††thanks:  0000-0003-3611-2437††thanks:  0000-0002-4892-2093††thanks:
0000-0002-2157-8982

# Leptonic Unitarity Triangles

Sebastian A. R. Ellis<EMAIL_ADDRESS>SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Kevin J. Kelly
<EMAIL_ADDRESS>Theoretical Physics Department, Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA Shirley Weishi Li
<EMAIL_ADDRESS>SLAC National Accelerator Laboratory, 2575 Sand
Hill Road, Menlo Park, CA 94025, USA

(December 4, 2020)

###### Abstract

We present a comprehensive analysis of leptonic unitarity triangles, using
both current neutrino oscillation data and projections of next-generation
oscillation measurements. Future experiments, sensitive to the degree of CP
violation in the lepton sector, will enable the construction of precise
triangles. We show how unitarity violation could manifest in the triangles and
discuss how they serve as unitarity tests. We also propose the use of Jarlskog
factors as a complementary means of probing unitarity. This analysis
highlights the importance of testing the unitarity of the leptonic mixing
matrix, an understanding of which is crucial for deciphering the nature of the
neutrino sector.

††preprint: FERMILAB-PUB-20-156-T, SLAC-PUB-17521

##  I Introduction

The discovery of neutrino oscillations confirmed that lepton flavor and mass
eigenstates are distinct. Their mixing is canonically parameterized by the
$3\times 3$ unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix Pontecorvo
(1968); Maki  _et al._ (1962), analogous to the Cabibbo-Kobayashi-Maskawa
(CKM) matrix Cabibbo (1963); Kobayashi and Maskawa (1973) for quarks. A
crucial difference between the leptonic and quark sectors, however, is our
knowledge of the origin of such mixing. The appearance of the known active
neutrinos in $SU(2)_{L}$ doublets means the flavor eigenstates are known.
However, the structure of the neutrino mass terms is unknown, since the
Standard Model as formulated does not contain right-handed singlet fermions,
and therefore does not allow for a renormalizable neutrino-Higgs Yukawa
interaction. The misalignment of the flavor and mass eigenstates in the lepton
sector, i.e., the origin of the PMNS matrix, therefore remains an open
question. Predictions of neutrino masses, e.g., those involving right-handed
neutrinos, often lead to a non-unitary $3\times 3$ leptonic mixing matrix
(LMM), which is a sub-matrix of a larger unitary matrix (see e.g. Wyler and
Wolfenstein (1983); Langacker and London (1988); Hewett and Rizzo (1989);
Buchmuller and Greub (1991); Ingelman and Rathsman (1993); Nardi (1993); Chang
_et al._ (1994); Tommasini  _et al._ (1995); Loinaz  _et al._ (2003)).
Searches for deviation from unitarity of the LMM therefore have the potential
to directly probe our fundamental understanding of neutrino masses. Throughout
this work, we refer to a general $3\times 3$ LMM as $U_{\rm LMM}$ and one
assumed to be unitary as $U_{\rm PMNS}$.

$U_{\rm PMNS}$ is parameterized by three angles,
$\theta_{12},~{}\theta_{13},~{}\theta_{23}$, and a phase $\delta_{\rm
CP}$.111If neutrinos are Majorana, two additional phases, which do not affect
oscillations, appear. We disregard them for the remainder of this work. The
degree to which the combination of charge C and parity P symmetry, CP, is
violated is proportional to the Jarlskog invariant Jarlskog (1985),

$J_{\rm PMNS}\equiv c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin\delta_{\rm
CP}\ ,$ (1)

where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$. This quantity is
relevant to understanding the baryon asymmetry of the universe: CP-violation
is one of the requirements for such an asymmetry to exist Sakharov (1967).
Indeed, studies of leptogenesis leading to baryogenesis have shown that the
Dirac phase $\delta_{\rm CP}$ that is measured by the Jarlskog invariant can
be directly involved in the generation of an asymmetry Branco  _et al._ (2001,
2002); Frampton  _et al._ (2002); Endoh  _et al._ (2002); Fujihara  _et al._
(2005); Abada  _et al._ (2006); Harigaya  _et al._ (2012); Barreiros  _et al._
(2020). Furthering our understanding of the PMNS matrix is therefore important
for probing fundamental questions.

In the quark sector, many experimental tests of the unitarity of the mixing
matrix have been performed Amhis  _et al._ (2017), with results often
visualized using unitarity triangles Hocker  _et al._ (2001); Bona  _et al._
(2006); CKM ; UTf ; Tanabashi  _et al._ (2018); Wolfenstein (1983); Buras  _et
al._ (1994); Charles  _et al._ (2005). In these, many measurements meet at a
point in the complex plane if the matrix is unitary. Areas of such triangles
are proportional to the Jarlskog invariant of the CKM matrix.

In the lepton sector, neutrino oscillation experiments can provide direct
tests of LMM unitarity. Assuming unitarity, several elements of $U_{\rm PMNS}$
have been measured to $\mathcal{O}$(10%) precision Gando  _et al._ (2011);
Adey  _et al._ (2018); Aharmim  _et al._ (2013); Vinyoles  _et al._ (2017);
Acero  _et al._ (2019); Agafonova  _et al._ (2018); Abe  _et al._ (2018a,
2020); Gando  _et al._ (2013); Cleveland  _et al._ (1998); Altmann  _et al._
(2005); Abdurashitov  _et al._ (2009); Bellini  _et al._ (2011); Abe  _et al._
(2016); Aartsen  _et al._ (2018, 2019); Jiang  _et al._ (2019). Next-
generation experiments will attain $\mathcal{O}$(1%) precision, and
importantly, will begin to measure the degree of CP-violation Acciarri  _et
al._ (2015); Abi  _et al._ (2020); de Gouvêa  _et al._ (2019); Ishihara
(2020); Capozzi  _et al._ (2019); Abe  _et al._ (2018b); An  _et al._ (2016).
This allows for precision tests of LMM unitarity Qian  _et al._ (2013); Parke
and Ross-Lonergan (2016), as well as the construction of leptonic unitarity
triangles Farzan and Smirnov (2002); He and Xu (2014); Gonzalez-Garcia  _et
al._ (2014); He and Xu (2017); Esteban  _et al._ (2017); nuf .

In this work, we present a comprehensive analysis of leptonic unitarity
triangles using current neutrino oscillation data and projections of future
experiments. Our main results are shown in Fig. 1, where we present six
unitarity triangles, and in Fig. 5, where we show nine Jarlskog factors and
the PMNS Jarlskog invariant. This set of measurements allows for a complete
understanding of the LMM in a compact form. In our companion paper Ellis  _et
al._ (2020), we discuss how well these datasets constrain all unitarity
conditions of $U_{\rm LMM}$. Our results show the importance of separately
analyzing appearance and disappearance data, and demonstrate the power of
future oscillation measurements to constrain fundamental physics related to
the neutrino sector.

Figure 1: Current and future 95% (dark contours) and 99% (light) credible
regions of six leptonic unitarity triangles. Green contours consist of all
current data Gando  _et al._ (2011); Adey  _et al._ (2018); Aharmim  _et al._
(2013); Vinyoles  _et al._ (2017); Abe  _et al._ (2018a); Acero  _et al._
(2019); Agafonova  _et al._ (2018). In red are the analogous contours
including also projections of future data from IceCube Aartsen  _et al._
(2019); Blot (2020), JUNO An  _et al._ (2016), DUNE Acciarri  _et al._ (2015);
Abi  _et al._ (2020); de Gouvêa  _et al._ (2019) and T2HK Abe  _et al._
(2018b). Blue contours include subsets of data that measure disappearance
probabilities, including reactor Gando  _et al._ (2011); Adey  _et al._
(2018); An  _et al._ (2016), solar Aharmim  _et al._ (2013) experiments, and
DUNE/T2HK $\nu_{\mu}\rightarrow\nu_{\mu}$. Orange contours include appearance
measurements, e.g.., DUNE & T2HK $\nu_{\mu}\rightarrow\nu_{e}$ and IceCube
$\nu_{\tau}$ appearance. This demonstrates the complementarity of these
measurements in constructing unitarity triangles. The black triangles point to
the best-fit point on the plane. Unitarity of the LMM is assumed. A subset of
triangles without unitarity assumed is shown in Fig. A1.

##  II The Leptonic Mixing Matrix and Unitarity Triangles

The LMM describing the mixing of leptonic flavor and mass eigenstates,

$U_{\rm LMM}=\begin{pmatrix}U_{e1}&U_{e2}&U_{e3}\\\ U_{\mu 1}&U_{\mu 2}&U_{\mu
3}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{pmatrix}\ ,$ (2)

is defined by its appearance in the charged current interaction. In full
generality, $U_{\rm LMM}$ may be a non-unitary $3\times 3$ sub-matrix of a
larger $M\times N$ complex matrix. If only three generations of leptons
exists, $U_{\rm LMM}\equiv U_{\rm PMNS}$. This mixing leads to neutrino
oscillation, the precise measurement of which is the focus of current and
future detection efforts. The primary measurements in oscillation experiments
are the appearance and disappearance probabilities, $P_{\alpha\beta}$ and
$P_{\alpha\alpha}$ ($\alpha\neq\beta$; $\alpha,\beta\in e,\mu,\tau$).
Distinguishing between these is important for testing unitarity.

For a $3\times 3$ unitary matrix, a set of six triangles may be defined from
the conditions $U^{\dagger}U=UU^{\dagger}=\mathbb{I}$, corresponding to the
closure of the products of cross-terms of the matrix:

$\begin{array}[]{c c}\displaystyle\sum_{i}U_{\alpha i}U_{\beta i}^{*}=0\ ,\\\
(\alpha\neq\beta,\text{rows})\end{array}\quad\begin{array}[]{cc}\displaystyle\sum_{\alpha}U_{\alpha
i}U_{\alpha j}^{*}=0\ .\\\ (i\neq j,\text{columns})\end{array}$ (3)

The elements $U_{\alpha i}$ are complex, so the above conditions can be shown
as closed triangles in complex planes. Unitarity triangles are constructed by
normalizing to one of the three terms in the sums of Eq. (3), and defining a
vertex of the triangle as $\rho_{xy}+i\eta_{xy}$, $(x\neq y)$,

$\rho_{xy}+i\eta_{xy}=\begin{cases}&-\displaystyle\frac{U_{xi}U_{yi}^{*}}{U_{xj}U_{yj}^{*}}\
,\quad\text{rows}\\\ \\\ &-\displaystyle\frac{U_{\alpha x}U_{\alpha
y}^{*}}{U_{\beta x}U_{\beta y}^{*}}\ ,\quad\text{columns}\end{cases}\ ,$ (4)

such that a closed triangle has vertices at the origin, $(1,0)$, and
$(\rho_{xy},\eta_{xy})$. There is ambiguity in the choice of the denominator,
and hence $(\rho_{xy},\eta_{xy})$ for a given row/column. We explain our
choices below, made in an attempt to cover measurements of all parameters of
$U_{\rm PMNS}$, and give the full definitions of $(\rho_{xy},\eta_{xy})$ in
Appendix A.

We define possible choices of row unitarity triangles as:

$\displaystyle(\rho_{\alpha\beta}+i\eta_{\alpha\beta})^{(1)}=T_{\alpha\beta}^{(1)}=-\frac{U_{\alpha
1}U_{\beta 1}^{*}}{U_{\alpha 3}U_{\beta 3}^{*}}\ ,$
$\displaystyle(\rho_{\alpha\beta}+i\eta_{\alpha\beta})^{(2)}=T_{\alpha\beta}^{(2)}=-\frac{U_{\alpha
2}U_{\beta 2}^{*}}{U_{\alpha 1}U_{\beta 1}^{*}},$
$\displaystyle(\rho_{\alpha\beta}+i\eta_{\alpha\beta})^{(3)}=T_{\alpha\beta}^{(3)}=-\frac{U_{\alpha
3}U_{\beta 3}^{*}}{U_{\alpha 2}U_{\beta 2}^{*}}\ ,$ (5) $\displaystyle
T_{\alpha\beta}^{(-m)}=\left(T_{\alpha\beta}^{(m)}\right)^{-1}\ ,$ (6)

where $\alpha,\beta\in[e,\mu,\tau]$. The column unitarity triangles are
defined as

$\displaystyle(\rho_{ij}+i\eta_{ij})^{(1)}=T_{ij}^{(1)}=-\frac{U_{\mu i}U_{\mu
j}^{*}}{U_{ei}U_{ej}^{*}}\ ,$
$\displaystyle(\rho_{ij}+i\eta_{ij})^{(2)}=T_{ij}^{(2)}=-\frac{U_{\tau
i}U_{\tau j}^{*}}{U_{\mu i}U_{\mu j}^{*}},$
$\displaystyle(\rho_{ij}+i\eta_{ij})^{(3)}=T_{ij}^{(3)}=-\frac{U_{ei}U_{ej}^{*}}{U_{\tau
i}U_{\tau j}^{*}}\ ,$ (7) $\displaystyle
T_{ij}^{(-m)}=\left(T_{ij}^{(m)}\right)^{-1}\ ,$ (8)

where $i,j\in[1,2,3]$.

We also define general Jarlskog factors $J_{\alpha i}$ as

$\varepsilon_{\alpha\beta\gamma}\varepsilon_{ijk}J_{\alpha
i}=\text{Im}\left(U_{\beta j}U_{\gamma k}U_{\beta k}^{*}U_{\gamma
j}^{*}\right)\ ,$ (9)

which are related to the areas $A_{T}$ of the possible triangles. For the row
triangles as defined above, we obtain the following relations between
$\text{Im}(T_{\alpha\beta}^{(m)})=\pm 2A_{T_{\alpha\beta}^{(m)}}$ and the
Jarlskog factors:

$\displaystyle\text{Im}(T_{\alpha\beta}^{(1)})=\frac{J_{\gamma 2}}{|U_{\alpha
3}|^{2}|U_{\beta 3}|^{2}}\ ,$
$\displaystyle\text{Im}(T_{\alpha\beta}^{(2)})=\frac{J_{\gamma 3}}{|U_{\alpha
1}|^{2}|U_{\beta 1}|^{2}}\ ,$
$\displaystyle\text{Im}(T_{\alpha\beta}^{(3)})=\frac{J_{\gamma 1}}{|U_{\alpha
2}|^{2}|U_{\beta 2}|^{2}}\ ,$ (10)

where $\alpha\neq\beta\neq\gamma$.

Repeating the same analysis for the column triangles as for the rows, we can
derive the following relations between the triangles and Jarlskog factors:

$\displaystyle\text{Im}(T_{ij}^{(1)})=\frac{J_{\tau
k}}{|U_{ei}|^{2}|U_{ej}|^{2}}\ ,$
$\displaystyle\text{Im}(T_{ij}^{(2)})=\frac{J_{ek}}{|U_{\mu i}|^{2}|U_{\mu
j}|^{2}}\ ,$ $\displaystyle\text{Im}(T_{ij}^{(3)})=\frac{J_{\mu k}}{|U_{\tau
i}|^{2}|U_{\tau j}|^{2}}\ ,$ (11)

where $i\neq j\neq k$.

The information contained in the triangles defined in Eqs. (II, II), is
duplicated by taking the reciprocal triangles of Eqs. (6, 8), so one need only
consider one set. If we do not assume unitarity when performing the triangle
analysis, we can see that 9 unitarity triangles must be constructed in order
to fully measure all 9 Jarlskog factors once in combination with all matrix
element norms twice.

If we assume unitarity when constructing triangles, we would find that all
$J$-factors are identical by definition, and equal to the Jarlskog invariant
in the PMNS parameterization. Thus constructing triangles allows us to measure
$J_{\rm PMNS}$ and products of norms. In this case, to display all the
information contained in the 18 possible triangles, we must pick 6 triangles
to cover all 9 norms with minimal redundancy. Any set of 6 triangles which
includes 9 separate norms actually contains 12 norms, such that there are
always three norms which are measured twice. An example of a set of triangles
which would encapsulate all possible information would be $T\supset
T_{e\mu}^{(1)},~{}T_{e\mu}^{(2)},~{}T_{e\mu}^{(3)},~{}T_{e\tau}^{(1)},~{}T_{e\tau}^{(2)},~{}T_{e\tau}^{(3)}$.
With this set, $|U_{ei}|$, $i=1,2,3$ would be repeated twice. A more “flavor-
democratic” approach to choosing a set of six triangles, and the one we use in
our analysis, is to choose one triangle from each row and column:

$T\supset
T_{e\mu}^{(1)},~{}T_{e\tau}^{(2)},~{}T_{\mu\tau}^{(3)},~{}T_{12}^{(-1)},~{}T_{13}^{(-2)},~{}T_{23}^{(-3)}\
.$ (12)

The full expressions for these triangles are given in Appendix A in terms of
the PMNS parameterization, and can be obtained in the LMM parameterization
from Eqs. (II) and (8).

In this way, we measure all information in the LMM matrix under the assumption
of unitarity, while only repeating measurements of $|U_{e3}|,~{}|U_{\mu 2}|$
and $|U_{\tau 1}|$. This specific set of choices is further motivated by the
discussion in Section IV around Fig. 3 of how to use unitarity triangles to
observe unitarity violation. Given the degree of non-unitarity allowed by
current measurements, which is then used to construct Fig. 3, it was
determined that the above set of 6 triangles was most instructive for
observing tension between the appearance and disappearance data in the various
$(\rho,\eta)$ planes.

Other choices of triangles have been made previously in Refs. Gonzalez-Garcia
_et al._ (2014); Esteban  _et al._ (2017); nuf . For the 1-3 triangle, their
choice of $(\rho,\eta)$ is

$\rho_{13}+i\eta_{13}=-\frac{U_{e1}U_{e3}^{*}}{U_{\mu 1}U_{\mu 3}^{*}}.$ (13)

This choice corresponds to $T_{13}^{(-1)}$ as defined above, and is therefore
measuring $J_{\tau 2}$. Note that this definition is the leptonic equivalent
of the $d-b$ CKM triangle that is commonly shown. When we adopt this
definition, our joint-fit region from current experiments is consistent with
the result of Refs. Esteban  _et al._ (2017); nuf , which can be broken down
to a disappearance circle that is centered at (0, 0) and an appearance region
that is more visibly radially oriented.

In order to fully characterize a potentially non-unitary LMM in a relatively
economical yet intuitive fashion, it is clear that we must account for the
fact that assuming unitarity sets all $J$-factors equal in the triangles. Thus
we must separately measure all nine possible $J$-factors. By showing an
appropriate set of 6 unitarity triangles when assuming unitarity (Fig. 1), 9
$J$-factors computed without assuming unitarity, and $J_{\rm PMNS}$ (Fig. 5),
we graphically represent all possible information in the LMM. There is an
added benefit to computing $J$-factors separately, as they include information
obtained from sterile searches that is not otherwise visible in the triangle
planes.

##  III Data Analysis and Methodology

We take global neutrino data and recast their joint measurements onto leptonic
triangles. Our goal is not to do the most precise, comprehensive global fit on
mixing parameters, so we interpret the majority of experimental results as
rate-only measurements. Concretely, we assume a given experiment measures an
oscillation probability $P_{\alpha\beta}$ at a fixed ($L$, $E_{\nu}$). This
reasonably reproduces the reported experimental results, so we apply it to T2K
Abe  _et al._ (2018a); Dunne (2020) and NOvA Acero  _et al._ (2019); Himmel
(2020) (which are sensitive to $\delta_{\rm{CP}}$), as well as Daya Bay Adey
_et al._ (2018), solar neutrino measurements Aharmim  _et al._ (2013, 2013);
Vinyoles  _et al._ (2017); Nakajima (2020), and OPERA Agafonova  _et al._
(2018). For KamLAND, we include a more detailed analysis that utilizes the
measured neutrino spectrum Gando  _et al._ (2011).

We also project the inclusion of future data in our simulations, namely, The
Deep Underground Neutrino Experiment (DUNE) $\nu_{\mu}$-disappearance and
$\nu_{e}$-appearance channels Acciarri  _et al._ (2015); Abi  _et al._ (2020);
Fields ; Formaggio and Zeller (2012); Berryman  _et al._ (2015); de Gouvêa and
Kelly (2016a, b), $\nu_{\tau}$-appearance channel de Gouvêa  _et al._ (2019),
The Jiangmen Underground Neutrino Observatory (JUNO) An  _et al._ (2016);
Capozzi  _et al._ (2014); An  _et al._ (2013); Huber (2011); Mueller  _et al._
(2011); Strumia and Vissani (2003), Tokai-to-Hyper-Kamiokande (T2HK) Abe  _et
al._ (2018b), and the IceCube Upgrade’s capabilities for measuring
$\nu_{\tau}$ appearance Aartsen  _et al._ (2019); Blot (2020). The following
subsections detail the current and future data included. We direct the reader
to Ref. Ellis  _et al._ (2020), where the current and future data included are
identical to those here, and more details are provided.

Using a given combination of data sets, we construct a likelihood function
depending on a set of oscillation parameters. Our fits in Figs. 1-3 depend on
six parameters: $\sin^{2}\theta_{12}$, $\sin^{2}\theta_{13},$
$\sin^{2}\theta_{23},$ $\delta_{\rm{CP}},$ $\Delta m_{21}^{2},$ $\Delta
m_{31}^{2}$. We include Gaussian priors on the mass-squared splittings from
respective experimental results when analyzing current data, and use the
Bayesian inference tool pyMultinest Feroz  _et al._ (2009); Buchner  _et al._
(2014) to construct credible regions in this parameter space. The posterior
distributions are then projected onto ($\rho_{xy}$, $\eta_{xy}$).

Analyzing current data, the maximum-likelihood parameters are
$\sin^{2}\theta_{12}=0.308$, $\sin^{2}\theta_{13}=0.0219$,
$\sin^{2}\theta_{23}=0.551$, $\delta_{\rm CP}=200^{\circ}$,
consistent222Previous versions of our analysis preferred a closer-to-maximal
value of $\delta_{\rm CP}$, in line with published global fits prior to the
latest release of data from T2K and NOvA Esteban  _et al._ (2019); de Salas
_et al._ (2018); Capozzi  _et al._ (2020). with global fits Esteban  _et al._
(2020). We assume these to be the true parameters when simulating future
experiments.

###  III.1 Current Experimental Results Included

Here we list the current experimental results that we include in our data
analysis, and specify to which parameters each experiment is most sensitive.
We also show the results of our data analysis (when unitarity is assumed) of
all of the current data included in measuring the combination of the
parameters $\delta_{\rm CP}$ and $\sin^{2}\theta_{23}$, validating our
approach.

Mass-Squared Splittings When mentioned in the following list, we allow the
mass-squared splittings $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$ to vary
independently, allowing the possibility of both mass orderings. Based on our
methods of incorporating existing measurements, KamLAND, Daya Bay, T2K, NOvA,
and OPERA are sensitive to mass-squared splittings. For each experiment, we
include a Gaussian prior on the relevant mass-squared splitting from the
experimentally reported 1$\sigma$ range. The resulting fit region of the two
mass-squared-splittings that we obtain is consistent with those from more
sophisticated global fits de Salas  _et al._ (2018); Esteban  _et al._ (2019);
Capozzi  _et al._ (2020); Esteban  _et al._ (2020). While tensions
exist333However, these tensions are smaller with the latest analysis from
Super-Kamiokande and SNO Nakajima (2020), which measure $\Delta
m_{21}^{2}=\left(6.11\pm 1.21\right)\times 10^{-5}$ eV2. We include this as a
prior in our analysis, even though the KamLAND measurement is significantly
more powerful. between different measurements of $\Delta m_{21}^{2}$, we find
that the analyses we perform do not change whether we also include a
measurement of $\Delta m_{21}^{2}$ from solar neutrino measurements or not Abe
_et al._ (2016); Nakajima (2020).

Normalization Many of the current (and future) experiments we consider infer a
neutrino oscillation probability by measuring a far-detector-to-near-detector
ratio, i.e., they measure the neutrino flux times cross sections of one flavor
$\alpha$ at the near detector and one flavor $\beta$ at the far detector. The
ratio of these two, up to cross section and flux effects, gives the
oscillation probability $P_{\alpha\beta}$. If the LMM is not unitary, however,
this is not completely true – zero distance effects lead to $P_{\alpha\alpha}$
not being $1$ at the near detector, but

$P(\nu_{\alpha}\to\nu_{\alpha};L=0)=\left(\sum_{i=1}^{3}\left\lvert U_{\alpha
i}\right\rvert^{2}\right)^{2}.$ (14)

When performing an analysis that does not assume unitarity, like those
surrounding Figs. 5 and A1, the inferred oscillation probability of an
experiment with a near detector must be normalized by the factor in Eq. (14).
This normalization factor has the same impact on an analysis as including an
overall systematic normalization uncertainty in an analysis, so as long as the
normalization uncertainty of a given experiment is larger than the uncertainty
on the quantity in Eq. (14), these effects are unimportant. Normalization
effects are described in much more detail in Ref. Ellis  _et al._ (2020).

KamLAND The reactor antineutrino experiment KamLAND measures the oscillation
probability $P(\overline{\nu}_{e}\to\overline{\nu}_{e})$ over a wide range of
baseline length $L$ divided by neutrino energy $E_{\nu}$. Appendix B of Ref.
Gando  _et al._ (2011) provides measurements of this oscillation probability
for different values of $L/E_{\nu}$. We use these measurements, which take
into account matter effects, as well as either the standard, three-neutrino
oscillation probability or a modified one to account for non-unitary mixing,
to place constraints on mostly $\sin^{2}\theta_{12}$. If the matrix is not
unitary, KamLAND is mostly sensitive to the product
$|U_{e1}|^{2}|U_{e2}|^{2}$. A more recent analysis is Ref. Gando  _et al._
(2013), but it does not contain enough information for us to reasonably
reproduce its results using this approach.

Daya Bay For the Daya Bay experiment, we include the most recent measurement
of $\sin^{2}\left(2\theta_{13}\right)=0.0856\pm 0.0029$ as a Gaussian prior in
our analysis Adey  _et al._ (2018). If the LMM is not unitary, the oscillation
probability at Daya Bay’s far detector is sensitive to the combination
$4|U_{e3}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2})$. Since Daya Bay uses a near and far
detector, its measurement of $\sin^{2}\left(2\theta_{13}\right)$ is actually a
measurement of
$4|U_{e3}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2})/(|U_{e1}|^{2}+|U_{e2}|^{2}+|U_{e3}|^{2})^{2}$,
as discussed above.

Solar Neutrinos The only solar neutrino experiments we include results from
are the Sudbury Neutrino Experiment (SNO) and Super-Kamiokande (Super-K). The
SNO two-flavor analysis ($\sin^{2}\theta_{13}\to 0$) yields
$\tan^{2}\theta_{12}=0.427^{+0.033}_{-0.029}$ Aharmim  _et al._ (2013). More
generically, we include the most up-to-date measurement of the solar charged-
current channel from a combined SNO and Super-K analysis, which reports
$|U_{e2}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2})+|U_{e3}|^{4}=0.2932\pm 0.0134$
Nakajima (2020).

SNO is also sensitive to neutral current scattering, which observes the
effective oscillation probability
$P_{\rm{NC}}=\sum_{i}|U_{ei}|^{2}_{\rm{prod.}}|U_{ei}|^{2}_{\rm{det.}}$. To
leading order (see Ref. Ellis  _et al._ (2020) for more detail), this becomes

$\displaystyle P_{\mathrm{NC}}=$
$\displaystyle\left(|U_{e1}|^{2}+|U_{e2}|^{2}\right)\left(|U_{e2}|^{2}+|U_{\mu
2}|^{2}+|U_{\tau 2}|^{2}\right)^{2}$
$\displaystyle+|U_{e3}|^{2}\left(|U_{e3}|^{2}+|U_{\mu 3}|^{2}+|U_{\tau
3}|^{2}\right)^{2}.$ (15)

This measurement is limited by theoretical uncertainties associated with the
Standard Solar Model Vinyoles  _et al._ (2017), so we conservatively assume
that SNO measures it at the 25% level.

T2K For the electron-neutrino appearance channels $P(\nu_{\mu}\to\nu_{e})$ and
$P(\overline{\nu}_{\mu}\to\overline{\nu}_{e})$ measured at T2K, we assume that
the experiment measures this probability for a fixed energy
$E_{\mathrm{T2K}}=600$ MeV (the mean energy of the J-PARC beam) at a distance
of $L=295$ km. We also assume a constant matter density of $\rho=2.6$ g/cm3
along the path of propagation. While this approach is an oversimplification
and does not include systematic uncertainties from T2K, we find that it
reproduces the results of Refs. Abe  _et al._ (2018a, 2020); Dunne (2020)
well. We use the predicted signal and background rates for the $\nu_{e}$ and
$\overline{\nu}_{e}$ appearance presented in Ref. Dunne (2020), the most up-
to-date results available. T2K also measures $\nu_{\mu}$ and
$\overline{\nu}_{\mu}$ disappearance. We interpret this measurement as
information on the quantity $4|U_{\mu 3}|^{2}(|U_{\mu 1}|^{2}+|U_{\mu
2}|^{2})$ to agree with the results of Ref. Abe  _et al._ (2018a). Matter
effects are much smaller in this channel, so we ignore them. If $U_{\rm LMM}$
is unitary, this translates effectively into a measurement of
$\sin^{2}\left(2\theta_{23}\right)$. We assume T2K measures $4|U_{\mu
3}|^{2}(|U_{\mu 1}|^{2}+|U_{\mu 2}|^{2})=1.00\pm 0.03$. We find that including
disappearance information in this way reproduces the measurement capability of
the experiment from Refs. Abe  _et al._ (2018a, 2020); Dunne (2020) better
than assuming a fixed-length, fixed-energy measurement in this channel. See
Ref. Ellis  _et al._ (2020) for more details of this analysis, as well as
validation of this approach for T2K.

NOvA Our methodology for NOvA is very similar to that of T2K: we assume a
fixed energy for the electron-neutrino appearance measurements of
$E_{\mathrm{NOvA}}=1.9$ GeV and $L=810$ km (as well as a constant matter
density of $2.84$ g/cm3). Expected signal event rates from Ref. Himmel (2020)
allow us to construct a log-likelihood as we vary oscillation parameters. Like
with T2K, we allow $\Delta m_{31}^{2}$ to vary within its prior for the NOvA
measurements. Since NOvA prefers a value of $\sin^{2}\theta_{23}$ slightly
away from maximal, we include its disappearance channel measurements in our
fit by assuming it measures $4|U_{\mu 3}|^{2}(|U_{\mu 1}|^{2}+|U_{\mu
2}|^{2})=0.99\pm 0.02$. We find good agreement between our simplified analysis
and the results of Ref. Acero  _et al._ (2019); Himmel (2020) for all
different oscillation parameters of interest. Again, Ref. Ellis  _et al._
(2020) provides details and validation of our NOvA analysis.

OPERA We include the OPERA collaboration’s measurement of tau neutrino
appearance via $P(\nu_{\mu}\to\nu_{\tau})$, where 10 $\nu_{\tau}$ signal
events were observed with an expected background of $2.0\pm 0.4$ events.
Assuming $\sin^{2}\left(2\theta_{23}\right)=1$ and $\Delta
m_{32}^{2}=2.5\times 10^{-3}$ eV2, OPERA expected $6.8\pm 1.4$ signal events.
We include this information, assuming a mono-energetic measurement at
$E_{\mathrm{OPERA}}=17$ GeV and $L=730$ km, giving results consistent with
those from OPERA Agafonova  _et al._ (2018). Matter effects are included, even
though they are small for $\nu_{\tau}$ appearance oscillation probabilities.

Sterile Neutrino Searches When unitarity is not assumed, sterile neutrino
searches provide additional constraints on the unitarity of the LMM.
Specifically, we include results from sterile neutrino searches at experiments
in regions where such new oscillations would have “averaged out” in the
experiment’s detector. This corresponds to the high $\Delta m_{41}^{2}$ mass-
squared splitting region of experimental sensitivities/exclusions and utilizes
the zero-distance effects. Searches for anomalous appearance of a different
neutrino flavor constrain triangle closure information, where searches for
anomalous disappearance constrain row normalizations. References Parke and
Ross-Lonergan (2016); Ellis  _et al._ (2020) provide further explanation of
these effects. We do not include sterile search constraints where mass-
squared-splitting information is utilized because they do not apply to a
generic unitarity violation scenario where we are agnostic about the mass
scale of the violation.

Neutrino Appearance Searches: Experiments that search for anomalous appearance
(such as NOMAD searching for anomalous $\nu_{\mu}\to\nu_{e}$ or
$\nu_{\mu}\to\nu_{\tau}$) are sensitive to these zero-distance effects, which,
if the LMM is not unitarity, correspond to the non-closure of a unitarity
triangle – $\sum_{i}U_{\alpha i}U_{\beta i}^{*}\neq 0$. We include results
from KARMEN Armbruster  _et al._ (2002), NOMAD Astier  _et al._ (2003, 2001),
and CHORUS Eskut  _et al._ (2008) in these analyses. The LSND Athanassopoulos
_et al._ (1996, 1998) and MiniBooNE Aguilar-Arevalo  _et al._ (2018)
experiments have famously observed an excess of electron-like events in a
$\nu_{\mu}$ beam, which can be interpreted as observing a non-closure between
the electron and muon rows of $\left\lvert\sum_{i}U_{ei}U_{\mu
i}^{*}\right\rvert^{2}\approx 2.6\times 10^{-3}$ Aguilar-Arevalo  _et al._
(2018). We do not include information from MiniBooNE and LSND in our analyses
due to the tension between these appearance searches and disappearance
searches – Ref. Ellis  _et al._ (2020) explores this in much more detail.

Neutrino Disappearance Searches: For muon neutrino disappearance, we include
results from the MINOS/MINOS+ experiments Adamson  _et al._ (2019) that
constrain the muon row normalization. In addition, hints for the existence of
a sterile neutrino with a new mass-squared splitting around $\Delta
m_{41}^{2}\approx 1$ eV2 have been observed in a variety of reactor
antineutrino experiments (see Refs. Dentler  _et al._ (2018); Diaz  _et al._
(2020); Böser  _et al._ (2020); Berryman and Huber (2020) for reviews of
these), which could point to the electron row being not properly normalized,
i.e., $\sum_{i}\left\lvert U_{ei}\right\rvert^{2}\neq 1$. However, in order to
interpret these results in terms of a constraint on unitarity, we must go to
the averaged-out regime of these experimental sensitivities, which is limited
by the predicted reactor antineutrino fluxes Mueller  _et al._ (2011); Huber
(2011). We therefore do not include these results due to the uncertainty
regarding reactor antineutrino flux predictions.

Fit Results: To demonstrate the validity of our methods, Fig. 2 (green)
displays the result of our fit (when unitarity is assumed) to all of the
current data discussed above. We show the fit as a measurement of the
parameters $\delta_{\rm CP}$ and $\sin^{2}\theta_{23}$, where the other,
unseen parameters have been marginalized. We find that our results are
consistent with more sophisticated global fits Esteban  _et al._ (2019); de
Salas  _et al._ (2018); Capozzi  _et al._ (2020). Specifically, we have
compared our result against the most recent fit from Ref. Esteban  _et al._
(2020) and find that our $1\sigma$ and $90\%$ regions are slightly more
conservative than theirs (likely due to the fact that their fits include all
current data where ours contain a subset) and our $99\%$ regions match very
well. Our best-fit point in this parameter space, $(0.55,3.49)$ is also very
close to that of Ref. Esteban  _et al._ (2020), $(0.57,3.40)$.

Figure 2: Results of our fit to all current data (green) discussed in Section
III.1 and including future projections (red) discussed in Section III.2 as
measurements of ($\sin^{2}\theta_{23}$, $\delta_{\rm CP}$) at 1$\sigma$, 90%,
and 99%. The star indicates the best-fit point of our analysis, as well as the
assumed-true combination of parameters for our future projections.

###  III.2 Future Experiment Simulations

In our analyses, we include simulations of the DUNE, T2HK, IceCube, and JUNO
experiments. In this subsection, we briefly describe how these simulations are
included and some of their details. Figure 2 includes our future projections
in measuring the combination of parameters $\sin^{2}\theta_{23}$ and
$\delta_{\rm CP}$ in red.

DUNE DUNE will utilize a wide-energy $\nu_{\mu}$ beam, with energies between
roughly $0.5$ and $10$ GeV, with a baseline distance of $1300$ km. This will
allow DUNE to study both $\nu_{\mu}$ disappearance and $\nu_{e}$ appearance,
and allow for a powerful measurement of $\delta_{\rm CP}$. Reference de Gouvêa
_et al._ (2019) demonstrated DUNE’s ability to use its beam to study
$\nu_{\tau}$ appearance as well. We include all three of these channels in our
simulations, assuming seven years of data collection (equal operation in
neutrino and antineutrino modes) with a 1.2 MW beam and 40 kt of far detector
fiducial mass.

Our simulation of both $\nu_{\mu}$ disappearance and $\nu_{e}$ appearance
channels follows those developed for Refs. Berryman  _et al._ (2015); de
Gouvêa and Kelly (2016a, b), and the analyses are designed to match the
official collaboration sensitivities and expected signal and background event
yields Acciarri  _et al._ (2015); Abi  _et al._ (2020). We take the neutrino
fluxes from Ref. Fields and neutrino cross sections from Ref. Formaggio and
Zeller (2012). We include energy uncertainty by using migration matrices,
assuming that the energy resolution is $\sigma_{E}=7\%\times
E_{\nu}{(\rm{GeV})}+3.5\%\sqrt{E_{\nu}(\rm{GeV})}$, consistent with the
analyses performed in Refs. Acciarri  _et al._ (2015); Abi  _et al._ (2020).
We include all of the different background channels discussed in Refs.
Acciarri  _et al._ (2015); Abi  _et al._ (2020) for the $\nu_{\mu}$
disappearance and $\nu_{e}$ appearance channels – the largest of which are due
to neutrino neutral-current scattering and beam contamination of opposite sign
or different flavor neutrinos. Efficiencies for both signal identification and
background rejection are both taken to be constant as a function of neutrino
energy, where we normalize our expected signal and background event rates to
those from Ref. Acciarri  _et al._ (2015).

Our simulation for $\nu_{\tau}$ appearance channel follows Ref. de Gouvêa  _et
al._ (2019). For a given true neutrino energy $E_{\nu}^{\mathrm{true}}$, we
assume that the reconstructed energy follows a Gaussian distribution with a
mean energy $bE_{\nu}^{\mathrm{true}}$ and an uncertainty
$\sigma_{E}=rE_{\nu}^{\mathrm{true}}$, where $b=45\%$ and $r=25\%$ de Gouvêa
_et al._ (2019). We include a 25% systematic normalization uncertainty to
account for uncertainties associated with the $\nu_{\tau}$ charged-current
cross section. We also assume a 30% signal identification efficiency for all
hadronically-decaying $\tau^{\pm}$ events, and that $0.5\%$ of neutral current
events will contribute to backgrounds in this search.

For all channels, we include a correlated systematic normalization uncertainty
on the muon neutrino flux for each beam mode (separate nuisance parameters for
neutrino and antineutrino modes) of 5%. As discussed in Section III.1, if we
do not assume the unitarity of the LMM and an experiment has a near detector,
the inferred measurement of an oscillation probability must be normalized by
the appropriate channel, i.e., the normalization of the muon row of the LMM in
this case. DUNE is one such experiment, however, since we include a 5%
normalization uncertainty on the muon neutrino flux, this effect is negligible
in our analyses. This is because the MINOS/MINOS+ sterile neutrino search
constrains this normalization effect to the $2.5\%$ level, so the systematic
uncertainty of 5% covers any impact of this type.

T2HK T2HK is the upcoming successor to T2K, which will include an upgraded
beam and a larger water Čerenkov detector Abe  _et al._ (2018b). Like with T2K
and NOvA, T2HK and DUNE will operate in similar ranges of $L/E_{\nu}$ but very
different ranges of $L$ and $E_{\nu}$. This results in matter effects being
much more relevant for DUNE than T2HK, although they are not negligible and we
therefore include them in our calculations.

We include simulations of both $\nu_{e}$ appearance and $\nu_{\mu}$
disappearance for both neutrino and antineutrino modes (we assume operation in
a $1:3$ ratio between $\nu:\overline{\nu}$ modes, as expected by the T2HK
collaboration). We assume seven years of data collection to be more consistent
with our projections for DUNE. References Kelly (2017); Ellis  _et al._ (2020)
provide further details on this simulation.

IceCube Upgrade The IceCube experiment has performed a number of measurements
of oscillation parameters using its atmospheric neutrino sample Aartsen  _et
al._ (2018, 2019), including a measurement of $\nu_{\tau}$ appearance that is
comparable in strength to the leading measurement from Super-K, and even
stronger (in terms of measuring the appearance normalization) as OPERA. Soon,
IceCube expects to be able to measure this appearance at the $10\%$ level, and
with the IceCube Upgrade, such a $10\%$ measurement should be quickly
attainable. We include this measurement in our future projections, where
IceCube is sensitive to the combination $4\lvert U_{\mu 3}\rvert^{2}\lvert
U_{\tau 3}\rvert^{2}/\left(\lvert U_{\mu 1}\rvert^{2}+\lvert U_{\mu
2}\rvert^{2}+\lvert U_{\mu 3}\rvert^{2}\right)^{2}$.

JUNO JUNO will measure the oscillation of reactor $\overline{\nu}_{e}$ of 2–8
MeV at a propagation distance of 53 km. Matter effects can cause
$\mathcal{O}(1\%)$ level modifications to this oscillation probability for
these energies and propagation distance. However, they do not impact
measurement sensitivity of the parameters of interest, so we do not include
them. This enables a measurement of the neutrino mass ordering, the primary
science goal for JUNO, as well as of $\theta_{12}$ and $\Delta m_{21}^{2}$.

Our analyses are designed to match the official collaboration sensitivities on
$\sin^{2}\theta_{12}$ An  _et al._ (2016). For reactor flux calculation, we
follow the strategy in Ref. Capozzi  _et al._ (2014), taking the fission
isotope fractions from Ref. An  _et al._ (2013), 235U, 239Pu, and 241Pu
spectra from Ref. Huber (2011), and 238U spectrum from Ref. Mueller  _et al._
(2011), and this leads to the following total reactor neutrino flux:

$\displaystyle\Phi(E_{\overline{\nu}_{e}})=0.60\exp($ $\displaystyle
4.367-4.577E_{\nu}+2.1E_{\nu}^{2}-0.5294E_{\nu}^{3}$
$\displaystyle+0.06186E_{\nu}^{4}-0.002777E_{\nu}^{5})$
$\displaystyle+0.27\exp($ $\displaystyle
4.757-5.392E_{\nu}+2.563E_{\nu}^{2}-0.6596E_{\nu}^{3}$
$\displaystyle+0.0782E_{\nu}^{4}-0.003536E_{\nu}^{5})$
$\displaystyle+0.07\exp($ $\displaystyle
2.611-2.284E_{\nu}+0.9692E_{\nu}^{2}-0.23679E_{\nu}^{3}$
$\displaystyle+0.025E_{\nu}^{4}-0.001E_{\nu}^{5})$ $\displaystyle+0.06\exp($
$\displaystyle 2.99-2.882E_{\nu}+1.278E_{\nu}^{2}-0.3343E_{\nu}^{3}$
$\displaystyle+0.03905E_{\nu}^{4}-0.001754E_{\nu}^{5})\ .$ (16)

We adopt the inverse beta decay cross sections from Ref. Strumia and Vissani
(2003):

$\displaystyle\sigma=10^{-43}\mathrm{cm}^{2}p_{e}E_{e}E_{\nu}^{-0.07056+0.02018\log
E_{\nu}-0.001953\log^{3}E_{\nu}},$ $\displaystyle
E_{e}=E_{\nu}-1.293~{}\mathrm{MeV}\ .$ (17)

For each event, the detected energy, which is smeared with an energy
resolution of 3%$\sqrt{E(\rm{MeV})}$ An  _et al._ (2016), is the total energy
of the positron plus its rest mass. We do not consider detector efficiencies
and backgrounds, and only match the total sample size to the CDR nominal
choice, 120k events (six years). For systematics, we include a correlated flux
uncertainty of 2%, an uncorrelated flux uncertainty of 0.8%, the spectrum
shape uncertainty of 1%, and the energy scale uncertainty of 1% An  _et al._
(2016).

Similarly to the case for DUNE, the oscillation probability that JUNO measures
depends on what one assumes of a near detector. Because electron row is well
constrained, this assumption has a negligible impact on our results, so we
perform our analysis assuming there will not be a near detector.

##  IV Results

Figure 3: Projected 1$\sigma$, 95%, and 99% regions with future measurements
of $(\rho_{e\mu},\eta_{e\mu})$ (left) and ($\rho_{{}_{23}},\eta_{{}_{23}})$
(right). Similarly to Fig. 1, the blue contours use disappearance analyses and
the orange appearance. The filled-in (open) star indicates the best-fit point
from the joint (appearance only) fit. Data are simulated with a non-unitary
LMM, but analyzed assuming it is.

Figure 1 shows the 95% and 99% credible regions of the six unitarity triangles
with all current data (green contours), and with the addition of future data
(red). These results assume unitarity of the LMM and therefore the PMNS
parameterization holds. We discuss the implications of assuming unitarity, and
how the results would differ without this assumption in Appendix B.

Figure 1 also shows the projections of future sensitivity to neutrino
disappearance ($P_{\alpha\alpha}$, blue contours) and appearance
($P_{\alpha\beta}$, orange) probabilities separately. This distinction between
the two measurements is crucial, both for understanding how the best fit
regions arise and for determining how non-unitarity manifests itself.

The intuitively tractable $e$-$\mu$ triangle (Fig. 1 top-left) can be
expressed in terms of PMNS parameters as,

$\displaystyle\rho_{e\mu}$ $\displaystyle=c_{12}^{2}+\cos\delta_{\rm
CP}\left(\frac{s_{12}c_{12}c_{23}}{s_{13}s_{23}}\right),$
$\displaystyle\eta_{e\mu}$ $\displaystyle=\sin\delta_{\rm
CP}\left(\frac{s_{12}c_{12}c_{23}}{s_{13}s_{23}}\right).$ (18)

Precise measurements of the disappearance channels $P_{ee}$ and $P_{\mu\mu}$
allow for the determination of $\theta_{12}$, $\theta_{13}$, and
$\theta_{23}$, which determine a circle (we discuss why the blue circles do
not close in the following paragraph). Long-baseline disappearance
measurements ($P_{\mu\mu}$) are only sensitive to $\sin^{2}(2\theta_{23})$,
leading to an octant degeneracy. This produces ambiguity in the radii of the
circles in the $e$-$\mu$ and $e$-$\tau$ planes, and results in the structure
in the 1-2 plane. Long-baseline $\nu_{e}$-appearance measurements can
determine the value of $\delta_{\rm CP}$, and therefore a preferred direction
in the $(\rho,\eta)$ plane. The definitions of $(\rho,\eta)$ affect the
appearance of these triangles, but the general features of disappearance
measurements giving ring-type structures and appearance selecting out a
direction are universal (For comparisons to other choices, see Refs. Gonzalez-
Garcia  _et al._ (2014); Esteban  _et al._ (2017); nuf and our discussion
above). $\nu_{\tau}$-appearance measurements, present or projected, are
insufficiently precise to provide further discriminatory power. In addition,
no planned measurement of $P_{\mu\tau}$ is actually sensitive to $\delta_{\rm
CP}$, even with improved precision de Gouvêa  _et al._ (2019); Ishihara
(2020). Note that the power of $\tau$-appearance measurement manifests clearly
when unitarity is not assumed; see Ref. Ellis  _et al._ (2020) for details. In
contrast with the CKM fits, where multiple observables independently measure
the CP-violating phase, in the neutrino sector there is only one present or
near-future observable, $P_{\mu e}$, sensitive to $\delta_{\rm CP}$.

Figure 4: Independent measurements of the parameters $\sin^{2}\theta_{13}$,
$\sin^{2}\theta_{23}$, and $\delta_{\rm CP}$ by the appearance dataset
(orange), disappearance dataset (blue), and their combination (red) when data
are simulated with a non-unitary LMM, but analyzed assuming it is. The
simulated data analyzed here are the same as those analyzed in Fig. 3.

From Eq. (18), it is apparent that some knowledge of $\delta_{\rm CP}$ is
required in order to prefer a direction in the $(\rho_{e\mu},\eta_{e\mu})$
plane, an effect seen in the blue regions throughout all six panels of Fig. 1.
Here, we see the combined measurement of disappearance channels, specifically
driven by the future DUNE, T2HK, and JUNO experiments, choosing a preferred
direction, i.e., being sensitive to the value of $\delta_{\rm CP}$. This is
driven by the $\nu_{\mu}$/$\overline{\nu}_{\mu}$ disappearance measurements
from DUNE/T2HK, which are sensitive to the quantities $\lvert U_{\mu
1}\rvert^{2}$ and $\lvert U_{\mu 2}\rvert^{2}$ independently at some level.
With the assumed true values of the mixing angles and $\delta_{\rm CP}$,

$\displaystyle\lvert U_{\mu 1}\rvert^{2}$
$\displaystyle=s_{12}^{2}c_{23}^{2}+c_{12}^{2}s_{23}^{2}s_{13}^{2}+2\cos\delta_{\rm
CP}s_{12}c_{12}s_{13}s_{23}c_{23}$ $\displaystyle\approx 0.083,$ (19)
$\displaystyle\lvert U_{\mu 2}\rvert^{2}$
$\displaystyle=c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{23}^{2}s_{13}^{2}-2\cos\delta_{\rm
CP}s_{12}c_{12}s_{23}s_{23}c_{23}$ $\displaystyle\approx 0.378.$ (20)

For $\delta_{\rm CP}\approx 0$, $2\pi$, these values approach roughly $0.215$
and $0.247$, respectively. While this is a minor effect at DUNE and T2HK,
their combination444Changing $\delta_{\rm CP}$ in this way leads to an
effective “tilt” in the expected number of muon-neutrino events in the
DUNE/T2HK disappearance channels, shifting the relative number of events
below/above the disappearance minimum in the oscillation probability. While
our simulations of T2HK and DUNE include normalization uncertainties, we do
not include a spectral shape uncertainty which would mask this effect. can
distinguish between these $\lvert U_{\mu 2}\rvert^{2}>\lvert U_{\mu
1}\rvert^{2}$ and $\lvert U_{\mu 1}\rvert^{2}\approx\lvert U_{\mu
2}\rvert^{2}$ scenarios at ${\sim}99\%$ confidence, leading to the non-closure
of our blue contours throughout Fig. 1. However, we note here that this does
not imply that these experiments are sensitive to CP violation, as they are
only measuring $\cos\delta_{\rm CP}$, a CP-symmetric quantity. Only with
$\nu_{e}$ appearance (or related channels) can the amount of CP violation in
the lepton sector, i.e., the Jarlskog invariant, be inferred. This is apparent
by the reflective symmetry of each of the blue contours in Fig. 1 about the
$\eta_{xy}=0$ axis in each panel.

In order to test how unitarity violation would appear in these triangles, we
simulate data with injected non-unitarity, but analyze it assuming the LMM is
unitary. To include effects of non-unitarity, we adopt a parameterization
beyond PMNS Fernandez-Martinez  _et al._ (2007); Escrihuela  _et al._ (2015);
Li and Luo (2016), which requires 13 parameters, as described in detail in
Appendix B. We inject non-unitarity by making $\sum_{i}U_{ei}U_{\mu
i}^{*}=0.01+0.04i$ and $\sum_{\alpha}U_{\alpha 2}U_{\alpha
3}^{*}=-0.004+0.017i$, on the edge of what is allowed by current data.

Figure 3 shows the constructed triangles in the $e$-$\mu$ and 2-3 planes,
where the fit incorrectly assumes the LMM is unitary. With a joint
disappearance and appearance analysis (red), there is no indication of
unitarity violation as the joint contours appear similar in shape and size to
those in Fig. 1. However, individual channel measurements reveal tension:
disappearance and appearance measurements disagree at over $95\%$ in the 2-3
plane. This demonstrates that separate analyses of disappearance and
appearance measurements can be a powerful probe of unitarity violation in the
lepton sector, and complementary to sterile neutrino searches Athanassopoulos
_et al._ (1996, 1998); Armbruster  _et al._ (2002); Astier  _et al._ (2003,
2001); Eskut  _et al._ (2008); Adamson  _et al._ (2019); Aguilar-Arevalo  _et
al._ (2018); Dentler  _et al._ (2018); Diaz  _et al._ (2020); Böser  _et al._
(2020); Berryman and Huber (2020).

Importantly, no tension is present (even at $1\sigma$) when viewed in terms of
measurements of the PMNS parameters even when disappearance and appearance
channels are separate. We demonstrate this in Fig. 4, where we show the same
sets of experiments measuring this simulated data in terms of the parameters
$\sin^{2}\theta_{13}$, $\sin^{2}\theta_{23}$, and $\delta_{\rm CP}$. Note that
all three of these measurements intersect. This shows an important advantage
of unitarity triangles as a test of LMM unitarity.

Testing whether certain subsets of data agree on these planes is one test of
unitarity. However, constructing the triangles assumes the LMM is unitary. To
numerically constrain non-unitarity, one needs to discard this assumption, and
include sterile neutrino searches. This leads to non-intuitive connections
between unitarity triangles and numerical results, e.g., the unitarity
violation in $e$-$\mu$ plane for Fig. 3 will be excluded by future searches at
over $2\sigma$, contrary to what the figure suggests, when sterile search
results are included. In what follows, we develop another intuitive means of
using oscillation measurements to probe LMM unitarity, accounting for
information loss when triangles are constructed under the assumption of
unitarity. Specifically, we consider the consistency between measurements of
the degree of CP violation in $U_{\rm LMM}$.

##  V Jarlskog Factors and the Jarlskog Invariant

For a unitary LMM, the Jarlskog invariant [Eq. (1)] is a measure of CP
violation and is related to the area of the unitarity triangles in Fig. 1. By
constructing the triangles as in Eq. (4), the area of each triangle is
$(J_{\rm PMNS}/2)/(|U_{\alpha i}|^{2}|U_{\beta j}|^{2})$. As long as
$\delta_{\rm CP}$ is not $0$ or $\pi$, $J_{\rm PMNS}$, and therefore the
triangle areas, are non-zero. In general, Jarlskog Factors $J_{\alpha i}$ can
be calculated by taking the cofactor of the $(\alpha,i)$ element of the LMM,
taking the complex conjugate of two elements. If the LMM is not unitary, these
nine different cofactors need not be the same Gandhi  _et al._ (2015).

The Jarlskog factors are defined as in Eq. (9), such that the nine $|J_{\alpha
i}|$ are the same and equal to $|J_{\rm PMNS}|$ when the matrix is unitary.
This condition is necessary, but not sufficient, for LMM unitarity. Without
the unitarity assumption, six triangles provide information on at most six
different $J_{\alpha i}$ (recall the discussion of Section II). Therefore, to
obtain a full characterization of the potentially non-unitary LMM, six
unitarity triangles do not suffice. One solution is to construct nine
triangles corresponding to the nine Jarlskog factors. However, as there are
only six closure relations (Eq. 3), this leads to some redundancy. Therefore,
a compact manner of representing all six closure relations, as well as
characterizing the full LMM is to show six unitarity triangles assuming
unitarity, and nine Jarlskogs without assuming unitarity.

Figure 5: Current (purple) and expected future (teal) 1, 2, and $3\sigma$
measurement ranges of the nine independent Jarlskog factors when not assuming
unitarity, $J_{\alpha i}$, compared with the allowed range when assuming
unitarity, $J_{\rm PMNS}$. An inconsistency between the measured $J_{\alpha
i}$ and $J_{\rm PMNS}$ would imply unitarity violation.

To compare the Jarlskog invariant and Jarlskog factors, we perform a fit to
all current and current plus future data while (not) assuming unitarity of the
LMM to construct $J_{\rm PMNS}$ ($J_{\alpha i}$). When not assuming unitarity,
we adopt the parameterization explained in Appendix B. Figure 5 shows the
result. Our current measurement of $J_{\rm PMNS}$ (bottom row, purple) is
consistent with the results of other more detailed fits Esteban  _et al._
(2020). Each independent $J_{\alpha i}$ measurement agrees, consistent with
the unitary LMM hypothesis. We see that $J_{\rm PMNS}$ and $J_{\alpha i}$ are
all consistent with zero at 1$\sigma$, consistent with the hypothesis that CP
is conserved in the lepton sector. When we simulate future data (assuming
$J_{\alpha i}$ are all equal and nonzero), the inclusion of future JUNO,
IceCube, DUNE, and T2HK projections allows us to exclude $J_{\rm PMNS}=0$ at
$3\sigma$. Moreover, many of the different $J_{\alpha i}$ will disfavor CP-
conservation at $3\sigma$. However, certain Jarlskog Factors, particularly
those involving knowledge of $|U_{\tau i}|$ (especially $J_{\mu 3}$), will
remain difficult to measure when unitarity is not assumed.

##  VI Conclusions

We have presented a comprehensive analysis of leptonic unitarity triangles
using all current and future neutrino oscillation data. Figure 1 displays how
the closure of six unitarity triangles is/will be constrained. By virtue of
the nature of these measurements, in contrast with the CKM matrix,
intersections of many measurements of PMNS matrix parameters are not possible.
Non-unitarity can nevertheless explicitly manifest itself as shown in Fig. 3,
though observation of non-unitarity requires distinguishing between appearance
and disappearance datasets. Figure 5 presents an alternative and complementary
visualization of constraints on LMM unitarity in terms of Jarlskog factors.
The allowed ranges of the $J_{\alpha i}$ are consistent with each other and
with non-zero CP violation in the lepton sector. If the LMM was not unitary,
these measurements would yield different $J_{\alpha i}$.

The Standard Model demands new physics to explain the origin of neutrino
masses and therefore oscillations. The impending era of precision experiments
will enable us to further understand the structure of the leptonic mixing
matrix, and constraints on the matrix’s unitarity serve as a probe of the
mechanism of neutrino masses. Meanwhile, the origin of the baryon asymmetry of
the universe will be better understood through studies of the degree of CP
violation in the lepton sector. Performing detailed studies of the leptonic
unitarity triangles therefore bears directly on both of these problems in the
Standard Model.

Acknowledgments

For helpful discussions, we are grateful to Francesco Capozzi, André de
Gouvêa, Peter Denton, Pedro Machado, Stephen Parke, Michael Peskin, Xin Qian,
and Natalia Toro. We acknowledge the 7th LCTP Spring Symposium: Neutrino
Physics. SARE and SWL are supported by the U.S. Department of Energy under
Contract No. DE-AC02-76SF00515. SARE is also supported in part by the Swiss
National Science Foundation, SNF project number P400P2$\\_$186678\. KJK is
supported by Fermi Research Alliance, LLC under contract DE-AC02-07CH11359
with the U.S. Department of Energy.

##  Appendix A Parameterization of the LMM

The unitarity conditions for the 3$\times$3 LMM are

$U^{\dagger}U=UU^{\dagger}=\mathbb{I}\ .$ (21)

From this, one can write down six real equations, requiring the normalization
of rows and columns of matrix-elements-squared $|U_{\alpha i}|^{2}$ to be 1:

$\begin{array}[]{c c}\displaystyle\sum_{i}|U_{\alpha i}|^{2}=0\ ,\\\
\text{(rows)}\end{array}\quad\begin{array}[]{cc}\displaystyle\sum_{\alpha}|U_{\alpha
i}|^{2}=0\ .\\\ \text{(columns)}\end{array}$ (22)

One can also write six complex equations corresponding to “closures” between
two different rows ($\alpha$ and $\beta$) or two different columns ($i$ and
$j$):

$\displaystyle\sum_{i=1}^{3}U_{\alpha i}U_{\beta i}^{*}=\,$ $\displaystyle
U_{\alpha 1}U_{\beta 1}^{*}+U_{\alpha 2}U_{\beta 2}^{*}+U_{\alpha 3}U_{\beta
3}^{*}=0,\quad\alpha\neq\beta$ $\displaystyle\text{(row closure)},$ (23)
$\displaystyle\sum_{\alpha=e}^{\tau}U_{\alpha i}U_{\alpha j}^{*}=\,$
$\displaystyle U_{ei}U_{ej}^{*}+U_{\mu i}U_{\mu j}^{*}+U_{\tau i}U_{\tau
j}^{*}=0,\quad i\neq j$ $\displaystyle\text{(column closure)}.$ (24)

From these closure relations, one can construct the familiar unitarity
triangles in the $(\rho,\eta)$ plane by dividing each term in the closure by
one of the sides. For a given row/column, there are 3 different triangles one
could define that are not related to one another by a simple inversion. The
triangles are defined as in the main text in terms of the LMM matrix elements.

The chosen set of $(\rho,\eta)$ used in the main text, under the assumption
that the LMM is unitary, can be expressed as: 555 A unitary LMM can be written
in the usual form in terms of the PMNS rotation angles and CP-violating phase
as: $\quad\ U_{\rm LMM}=U_{\rm
PMNS}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm CP}}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta_{\rm
CP}}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta_{\rm CP}}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta_{\rm
CP}}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta_{\rm
CP}}&c_{13}c_{23}\end{pmatrix}\ .$

$\displaystyle\rho_{e\mu}=c_{12}^{2}+\cos\delta_{\rm
CP}\left(\frac{s_{12}c_{12}}{s_{13}t_{23}}\right),$
$\displaystyle\eta_{e\mu}=\sin\delta_{\rm
CP}\left(\frac{s_{12}c_{12}}{s_{13}t_{23}}\right),$ (25)
$\displaystyle\rho_{e\tau}=\frac{1}{2}\left(\frac{2s_{12}^{2}(s_{23}^{2}-c_{23}^{2}s_{13}^{2})-t_{12}s_{13}\sin
2\theta_{23}\cos 2\theta_{12}\cos\delta_{\rm
CP}}{c_{12}^{2}s_{13}^{2}c_{23}^{2}+s_{12}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}}\right)\ ,$
$\displaystyle\eta_{e\tau}=-\frac{1}{2}\left(\frac{t_{12}s_{13}\sin
2\theta_{23}\sin\delta_{\rm
CP}}{c_{12}^{2}s_{13}^{2}c_{23}^{2}+s_{12}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}}\right)\ ,$ (26)
$\displaystyle\rho_{\mu\tau}=\frac{c_{13}^{2}}{4}\left(\frac{\sin^{2}2\theta_{23}\left(c_{12}^{2}-s_{12}^{2}s_{13}^{2}\right)+4\cos
2\theta_{23}\Delta\cos\delta_{\rm
CP}}{\left(c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{13}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}\right)\left(c_{12}^{2}s_{23}^{2}+s_{12}^{2}s_{13}^{2}c_{23}^{2}+2\Delta\cos\delta_{\rm
CP}\right)}\right)\ ,$
$\displaystyle\eta_{\mu\tau}=c_{13}^{2}\frac{\Delta\sin\delta_{\rm
CP}}{\left(c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{13}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}\right)\left(c_{12}^{2}s_{23}^{2}+s_{12}^{2}s_{13}^{2}c_{23}^{2}+2\Delta\cos\delta_{\rm
CP}\right)}\ ,$ (27)
$\displaystyle\rho_{12}=\frac{c_{13}^{2}}{4}\left(\frac{\sin^{2}2\theta_{12}\left(c_{23}^{2}-s_{13}^{2}s_{23}^{2}\right)+4\cos
2\theta_{12}\Delta\cos\delta_{\rm
CP}}{\left(s_{12}^{2}c_{23}^{2}+c_{12}^{2}s_{13}^{2}s_{23}^{2}+2\Delta\cos\delta_{\rm
CP}\right)\left(c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{13}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}\right)}\right)\ ,$
$\displaystyle\eta_{12}=-c_{13}^{2}\frac{\Delta\sin\delta_{\rm
CP}}{\left(s_{12}^{2}c_{23}^{2}+c_{12}^{2}s_{13}^{2}s_{23}^{2}+2\Delta\cos\delta_{\rm
CP}\right)\left(c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{13}^{2}s_{23}^{2}-2\Delta\cos\delta_{\rm
CP}\right)}\ ,$ (28)
$\displaystyle\rho_{13}=\frac{1}{2}\left(\frac{2s_{23}^{2}\left(s_{12}^{2}-s_{13}^{2}c_{12}^{2}\right)-t_{23}s_{13}\sin
2\theta_{12}\cos 2\theta_{23}\cos\delta_{\rm
CP}}{s_{12}^{2}s_{23}^{2}+c_{12}^{2}s_{13}^{2}c_{23}^{2}-2\Delta\cos\delta_{\rm
CP}}\right)\ ,$
$\displaystyle\eta_{13}=\frac{1}{2}\left(\frac{t_{23}s_{13}\sin
2\theta_{12}\sin\delta_{\rm
CP}}{s_{12}^{2}s_{23}^{2}+c_{12}^{2}s_{13}^{2}c_{23}^{2}-2\Delta\cos\delta_{\rm
CP}}\right)\ ,$ (29)
$\displaystyle\rho_{23}=c_{23}^{2}\left(1+\frac{t_{23}\cos\delta_{\rm
CP}}{t_{12}s_{13}}\right)\ ,$
$\displaystyle\eta_{23}=-c_{23}^{2}\left(\frac{t_{23}\sin\delta_{\rm
CP}}{t_{12}s_{13}}\right)\ ,$ (30)

where $4\Delta\equiv s_{13}\sin 2\theta_{12}\sin 2\theta_{23}$.

##  Appendix B Non-Unitarity Parameterization & Effects of Assuming Unitarity

In this appendix we give more detail regarding the $U_{\rm LMM}$
parameterization that is used when unitarity is not manifestly assumed (as in
the PMNS parameterization). We also show how such assumptions impact the
measurements shown in unitarity triangles like those in Fig. 1.

When we do not assume that the LMM is unitarity, we assume that the it takes
the form

$U_{\rm LMM}=\left(\begin{array}[]{l l l}\left\lvert
U_{e1}\right\rvert&\left\lvert U_{e2}\right\rvert e^{i\phi_{e2}}&\left\lvert
U_{e3}\right\rvert e^{i\phi_{e3}}\\\ \left\lvert U_{\mu
1}\right\rvert&\left\lvert U_{\mu 2}\right\rvert&\left\lvert U_{\mu
3}\right\rvert\\\ \left\lvert U_{\tau 1}\right\rvert&\left\lvert U_{\tau
2}\right\rvert e^{i\phi_{\tau 2}}&\left\lvert U_{\tau 3}\right\rvert
e^{i\phi_{\tau 3}}\\\ \end{array}\right)\ ,$ (31)

where the 13 free parameters (ignoring the potentially physical Majorana
phases) are necessary to describe a $3\times 3$ complex matrix after
accounting for rephasing of the charged lepton fields. Alternative
parameterizations are commonly adopted in the literature Fernandez-Martinez
_et al._ (2007); Escrihuela  _et al._ (2015); Li and Luo (2016), all with the
same number of free parameters. We motivate the use of our parameterization,
and discuss maps between this and others in the literature in Ref. Ellis  _et
al._ (2020). While the complex phases in $U_{\rm LMM}$ appear on different
matrix elements than in $U_{\rm PMNS}$, the two parameterizations are related
(if $U_{\rm LMM}$ satisfies the unitarity conditions) by rephasing of the
neutrino fields. For any set of PMNS parameters, an equivalent set of LMM
parameters may be determined uniquely.

We obtain best-fit values for the 13 LMM parameters by making use of the
observation that the LMM fit must match the global fit for the PMNS parameters
when the LMM is unitary. Analyzing current data when assuming unitarity yields
the maximum-likelihood values of $\sin^{2}\theta_{12}=0.308$,
$\sin^{2}\theta_{13}=0.0219$, $\sin^{2}\theta_{23}=0.551$, and $\delta_{\rm
CP}=200^{\circ}$. These four best-fit values may then be used in conjunction
with the 9 constraints applicable to a $3\times 3$ unitary matrix to solve for
the 13 LMM elements, yielding

$\displaystyle\left\lvert U_{\rm
LMM}\right\rvert=\begin{pmatrix}0.823&0.549&0.148\\\ 0.288&0.615&0.734\\\
0.490&0.555&0.663\\\ \end{pmatrix}\ ,$
$\displaystyle\begin{array}[]{cc}\phi_{e2}=172^{\circ},&\phi_{e3}=333^{\circ},\\\
\phi_{\tau 2}=346^{\circ},&\phi_{\tau 3}=170^{\circ}.\end{array}$ (34)

When we study the nine separate Jarlskog factors $J_{\alpha i}$ cf. Fig. 5, we
use this set of 13 parameters, projecting down to relevant combinations for
the allowed regions of different $J_{\alpha i}$.

We also use this parameterization to simulate future data for JUNO IceCube,
T2HK, and DUNE assuming $U_{\rm LMM}$ is not unitary, i.e.,
$U^{\dagger}U\neq\mathbb{I}$. In generating the data that are analyzed for
Fig. 3, we modify the input values of $\phi_{e2}$ and $\phi_{e3}$ from those
in Eq. (B) by $\Delta\phi_{e2}=-5.30^{\circ}$ and
$\Delta\phi_{e3}=7.23^{\circ}$. This preserves the unitarity constraint that
the individual rows and columns of $U_{\rm LMM}$ are properly normalized,
$\sum_{i}|U_{\alpha i}|^{2}=1$, $\sum_{\alpha}|U_{\alpha i}|^{2}=1$, but
causes non-closure of the triangles in several planes.

Figure A1: Current (purple and green) and expected future (pale blue and red)
measurements 95% (dark colors) and 99% confidence level (light) of two
different unitarity triangles – $\rho_{e\mu}$ vs. $\eta_{e\mu}$ (left) and
$\rho_{23}$ vs. $\eta_{23}$ (right). We contrast two assumptions in this
figure, showing the resulting measurements when the unitarity of the leptonic
mixing matrix is or is not assumed. Purple and light blue contours display the
results when unitarity is not assumed, where green and red contours show the
results when it is assumed. The filled-in (open) star indicates the best-fit
point of the analysis of current data when unitarity is (not) assumed,
corresponding to the green (purple) contours.

One non-closure is in the $e$-$\mu$ plane, with $\sum_{i}U_{ei}U_{\mu
i}^{*}=0.01+0.04i$. Sterile neutrino searches that look for zero-distance
neutrino oscillation (as described in Section III.1) are sensitive to the
absolute value squared of the non-closure, and this level is at the upper end
of what is currently allowed by data. In addition, there is also non-closure
in the 2-3 plane, with $\sum_{\alpha}U_{\alpha 2}U_{\alpha
3}^{*}=-0.004+0.017i$. This is shown in Fig. 3.

Triangles when Unitarity is not assumed: In the results shown in Fig. 1 and
Fig. 3, fits to existing/future data were performed with the PMNS mixing
angles as free parameters ($\sin^{2}\theta_{12}$, $\sin^{2}\theta_{13}$,
$\sin^{2}\theta_{23}$, $\delta_{\rm CP}$, and mass-squared splittings), such
that unitarity was explicitly assumed. Confidence level contours were then
constructed by mapping these parameters onto those that enter each unitarity
triangle, using the PMNS parameterization. Here we perform a fit using the
parameterization described above, where the unitarity of $U_{\rm LMM}$ is not
guaranteed.

The difference between these two fits is most apparent in triangles that
depend on mixing matrix elements that are not powerfully measured on their own
in experiments, but can be inferred by other measurements if unitarity is
assumed. Specifically, that is the case for the mixing matrix elements
$U_{\tau i}$. For instance, in the PMNS parameterization, $U_{\tau
3}=\cos{\theta_{23}}\cos{\theta_{13}}$, which can be constrained fairly well
by atmospheric and reactor experiments. Without unitarity, the strongest
current measurement of $U_{\tau 3}$ in our fit comes from OPERA’s
$\nu_{\mu}\to\nu_{\tau}$ appearance, a significantly less precise measurement.

We show this difference in Fig. A1, focusing on two different triangle planes,
($\rho_{e\mu}$, $\eta_{e\mu}$), where the differences are small, and
($\rho_{23}$, $\eta_{23}$), where the differences are the largest. All
contours shown are 95% (dark contours) and 99% confidence level (light). Here,
the purple and green regions correspond to current data fits, where the purple
(green) region is the result of the fit without (with) assuming unitarity.
Likewise, light blue (unitarity not assumed) and red (unitarity assumed)
regions are from fits including current and future data. The green and red
regions here correspond with those of the same color shown in the appropriate
panels of Fig. 1. In each of the two panels, the filled-in star denotes the
best-fit point in this parameter space obtained by analyzing all current data
when unitarity is assumed (green dataset), where the open star indicates the
best fit point when unitarity is not assumed (purple dataset).

Most notable here is the difference in the size of allowed regions between
when unitarity is or is not assumed for ($\rho_{23}$, $\eta_{23}$). As
mentioned above, this is largely due to the uncertainty regarding the
magnitude of the elements $|U_{\tau 2}|$ and $|U_{\tau 3}|$. We also see that
the current data prefer a much larger triangle in this plane if unitarity is
not assumed – this is due to the preference for large $|U_{\tau 3}|$ from the
OPERA experiment Acero  _et al._ (2019). We also highlight the second island
of allowed parameter space in the future projections of ($\rho_{e\mu}$,
$\eta_{e\mu}$) when unitarity is not assumed (light blue) – this comes about
because, when unitarity is not assumed, future experiments cannot definitively
determine whether $\left\lvert U_{\mu 1}\right\rvert^{2}$ is smaller or larger
than $\left\lvert U_{\mu 2}\right\rvert^{2}$.

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