sigma$ and only low values of $\sin^{2}\theta_{12}$ and high
values of $\sin^{2}\theta_{12}$ are still viable. Similarly for GRc mixing
(bottom-left panel), with $\cos\theta_{12}^{\nu}=\varphi/3$, the viable
parameter space is very tight, only for maximal values of $\sin\theta_{13}$
and minimal values of $\sin\theta_{12}$ and $\sin\theta_{23}$ we can obtain
physical results for the CP phase. For TB mixing (top-left panel) with
$\sin\theta_{12}^{\nu}=1/\sqrt{3}$ in the neutrino sector with charged lepton
correction lead consistent results in all parameters space, with the
prediction for $\cos\delta$ that shows an approximately linear dependence on
$\sin^{2}\theta_{12}$ as understood by the leading order term in the sum rules
in Equation (53). The prediction for the CP phase lies in the
$0.52\lesssim\cos\delta\lesssim 0.12$ range. The yellow and green bands are
the $1\sigma$ range respectively of $\sin^{2}\theta_{12}$ and $\cos\delta$ and
we notice how these ranges favor GRa and GRb mixing. For both these models we
see that the prediction of $\cos\delta$ are in the negative plane. For GRa
(center-left panel), with $\tan\theta_{12}^{\nu}=1/\varphi$, the whole
parameter space leads to physical prediction of $\cos\delta$. For GRb (center-
right panel), with $\theta_{12}^{\nu}=\pi/5$ mixing, larger values
$\sin\theta_{23}$ are excluded for small values of $\sin^{2}\theta_{12}$. We
finally notice that TM and HEX are the only model predicting positive values
of $\cos\delta$ and HEX (bottom-right panel), with $\theta_{12}^{\nu}=\pi/6$
in particular the only predicting values of $\cos\delta\gtrsim 0.2$. Of the
mixing pattern we studied GRa and GRb are favoured by the current $1\sigma$
ranges and BM and GRc are much disfavoured and only consistent with the far
corners of the parameter space with a prediction of $|\cos\delta\;|\approx 1$.

##  4 Atmospheric sum rules

In this section we discuss the second possibility, that is to have the T
generator unbroken, therefore the charged lepton mixing matrix is exactly
diagonal. In this case the correction to the PMNS matrix predicted from the
group $G$ comes from the neutrino sector and it provides a non zero reactor
angle. For each group there are two possible corrections achieved either
breaking U and preserving S or with S and U broken and SU preserved. Therefore
for each discrete symmetry we will study two mixing pattern Ballett:2013wya ;
Hernandez:2012ra ; Luhn:2013lkn .

Let us consider again $G=S_{4}$ and the TB mixing in Equation (12) as an
example. If we break $S$ and $U$ but preserve $SU$ the first column of the TB
matrix is preserved and we have the so-called TM1 mixing pattern
Albright:2008rp ; Albright:2010ap

$\displaystyle
U_{\mathrm{TM}1}\approx\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&-&-\\\
-\frac{1}{\sqrt{6}}&-&-\\\ \frac{1}{\sqrt{6}}&-&-\end{array}\right),$ (58)

if instead $S$ is unbroken the second column is preserved and we have the
second mixing pattern TM2

$\displaystyle
U_{\mathrm{TM}2}\approx\left(\begin{array}[]{ccc}-&\sqrt{\frac{1}{3}}&-\\\
-&\sqrt{\frac{1}{3}}&-\\\ -&-\sqrt{\frac{1}{3}}&-\end{array}\right).$ (62)

We can explicitly check this noticing that

$\displaystyle S\,\left(\begin{array}[]{c}\sqrt{\frac{1}{3}}\\\
\sqrt{\frac{1}{3}}\\\
\sqrt{\frac{1}{3}}\end{array}\right)=\left(\begin{array}[]{c}\sqrt{\frac{1}{3}}\\\
\sqrt{\frac{1}{3}}\\\ \sqrt{\frac{1}{3}}\end{array}\right),$ (69)

meaning that the second column of the TB mixing matrix is an eigenvector of
the S matrix. Similarly for the first column with the $SU$ matrix. In this
second case where the second column of TB matrix is conserved we have

$\displaystyle\left|U_{e2}\right|=\left|U_{\mu 2}\right|=\left|U_{\tau
2}\right|=\frac{1}{\sqrt{3}},$ (70)

and given the parametrisation in Equation (21) we have

$\displaystyle\left|U_{e2}\right|=|s_{12}$ $\displaystyle
c_{13}|,\quad\left|U_{\mu
2}\right|=|c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}|,$ (71)
$\displaystyle\left|U_{\tau
2}\right|=|-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}|.$ (72)

Figure 7: The red band is the allowed region of the exact TM2 sum rules using
the 3$\sigma$ range of r (i.e. the deviation of $\sin\theta_{13}$ and
$\sin\theta_{23}$ from the TB value). The red band is given by the linearised
sum rule. On the right we zoom on the region $-0.1<a<0$.

Using the first equation $\left|U_{e2}\right|=|s_{12}c_{13}|$ we have the
first atmospheric sum rule

$\displaystyle s_{12}^{2}=\frac{1}{3\,c^{2}_{13}},$ (73)

that allows us to write $\theta_{12}$ in terms of $\theta_{13}$ and removing a
parameter in our description and gives a prediction that can be tested. Using
Equation (73) and
$|c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}|^{2}=\frac{1}{3}$ we obtain the
second atmospheric sum rule Albright:2008rp ; Albright:2010ap

$\displaystyle\cos\delta=\frac{2c_{13}\cot 2\theta_{23}\cot
2\theta_{13}}{\sqrt{2-3s_{13}^{2}}}.$ (74)

For the other models the discussion is similar where we call $X_{1}$ and
$X_{2}$ the atmospheric sum rules respectively derived by preserving the first
and second column of the unbroken group with mixing $X$. In terms of the
deviation parameters for TM2 we have the sum rule

$\displaystyle\cos\delta=\frac{2a(2+a)\left(-1+r^{2}\right)}{(1+a)\sqrt{1-2a-a^{2}}r\sqrt{4-3r^{2}}}.$
(75)

We can expand this expression for small deviation parameters and at the zero-
th order we have Ballett:2013wya

$\displaystyle\cos\delta=-\frac{2a}{r}$ (76)

and in Figure 7 we test this approximation against the exact sum rules using
the experimental constraint in (24). We can see that given the updated data
the linear approximation is now insufficient to describe the exact expression
as it was instead in previous studies Ballett:2013wya . Similarly for TM1, as
seen in Figure 8. This is true for the other model we will discuss later and
therefore we provide the higher order expansions that agrees with the exact
sum rule in Equation (75) given the current data and is

$\displaystyle\cos\delta=-\frac{2a}{r}-\frac{a^{2}}{r}$ (77)

For the TM2 example we see in Figure 7 that the second order expansion is a
good description of the exact sum rule. For TM1 instead, as shown in Figure 8
the third order expansion is needed.

Figure 8: The red band is the allowed region of the exact TM2 sum rules using the 3$\sigma$ range of r (i.e. the deviation of $\sin\theta_{13}$ and $\sin\theta_{23}$ from the TB value). The blue band is given by the second order sum rule. On the right we zoom on the region $-0.1<a<0$.  | Exact sum rule | Approximated sum rule
---|---|---
TM1 | $\cos\delta=-\frac{\cot 2\theta_{23}\left(1-5\sin\theta_{13}^{2}\right)}{2\sqrt{2}\sin\theta_{13}\sqrt{1-3\sin\theta_{13}^{2}}}$ | $\cos\delta=\frac{a}{r}+\frac{a^{2}}{2r}+\frac{2a^{3}}{r}-\frac{7ar}{4}$
TM2 | $\cos\delta=\frac{2\cos\theta_{13}\cot 2\theta_{23}\cot 2\theta_{13}}{\sqrt{2-3\sin\theta_{13}^{2}}}$ | $\cos\delta=-\frac{2a}{r}-\frac{a^{2}}{r}$
GRa2 | $\cos\delta=\frac{(1-\tan^{2}\theta_{23})\csc\theta(1-3\sin^{2}\theta_{13}+(1+\sin^{2}\theta_{13})\cos 2\theta)}{8\sin\theta_{13}\cos\theta_{23}\sqrt{\cos^{2}\theta_{13}-\sin^{2}\theta}}$ | $\cos\delta=a\frac{\sqrt{1+\cos 2\theta}\csc\theta}{r}\left(1+\frac{a}{2}\right)$
Table 1: Exact and approximated sum rules for the experimentally viable
models, where $\theta=\arctan\frac{1}{\phi}$ and $\phi=\frac{1+\sqrt{5}}{2}$.

Since the second exact sum rules are quite involved having an approximated
expression is of help to understand the physical meaning of it and to
understand the difference with respect to the TB model. We present in Table 1
the exact and approximated second sum rule for TM1, TM2 and GRa2 that as we
will see later are the viable atmospheric mixing. Note that the approximated
lead to simple results for TM1 and TM2 because the parameters $a$, $r$ and $s$
are built as deviation parameters from the TB mixing and beyond the first
order expansion may not bring new insight for other mixing. We present in
Table 2 the first atmospheric sum rules used in Figure 9. These results were
derived using the normal ordered data without SK atmospheric results, the
discussion regarding linearisation is the same including SK or considering the
inverted ordering since $\sin\theta_{13}$ is very constrained and it does not
change much in the different case considered.

TM1 | $\cos\theta_{12}=\sqrt{\frac{2}{3}}\frac{1}{\cos\theta_{13}}$ | TM2 | $\sin\theta_{12}=\frac{1}{\sqrt{3}\cos\theta_{13}}$
---|---|---|---
BM1 | $\cos\theta_{12}=\frac{1}{\sqrt{2}\cos\theta_{13}}$ | BM2 | $\cos\theta_{12}=\frac{1}{\sqrt{2}\cos\theta_{13}}$
GRa1 | $\cos\theta_{12}=\frac{\cos\theta}{\cos\theta_{13}}$ | GRa2 | $\cos\theta_{12}=\frac{\sin\theta}{\cos\theta_{13}}$
GRb1 | $\cos\theta_{12}=\frac{1+\sqrt{5}}{4\cos\theta_{13}}$ | GRb2 | $\sin\theta_{12}=\frac{\sqrt{5+\sqrt{5}}}{4\cos\theta_{13}}$
GRc1 | $\cos\theta_{12}=\frac{1+\sqrt{5}}{2\sqrt{3}\cos\theta_{13}}$ | GRc2 | $\sin\theta_{12}=\frac{1+\sqrt{5}}{2\sqrt{3}\cos\theta_{13}}$
HEX1 | $\cos\theta_{12}=\frac{\sqrt{3}}{2\cos\theta_{13}}$ | HEX2 | $\sin\theta_{12}=\frac{1}{2\sqrt{2}\cos\theta_{13}}$
Table 2: Exact sum rules plotted in Figure 9. where
$\theta=\arctan\frac{1}{\phi}$ and $\phi=\frac{1+\sqrt{5}}{2}$.  Figure 9: We
present the $3\sigma$ allowed region in green. The pink, blue, red, orange and
black curves are respectively the predictions for TM1, TM2, GRa1, GRa2 and
GRb1 mixing patterns. Figure 10: We present with the blue band the sum rule
prediction for TM2 for $\cos\delta$ letting $\sin\theta_{13}$ vary in its
$3\sigma$ range. In orange and purple the sum rules for GRa2 and TM1. The
yellow and gray regions are respectively the $1\sigma$ range of
$\sin\theta_{23}$ and $\cos\delta$, while the plot covers the whole $3\sigma$
range.

In Figures 9 and 10 we study the exact atmospheric sum rules for models
obtained modifying TB, BM, GRa, GRb, GRc and HEX. In Figure 9 we present first
atmospheric sum rule in Table 2, where the green band is the $3\sigma$ range
for $\sin^{2}\theta_{12}$. The models that do not appear are already excluded
and far from the $3\sigma$ region. Therefore BM1, BM2, GRa1, GRb2, GRc1, GRc2,
HEX1 and HEX2 are already excluded. In red we show GRa1 that is excluded a
$3\sigma$ and in blue TM2, that is still not excluded only in a narrow
parameter space, for high values of the solar and atmospheric angle. TM1 is
showed in purple, GRa2 in orange and GRb1 in black.

In Figure 10 we show the exact atmospheric sum rules (Table 1) and the
corresponding equations for other models that are still allowed from Figure 9.
We plot $\cos\delta$ against $\sin\theta_{23}$ and letting $\sin\theta_{13}$
vary in its $3\sigma$ range, this gives the width of the different bands, in
yellow and gray respectively are the $1\sigma$ band for $\sin^{2}\theta_{23}$
and $\cos\delta$. The GRb1 mixing do not appear in the plot because it lays in
unphysical values of $\cos\delta$. In purple, blue and orange we present TM1,
TM2 and GR12. We can see that given the $1\sigma$ bands, the GRa2 mixing is
favoured when considering normal ordering and without the SK data, since TM2
is allowed only on a small portion of the parameter space as shown in Figure
9.

##  5 Littlest Seesaw

The Littlest Seesaw (LS) mechanism is the most economic neutrino mass
generation mechanism that is still consistent with the experimental neutrino
data King:2013iva ; King:2015dvf . We will show that after the choice of a
specific $n$ value, all the neutrino observables are fixed by two free
parameters. Different values of $n$ can be realised by different discrete
symmetry groups. The LS introduces two new Majorana right-handed (RH)
neutrinos $N_{R}^{atm}$ and $N_{R}^{sol}$ that will be mostly responsible for
providing the atmospheric and solar neutrino mass respectively and the
lightest SM neutrino is approximately massless; this is the idea of sequential
dominance (SD) of RH neutrinos combined with the requirement for the
$N_{R}^{atm}$ \- $\nu_{e}$ interaction to be zero King:2002nf . The Majorana
neutrino mass matrix is given by the standard type I seesaw equation

$\displaystyle m^{\nu}=-m^{D}M_{R}^{-1}m^{D^{T}},$ (78)

where the RH neutrino mass matrix $M_{R}$ is a $2\times 2$ diagonal matrix

$\displaystyle M_{R}=\left(\begin{array}[]{cc}M_{\mathrm{atm}}&0\\\
0&M_{\mathrm{sol}}\end{array}\right),\quad
M_{R}^{-1}=\left(\begin{array}[]{cc}M_{\mathrm{atm}}^{-1}&0\\\
0&M_{\mathrm{sol}}^{-1}\end{array}\right),$ (83)

that comes from the Lagrangian term

$\displaystyle\mathcal{L}_{\mathrm{LS}}\supset-\frac{1}{2}M_{\mathrm{atm}}\bar{N}_{R}^{\mathrm{atm}}N_{R}^{\mathrm{atm}}-\frac{1}{2}M_{\mathrm{sol}}\bar{N}_{R}^{\mathrm{sol}}N_{R}^{\mathrm{sol}}+h.c.\;.$
(84)

The Dirac mass matrix is instead a $3\times 2$ matrix with arbitrary entries

$\displaystyle m^{D}=\left(\begin{array}[]{cc}d&a\\\ e&b\\\
f&c\end{array}\right),\quad\left(m^{D}\right)^{T}=\left(\begin{array}[]{ccc}d&e&f\\\
a&b&c\end{array}\right),$ (90)

where the entries are the coupling between the Majorana RH neutrinos and the
SM neutrinos. The first column describe the interaction of the neutrinos in
the flavour basis with the atmospheric RH neutrino and the second with the
solar RH neutrino. The SD assumptions are that $d=0$, $d\ll e,f$, and

$\displaystyle\frac{(e,f)^{2}}{M_{\mathrm{atm}}}\gg\frac{(a,b,c)^{2}}{M_{\mathrm{sol}}},$
(91)

these, together with the choice that of the almost massless neutrino to be the
first mass eigenstate $m_{1}$, leads to $m_{3}\gg m_{2}$ and therefore a
normal mass hierarchy. This description can be further constrained choosing
exactly $e=f$, $b=na$ and $c=(n-2)a$ giving a simplified Dirac matrix

$\displaystyle m^{D}=\left(\begin{array}[]{cc}0&a\\\ e&na\\\
e&(n-2)a\end{array}\right),$ (95)

that is called constrained dominance sequence (CSD) for positive integer $n$
King:2005bj ; Antusch:2011ic ; King:2013iva . Following the literature we will
refer to models with $n$ real as LS models King:2015dvf . It has been shown
that the reactor angle is King:2015dvf

$\displaystyle\theta_{13}\sim(n-1)\frac{\sqrt{2}}{3}\frac{m_{2}}{m_{3}},$ (96)

therefore this can provide non-zero and positive angle for $n>1$ and also
excludes already models with $n\geq 5$ since they do not fit the experimental
value.

The littlest seesaw Lagrangian unifies in one triplet of flavour symmetry the
three families of electroweak lepton doublets while the two extra right-handed
neutrinos, $\nu_{R}^{\mathrm{atm}}$ and $\nu_{R}^{\mathrm{sol}}$ are singlets
and reads King:2015dvf

$\displaystyle\mathcal{L}=-y_{\mathrm{atm}}\bar{L}\cdot\phi_{\mathrm{atm}}\nu_{R}^{\mathrm{atm}}-y_{\mathrm{sol}}\bar{L}\cdot\phi_{\mathrm{sol}}\nu_{R}^{\mathrm{sol}}-\frac{1}{2}M_{\mathrm{atm}}\nu_{R}^{\bar{c}}{}^{\mathrm{atm}}\nu_{R}^{\mathrm{atm}}-\frac{1}{2}M_{\mathrm{sol}}\nu_{R}^{\mathrm{c}}{}^{\mathrm{sol}}\nu_{R}^{\mathrm{sol}}+h.c.\;,$
(97)

which can be enforced by a $Z_{3}$ symmetry and where $\phi_{\mathrm{atm}}$
and $\phi_{\mathrm{sol}}$ can be either Higgs-like triplets under the flavour
symmetry or a combination of Higgses electroweak doublets and flavons
depending on the specific choice of symmetry to use. In both cases the
alignment should follow

$\displaystyle\phi_{\mathrm{atm}}^{T}\propto(0,1,1),\quad\phi_{\mathrm{sol}}^{T}\propto(1,n,n-2),$
(98)

or

$\displaystyle\phi_{\mathrm{atm}}^{T}\propto(0,1,1),\quad\phi_{\mathrm{sol}}^{T}\propto(1,n-2,n).$
(99)

We will refer to the first possibility in Equation (98) as the normal case and
the second, in Equation (99) as the flipped case. The predictions for $n$ in
the flipped case are related to the normal one, as we will discuss later, by

$\tan\theta_{23}\rightarrow\cot\theta_{23}\quad\quad\quad\delta\rightarrow\delta+\pi,$
(100)

therefore we will discuss them together as one single $n$ case.

There is an equivalent convention that can be found in the literature
deMedeirosVarzielas:2022fbw , where the alignment is chosen to be

$\displaystyle\phi_{\mathrm{atm}}^{T}\propto(0,1,-1),\quad\phi_{\mathrm{sol}}^{T}\propto(1,n,2-n).$
(101)

or

$\displaystyle\phi_{\mathrm{atm}}^{T}\propto(0,1,-1),\quad\phi_{\mathrm{sol}}^{T}\propto(1,2-n,n).$
(102)

that leads to the same results as the previous two cases respectively. In the
neutrino mass matrix there will appear a $(-1)$ factor that is only a non-
physical phase that can therefore be neglected. In particular the case
$n=1+\sqrt{6}$ that can be obtained with modular symmetry in
deMedeirosVarzielas:2022fbw  is still $n=1+\sqrt{6}$ in our convention using
the Equation (98).

We will compute everything following the derivation in King:2015dvf  and using
Equation (98) and deriving the flipped result with Equation (100). We will
consider $n=2.5$, $3$ and $1+\sqrt{6}\approx 3.45$ and their flipped cases.
The mass matrix then in the diagonal charged lepton basis is given by

$\displaystyle m^{\nu}=m_{a}\left(\begin{array}[]{ccc}0&0&0\\\ 0&1&1\\\
0&1&1\end{array}\right)+m_{b}e^{i\eta}\left(\begin{array}[]{ccc}1&n&n-2\\\
n&n^{2}&n(n-2)\\\ 1&n(n-2)&(n-2)^{2}\end{array}\right),$ (109)

where we used Equations (78), (83) and (95)

$\displaystyle m_{a}=\frac{|e|^{2}}{M_{\mathrm{atm}}}\quad\quad\quad
m_{b}=\frac{|a|^{2}}{M_{\mathrm{sol}}},$ (110)

and the only relevant phase is $\eta=\arg(a/e)$. At this point we notice that,
in the diagonal charged lepton mass basis where we work, the PMNS mixing
matrix is fully specified by the choice of $n$ and the parameters
$m_{a}/m_{b}$ and $\eta$. Indeed it is possible to derive exact analytic
results for the masses and mixing angles King:2015dvf , and hence obtain the
LS prediction for the neutrino oscillation observables.

We first observe that

$\displaystyle m^{\nu}\left(\begin{array}[]{c}\sqrt{\frac{2}{3}}\\\
-\sqrt{\frac{1}{3}}\\\
\sqrt{\frac{1}{3}}\end{array}\right)=\left(\begin{array}[]{l}0\\\ 0\\\
0\end{array}\right),$ (117)

where the vector
$(\sqrt{\frac{2}{3}},-\sqrt{\frac{1}{3}},\sqrt{\frac{1}{3}})^{T}$ is the first
column of the TB matrix in Equation (12) and is then an eigenvector of the
neutrino mass matrix with eigenvalue $0$ and it corresponds to the massless
neutrino eigenstate. This means that for a generic $n$ we get a TM1 mixing,
Equation (58), where the first column of the TB matrix is preserved and the
other two can change. Therefore we can think of the LS as a special case of
the atmospheric sum rules for the TB mixing. Since the atmospheric sum rules
were derived only using the fact that the first column of the TB matrix is
preserved all LS implementations also follow the TM1 sum rules in Equation
(58). Once we have noticed this it is clear that $m_{\nu}$ can be block
diagonalised using the TB matrix

$\displaystyle m_{\text{block
}}^{\nu}=U_{\mathrm{TB}}^{T}m^{\nu}U_{\mathrm{TB}}=\left(\begin{array}[]{ccc}0&0&0\\\
0&x&y\\\ 0&y&z\end{array}\right),$ (121)

with

$\displaystyle x=3m_{b}e^{i\eta},\quad y=\sqrt{6}m_{b}e^{i\eta}(n-1),\quad
z=|z|e^{i\phi_{z}}=2\left[m_{a}+m_{b}e^{i\eta}(n-1)^{2}\right].$ (122)

Finally we diagonalise $m_{\text{block }}^{\nu}$ to obtain a matrix
$\operatorname{diag}\left(0,m_{2},m_{3}\right)$

$\displaystyle U_{\text{block }}^{T}m_{\text{block }}^{\nu}U_{\text{block
}}=P_{3\nu}^{*}R_{23\nu}^{T}P_{2\nu}^{*}m_{\text{block
}}^{\nu}P_{2\nu}^{*}R_{23\nu}P_{3\nu}^{*}=m_{\text{diag
}}^{\nu}=\operatorname{diag}\left(0,m_{2},m_{3}\right),$ (123)

where the phases are

$\displaystyle\begin{gathered}P_{2\nu}=\left(\begin{array}[]{ccc}1&0&0\\\
0&e^{i\phi_{2}^{\nu}}&0\\\ 0&0&e^{i\phi_{3}^{\nu}}\end{array}\right),\\\
P_{3\nu}=\left(\begin{array}[]{ccc}e^{i\omega_{1}^{\nu}}&0&0\\\
0&e^{i\omega_{2}^{\nu}}&0\\\
0&0&e^{i\omega_{3}^{\nu}}\end{array}\right),\end{gathered}$ (132)

and the angle we use to diagonalise

$\displaystyle R_{23\nu}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{23}^{\nu}&\sin\theta_{23}^{\nu}\\\
0&-\sin\theta_{23}^{\nu}&\cos\theta_{23}^{\nu}\end{array}\right)\equiv\left(\begin{array}[]{ccc}1&0&0\\\
0&c_{23}^{\nu}&s_{23}^{\nu}\\\
0&-s_{23}^{\nu}&c_{23}^{\nu}\end{array}\right),$ (139)

with the angle being fully specified by the free parameters $m_{a}/m_{b}$ and
$\eta$

$\displaystyle t\equiv\tan 2\theta_{23}^{\nu}=\frac{2|y|}{|z|\cos(A-B)-|x|\cos
B},$ (140)

where

$\displaystyle\tan
B=\tan\left(\phi_{3}^{\nu}-\phi_{2}^{\nu}\right)=\frac{|z|\sin A}{|x|+|z|\cos
A},$ (141)

and

$\displaystyle
A=\phi_{z}-\eta=\arg\left[m_{a}+m_{b}e^{i\eta}(n-1)^{2}\right]-\eta.$ (142)

The PMNS matrix is therefore the product of the TB and the
$U_{\mathrm{block}}$ matrices

$\displaystyle U_{\mathrm{PMNS}}=U_{\mathrm{block}}^{T}U_{\mathrm{TB}}^{T}.$
(143)

$\displaystyle\begin{aligned}
\sin\theta_{13}&=\frac{1}{\sqrt{3}}s_{23}^{\nu}=\frac{1}{\sqrt{6}}\left(1-\sqrt{\frac{1}{1+t^{2}}}\right)^{1/2}\\\
\tan\theta_{12}&=\frac{1}{\sqrt{2}}c_{23}^{\nu}=\frac{1}{\sqrt{2}}\left(1-3\sin^{2}\theta_{13}\right)^{1/2}\\\
\tan\theta_{23}&=\frac{\left|\frac{e^{iB}}{\sqrt{2}}c_{23}^{\nu}+\frac{1}{\sqrt{3}}s_{23}^{\nu}\right|}{\left|\frac{e^{iB}}{\sqrt{2}}c_{23}^{\nu}-\frac{1}{\sqrt{3}}s_{23}^{\nu}\right|}=\frac{\left|1+\epsilon_{23}^{\nu}\right|}{\left|1-\epsilon_{23}^{\nu}\right|},\end{aligned}$
(144)

with

$\displaystyle\epsilon_{23}^{\nu}\equiv\sqrt{\frac{2}{3}}\tan\theta_{23}^{\nu}e^{-iB}=\sqrt{\frac{2}{3}}t^{-1}\left[\sqrt{1+t^{2}}-1\right]e^{-iB}.$
(145)

The neutrino masses can be computed from $m_{\text{block }}^{\nu}$ and they
are

$\displaystyle H_{\text{block }}^{\nu}=m_{\text{block }}^{\nu}m_{\text{block
}}^{\nu\dagger}=\left(\begin{array}[]{ccc}0&0&0\\\
0&|x|^{2}+|y|^{2}&|x||y|+|y|e^{i\eta}z^{*}\\\
0&|x||y|+|y|e^{-i\eta}z&|y|^{2}+|z|^{2}\end{array}\right),$ (149)

and after diagonalisation we are left with

$\displaystyle\begin{aligned}
m_{2}^{2}+m_{3}^{2}&=T\equiv|x|^{2}+2|y|^{2}+|z|^{2},\\\
m_{2}^{2}m_{3}^{2}&=D\equiv|x|^{2}|z|^{2}+|y|^{4}-2|x||y|^{2}|z|\cos
A,\end{aligned}$ (150)

and finally

$\displaystyle\begin{aligned}
m_{3}^{2}&=\frac{1}{2}T+\frac{1}{2}\sqrt{T^{2}-4D},\\\
m_{2}^{2}&=D/m_{3}^{2},\\\ m_{1}^{2}&=0,\end{aligned}$ (151)

and the CP phase is

$\displaystyle\delta=-\arg\left(\operatorname{sign}(t)e^{i\eta}\left(4\left(\sqrt{t^{2}+1}-1\right)+\left(-2+3e^{2iB}\right)t^{2}\right)\right).$
(152)

Therefore we notice that by just specifying two parameters, the phase $\eta$
and the ratio of the masses $r=m_{a}/m_{b}$ and choosing the $n$ that
eventually is determined by the choice of flavour symmetry, LS predicts all
the neutrino oscillation observables, that is why it is called littlest
seesaw.

Figure 11: The parameters $\eta$ and $r$ are constrained to a good degree by
only two experimental observables, namely $\theta_{13}$ and the mass ratio
$m_{2}^{2}/m_{3}^{2}$. The $3\sigma$ allowed region for $\theta_{13}$ and the
mass ratio are respectively the orange and blue band. The area of intersection
is the allowed parameter space for $\eta$ and $r$. From the left to the right
we present, $n=2.5$, $3$ and $1+\sqrt{6}$.

Let us consider $n=3$ and the correspondent flipped case, which was realised
successfully via $S_{4}$ symmetry in King:2015dvf . We can plot the
constraints on the parameter space given by the experimental ranges of
$\theta_{13}$ and the mass ratio $m_{2}^{2}/m_{3}^{2}$. In Figure 11 we notice
that imposing these two experimental constraint leave only two small allowed
parameter regions in the plane $r-\eta$. The allowed range we consider in $r$
and $\eta$ are given by the maximal and minimal values of them in the
intersection of the blue and orange bands. Once we have the value of $r$ and
$\eta$, thanks to the high predictivity of the model we can derive all the
physical parameters and we can test them against the observed values. We do
this for different values of $n$ in Table 3 to 5. We do not present the plot
for the flipped case since it is exactly the same. In fact it involves only
the mass ration and $\theta_{13}$. This fact can be understood easily studying
the parameter $t$ for example in the case $n=1+\sqrt{6}$ and the flipped. In
this case going from $n$ to the flipped changes sign of t in Equation (140).
The prediction for the mass ratio, $\theta_{13}$, $\theta_{12}$ are
independent of this sign while $\theta_{23}$ and $\delta$ are affected by it,
we can see this in Equations (144) and (152). The predictions as anticipated
before are related by $\tan\theta_{23}\rightarrow\cot\theta_{23}$ and
$\delta\rightarrow\delta+\pi$.

| $n=3$ | $\eta=2.1\pm 0.3$ | $\eta=4.2\pm 0.3$ | Exp. range
---|---|---|---|---
| $\theta_{12}\;[^{\circ}]$ | $34.3{}^{+0.4}_{-0.7}$ | $34.3{}^{+0.4}_{-0.7}$ | $31.27-35.86$
normal | $\theta_{23}\;[^{\circ}]$ | $46{}^{+4}_{-5}$ | $46{}^{+4}_{-5}$ | $39.5-52.0$
normal | $\delta\;[^{\circ}]$ | $93^{+12}_{-13}$ | $255^{+12}_{-13}$ |  $0-45$ & $105-360$
flipped | $\theta_{23}\;[^{\circ}]$ | $44{}^{+4}_{-5}$ | $44{}^{+4}_{-5}$ | $39.5-52.0$
flipped | $\delta\;[^{\circ}]$ | $273^{+12}_{-13}$ | $75^{+12}_{-13}$ |  $0-45$ & $105-360$
Table 3: 3$\sigma$ ranges of the predicted parameters and experimental ranges
for $n=3$. With $r=0.10\pm 0.02$.

In Table 3 we studied the $n=3$ and its flipped case. We present the
theoretical prediction and its uncertainty coming from the allowed region in
Figure 11 and the experimental bound. We notice that $\eta=2.1$ is still
allowed only in the lower part of the $\delta$ parameter space for the normal
case and similarly $\eta=4.2$ for the flipped case. In Table 4 for
$n=1+\sqrt{6}$, which can be realised with a modular symmetry
deMedeirosVarzielas:2022fbw , we notice that the region with $\eta=2.40$ is
excluded thanks to the experimental bounds on $\delta$ while the one with
$\eta=3.88$ is well within the $3\sigma$ range. For the flipped case instead
$\eta=3.88$ is close the lower boundary of the $3\sigma$ region in $\delta$.
For $n=2.5$ in Table 5 we notice that $\eta=4.7$ is excluded for normal case
while for the flipped both values are allowed in the $3\sigma$ range. This
case is also known in the literature as $n=-1/2$ using the convention in
Equation (102). But it is more consistent to refer to it as $n=2.5$ in our
notation. And in fact we notice that the LS that are still allowed by the
data, $n=2.5$ and $n=1+\sqrt{6}$ have $n\sim 3$ which is the original CDS that
worked.

| $n=1+\sqrt{6}$ | $\eta=2.40\pm 0.04$ | $\eta=3.88\pm 0.04$ | Exp. range
---|---|---|---|---
| $\theta_{12}\;[^{\circ}]$ | $34.4^{+0.4}_{-0.6}$ | $34.4^{+0.4}_{-0.6}$ | $31.27-35.86$
normal | $\theta_{23}\;[^{\circ}]$ | $41.7^{+0.19}_{-0.12}$ | $41.6^{+0.19}_{-0.12}$ | $39.5-52.0$
normal | $\delta\;[^{\circ}]$ | $75\pm 3$ | $285\pm 3$ |  $0-45$ & $105-360$
flipped | $\theta_{23}\;[^{\circ}]$ | $48.3^{+0.19}_{-0.12}$ | $48.4^{+0.19}_{-0.12}$ | $39.5-52.0$
flipped | $\delta\;[^{\circ}]$ | $255\pm 3$ | $103\pm 3$ |  $0-45$ & $105-360$
Table 4: 3$\sigma$ ranges of the predicted parameters and experimental ranges for $n=1+\sqrt{6}$. With $r=0.072\pm 0.004$.  | $n=2.5$ | $\eta=1.5\pm 0.4$ | $\eta=4.7\pm 0.4$ | Exp. range w/o SK
---|---|---|---|---
| $\theta_{12}\;[^{\circ}]$ | $35.0\pm 0.1$ | $35.0\pm 0.1$ | $31.27-35.86$
normal | $\theta_{23}\;[^{\circ}]$ | $47.0{}^{+3}_{-2}$ | $46.5{}^{+4}_{-2}$ | $39.5-52.0$
normal | $\delta\;[^{\circ}]$ | $289\pm 11$ | $76\pm 11$ |  $0-45$ & $105-360$
flipped | $\theta_{23}\;[^{\circ}]$ | $43.0{}^{+3}_{-2}$ | $43.5{}^{+4}_{-2}$ | $39.5-52.0$
flipped | $\delta\;[^{\circ}]$ | $109\pm 11$ | $256\pm 11$ |  $0-45$ & $105-360$
Table 5: 3$\sigma$ ranges of the predicted parameters and experimental ranges
for $n=2.5$. With $r=0.15\pm 0.02$.

##  6 Conclusions

In the past decades many attempts have been made to explain the flavour
structure of the PMNS matrix by imposing symmetry on the leptonic Lagrangian.
These symmetries imply correlations among the parameters that are called sum
rules. We have studied two types of sum rules: solar and atmospheric sum
rules. The former breaks the $T$ generator of a given symmetry group in the
charged lepton sector in order to generate a non-zero reactor angle
$\theta_{13}$. This leads with prediction for $\cos\delta$ that can be tested
against the experimental data. These in turn show a preference for GRa and GRb
mixing while BM and GRc are constrained to live in a very small window of the
parameter space.

The atmospheric sum rules instead come from either the breaking of both $S$
and $U$ in the neutrino sector while preserving $SU$ or by breaking $S$ and
preserving $U$. In this case we have two relations among the parameters that
can be tested. We noticed that only TM1, TM2 and GRa2 are still allowed by the
neutrino oscillation data with a preference for GRa2 and with TM2 very close
to be excluded.

We also studied LS models that follow the constrained sequential dominance
idea with $n$ allowed to be real. These models are very predictive with only
two free real parameters fixing all the neutrino oscillation observables. It
is the most economical framework that explains the neutrino masses that is
still compatible with data. Despite the great predictivity and the very small
parameter space in the $r$ \- $\eta$ plane allowed by the data all the LS
examples we considered are still allowed in one part of the parameter space.

Finally, we note that the recent global fits to experimental data have
provided significantly improved constraints on all of the above models and
with future improvement in the neutrino oscillation data we will be able to
restrict the pool of viable models still further. In particular advancements
in the measurement of the leptonic CP violating Dirac phase $\delta$ can
strongly constrain more models in the near future. This is particularly true
in LS models where already the experimental bounds on the CP phase is removing
the degeneracy in the $r$ \- $\eta$ parameter space and given the very precise
theoretical prediction can strongly constrain these LS models. Future
precision neutrino experiments are of great importance to continue to narrow
down the possible PMNS flavour models and lead to a deeper understanding of
this part of the flavour puzzle of the SM.

###### Acknowledgements.

The work is supported by the European Union Horizon 2020 Research and
Innovation programme under Marie Sklodowska-Curie grant agreement HIDDeN
European ITN project (H2020- MSCA-ITN-2019//860881-HIDDeN). SFK acknowledges
the STFC Consolidated Grant ST/L000296/1.

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