#### DUNE sensitivities to the mixing between sterile and tau neutrinos

Received: July
DUNE sensitivities to the mixing between sterile and
Pilar Coloma 0 1 2 4
David V. Forero 0 1 2 3
Stephen J. Parke 0 1 2 4
0 Blacksburg , VA 24061 , U.S.A
1 13083-859 , Campinas, SP , Brazil
2 P. O. Box 500, Batavia, IL 60510 , U.S.A
3 Center for Neutrino Physics , Virginia Tech , USA
4 Theoretical Physics Department, Fermi National Accelerator Laboratory
Light sterile neutrinos can be probed in a number of ways, including electroweak decays, cosmology and neutrino oscillation experiments. At long-baseline experiments, the neutral-current data is directly sensitive to the presence of light sterile neutrinos: once the active neutrinos have oscillated into a sterile state, a depletion in the neutral-current data sample is expected since they do not interact with the Z boson. This channel o ers a direct avenue to probe the mixing between a sterile neutrino and the tau neutrino, which is currently only weakly constrained by current data from SuperK, IceCube and NOvA, however, these constrains will continue to improve as more data is collected by these experiments. In this work, we study the potential of the DUNE experiment to constrain the mixing angle which parametrizes this mixing, 34, through the observation of neutral-current events at the far detector. We nd that DUNE will be able to improve signi cantly over current constraints thanks to its large statistics and excellent discrimination between neutral- and charged-current events.
Beyond Standard Model; Neutrino Physics; CP violation
Oscillation probabilities in the 3 + 1 framework
Rejection power for the three-family hypothesis, for 24; 34 6= 0
Expected allowed regions in the 24
Summary and conclusions
A Complete expressions for the relevant mixing matrix elements in our
1 Introduction 2 3 4
Simulation
Results
4.1
4.2
4.3
parametrization
B
2-function
1
Introduction
In the past decade, a tremendous experimental e ort has been carried out in order to
constrain scenarios with additional neutrinos with masses below the electroweak scale.
LEP data places severe constraints on the invisible decay of the Z. Hence, if there are
additional neutrinos below the electroweak scale, they cannot couple to the Standard Model
weak bosons (i.e., they should be sterile). Light sterile neutrinos can lead to observable
phenomena in a number of electroweak processes through their impact on the unitarity
of the leptonic mixing matrix, including meson decays, muon decay, neutrinoless double
beta decay and charged lepton
avor violating transitions (see e.g., refs. [1, 2] for recent
global ts using these observables). Nevertheless, if their masses are light enough so that
they are kinematically accessible in these processes, unitarity is e ectively restored at low
energies and the bounds from electroweak processes fade away. In this case the best limits
are derived from oscillation data [3{8], see e.g., refs. [
9, 10
] for a detailed discussion of these
constraints.
In recent years, the eV-scale has recently been put on the spot due to a set of
experimental anomalies independently reported in LSND [11], MiniBooNE [12, 13], reactor [14, 15]
and Gallium experiments [16]. The current and next generation of oscillation experiments
will attempt to refute or con rm these hints. The Icecube experiment has recently put
impressive limits on the mixing between sterile neutrinos and muon neutrinos U 4 [17, 18],
{ 1 {
to constrain the cross-product jUe4j2jU 4j2 [23]. Conversely, placing equally competitive
limits on the mixing with tau neutrinos is a much more di cult task, due to the technical
challenges associated to the production and detection of a
beam.
Indirect constraints on the mixing with
can be derived from the observation of
matter e ects in atmospheric neutrino oscillations. For example, the IceCube experiment
90% CL) for an active-sterile mass splitting above 0:1 eV2 [4].1 A non-zero
have set the limit jU 4j2 < 0:15 (at 90% CL) for an active-sterile mass splitting equal to
1 eV2 [18] while the Super-Kamiokande experiment have set the bound jU 4j2 < 0:18 (at
24 and 34
active-sterile mixing produces striking signatures in the zenith and energy distribution of
cascade events in IceCube DeepCore, and after some years of data taking it is possible
to probe the 34 parameter space [25]. On the other hand, a more direct test for the
mixing between sterile neutrinos and tau neutrinos can be performed using long-baseline
experiments. At long-baseline experiments most of the initial
ux has oscillated into
tau neutrinos by the time it reaches the far detector, thanks to
by the atmospheric mass-squared splitting. The OPERA experiment has constrained the
impact of sterile neutrinos on this oscillation channel, using charged-current
events at the
far detector, setting the bound 4jU 4j2jU 4j2 < 0:116 (at 90% CL) for an active-sterile
masssquared splitting above 0:1 eV2 [26]. However, their results are severely limited by statistics,
since the
charged-current cross section is still low at multi-GeV neutrino energies.
Alternatively, the mixing between sterile neutrinos and tau neutrinos can be tested
at long-baseline experiments searching for a depletion in the neutral-current event rates
at the far detector. In fact, both the MINOS and the NOvA experiments have provided
competitive constraints using this approach [27, 28]. Future long-baseline experiments, with
larger detectors, more powerful beams and a better control of systematic uncertainties, may
be able to push these limits even further. In this work, we focus on the potential of the
DUNE experiment [29]. Previous studies of sterile neutrino oscillations using the DUNE far
detector data can be found, e.g., in refs. [9, 30{36].2 However, to the best of our knowledge
the neutral-current data sample has not been considered in any of these works. The liquid
Argon detector technology has excellent particle identi cation capabilities and therefore a
very good discrimination power between charged- and neutral-current events. In addition,
the statistics collected at DUNE will exceed considerably (by a rough order of magnitude)
the number of events collected at MINOS or NOvA. Thus, DUNE o ers an excellent
benchmark to conduct a search for sterile neutrino mixing using neutral-current data.
!
oscillations driven
1An important constraint on the tau-sterile mixing angle 34 has been obtained by combining IceCube
DeepCore data [18] and short baseline data in ref. [24]. However, the constraint is given for a speci c value
of the sterile mass squared di erence larger than 1 eV2.
2For a sensitivity study using the DUNE near detector to probe sterile neutrino oscillations at
m421
1eV2 we refer the reader to ref. [37].
{ 2 {
HJEP07(218)9
Although current hints of sterile-active neutrino mixing with e and
occurs for a
m2 of 0.1 eV2, in this paper we consider a broader range of
m2 's similar to what Daya
Bay has performed for the e disappearance search for sterile neutrinos, see ref. [19]. If a
sterile neutrino only mixes with
, then searches using e and
disappearance as well as
e appearance in a
beam will not constrain such sterile-tau mixing.
The manuscript is organized as follows. In section 2 we derive the oscillation
probabilities in the
! s and
! s oscillation channels at the far detector of long-baseline
experiments, and discuss the di erent limits of interest depending on the active-sterile
mass-squared splitting. Section 3 summarizes the main features of the DUNE experiment
and the details relevant to our numerical simulations. Our results are presented in section 4,
and in section 5 we summarize and draw our conclusions. Some useful expressions for the
elements of the mixing matrix using our parametrization can be found in appendix A.
2
Oscillation probabilities in the 3 + 1 framework
In this section we derive approximate expressions for the oscillation probabilities, which will
be useful in understanding the results of our numerical simulations later on. The mixing
matrix U that changes from the avor to the mass basis in the 3 + 1 neutrino framework
is a 4
4 unitary matrix:
= U i i ;
where
e; ; ; s and i
1; 2; 3; 4. In this work we are interested in the e ect of
oscillations into sterile states on the event rates measured at the DUNE far detector.
Assuming that no oscillations have taken place at the near detector, this can be done
searching for a depletion in the number of neutral-current (NC) events at the far detector
with respect to the prediction obtained using near detector data. For a perfect beam of
muon neutrinos with
ux
(i.e., assuming no beam contamination from other neutrino
avors), the number of NC events at the far detector can be expressed as:
NNC = N NeC + NNC + NNC =
NC fP (
NC 1
f
=
and is therefore sensitive to oscillations in the
! s channel. Here, NC is the
neutralcurrent cross section for the active neutrinos, which is independent of the neutrino avor.
In the absence of a sterile neutrino, the NC event rates should be the same at the far
and near detectors up to a known normalization factor coming from the di erent distance,
detector mass, e ciency, and the di erent geometric acceptance of the beam at the two
sites. In fact, the combined t between near and far detector data should provide a very
e cient cancellation of systematic errors associated to the ux and cross section in this
channel [28].
In addition to the standard solar and atmospheric mass-squared di erences, in the 3+1
framework the oscillation probabilities depend on three new splittings
with k = 1; 2; 3. Given the values of the neutrino energy and distance corresponding to
the far detector at DUNE, for illustration purposes we can e ectively neglect the solar
m24k
m24
m2k,
{ 3 {
mass splitting and focus on the e ects of the oscillation due to the atmospheric and the
sterile mass-squared splittings.3 Under the approximation
probability in the
! s channel is given (in vacuum) by:
21
31; 41, the oscillation
P s
P (
! s) = 4jU 4j2jUs4j2 sin2
41 + 4jU 3j2jUs3j2 sin2
31
+ 8 Re U 4Us4U 3Us3 cos 43 sin 41 sin 31
+ 8 Im U 4Us4U 3Us3 sin 43 sin 41 sin 31;
where we have de ned
ij
mi2j L=4E.
The probability in eq. (2.2) is completely general, but does not allow to see the
number of independent parameters which enter the oscillation probability. A 4
4 unitary
matrix U can be parametrized in terms of six mixing angles and three Dirac CP-violating
phases.4 In the following, we choose to parametrize it as the product of the following
consecutive rotations:
U = O34V24V14O23V13O12:
Here, Oij denotes a real rotation with an angle ij a ecting the i and j sub-block of the
mixing matrix, while Vij denotes a similar rotation but this time including a complex phase.
For example:
0
c24
0
0
cos ij . In this notation, i4 are the new mixing angles with the
fourth state, and 14; 24 are the two new CP-violating phases. In this parametrization, the
complex phase associated with the V13 rotation corresponds to the standard CP-violating
phase in three-families, 13
CP , and the 3
3 sub-block of the matrix shows only
small deviations from a unitary matrix, which at leading order are proportional to sj24 and
therefore within current bounds [8].
For simplicity, from now on we consider 14 = 0, which is a valid approximation given
the strong constraints set by reactor experiments in the range of
work [23]. In this case there is no sensitivity to the 14 phase, which disappears from the
mixing matrix, and the relevant elements of the mixing matrix read
m241 considered in this
3In our numerical simulations the full Hamiltonian is diagonalized to extract the oscillation probabilities
4If neutrinos are Majorana, additional CP-phases enter the matrix. However, neutrino oscillations are
{ 4 {
see eq. (A.1). Then we can rewrite the
! s oscillation probability, eq. (2.2), as
where the dependence with the new CP-violating phase 24 phase is now evident.
Depending on the value of the new mass-squared splitting,
can be considered for the probability in eq. (2.6):
m241, the following three limiting cases
1. The oscillations due to the active-sterile mass-squared splitting have not developed
distance to the far detector (i.e.,
41
31):
P s = 4 U 4Us4 + U 3Us3 2 sin2
31
= 2c143s223c224 2c223s324 + sin 2 23 sin 2 34s24 cos 24 + 2s223s224c324 sin2
31 :
2. The oscillation maximum due to the active-sterile mass-squared splitting matches the
= 4 jU 4j2jUs4j2 + jU 3j2jUs3j2 + 2 Re[U 4Us4U 3Us3] sin2
31
=
c413 sin2 2 23c224s324 + c324 sin2 2 24(1
c213s223)2
c213c24 sin 2 23 sin 2 24 sin 2 34(1
c213s223) cos 24 sin2
0 there is a signi cant cancellation in the probability.
This will be discussed in more detail later in this section.
3. The oscillations due to the active-sterile mass-splitting are already averaged-out at
the far detector5 (i.e.,
41
31):
P s = 2 jU 4j2jUs4j2 + 4 jU 3j2jUs3j2 + Re[U 4Us4U 3Us3] sin2
31
+ 2 Im[U 4Us4U 3Us3] sin 2 31
=
21 c324 sin2 2 24
41 c123c24 sin 2 23 sin 2 24 sin 2 34 sin 24 sin 2 31:
5A similar expression in this limit, but assuming a real mixing matrix, can be found in ref. [38].
Δm241=0.002 eV2
Δm241=0.004 eV2
Δm241=Δm231
5
10
spond to di erent values of the new CP-violating phase 24, while the di erent lines shown in each
panel correspond to di erent values of the active-sterile mass splitting
end. The rest of the oscillation parameters have been xed to:
0:5 ; sin2 2 13 = 0:084 ; and sin2 24 = sin2 34 = 0:1.
m231 = 2:48 10 3 eV2 ; sin2 23 =
m241, as indicated in the
leg
As mentioned above, a destructive interference between the standard and non-standard
contributions to the oscillation amplitude is possible for certain values of the active-sterile
mixing parameters and, in particular, for certain values of the CP phase 24. This is shown
in gure 1 for di erent values of
m241 around the atmospheric scale, when the oscillation
probability simpli es to eq. (2.8). The solid lines in all panels have been obtained for
m241 =
m231: notice that a cancellation of the oscillation amplitude takes place in this
case for 24 = 0, as shown in the left panel in
gure 1. In this case, the contribution
from the interference (last term in eq. (2.8)) is negative and cancels almost exactly the
two other contributions to the oscillation probability. In fact, it is straightforward to show
that, in the limit c13 = c24 = c34 = 1, the amplitude of the oscillation is proportional
to c223js24c23
s34s23ei 24 j2, which vanishes exactly if 24 = 0 and s24c23 = s34s23. This
cancellation is only partial (or negligible) for other values of the CP phase, as expected,
and this can be seen from the middle and right panels in the gure. For other values of the
active-sterile mass splitting the oscillation pattern is more complex, as shown by the dotted
blue and dashed yellow lines in
gure 1. In the most general case, the dependence of the
probability with the energy becomes non-trivial due to the interference of di erent terms
oscillating at di erent frequencies. Moreover, as we will see in section 4 the cancellation in
the probability can also be severe in the limit
m241
m231.
Given the strong limits that have been set on the 24 angle by the oscillation
experiments looking for oscillations involving a sterile neutrino in the eV scale, it is worth to
address explicitly the case when 24 ! 0. Under this assumption, the probability simpli es
considerably with respect to the expression in eq. (2.6):
P s( 24 ! 0) = c143 sin2 2 23s324 sin2
31:
(2.10)
In contrast with eq. (2.6), in this case there is no sensitivity to 24 and, most importantly,
there is no dependence with the sterile mass-squared splitting. The oscillations in this
case are solely driven by the atmospheric mass-squared splitting, and the size of the e ect
{ 6 {
parameters goes as c413 sin2 2 23
O(1).
is directly proportional to s234. Moreover, the dependence with the standard oscillation
Finally, it is worth to mention that matter e ects will modify the oscillation probability
in eq. (2.6). We have checked that the size of these modi cations is relatively small and,
therefore, the vacuum probabilities are precise enough to understand the behavior of the
numerical simulations in the following sections. However, in our numerical analysis, matter
e ects have been properly included using a constant matter density of 2:96 g cm 3
:
3
In contrast to usual analyses searching for signals of sterile neutrino oscillations at short
distances, in this work we want to take advantage of the capabilities of the DUNE far
detector, located at a distance of L = 1300 km from the source. In particular, we focus
on the potential of NC measurements to discriminate between the 3- avor and 4- avor
scenarios. To this end, we rely on the excellent capabilities of the DUNE far detector
to discriminate between charged-current (CC) and NC events. All the simulations in the
current work have been performed using a modi ed version of the GLoBES [39, 40] library
which includes a new implementation of systematic errors as described in ref. [41]. The
neutrino oscillation probabilities in a 3+1 scenario have been implemented using the new
physics engine available from ref. [42].
In our simulation of the signal, we have computed separately the contributions to the
total number of events coming from
e
,
and
NC interactions at the detector. For
simplicity, we have assumed a 90%
at e ciency as a function of the reconstructed visible
energy. The experimental observable for a NC event is a hadronic shower with a certain
visible energy (energy deposited in the detector in the form of a track and scintillation
light). The correspondence between a given incident neutrino energy and the amount of
visible energy deposited in the detector has to be obtained from the simulation of
neutrino interactions and detector reconstruction of the particles produced in the nal state.
To this end, we use the migration matrices provided by the authors of ref. [43], which
were obtained using the LArSoft simulation software [
44
] accounting for the far detector
geometry, neutrino-argon interactions and propagation of the
nal state particles in the
detector active volume. The authors of ref. [43] used bins in visible energy of 50 MeV for
the reconstructed energy of the hadron shower, as opposed to the DUNE CDR studies
where wider bins of 125 MeV were considered [29]. In the present work we have considered
two sets of matrices: the original set provided by the authors of ref. [43], with 50 MeV
bins, and a (more conservative) rebinned version of these matrices where the bin size was
increased to 250 MeV. We performed our simulations for the two options (with 50 MeV bins
and 250 MeV bins) and found similar results for the two sets of matrices. Therefore, in the
following we will adopt the more conservative 250 MeV bin size as our default con guration.
The main backgrounds for this search would be e
and
CC events that might
be mis-identi ed as NC events. We have assumed that the background rejection e ciency
for CC events is at the level of 90%. However, this is probably a conservative estimate:
for instance, muons leave long tracks in liquid Argon (LAr) that are di cult to misidentify
{ 7 {
HJEP07(218)9
mode
mode
(NNeC + NNC + NNC )
6489
2901
Background
NCeC
129
22
NCC
751
301
NCC
140
39
8 GeV at the DUNE far detector. The number of events is shown for the signal and background
contributions separately. This corresponds to 7 yrs of data taking (equally split between neutrino
and antineutrino running modes) with a 40 kton detector and 1.07 MW beam power, yielding a
accounted for. In the case of NCC , the number of events already includes the branching ratio for
hadronic
decays. Usual oscillations (in the three-family scenario) have been considered in the
computation of the backgrounds, setting 23 = 42 and the rest of the oscillation parameters in
agreement with their current best- t values.
as NC events, except when they have very low energies or are not completely contained in
the detector. On the other hand, the active neutrino avors would be a ected by standard
oscillations. Consequently, the number of
CC events would be largely suppressed since
most of the initial muon neutrinos have oscillated to tau neutrinos by the time they reach
the detector. Given the energetic neutrino ux at DUNE, some of the oscillated
ux will
interact at the detector via CC, producing
leptons. In most of the cases (
65%), the
decays hadronically producing a shower: these events constitute an irreducible background
and consequently no rejection e ciency has been assumed in this case. We have assumed a
Gaussian energy resolution function for the
and e background contributions, following
the values derived in ref. [43] from LArSoft simulations, while the hadronic showers
produced from hadronic tau decays have been smeared using the same migration matrices as
for the NC signal.
The expected total number of signal and background events is summarized in table 1,
where the di erent background contributions are shown separately for clarity. As can be
seen from this table, the largest background contribution comes from
CC events
misidenti ed as NC, due to the large ux available at the far detector, while the contributions
coming from
e and
CC events are much smaller and approximately of equal size. In
all cases, both signal and backgrounds receive contributions from right- and wrong-sign
neutrino events due to the intrinsic contamination of the beam. The number of events
has been computed for visible energies between 0.5 GeV and 8 GeV, which is the region
used in our analysis, using the beam con guration with 80 GeV protons as in ref. [45].
Additional experimental details for the DUNE setup considered in this work can be found
in refs. [29, 45].
The expected NC event distributions are shown in gure 2, as a function of the
(reconstructed) visible energy, for the three-family scenario (white histogram) and for the case
when there is a sizable mixing angle with the sterile neutrino (blue/light gray histogram).
As expected, a depletion in the number of events can be observed in the 3 + 1 case with
{ 8 {
Erec [GeV]
visible energy, after e ciencies and detector reconstruction. The white histogram shows the
expected number of NC events in the 3-family standard scenario, while the blue (light gray) histogram
shows the expected number of NC events for sin2 34 = 0:1, 14 = 24 = 0. The expected distribution
for background events (CC mis-identi ed as NC) is given by the green (dark gray) histogram.
respect to the three-family scenario. Moreover, the events pile up at low energies due to
the energy carried away by the outgoing neutrino in the nal state. One can also see that
the energy distribution of the background (shown by the green/dark gray histogram) is
dictated by the standard oscillations su ered by the active neutrinos as they propagate to
the far detector, which is well-known. In this case, all particles in the
nal state would
be observed, and there is practically no pile-up at low energies. Due to this, the
sensitiv!
ity to oscillations in the
s channel is enhanced when some energy information is
included in the t, as we will see in the next section. This is exploited in our numerical
2 in the visible (deposited) energy in the detector, with
analysis implementing a binned
a 2 function de ned in eq. (B.1).
The e ect of systematic uncertainties is accounted for through the addition of
pullterms to the
2, as speci ed in the appendix B. In addition to an overall normalization
uncertainty for the signal and background (which is bin-to-bin correlated), a shape
uncertainty for the signal (bin-to-bin uncorrelated) has been included to account for possible
systematic uncertainties related to the shape of the event distributions. Moreover, all nuisance
parameters are taken to be uncorrelated between the neutrino and antineutrino channels as
well as between the di erent contributions to the signal and/or background events. Unless
otherwise stated, the
nal 2 is obtained after marginalization over the nuisance
parameters and the relevant standard oscillation parameters (sin2 2 23; sin2 2 13;
m231) within
current experimental uncertainties [46{48]. Speci cally, we consider the following
Gaussian priors:
(sin2 2 13) = 0:005, (sin2 2 23) = 0:05 and ( m231)=
m231 = 0:04. Unless
otherwise speci ed, we have assumed a conservative 10% Gaussian prior for all nuisance
2
parameters, included as pull-terms in the
. In practice, however, the cancellation of
{ 9 {
systematic errors in the NC channels is expected to be extremely e cient, since the near
detector can be used to measure the same convolution of the ux and cross section as in the
far detector. This contrasts with oscillation measurements in appearance mode (
using CC data, where the initial and
nal neutrino
ux spectrum (and
avor) di er due
to the impact of standard oscillations, making the cancellation of systematic uncertainties
extremely challenging.6 In spite of these di culties, the DUNE collaboration expects to
!
)
reach a precision at the percent level in the
! e and
! e appearance channels. In
view of this, we expect the 5%{10% values considered in this work for the NC sample to
be conservative.
Before concluding this section, let us comment on the relevance of the near detector
data and its possible impact on the t. In this work, we have not simulated the near
detector explicitly: its design is still undecided and its expected performance is therefore
unclear yet. A detailed simulation of the near-far detector data combination is beyond the
scope of this work and can ultimately be performed only by the experimental collaboration.
In this work, instead, we have assumed that the oscillations due to the new state have not
developed yet at the near detector. For neutrino energies in the region around 2-3 GeV, and
for a near detector located at a distance of L
O(500) m, this is a valid approximation as
long as
m241 < 1 eV2. Under this assumption, the near detector measurements will provide
a clean determination of the convolution of the NC cross section and the muon neutrino
ux, which can then be extrapolated to the far detector with a small uncertainty. At this
point, it should be mentioned that our assumed prior uncertainties for the systematic errors
in the t would correspond to the values used for the analysis of the far detector event
rates. Thus, they correspond to estimates on the size of the
nal systematic errors that
have to be propagated to the far detector, once the near detector data has already been
accounted for. Finally, it should also be stressed that in the case that 14 = 24 = 0 there
would be no e ect on the near detector data regardless of the new mass-squared splitting.
The reason is that, as it was shown in eq. (2.10), the dependence with
the oscillation probabilities: this guarantees no e ect at the near detector, while at the far
detector data the oscillation would be driven by the atmospheric scale. Thus, in this case
m241 drops from
the e ect in the oscillation would be observable for large enough 34.
4
Results
In this section we show our numerical results for the expected sensitivities to the new
mixing parameters in the di erent scenarios discussed in section 2. By the time DUNE
starts taking data the constraints on the sterile mixing angles 14 and 24 might be very
tight. Nevertheless DUNE is also sensitive to the 34 sterile mixing angle, which is currently
the less constrained among the three sterile-active mixing angles. Therefore, we initially
consider the simpler case where two of the new mixing angles xed to zero, 14 = 24 = 0
and study the sensitivity of the DUNE experiment to 34. Next we proceed to turn on the
mixing angle 24 and determine for which values of 24
m241 the three-family hypothesis
6For a recent review of the challenges that long-baseline experiments have to meet regarding systematic
uncertainties see ref. [49].
2 90% C.L
shape = norm = 5%
shape =
norm = 10%
norm = 5%, Rate only
norm = 10%, Rate only
14
12
10
8
6
4
2
0
lines correspond to di erent assumptions of systematical uncertainties, see text for details. Right
panel: 34-discovery reach analysis where 4- avor event rates were calculated in `data' and t, with
all the 4- avor parameters
xed to the values in the plot, except for 34. Also, as shown in the
plot, we
xed the systematical errors to 5%. The shaded region is disfavored at 90% C.L. from
Super-Kamiokande atmospheric data [4] and whose limit on jU 4j2 < 0:15 (at 90% CL) translates
into the constraint sin2 34 < 0:15, for 14 = 24 = 0. The horizontal dotted line indicates the value
of the
2 corresponding to 90% C.L. for 1 d.o.f..
could be rejected. We nalize this section by showing the expected limits that could be
derived simultaneously on the two mixing angles 24 and 34, for di erent values of the
active-sterile mass-squared splitting.
4.1
Sensitivity to 34, for 24 = 0
Under the assumption 24 = 14 = 0, the expression for the vacuum sterile neutrino
appearance probability is given by eq. (2.10) and does not depend on any of the new oscillation
frequencies induced by the sterile, nor any of the CP-violating phases. An interesting
question to ask in this case is if DUNE will be able to improve over current constraints on
34, assuming that the experiment will measure event distributions in agreement with the
expectation in the three-family scenario. In this case, the \observed" event distributions
are simulated setting all i4 = 0, and are then tted using increasing values of 34.
The sensitivity to 34 is shown in the left panel of gure 3. As seen in the
gure,
our results show a considerable dependence on the size and implementation of systematic
errors. Assuming a (conservative) 10% systematic error on both normalization ( norm) and
shape ( shape), we
nd that DUNE will be sensitive down to values of sin2 34
0:12, at
90% C.L. (1 d.o.f.). For comparison we also show the limit on this mixing angle obtained
from atmospheric neutrino data collected by the Super-Kamiokande (SK) collaboration [4],
for
m241 > 0:1 eV2. If prior uncertainties could be reduced to the 5% level for both
normalization and shape errors, we nd that DUNE would be able to improve over the SK
constraint by more than a factor of two. It should be stressed that the DUNE constraint
would be valid for any value of
m241, as long as 24; 14 ' 0. In the next subsections we
will study in detail the phenomenology in case 24 6= 0.
The lines labeled as \Rate only" in the left panel of gure 3 do not include a binned
2 and only consider the total event rates in the computation of the
2
. The change
in sensitivity can be appreciated from the comparison between the dashed pink and
dotdashed red lines, for 10% systematic errors (or between the dot-dot-dashed green and solid
blue lines, for 5% systematic errors). As can be seen, the inclusion of energy information
leads to a noticeable improvement in the results. Therefore, in the rest of this section we
will only consider a binned
2, using equally-sized bins in visible energy, as described in
section 3.
Finally, we comment on the analysis shown in the right panel of gure 3. Di erent
to the analysis shown in the left panel, in the right panel we performed a discovery reach
analysis for 34 taking all systematical errors at the 5% level. In this case, we assume that
the experiment will measure event distributions in agreement with the expectation in the
four-family scenario. In order to quantify the impact of also having a nonzero 14 and 24,
for simplicity, the four- avor parameters where xed to their `true' values (except for 34)
with the values in the plot labels. In this case (for 14 6= 0 or 24 6= 0) there is a dependence
with the sterile mass squared di erence, and we have
xed its value to
which is one of the values considered in our results in
gure 5. For
are therefore in the regime where the sterile oscillation is averaged-out at the far detector,
in relation to eq. (2.9). In fact, is in this regime where constraints in the 24
34 plane
are reported by di erent the collaborations (as will be addressed in section 4.3). It is
then worth to mention that 14 is tightly constrained by reactor experiments for the
considered [19] and therefore its impact (even for 24 6= 0) is marginal, as shown in the
right panel of gure 3. Thus, since by the time DUNE will be running smaller values of
m241
14 and 24 are expected, the case when 14
24
0 is of particular relevance. In this
last case DUNE, with the considered con guration, will produce a `indication' of a nonzero
34 (i.e. if it happens to be as large as
18 ) with a signi cance of
2 for the assumed
systematical errors.
Rejection power for the three-family hypothesis, for 24; 34 6= 0
The scenario where 24 6= 0 leads to a more interesting phenomenology, since in this case
the oscillation probability also depends on the active-sterile mass-squared splitting. In this
case, assuming as our true hypothesis a 3+1 with nonzero 34 and 24, it is relevant to
ask if the experiment would be able to reject the three-family hypothesis. This is shown
in
gure 4, as a function of the possible true values of
m241 and sin2 24. The true value
of 34 is set to be nonzero, while 14 = 0 is assumed for simplicity. In all panels, the
expected events distributions are computed using the indicated values as true input values.
The obtained \observed" event distributions are then compared to the expected result in
the three-family scenario, i.e., in absence of a sterile neutrino. The contours indicate the
sets of true values ( 24,
m241) for which the three-family hypothesis would be successfully
rejected at 90% C.L.. The di erent panels in gure 4 show the dependence of our results
10 2
10% syst.
10 2
sin2 2410 1
100
m241 and sin2 24. The true value of 34 has been set to a non-zero value in all cases, as
indicated in the labels, while 14 = 0 for simplicity. The contours indicate the sets of true values
( 24,
m241) for which the three-family hypothesis would be successfully rejected at 90% C.L.. In
other words, they indicate the fraction of parameter space where the SM hypothesis (namely, the
point i4 = 0) would be disfavored with a
2 > 2:71. Left panel: dependence of the results with
the true value of 24. Central panel: dependence of the results with the true value of sin2 34. Right
panel: dependence of the results with the assumed priors for the systematic uncertainties.
with respect to di erent parameters: the true value of 24 (left panel), the true value of 34
(central panel); and the assumed priors for the systematic uncertainties (right panel).
As explained in section 2, if both 24 and 34 are di erent from zero, the oscillation
probability P s also depends on the value of the CP phase 24. Such dependence can be
appreciated by comparing the three lines shown in the left panel in gure 4, corresponding
to di erent true values of 24. The same true value of 34 and the same implementation
of systematic uncertainties have been assumed for all lines (indicated by the top label).
gure 1 (see also eq. (2.8)), for values of
interference between the di erent contributions to the oscillation amplitude, depending on
the value of 24. For values of
m231, this leads to a decreased sensitivity in this
region of the parameter space for 24 = 0 with respect to the results obtained for 24 = .
The interference has the opposite e ect in the region
m241
m231: for negative values
of cos 24 the second term in eq. (2.7) is negative and suppresses the probability, leading
to worse results for 24 =
. In fact, it can be easily shown that, in the limit 23 =
=4,
c13 = 1 and at the rst oscillation maximum (sin2
2
eq. (2.7) approximates to
31 = 1) the oscillation probability in
P s
c224(s324 + 2s24s34c34 cos + s224c234) ;
(4.1)
where the e ect of the interference term can be easily appreciated.
Conversely, in the limit where the new frequency is averaged-out ( m241
m231) the
results show a very mild dependence with the value of 24. This can be easily explained
from the expression in eq. (2.9), which shows two terms that depend on the value of 24: the
rst one is directly proportional to (c213s23
2
1=2) ' O( 23 s213=2), where
23
23
=4,
and is therefore very suppressed; while the second term is proportional to sin 2 31 and it is
completely o -peak at the rst oscillation maximum. In fact, in the same limit ( 23 = =4,
c123 = 1) and at the rst oscillation maximum it is easy to show that the term proportional
to cos 24 in the oscillation probability in eq. (2.9) is additionally suppressed with cos 2 23,
which is small for 23 near maximal mixing.
The central panel in
gure 4 shows the dependence of the results with the true value
of 34. In this case, all priors for the systematic uncertainties are set at the 10% and we
have xed 24 = 0. As shown in the gure, in the region where
m241
m231 there is a
strong dependence of the results with the true value of 34, while the contours do not show
large variations for larger mass splittings. This behavior can again be easily traced back
to the approximate oscillation probabilities in section 2.
Finally, the right panel in gure 4 shows the dependence of the results with the assumed
priors for the systematic uncertainties. In this panel, the true values of 24 and 34 have
been set as indicated in the top label. The solid line uses our default implementation
for the systematic uncertainties, where all priors are set to 10% for both the shape and
normalization and for both signal and background. The dot-dashed line, on the other
hand, shows the room for improvement if all prior uncertainties can be reduced down to
5%. As can be seen from the gure, the improvement is dramatic and leads to a successful
rejection of the three-family hypothesis in practically all the parameter space, with the sole
exception of the region around
m231 (which is very di cult to reject, since this
is the region where signi cant cancellations can take place for 24 = 0).
4.3
Expected allowed regions in the 24 34 parameter space
If the observed event distributions show an agreement with the three-family expectation,
one would proceed to derive a limit on the mixing angles 24 and 34. However, as we saw in
section 2 the oscillation probabilities show a large dependence with the new CP-violating
phase 24, and strong cancellations between the di erent contributions may occur. The
e ect of the cancellations is much more severe in the limit
m241 ! 0 than for larger
values of the active-sterile mass splitting and, therefore, we expect very di erent results as
a function of this parameter.
Figure 5 shows the expected allowed regions in the 24 and 34 plane if the observed
event distributions are found to be in agreement with the three-family hypothesis. In this
case, the \observed" event distributions are simulated assuming the three-family
hypothesis, and tted in a 3+1 scenario. The value of the
2 function, for a given pair of test
values 24
34, is obtained after minimization over the new CP-violating phase 24 and
over all nuisance parameters. As for the mass splitting
m241, it has been kept xed during
the t to the test value indicated in each panel to show the di erence in the results. For
simplicity, we have also kept all the standard parameters
xed during the minimization
procedure; however, minimization over the standard parameters is not expected to a ect
signi cantly the results shown here.
gure 5, the resulting allowed regions are very di erent if the results are
tested using
m241
m231 or a
m241 in the averaged-out regime. In the former case, a
strong cancellation in the oscillation probability can always be achieved setting the value
of 24
, as outlined in section 2 and section 4.2. Therefore, in this case it is not possible
to disfavor large values of the new mixing angles. Only if the two mixing angles have very
di erent values (e.g., in the region 24 ! 0; 34 & 25 ) the interference term would not be
10%
5
%
sy
s
y
10% sys.
DUNE
15
34( )
correspond to the expected con dence regions allowed at 90% C.L. (2 d.o.f.), for a simulation
assuming i4 = 0 as true input values. The lines labeled as \10% sys" (\5% sys") have been
obtained assuming 10% (5%) prior uncertainties for the signal (both shape and normalization) and
10% for the background (normalization only). For comparison, the right panel shows the latest
results from the NOvA experiment from a NC search [28] (gray region), and from atmospheric data
from the Super-Kamiokande experiment [4] and IceCube DeepCore data [18] (darker gray regions
labeled with red lines), also at the 90% C.L..
large enough to allow for an e cient cancellation in the probability. Thus, in this regime
DUNE could disfavor just the upper left and lower right corner of the parameter space.
Conversely, in the limit
m241
m231 the impact of the new CP-violating phase 24 is
much milder and does not allow for a cancellation in the oscillation probability. A closed
contour is therefore obtained in this case.
NOvA has observed 95 neutral current events at the far detector while 83:5 9:7(stat.)
9:4(syst.) events where expected in the three- avor case. Since no evidence for an sterile
neutrino oscillation was found, they placed the following constraints (assuming cos2 14 = 1)
for the active-sterile mixings: 24 < 20:8 and 34 < 31:2 at 90% of C.L for a
patible with no oscillation at the near detector (0:05 eV2
m241
0:5 eV2). This results
m241
comcorrespond to an exposure-equivalent of 6:05
1020 POT and a total systematical errors
12% ref. [28]. Experiments observing atmospheric neutrinos like the Super-Kamiokande
experiment have also constrained the tau-sterile mixing angle. SK, after an analysis of
4; 438 live-days of data, found no evidence for sterile neutrinos constraining jU 4j2 < 0:041
and jU 4j2 < 0:18 for
m241 > 0:1 eV2 at 90% of C.L [4]. Similarly, IceCube, by the use of
jU 4j2 < 0:15 for
three years of atmospheric neutrino data from the DeepCore detector, which was consistent
with three- avor neutrinos, placed a bound on the active-sterile mixing: jU 4j2 < 0:11 and
m241 = 1 eV2 at 90% of C.L [18]. For comparison, in the right panel of
gure 5 we show the currently allowed NOvA regions from ref. [28] as well as from
atmospheric data from the Super-Kamiokande experiment [4] and IceCube DeepCore data [18].7
7It is worth to notice that all constraints shown in the right panel of gure 5 are valid for an sterile mass
squared di erence
m421 > 0:1 eV2 and therefore they do not apply to the case shown in left panel.
As shown in the gure, DUNE is expected to improve over a factor of two with respect to
the current allowed region set by NOvA with a good control of systematics below 5%. With
5% systematics DUNE will also improve over the current IceCube constraint for 24 < 9 .
Finally, we want to comment on the impact of having a nonzero electron-sterile mixing
in the results shown in
gure 5.
Even in the case where 14 current constraint were
completely relaxed in the analysis (i.e. `free'), our result for 34 does not change at all.
Only the 24 bound (for 34
Bay experiment [19] for the
1) is a ected. However, 14 is tightly constrained by Daya
m241 range where the current limits on 34, shown in the right
panel of gure 5, apply. On the other hand, in the left panel of the same gure (with 14
unconstrained) DUNE is not able to exclude the small window 24 . 24 . 30 with 10%
systematical errors. This, also reinforces our conclusion about the loose of constraining
power for
m241
m231 due to cancellations in the probability.
5
Summary and conclusions
The experimental anomalies independently reported in LSND, MiniBooNE, reactor and
Gallium experiments have put the possible existence of an eV-scale sterile neutrino under
intense scrutiny. In the near future a new generation of short-baseline experiments will
come online to refute or con rm these hints, and will place strong constraints on the
mixing of a light sterile neutrino with electron and muon neutrinos. Achieving similar
bounds on the mixing with tau neutrinos is a much more di cult task, given the technical
!
challenges associated to the production and detection of
. At long-baseline experiments,
however, oscillations in the
channel guarantee that most of the beam will have
oscillated into
by the time it reaches the far detector, thanks to the atmospheric
masssquared splitting. By searching for a depletion in the number of neutral-current (NC)
events measured at the far detector, experiments like NOvA or MINOS have been able to
probe the mixing between
and a fourth neutrino.
In this work, we have studied the potential of the future DUNE experiment to conduct a
search for sterile neutrinos using the NC data expected at the far detector, taking advantage
of the excellent capabilities of liquid Argon to discriminate between charged-current and
NC events. For simplicity, we have focused on a 3 + 1 scenario, where only one extra sterile
neutrino is introduced. In this case, the mixing matrix has to be extended including three
additional mixing angles ( 14, 24 and 34) and two CP-violating phases 14 and 24 (our
parametrization is given by eq. (2.3)). The oscillation probabilities will generally depend
on an additional oscillation frequency dictated by the mass-squared splitting between the
active and sterile states,
m241. First, we have derived the oscillation probabilities in
di erent regimes paying particular attention to the dependence with the new CP-violating
phases. Unlike in other studies where the mass of the sterile was required to be at (or
around) the eV scale, here we have allowed it to vary between 10 5 eV2 and 10 1 eV2;
thus, in eqs. (2.7){(2.9) we provide approximate expressions for the oscillation probabilities
in three di erent regimes, depending on the mass of the sterile state: (i)
m241 ! 0; (ii)
We have then proceeded to simulate the expected sensitivity of the DUNE experiment
using the expected NC events collected at the far detector. We have studied the variation
of our results with the implementation and size of the systematic errors. The details of our
numerical simulations and the
2 implementation can be found in section 3.
First, working under the assumption 24 = 14 = 0, we have determined the sensitivity
of the DUNE experiment to the third mixing angle 34. In this case, the oscillation
probability is independent of the new CP-violating phases; furthermore, oscillations are solely
driven by
m231, see eq. (2.10). We nd that DUNE will be able to improve over current
constraints on this parameter set by the SK experiment, and will be sensitive to values of
0:12 (at 90% CL) for our default implementation of systematic uncertainties. If
systematic errors could be reduced down to 5%, the experimental sensitivity would reach
0:07 (at 90% CL).
Next we proceeded to study the case where 24 6= 0. In this case, the oscillation
probabilities depend on the active-sterile mass-squared splitting. The phenomenology becomes
more complicated and, in particular, strong cancellations in the probability can take place
for certain values of 24 and
m241. First, we considered the 3+1 scenario as the true
hypothesis, and determined for which values of the mixing parameters DUNE would be
able to reject the three-family scenario. Our results are summarized in
gure 4, where
we show the dependence of the sensitivity with the CP phase 24, the mixing angle 34
and the size of the systematic errors. We found that the sensitivity of the experiment to
the presence of a sterile neutrino, measured as its ability to reject the three-family
scenario, depends heavily on the value of the CP phase. For example, for
sin2 34 = 0:1 and 24 = 0, DUNE would be able to reject the three-family scenario for
sin2 24 . 4
10 3 eV2; conversely, for 24 =
(and assuming the same value for 34 and
m241), 24 could be almost two orders of magnitude larger and the three-family scenario
would not be rejected by the data. The behavior of our results can be easily understood
m241 = 10 4 eV2,
in terms of the oscillation probabilities, as explained in detail in section 4.2.
Finally, we considered the opposite situation, and assumed that the experiment will nd
a result that is in agreement with the three-family expectation. In this case, we determined
the allowed con dence regions that would turn from the analysis of the simulated data. Our
results are shown in gure 5. The simulated data were tested using two very di erent values
of the active-sterile mass-squared splitting. In the averaged-out regime (
a closed contour is obtained; we
nd that DUNE would be able to improve over NOvA
constraints in this place by a factor of two or more, depending on the size of the systematic
m241
m231),
errors assumed. Conversely, in the case of
m241
m231 the experimental results would
allow values of 24 and 34 to be as large as 30 . The reason is, again, the possibility
of having a strong cancellation in the oscillation probability, which could lead to a
nonobservable e ect in the event distributions even in presence of very large mixing angles.
The DUNE experiment has unprecedented discrimination between neutral-current and
charged-current events for a long-baseline experiment: this will allow for a measurement
or constraint of the
fraction of a possible sterile neutrino(s).
Given the di culties
associated to the production and detection of
's, measurement or limiting this fraction
by other means is very challenging. In this paper, we show that the DUNE experiment
can provide an excellent constrain or discover a sterile neutrino that primarily mixes with
only the
.
Acknowledgments
We warmly thank Michel Sorel for providing us with the smearing matrices needed to
simulate the liquid Argon detector reconstruction for neutral-current events. PC also thanks
Enrique Fernandez-Martinez for useful discussions. DVF is thankful for the support of S~ao
Paulo Research Foundation (FAPESP) funding Grant No. 2014/19164-6 and
2017/017496., and also for the URA fellowship that allowed him to visit the theory department at
Fermilab where this project started. DVF was also supported by the U.S. Department Of
Energy under contracts DE-SC0013632 and DE-SC0009973. This work has received
partial support from the European Union's Horizon 2020 research and innovation programme
under the Marie Sklodowska-Curie grant agreement No. 674896. This manuscript has been
authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy, O
ce of Science, O
ce of High Energy Physics. The
United States Government retains and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-exclusive, paid-up,
irrevocable, world-wide license to publish or reproduce the published form of this manuscript,
or allow others to do so, for United States Government purposes.
A
Complete expressions for the relevant mixing matrix elements in our parametrization
Starting from the parametrization in eq. (2.3), the mixing matrix elements needed for the
calculation of the sterile appearance probability are given by:
(A.1)
(A.2)
(B.1)
U 4 = e i 24 c14s24 ;
For 14 = 0, and using eq. (A.1), we nd the following useful expressions:
jUs3j2 = c123 c223s234 +
sin 2 23s24 sin 2 34 cos 24 + s223s224c324
;
8 U 4Us4U 3Us3 =
c213c24 sin 2 23 sin 2 24 sin 2 34ei 24
2c123s223c324 sin2 2 24:
B
2-function
The results of our di erent analyses, including spectral information, have been performed
with the following Poissonian
2-function:
Oi
Ti
1
2
n-bins
X
i
where T are the theoretical events (depending on the model parameters) while O
corresponds to the `observed' events. T is the result of the sum of signal s(a; c) plus background
bg(b), where the systematical errors where included in the usual form:
si(a; c) :=
bgi(b) :=
k
a and b are total normalization systematical errors in signal and background,
respectively. For simplicity we have assumed
a =
b =
norm. ci are the bin-to-bin
uncorrelated systematics with error shape. The last four terms in eq. (B.1) are penalties to
the 2 function due to the systematics included in eq. (B.2), and also due to the standard
oscillation parameters
that are marginalized over assuming they have been measured
as j
j .
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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