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# Neutrino Energy Estimation in Neutral Current Interactions and
Prospects for Sterile Neutrino Searches

A. P. Furmanski<EMAIL_ADDRESS>School of Physics and Astronomy, University
of Minnesota, Minneapolis, MN 55455, USA  C. Hilgenberg<EMAIL_ADDRESS>School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA

###### Abstract

Modern neutrino detectors, particularly the large liquid argon time projection
chambers (LAr TPCs) at SBN and DUNE, provide an unprecedented amount of
information about GeV-scale interactions. By taking advantage of the excellent
spatial and calorimetric resolution as well as the low tracking thresholds
provided by LAr TPCs, we present a novel method of estimating the neutrino
energy in neutral current interactions. This method has potential implications
for the search for a sterile neutrino; it allows for the potential observation
of spectral distortions due to sterile neutrino-induced oscillations in the
neutral current neutrino energy spectrum. As an example, we use this method to
perform an analysis of the statistics-only sensitivity to sterile neutrinos in
the neutral current channel at SBN under a 3+1 model.

††preprint: APS/123-QED

##  I Introduction

Accurately and precisely reconstructing the incident neutrino energy in
neutrino interactions is notoriously difficult. The level of difficulty
depends on whether the neutrino interaction is charged current (CC) or neutral
current (NC). CC interactions provide a means of identifying the scattering
neutrino’s flavor. In addition, in the few-GeV neutrino energy regime where
quasielastic (QE) scattering is the dominant interaction mode, the kinematics
of the outgoing lepton are sufficient for determining the neutrino energy.
This gets more challenging at higher energies where resonant production and
deep inelastic scattering dominate and final state interactions (FSI) are more
prevalent. Nevertheless, the resolution on the neutrino energy is typically at
the level of tens of percent.

Neutral current interactions provide neither the neutrino flavor nor knowledge
of the energy carried away by the outgoing lepton. The only handle on the
neutrino energy comes from the final state hadronic system.

The typical method employed in estimating the incident neutrino energy in NC
interactions is to rely on calorimetry alone, summing all visible energy in
the detector to provide a lower bound. This method yields a poor energy
resolution relative to CC interactions while also introducing a significant
bias toward lower energies. The method’s performance depends on the detector
tracking thresholds and calorimetric resolution.

Compared to water Cherenkov and scintillator based detectors, liquid argon
time projection chambers (LAr TPCs) provide low tracking thresholds and
precise energy and angular resolution for charged particles. This detector
technology has been adopted by the Short-Baseline Neutrino Program (SBN) [1]
and the future Deep Underground Neutrino Experiment (DUNE) [2][3][4][5]. LAr
TPCs, in combination with intense accelerator based neutrino sources, are
providing and will provide unprecedented detail about neutrino interactions in
the few-GeV neutrino energy regime.

With the unprecedented level of detail about the hadronic final state provided
by LAr TPCs, it might be possible to improve neutrino energy estimation in NC
interactions. In the first half of this paper, we revisit the visible energy
based method and introduce a new method based on the final state hadronic
system kinematics. We present a toy analysis that demonstrates the proof of
concept of our kinematic method.

In the second half of this paper, we explore one possible application of
neutral current interactions reconstructed with the kinematic method: the
sterile neutrino search at SBN. If the incident neutrino energy can be
reconstructed sufficiently well, NC interactions can be used to probe neutrino
oscillation parameters that are challenging or impossible to probe with CC
channels. We present a statistics-only toy analysis of NC disappearance
sensitivities, illustrating that the addition of our reconstruction method has
the potential to improve sensitivity.

Figure 1: A neutrino scatters off a nucleon, producing a final state hadronic
system that can be used to determine the incident neutrino energy.

##  II Neutrino Energy Reconstruction in Neutral Current Interactions

In NC interactions, the scattering neutrino carries away some fraction of its
initial energy as illustrated in Figure 1. A trivial lower bound on the
neutrino energy can be obtained by summing over all visible deposited energy
associated with a neutrino vertex candidate. This calorimetric method leads to
a significant underestimation of the neutrino energy.

Significant improvements in neutral current neutrino energy reconstruction
would be beneficial for several different kinds of measurements: improved
characterization of NC backgrounds, more precise measurements of NC cross
sections, gleaning shape information for oscillation searches, etc. This
motivates the search for alternatives to the conventional calorimetric method.
In what follows, we present a comparison of the conventional calorimetric
method and a new reconstruction method based on the final state hadronic
system kinematics.

We evaluate reconstruction performance using a sample of five million
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions on argon, each generated using
GENIE v3.00.06 [6] with the G18_10a configuration and the G18_10a_02_11a tune.
This configuration includes a local Fermi gas nuclear model and an empirical
meson exchange current model.

For this study, we use the SBN Booster Neutrino Beam (BNB) neutrino flux,
taken from [1]. The flux is broad-band, peaking at 600 MeV (see Figure 7). The
flux prediction for the neutrino running configuration (forward horn current)
at each SBN detector is used to generate detector-specific, SBN-like samples.
A combination of the flux and detector active mass is used to generate a
normalization factor for each flux component in terms of protons on target
(POT). Detector baselines, active masses, and POT are provided in Section III.

###  II.1 Neutrino Energy from Visible Energy

Relying on calorimetry alone, to measure the total energy deposited in a
detector following a NC interaction, does not account for the energy carried
away by the scattering neutrino. This feature combined with threshold effects
leads to a significant underestimation of the initial neutrino energy. In our
case, this underestimation is at the level of 90%. This bias can be simulated,
and the effect can be corrected, usually by using Monte Carlo templates or
unfolding techniques. However, model based corrections cannot improve poor
energy resolution. Furthermore, reliance on such corrections can introduce
systematic uncertainties into the analysis. Despite these limitations, this
calorimetric method has been used with success, by MINOS/MINOS+[7] and
NO$\nu$A [8][9] for example.

The initial neutrino energy, reconstructed with the calorimetric method, is
plotted against the true neutrino energy in Figure 2(a) for a NC0$\pi^{+/-}$
selection; the reason for this choice will become clear in Section III. The
reconstructed energy includes approximate detector effects, discussed later.
The impact of the missing energy is evident. The mean reconstructed energy
profile is also shown, approximately linear in true energy.

(a) No correction.

(b) With correction.

Figure 2: Visible hadronic energy (y-axis) in NC0$\pi^{+/-}$ interactions vs.
true neutrino energy (x-axis) shows that the calorimetric method significantly
underestimates the neutrino energy (a). A correction derived from Monte Carlo
can remove the bias (as shown by the reconstructed energy profile), but it
cannot improve the resolution (b).

In order to make a fair comparison between the calorimetric and the kinematic
method, discussed next, we attempt to correct the reconstructed energy. A fit
to the ratio of the true to the reconstructed neutrino energy as a function of
reconstructed energy is used to calculate a correction on an event-by-event
basis. By construction, this correction makes the median bias zero. The
corrected version of Figure 2(a) is shown in Figure 2(b). Profiles based on
the mean show the impact of the highly asymmetric resolution, leading to large
biases remaining in the mean corrected energy. Additionally, while a
correction can significantly reduce the bias, the resolution remains poor.

###  II.2 Neutrino Energy from Final State Kinematics

The capabilities of LAr TPCs motivate a new reconstruction method based on the
kinematics of the final state hadronic system. We have developed a kinematic
model that, using some simplifying assumptions, uniquely determines the
initial neutrino energy, accounting for the energy carried away by the
neutrino. We assume that the neutrino scatters off a single nucleon at rest.
If we ignore nuclear effects, including nucleon-nucleon correlation, nuclear
recoil, and binding energy, the incoming neutrino energy can be determined
from the energy and momentum of the final state hadronic system.

(a) No correction.

(b) With correction.

Figure 3: Reconstructed neutrino energy using the kinematic method (y-axis)
vs. true neutrino energy (x-axis) for NC1p events with neutron tagging shows
reasonable agreement, peaking along the diagonal (a). An overall scale
correction is applied to remove bias (b). The reconstructed energy profile
shows minimal bias at or around the BNB flux peak.

In the lab frame, the initial state neutrino and nucleon four momenta are
given by $p_{\nu,i}^{\mu}=(E_{\nu},0,0,p_{\nu})$ and
$p_{N,i}^{\mu}=(m_{N},0,0,0)$ with $m_{N}$ taken to be the average of the
neutron and proton masses. The z-axis is aligned with the neutrino beam
direction. In the final state, also in the lab frame, the outgoing neutrino is
unmeasured, but the four momentum of the hadronic system can be described by
$p_{h}^{\mu}=(E_{h},\bm{p_{h}})$. From this, we calculate the initial neutrino
energy in terms of the final state hadronic system kinematics, shown in
Equation 1 where $\theta_{h}$ is the hadronic system angle with respect to the
beam direction ($\cos\theta_{h}=\mathbf{p_{h}}\cdot\hat{z}/|\mathbf{p_{h}}|$).
The hadronic four-momentum is calculated by summing the four-momenta of all
visible particles. This method is analogous to the CC energy reconstruction
used by T2K and MiniBooNE, based on the quasi-elastic assumption when only the
final state lepton is observed.

$E_{\nu}^{\text{reco}}=\frac{p_{h}^{2}-(E_{h}-m_{N})^{2}}{2(m_{N}+p_{h}\text{cos}\theta_{h}-E_{h})}$
(1)

Figure 3(a), with the reconstructed neutrino energy plotted as a function of
the true neutrino energy, demonstrates the validity of our method. The figure
was obtained from a NC1p selection with neutron tagging, discussed in the next
subsection. In contrast to the calorimetric method, the distribution obtained
with our method is peaked along the diagonal without any Monte Carlo
corrections. For consistency, as we apply a correction to the kinematic
method, we apply a small scale correction to remove any bias. The correction
is obtained by fitting the absolute energy resolution with a Gaussian to
extract the peak value. The mean reconstructed energy does still show a bias,
as in the calorimetric case, but the resolution is significantly better.

Having introduced both reconstruction methods, we now discuss how we account
for detector effects in our analysis.

###  II.3 Adding Detector Effects

For simplicity, we forgo a full detector simulation and assume Gaussian
angular and energy resolutions for the final state particles. Table 1
summarizes a reasonable range of resolutions by particle type. In addition,
the tracking thresholds, applied to the true kinetic energy, are provided. The
ranges of possible reconstruction performances are taken from published
ArgoNeut [10] and MicroBooNE analyses [11][12][13][14].

Table 1: Assumed performances for LAr TPCs are given for species of interest. Charged pions with momenta below 300 MeV/c are assumed to have similar reconstruction performances as muons. The threshold is set by the requirement that two or more TPC wires are crossed and is applied to the true kinetic energy.  Species | Threshold | Energy | Angular
---|---|---|---
| [MeV] | Resolution | Resolution [deg]
p [10][11] | 25-50 | 60 MeV | 5-10
$\pi^{+/-}$ [12][13] | 10-20 | 10-20% | 2-5
$\gamma$ [14] | 30 | 10-20% | 5-10

If particle species not listed in Table 1 are present in a given NC event,
with the exception of neutrons or neutral pions, the event is excluded from
the analysis. This results in a negligible loss in sample size while making
the interpretation of our results more straightforward. Neutral pions are not
reconstructed directly, by calculating the invariant mass of the decay photons
for example; only the decay photons are used.

In comparing reconstruction performances between the calorimetric and
kinematic methods, discussed next, we adopt the most optimistic choice of the
performances listed in Table 1.

###  II.4 Evaluating Reconstruction Performance

In order to evaluate the reconstruction performance of each reconstruction
method, we first compare the neutrino energy resolutions. Next, we compare the
reconstruction efficiencies. In both cases, we consider different sample
selections. We adopt a NC0$\pi^{+/-}$ selection for the calorimetric method.

(a) No correction.

(b) With correction.

Figure 4: Absolute energy resolution obtained with different reconstruction
methods shows that the kinematic method has better energy resolution and lower
bias than the calorimetric method. Each distribution has been area normalized.
Results for no Monte Carlo correction (a) and with the correction (b) show
that, while the correction removes bias, it does not improve resolution.

For the kinematic method, we investigated several exclusive final state
topologies. Here, we show results for NC0$\pi^{+/-}$ as well as NC1p, the
dominant topology at BNB energies. The other topologies we investigated were
statistically limited. While other topologies are not considered here, it is
worth noting that if the kinematic method were to be applied at DUNE, where
the neutrino flux peaks in the few-GeV range, other final state topologies may
be relevant.

(a) No correction.

(b) With correction.

Figure 5: Fractional energy resolution obtained with different reconstruction
methods shows that the kinematic method has better energy resolution
($\sim$20%) than the calorimetric method ($\sim$90%). Each distribution has
been area normalized. Results for no Monte Carlo correction (a) and with the
correction (b) show that, while the correction removes bias, it does not
improve resolution.

One of the most significant drivers for underestimating the incident neutrino
energy in CC or NC interactions is neutrons. LAr TPCs have some ability to
observe neutrons either through deexcitation photons produced in neutron
capture or through the production of charged hadrons in inelastic scatters. We
neglect deexcitation photons and conservatively estimate that we can tag
neutrons (n-tagging) having kinetic energies above 50 MeV via inelastic
scatters with 50% efficiency. If one or more neutrons are tagged, we make no
attempt to estimate the energy of the scattering neutron(s) and reject the
whole event. This cut has been applied for the kinematic method with NC1p
selection only.

The absolute reconstructed energy resolution is plotted in Figure 4 while the
fractional reconstructed resolution is plotted in Figure 5. Resolutions are
shown for calorimetric NC0$\pi^{+/-}$, kinematic NC0$\pi^{+/-}$, and kinematic
NC1p with n-tagging methods and selections. From both figures, it is evident
that the kinematic method provides a less biased estimate and better
resolution ($\sim$20% vs. $\sim$90%) of the neutrino energy. While Monte Carlo
corrections are not necessary for the kinematic method, a desirable feature to
be sure, a more fair comparison is between the Monte Carlo-corrected
resolutions. With or without corrections, the energy resolution obtained by
the kinematic method is significantly better.

Table 2 summarizes the energy resolution and bias associated with each
reconstruction method and sample selection discussed above. These results
quantitatively demonstrate that the kinematic method provides 80% lower bias
and 25% lower full width at half maximum (FWHM) compared to the calorimetric
method without any corrections. With corrections, the kinematic method
provides 75% lower bias and 45% lower FWHM.

Table 2:  The neutrino energy bias and resolution is summarized for the two different reconstruction methods with different sample selections. Values are shown for both uncorrected (inside parentheses) and corrected (outside parentheses) cases. Method and Selection | Bias [MeV] | FWHM [MeV]
---|---|---
Calorimetric NC0$\pi^{+/-}$ | 108 (662) | 1198 (824)
Kinematic NC0$\pi^{+/-}$ | 20 (133) | 748 (711)
Kinematic NC1p + n-tag | 34 (109) | 524 (524)
Figure 6: The reconstruction efficiency (y-axis) is plotted as a function of
the true neutrino energy (x-axis) for the two different reconstruction
methods, calorimetric or kinematic, and different sample selections: NC
inclusive, NC0$\pi^{+/-}$, or NC1p with neutron tagging. The efficiency values
are most significant near the BNB flux peak at $\sim$600 MeV.

While the kinematic method provides superior neutrino energy resolution
compared to the conventional calorimetric method, it incurs inefficiencies
when the assumptions of the model fail. The model assumptions are valid
primarily for NC quasi-elastic scattering, the dominant process at BNB
energies, as evidenced by the best neutrino energy resolution being obtained
with the NC1p with n-tagging selection. However, sub-leading processes are
significant such as meson exchange current, resonance production, and deep
inelastic scattering.

There is a long tail in the neutrino energy fractional resolution distribution
where the energy is overestimated. This is primarily driven by the waning
validity or outright failure of our model assumptions, the nucleon-nucleon
correlations and the nucleon Fermi momentum in particular. In some cases,
invalid model assumptions result in a negative reconstructed neutrino energy;
these events are rejected, resulting in reduced efficiency with respect to the
calorimetric method.

The energy dependence of the reconstruction efficiency for both methods with
different sample selections is shown in Figure 6. Here, the efficiency is
calculated relative to the true number of NC inclusive events. The efficiency
values are most important near the BNB flux peak.

For NC inclusive selections, the kinematic method has an approximately 15%
lower integrated efficiency. It should be noted, however, that LAr TPC
detectors at these energies struggle to differentiate between muons and
charged pions, each having nearly identical energy loss as a function of
range. It is not unreasonable to assume, therefore, that to eliminate
$\nu_{\mu}$CC background contributions, events with charged pions in the final
state may need to be rejected. This motivates the choice of the NC0$\pi^{+/-}$
selection.

Table 3: Efficiencies integrated over the full range of neutrino energy are provided for the different reconstruction methods, calorimetric or kinematic, and event selections: NC inclusive, NC0$\pi^{+/-}$, or NC1p with neutron tagging.  Method and Selection | Integrated Efficiency
---|---
Calorimetric NCinc | 0.60
Calorimetric NC0$\pi^{+/-}$ | 0.47
Kinematic NCinc | 0.45
Kinematic NC0$\pi^{+/-}$ | 0.37
Kinematic NC1p + n-tag | 0.16

In adopting a NC0$\pi^{+/-}$ selection, the efficiency of the calorimetric
method is more significantly affected than that for the kinematic method as
demonstrated by the integrated efficiencies for the different reconstruction
methods and event selections, shown in Table 3. The difference in efficiency
is reduced to 10%.

Having demonstrated the benefits and limitations of our new reconstruction
method, we now consider one possible use case for the kinematic method:
searching for NC disappearance as a part of SBN’s broader sterile neutrino
search.

##  III Utilizing Neutral Current Interactions in Sterile Neutrino Searches

Over the last few decades, a series of anomalous neutrino flavor oscillation
measurements [15][16][17][18] have been made at short baselines ($L/E\sim$ 1
m/MeV) that could be explained by the existence of one or more eV-mass scale
neutrinos that do not interact via any Standard Model process, so-called
sterile neutrinos. To date, no long-baseline oscillation experiments have
found evidence for sterile neutrinos [7][19][20]. Global analyses [21][22][23]
have shown tension at the level of 4$\sigma$ between short-baseline $\nu_{e}$
appearance measurements and $\nu_{e}$/$\nu_{\mu}$ disappearance measurements.

In the case of one eV-mass scale, stable sterile neutrino added to the
Standard Model - known as a 3+1 model - at short baselines and hundred-MeV- to
GeV-scale neutrino energies, before standard neutrino oscillations become
significant, the disappearance and survival probabilities are described by
Equation 2 and 3 respectively. In this regime, CP-violating phases can be
ignored. This is not the case at long baselines (e.g. MINOS/MINOS+ and
NO$\nu$A) as discussed in the next subsection. The mixing is dominated by
$\Delta m^{2}_{41}$ and the last column of the 4x4 extended PMNS matrix
elements, $U_{\alpha 4}$, where $\alpha$ refers to the neutrino flavors,
$\alpha\in\\{e,\mu,\tau,s\\}$. Here, s refers to the new sterile flavor. The
effective mixing angles are related to the 4x4 PMNS matrix elements as shown
in Equations 4 and 5. $\Delta m_{41}^{2}$ is the mass-squared difference
associated with the new sterile mass state. $L$ and $E_{\nu}$ are the
propagation distance and energy of the neutrino respectively.

$P_{\nu_{\alpha}\rightarrow\nu_{\alpha}}\simeq
1-\sin^{2}(2\theta_{\alpha\alpha})\sin^{2}\left(1.27\Delta
m_{41}^{2}\frac{L}{E_{\nu}}\right)$ (2)

$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}\simeq\sin^{2}(2\theta_{\alpha\beta})\sin^{2}\left(1.27\Delta
m_{41}^{2}\frac{L}{E_{\nu}}\right),\hskip 3.0pt\alpha\neq\beta$ (3)

$\sin^{2}(2\theta_{\alpha\alpha})\equiv 4|U_{\alpha 4}|^{2}(1-|U_{\alpha
4}|^{2})$ (4)

$\sin^{2}(2\theta_{\alpha\beta})\equiv 4|U_{\alpha 4}U_{\beta 4}|^{2},\hskip
3.0pt\alpha\neq\beta$ (5)

For readers who are more accustomed to the parameterization more broadly used
in oscillation experiments, we also provide the relations between
$\theta_{\alpha\beta}$ and $\theta_{ij}$, $i,j=1,2,3,4$, for mixing angles
that are relevant for this discussion.

$\begin{gathered}\sin^{2}2\theta_{\mu\mu}=\sin^{2}2\theta_{24}\cos^{2}\theta_{14}+\sin^{4}\theta_{24}\sin^{2}2\theta_{14}\\\
\sin^{2}2\theta_{\mu e}=\sin^{2}2\theta_{14}\sin^{2}\theta_{24}\\\
\sin^{2}2\theta_{\mu\tau}=\cos^{4}\theta_{14}\sin^{2}2\theta_{24}\sin^{2}\theta_{34}\\\
\sin^{2}2\theta_{\mu
s}=\cos^{4}\theta_{14}\sin^{2}2\theta_{24}\cos^{2}\theta_{34}\end{gathered}$
(6)

Motivated by the need for a definitive resolution to the short-baseline
anomalies, the Short-Baseline Neutrino Program (SBN), hosted at Fermilab, was
proposed in 2015 [1]. SBN consists of three, hundred-ton scale liquid argon
time projection chambers (LAr TPCs) located along the Booster Neutrino Beam
(BNB) axis at distances of hundreds of meters from the BNB target: a near
detector, SBND; an intermediate detector, MicroBooNE; and a far detector,
ICARUS. Table 4 summarizes the detector masses and positions. The detector
positions are optimized for sterile neutrino induced oscillations with a mass-
squared difference of order 1 eV2. SBN sensitivity studies have focused on
$\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance and $\nu_{e}(\bar{\nu}_{e})$
appearance, identified via charged current (CC) interactions. The most recent
sensitivity study [24] shows that SBN is positioned to cover the LSND 90% C.L.
allowed region and most of the globally allowed regions with 5$\sigma$
significance.

Table 4: SBN detector parameters used in this analysis. All values are taken from [1]. Detector | Active Mass | BNB Target | Exposure
---|---|---|---
| [tons] | Distance [m] |  $\times 10^{20}$[POT]
SBND | 112 | 110 | 6.6
MicroBooNE | 89 | 470 | 13.2
ICARUS | 476 | 600 | 6.6

A study evaluating SBN’s sensitivities over a larger range of models than the
minimal 3+1 model, including 3+2 and 3+3 with a consideration of CP-violating
phases, can be found in [25]. There, as in the SBN proposal, only CC
interactions were considered.

We present a statistics only sensitivity analysis in the context of a 3+1
model, a commonly used benchmark in sterile neutrino search sensitivity
studies. Note that sensitivities in the 3+1 case roughly translate into
sensitivities in a 3+N scenario.

Figure 7: The BNB flux prediction at the SBN intermediate detector
(MicroBooNE) [1] is shown for the neutrino running configuration (forward horn
current).

As shown in Figure 7, the BNB flux peaks at about 600 MeV. The flavor content
is about 93.6% $\nu_{\mu}$ and 5.9% $\bar{\nu}_{\mu}$ with contamination from
$\nu_{e}$ and $\bar{\nu}_{e}$ at the level of 0.5% below energies of 1.5 GeV.
Given the low level of contamination from $\nu_{e}$’s and $\bar{\nu}_{e}$’s,
their contribution to the NC disappearance signal will be neglected in our
analysis. This flux is produced with the BNB forward horn current
configuration. Note that the BNB can also be run in a
$\bar{\nu}_{\mu}$-dominant mode by inverting the focusing horn polarity
(reverse horn current). The reverse horn current configuration is not
considered in this analysis.

In a sterile neutrino search based on NC interactions, the signal is the
disappearance of any active neutrinos. NC disappearance is an interesting
channel as it is agnostic to active flavors; NC disappearance is sterile
neutrino appearance. NC disappearance, quantified by
$1-P_{\nu_{\mu}\rightarrow\nu_{s}}$, provides the only means of directly
constraining $|U_{s4}|$. Note that the disappearance probability is the same
for $\bar{\nu}_{\mu}$’s. The corresponding effective mixing angle,
$\sin^{2}2\theta_{\mu s}$, can be related to other mixing angles being probed
at SBN by imposing unitarity on the 4x4 PMNS matrix (Equation 7), yielding the
relation shown in Equation 8, providing a link between the
$\nu_{\mu}$/$\bar{\nu}_{\mu}$ disappearance,
$\nu_{\mu}\rightarrow\nu_{e}$/$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$
appearance, and
$\nu_{\mu}\rightarrow\nu_{\tau}$/$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\tau}$
appearance mixing angles.

$1=\sum_{i=e,\mu,\tau,s}|U_{i4}|^{2}$ (7) $\sin^{2}2\theta_{\mu
s}=\sin^{2}2\theta_{\mu\mu}-\sin^{2}2\theta_{\mu e}-\sin^{2}2\theta_{\mu\tau}$
(8)

###  III.1 Constraints on PMNS Matrix Elements

Equation 8 demonstrates the additional physics reach that could be provided to
SBN by the addition of a NC disappearance search. In addition to providing
complimentary information to the SBN $\nu_{\mu}$ disappearance and $\nu_{e}$
appearance analyses, a NC disappearance search at SBN can provide unique
constraints on the 3+1 phase space.

Past sterile neutrino searches have contributed to a growing collection of
data sets, providing constraints on $|U_{e4}|$ and $|U_{\mu 4}|$. There are
relatively few constraints on $|U_{\tau 4}|$ due to the lack of a $\nu_{\tau}$
source and the few-GeV energy threshold for $\nu_{\tau}$ CC interactions,
nearly beyond the reach of most neutrino sources. To date, the most stringent
constraint on $|U_{\tau 4}|$ comes from atmospheric neutrino oscillations
observed in IceCube [26]. The IceCube measurements utilize the MSW effect to
probe $|U_{\tau 4}|$ via CC interactions.

We are aware of only two experiments that have set limits in the case of a
sterile neutrino NC disappearance search, NO$\nu$A [8][9] and MINOS/MINOS+
[27]. These two experiments and IceCube each set mutually consistent limits in
the $(|U_{\mu 4}|,|U_{\tau 4}|)$ plane with the IceCube limit being more
stringent.

Since MINOS/MINOS+ and NO$\nu$A operate at long baselines, the NC
disappearance probability dependence on $\Delta m_{31}^{2}$ becomes important
for describing oscillations at the far detector, introducing more free
parameters into the oscillation fit. As is shown in Equation 9, there are two
corrections to the short-baseline limit that are driven by $\Delta
m_{31}^{2}$. These additional terms introduce two additional parameters to the
probability calculation: one mixing angle from standard oscillations,
$\theta_{23}$ and one new CP-violating phase, $\delta_{24}$. These challenges
were addressed by incorporating the near detector into a simultaneous two-
detector fit, for example see a recent result from MINOS+ [7]. Oscillations
occurring in the near detector are still well described in the short-baseline
limit. Such measurements are complimentary to SBN, where all three detectors
operate in the short-baseline limit. With a NC disappearance analysis, SBN
should be able provide one of the most stringent constraints on $|U_{\tau 4}|$
to date.

$\begin{gathered}1-P_{\nu_{\mu}\rightarrow\nu_{s}}\approx\\\
1-\cos^{4}\theta_{14}\cos^{2}\theta_{34}\sin^{2}2\theta_{24}\sin^{2}\left(\frac{\Delta
m_{41}^{2}L}{4E}\right)\\\
-\sin^{2}\theta_{34}\sin^{2}2\theta_{23}\sin^{2}\left(\frac{\Delta
m_{31}^{2}L}{4E}\right)\\\ +\frac{1}{2}\sin\delta_{24}\sin\theta_{24}\sin
2\theta_{23}\sin\left(\frac{\Delta m^{2}_{31}L}{2E}\right)\end{gathered}$ (9)

In evaluating the impact of a NC analysis at SBN, it is useful to compare our
sensitivities to the globally allowed phase space. However, this is difficult
in that, to our knowledge, a global analysis of sterile neutrino induced NC
disappearance does not exist. In order to provide some baseline with which we
can benchmark our performance, we used the globally allowed values of the PMNS
matrix elements, specified for two values of $\Delta m_{41}^{2}$, found in
[26]. This supplies four points in the $(\sin^{2}2\theta_{\mu s},\Delta
m_{41}^{2})$ plane after applying Equations 4, 5, and 8. While the allowed
region that this procedure produces is likely not complete, it does provide a
reasonable set of values to which we can compare limits that we produce.
Global analyses of sterile neutrino searches and likewise calculating globally
allowed regions in phase space is difficult. For more discussion, [28] gives a
summary of the current status and challenges of sterile neutrino searches.

##  IV Applying Neutral Current Reconstruction at SBN

We have established that the conventional calorimetric method is better in
terms of efficiency while our new kinematic method performs better in
accurately and precisely reconstructing the neutrino energy. In this section,
we apply each reconstruction method to the search for sterile neutrinos at SBN
via a NC disappearance search in the context of the minimal 3+1 model.

We demonstrate that the calorimetric and kinematic methods are each sufficient
for observing spectral features that are characteristic of neutrino
disappearance. This is important not only for observing disappearance but also
for determining the values of the mixing parameters, $\Delta m^{2}_{41}$ and
$\sin^{2}2\theta_{\mu s}$. We present a comparison of the true and
reconstructed neutrino energy spectra at each SBN-like detector. Next, we
quantify our sensitivity to eV-scale sterile neutrinos in the context of the
minimal 3+1 model. We show exclusion limits followed by allowed regions for
selected nonzero mixing parameters.

For this analysis, each of the detector specific samples is scaled to the
expected exposure provided in the SBN proposal: 6.6 x 1020 POT for the near
and far detectors, 13.2 x 1020 POT for the intermediate detector. We consider
only exclusive NC0$\pi^{+/-}$ samples, identified by the reconstruction, for
this sensitivity analysis. $\nu_{\mu}$CC events where the final state muon and
any charged pions have momenta blow the tracking threshold contribute to event
rates at the percent-level so are neglected from this analysis. We do
anticipate some background from beam induced activity in materials surrounding
the argon active volume, so-called dirt events. These are not expected to
comprise a significant fraction of our NC events. However, all possible
sources of background will be addressed in followup studies using a full
detector simulation and reconstruction.

We present sensitivities for the calorimetric method with NC0$\pi^{+/-}$
selection, the kinematic method with NC0$\pi^{+/-}$ selection, and the
kinematic method with NC1p and n-tagging selection. For these sensitivities,
we only present results obtained without Monte Carlo corrections to the
reconstructed neutrino energy. We verified that the sensitivities are not
affected by the introduction of such corrections.

###  IV.1 Neutrino Energy Spectra

(a) $\Delta m^{2}_{41}$ = 1 eV2, $\sin^{2}2\theta_{\mu s}$ = 0.04

(b) $\Delta m^{2}_{41}$ = 7 eV2, $\sin^{2}2\theta_{\mu s}$ = 0.04

Figure 8: NC neutrino energy spectra with and without oscillations are shown
for each SBN-like detector: near (left column), intermediate (center column),
and far (right column). There is one row per method: true (top row), kinematic
NC1p + n-tag (center row), and calorimetric NC inclusive (bottom row).

To illustrate the power of resolving spectral features in observing
oscillation effects, Figure 8 shows NC neutrino energy spectra for each SBN-
like detector. The null hypothesis, absence of short-baseline oscillations, is
compared to a case with $\sin^{2}2\theta_{\mu s}=0.0403$ and $\Delta
m^{2}_{41}=$ 1 eV2 (8(a)) or $\Delta m^{2}_{41}=$ 7 eV2 (8(b)). These values
are taken from the edge of the globally allowed region in the 3+1 phase space.

The oscillation probabilities are evaluated at a single, fixed baseline for
each detector, set by the detector distance to the BNB target (see Table 4).
In reality, the precise production point of the neutrino is smeared out by the
length of the decay pipe, introducing a systematic uncertainty that will cause
smearing of the oscillation features. This will be addressed in a future
study.

The $L/E_{\nu}$ behavior of the oscillations is evident. A clear, energy
dependent depletion in events is observable in the near detector for both
values of $\Delta m^{2}_{41}$ while the energy dependent depletion at the
intermediate and far detectors is primarily only resolvable at lower values.

Comparing the calorimetric and kinematic methods, we see that the kinematic
method performs better in pinpointing the location of the oscillation maximum.
While the calorimetric method yields some shape information, it does not
directly provide the oscillation peak location, however it has the sizeable
advantage of significantly higher statistics. Comparing the low- and
high-$\Delta m^{2}_{41}$ cases shows how both methods have sensitivity to the
true energy spectral distortions.

In the following subsection, we describe how the advantages of both methods
may be combined to boost the oscillation sensitivity.

###  IV.2 Oscillation Sensitivity

In order to evaluate the sensitivity of our method in the context of SBN, we
adopt a binned Poisson likelihood approach to evaluate the statistical
significance of a 3+1 signal compared to the null hypothesis with no sterile
neutrinos. We use the Asimov data set and construct a log likelihood ratio
($\mathcal{L}$) as shown in Equation 10, where $N^{exp}$ is the expected
number of events given some test hypothesis, and $N^{obs}$ is the observed
number of events. Assuming Wilk’s Theorem applies, we can obtain the
statistical significance $\sigma=\sqrt{\Delta\chi^{2}}=\sqrt{\mathcal{L}}$.
$\Delta\chi^{2}$ is the $\chi^{2}$ difference from the global minimum,
corresponding to the maximum likelihood that is achieved with the Asimov data
set. We use an approximate form of $\mathcal{L}$ obtained through the
application of Sterling’s approximation for sensitivity calculations, shown in
Equation 11. The index runs over bins of reconstructed neutrino energy across
all three SBN-like detectors. For consistency, we verified that our analysis
methods produce a similar result for the 3+1 statistics-only $\nu_{\mu}$
disappearance sensitivity analysis as in [25].

$\mathcal{L}=-2\ln\left[\prod_{i}^{N}\frac{(N_{i}^{obs})^{N_{i}^{exp}}}{N_{i}^{exp}!}\text{e}^{-N_{i}^{obs}}\right]$
(10)

$\mathcal{L}\approx
2\sum_{i}^{N}\left[N_{i}^{obs}-N_{i}^{exp}+N_{i}^{exp}\text{ln}\frac{N_{i}^{exp}}{N_{i}^{obs}}\right]$
(11)

Figure 9: Using the kinematic reconstruction method with NC1p and n-tagging
selection, exclusion contours are shown for a SBN-like, three-detector fit
including statistical uncertainties only. 1$\sigma$, 3$\sigma$, and 5$\sigma$
contours are shown along with the globally allowed 90% C.L. that includes both
statistical and systematic uncertainties [26].

The neutral current disappearance sensitivity for the kinematic method using a
NC1p with n-tagging selection is shown in Figure 9 with 1$\sigma$, 3$\sigma$,
and 5$\sigma$ contours drawn in $(\sin^{2}2\theta_{\mu s},\Delta m^{2}_{41})$
space. The strength of the limit increases as $\Delta m^{2}_{41}$ increases
due to the oscillation maximum moving closer to the far then intermediate
detectors. The enhancement at $\Delta m^{2}_{41}\sim$ 8 eV2 is where the
oscillation maximum occurs in the near detector.

The contours in this analysis include statistical uncertainties only. The
addition of systematic uncertainties is expected to reduce sensitivity
primarily for $\Delta m^{2}_{41}>$ 1 eV2 where the oscillation probability
begins to vary rapidly with neutrino energy. For comparison, globally allowed
values from [26], which include statistical and systematic uncertainties, are
shown to be covered with $5\sigma$ significance.

Figure 10: 1$\sigma$ exclusion contours are shown for a SBN-like, three-
detector fit including statistical uncertainties only for the different
reconstruction methods and sample selections.

While our kinematic reconstruction method is sufficient to cover the globally
allowed region with 5$\sigma$ significance, its limited efficiency yields a
weaker NC disappearance sensitivity compared to the calorimetric method. In
setting limits in our case, statistics are the limiting factor. This motivates
a multi-sample approach where we combine the kinematic and calorimetric
methods - we want to extract the more accurate and precise shape information
with the kinematic method while maintaining the same reconstruction efficiency
as the calorimetric method.

For the multi-sample approach, we construct three statistically independent
samples of reconstructed neutrino energy using a combination of reconstruction
methods and event selections: kinematic NC1p with n-tagging, kinematic
NC0$\pi^{+/-}$, and calorimetric NC0$\pi^{+/-}$. First, the kinematic method
with a NC1p and n-tagging selection is applied. Any events rejected by the
selection are passed to the kinematic method with a NC0$\pi^{+/-}$ selection.
Finally, any events rejected in this step, those with a negative reconstructed
energy, are passed to the calorimetric method with a NC0$\pi^{+/-}$ selection.

1$\sigma$ exclusion limits for the four different methods and selections are
shown in Figure 10. The statistical power of the calorimetric method is
evident compared to kinematic method selections. In the multi-sample selection
that maintains the same overall sample size as the calorimetric method, the
added shape information from the kinematic method improves the sensitivity by
approximately 10%.

Figure 11: For the case with $\Delta m^{2}_{41}$ = 1 eV2 and
$\sin^{2}2\theta_{\mu s}$ = 0.04, 1$\sigma$ allowed regions are shown for a
SBN-like, three-detector fit including statistical uncertainties only for the
different reconstruction methods and sample selections. Figure 12: For the
case with $\Delta m^{2}_{41}$ = 7 eV2 and $\sin^{2}2\theta_{\mu s}$ = 0.04,
1$\sigma$ allowed regions are shown for a SBN-like, three-detector fit
including statistical uncertainties only for the different reconstruction
methods and sample selections.

Note that it is possible to improve the limits shown in Figure 10 in some
regions of phase space by increasing the sample size at the far detector. For
larger values of $\Delta m^{2}_{41}$, the limits are expected to be
systematics limited, driven by the shape measurement in the near detector. For
smaller $\Delta m^{2}_{41}$, however, the limits are constrained by the far
detector statistics. For example, a doubling of the planned exposure at the
far detector results in an approximate 50% sensitivity improvement for 0.5
eV${}^{2}<\Delta m^{2}_{41}<$ 4 eV2, where the oscillation maximum is in close
proximity to the far detector.

The power of the multi-sample approach becomes more evident when considering
the ability to pinpoint the oscillation parameters in the presence of a
disappearance signal where shape information is crucial. Figures 11 and 12
show the 1$\sigma$ allowed regions for the same mixing parameter values shown
in Figures 8(a) and 8(b). In both cases, the multi-sample fit clearly
outperforms the calorimetric method alone. For both cases, with
$\sin^{2}2\theta_{\mu s}=0.0403$ and $\Delta m^{2}_{41}=1~{}\text{eV}^{2}$ or
$\Delta m^{2}_{41}=7~{}\text{eV}^{2}$, the multi-sample approach produces a
30% smaller allowed region compared to the calorimetric method with
NC0$\pi^{+/-}$ selection.

Additionally, we expect the use of two different energy estimation methods
will provide a way of cross-checking biases from interaction modelling
uncertainties, reducing the final systematic uncertainty. While the selected
distributions are not identical, there is substantial overlap.

##  V Conclusions and Outlook

Estimating the incident neutrino energy in neutral current interactions is
challenging. The conventional method of relying on calorimetry alone,
measuring the total visible energy, provides a lower bound on the neutrino
energy, but the accuracy and precision is poor relative to methods used in
charged current interactions. The excellent calorimetry and tracking
capabilities and low tracking thresholds of liquid argon time projection
chambers (LAr TPCs) motivate a new method for reconstructing the incoming
neutrino energy based on the kinematics of the final state hadronic system.

While our new kinematic method provides significantly better energy resolution
and reduced bias compared to the conventional calorimetric method, it suffers
from low efficiency. In combining our method with the conventional one in a
multi-method, multi-sample approach, we showed that we can push oscillation
sensitivities beyond what is possible with the conventional method alone.

Our toy study illustrates how neutral current interactions can be utilized in
the context of sterile neutrino searches, the 3+1 model in particular. As a
relevant example of how our approach can be applied, we showed that the
addition of a neutral current disappearance search to the Short-Baseline
Neutrino Program (SBN) can potentially enhance its physics reach, providing
both complimentary and redundant information that can serve to overconstrain
the sterile neutrino phase space in the currently considered $\nu_{\mu}$
disappearance and $\nu_{e}$ appearance channels. In addition, our approach
could set unique limits on the poorly constrained 3+1 PMNS matrix element,
$U_{\tau 4}$.

We emphasize that this study is a toy study with several caveats. First, we
adopt optimistic reconstruction performance parameters. Next, we assume we can
reduce charged current backgrounds to negligible levels by rejecting any
neutral current event candidates with a muon-like or charged pion-like track.
We further assume that other sources of background are negligible. In
addition, our study neglects the distribution of neutrino production points in
the decay pipe, an effect that smears out spectral oscillation features.
Finally, we do not account for systematic uncertainties. These points need to
be addressed in a more thorough, detailed analysis.

With this proof of concept, we will implement our neutral current analysis
approach in the SBN simulation and reconstruction framework. We will test our
method using a full simulation and reconstruction and reevaluate the
performances presented in this work. Finally, the neutral current analysis
will be integrated into the oscillation sensitivity analysis framework to
provide the most realistic sensitivity projections.

While we wait for SBN to accumulate data, we can take advantage of the data
set obtained by MicroBooNE to test our reconstruction method. This is a
crucial step as the performance we have presented here is model dependent. For
example, if nuclear effects are more or less significant, this will have an
impact on the energy resolution and selection efficiency of the neutral
current analysis and, therefore, the sensitivities presented in this work. To
this end, we can make use of charged current interactions, replacing the
outgoing neutrino in our model with the outgoing lepton. This will provide a
reliable test of using the kinematics of the final state hadronic system as an
estimator of the incoming neutrino energy.

Looking further into the future, the Deep Underground Neutrino Experiment
(DUNE) near detector will be exposed to what will be the world’s most intense
neutrino beam. Along with state-of-the-art detectors, including a LAr TPC and
a magnetized high pressure gaseous argon TPC, the performance of our neutral
current analysis approach should be improved. In a future study, we will
investigate the application of our approach to neutral current inclusive and
exclusive cross section measurements at DUNE.

###### Acknowledgements.

This work was supported by funding from the University of Minnesota.

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