aainstitutetext: High Energy Theory Group, Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USAbbinstitutetext: Physics Department,
Colorado State University, Fort Collins, CO 80523, USAccinstitutetext:
Theoretical Physics Department, CERN, 1 Esplanade des Particules, 1211 Geneva
23, Switzerland

# A Survey of Neutrino Flavor Models and the Neutrinoless Double Beta Decay
Funnel

Peter B. Denton1110000-0002-5209-872X b, c, a, 2 and Julia
Gehrlein2220000-0002-1235-0505<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The neutrinoless double beta decay experimental effort continues to make
tremendous progress with hopes of covering the inverted neutrino mass
hierarchy in coming years and pushing from the quasi-degenerate hierarchy into
the normal hierarchy. As neutrino oscillation data is starting to suggest that
the mass ordering may be normal, we may well be faced with staring down the
funnel of death: a region of parameter space in the normal ordering where for
a particular cancellation among the absolute neutrino mass scale, the Majorana
phases, and the oscillation parameters, the neutrinoless double beta decay
rate may be vanishingly small. To answer the question if this region of
parameter space is theoretically preferred, we survey five broad categories of
flavor models which make various different predictions for parameters relevant
for neutrinoless double beta decay to determine how likely it is that the rate
may be in this funnel region. We find that a non-negligible fraction of flavor
models are at least partially in the funnel region. Our results can guide
model builders and experimentalists alike in focusing their efforts on
theoretically motivated regions of parameter space.

††preprint: CERN-TH-2023-160

##  1 Introduction

Neutrino oscillations Super-Kamiokande:1998kpq ; SNO:2001kpb ; SNO:2002tuh
provide one of the few strong motivations for physics beyond the Standard
Model (SM) as it requires at least two massive, active neutrinos. Oscillation
experiments, however, neither tell us about the absolute neutrino mass scale
nor about the nature of the neutrino mass: Dirac vs Majorana. One possible
means of probing the latter question is to consider lepton number violating
processes which provide a clear and striking signature that neutrinos are
Majorana particles Schechter:1981bd . The most experimentally promising lepton
number violating process is neutrinoless double beta decay (0$\nu\beta\beta$)
which is the transition of a nucleus with (A, Z) atomic numbers to (A, Z + 2),
accompanied by the emission of two electrons, but without the emission of two
anti-neutrinos Furry:1939qr . The observation of neutrino oscillations has
already demonstrated that the lepton number of individual flavors is not
conserved; 0$\nu\beta\beta$ could go one step farther marking the first
observation that total lepton number is not a conserved symmetry of nature
either333Note that the converse need not be true. That is, the non-observation
of 0$\nu\beta\beta$, e.g. if the atmospheric mass ordering was found to be
inverted, does not guarantee that neutrinos are Dirac. One such scenario is
pseudo-Dirac neutrinos Beacom:2003eu ; Wolfenstein:1981kw ; Bilenky:1983wt
where neutrinos are actually Majorana but $|m_{\beta\beta}|$ may be quite
small even in the inverted ordering.. This process is experimentally
challenging to measure (for reviews see Vergados:2016hso ; DellOro:2016tmg ;
Dolinski:2019nrj ; Agostini:2022zub ; Cirigliano:2022oqy ), however
experiments continue to make tremendous progress covering more and more
parameter space pushing into the region suggested by neutrino oscillations.
Indeed, thanks to the progress in neutrino oscillation experiments, all
neutrino mixing angles and mass splittings are now measured to a good accuracy
Denton:2022een  which allows for improved predictions of the theoretically
allowed regions of parameter space for $0\nu\beta\beta$ experiments.

Somewhat surprisingly, the observed leptonic mixing pattern seems to be in
considerable contrast to the quark mixing matrix, a difference that could
imply a nontrivial connection between the two sectors. Many models attempting
to make sense of the so-called “flavor puzzle” of the SM make predictions for
the mixing parameters, including the so-called Majorana phases and absolute
neutrino mass scale, which can be compared with experimental data. These
models provide experimental targets for a wide variety of neutrino experiments
and can be used to plan experimental stages or requested benchmark
sensitivities IntensityFrontierNeutrinoWorkingGroup:2013sdv ; Gehrlein:2022nss
. Making precise predictions is challenging, however, due to the very large
number of flavor models considered in the literature that still provide
acceptable fits to existing neutrino (and possibly quark) data as they make a
wide variety of predictions for the remaining neutrino parameters.

In this paper we will focus on the predictions from flavor models applied to
the neutrinoless double beta decay rate observable, motivated by the
predicivity of flavor models for several parameters which enter this
observable. Taking the exchange of three light Majorana neutrinos as the
dominant contribution to 0$\nu\beta\beta$, the predicted ranges for the
particle physics observable $|m_{\beta\beta}|$ depend critically on the
neutrino mass ordering which is largely undetermined by oscillation data:
normal (NO) with $m_{1}<m_{2}<m_{3}$ or inverted (IO) with
$m_{3}<m_{1}<m_{2}$444We define the neutrino mass eigenstates in the usual way
with decreasing amount of $\nu_{e}$ fraction: $|U_{e1}|>|U_{e2}|>|U_{e3}|$,
see e.g. Denton:2020exu .. Of particular interest is the region in the NO
which leads to immeasurably small rates of 0$\nu\beta\beta$ due to a precise
cancellation among the absolute neutrino mass scale, the Majorana phases, and
the oscillation parameters555While this region is essentially impossible to
probe experimentally, it does have the advantage that if it was known that
$|m_{\beta\beta}|$ were in the funnel, then we would have good knowledge on
both Majorana phases Ge:2016tfx ; Ge:2019ldu , something that is otherwise
essentially impossible.; this region is known as the funnel and is often
quantified as values of $|m_{\beta\beta}|<1$ meV for concreteness. In this
manuscript we will study this region of parameter space from a theoretical
point of view in the context of a wide range of flavor models (see
Jenkins:2008ex  for an earlier study).

We aim to provide a comprehensive study of viable categories of conceivable
flavor models which make predictions for $|m_{\beta\beta}|$ and determine the
fractions of predicted parameter space which fall into the funnel region
within the constraints of the latest neutrino oscillation data. That is, we
are investigating whether or not categories of flavor models that have been
studied in the literature containing any conceivable models prefer to be in
the funnel. Even though a particular focus of our work is the funnel region we
will also present a global overview of the preferred regions of parameter
space in these categories of models to demonstrate the existence and location
of theoretically motivated regions of observables. The advantage of doing such
a study before experimental limits reach the normal hierarchy is to understand
what ranges of observables models predict _before_ the measurements are made.
This can guide future work both from the experimental and theoretical side as
we identify preferred regions of parameter space which can serve as targets to
focus experimental efforts on. From the theoretical side we give a detailed
overview of different categories of flavor models which make predictions for
$0\nu\beta\beta$, asses their validity by comparing their predictions to
current knowledge of the mixing parameters and bounds on the absolute mass
scale, and calculate their preferred regions of parameter space. Our work can
thereby provide important guidance for future model building work.

This paper is organized as follows: we will start with a short introduction to
0$\nu\beta\beta$ in sec. 2, then we explain and discuss the categories of
models we consider including our results in secs. 3, 4, and conclude in sec.
5.

##  2 Neutrinoless double beta decay review

We start with a short review about neutrinoless double beta decay. We make the
oft-used assumption that the dominant contribution to 0$\nu\beta\beta$ arrives
from the exchange of three light ($m_{\nu}\lesssim$ 100 MeV Blennow:2010th )
Majorana neutrinos; see Deppisch:2012nb ; Deppisch:2015qwa ; Peng:2015haa ;
Fuks:2020zbm ; Mitra:2011qr ; Lopez-Pavon:2012yda ; Blennow:2010th ;
Tello:2010am ; Li:2020flq  for other new physics scenarios which give rise to
0$\nu\beta\beta$. The observable in neutrinoless double beta decay is the
decay half-life which is a function of various physics parameters,

$\displaystyle(T_{1/2}^{0\nu\beta\beta})^{-1}=G_{0\nu\beta\beta}(Q,Z)|\mathcal{M}_{0\nu\beta\beta}(A,Z)|^{2}|m_{\beta\beta}|^{2}~{},$
(1)

where $G_{0\nu\beta\beta}(Q,Z)$ is the phase-space factor of the particular
transition which depends on the isotope’s $Q$ value and is well known,
$|\mathcal{M}_{0\nu\beta\beta}(A,Z)|^{2}$ is the nuclear matrix element which
currently presents a considerable source of theoretical uncertainty
Engel:2016xgb ; Agostini:2022zub ; Belley:2023btr , and $|m_{\beta\beta}|$ is
the effective neutrino mass defined as Bilenky:1987ty

$\displaystyle|m_{\beta\beta}|=\left|\sum_{i=1}^{3}U_{ei}^{2}m_{i}\right|\,.$
(2)

The effective neutrino mass contains the particle physics information of
interest relevant for understanding neutrino masses and mixings and is the
focus of this paper. We write it in terms of mixing parameters in the standard
parametrization of the neutrino mixing matrix Zyla:2020zbs , the PMNS matrix
Pontecorvo:1957cp ; Maki:1962mu , where we choose to assign the Dirac CP phase
to the second row of the matrix such that eq. (2) is independent of it666It is
clear from eq. (2) that there can be only two physical phases; the inclusion
of $\delta$ in the first row leads to an additional phase redundancy which
does not affect observables.. In this case the PMNS matrix reads

$\displaystyle
U_{\text{PMNS}}=\begin{pmatrix}c_{12}c_{13}\text{e}^{\text{i}\alpha/2}&s_{12}c_{13}\text{e}^{\text{i}\beta/2}&s_{13}\\\
(-c_{23}s_{12}-c_{12}s_{13}s_{23}\text{e}^{\text{i}\delta})\text{e}^{\text{i}\alpha/2}&(c_{12}c_{23}-s_{12}s_{13}s_{23}\text{e}^{\text{i}\delta})\text{e}^{\text{i}\beta/2}&c_{13}s_{23}\text{e}^{\text{i}\delta}\\\
(-c_{12}c_{23}s_{13}+\text{e}^{-\text{i}\delta}s_{12}s_{23})\text{e}^{\text{i}\alpha/2}&(-c_{23}s_{12}s_{13}-\text{e}^{-\text{i}\delta}c_{12}s_{23})\text{e}^{\text{i}\beta/2}&c_{13}c_{23}\end{pmatrix}\,,$
(3)

where we use the short hand notation $c_{ij}=\cos\theta_{ij}$,
$s_{ij}=\sin\theta_{ij}$ and the Majorana phases $\alpha,\beta$ where one of
them can be between $\in[0,\pi]$, the other one $\in[0,2\pi]$. Thus we can see
that $|m_{\beta\beta}|$ is a function of seven free parameters: the three
neutrino masses, two mixing angles and two Majorana phases.

A non-trivial feature of the seven parameters is that the predicted range of
0$\nu\beta\beta$ also depends on the neutrino mass ordering. Oscillation
experiments are starting to provide hints for the neutrino mass ordering, in
particular when combined in global fits which currently show a preference for
the NO deSalas:2020pgw ; Esteban:2020cvm ; Capozzi:2021fjo 777The hints coming
from long baseline accelerator neutrino experiments, however, might be an
indication of new physics Kelly:2020fkv ; Denton:2020uda ; Chatterjee:2020kkm
.. Using the measured values of the neutrino mass splittings and mixing
angles, $|m_{\beta\beta}|$ depends on only three unknown parameters: the
absolute neutrino mass scale which is constrained to be at most somewhat
light, and two Majorana phases which are completely unconstrained. Note that
$|m_{\beta\beta}|$ only constrains at most one combination of the two phases;
unless $|m_{\beta\beta}|\approx 0$ it is not possible to determine both
Majorana phases using oscillation data with a detection of $0\nu\beta\beta$
alone Ge:2016tfx ; Ge:2019ldu . Furthermore, the Majorana phases do not lead
to manifest CP violation in $0\nu\beta\beta$ deGouvea:2002gf ; Barger:2002vy
888Note that the Majorana phases can lead to CP violating phenomena in other
observables, for example in the leptogenesis scenario where a lepton asymmetry
is generated via the decay of heavy, right-handed neutrinos which depends on
the Majorana phases of these neutrinos Fukugita:1986hr ., they affect the
$0\nu\beta\beta$ amplitude in a CP-even way, which excludes the possibility to
determine them by considering $0\nu\beta\beta$ with the emission of two
electrons and its CP conjugated process with the emission of two positrons.

Figure 1: The current knowledge on the absolute masses of the three neutrinos.
The data included is the average of the three global fits for $\Delta
m^{2}_{21}$, $|\Delta m^{2}_{31}|$, and the preference for the normal ordering
Esteban:2020cvm ; deSalas:2020pgw ; Capozzi:2021fjo , as well as the
cosmological constraint on the sum of the neutrino masses that does not yet
show evidence for neutrino masses DiValentino:2021hoh . Left: The
$\Delta\chi^{2}$ for parameter estimation where we also note that the minimum
$\chi^{2}$ for each state is $\sim 1.75$ and thus is an acceptable fit to the
data. Right: The 1, 2, 3 $\sigma$ preferred regions for each mass state,
individually. The thicker regions are more preferred.

The absolute mass scale can in principle be constrained by beta decay end-
point experiments such as KATRIN KATRIN:2019yun , but the best constraints up
to now come from cosmology. Constraints vary from $\sum m_{\nu}<[87,90]$ meV
at 95% CL DiValentino:2021hoh ; diValentino:2022njd . We take 90 meV
DiValentino:2021hoh  as our fiducial number which then maps onto
$m_{1}\lesssim 17$ meV in the normal ordering for the best fit oscillation
parameters. Current cosmological data seems to be incompatible with the
inverted ordering at 95% CL, although the details of this constraint depend
considerably on one’s choice of priors Gariazzo:2018pei ; Gariazzo:2022ahe ;
Gariazzo:2023joe . The currently preferred region for the neutrino masses are
shown in fig. 1 using oscillation data, the oscillation preference for the
normal ordering (not used in the statistical tests elsewhere in this paper),
and the cosmological constraint on the sum of neutrino masses (also not used
in the statistical tests elsewhere in this paper). Note that the different
preferred regions for each mass are correlated with one another. We see that
$m_{2}$ has both an upper and lower limit while either $m_{1}$ or $m_{3}$ can
be zero. We also see that each mass state has two disjoint preferred regions
due to the different mass orderings as well as the important constraint from
cosmology.

The current best limit on $|m_{\beta\beta}|$ is from KamLAND-Zen with 136Xe
KamLAND-Zen:2022tow

$|m_{\beta\beta}|^{\text{exp}}<(36–156)\text{ meV}\,,$ (4)

where the range of values is due to the range of predictions for the nuclear
matrix element. The most optimistic matrix element values indicate that this
constraint starts to push into the inverted hierarchy while future experiments
DAndrea:2021gqg  will further probe a large part of this region, subject to
nuclear matrix element uncertainties. In addition, future constraints on the
neutrino mass scale from cosmology have important implications for
$0\nu\beta\beta$ Ettengruber:2022mtm .

In fig. 2 we show the allowed regions in the
$|m_{\beta\beta}|-m_{\text{lightest}}$ plane based on our knowledge of
oscillation data, as well as the constraints on the lightest neutrino mass and
upper limits on $|m_{\beta\beta}|$. The regions are drawn at the $3\sigma$
limit which means we impose that the total $\Delta\chi^{2}$, understood as the
sum of all $\Delta\chi^{2}$ of the mixing angles and mass splittings, is equal
to 11.83 which is $3\sigma$ with 2 dof999The choice of the number degrees of
freedom is here is non-trivial. Our choice is based on the fact that since
there are two physics parameters: $|m_{\beta\beta}|$ and $m_{\rm lightests}$
this corresponds to two degrees of freedom. While they are clearly related to
each other, even with known oscillation parameters, the additional freedom
from the two Majorana phases more than ensures that $|m_{\beta\beta}|$ is a
distinct degree of freedom.. This is different from what is commonly done in
the literature where each individual oscillation parameter is allowed to
increase to some critical threshold without consideration for the total test
statistic. We do not impose information in the test statistic for the lightest
mass from cosmological measurements or from $|m_{\beta\beta}|$, although since
the latter only pushes into the inverted hierarchy it would not affect a
discussion of the funnel. We also do not include a penalty factor for current
preference from the oscillation data for the normal ordering over the inverted
ordering, although this also would not affect the funnel discussion. We avoid
those constraints because they make the distinction between the normal
ordering and the inverted ordering complicated in a way that depends quite
sensitively on the precise statistical test performed. Finally, we do not
include any information about $\delta$ (even though $\delta$ does not affect
0$\nu\beta\beta$, it is relevant for specific classes of flavor models) from
long-baseline oscillation data as there is a mild tension among the two
relevant data sets, NOvA and T2K Denton:2020uda ; Chatterjee:2020kkm  and most
values are allowed in any case. The yellow regions show the allowed region if
all oscillation parameters are known perfectly at the best fit values from
Esteban:2020cvm . The only free parameters in the yellow region are the two
Majorana phases. The blue region shows the enlarged region that we can
expected with the expected precision of the oscillation parameters from DUNE
and JUNO JUNO:2022mxj ; DUNE:2020ypp . This shows that the future measurements
of the oscillation parameters by DUNE and JUNO will take us very nearly to the
perfect knowledge case. The red regions show the additional parameter space
due to the current oscillation uncertainties, also taken from Esteban:2020cvm
. We see that future oscillation experiments will constrain the parameter
space further close to the relevant limit of perfect information, however the
oscillation parameters are already measured rather precisely such that the
Majorana phases present the largest uncertainty in the allowed regions of
parameter space. Therefore models which are in agreement with the oscillation
data but additionally predict the Majorana phases are of particular
phenomenological interest as they prefer only parts of the generally allowed
parameter space.

We are specifically interested in the funnel region in NO which we define as
$|m_{\beta\beta}|<10^{-3}$ eV, consistent with other analyses in the
literature, e.g. Ge:2016tfx ; Ge:2019ldu . Such small values of
$|m_{\beta\beta}|$ can only be achieved if the atmospheric mass ordering is
normal and for $m_{1}\in[6\times 10^{-4},8\times 10^{-3}]$ eV assuming the
best fit values of the neutrino parameters from oscillations Esteban:2020cvm .
To understand the cancellation we interpret the expression for
$|m_{\beta\beta}|$ as an quadrilateral in the complex plane, see fig. 3 (see
Vissani:1999tu  for an alternative graphical representation of
$|m_{\beta\beta}|$). If $|m_{\beta\beta}|\approx 0$, the quadrilateral reduces
to a triangle. Since $m_{1}|U_{e1}^{2}|$ grows faster with $m_{1}$ than
$m_{2}|U_{e2}^{2}|$ and $m_{3}|U_{e3}^{2}|$ in the NO, there are values of
$m_{1}$ where the sum or difference of $m_{2}|U_{e2}^{2}|$ and
$m_{3}|U_{e3}^{2}|$ correspond on $m_{1}|U_{e1}^{2}|$. The situation is
different in IO as the inequalities
$m_{1}|U_{e1}^{2}|>m_{2}|U_{e2}^{2}|>m_{3}|U_{e3}^{2}|$ are satisfied for all
values of $m_{3}$ and the currently allowed values for the mixing matrix
elements from Esteban:2020cvm . There are no values of the lightest mass where
two sides of the quadrilateral sum up another side. Therefore the
quadrilateral never collapses to a triangle and the minimum of
$|m_{\beta\beta}|$ in the IO is $|m_{\beta\beta}|=19.8$ meV with $m_{3}=2.98$
meV. These values for the absolute neutrino mass scale which lead to
$|m_{\beta\beta}|<10^{-3}$ eV can be tested with the next generation of
laboratory based experiments like the ECHo experiment Gastaldo:2017edk ,
Project 8 Project8:2017nal , and the PTOLEMY experiment PTOLEMY:2019hkd  as
well as cosmological experiments Font-Ribera:2013rwa ;
SimonsObservatory:2018koc ; CMB-S4:2016ple ; Brinckmann:2018owf  which will be
sensitive down to neutrino masses in the $\mathcal{O}(10~{}\text{meV})$
region. This, combined with the preference for the NO over the IO from
oscillation data, makes a study into the funnel region most timely.

Figure 2: The currently allowed region in the
$|m_{\beta\beta}|-m_{\text{lightest}}$ plane for both mass hierarchies (the
upper band corresponds to IO, the lower region to NO). The yellow region is
the expected allowed region with perfect knowledge of the oscillation
parameters and the blue region indicates the increased region allowed due to
the expected future precision in the oscillation parameters from DUNE and
JUNO. The red region indicates the increased region including the current
uncertainties on the oscillation parameters. All contours are drawn at true
$3\sigma$. The current upper limit on $|m_{\beta\beta}|$ from KamLAND-Zen is
shown as horizontal gray bands where the darker and lighter gray region assume
different determinations of the nuclear matrix element. The upper bound from
cosmology on the absolute neutrino mass scale in NO DiValentino:2021hoh  is
shown as vertical gray band. Figure 3: A visual representation of
$m_{\beta\beta}$ on the complex plane in the normal ordering for some choice
of the Majorana phases and the mass of the lightest neutrino. Since the three
legs that make up $m_{\beta\beta}$ nearly close in this example,
$|m_{\beta\beta}|$ is small enough to be in the funnel.

##  3 Results for model categories

In this section we will introduce the model categories we study, provide an
overview of the underlying theory, and present the preferred regions of
parameter space and fractions in the funnel.

We start by providing a complete phenomenological study of categories of
models which make predictions for observables which enter $|m_{\beta\beta}|$.
These categories of models can be further subdivided into groups of models
which have the same predictions. These groups of models are defined to cover
existing individual models that are studied in the literature, but are
expanded to include other conceivable models with different combinations of
the same input parameters. An important condition of our analysis is that we
will not be concerned with whether or not these models can be fully realized
in concrete scenarios, and simply consider the option that they could be,
thereby providing a phenomenological starting point for model builders by
scanning over conceivable models.

The model categories and the neutrino parameters they predict are
schematically shown in fig. 4. Starting from the phenomenological point of
view from model predictions for observables entering $|m_{\beta\beta}|$ we
consider categories of models which make predictions for one or several of
them. Indeed, each phenomenological category makes predictions at a certain
level in the mass matrix. Predictions for the mixing parameters are generally
driven by the structure of the neutrino mass matrix while predictions for the
neutrino masses depend on the number of free parameters in the neutrino mass
matrix. Models which affect the neutrino masses typically also predict a lower
(and upper) limit on the lightest neutrino mass allowing an additional probe
of these categories via experiments sensitive to the absolute neutrino mass
scale.

Figure 4: Overview and categorization of the models studied and the parameters
they predict. The grayed parameters, $\delta$ and $\theta_{23}$ do not affect
neutrinoless double beta decay and not all flavor models (specifically sum
rules) that predict the Majorana phases $\alpha$ and $\beta$ also predict
$\delta$. Each category contains groups of models for which we derive results.
The groups of models contain individual models realized in complete scenarios,
for example based on underlying symmetries.

The five model categories we consider in the following are:

  * •

Generalized CP (§ 3.2) which makes predictions for all three complex phases
only.

  * •

Sum rules (§ 3.3) which make predictions for the masses and the Majorana
phases only101010We will consider sum rules for the masses here. There exists
another category of sum rules which involves the angles, these typically arise
in models with charged lepton corrections, see sec. 3.4..

  * •

Charged lepton corrections (§ 3.4) which make predictions for the mixing
angles and complex phases.

  * •

Texture zeros (§ 3.5) which make predictions for all nine parameters in the
mass matrix.

  * •

Modular symmetries (§ 3.6) which also make predictions for all nine parameters
in the mass matrix.

To better understand the predictions of theses models we compare the number of
constraints to the number of free parameters in the neutrino sector. As we are
interested in neutrinoless double beta decay which can only happen for
Majorana neutrinos, we focus on this case only, even though the model
categories we study here also allow for Dirac neutrinos. The complex,
symmetric Majorana mass matrix has twelve parameters of which three phases can
be absorbed into the three flavor eigenstates. Therefore we are left with nine
free parameters. The Majorana mass matrix $M_{\nu}$ is diagonalized as

$\displaystyle M_{\nu}=U_{\text{PMNS}}~{}D_{\nu}~{}U_{\text{PMNS}}^{T}$ (5)

with the PMNS matrix $U_{\text{PMNS}}$ Pontecorvo:1957cp ; Maki:1962mu  and
the diagonal mass matrix $D_{\nu}$ which contains the eigenvalues of $M_{\nu}$
which are the light neutrino masses including the Majorana phases
$D_{\nu}=\text{diag}(m_{1}\text{e}^{\text{i}\alpha},~{}m_{2}\text{e}^{\text{i}\beta},m_{3})$
which amounts to nine free parameters. In appendix A, we give the expressions
for the mass matrix elements as a function of the mixing parameters and mass
eigenvalues. Both sides of eq. (5) are parametrized with the same number of
parameters: nine, nevertheless there is no one-to-one mapping of the mixing
parameters to the matrix elements, all mass matrix elements depend on a
combination of mixing parameters.

We point out that the nine parameters in the mass matrix need not be split up
in terms of masses (eigenvalues), mixing angles, and complex phases in the
usual way; there are other viable parameterizations of the degrees of freedom
of the mass matrix. One such example is with SU(3) generators (e.g. Gell-Mann
matrices), see appendix B.

Focused on the usual parameterizations, the nine free parameters in the
neutrino sector, assuming Majorana neutrinos, are the three neutrino masses,
three neutrino mixing angles, and three CP phases. Out of these nine
parameters five have been measured at neutrino oscillation experiments (three
angles, two mass splitting) while a sixth parameter, the Dirac CP phase will
be measured in the future DUNE:2020ypp ; Hyper-Kamiokande:2018ofw . Out of
these five measured parameters, only four impact $0\nu\beta\beta$ as
$|m_{\beta\beta}|$ does not depend on $\theta_{23}$ (it also does not depend
on $\delta$). Experiments sensitive to the absolute neutrino mass can
constrain one parameter which also plays a role in $0\nu\beta\beta$.

Each different class of models not only impacts different sets of the physical
parameters as shown in fig. 4, but also constrains those parameters at
different levels. Some such as generalized CP or sum rules provide only a
small number of constraints while others like modular symmetries provide a
large number of interconnected constraints among all the parameters.

###  3.1 Numerical approach

In order to quantify the validity of a given model and also its interplay with
the funnel, we perform careful numerical studies, the methods of which are
outlined here. While there are some necessary choices to be made about the
nature of the analyses, they have been made in such a way as to allow for a
direct comparison among the different models and model classes and a
representative numerical picture of the relationship between flavor models
compatible with oscillation data and the funnel.

We study the predictions of the flavor models requiring that the model
predictions for the mixing parameters are in agreement with the experimental
data, i.e. these flavor models correctly describe leptonic mixing and are
hence not ruled out111111We only consider priors on the three mixing angles
and two mass splittings, but not on the sign of $\Delta m^{2}_{31}$, despite
some evidence that it is positive. In case a model also predicts $\delta$ we
do not include a prior. Similarly, we do not include any prior on the absolute
neutrino mass scale.. We will use the current global fit data for the mixing
angles from Esteban:2020cvm  to derive the allowed values for
$|m_{\beta\beta}|$. As discussed in the previous section, we consider the true
$3\sigma$ allowed regions of parameter space which corresponds to a total
$\Delta\chi^{2}=11.83$ ($3\sigma$ for 2 dof) interpreted as the sum of the
individual $\Delta\chi^{2}$ of the mixing angles and mass splittings. This
approach is different to what is commonly done in the literature where the
allowed regions in the $|m_{\beta\beta}|-m_{\text{lightest}}$ plane (either in
general or for a specific model) are derived by varying the individual mixing
parameters in their $3\sigma$ ranges which leads to a total $\Delta\chi^{2}$
larger than it should be. This difference in the statistical approach leads to
a difference in our results compared to results in the literature with other
allowed regions being artificially looser than the quoted statistical
significance implies. Another crucial difference arises from our usage of up-
to-date global fit results of the mixing parameters. In particular, the
uncertainty on $\Delta m_{31}^{2}$ decreased in the past 5 years from 4% to 1%
while after 2013 the uncertainty on $\Delta m^{2}_{21}$ and $\theta_{12}$
remained similar and the improvements in the precision of $\theta_{13}$
Denton:2022een  do not have a huge impact on the uncertainties for
0$\nu\beta\beta$.

To determine the fraction of models in the funnel we follow several steps for
each class of models:

  1. 1.

We first calculate the number of models which are viable. These are the models
that are in agreement with the oscillation data.

  2. 2.

Then we determine which of those have any fraction within the funnel which we
define to be $m_{\beta\beta}<10^{-3}$ eV.

  3. 3.

Then we determine the fraction of each model that is within the funnel as
outlined below.

Different classes of models structure their predictions differently; some
provide constraint equations while others also introduce new underlying
parameters of the model. Thus there is not a straightforward means to
consistently sample the model space; a study of one model (or one class of
models) might prefer a different statistical test and come to slightly
different conclusions. Instead, we use a simple phenomenologically motivated
definition that will be equally representative for all models, although we
caution the reader that even still some regions of parameter space may be
over-/under-represented compared to the representative size of the underlying
parameters. We define the fraction within the funnel as,

$f=\frac{\int_{\rm funnel}d\log m_{\rm lightest}d\log m_{\beta\beta}}{\int
d\log m_{\rm lightest}d\log m_{\beta\beta}}\,,$ (6)

where the integral are over the allowed parameter space for a given model. For
the denominator we only take the NO into account. We also consider the same
expression with a linear distribution on the masses ($d\log m\to dm$). We also
bound the integral $m_{\rm lightest}\in[10^{-4},10^{-1}]$ eV and
$m_{\beta\beta}\in[10^{-4},10^{0}]$ eV as shown in fig. 2. In some cases this
affects the numerical results somewhat artificially, but these numbers are
well motivated by existing limits on the lightest neutrino mass and
$|m_{\beta\beta}|$ and the general narrative does not change much. In addition
to the fraction within in the funnel we also show probability density
functions (PDFs) of each category in the
$|m_{\beta\beta}|-m_{\text{lightest}}$ plane where the darker the color, the
higher the PDF. Due to common choice to present these plots in log-log scale
the regions covered in these models are not necessary uniform in the colored
regions.

We now turn to the five model classes in the following subsections.

###  3.2 Models with generalized CP

In models where CP is a conserved quantity the values of the CP violating
phases are constrained to be 0 or $\pi$ Wolfenstein:1981rk ; Kayser:1984ge ;
Bilenky:1984fg ; Branco:1986gr ; Feruglio:2012cw ; Holthausen:2012dk . On the
other hand, the phases can have non-trivial phases if a discrete symmetry is
combined with a generalized CP symmetry Feruglio:2012cw . Apart from the CP
conserving values possible predictions for the Majorana phases are
$\pi/2,~{}3\pi/2$ Ding:2013nsa ; Ding:2014hva ; King:2014rwa ; Ding:2014ssa ;
Hagedorn:2014wha ; Ding:2014ora ; Ding:2015rwa . Similar to Penedo:2018kpc  we
consider 16 combinations of values for the Majorana phases
$(\alpha,\beta)\in\\{0,\pi/2,\pi,3\pi/2\\}$. Out of the 16 combinations,
several map onto each other (see appendix C) such that there are only 10
independent combinations. All of them are viable since they only predict the
Majorana phases and the ones with $(0,~{}\pi),~{}(\pi,~{}0)$ predict a region
in the funnel, see fig. 5. We find a $\sim 50\%$ probability (using a log
prior) that these two models are in the funnel. Furthermore, these models
cover much of the whole allowed region for $m_{\beta\beta}$. Since the PDF is
not uniform, however, there is a preference in these models for
$m_{\beta\beta}$ values close to the lower allowed bound in IO and towards
small values of the lightest mass in NO, even though all models are compatible
with all values of $m_{\text{lightest}}$ as they do not predict a lower limit
on the absolute neutrino mass.

Figure 5: Left: The PDF of all 10 different models within the generalized CP
classification using the latest constraints from the oscillation data. Right:
The histograms showing how much each model is in the funnel. We see that all
10 models are consistent with oscillation data and two of them are in the
funnel. Using a log prior on $m_{\rm lightest}$ there is a $\sim 50\%$
probability that they are in the funnel, while with a linear prior the
probability is much less.

###  3.3 Models with mass sum rules

Mass sum rules are relations between the three complex neutrino eigenvalues
$m_{i}\text{e}^{\text{i}\alpha_{i}}$ (for overviews see Barry:2010yk ;
King:2013psa ; Gehrlein:2020jnr ) and arise in flavor models where the
neutrino mass matrix depends on two complex parameters only Gehrlein:2017ryu .
Then the three eigenvalues of the mass matrix are not independent but are
related by a sum rule. As one complex neutrino mass eigenvalue can be
expressed as a function of the other two, these models constrain two
parameters in the mass matrix and therefore these models predict two
parameters of interest in $|m_{\beta\beta}|$.

Mass sum rules can be parameterized with 5 free real model parameters

$\displaystyle
c_{1}\text{e}^{\text{i}\chi_{1}}(m_{1}\text{e}^{\text{i}\alpha})^{d}+c_{2}\text{e}^{\text{i}\chi_{2}}(m_{2}\text{e}^{\text{i}\beta})^{d}+m_{3}^{d}=0\,,$
(7)

where $d$ is the power of the sum rule, $c_{1},c_{2}$ are the real
coefficients of the sum rule, and $\chi_{1},\chi_{2}$ are the phases. Note
that we have set the coefficient and phase of $m_{3}$ to be 1 and 0,
respectively.

Up to now 12 mass sum rules have been identified in over 60 different models
Barry:2010yk ; Bazzocchi:2009da ; Ding:2010pc ; Ma:2005sha ; Ma:2006wm ;
Kang:2015xfa ; Honda:2008rs ; Brahmachari:2008fn ; Altarelli:2005yx ;
Chen:2009um ; Chen:2009gy ; Cooper:2012bd ; Altarelli:2009kr ;
Altarelli:2008bg ; Hirsch:2008rp ; Bazzocchi:2009pv ; Everett:2008et ;
Boucenna:2012qb ; Mohapatra:2012tb ; Altarelli:2005yp ; Altarelli:2006kg ;
Ma:2006vq ; Bazzocchi:2007na ; Bazzocchi:2007au ; Lin:2008aj ; Ma:2009wi ;
Ciafaloni:2009qs ; Bazzocchi:2008ej ; Feruglio:2013hia ; Chen:2007afa ;
Ding:2008rj ; Chen:2009gf ; Feruglio:2007uu ; Merlo:2011hw ; Luhn:2012bc ;
Fukuyama:2010mz ; Ding:2013eca ; Lindner:2010wr ; Hashimoto:2011tn ;
Ding:2011cm ; Morisi:2007ft ; Adhikary:2008au ; Lin:2009bw ; Csaki:2008qq ;
Hagedorn:2009jy ; Burrows:2009pi ; Ding:2009gh ; Mitra:2009jj ;
delAguila:2010vg ; Burrows:2010wz ; Ahn:2014zja ; Karmakar:2014dva ;
Ahn:2014gva ; He:2006dk ; Berger:2009tt ; Kadosh:2010rm ; Lavoura:2012cv ;
King:2012in ; Adulpravitchai:2009gi ; Dorame:2011eb ; Dorame:2012zv . These
previously studied sum rules have parameters within certain typical ranges:
$c_{1},~{}c_{2}\sim\mathcal{O}(1),~{}d=\pm 1,\pm
1/2,\chi_{1},\chi_{2}=0,\pi,\pm\pi/2$. However other values for these
parameters are possible.

For this study we will remain agnostic about the model realizations of the
mass sum rules and study mass sum rules with
$c_{1},~{}c_{2}\in[1/6,2/6,3/6,4/6,5/6,1,2]$, $d=\pm 1,~{}\pm 1/2$,
$\chi_{1},\chi_{2}=0,\pi,\pi/2,3\pi/2$. These choices include 11 realized mass
sum rules, additionally we consider
$c_{1}=2/(\sqrt{3}+1),~{}c_{2}=(\sqrt{3}-1)/(\sqrt{3}+1),~{}d=1,~{}\chi_{1}=0,~{}\chi_{2}=\pi$
to fully cover the parameter space of allowed mass sum rules with constant
coefficients. For the sake of concreteness we do not include $d=\pm 1/4,\pm
1/3$ as they have not appeared yet in realized sum rules in the literature.
Our choice of parameters to study covers existing models in the literature. It
is conceivable that also other models could be realized with ratios of larger
integers; in order to retain some amount of predictivity, we truncate the
parameters at the level of existing models in the literature.

Figure 6: The allowed region at $\Delta\chi^{2}=11.83$ for the current
constraint on the oscillation parameters is shown in red, the region for the
expected future precision at $\Delta\chi^{2}=11.83$ is shown in blue, and
perfect precision is shown in orange. The large regions are with no model
constraints while the smaller regions are for various sum rules
$(c_{1},c_{2},d,\chi_{1},\chi_{2})$. A: $(1,2,\frac{1}{2},\pi,\frac{\pi}{2})$,
B: $(\frac{1}{2},\frac{1}{2},-\frac{1}{2},\pi,\pi)$, and C: $(1,2,1,\pi,0)$.

A mass sum rule can be interpreted as a triangle in the complex plane which
closes if the sum rule is fulfilled, this leads to prediction for the Majorana
phases depending on the light neutrino masses. Furthermore, there is a
lightest neutrino mass for which the triangle can close, in some cases there
is also an upper limit on the neutrino masses. For some coefficients the mass
sum rule can never be fulfilled like
$-2\sqrt{m_{1}\text{e}^{\text{i}\alpha}}+1/2\sqrt{m_{2}\text{e}^{\text{i}\beta}}-\sqrt{m_{3}}=0$,
while in other cases the mass sum rule can only be fulfilled for one neutrino
mass ordering but not for the other like
$m_{1}\text{e}^{\text{i}\alpha}-2m_{2}\text{e}^{\text{i}\beta}-m_{3}=0$ which
can only be fulfilled in the NO. All of these predictions affect
$|m_{\beta\beta}|$ making this observable the ideal probe of the existence and
type of mass sum rules. In general mass sum rules only allow a small range in
the $|m_{\beta\beta}|-m_{\text{lightest}}$ parameter space Cirigliano:2022oqy
. In fig. 6 we show the allowed ranges for several representative sum rules.
We see that predictions from sum rules can be very different and while all sum
rules predict a lower bound on the lightest mass some of them also predict an
upper bound.

Figure 7: The same as fig. 5 but for sum rules.

In fig. 7 we show the PDF of models with sum rules. Out of the 3137 models,
1968 are viable of which 14% are in the funnel. None of the 12 models realized
in the literature have a fraction in the funnel121212This statement seemingly
contradicts previous results King:2013psa ; Gehrlein:2016wlc  however this
discrepancy arises due to the different choice of $\chi^{2}$ contours.. There
are 17 models with at least 50% in the funnel and they are enumerated in
appendix D131313We provide a text file containing all sum rule models at
peterdenton.github.io/Data/0nubb_Survey.

From fig. 7 we also see that sum rules cover the whole parameter space rather
uniformly however none of the models we studied allows for $m_{1}<10^{-4}$ eV
in NO while there is no lower bound in IO. This can be understood as in NO
there is a hierarchy between the masses even for small $m_{1}$ which requires
larger coefficients than we study to fulfill the sum rule for smaller masses.
In IO on the other hand $m_{1}$ and $m_{2}$ are nearly degenerate such that
cancellations between them can occur which allows to fulfill the sum rule.

We further investigated the roles the five individual sum rule parameters play
in the behavior of the models as shown in fig. 8. These figures show, for each
value of each parameter, the fraction of all models that are either not
consistent with oscillation data (orange), consistent with oscillation data
but never in the funnel (green), or consistent with oscillation data and some
fraction in the funnel (blue). Interesting trends appear. We see that sum rule
models are more likely to be in the funnel for small $c_{1}$ and large
$c_{2}$. The exponent $d$ also plays an important role, particularly that
$d=1$ is never in the funnel as in this case the sum rule always leads to
values of $m_{3}$ so large that the quadrilateral for $m_{\beta\beta}$ cannot
collapse to a triangle. On the other hand, models with $d=-1/2$ have the
largest fraction ($\approx$ 20%) in the funnel. The values of
$\chi_{1},~{}\chi_{2}$ individually do not drastically impact the validity or
fraction in the funnel while we find that for $d=-1$ over 80% of the models
are viable but less than 40% for $d=1/2$.

We generally find that more than 50% of the models studied are viable. For the
coefficients $c_{1},~{}c_{2}$ we find that the larger they are the more viable
models we find however $c_{1}=2$ again leads to fewer viable models.

Furthermore, if both coefficients are small there is only a small fraction of
valid models which we also show in fig. 9. Finally, even though the values of
$\chi_{1},~{}\chi_{2}$ individually are not very important for the validity or
fraction in the funnel, we find a correlation between them and larger values
of both are preferred to find valid models, see fig. 9. For the other
parameters we do not find strong correlations among them.

Thus if the data indicates that we could be in the funnel or if one wants to
build specific models that map onto sum rules that are consistent with current
oscillation data and are or are not in the funnel, this can give some guidance
about what kinds of parameters are likely to achieve those goals.

Figure 8: The 3137 considered sum rule models split into the values of each of
the five parameters with one panel for each of the five parameters. The bars
indicate the fraction of models with a specific value of one parameter that
are either inconsistent with oscillation data (orange), consistent with
oscillation data but never in the funnel (green), or consistent with
oscillation data and some fraction in the funnel (blue). We only study the
case of NO.

Figure 9: Fraction of valid models with sum rules for a set of two model
parameters where we find correlations.

###  3.4 Models with discrete symmetries in the neutrino sector and non-zero
charged lepton mixing

Many flavor models based on discrete symmetries predict $\theta_{13}=0$ which
is in strong contrast to the experimental data which prefers
$\theta_{13}^{\text{exp}}\approx 8.5^{\circ}$ DayaBay:2018yms ; RENO:2015ksa .
Therefore these predictions from discrete symmetries need to be corrected. A
way to do so is by introducing a non-diagonal charged lepton mixing matrix as
the measureable PMNS matrix is the product of the neutrino mixing matrix and
the charged lepton mixing matrix $U_{\text{PMNS}}=U_{e}^{\dagger}U_{\nu}$. The
introduction of a non-diagonal charged lepton mixing matrix leads to relations
between the observable mixing parameters, including the Majorana phases. These
relations are called mixing sum rules Ge:2011ih ; Ge:2011qn ; Marzocca:2013cr
; Petcov:2014laa ; Girardi:2014faa ; Girardi:2015zva ; Girardi:2015vha ;
Girardi:2016zwz ; Girardi:2015rwa ; Gehrlein:2022nss  (for reviews, see
King:2013eh ; King:2014nza ; Petcov:2017ggy ) and are similar to the relations
between the mixing parameters which arise in models with modular symmetries
described below. A non-diagonal charged lepton mixing matrix could for example
originate in grand unified theories based on SU(5) Georgi:1974sy  or SO(10)
Georgi:1974my ; Fritzsch:1974nn  where the structures of the mass matrices for
the charged lepton mass matrix and down quarks coincide Antusch:2011qg ;
Marzocca:2011dh ; Antusch:2009gu ; Antusch:2012fb  such that the charged
lepton sector exhibits CKM like mixings Datta:2005ci . In Petcov:2014laa ;
Girardi:2016zwz  a detailed, systematic study of various forms of
$U_{\nu},~{}U_{e}$ in flavor models has been conducted and the expressions for
the Majorana phases, as well as for the mixing parameters, have been derived.

We will consider the cases of two or three rotations in the neutrino sector
and one or two rotations in the charged lepton sector. We use for the neutrino
mixing angles $\theta_{23}^{\nu}=45^{\circ}$, and several cases for
$\theta_{12}^{\nu}$ motivated by different popular symmetry forms of the
neutrino mixing matrix, i.e. $\sin^{2}\theta_{12}^{\nu}=1/3$, $1/2$,
$1/(2+\phi_{g})$, $(3-\phi_{g})/4$, and $1/4$ with $\phi_{g}=(1+\sqrt{5})/2$
the golden ratio. We call these models TBM, BM, GRA, GRB, and HG standing for
tri-bi-maximal mixing, bi-maximal mixing, golden ratio A form, golden ratio B
form, and hexagonal form respectively. Additionally, we consider models with
three neutrino rotations with $\theta_{13}^{\nu}=\pi/10$, $\pi/20$, and
$\arcsin(1/3)$, which we call T13-1, T13-2, and T13-3 respectively, motivated
by existing models in the literature Bazzocchi:2011ax ; Rodejohann:2014xoa ;
deAdelhartToorop:2011nfg ; Ding:2012xx ; King:2012in . The rotations in the
charged lepton sector are free and are effectively constrained the measured
mixing angles. In fact, for models with $\theta_{13}^{\nu}=0$ the charged
lepton corrections are crucial to reproduce the observed mixing angles.
However charged lepton corrections also impact the predictions for the other
mixing angles such that also deviations from maximal $\theta_{23}$ can be
achieved. Therefore we include also models with two charged lepton rotations.
However we constrain ourselves to a maximum of a total of four rotations split
between the neutrino and charged lepton sector as they provide sufficient
freedom to reproduce the three measured mixing angles. More rotations or
different predicted values of the neutrino or charged lepton mixing angles
might arise however in concrete models Frampton:2004ud .

To summarize the previous paragraph, we consider one or two charged lepton
corrections where we study all combinations of
$\theta_{13}^{e},~{}\theta_{12}^{e},~{}\theta_{23}^{e}$ rotations141414We do
the rotations in the standard order
$U_{\text{PMNS}}=(U_{12}^{e})^{\dagger}(U_{13}^{e})^{\dagger}(U_{23}^{e})^{\dagger}\Psi
U_{23}^{\nu}U_{13}^{\nu}U_{12}^{\nu}Q$.. Additionally, we also consider phases
in the charged lepton mixing matrix which are necessary to obtain predictions
on the phases in the PMNS matrix. We introduce the phase matrices
$\Psi=\text{diag}(1,~{}\text{e}^{-\text{i}\psi_{1}},~{}\text{e}^{-\text{i}\psi_{2}})$
and
$Q=\text{diag}(1,~{}\text{e}^{\text{i}\xi_{1}/2},~{}\text{e}^{\text{i}\xi_{2}/2})$.
Explicitly we derive results for the following scenarios

  * •

two rotations in the neutrino sector, one charged lepton rotation (15 cases)

$\displaystyle U_{\text{PMNS}}$ $\displaystyle=(U_{ij}^{e})^{\dagger}\Psi
U_{23}^{\nu}(\pi/4)U_{12}^{\nu}(\theta_{12}^{\nu,k})Q$
$\displaystyle\text{where }(ij)\in\\{12,~{}13,~{}23\\}$
$\displaystyle\text{and }k\in\text{\\{TBM, BM, GRA, GRB, HG\\}}$ (8)

  * •

two rotations in the neutrino sector, two charged lepton rotation (15 cases)

$\displaystyle U_{\text{PMNS}}$
$\displaystyle=(U_{ij}^{e})^{\dagger}(U_{lm}^{e})^{\dagger}\Psi
U_{23}^{\nu}(\pi/4)U_{12}^{\nu}(\theta_{12}^{\nu,k})Q$
$\displaystyle\text{where
}(ij)\in\\{12,~{}13\\},~{}(lm)\in\\{13,~{}23\\},~{}(ij)\neq(lm)~{}$
$\displaystyle\text{and }k\in\text{\\{TBM, BM, GRA, GRB, HG\\}}$ (9)

  * •

three rotations in the neutrino sector, one charged lepton rotation (45 cases)

$\displaystyle U_{\text{PMNS}}$ $\displaystyle=(U_{ij}^{e})^{\dagger}\Psi
U_{23}^{\nu}(\pi/4)U_{13}^{\nu}(\theta_{13}^{\nu,p})U_{12}^{\nu}(\theta_{12}^{\nu,k})Q~{}$
$\displaystyle\text{where }(ij)\in\\{12,~{}13~{},~{}23\\}$ (10)
$\displaystyle\text{and }k\in\text{\\{TBM, BM, GRA, GRB, HG\\}}$
$\displaystyle\text{and }p\in\text{\\{T13-1, T13-2, T13-3\\}}$

For the phases contained in $\Psi,~{}Q$ we remain agnostic about their values
and we vary them freely151515Simultaneously employing, for example, a
generalized CP symmetry allows to fix the values of the phases like in
Girardi:2016zwz .. Notice that not in all cases are all four phases physical.
In fact, only for the case of two charged lepton rotations with
$\theta_{12}^{e},\theta_{13}^{e}\neq 0$ do all four phases play a role. The
number of free parameters thus varies in the different scenarios. The case of
three neutrino rotations and one charged lepton rotation has the same number
of rotations as the case with two of each, yet since the free mixing angles
are always contained in the charged lepton sector the case of two neutrino
rotations and two charged lepton rotations has the most freedom.

These models make predictions for the mixing angles and the Majorana phases
therefore they can be tested in 0$\nu\beta\beta$ experiments. In addition
these models also predict the CP phase $\delta$, however we do not include a
prior on $\delta$ in our analysis. Nevertheless this prediction also presents
a crucial test of these class of models Petcov:2014laa ; Girardi:2014faa . On
the other hand these models do not predict a lower bound on the lightest
neutrino mass.

For the models with two rotations in the neutrino sector and one charged
lepton rotation we find that 8 out of 15 models are viable. The BM mixing
pattern in the neutrino sector cannot be brought in agreement with
experimental data with one charged lepton rotation. Other mixing patterns are
viable assuming a 1-2 or 1-3 rotation in the charged lepton sector. All models
studied with two charged lepton rotations are viable. In particular for BM
mixing, two charged lepton rotations are required to correct both vanishing
$\theta_{13}$ and maximal $\theta_{12}$. For three neutrino rotations and one
charged lepton rotation we find that only 8 out of 45 models are viable.

These results are in general agreement with results from the literature
Girardi:2015vha ; Girardi:2016zwz . Nevertheless, we notice that improved
precision on the oscillation parameters in comparison to the time were these
studies were done disfavors now some models which were previously allowed. In
total we find that out of the total 75 cases, 31 models are viable.

The predictions for $0\nu\beta\beta$ experiments are shown in fig. 10. We find
that many models predict a region in the funnel. As this category of models
does not predict the mass scale the regions extend to small masses and cover
the quasi-degenerate region disfavored by cosmology as well. The funnel
fractions are very similar in all models and between 20%-50%, demonstrating
that in this category of models up to a third of the parameter space can be
contained in the funnel161616We provide a text file containing all the charged
lepton correction models at peterdenton.github.io/Data/0nubb_Survey.

Figure 10: The same as fig. 5 but for models with discrete symmetries and
charged lepton corrections.

###  3.5 Models with texture zeros

In models with texture zeros it is assumed that the complex symmetric Majorana
mass matrix has some vanishing entries171717Note that one can also consider
the case where there are zeros in the charged- and neutral-lepton mass
matrices separately Ramond:1993kv ; Benavides:2022hca ; we will not consider
these scenarios..

Of particular importance for this paper is the 1-1 element of the Majorana
mass matrix which coincides with the observable $|m_{\beta\beta}|$, see
Merle:2006du ; Lashin:2011dn ; Dev:2006qe ; Verma:2020gpl ; Xing:2003jf ;
Xing:2003ic ; BenTov:2011tj ; Liu:2012axa . A symmetry realizations of texture
zeros can be come from an extended scalar sector and suitable Abelian
symmetries Grimus:2004hf . Here however we will remain agnostic of any
underlying symmetry behind texture zeros as well as about the origin of the
neutrino mass term181818There are other models which constrain the number of
free parameters in the mass matrix by imposing that the trace or the minor of
the mass matrix is zero Branco:2002ie ; Black:2000bk ; Singh:2018tqu ;
Dey:2023tsk  which we won’t consider here..

Majorana mass matrices with three or more independent texture zeros are
already ruled out by current oscillation data Frampton:2002yf  as in this case
there are more observables than free parameters. Therefore we will focus on
one- and two-texture zero mass matrices191919The case with no texture zeros is
not predictive as there are more free parameters than observables..

For vanishing mass matrix element $M_{\alpha\beta}=0$ the conditions

$\displaystyle\sum_{i=1}^{3}U_{\alpha i}U_{\beta i}D_{i}=0\,,$ (11)

applies where $D_{i}$ stands for the elements of the diagonal matrix $D$ and
$\alpha,~{}\beta$ run over the flavor indices $e,~{}\mu,~{}\tau$. This
condition takes the form of a mass sum rule, similar to sec. 3.3, where the
coefficients are the mixing matrix elements. We show explicitly the
expressions for vanishing mass matrix elements in appendix A. In the case of
one-texture zero mass matrices all six possible matrices are in agreement with
experimental data Lashin:2011dn , although in some case only one mass ordering
is allowed, see tabs. 1, 2.

Table 1: The fraction of each model that is in the funnel for the 1-texture zero cases as defined in the text using a log prior assuming the NO. All six models are viable in some region of parameter space. | Fraction in funnel
---|---
$M_{ee}$ | 1
$M_{e\mu}$ | 0.31
$M_{e\tau}$ | 0.30
$M_{\mu\mu}$ | 0
$M_{\mu\tau}$ | 0
$M_{\tau\tau}$ | 0
Table 2: The fraction of each model in the funnel for the 2-texture zeros cases as defined in the text assuming the NO. Models with an X are not viable anywhere in parameter space at $3\sigma$.  | $M_{e\mu}$ | $M_{e\tau}$ | $M_{\mu\mu}$ | $M_{\mu\tau}$ | $M_{\tau\tau}$
---|---|---|---|---|---
$M_{ee}$ | 1 | 1 | X | X | X
$M_{e\mu}$ |  | X | 0 | X | 0
$M_{e\tau}$ |  |  | 0 | X | 0
$M_{\mu\mu}$ |  |  |  | X | 0
$M_{\mu\tau}$ |  |  |  |  | X

Figure 11: The same as fig. 5 but for 1 texture zeros. Note that the
$M_{ee}=0$ model predicts that $|m_{\beta\beta}|=0$ and is thus at the bottom
of the left panel, hence the presence of a model that predicts 100% of the
model space in the funnel.

There are in total 15 two-texture zero matrices of which seven are in
agreement with experimental data Frampton:2002yf ; Guo:2002ei ; Xing:2002ta ;
Desai:2002sz ; Xing:2002ap ; Dev:2006if ; Dev:2006xu ; Fritzsch:2011qv ;
Honda:2003pg . Two of them feature a vanishing $e-e$ mass matrix element
($M_{ee}=0,~{}M_{e\mu}=0$ or $M_{ee}=0,~{}M_{e\tau}=0$) and predict therefore
100% of the parameter space in the funnel. Upon imposing two vanishing mass
matrix elements we obtain four relations between the mixing matrix elements
and observables. The two-texture zero mass matrices in agreement with
experimental data have one vanishing diagonal element of $M$,
$M_{\eta\eta}=0,~{}\eta=(e,~{}\mu,~{}\tau)$, and one of off-diagonal elements
in the electron row vanishes, $M_{e\gamma}=0$ with $\gamma={\mu,~{}\tau}$.
Lastly, the case with $M_{\mu\mu}=M_{\tau\tau}=0$ is also in agreement with
current data.

Figure 12: The same as fig. 5 but for 2 texture zeros. Note that as in fig.
11, the two $M_{ee}=0$ models predict that $|m_{\beta\beta}|=0$ and is thus at
the bottom of the left panel, hence the presence of two models that predict
100% of the model space in the funnel. On the left panel there is a small
sliver of predicted space in the quasi-degenerate region on the top right in
the cosmologically disfavored region.

The one-texture zero case leads to two predictions for $M_{\nu}$ as it
constrains the real and imaginary part of this mass matrix element202020Even
if the mass matrix element is chosen to be real there are still two
constraints on the combination of mixing matrix elements and mass
eigenvalues.. Two constraints also apply to oscillation and $0\nu\beta\beta$
experiments. In the two-texture zero case the number vanishing mass matrix
elements double therefore the number of constraints is now four. In both one-
and two-texture zero cases, one can derive expressions for masses, see
appendix A.

In fig. 11 we show the results for the one-texture zero case. We find that the
three models are in the funnel of which the model with $M_{ee}=0$ is 100% in
the funnel and $M_{e\mu}$ and $M_{e\tau}$ are partially in the funnel.
Additionally, both models in the funnel predict a lower bound on $m_{1}\gtrsim
4\times 10^{-3}$ eV in NO.

In fig. 12 we show the results for the two-texture zero case. We find that
only $46\%$ (7/15) of the models are viable and 28% (2/7) of the viable models
are in the funnel, specifically the two $M_{ee}=0$ models. Furthermore, the
five non-funnel viable models predict large values of the lightest mass
$m_{\text{lightest}}>3\times 10^{-2}$ eV which is in the quasi-degenerate
region and is already ruled out by cosmology. This means the only actual
viable models, when also including cosmological data, for two-texture zeros
are the models with $M_{ee}=0$ and one of either $M_{e\mu}=0$ or $M_{e\tau}=0$
which also predict $m_{\beta\beta}=0$. This is a new result.

There is another study which looked at a unique class of models which can be
described as texture zeros with rotational corrections. This study concluded
that it was possible for models to go well into the funnel, although it is
important to note that $\theta_{13}$ was not known at the time of this study

###  3.6 Modular symmetries with fixed modulus

In Gehrlein:2020jnr  models based on modular symmetries with a fixed modulus
were studied. In these models only one field is introduced which, upon
obtaining a vacuum expectation value, breaks the flavor symmetry
Feruglio:2017spp  (for a review see also Kobayashi:2023zzc ). In comparison to
models with discrete symmetries where multiple fields are introduced, a
reduction of free parameters is achieved which leads to more correlations
between physical parameters. So far, five models with the most amount of
correlations have been identified in the literature. In these models the
symmetric mixing matrix gets corrected by a 1-2 or 1-3 rotation, similar to
the case of one charged lepton rotation. Then the three mixing angles and the
Dirac CP phase are determined by two free model parameters only. These models
also lead mass sum rules similar to those discussed in subsection 3.3. In this
case, however, the coefficients of the mass sum rule are not constant but they
depend on the two free model parameters which leads to a correlation among the
neutrino masses, Majorana phases, and mixing parameters. The expressions for
the mass sum rules and the mixing angles can be found in Gehrlein:2020jnr ,
for convenience we quote them again in appendix E. Similar to the case of mass
sum rules these models predict a lower and an upper bound on the lightest
mass, see sec. 3.3.

Our results are shown in fig. 13. All five models are viable, although two of
them are only valid in the high mass region that is disfavored by cosmological
data. For the five models present in the literature we find that two are in
the funnel at only the 5% or 7% level (log prior).

It is likely that more models with such correlations exist. Their predictivity
of different neutrino observables makes them an interesting target for future
neutrino experiments, even beyond neutrinoless double beta decay
Gehrlein:2022nss .

Figure 13: The same as fig. 5 but for models with modular symmetries.

##  4 Discussion

In table 3 we give an overview of the number of models contained in each
category, the number of allowed models, and we provide the fraction of models
in the funnel. We see that some model categories have a sizable fraction in
the funnel, however we caution the reader that this could be an over
representation due to the measure chosen, see eq. (6). Finally, some model
categories only feature a small number of viable models therefore the total
numbers of models in the funnel is not so big.

Among the model categories surveyed we find that funnel fractions of $\approx$
20–100% are possible, making probes of the funnel region a crucial target to
comprehensively test different flavor models. Interestingly, most of the
models studied which feature a fraction of parameter space in the funnel also
predict parameter space outside of the funnel. This allows $0\nu\beta\beta$
experiments to narrow down the parameter spaces of the models in the near
future, even without penetrating the funnel region. An exception to this are
texture zero models with $M_{ee}=0$ which predict $m_{\beta\beta}=0$ exactly
and are therefore fully contained in the funnel212121Potentially some sum
rules with coefficients $c_{i}>2$ or different values of $d$ might also be
fully contained in the funnel..

Table 3: An overview of the number of total model groups contained in each model category, the number of valid model groups given oscillation data, and the fraction of valid model groups which penetrate the funnel region in the NO. Model | Total | Viable | Fraction of viable models in funnel
---|---|---|---
Generalized CP | 10 | 10 | 0.20
Mass sum rules | 3137 | 1968 | 0.14
Charged lepton corrections | 75 | 31 | 1.00
1-Texture zeros | 6 | 6 | 0.50
2-Texture zeros | 15 | 7 | 0.28
Modular symmetries | 5 | 5 | 0.40

Here we focused on models based on symmetries. Another approach, referred to
as “anarchy” assumes that the leptonic mixing matrix can be described as the
result of a random draw from an unbiased distribution of unitary three-by-
three matrices Hall:1999sn ; Haba:2000be ; deGouvea:2012ac . In the past it
has been shown that the probability for $|m_{\beta\beta}|<10^{-3}$ eV is
small, around $5\%$ deGouvea:2012ac , see also Jenkins:2008ms . Therefore
flavor models based on symmetries can be more likely to predict a region in
the funnel in some cases.

Additionally, several models like sum rules, modular symmetries, and
1-texture-zeros prefer large values for the lightest mass and present a lower
limit on $m_{\text{lightest}}$ within the reach of near future
0$\nu\beta\beta$ experiments Agostini:2022zub  such that the whole region of
parameter space can be probed with cosmology very soon. Generalized CP and
charged lepton corrections, on the other hand, do not predict the absolute
mass scale such that these models will remain viable independent of a future
measurement of $m_{\text{lightest}}$. We note that these models also predict
other observables which allow to test these models. In fact, these predictions
are crucial in our assessment of the validity of these models. We find that
for charged lepton corrections, and 2-texture zeros only roughly half of the
total models are viable due to their predictions for the mixing angles. In
comparison to previous studies in the literature, an important change is that
the precision on $\Delta m_{31}^{2}$ has improved from 4% to 1% Denton:2022een
which has a big effect on the results and in particular the validity of
models. Future measurements of the oscillation parameters will further test
models, in particular improvements on $\delta$ and $\theta_{12}$ will probe
and distinguish different models Gehrlein:2022nss  which will further narrow
down the number of valid flavor models. Even though neither $\delta$ nor
$\theta_{23}$, the two oscillation parameters which are currently most
uncertain, do not play a role for $0\nu\beta\beta$ their measurements
indirectly affect our results here as these measurements constrain the model
parameters in that they tell us which models are valid. For this study we did
not include a prior on $\delta$ however models based on generalized CP, with
charged lepton corrections, texture zeros and models with modular symmetries
also predict $\delta$ and their validity will be tested with the next
generation of experiments which in turn will change the 0$\nu\beta\beta$
landscape. Therefore flavor models provide a rich model space to test with
upcoming experiments, including also oscillation experiments and cosmology.

Finally, we assume that the predictions in the models are exact since we
remain agnostic about underlying model realizations. In a realistic model,
however, potential corrections to the predictions can arise, for example from
running effects or higher order operators. In the case of sum rules it has
been shown that potential corrections can affect the predictions on
$|m_{\beta\beta}|$ drastically Gehrlein:2015ena ; Gehrlein:2016wlc . Also in
the case of random values of the phases Brdar:2015jwo  running effects can
have a sizable impact. It has also been shown that some two texture zero mass
matrices can be brought into agreement with data when considering running
effects Hagedorn:2004ba .

##  5 Conclusions

An observation of neutrinoless double beta decay will have an tremendous
impact on our understanding of nature. Apart from proving that lepton number
is not a conserved symmetry of nature it can also provide valuable insights
into other open problems of the SM like the flavor puzzle. Motivated by the
current and anticipated experimental progress of various neutrino experiments
we have studied the predicted ranges of $m_{\beta\beta}$ and
$m_{\text{lightest}}$ of several classes of flavor models. In particular we
focused on the funnel region in normal mass ordering with $|m_{\beta\beta}|<1$
meV which is experimentally challenging to probe to answer the question how
likely a model prediction is only realized in the funnel which would require a
massive leap in experimental progress.

We have considered five broad classes of flavor models based on different
symmetries. After assessing their validity by comparing their predictions to
our up-to-date experimental knowledge from oscillation experiments, we
calculated the funnel fractions of the valid models. Our studies show that all
of studied model classes feature models with parameter space in the funnel.
Indeed, the fractions of viable models that are in the funnel range from
5–100%. Thus flavor models may well be more likely to predict that
$|m_{\beta\beta}|$ is in the funnel than in the case of random neutrino mixing
matrices, anarchy, where the funnel probability is around 5%. Additionally, we
have provided PDFs of the predicted $m_{\beta\beta}-m_{\text{lightest}}$
regions of the classes of flavor models. We find that models which predict the
absolute mass scale generally predict larger neutrino masses such that
cosmological observatories can test them as well in the near future in
addition crucial test of the predicted values for the mixing angles to
upcoming oscillation experiments.

Our results can be used to plan the target sensitivity of upcoming
neutrinoless double beta decay experiments with the goal to probe the most of
the parameter space motivated by flavor models (see Agostini:2015dna  for a
similar study focusing on existing sum rules in flavor models).

In this study we focused on light Majorana neutrino exchange as the underlying
scenario for $0\nu\beta\beta$. Other scenarios when new particles are
introduced could also predict a region in the funnel. For example models with
a sterile neutrino allow for vanishing rates for 0$\nu\beta\beta$
Abada:2018qok ; Zhao:2022trb ; Asaka:2020wfo ; Asaka:2020lsx ; Asaka:2021hkg ;
Pascoli:2013fiz . In particular, depending on the sterile parameters a funnel
in IO opens up. However a Bayesian analysis of eV steriles showed the
posterior probability that $|m_{ee}|$ falls into the funnel region is very
small $<0.3\%$ Huang:2019qvq . On the other hand models based on a left-right
symmetry do not predict a region in the funnel Tello:2010am ; Li:2020flq ,
neither does a model where a new scalar interaction Graf:2020cbf  is
introduced.

As neutrinoless double beta decay experiments continue to push the limits down
into the inverted mass ordering region understanding the theoretically favored
regions of parameter space is important to plan for experimental upgrades.
This progress needs to be accompanied by additional work to unambiguously
interpret measurement to improvement in nuclear matrix calculation is needed
as well.

###### Acknowledgements.

The authors acknowledge support by the United States Department of Energy
under Grant Contract No. DE-SC0012704. Part of this research was completed
while at the Center for Theoretical Underground Physics and Related Areas
(CETUP*).

##  Appendix A Expressions for the elements of the mass matrix

Here we give the expressions for the elements of the mass matrix as a function
of the mixing parameters and the mass eigenvalues. The number of free
parameters in the Majorana mass matrix and in the mixing matrix together with
the light mass eigenvalues, nine, coincide as expected. Realistically we are
only able to measure eight out of the nine free parameters in the mass matrix
as we have no observable which depends on the individual Majorana phases.

$\displaystyle m_{ee}={}$ $\displaystyle
m_{1}\text{e}^{\text{i}\alpha}c_{12}^{2}c_{13}^{2}+m_{2}\text{e}^{\text{i}\beta}c_{13}^{2}s_{12}^{2}+m_{3}s_{13}^{2}$
(12) $\displaystyle m_{e\mu}={}$ $\displaystyle
m_{1}\text{e}^{\text{i}\alpha}c_{12}c_{13}(-c_{23}s_{12}-\text{e}^{\text{i}\delta}c_{12}s_{13}s_{23})+m_{2}\text{e}^{\text{i}\beta}c_{13}s_{12}(c_{12}c_{23}-\text{e}^{\text{i}\delta}s_{12}s_{13}s_{23})$
$\displaystyle+m_{3}\text{e}^{\text{i}\delta}c_{13}s_{13}s_{23}$ (13)
$\displaystyle m_{e\tau}={}$ $\displaystyle
m_{1}\text{e}^{\text{i}\alpha}c_{12}c_{13}(-c_{12}c_{23}s_{13}+\text{e}^{-\text{i}\delta}s_{12}s_{23})+m_{2}\text{e}^{\text{i}\beta}c_{13}s_{12}(-c_{23}s_{12}s_{13}-\text{e}^{-\text{i}\delta}c_{12}s_{23})$
$\displaystyle+m_{3}c_{13}c_{23}s_{13}$ (14) $\displaystyle m_{\mu\mu}={}$
$\displaystyle
m_{1}\text{e}^{\text{i}\alpha}(-c_{23}s_{12}-\text{e}^{\text{i}\delta}c_{12}s_{13}s_{23})^{2}+m_{2}\text{e}^{\text{i}\beta}(c_{12}c_{23}-\text{e}^{\text{i}\delta}s_{12}s_{13}s_{23})^{2}+m_{3}\text{e}^{2\text{i}\delta}c_{13}^{2}s_{23}^{2}$
(15) $\displaystyle m_{\mu\tau}={}$ $\displaystyle
m_{1}\text{e}^{\text{i}\alpha}(-c_{12}c_{23}s_{13}+\text{e}^{-\text{i}\delta}s_{12}s_{23})(-c_{23}s_{12}-\text{e}^{\text{i}\delta}c_{12}s_{13}s_{23})$
$\displaystyle+m_{2}\text{e}^{\text{i}\beta}(-c_{23}s_{12}s_{13}-\text{e}^{-\text{i}\delta}c_{12}s_{23})(c_{12}c_{23}-\text{e}^{\text{i}\delta}s_{12}s_{13}s_{23})+m_{3}\text{e}^{\text{i}\delta}c_{13}^{2}c_{23}s_{23}$
(16) $\displaystyle m_{\tau\tau}={}$ $\displaystyle
m_{1}\text{e}^{\text{i}\alpha}(-c_{12}c_{23}s_{13}+\text{e}^{-\text{i}\delta}s_{12}s_{23})^{2}+m_{2}\text{e}^{\text{i}\beta}(-c_{23}s_{12}s_{13}-\text{e}^{-\text{i}\delta}c_{12}s_{23})^{2}$
$\displaystyle+m_{3}c_{13}^{2}c_{23}^{2}$ (17)

For one vanishing matrix element $M_{\alpha\beta}=0$ the expression for the
neutrino mass ratios are Lashin:2011dn

$\displaystyle\frac{m_{1}}{m_{3}}$ $\displaystyle=\frac{\text{Re}(U_{\alpha
3}U_{\beta 3})\text{Im}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})-\text{Re}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})\text{Im}(U_{\alpha 3}U_{\beta
3})}{\text{Re}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})\text{Im}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})-\text{Im}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})\text{Re}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})}$ (18) $\displaystyle\frac{m_{2}}{m_{3}}$
$\displaystyle=\frac{\text{Re}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})\text{Im}(U_{\alpha 3}U_{\beta
3}\text{e}^{\text{i}\beta})-\text{Im}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})\text{Re}(U_{\alpha 3}U_{\beta
3}\text{e}^{\text{i}\beta})}{\text{Re}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})\text{Im}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})-\text{Im}(U_{\alpha 2}U_{\beta
2}\text{e}^{\text{i}\beta})\text{Re}(U_{\alpha 1}U_{\beta
1}\text{e}^{\text{i}\alpha})}$ (19)

The condition of two vanishing mass matrix elements
$M_{\alpha\beta},~{}M_{\delta\gamma}$, $(\alpha\beta)\neq(\delta\gamma)$ can
be translated to expressions for the neutrino masses and Majorana phases
Xing:2002ta

$\displaystyle\frac{m_{1}}{m_{3}}$ $\displaystyle=\left|\frac{U_{\gamma
3}U_{\delta 3}U_{\alpha 2}U_{\beta 2}-U_{\gamma 2}U_{\delta 2}U_{\alpha
3}U_{\beta 3}}{U_{\gamma 2}U_{\delta 2}U_{\alpha 1}U_{\beta 1}-U_{\gamma
1}U_{\delta 1}U_{\alpha 2}U_{\beta 2}}\right|\;,$
$\displaystyle\frac{m_{2}}{m_{3}}$ $\displaystyle=\left|\frac{U_{\gamma
1}U_{\delta 1}U_{\alpha 3}U_{\beta 3}-U_{\gamma 3}U_{\delta 3}U_{\alpha
1}U_{\beta 1}}{U_{\gamma 2}U_{\delta 2}U_{\alpha 1}U_{\beta 1}-U_{\gamma
1}U_{\delta 1}U_{\alpha 2}U_{\beta 2}}\right|\;.$ (20)

$\displaystyle\alpha$ $\displaystyle=\arg\left[\frac{U_{\gamma 3}U_{\delta
3}U_{\alpha 2}U_{\beta 2}-U_{\gamma 2}U_{\delta 2}U_{\alpha 3}U_{\beta
3}}{U_{\gamma 2}U_{\delta 2}U_{\alpha 1}U_{\beta 1}-U_{\gamma 1}U_{\delta
1}U_{\alpha 2}U_{\beta 2}}\right]\;,$ $\displaystyle\beta$
$\displaystyle=\arg\left[\frac{U_{\gamma 1}U_{\delta 1}U_{\alpha 3}U_{\beta
3}-U_{\gamma 3}U_{\delta 3}U_{\alpha 1}U_{\beta 1}}{U_{\gamma 2}U_{\delta
2}U_{\alpha 1}U_{\beta 1}-U_{\gamma 1}U_{\delta 1}U_{\alpha 2}U_{\beta
2}}\right]\;.$ (21)

We see that the Majorana phases depend on the value of Dirac CP phase in the
PMNS matrix contained in matrix elements $U_{\mu i}$, $i\in[1,3]$, $U_{\tau
1},~{}U_{\tau 2}$. Furthermore, from eq. (20) we see that the ratios of the
neutrino masses depend on the values of the matrix elements. With the known
values for the mass splittings we obtain a lower bound on the lightest mass
depending on which matrix elements are zero.

From these expressions we see that there is no one-to-one correspondence
between mass eigenvalues and observables in experiments. This means that only
with a combination of measurements (i.e. different oscillation channels and an
observation of neutrinoless double beta decay) one can reconstruct the
neutrino mass matrix. This situation is similar considering only one
measurement at oscillation experiments. In one channel one is only sensitive
to a certain combination of parameters. Only a combination of measurements can
tell us the values of the mixing angles.

Also for absolute neutrino mass measurements like from KATRIN or cosmology and
neutrinoless double beta decay one is left with one measurement of one
combination of parameters assuming no prior knowledge of the results from
other experiments one can therefore predict something for oscillation
experiments as well. In reality, we have already measurements from
oscillations such that predictions from a absolute mass measurement do not
contribute new knowledge for oscillation experiments.

##  Appendix B Gell-Man $SU(3)$ generators and the mass matrix

The mass matrix need not be parameterized as three masses, three mixing
angles, and three phases. Other parameterizations are possible. One such
explicit example is with $SU(3)$ generators, such as the Gell-Mann matrices,
see e.g. Krivoruchenko:2023npj . That is, the mass matrix from eq. (5) can be
written as

$M=M_{\rm scale}\prod_{i=1}^{8}\exp(a_{i}\lambda_{i})\,,$ (22)

where the eight $a_{i}\in\mathbb{R}$ are free parameters as is $M_{\rm scale}$
which sets the dimensionful scale and the $\lambda_{i}$ are some traceless
representation of $SU(3)$ such as the Gell-Mann matrices. The dimensionful
scale parameter can also be thought of as the trace part of $M$. This could
imply a novel flavor structures similar to texture zeros by requiring some
subset of the $a_{i}$ to be zero. One could also consider representations
other than the Gell-Mann matrices, such as cyclic representations
Harrison:2014epa . Investigating the phenomenology of such flavor models is
beyond the scope of this work.

##  Appendix C Independent generalized CP models

In table 4 we list the phase combinations which are independent for models
with generalized CP. We see that it is sufficient to constrain $\alpha$ to be
between $[0,\pi]$ and $\beta\in[0,2\pi]$ to cover the whole parameter space.

Table 4: Pairs of values for the Majorana phases $\alpha,~{}\beta$ in models
with generalized CP which lead to different results for $|m_{\beta\beta}|$.
Some pairs are equivalent to others, these are in the table to the right. The
bolded pairs are the ones which predict a region in the funnel.

$(\alpha,~{}\beta$)
---
$\mathbf{(0,\boldsymbol{\pi})}$
$\mathbf{(\boldsymbol{\pi},0)}$
$(0,0)$
$(\pi,\pi)$

$(\alpha,~{}\beta$)
---
$(0,\pi/2)$ or $(0,3\pi/2)$
$(\pi/2,3\pi/2)$ or $(3\pi/2,\pi/2)$
$(\pi,\pi/2)$ or $(\pi,3\pi/2)$
$(\pi/2,0)$ or $(3\pi/2,0)$
$(\pi/2,\pi/2)$ or $(3\pi/2,3\pi/2)$
$(\pi/2,\pi)$ or $(3\pi/2,\pi)$

##  Appendix D Funnel models with sum rules

In table 5 we show the parameters of sum rules which lead to at least $50\%$
fraction in the funnel with a log prior.

Table 5: Parameters of sum rules which lead to at least 50% fraction in the funnel with a log prior.  $c_{1}$ | $c_{2}$ | $d$ | $\chi_{1}$ | $\chi_{2}$ | Fraction in funnel
---|---|---|---|---|---
1 | 2 | $-1/2$ | $\pi/2$ | 0 | 0.74
1 | 2 | $-1/2$ | $3\pi/2$ | 0 | 0.74
4/6 | 1 | $-1/2$ | $3\pi/2$ | 0 | 0.67
4/6 | 1 | $-1/2$ | $\pi/2$ | 0 | 0.62
5/6 | 1 | $-1/2$ | $3\pi/2$ | 0 | 0.59
5/6 | 1 | $-1/2$ | $\pi/2$ | 0 | 0.58
5/6 | 2 | $-1/2$ | $\pi/2$ | 0 | 0.58
5/6 | 2 | $-1/2$ | $3\pi/2$ | 0 | 0.58
1 | 2 | $-1/2$ | 0 | $\pi/2$ | 0.58
1 | 2 | $-1/2$ | 0 | $3\pi/2$ | 0.58
1/3 | 1 | $-1/2$ | 0 | $\pi/2$ | 0.56
4/6 | 5/6 | $-1/2$ | $\pi/2$ | 0 | 0.54
4/6 | 5/6 | $-1/2$ | $3\pi/2$ | 0 | 0.54
1/6 | 1/6 | $-1/2$ | $\pi$ | $\pi$ | 0.54
1/3 | 1 | $-1/2$ | 0 | $3\pi/2$ | 0.54
1/6 | 1/2 | $-1$ | 0 | 0 | 0.51
1/6 | 4/6 | $-1$ | 0 | 0 | 0.51

##  Appendix E Expressions for physical parameters in models with modular
symmetries

Here we give the expressions for the oscillation parameters and the sum rule
in models with modular symmetries, first derived in Gehrlein:2020jnr . The
parameters $\theta,~{}\phi$ are free model parameters.

  * •

A model based on $A_{4}$ symmetry was studied in Novichkov:2018yse . Two cases
arise, depending on the assumption on the charged lepton mixing matrix. The
expressions for the mixing parameters remain the same in both scenarios.

$\displaystyle\sin^{2}\theta_{12}(\theta)$
$\displaystyle=\frac{1}{3-2\sin^{2}\theta}\;,$ (23)
$\displaystyle\sin^{2}\theta_{13}(\theta)$
$\displaystyle=\frac{2}{3}\sin^{2}\theta\;,$ (24)
$\displaystyle\sin^{2}\theta_{23}(\theta,\phi)$
$\displaystyle=\frac{1}{2}+\frac{\sin\theta_{13}(\theta)}{2}\frac{\sqrt{2-3\sin^{2}\theta_{13}(\theta)}}{1-\sin^{2}\theta_{13}(\theta)}\cos\phi\;,$
(25) $\displaystyle\delta(\theta,\phi)$
$\displaystyle=\arcsin\left(-\frac{\sin\phi}{\sin
2\theta_{23}(\theta,\phi)}\right)\;,$ (26)

The parameters in the sum rule read (see eq. (7)) in case I are

$\displaystyle c_{1}$
$\displaystyle=-\text{e}^{-2\operatorname{\text{i}}\phi}-\operatorname{\text{i}}\text{e}^{-\operatorname{\text{i}}\phi}f_{2}\sin\phi\;,$
$\displaystyle=-\text{e}^{-2\operatorname{\text{i}}\phi}\frac{\sqrt{3}\sin(2\theta)-\cos\phi\cos(2\theta)-\operatorname{\text{i}}\sin\phi}{\sqrt{3}\sin(2\theta)-\cos\phi\cos(2\theta)+\operatorname{\text{i}}\sin\phi}$
(27) $\displaystyle c_{2}$
$\displaystyle=-\text{e}^{-\operatorname{\text{i}}\phi}\frac{2}{\sqrt{3}\sin(2\theta)-\cos\phi\cos(2\theta)+\operatorname{\text{i}}\sin\phi}\;$
(28) $\displaystyle d$ $\displaystyle=1$ (29)

In the second scenario the coefficients of the mass sum rule are related to
the coefficients in the first case by

$\displaystyle c_{1}^{(\text{II})}$
$\displaystyle=c_{1}^{(\text{I})}\text{e}^{-4\operatorname{\text{i}}\phi}\;,$
(30) $\displaystyle c_{2}^{(\text{II})}$
$\displaystyle=-c_{2}^{(\text{I})}\text{e}^{2\operatorname{\text{i}}\phi}\;.$
(31)

  * •

A model based on two modular $S_{4}$ symmetries has been studied in
King:2019vhv :

$\displaystyle\sin\theta_{13}$ $\displaystyle=\frac{\sin\theta}{\sqrt{3}}~{},$
(32) $\displaystyle\tan\theta_{12}$
$\displaystyle=\frac{\cos\theta}{\sqrt{2}}~{},$ (33)
$\displaystyle\tan\theta_{23}$
$\displaystyle=\left|\frac{\cos\theta+\sqrt{\frac{2}{3}}\text{e}^{\operatorname{\text{i}}\phi}\sin\theta}{\cos\theta-\sqrt{\frac{2}{3}}\text{e}^{\operatorname{\text{i}}\phi}\sin\theta}\right|~{}.$
(34) $\displaystyle\tan\delta$
$\displaystyle=-\frac{5+\cos(2\theta)}{1+5\cos(2\theta)}\tan\phi\;.$ (35)

The parameters in the sum rule read

$\displaystyle c_{1}$
$\displaystyle=\frac{1}{\cos^{2}\theta-\text{e}^{\operatorname{\text{i}}\phi}\sin(2\theta)}\;,$
(36) $\displaystyle c_{2}$
$\displaystyle=-\frac{\tan\theta+2\,\text{e}^{\operatorname{\text{i}}\phi}}{2\,\text{e}^{3\operatorname{\text{i}}\phi}-\text{e}^{2\operatorname{\text{i}}\phi}\cot(\theta)}\;,$
(37) $\displaystyle d$ $\displaystyle=-1\;.$ (38)

  * •

In Novichkov:2018ovf  a model with a modular $S_{4}$ symmetry has been
investigated:

$\displaystyle\sin\theta_{13}$ $\displaystyle=\frac{1}{\sqrt{3}}\sin\theta\;,$
(39) $\displaystyle\tan\theta_{12}$
$\displaystyle=\frac{1}{\sqrt{2}}\cos\theta\;,$ (40)
$\displaystyle\tan\theta_{23}$
$\displaystyle=\left|\frac{2\,\text{e}^{\operatorname{\text{i}}\phi}\tan\theta+\sqrt{3/2}\left(1+\operatorname{\text{i}}\sqrt{3}\right)}{3\sqrt{2/3}-\left(1-\sqrt{3}\operatorname{\text{i}}\right)\text{e}^{\operatorname{\text{i}}\phi}\tan\theta}\right|\;,$
(41) $\displaystyle\tan\delta$
$\displaystyle=-\frac{(\cos(2\theta)+5)\left(\sqrt{3}\sin\phi-3\cos\phi\right)}{(5\cos(2\theta)+1)\left(\sqrt{3}\cos\phi+3\sin\phi\right)}\;.$
(42)

The parameters of the sum rule are

$\displaystyle f_{1}$
$\displaystyle=\frac{2/(\cos\theta\sin\theta)}{(-2-2\operatorname{\text{i}}\sqrt{3})\text{e}^{\operatorname{\text{i}}\phi}+\operatorname{\text{i}}(\operatorname{\text{i}}+\sqrt{3})\cot\theta}~{},$
(43) $\displaystyle f_{2}$
$\displaystyle=-\frac{(\operatorname{\text{i}}+\sqrt{3}+2(-\operatorname{\text{i}}+\sqrt{3})\text{e}^{\operatorname{\text{i}}\phi}\cot\theta)\tan\theta}{2(-\operatorname{\text{i}}+\sqrt{3})\text{e}^{3\operatorname{\text{i}}\phi}-(\operatorname{\text{i}}+\sqrt{3})\text{e}^{2\operatorname{\text{i}}\phi}\cot\theta}~{},$
(44) $\displaystyle d$ $\displaystyle=-1\;.$ (45)

  * •

The model studied in Novichkov:2018nkm  is based on a $A_{5}$ symmetry which
leads to the following expressions with $\phi_{g}=(1+\sqrt{5})/2$ the golden
ratio:

$\displaystyle\sin\theta_{13}$
$\displaystyle=\sqrt{\frac{1}{10}(5+\sqrt{5})}\sin\theta\;,$ (46)
$\displaystyle\tan\theta_{12}$
$\displaystyle=\frac{2}{1+\sqrt{5}}\frac{1}{\cos\theta}\;,$ (47)
$\displaystyle\tan\theta_{23}$
$\displaystyle=\left|\frac{\sqrt{\sqrt{5}\phi_{g}}-\text{e}^{-\operatorname{\text{i}}\phi}\tan\theta}{\sqrt{\sqrt{5}\phi_{g}}+\text{e}^{-\operatorname{\text{i}}\phi}\tan\theta}\right|\;,$
(48) $\displaystyle\tan\delta$
$\displaystyle=\frac{4\sqrt{5+\sqrt{5}}\sin(\phi)\left(2\left(\sqrt{5}+2\right)\cos^{2}(\theta)+1+\sqrt{5}\right)}{D_{\delta}}\;,$
(49) $\displaystyle D_{\delta}$
$\displaystyle=4\sqrt{5+\sqrt{5}}\cos(\phi)\cos(2\theta)\left((\sqrt{5}+2)\cos(2\,\theta)+3+2\sqrt{5}\right)$
$\displaystyle+\sqrt{2}\sin(2\,\theta)\left((5\sqrt{5}+11)\cos(2\theta)+19+9\sqrt{5}\right)\cos(2\,\theta_{23})\;.$
(50)

The coefficients of the sum rule are

$\displaystyle c_{1}$
$\displaystyle=\text{e}^{-2\operatorname{\text{i}}\phi}\frac{\left(1-\sqrt{5}\right)\text{e}^{2\operatorname{\text{i}}\phi}\cot\theta+\left(\sqrt{5}+1\right)\tan\theta-8\,\text{e}^{\operatorname{\text{i}}\phi}}{\left(1-\sqrt{5}\right)\text{e}^{2\operatorname{\text{i}}\phi}\tan\theta+\left(\sqrt{5}+1\right)\cot\theta+8\,\text{e}^{\operatorname{\text{i}}\phi}}\;,$
(51) $\displaystyle c_{2}$
$\displaystyle=\frac{10}{\left(\sqrt{5}-5\right)\text{e}^{2\operatorname{\text{i}}\phi}\sin^{2}\theta+4\sqrt{5}\,\text{e}^{\operatorname{\text{i}}\phi}\sin(2\theta)+\left(5+\sqrt{5}\right)\cos^{2}\theta}\;,$
(52) $\displaystyle d$ $\displaystyle=1\;.$ (53)

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