# Nonunitarity of the lepton mixing matrix at the European Spallation Source

Sabya Sachi Chatterjee<EMAIL_ADDRESS>Institut de Physique
Théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France
Institute for Particle Physics Phenomenology, Department of Physics, Durham
University, Durham, DH1 3LE, UK  O. G. Miranda<EMAIL_ADDRESS>Departamento de Física, Centro de Investigación y de Estudios Avanzados del
IPN,
Apartado Postal 14-740 07000 Mexico, Distrito Federal, Mexico  M. Tórtola
<EMAIL_ADDRESS>Departament de Física Teórica, Universitat de València,
Burjassot 46100, Spain  AHEP Group, Institut de Física Corpuscular –
CSIC/Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain  J. W. F.
Valle<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular –
CSIC/Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain

###### Abstract

If neutrinos get mass through the exchange of lepton mediators, as in seesaw
schemes, the neutrino appearance probabilities in oscillation experiments are
modified due to effective nonunitarity of the lepton mixing matrix. This also
leads to new CP phases and an ambiguity in underpinning the “conventional”
phase of the three-neutrino paradigm. We study the CP sensitivities of various
setups based at the European spallation source neutrino super-beam (ESSnuSB)
experiment in the presence of nonunitarity. We also examine its potential in
constraining the associated new physics parameters. Moreover, we show how the
combination of DUNE and ESSnuSB can help further improve the sensitivities on
the nonunitarity parameters.

Neutrino Oscillation, Long-baseline, Nonunitarity, ESSnuSB

††preprint: t21/073

##  I Introduction

The discovery of neutrino oscillations McDonald:2016ixn ; Kajita:2016cak  has
brought neutrinos to the center of particle physics. The current experimental
data mainly converge into a consistent global picture in which the oscillation
parameters are pretty well determined. However, three challenges still remain,
namely, to determine the CP phase, the atmospheric octant and the ordering of
the neutrino mass spectrum deSalas:2020pgw ; 10.5281/zenodo.4726908 . These
will be the target of a number of future experiments, such as DUNE
DUNE:2015lol . A fourth item must be added to this list, namely probing the
robustness of the interpretation, such as testing the unitarity of the lepton
mixing matrix. This is crucial because it undermines the efforts of
underpining the CP phase $\delta_{\rm CP}$ Miranda:2016wdr ;
Escrihuela:2016ube .

This task is well justified also on theory grounds. Indeed, one of the most
attractive ways to generate neutrino mass is through the mediation of heavy
neutral leptons. While these emerge in many gauge extensions of the standard
model, they can be postulated directly at the $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ level, as the neutrino mass generation mediators.

This, in fact, provides the most general realization of the seesaw mechanism
and many of its variants Schechter:1980gr . For generality here we focus
exclusively on this case, namely, the standard $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ seesaw mechanism. The resulting lepton mixing
matrix is in general quite complex when compared with Cabibbo-Kobayashi-
Maskawa (CKM) mixing. First, lepton mixing contains extra phases that can not
be eliminated by field redefinitions Schechter:1980gr  and are therefore
physical Schechter:1980gk , crucially affecting lepton number violating
processes. However, they do not affect conventional oscillations, so we will
ignore them in what follows. On the other hand the lepton mixing matrix must
in general take into account the admixture of the heavy lepton seesaw
mediators with the light active neutrinos Schechter:1981cv . These are usually
neglected, as the smallness of neutrino masses indicated by neutrino
experiments suggests a very high seesaw scale.

Nonetheless, the seesaw mechanism can also be realized at low scales. The
template for this is a scenario where two SM-singlet leptons are added
sequentially, instead of just one. If lepton number symmetry is imposed, then
all three active neutrinos are massless, as in the standard model. In contrast
to the Standard Model, however, lepton flavor is violated, and similarly,
leptonic CP symmetry. This shows that flavor and CP violation can exist in the
leptonic weak interaction despite the masslessness of neutrinos, implying that
such processes need not be suppressed by the small neutrino masses, and hence
can be large Bernabeu:1987gr ; Branco:1989bn ; Rius:1989gk ; Dittmar:1989yg ;
Deppisch:2004fa ; Deppisch:2005zm .

Over such basic template one can build genuine “low-scale” realizations of the
seesaw mechanism in which lepton number symmetry is restored at low, instead
of high, values of the lepton number violating scale. The models are natural
in t’Hooft sense, and lead to small, symmetry-protected neutrino masses. Such
“low-scale” seesaw realizations include the inverse Mohapatra:1986bd ;
GonzalezGarcia:1988rw  as well as the linear seesaw mechanisms Akhmedov:1995ip
; Akhmedov:1995vm ; Malinsky:2005bi . In all of these we expect potentially
sizeable unitarity violation in the leptonic weak interaction. This paper is
dedicated to probing such effects at the European Spallation Source neutrino
Super-Beam (ESSnuSB) experiment. As far as we can tell, this is the first
study of this kind for ESSnuSB. In addition, we also review the sensitivities
of the DUNE experiment to the nonunitarity parameters and show how DUNE and
ESSnuSB can play a complementary role to each other. Sensitivity studies of
nonunitarity at other future neutrino facilities can be found in Refs.
Escrihuela:2016ube ; Ge:2016xya ; Meloni:2009cg ; Blennow:2016jkn ;
Miranda:2018yym ; Soumya_2018 ; Miranda:2020syh ; Coloma:2021uhq ;
Agarwalla:2021owd .

We briefly describe the theoretical framework for unitarity violation in the
charged current (CC) leptonic weak interaction in Sec. II, and the matter
three-neutrino oscilation probabilities with and without unitarity violation
in Sect. III. Next, in Sec. IV, we describe the experimental setups of
interest, and also present the details of the simulation we have performed.
Our results are given in Sec. V and include our calculated ESSnuSB and DUNE
sensitivities to nonunitary (NU) neutrino mixing in Sec. V.1, the CP violation
discovery potential in the presence of unitarity violation is given in V.2,
and the CP reconstruction capabilities both for the standard phase as well as
the seesaw phase of $\alpha_{21}$ in V.3. Finally we briefly summarize in Sec.
VI.

##  II Theoretical framework

In the standard $3\times 3$ oscillation picture, the neutrino mixing matrix is
described symmetrically by a product of three mixing matrices

$U=\omega_{2\,3}\>\omega_{1\,3}\>\omega_{1\,2}\,,$ (1)

where each $\omega_{i,j}$ describes an effective $2\times 2$ complex rotation,
characterized by a mixing angle and its phase. This symmetrical form
complements the original description Schechter:1980gr  by specifying the most
convenient factor ordering. In explicit form, the standard $3\times 3$
leptonic mixing matrix is given by

$U=\left(\begin{array}[]{ccccc}1&0&0\\\ 0&c_{23}&e^{-i\varphi_{23}}s_{23}\\\
0&-e^{i\varphi_{23}}s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccccc}c_{13}&0&e^{-i\varphi_{13}}s_{13}\\\
0&1&0\\\
-e^{i\varphi_{13}}s_{13}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccccc}c_{12}&e^{-i\varphi_{12}}s_{12}&\\\
-e^{i\varphi_{12}}s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right),$ (2)

where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$, and $\varphi_{ij}$
is the corresponding phase. One sees the appearance of two extra physical
phases with no counterpart in the quark sector: the so-called Majorana phases
Schechter:1980gr . Note that the above parameterization of the neutrino mixing
matrix is equivalent to the oscillation-sensitive part of the PDG form
Zyla:2020zbs  with
$\varphi_{13}-\varphi_{12}-\varphi_{23}\equiv\delta_{\mathrm{CP}}$
Rodejohann:2011vc  so that when $\varphi_{12}=\varphi_{23}=0$ one has
$\varphi_{13}=\delta_{\mathrm{CP}}$.

Apart from the presence of these new physical Schechter:1980gk  phases, the
leptonic CC interaction will in general also contain the mixing of neutral
heavy leptons that mediate neutrino mass generation, as in the so-called
type-I seesaw mechanism. These two facts make the mixing of massive neutrinos
substantially richer in structure than that which describes the quark weak
interactions Schechter:1980gr . As a result, in this general neutrino
framework, $U^{n\times n}$ can be expressed as the product of the new physics
(NP) piece, times the Standard Model (SM) piece

$U^{n\times n}=U^{NP}\,U^{SM}.$ (3)

Thinking in terms of the seesaw mechanism it is convenient to express the full
$U$ matrix as four submatrices. Here we label them as in Hettmansperger:2011bt
111The form of the matrices N, S, T and V within the full seesaw expansion was
given in Ref. Schechter:1981cv . They correspond, respectively, to $U_{a}$,
$U_{b}$, $U_{c}$ and $U_{d}$ of Eqs. (2.8) and (3.5) of the above reference.,
i.e.

$U^{n\times n}=\left(\begin{array}[]{cc}N&S\\\ T&V\end{array}\right).$ (4)

Notice that we have a block, $N$, relating the light neutrino sector with the
three active neutrino flavors. Here $V$ will be a $(n-3)\times(n-3)$
submatrix, while $S$ and $T$ will be, in general, rectangular matrices.

Clearly, in this general case, the full unitarity condition will take the form

$\displaystyle NN^{\dagger}+SS^{\dagger}=I,$ $\displaystyle
TT^{\dagger}+VV^{\dagger}=I.$ (5)

Therefore, the $3\times 3$ matrix $N$ describing the mixing of light neutrinos
will no longer be unitary. One can show Escrihuela:2015wra  that in the most
general case $N$ can be parametrized as

$N=N^{NP}\,U^{3\times 3}=\left(\begin{array}[]{ccc}\alpha_{11}&0&0\\\
\alpha_{21}&\alpha_{22}&0\\\
\alpha_{31}&\alpha_{32}&\alpha_{33}\end{array}\right)\>U^{3\times 3},$ (6)

where the diagonal $\alpha$’s are real and close to 1, while the off-diagonals
are small but complex. Indeed, for any number of additional neutrino states,
we will have for the diagonal entries of this matrix that

$\alpha_{jj}\>=\>\prod_{i=4,n}\>\cos\theta_{ji}\,,$ (7)

with no sum over $j$. For small mixings, the non-diagonal entries are given as

$\alpha_{ji}\>\simeq\>-\sum_{k=4,n}\theta_{jk}\theta_{ik}e^{-i(\varphi_{jk}-\varphi_{ik})}\,\,\,;\,\,\,i<j\,\,\,.$
(8)

From this last expression, one can see that

$|\alpha_{21}|^{2}\leq\sum^{N}_{i=4}|\theta_{2i}\theta_{1i}e^{-i(\varphi_{2i}-\varphi_{1i})}|^{2}~{}=~{}\sum^{N}_{i=4}\theta^{2}_{2i}\theta^{2}_{1i},$
(9)

and similar equations for the other two non-diagonal terms. Using the triangle
inequality, one can now derive the consistency relations 222For a general
derivation of this expression without the assumption of small mixing angles
see Forero:2021azc . ,

$|\alpha_{ji}|\leq\sqrt{(1-\alpha_{jj}^{2})(1-\alpha_{ii}^{2})}.\\\ $ (10)

The muon neutrino appearance probability will be given as

$\displaystyle P_{\mu
e}=\alpha_{11}^{2}|\alpha_{21}|^{2}-4\sum\limits^{3}_{j>i}Re\left[N^{*}_{\mu
j}N_{ej}N_{\mu i}N^{*}_{ei}\right]\sin^{2}\left(\frac{\Delta
m^{2}_{ji}L}{4E}\right)$
$\displaystyle+2\sum\limits^{3}_{j>i}Im\left[N^{*}_{\mu j}N_{ej}N_{\mu
i}N^{*}_{ei}\right]\sin\left(\frac{\Delta m^{2}_{ji}L}{2E}\right)\,,$ (11)

where $N$ is given in terms of the $\alpha$’s as in Eq. (6).

For the case of vacuum oscillations, the parameters characterizing unitarity
violation in the $\mu$-e sector are $\alpha_{11}$, $\alpha_{22}$, and
$\alpha_{21}$. In the presence of matter effects, the appearance probability
could also involve the third neutrino type, since the charged and neutral
current potential will modify the effective form of the matrix $N$. The
charged current potential for the nonunitary case will be given by

$V^{\alpha\beta}_{CC}=\sqrt{2}\,G_{F}N_{e}\left(NN^{\dagger}\right)_{\alpha
e}\left(NN^{\dagger}\right)_{e\beta}\,,$ (12)

where $G_{F}$ is the Fermi constant and $N_{e}$ the number density of
electrons in the medium. The matrix product will be written, in terms of the
$\alpha$’s, as Escrihuela:2016ube :

$(NN^{\dagger})_{\alpha
e}(NN^{\dagger})_{e\beta}=\alpha_{11}^{2}\left(\begin{array}[]{ccc}\alpha_{11}^{2}&\alpha_{11}\alpha_{21}^{*}&\alpha_{11}\alpha_{31}^{*}\\\
\alpha_{11}\alpha_{21}&|\alpha_{21}|^{2}&\alpha_{21}\alpha_{31}^{*}\\\
\alpha_{11}\alpha_{31}&\alpha_{21}^{*}\alpha_{31}&|\alpha_{31}|^{2}\\\
\end{array}\right)\,.$ (13)

The corresponding potential for neutral currents will be

$\displaystyle V^{\alpha\beta}_{NC}=$
$\displaystyle-\sqrt{2}G_{F}\frac{N_{n}}{2}\sum_{\rho}(NN^{\dagger})_{\alpha\rho}(NN^{\dagger})_{\rho\beta}$
$\displaystyle=$
$\displaystyle-\sqrt{2}G_{F}\frac{N_{n}}{2}\left[(NN^{\dagger})^{2}\right]_{\alpha\beta}\,,$
(14)

where the matrix product $(NN^{\dagger})^{2}$, at leading order in the non-
diagonal $\alpha$’s, takes the form Escrihuela:2016ube

$\leavevmode\resizebox{212.47617pt}{}{$\left(\begin{array}[]{ccc}\alpha_{11}^{4}&\alpha_{11}\alpha^{*}_{21}\left(\alpha_{11}^{2}+\alpha_{22}^{2}\right)&\alpha_{11}\alpha^{*}_{31}\left(\alpha_{11}^{2}+\alpha_{33}^{2}\right)\\\
\alpha_{11}\alpha_{21}\left(\alpha_{11}^{2}+\alpha_{22}^{2}\right)&\alpha_{22}^{4}&\alpha_{22}\alpha^{*}_{32}\left(\alpha_{22}^{2}+\alpha_{33}^{2}\right)\\\
\alpha_{11}\alpha_{31}\left(\alpha_{11}^{2}+\alpha_{33}^{2}\right)&\alpha_{22}\alpha_{32}\left(\alpha_{22}^{2}+\alpha_{33}^{2}\right)&\alpha_{33}^{4}\\\
\end{array}\right)\,$}.$ (15)

Neglecting cubic terms in $\alpha_{21}$, $\sin\theta_{13}$, and $\Delta
m^{2}_{21}$, one finds that, in the vacuum case limit, the main contribution
to the conversion probability will be given by

$P_{\mu e}=(\alpha_{11}\alpha_{22})^{2}P^{3\times 3}_{\mu
e}+\alpha_{11}^{2}\alpha_{22}|\alpha_{21}|P^{I}_{\mu
e}+\alpha_{11}^{2}|\alpha_{21}|^{2},$ (16)

where $P^{3\times 3}_{\mu e}$ denotes the usual three-neutrino conversion
probability,

$\displaystyle\tiny P^{3\times 3}_{\mu e}$
$\displaystyle=4\bigg{[}\cos^{2}\theta_{12}\cos^{2}\theta_{23}\sin^{2}\theta_{12}\sin^{2}\left(\frac{\Delta
m^{2}_{21}L}{4E_{\nu}}\right)$
$\displaystyle+\cos^{2}\theta_{13}\sin^{2}\theta_{13}\sin^{2}\theta_{23}\sin^{2}\left(\frac{\Delta
m^{2}_{31}L}{4E_{\nu}}\right)\bigg{]}$ $\displaystyle+\sin
2\theta_{12}\sin\theta_{13}\sin 2\theta_{23}\sin\left(\frac{\Delta
m^{2}_{21}L}{2E_{\nu}}\right)\times$ $\displaystyle\sin\left(\frac{\Delta
m^{2}_{31}L}{4E_{\nu}}\right)\cos\left(\frac{\Delta
m^{2}_{31}L}{4E_{\nu}}+\delta_{\mathrm{CP}}\right)\,,$ (17)

where $P^{I}_{\mu e}$ is the interference term

$\displaystyle P^{I}_{\mu e}$
$\displaystyle=-2\bigg{[}\sin(2\theta_{13})\sin\theta_{23}\sin\left(\frac{\Delta
m^{2}_{31}L}{4E_{\nu}}\right)\times$ $\displaystyle\hskip
71.13188pt\sin\left(\frac{\Delta
m^{2}_{31}L}{4E_{\nu}}+\delta_{\mathrm{CP}}-\phi_{21}\right)\bigg{]}$
$\displaystyle+\cos\theta_{13}\cos\theta_{23}\sin
2\theta_{12}\sin\phi_{21}\sin\left(\frac{\Delta
m^{2}_{21}L}{2E_{\nu}}\right),$ (18)

with $\phi_{21}=\text{arg}(\alpha_{21})$.

##  III Three neutrino oscillation probabilities

Before coming to our numerical results, in this section we discuss the
behaviour of the appearance (anti)neutrino probabilities. To this end, we show
in the left (right) panel of Fig. 1 the $\nu_{\mu}\to\nu_{e}$
($\bar{\nu}_{\mu}\to\bar{\nu}_{e}$) oscillation probabilities as a function of
the neutrino energy. The upper, middle, and lower panels correspond to 540 km,
360 km, and 200 km baselines, respectively. We show the conversion probability
in the standard unitary framework as a solid line, while the dashed one
represents the nonunitary case. For the unitary case, we consider the values
of the neutrino oscillation parameters given in Table 1 with $\delta_{\rm
CP}=-90^{\circ}$. For the nonunitary case, besides these values, we fix the
nonunitary $\alpha$-parameters to be $\alpha_{11}=0.97$, $\alpha_{22}=0.99$,
$\alpha_{33}=1$, $|\alpha_{21}|=0.02$, $\phi_{21}=90^{\circ}$,
$|\alpha_{31}|=0$, and $|\alpha_{32}|=0$. In Fig. 1 we also include in green
color (arbitrary units) the unoscillated (anti)neutrino $\nu_{\mu}$ flux times
the (anti)$\nu_{e}$-nucleus cross section Messier:1999kj ; Paschos:2001np .
This provides a bird-eye view of the real analysis at the event level. The
main region of interest extends from $0.1\,\rm GeV$ to $0.7\,\rm GeV$, with a
peak around $0.30\,\rm GeV$. One can see the relative location of this peak
with respect to the second oscillation maximum for the three baselines of
$540$ km and $360$ km and the $200$ km case, see also Table 2. The true
expected event number for the appearance signal must, of course, take into
account the convolution of the cross section, the appearance probability and
the unoscillated neutrino energy spectrum at the detector, which depends on
its distance to the source. Thus, the total unoscillated neutrino flux for the
$200$ km case will be approximately three times that for a $360$ km baseline,
and seven times the flux of the $540$ km case. As a result, the shortest the
baseline, the higher the expected number of events, as we will see in Fig. 3.

Figure 1: The left panels represent the neutrino appearance probabilities for
the unitary (solid line) and nonunitary (dashed line) oscillation case, for
three different baselines, as indicated. The right panels show the
corresponding antineutrino probabilities. The product of the
$\nu_{\mu}$($\overline{\nu}_{\mu}$) fluxes and the
$\nu_{e}$($\overline{\nu}_{e}$)-nucleus cross sections (in arbitrary units) is
also shown as a green shaded area.

$3\nu$ parameters $\sin^{2}\theta_{12}$ $\sin^{2}\theta_{13}$
$\sin^{2}\theta_{23}$ $\delta_{\rm CP}$ $\frac{\Delta
m_{21}^{2}}{10^{-5}\,\rm{eV^{2}}}$ $\frac{\Delta
m_{31}^{2}}{10^{-3}\,\rm{eV^{2}}}$ Benchmark values $0.318$ $0.022$ $0.57$
$[-180,\,180]$ $7.5$ $2.55$ Status fixed marginalized marginalized
marginalized fixed marginalized

Table 1: Benchmark values for the standard three-neutrino oscillation parameters taken from the current global fit analysis deSalas:2020pgw , along with their assumed marginalization status within our analyses, except $\delta_{\rm CP}$. For simplicity, normal mass ordering (NO) has been assumed.  baseline (km) | 1st osc. max. (GeV) | 2nd osc. max. (GeV)
---|---|---
540 | 1.05 | 0.35
360 | 0.70 | 0.23
200 | 0.39 | 0.13
Table 2: Possible baselines of the ESSnuSB project Baussan:2013zcy ;
Dracos:2016wso ; Dracos:2018jsn ; Dracos:2018syh , along with the
corresponding values of the neutrino energy for the first and second
oscillation maxima.

##  IV Experimental setup and simulation

In this section, we briefly discuss the experimental specifications of the
ESSnuSB and DUNE setups used in this work, followed by a description of our
simulation procedure.

###  IV.1 Experimental setup options

The ESSnuSB project is a proposed accelerator neutrino experiment sourced at
Lund (Sweden), where the ESS linac facility is currently under construction.
The original ESSnuSB proposal was to use of a very intense proton beam of $2$
GeV energy and an average beam power of 5 MW, resulting in $2.7\times 10^{23}$
protons on target (POT) per year ($208$ effective days) Baussan:2013zcy ;
Dracos:2016wso ; Dracos:2018jsn ; Dracos:2018syh . Here we will adopt this
configuration. It is expected that the future linac upgrade can increase the
proton energy up to 3.6 GeV. The neutrino and antineutrino fluxes arising from
the 2 GeV proton beam peak around $0.25$ GeV enrique . These (anti)neutrinos
will be detected by a 500 kton fiducial mass Water Cherenkov detector similar
to the MEMPHYS project Agostino:2012fd ; luca . Since the baseline of the far
detector has not been finalized yet, we have considered in this work three
possible baselines Baussan:2013zcy , which are 200 km, 360 km, and 540 km
respectively. It has been shown in Baussan:2013zcy  that if the detector is
placed in any of the existing mines in between 200 km to 600 km from the
ESSnuSB site Lund, a $3\sigma$ evidence of CP violation could be achieved for
60% coverage of the full $\delta_{\rm CP}$ range. Our simulation matches the
event numbers of Table 3 and all other results given in Baussan:2013zcy . In
all the numerical results presented here, we have assumed $2$ years of
neutrino and $8$ years of antineutrino running with an optimistic assumption
of uncorrelated $5$% signal normalization and 10% background normalization
error for both neutrino and antineutrino appearance and disappearance
channels, respectively. For more details about the accelerator facility,
beamline design, detector and baseline positions of this setup, see
Baussan:2013zcy . Note that, while working on this paper, an updated analysis
from the collaboration has come out Alekou:2021coj  333Ref. Alekou:2021coj
has not considered the 200 km baseline option which, as we will see, actually
turns out to be somewhat more promising for certain tasks, e.g. probing
$\alpha_{21}$. Enhanced sensitivities to unitarity violation might be expected
for the updated setup of the proposal. This highly potential and ambitious
facility is expected to start taking data around the year 2030.

DUNE is a future long-baseline accelerator-based neutrino experiment with a
baseline of 1300 km from the source at Fermilab to the far detector placed
deep underground at the Sanford Laboratory site in South Dakota. DUNE will use
a 40 kton LArTPC detector and a 120 GeV proton beam with 1.2 MW beam power
resulting in $1.1\times 10^{21}$ POT/year. For the numerical simulations, we
have followed the experimental configurations provided by the collaboration in
the Technical Design Report (TDR) Abi:2020qib ; Abi:2021arg , assuming equal
runtime of 3.5 years in neutrino and antineutrino mode, which results in 336
kton-MW-year exposure for the TDR setup. More details on the systematic
errors, efficiencies and energy resolutions can be found in Refs.
Alion:2016uaj ; Abi:2021arg .

###  IV.2 Simulation procedure

In order to assess the statistical sensitivity of the ESSnuSB facility to
neutrino oscillations, we have made use of the built-in $\chi^{2}$ function of
the GLoBES package Huber:2004ka ; Huber:2007ji , which incorporates the
systematic errors through the pull terms Huber:2002mx . To perform the
nonunitarity analysis we have used the modified version of Kopp_NSI . The
total $\chi^{2}$ is a sum of all the contributions coming from different
channels,

$\displaystyle\underset{total}{\chi^{2}}=\underset{\nu_{\mu}\to\nu_{e}}{\chi^{2}}+\underset{\bar{\nu}_{\mu}\to\bar{\nu}_{e}}{\chi^{2}}+\underset{\nu_{\mu}\to\nu_{\mu}}{\chi^{2}}+\underset{\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}}{\chi^{2}}.$
(19)

Unless stated otherwise, the benchmark choices of the standard three-neutrino
unitary oscillation parameters and their marginalization status in our
analysis are given in Table 1. Our benchmark choices closely follow the
current global fit analysis deSalas:2020pgw . Following the same analysis, we
have adopted a $1\%$ uncertainty on the atmospheric mass-squared splitting
$\Delta m_{31}^{2}$ and a $3.2\%$ uncertainty on the reactor mixing angle
$\sin^{2}\theta_{13}$. We have freely marginalized over the atmospheric
parameter $\sin^{2}\theta_{23}$ from $0.35$ to $0.65$.

For the case of ESSnuSB, we have considered a line-averaged constant matter
density $\rho=2.8\,\rm g/cm^{3}$ following the PREM profile DZIEWONSKI1981297
; stacey:1977 . For DUNE, we have also assumed the same matter density but
with a $5\%$ uncertainty due to the longer baseline. For definiteness, we have
assumed the currently preferred case of normal mass ordering (NO) throughout
all of our analyses. Whenever appropriate, we have also marginalized over the
NU parameters, along with their associated CP phases, implementing the current
$3\sigma$ bounds shown in Table 3. These come essentially from short-baseline
oscillation searches such as from NOMAD NOMAD:2001xxt ; NOMAD:2003mqg  and
CHORUS CHORUS:1997wxi ; CHORUS:2007wlo  and the long-baseline experiments T2K
T2K:2021xwb , NOvA NOVA2020  and, most importantly, MINOS/MINOS+ MINOS:2017cae
.

NU parameters $|\alpha_{21}|$ $|\alpha_{31}|$ $|\alpha_{32}|$ $\alpha_{11}$
$\alpha_{22}$ $\alpha_{33}$ 3$\sigma$ bounds $<0.025$ $<0.077$ $<0.020$
$>0.929$ $>0.987$ $>0.715$

Table 3:  Current neutrino constraints on the nonunitary parameters from Ref.
Forero:2021azc .

##  V Results

In this section we discuss in detail the numerical findings of our analyses,
where we explore the sensitivity of the ESSnuSB facility to the nonunitary
neutrino mixing, as well as the impact of nonunitarity on the measurement of
the standard three-neutrino oscillation parameters, with emphasis on the CP-
violating phase, $\delta_{\rm CP}$.

###  V.1 Probing nonunitary neutrino mixing at ESSnuSB

Figure 2: ESSnuSB sensitivity to the nonunitarity scenario in the
$(|\alpha_{21}|,\,\delta_{CP})$ plane. Left, middle, and right panels
correspond to 540, 360, and 200 km baselines, respectively. Contours are shown
at $1\sigma$, $2\sigma$, and $3\sigma$ for 2 d.o.f. The red vertical lines
indicate the current $3\sigma$ upper limit on $|\alpha_{21}|$ from neutrino
data. We have assumed $\delta_{\rm CP}(\rm true)=-90^{\circ}$ and normal mass
ordering. The magenta star marked as benchmark in each panel has been used
later to produce dashed spectra in the upper panel of Fig. 3.

We start our discussion from Fig. 2, where we show the ESSnuSB sensitivity to
nonunitarity in the $(|\alpha_{21}|,\,\delta_{\rm CP})$ plane. Left, middle
and right panels show the results for 540, 360, and 200 km baselines,
respectively. The light, medium, and dark green contours in each panel
correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ allowed regions for 2
degrees of freedom (d.o.f.) i.e., $\Delta\chi^{2}=2.3,\,6.18,$ and $11.83$,
respectively. In this figure we have assumed the standard unitary framework as
the true hypothesis, and then we have fitted the nonunitary hypothesis against
it. Normal mass ordering has been assumed all over the analyses. The true data
have been generated assuming the benchmark choices of the standard unitary
oscillation parameters in Table 1 with $\delta_{\rm CP}(\rm
true)=-90^{\circ}$, while for the reconstruction we have fixed the solar
oscillation parameters and marginalized over $\theta_{13}$, $\theta_{23}$, and
$\Delta m_{31}^{2}$. In addition, we have also marginalized over the NU
parameters $\alpha_{11}$, $\alpha_{22}$, and $\alpha_{33}$ within their
allowed $3\sigma$ ranges as given in Table 3. We have also freely varied
$\phi_{21}$ from $-\pi$ to $+\pi$. We have assumed zero values of the other
non-diagonal NU parameters, $|\alpha_{31}|$ and $|\alpha_{32}|$. The red
vertical lines in each panel indicate the current $3\sigma$ upper limit on
$|\alpha_{21}|$ from neutrino data. One sees that the 200 km baseline gives
the best $1\sigma$, $2\sigma$, and $3\sigma$ sensitivities on $|\alpha_{21}|$,
in comparison to the $360$ km and $540$ km baseline option. Quantitatively,
the attainable upper limits for $|\alpha_{21}|$ at $1\sigma$ are
$0.044,\,0.024,\,\rm{and}\,\,0.02$ for 540 km, 360 km, and 200 km baselines,
respectively. Conversely, we have checked that, there is basically no
sensitivity to the new CP phase $\phi_{21}$ for any of the baselines. On the
other hand, the measurement on $\delta_{\rm CP}$ is not affected much by the
presence of nonunitarity. One can also see that the uncertainty on the
measurement of $\delta_{\rm CP}$ is the lowest for 200 km and the highest for
540 km.

Figure 3:  Upper panels: $\nu_{\mu}\to\nu_{e}$ and
$\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ appearance event-spectra as a function of
reconstructed neutrino energy at ESSnuSB for different baselines, as
indicated. Solid lines represent the standard three-neutrino spectra generated
with the standard oscillation parameter values from Table 1 with $\delta_{\rm
CP}=-90^{\circ}$ and dashed lines correspond to the NU scenario taking the
optimized values of the standard oscillation parameters (not shown explicitly)
and the new physics nonunitarity parameters from Table 4. The lower panels
represent the absolute difference in the appearance events between the
standard unitary (UN) and the NU scenarios divided by the statistical
uncertainty in each bin.

baseline (km) $\alpha_{11}$ $\alpha_{22}$ $\alpha_{33}$ $\phi_{21}$
$|\alpha_{31}|$ $|\alpha_{32}|$ $540$ $0.95$ $0.99$ $0.73$ $155^{\circ}$ $0$
$0$ $360$ $0.94$ $0.99$ $0.73$ $140^{\circ}$ $0$ $0$ $200$ $0.94$ $0.99$
$0.96$ $75^{\circ}$ $0$ $0$

Table 4: Nonunitarity parameters used for obtaining the nonunitary spectra
(dashed line) in Fig. 3, see text for more details.

To understand better the previous result one can select a point allowed at
$1\sigma$, $2\sigma$, and $3\sigma$ for a $540$ km, $360$ km, and $200$ km
baseline, respectively. We show in Fig. 3 the appearance event spectra for the
neutrino (upper-left panel) and antineutrino mode (upper-right panel). For
each of the three different baselines we choose a benchmark point (magenta
star in Fig. 2) given by the NU parameter $|\alpha_{21}|=0.04$ and the
standard CP phase $\delta_{\rm CP}=-90^{\circ}$. For this benchmark choice,
the other NU parameters arising from the $\chi^{2}$ marginalization take the
values shown in Table 4. Note that the optimized values of the standard
oscillation parameters are not shown here explicitly. We use all of them as
input to generate the event spectra in Fig. 3. These are shown as dashed
lines, while the unitary scenario is represented with solid lines. In order to
better compare the standard unitary (UN) with the NU case, we show in the
lower panels of Fig. 3 the absolute differences of the number of events
divided by the statistical uncertainty in each bin i.e.,
$\rm\rm\Delta=|N_{UN}-N_{NU}|/\sqrt{N_{UN}}$. This provides a crude
measurement of the statistical significance of our unitarity test. One sees
that the best sensitivities to the standard parameter $\delta_{\rm CP}$ and
the NU parameter $|\alpha_{21}|$ are achieved for the $200$ km baseline, as
already shown in Fig. 2. Notice also that the $360$ km baseline does somewhat
better than the $540$ km baseline. This is also reflected in the left panels
of Fig. 4.

Figure 4:  Upper panels: one dimensional $\Delta\chi^{2}$ projection on the NU
parameters $|\alpha_{21}|$ (left), $\alpha_{11}$ (middle), and $\alpha_{22}$
(right panel) for the three different baselines considered for ESSnuSB (red,
blue and green lines, see the legend) and for DUNE (black line). Lower panels:
combined sensitivities for DUNE and ESSnuSB for the three baselines
considered. Normal mass ordering has been assumed in the analysis.

In Fig. 4, we show the one-dimensional projection of the $\Delta\chi^{2}$ as a
function of the NU parameters $|\alpha_{21}|$ (left panel), $\alpha_{11}$
(middle panel), and $\alpha_{22}$ (right panel). The red, blue, and green
curves in the upper panels show the sensitivities corresponding to the three
ESSnuSB baselines, 540 km, 360 km, and 200 km, respectively, while the black
curve corresponds to the DUNE sensitivity. In the lower panels, the same color
code is used to represent the sensitivities for the combined analysis of DUNE
and the ESSnuSB for baselines of 540, 360, and 200 km.

As before, this figure is obtained assuming the standard unitary framework as
the true hypothesis, and testing the NU framework against it, i.e.
$\Delta\chi^{2}=\chi^{2}_{\rm NU}-\chi^{2}_{\rm UN}$. The true values of the
oscillation parameters are taken from Table 1, fixing the two solar parameters
$\theta_{12}$ and $\Delta m_{21}^{2}$ at their benchmark choices, and
marginalizing over $\theta_{13}$, $\theta_{23}$, and $\Delta m_{31}^{2}$.
Since the exact value of the standard CP phase $\delta_{\rm CP}$ is currently
not accurately known, we have marginalized over its true and test values
within its full range.

For the sensitivity to $|\alpha_{21}|$, shown in the left panels of Fig. 4, we
have freely varied the NU parameters $\alpha_{11}$, $\alpha_{22}$, and
$\alpha_{33}$ within their allowed $3\sigma$ ranges as given in Table 3 and
the phase $\phi_{21}$ from $-\pi$ to $+\pi$. The other non-diagonal NU-
parameters $|\alpha_{31}|$ and $|\alpha_{32}|$ have been set to zero. From the
upper left panel we can see that the best sensitivity for the ESSnuSB comes
from the 200 km baseline, followed by the 360 km and 540 km baselines. The
expected $3\sigma$ upper limits on $|\alpha_{21}|$ corresponding to 540 km,
360 km, and 200 km baselines are 0.16, 0.075, and 0.04, respectively. These
limits would be independent and complementary to those given in Table 3 and
illustrate the ESSnuSB potential in probing new physics. For the case of DUNE,
the upper limit at $3\sigma$ would be 0.046. One sees the sensitivities
expected at ESSnuSB are competitive and, for the smaller baseline, it performs
slightly better than DUNE Escrihuela:2016ube . In the lower left panel of Fig.
4, we show how the combined sensitivities of DUNE and ESSnuSB, improve with
respect to the individual sensitivities. The $3\sigma$ upper limits
corresponding to the DUNE+ESS540, DUNE+ESS360, and DUNE+ESS200 combinations
would be 0.04, 0.03, and 0.022, respectively. It is encouraging to see that
the combination of DUNE and the 200 km baseline of ESSnuSB would be able to
set a constraint on $|\alpha_{21}|$ which is better than the current $3\sigma$
upper bound given in Table 3.

The middle panel of Fig. 4 shows the sensitivities to the diagonal NU
parameter $\alpha_{11}$. The analysis method is very similar to the previous
one as far as the standard unitarity-parameters are concerned. However, for
the NU parameters, we marginalize over the parameters $\alpha_{22}$,
$|\alpha_{21}|$, and its associated new CP-phase $\phi_{21}$. As before, the
off-diagonal parameters $\alpha_{31}$ and $\alpha_{32}$ have been set to zero
and $\alpha_{33}$ was set to unity. One can see that the $3\sigma$
sensitivities on $\alpha_{11}$ corresponding to the 540, 360, and 200 km
baselines are lower than for DUNE, i.e. $0.87$ versus $0.95$. From the lower
middle panel one sees that the combined sensitivities corresponding to
DUNE+ESS540, DUNE+ESS360, and DUNE+ESS200 are very similar and approximately
equal to $0.955$, which would somewhat improve the current sensitivity on
$\alpha_{11}$.

The sensitivities on $\alpha_{22}$ are shown in the right panels of Fig. 4. In
this case, we follow exactly the same steps as in the $\alpha_{11}$ analysis
but marginalizing over $\alpha_{11}$. Our results show that the best
performance is expected from DUNE, followed by the ESSnuSB with baselines 360
km, 540 km, and 200 km. The expected sensitivities for the individual setups
do not improve over the current lower bound. However, one sees that the
combined sensitivity of DUNE and any ESSnuSB baseline gives a slightly lower
sensitivity compared to the current $3\sigma$ lower bound. One concludes that
all these sensitivities, specially those coming from the combined analyses,
are encouraging and will play an important role especially when combined with
the already available data. For the sake of convenience we quote all our
bounds as Table 5. Note also that these NU parameter sensitivities at the
ESSnuSB may be complemented by additional information coming from future
experiments such as T2HK or JUNO Dutta:2019hmb ; JUNO:2015zny .

3$\sigma$ sensitivity |  $|\alpha_{21}|$ |  $\alpha_{11}$ |  $\alpha_{22}$
---|---|---|---
ESS540 | $<0.160$ | $>0.850$ | $>0.964$
ESS360 | $<0.075$ | $>0.860$ | $>0.973$
ESS200 | $<0.040$ | $>0.870$ | $>0.967$
DUNE | $<0.046$ | $>0.945$ | $>0.975$
DUNE+ESS540 | $<0.040$ | $>0.951$ | $>0.981$
DUNE+ESS360 | $<0.030$ | $>0.952$ | $>0.982$
DUNE+ESS200 | $<0.022$ | $>0.954$ | $>0.979$
Table 5: ESSnuSB, DUNE, and combined $3\sigma$ sensitivities on the NU
parameters $|\alpha_{21}|$, $\alpha_{11}$, and $\alpha_{22}$ obtained in this
work.

###  V.2 CP violation discovery potential

Now we turn back to the “conventional” CP violation discovery potential within
our generalized nonunitary framework. The CP violation (CPV) discovery
potential of the ESSnuSB and DUNE setup is summarized in Figs. 5, 6 and 7. Our
results are given both for the unitary and the nonunitary framework. As in the
standard $\delta_{\rm CP}$ sensitivity study DUNE:2015lol , the CP-violating
hypothesis is tested against a CP-conserving scenario through
Escrihuela:2016ube

$\displaystyle\Delta\chi^{2}(\delta_{\rm CP}^{\rm true})=$
$\displaystyle\rm{Min}\left[\Delta\chi^{2}\left(\delta_{\rm CP}^{\rm
true},\delta_{\rm CP}^{\rm test}=0\right),\,\,\Delta\chi^{2}\left(\delta_{\rm
CP}^{\rm true},\delta_{\rm CP}^{\rm test}=\pm\pi\right)\right].$ (20)

This way, we obtain the significance with which one can reject the test
hypothesis of no CP violation. We have assumed five nonzero NU parameters: the
three diagonal ones (marginalized over the true and test values), plus one
non-diagonal parameter, either $|\alpha_{21}|$ (left panels) or
$|\alpha_{31}|$ (right panels), with the associated complex CP phase
($\phi_{21}$ or $\phi_{31}$), also marginalized. Concerning the standard
three-neutrino parameters (both in the unitary and nonunitary scenarios), we
have marginalized over the two mixing angles $\theta_{13}$ and $\theta_{23}$
and the “atmospheric” mass-squared splitting $\Delta m_{31}^{2}$. The left
(right) column of Fig. 5 represents the CPV discovery potential of ESSnuSB for
the three baseline choices and three different values of the NU parameter
$|\alpha_{21}|$ ($|\alpha_{31}|$). The black solid line in each plot
corresponds to the standard three-neutrino unitary framework. Upper, middle,
and lower panels represent the results obtained for 540, 360, and 200 km
baselines, respectively. The red, green, and blue dashed curves in the left
plots represent the CPV discovery sensitivities in the NU framework
corresponding to different values of $|\alpha_{21}|$,
$0.01,\,0.02,\,\rm{and}\,0.03$, respectively (the latter, relatively large
value, is taken mainly for comparison). One sees that the CPV sensitivity in
the standard unitary neutrino oscillation framework always lies around
$8\sigma$ for $\delta_{\rm CP}(\rm true)=\pm 90^{\circ}$ for all three
baselines, see the solid black line in all the panels. This agrees with the
results presented in Refs. Baussan:2013zcy ; Agarwalla:2014tpa ;
Alekou:2021coj . All baselines have more or less similar sensitivities, except
for the fact that the $\delta_{\rm CP}$ range over which CPV can be
established for 540 km and 360 km is slightly bigger than for 200 km. This
fact is also confirmed in Ref. Baussan:2013zcy . However, we will see the
merits of the 200 km baseline in what follows. As far as the NU framework is
concerned, two of our benchmark values, $|\alpha_{21}|=0.01,\,\rm{and}\,0.02$
lie within the current $3\sigma$ limit, whereas $|\alpha_{21}|=0.03$ lies
slightly outside the current allowed limit. 444As already mentioned, the line
with 0.03 is included mainly for comparison, to have a broader view of
degrading behaviour with respect to the non-diagonal NU parameters..

Figure 5: CPV sensitivities at ESSnuSB for various baselines, as indicated.
The solid black lines correspond to unitary neutrino mixing, while dashed
lines show the sensitivities in the presence of unitarity violation. The value
of $|\alpha_{21}|$ ($|\alpha_{31}|$) has been fixed in the left (right) panel
in the data as well as in the theory. We have marginalized over $\theta_{13}$,
$\theta_{23}$, $\Delta m_{31}^{2}$, as well as over the true and test values
of $\alpha_{11}$, $\alpha_{22}$, $\alpha_{33}$, and $\phi_{21}$ (in the left
panel) and $\phi_{31}$ (in the right panel), within their allowed ranges.

We stress that the CPV discovery sensitivity is degraded with respect to the
unitary case. This is expected, due to the presence of new phases associated
to unitarity violation Miranda:2016wdr . Clearly, the CPV discovery
sensitivity decreases with the increasing values of $|\alpha_{21}|$, specially
for 540 km, leading to a minimum sensitivity of $4.6\sigma$ for $\delta_{\rm
CP}(\rm true)=\pm 90^{\circ}$. The deterioration of the CPV sensitivity is
smaller for 360 km and, with a minimum sensitivity of around $5.7\sigma$ for
$\delta_{\rm CP}(\rm true)=\pm 90^{\circ}$. For the 200 km baseline, the
deterioration further reduces, with a minimum sensitivity of $6.1\sigma$ for
all three benchmark choices, at $\delta_{\rm CP}(\rm true)=\pm 90^{\circ}$.
All in all, one sees that the degrading in CP sensitivity is not as large as
one might expect, showing the robustness of the oscillation picture with
respect to unitarity violation. The best sensitivities to $\delta_{\rm CP}$
and the nonunitary parameter $|\alpha_{21}|$ are achieved for a 200 km
baseline.

We now turn to the right panels of Fig. 5. There, we repeated the same
analysis for the non-diagonal parameter $|\alpha_{31}|$ and its associated CP
phase, $\phi_{31}$. The three benchmark choices considered for $|\alpha_{31}|$
are $0.03,\,0.07,\,\rm{and},\,0.10$555Note that, as in the previous case, the
third value of $|\alpha_{31}|$ lies slightly outside the 3$\sigma$ current
limit, and is taken only for comparison.. In this case, one finds a mild
deterioration of the CPV sensitivity in comparison to the unitary framework
for all baselines, so the impact of $|\alpha_{31}|$ is not significant for 540
km, and negligible for 360 km and 200 km baselines. This is due to the fact
that $|\alpha_{31}|$ does not appear in the vacuum appearance probability, Eq.
(16), and also because of the lower matter effects for ESSnuSB with respect to
DUNE Escrihuela:2016ube ; DUNE:2020fgq . As expected, one finds a negligible
impact on the CPV sensitivity arising from unitarity violation in this case.
Note, however that this will not be the case for DUNE and, as a result, the
combined analysis with ESSnuSB will be useful to recover the maximum CPV
sensitivity in this case.

Figure 6: CPV sensitivities at DUNE. Conventions and analysis procedure are
the same as in Fig. 5, see its caption for details.

Next, in Fig. 6, we show the CPV sensitivities for DUNE. As before, the black
solid lines in each panel correspond to the CPV sensitivity in the standard
unitary framework, showing that DUNE can establish CPV discovery above
$5\sigma$ for $\delta_{\rm CP}\,(\rm true)=\pm 90^{\circ}$. The red, green,
and blue dashed curves in the left panel represent the CPV discovery
sensitivities in the NU framework corresponding to different values of
$|\alpha_{21}|$: $0.01,\,0.02,\,\rm{and}\,0.03$, respectively. Similarly, in
the right panel, these lines correspond to the CPV sensitivities for
$|\alpha_{31}|$ equal to $0.03,\,0.07,\,\rm{and}\,0.10$. The simulation method
followed in both the panels is the same as for Fig. 5. From the left panel, we
see that the CPV sensitivity in the presence of NU is degraded with respect to
the unitary framework, according to the value of $|\alpha_{21}|$. As a result,
for the maximal values of $\delta_{\rm CP}$(true) one can only ensure a
minimum CPV sensitivity of $3.6\sigma$. Similarly, from the right panel, we
can see that the CPV sensitivity decreases with increasing $|\alpha_{31}|$
with a minimum sensitivity of $3.6\sigma$ for $\delta_{\rm CP}\,(\rm true)=\pm
90^{\circ}$.

Comparing Figs. 5 and 6 one sees that, for the conventional unitary scenario
the CPV sensitivity of ESSnuSB is better than that of DUNE. Also the
deterioration due to NU effects is smaller than for DUNE, where the matter
effect is large. Hence, in this case, ESSnuSB will play a complementary role
to DUNE in achieving a robust CP violation sensitivity in the presence of
nonunitarity.

Figure 7:  CPV sensitivities at ESSnuSB for the three baselines under
consideration and for DUNE. The red, green, and blue dashed lines in each
panel correspond to the CPV sensitivities for the fixed values of
$\alpha_{11}=0.94$, $\alpha_{22}=0.99$, and $\alpha_{33}=0.90$, respectively,
in the data as well as in the theory. For simulation details, see the text.

So far we have analyzed the effect of the non-diagonal NU parameters on the
CPV sensitivity of the experimental setups under consideration. Now we will
investigate the role of the diagonal nonunitary parameters, $\alpha_{11}$,
$\alpha_{22}$, and $\alpha_{33}$. The corresponding CPV sensitivities in this
case for ESSnuSB and DUNE are shown in the four panels of Fig. 7. The results
for the standard unitary case (solid black line) for both the ESSnuSB and DUNE
have been already described in Fig. 5 and Fig. 6. The red, green, and blue
dashed lines in each panel correspond to the CPV sensitivities for fixed
values of $\alpha_{11}=0.94$, $\alpha_{22}=0.99$, and $\alpha_{33}=0.90$,
respectively, in the data as well as in the theory. The procedure we have
followed for this analysis is slightly different from what we have used for
the non-diagonal parameters. In order to evaluate the CPV sensitivity in the
NU framework for $\alpha_{11}=0.94$, we have marginalized over $\alpha_{22}$,
$|\alpha_{21}|$, and $\phi_{21}$ within their allowed $3\sigma$ ranges. All
other off-diagonal parameters are assumed to be zero, and we have also fixed
$\alpha_{33}=1$ in the data as well as in the theory. The analysis method for
$\alpha_{22}$ is analogous to the previous one, where we have marginalized
over $\alpha_{11}$, $|\alpha_{21}|$, and $\phi_{21}$. On the other hand for
$\alpha_{33}$, we marginalized over $\alpha_{11}$, $|\alpha_{31}|$, and
$\phi_{31}$, fixing $\alpha_{22}=1$.

One sees that ESSnuSB performs better than DUNE for the overall CPV discovery
sensitivity. Of course, in all cases these sensitivities are degraded
substantially with respect to those of the unitary case. In fact, the CPV
discovery for DUNE falls below $4\sigma$ for any value of $\delta_{\rm
CP}\,(\rm true)$. As seen in Fig. 7 the best result is obtained for the 200 km
baseline, followed by 360 km of ESSnuSB.

In short, the ESSnuSB will play a crucial role in securing a robust CPV
discovery sensitivity in the presence of unitarity deviations close to the
current upper bound.

###  V.3 CP reconstruction

In our simplest scenario there are two relevant CP phases, the standard three-
neutrino Dirac phase $\delta_{\rm CP}$ and the phase $\phi_{21}$ associated to
nonunitarity 666Note that $|\alpha_{31}|$ and $|\alpha_{32}|$ enter only in
the appearance neutrino probability through matter effects, strongly
suppressing the sensitivity to the associated phases $\phi_{31}$ and
$\phi_{32}$.. One can therefore have four “CP conserving” cases, when either
of them equals 0 or $\pi$. Likewise, four cases in which one has “maximal” CP
violation, defined by having the modulus of any of them equal to $\pi/2$. In
this section, we discuss how well the European spallation source setups can
reconstruct the standard CP phase $\delta_{\rm CP}$ as well as the
nonunitarity phase $\phi_{21}$ for a few selected benchmarks.

The results of our analysis are shown in Fig. 8. The upper two panels
correspond to the two CP conserving cases $(0,0)$ and $(\pi,\pi)$, and the
lower two panels to the two CP violating scenarios $(-\pi/2,-\pi/2)$ and
$(\pi/2,\pi/2)$. The red, green, and cyan contours in each panel correspond to
540, 360, and 200 km baselines of the ESSnuSB experiment, respectively,
whereas the orange contours represent the sensitivity expected in DUNE. All
contours correspond to the $2\sigma$ level for 2 d.o.f. For this analysis, we
have fixed $|\alpha_{21}|=0.02$, which lies within the current $3\sigma$
allowed boundary. All the other off-diagonal NU parameters have been kept
fixed to zero. We have marginalized over the mixing angles $\theta_{13}$ and
$\theta_{23}$ and the atmospheric mass-squared splitting, $\Delta m_{31}^{2}$,
as well as over the true and test values of the NU parameters $\alpha_{11}$,
$\alpha_{22}$, $\alpha_{33}$ within their allowed ranges.

We have checked that the expected $1\sigma$ uncertainties on $\delta_{\rm CP}$
($\phi_{21}$) for the CP-violating scenarios are $16^{\circ}$ ($100^{\circ}$)
for 540 km, and $13^{\circ}$ ($70^{\circ}$) for 360 km. For the CP-conserving
benchmarks, the uncertainties on $\delta_{\rm CP}$ are $10^{\circ}$ for 360
km, and $12^{\circ}$ for 540 km respectively, whereas no sensitivity on
$\phi_{21}$ is found for these two baselines. On the other hand, for the 200
km baseline, the typical $1\sigma$ level uncertainty on $\delta_{\rm CP}$
($\phi_{21}$) is $10^{\circ}$ ($40^{\circ}$). One sees that the best
performance would be obtained for the shortest baseline, 200 km. For
comparison, we have also projected the sensitivity of the DUNE experiment in
the same plot. For DUNE, the typical $1\sigma$ level uncertainty on the
reconstructed CP phase $\delta_{\rm CP}$ ($\phi_{21}$) is $21^{\circ}$
($45^{\circ}$).

In short, one can see that the $\delta_{\rm CP}$ reconstruction capability of
ESSnuSB does not get too much impaired by the presence of unitarity violation,
while the sensitivity to the NU phase $\phi_{21}$ is competitive with that in
DUNE and even better for 200 km baseline. Certainly, a future combined
analysis of the actual results of these experiments would improve the standard
and new CP-phase reconstruction sensitivities.

Figure 8:  CP reconstruction in the plane $\left[\delta_{\rm
CP},\,\phi_{21}\right]$ (test) for different ESSnuSB baselines at $2\sigma$
level for 2 d.o.f. We have fixed $|\alpha_{21}|=0.02$ and normal mass ordering
in the data as well as in the theory. The red, green, and cyan contours
correspond to 540, 360, 200 km baselines of the ESSnuSB experiment,
respectively, whereas the orange contours denote the sensitivity expected in
DUNE.

##  VI Summary and outlook

Here we have explored the physics potential of the proposed European
Spallation Source facility in the presence of nonunitarity of the lepton
mixing matrix, as generally expected within the seesaw paradigm. We have also
explored the DUNE physics potential in this framework as well as the combined
result of both DUNE and ESSnuSB experiments. First, we presented in detail the
theory framework of neutrino oscillations with effective nonunitary neutrino
mixing, discussing in Fig. 1 the resulting neutrino and antineutrino
appearance oscillation probabilities. Throughout the paper we have assumed
normal mass ordering, and considered three reference baseline choices of 540,
360, and 200 km for the ESSnuSB setup. In Fig. 2 we have presented the
sensitivity contours in the $(|\alpha_{21}|,\,\delta_{\rm CP})$ plane. The
promising results for the 200 km baseline were understood in terms of the
expected $\nu_{\mu}\to\nu_{e}$ and $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$
appearance event spectra. These are given in Fig. 3 as a function of the
reconstructed neutrino energy. We found encouraging ESSnuSB and DUNE
sensitivities for the off-diagonal NU parameter $|\alpha_{21}|$, and the two
diagonal NU parameters $\alpha_{11}$ and $\alpha_{22}$, as seen in Fig. 4. One
also sees how the combined analysis of DUNE and ESSnuSB can help improving the
sensitivity on the above mentioned NU parameters beyond the current $3\sigma$
bounds 777The results we have obtained are also competitive with the
sensitivity expected at future coherent elastic neutrino-nucleus scattering
experiments Miranda:2020syh ..

More remarkable, perhaps, is the ESSnuSB CPV discovery potential, which is
better than that of DUNE, as illustrated in Figs. 5, 6, and 7. One appreciates
also a milder degrading of the ESSnuSB CP violation sensitivities with respect
to the standard unitary mixing scenario in comparison with DUNE, where the
effect is stronger. ESSnuSB would therefore contribute to establishing the
robustness of CP determination against small departures from unitarity
arising, say, from the seesaw mechanism. Finally, we have also obtained a
promising CP reconstruction potential, both for the standard CP phase of the
three-neutrino paradigm, as well as for the phase associated to nonunitarity,
see Fig. 8. Altogether, within the generalized nonunitary neutrino mixing
framework, we have found that the proposed ESSnuSB facility is competitive and
complementary to DUNE, not only for leptonic CP violation studies, but also
for probing new physics parameters associated to unitarity violation.

### ACKNOWLEDGMENTS

This work is supported by Spanish grants Agencia Estatal de Investigación
under grant no. PID2020-113775GB-I00 (AEI/10.13039/501100011033) and Prometeo
CIPROM/2021/054 (Generalitat Valenciana), in part by the European Union
Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-
Curie grant agreement No. 860881-HIDDeN, and by the LabEx P2IO
(ANR-10-LABX-0038 - Project “BSMNu”) in the framework of the “Investissements
d’Avenir” (ANR-11-IDEX-0003-01 ) managed by the Agence Nationale de la
Recherche (ANR), France, and by CONACYT-Mexico under grant A1-S-23238. O. G.
M. has been supported by SNI (Sistema Nacional de Investigadores, Mexico).

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parameters.

  2. 2.

The winning state paths corresponding to $\mathcal{C}_{+}$ and
$\mathcal{C}_{-}$ are

$\displaystyle\tau_{\mathcal{C}_{+}}=(\ket{0},\ket{+},\ket{0})\quad\text{and}\quad\tau_{\mathcal{C}_{-}}=(\ket{0},\ket{-},\ket{0})\
.$ (A.36)

  3. 3.

Picard has no winning strategy.

###### Proof of Theorem 5.9.

  1. 1.

The two classes of winning strategies of Q, $\mathcal{C}_{+}$ and
$\mathcal{C}_{-}$, which were establish by Theorem 5.2, are still present in
$U(2)$. However, they now contain infinitely many equivalent strategies. To
see why this is so, let us consider a winning strategy $\sigma=(A_{1},A_{2})$
in $\mathcal{C}_{+}$. We may associate to this strategy the collection of
infinitely many strategies $(A_{1}(\theta_{1}),A_{2}(\theta_{2}))$, where
$\theta_{1},\theta_{2}\in\mathbb{R}$. Every strategy in this collection of
strategies is equivalent to $(A_{1},A_{2})$ because the action of every
operator $A\in U(2)$ on a ket $\ket{\psi}$ is the same as the action of
$e^{i\theta}A\in U(2)$ on $\ket{\psi}$. The same holds for every strategy in
$\mathcal{C}_{-}$. Hence, we may conclude that in $U(2)$ the two classes
$\mathcal{C}_{+}$ and $\mathcal{C}_{-}$ contain infinitely many equivalent
strategies.

Let us assume that there exists a third class $\mathcal{C}^{\prime}$ and
consider a strategy $\sigma=(A_{1},A_{2})$ in this class. Then the action of
$A_{1}$ must drive the coin into some state other than $\ket{+}$ or $\ket{-}$.
However, (A.9) asserts that $A_{1}\ket{0}\in Fix(\\{F\\})$, which, in view of
(A.34), implies that $A_{1}\ket{0}\in\\{\ket{+},\ket{-}\\}$, a contradiction.
Consequently, there are just two classes of winning strategies
$\mathcal{C}_{+}$ and $\mathcal{C}_{-}$.

It remains to prove that $\mathcal{C}_{+}$ and $\mathcal{C}_{-}$ do not
contain any new winning strategy that are not of the form stated in (A.35). To
arrive at a contradiction, let us suppose that $\sigma=(A_{1},A_{2})$ is a
“new winning” strategy. If $A_{1}$
$=\begin{bmatrix}\begin{array}[]{lr}z_{1}&w_{1}\\\
z_{2}&w_{2}\end{array}\end{bmatrix}$, then its columns form an orthonormal
basis for $\mathcal{H}_{2}$ because it is a unitary operator. This means that
its rows and columns satisfy the following relations:

$\displaystyle z_{1}^{*}z_{1}+z_{2}^{*}z_{2}$ $\displaystyle=1\ ,$ ( 5.9.i )
$\displaystyle w_{1}^{*}w_{1}+w_{2}^{*}w_{2}$ $\displaystyle=1\
,\quad\text{and}$ ( 5.9.ii ) $\displaystyle z_{1}^{*}w_{1}+z_{2}^{*}w_{2}$
$\displaystyle=0\ .$ ( 5.9.iii )

First, we investigate the possibility that $A_{1}$ sends the coin to state
$\ket{+}$, assuming that $A_{1}$ is other than
$H(\theta_{1}),R_{\frac{2\pi}{8}}(\theta_{1}),S_{\frac{5\pi}{8}}(\theta_{1})$
or $R_{\frac{10\pi}{8}}(\theta_{1})$. In this case,

$\displaystyle\begin{bmatrix}\begin{array}[]{lr}z_{1}&w_{1}\\\
z_{2}&w_{2}\end{array}\end{bmatrix}\begin{bmatrix}\begin{array}[]{lr}1\\\
0\end{array}\end{bmatrix}=e^{i\theta_{1}}\begin{bmatrix}\begin{array}[]{lr}\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}\end{array}\end{bmatrix}\Rightarrow\begin{bmatrix}\begin{array}[]{lr}z_{1}\\\
z_{2}\end{array}\end{bmatrix}=e^{i\theta_{1}}\begin{bmatrix}\begin{array}[]{lr}\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}\end{array}\end{bmatrix}\ ,\ \text{for \ some}\
\theta_{1}\in\mathbb{R}\ .$ ( 5.9.iv )

If we combine (5.9.iv) with (5.9.iii) we derive that

$\displaystyle
z_{1}^{*}w_{1}+z_{2}^{*}w_{2}=0\Rightarrow\frac{e^{-i\theta_{1}}}{\sqrt{2}}(w_{1}+w_{2})=0\Rightarrow
w_{2}=-w_{1}\ .$ ( 5.9.v )

In view of (5.9.v), (5.9.iii) becomes

$\displaystyle 2|w_{1}|^{2}=1\Rightarrow|w_{1}|=\frac{1}{\sqrt{2}}\ .$ (
5.9.vi )

All complex numbers of the form $e^{i\varphi}\frac{1}{\sqrt{2}}$, where
$\varphi\in\mathbb{R}$, are solutions of the equation (5.9.vi). We may
therefore choose $\varphi=\theta_{1}$ and set
$w_{1}=\frac{e^{i\theta_{1}}}{\sqrt{2}}$, in which case, $w_{2}$ becomes
$-\frac{e^{i\theta_{1}}}{\sqrt{2}}$. Hence, $A_{1}$ is in fact
$e^{i\theta_{1}}\begin{bmatrix}\begin{array}[]{lr}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{array}\end{bmatrix}$, which means
that $A_{1}$ is one of
$H(\theta_{1}),R_{\frac{2\pi}{8}}(\theta_{1}),S_{\frac{5\pi}{8}}(\theta_{1})$
or $R_{\frac{10\pi}{8}}(\theta_{1})$, in stark contrast to our initial
assumption.

We arrive at similar contradictions if we assume that $A_{1}$ drives the coin
to state $\ket{-}$. Thus, we conclude that, other than the strategies
described by (A.35), there are no more winning strategies for Q.

  2. 2.

Based on the above analysis it is straightforward to see that (A.36) holds.

  3. 3.

By Definition 2.1, Picard has no winning strategy because if Q employs one of
his winning strategies, Picard has $0.0$ probability to win the game.

∎

###  A.4 Proofs for Section 6

We now prove the important Theorem 6.1.

######  Theorem 6.1 (Picard lacks a winning strategy).

Picard does not have a winning strategy in any $n$-round game, $n\geq 2$, as
long as Q makes at least one move.

###### Proof of Theorem 6.1.

Let us first note that according to Definition 6.1 and our assumptions at the
beginning of Section 6, in every $n$-round game with $n\geq 2$, Q makes at
least one move. As a matter of fact, any $n$-round game has one of the
following forms.

  1. 1.

$(Q,P,\dots,Q,P)$, in which case if Q employs the strategy $(H,I,\dots,I)$,
the state of the coin prior to measurement will either be $\ket{+}$, if the
initial state of the coin is $\ket{0}$, or $\ket{-}$, if the initial state of
the coin is $\ket{1}$. This is because $Fix(\\{I,F\\})=\\{\ket{+},\ket{-}\\}$,
so the coin will stay in one of these states no matter which strategy Picard
uses. When the coin is measured, Picard will have exactly $0.5$ probability to
win irrespective of which is his target state. Thus, Picard does not possess a
winning strategy because, by Definition 2.1, a winning strategy means that he
wins the game with probability $1.0$.

  2. 2.

$(P,Q,\dots,P,Q)$, where again, if the same strategy $(H,I,\dots,I)$ is used
by Q, will prevent Picard from surely winning the game. The coin will be in
one of its basis states (which one depends on the initial state and Picard’s
first move) when Q acts on it for the first time. His action will drive the
coin to state $\ket{+}$ (if the coin was at state $\ket{0}$) or $\ket{-}$ (if
the coin was at state $\ket{1}$). This will be the state of the coin prior to
measurement no matter which strategy Picard uses because
$Fix(\\{I,F\\})=\\{\ket{+},\ket{-}\\}$. When the coin is measured, Picard will
have exactly $0.5$ probability to win irrespective of which is his target
state. Hence, Picard does not have a winning strategy because, by Definition
2.1, a winning strategy means that he wins the game with probability $1.0$.

  3. 3.

$(Q,P,\dots,Q,P,Q)$, in which case Q has a winning strategy. If the initial
state is the same as Q’s target state, then Q’s winning strategy is
$(H,I,\dots,I,H)$; if it is different then Q’s winning strategy is
$(H,I,\dots,I,FH)$. In this case Picard has precisely $0.0$ probability to
win, so he certainly does not possess a winning strategy.

  4. 4.

$(P,Q,\dots,P,Q,P)$, where once more the strategy $(H,I,\dots,I)$ can be used
by Q to prevent Picard from surely winning the game. The coin will be in one
of its basis states (which one depends on the initial state and Picard’s first
move) when Q acts on it for the first time. His action will drive the coin to
state $\ket{+}$ (if the coin was at state $\ket{0}$) or $\ket{-}$ (if the coin
was at state $\ket{1}$). This will be the state of the coin prior to
measurement no matter which strategy Picard uses because
$Fix(\\{I,F\\})=\\{\ket{+},\ket{-}\\}$. When the coin is measured, Picard will
have exactly $0.5$ probability to win irrespective of which is his target
state. Therefore, Picard does not have a winning strategy because, by
Definition 2.1, a winning strategy means that he wins the game with
probability $1.0$.

∎

The next couple of theorems give important negative results for Q by
specifying the games in which he cannot surely win.

######  Theorem 6.2 (Q lacks a winning strategy when Picard plays last).

Q does not have a winning strategy in any $n$-round game, $n\geq 2$, in which
Picard makes the last move.

###### Proof of Theorem 6.2.

Any $n$-round game in which Picard makes the last move has one of the
following two forms.

  1. 1.

$(Q,P,\dots,Q,P)$, where $n$ is even and both Picard and Q make $\frac{n}{2}$
moves. Let us assume to the contrary that there exists a winning strategy
$\sigma_{Q}=(A_{1},\dots,A_{\frac{n}{2}})$ for Q. According to Definition 2.1,
the fact that $\sigma_{Q}$ is a winning strategy means that for every strategy
of Picard, Q wins the game with probability $1.0$. Since this holds for every
strategy of Picard, it must also hold for the strategies
$\sigma_{P}=(I,\dots,I,I)$ and $\sigma^{\prime}_{P}=(I,\dots,I,F)$. The former
implies that after Q’s last action the coin must be at the basis state
$\ket{q_{Q}}$, whereas the latter implies that after Q’s last action the coin
must be at the opposite basis state, which is absurd. Thus, Q does not possess
a winning strategy.

  2. 2.

$(P,Q,\dots,P,Q,P)$, where $n$ is odd, Q makes $\frac{n}{2}$ moves and Picard
make $\frac{n}{2}+1$ moves. In order to arrive at a contradiction, we assume
to the contrary that there exists a winning strategy
$\sigma_{Q}=(A_{1},\dots,A_{\frac{n}{2}})$ for Q. In view of Definition 2.1,
if Q employs $\sigma_{Q}$, he will win the game with probability $1.0$ no
matter which strategy Picard chooses. If Picard uses
$\sigma_{P}=(I,\dots,I,I)$, then the fact that $\sigma_{Q}$ is a winning
strategy means that after Q’s last action the coin must be at the basis state
$\ket{q_{Q}}$. On the other hand, if Picard uses $\sigma_{P}=(I,\dots,I,F)$,
then the fact that $\sigma_{Q}$ is a winning strategy means that after Q’s
last action the coin must be at the opposite basis state. This contradiction
proves that Q does not possess a winning strategy.

∎

######  Theorem 6.3 (Q lacks a winning strategy when Picard plays first).

Q does not have a winning strategy in any $n$-round game, $n\geq 2$, in which
Picard makes the first move.

###### Proof of Theorem 6.3.

Any $n$-round game in which Picard makes the first move has one of the
following two forms.

  1. 1.

$(P,Q,\dots,P,Q)$, where $n$ is even and both Picard and Q make $\frac{n}{2}$
moves. Let us assume to the contrary that there exists a winning strategy
$\sigma_{Q}=(A_{1},\dots,A_{\frac{n}{2}})$ for Q. According to Definition 2.1,
the fact that $\sigma_{Q}$ is a winning strategy means that for every strategy
of Picard, Q wins the game with probability $1.0$. Since this holds for every
strategy of Picard, it must also hold for the strategies
$\sigma_{P}=(I,\dots,I,I)$ and $\sigma^{\prime}_{P}=(F,\dots,I,I)$. The former
implies that

$\displaystyle A_{\frac{n}{2}}\dots A_{1}\ket{q_{0}}=\ket{q_{Q}}\Rightarrow
C\ket{q_{0}}=\ket{q_{Q}}\ ,$ ( 6.3.i )

where $C=A_{\frac{n}{2}}\dots A_{1}$. Since the composition of unitary
operators produces a unitary operator, we know that $C$ is unitary. On the
other hand, the latter implies that

$\displaystyle A_{\frac{n}{2}}\dots A_{1}F\ket{q_{0}}=\ket{q_{Q}}\Rightarrow
CF\ket{q_{0}}=\ket{q_{Q}}\ .$ ( 6.3.ii )

If $\ket{q_{0}}=\ket{0}$, then $F\ket{q_{0}}=\ket{1}$, whereas if
$\ket{q_{0}}=\ket{1}$, then $F\ket{q_{0}}=\ket{0}$. If we combine this last
result with (6.3.i) and (6.3.ii), we conclude that

$\displaystyle C\ket{0}=C\ket{1}=\ket{q_{Q}}\ ,$ ( 6.3.iii )

which is of course impossible because $C$ is unitary. Thus, Q does not possess
a winning strategy.

  2. 2.

$(P,Q,\dots,P,Q,P)$, where $n$ is odd, Q makes $\frac{n}{2}$ moves and Picard
make $\frac{n}{2}+1$ moves. In order to arrive at a contradiction, we assume
to the contrary that there exists a winning strategy
$\sigma_{Q}=(A_{1},\dots,A_{\frac{n}{2}})$ for Q. In view of Definition 2.1,
if Q employs $\sigma_{Q}$, he will win the game with probability $1.0$ no
matter which strategy Picard chooses. If Picard uses
$\sigma_{P}=(I,\dots,I,I)$, then the fact that $\sigma_{Q}$ is a winning
strategy means that after Q’s last action the coin must be at the basis state
$\ket{q_{Q}}$. On the other hand, if Picard uses $\sigma_{P}=(I,\dots,I,F)$,
then the fact that $\sigma_{Q}$ is a winning strategy means that after Q’s
last action the coin must be at the opposite basis state. This contradiction
proves that Q does not possess a winning strategy.

∎

The next Theorem 6.4 gives a positive answer to the question of whether Q, and
in effect the quantum player, can surely win and under what circumstances.

######  Theorem 6.4 (When Q possesses a winning strategy).

In any $n$-round game, $n\geq 2$, Q has a winning strategy iff Q makes the
first and the last move.

###### Proof of Theorem 6.4.

The one direction, i.e., if Q has a winning strategy then he must make the
first and the last move, is an immediate consequence of Theorems 6.2 and 6.3.

It remains to prove the other direction, that is if Q makes the first and the
last move, then Q possesses a winning strategy. If the initial state of the
coin $\ket{q_{0}}$ is the same as the target state of Q, then
$\sigma_{Q}=(H,I,\dots,I,H)$ a winning strategy for Q. Q’s first action will
drive the coin to either $\ket{+}$ (if the initial state is $\ket{0}$) or
$\ket{-}$ (if the initial state is $\ket{1}$). In any case both
$\\{\ket{+},\ket{-}\\}$ are fixed by $\\{I,F\\}$, as Theorem 5.8 asserts. Q’s
last move will send the coin back to $\ket{q_{0}}$. If the initial state of
the coin $\ket{q_{0}}$ is different from the target state of Q, then it is
trivial to check that $\sigma_{Q}=(H,I,\dots,I,FH)$ a winning strategy for Q.
∎

######  Corollary 6.5 (The impact of initial and target states).

In any $n$-round game, $n\geq 2$, if Q has a winning strategy, then he has a
winning strategy for every combination of initial and target states.

###### Proof of Corollary 6.5.

An immediate consequence of Theorem 6.4.  ∎

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