††institutetext: Leinweber Center for Theoretical Physics, Department of
Physics, University of Michigan Ann Arbor, MI 48109 USA

# Probing Non-Standard Neutrino Interactions with Supernova Neutrinos at
Hyper-K

Minjie Lei  Noah Steinberg  James D. Wells

###### Abstract

Non-standard neutrino self interactions (NSSI) could be stronger than Fermi
interactions. We investigate the ability to constrain these four-neutrino
interactions by their effect on the flux of neutrinos originating from a
galactic supernova. In the dense medium of a core collapse supernova, these
new self interactions can have a significant impact on neutrino oscillations,
leading to changes at the flavor evolution and spectra level. We use
simulations of the neutrino flux from a 13 solar mass, core collapse supernova
at 10 kpc away, and numerically propagate these neutrinos through the stellar
medium taking into account vacuum/MSW oscillations, SM $\nu-\nu$ scattering as
well as $\nu-\nu$ interactions that arise from NSSI. We pass the resulting
neutrino flux to a simulation of the future Hyper-Kamiokande detector to see
what constraints on NSSI parameters are possible when the next galactic
supernova becomes visible. We find that these constraints depend strongly on
the neutrino mass hierarchy and if the NSSI is flavor-violating or preserving.
Sensitivity to NSSI in the normal hierarchy (NH) at Hyper-K is limited by the
experiment’s ability to efficiently detect $\nu_{e}$, but deviations from no
NSSI could be seen if the NSSI is particularly strong. In the inverted
hierarchy (IH) scenario, Hyper-K can significantly improve constraints on
flavor-violating NSSI down to $\mathcal{O}(10^{-1})G_{F}$.

##  1 Introduction

The standard picture of three neutrino flavor oscillations has become well
established in the last 20 years, with many details, however, yet to be
determined. In this picture, neutrinos behave according to the SM supplemented
by some yet to be determined mass generation mechanism. Though this picture is
rather complete, there is room for new physics to appear, among other ways, in
the form of non-standard neutrino interactions (NSI), i.e. higher dimensional
operators involving neutrinos and matter. NSI of the form

$\sqrt{2}G_{F}\varepsilon_{\alpha\beta}(\bar{\nu_{\alpha}}\gamma^{\rho}\nu_{\beta})(\bar{f}\gamma_{\rho}f),$
(1)

where $f=e,u,d$ and $\varepsilon_{\alpha\beta}$ is a $3\times 3$ matrix of
couplings in flavor space, have recently been the focus of intense scrutiny
Rius:2018cjf ; Experiments ; Altmannshofer:2018xyo ; Status . In long baseline
oscillation experiments (T2K, IceCube, DUNE, etc.) these non-standard
interactions modify the matter potential through which neutrinos traverse and
can greatly obscure the extraction of neutrino oscillation parameters
($\sin^{2}(2\theta_{ij})$, $\delta_{\text{CP}}$).

In contrast, somewhat less attention has been paid to non-standard self
interactions (NSSI). In the Standard Model, neutrino self interactions occur
through $Z$ boson exchange with an effective Hamiltonian given by:

$\mathcal{H}_{SM}^{\nu-\nu}=\frac{G_{F}}{\sqrt{2}}\sum_{l,l^{\prime}=e,\mu,\tau}(\bar{\nu}_{lL}\gamma_{\alpha}\nu_{lL})(\bar{\nu}_{l^{\prime}L}\gamma^{\alpha}\nu_{l^{\prime}L}),$
(2)

where $G_{F}$ is the Fermi constant. Because of the difficulty of directly
measuring neutrino-neutrino scattering, NSSI of the schematic form
$F(\bar{\nu}\gamma_{\alpha}\nu)(\bar{\nu}\gamma^{\alpha}\nu)$ where
$F\gg\frac{G_{F}}{\sqrt{2}}$ is possible. Indirect bounds on $F$ were obtained
from the decays $\pi^{+}\rightarrow e^{+}\nu_{e}\bar{\nu}\nu$ and
$K^{+}\rightarrow l^{+}\nu_{l}\bar{\nu}\nu(l=e/\mu)$ Manohar:1987ec ;
Bilenky:1980ym  as well as from comparing the expected diffusion time of
neutrinos exiting SN1987A with the time period over which the neutrinos were
detected BialynickaBirula:1964zz . These lead to bounds of roughly
$F<(10^{3}-10^{7})G_{F}$ where the lower bounds come from SN1987A and the
higher bounds from meson decay. The tightest bounds on $F$ come from the
1-loop contributions of the above NSSI operator to the invisible width of the
$Z$ boson Bilenky:1994ma , which is known to $\sim 0.3\%$ Tanabashi:2018oca .
Here, the new 1-loop amplitude interferes with the SM amplitude with the same
final state, and the authors obtain roughly $F<\mathcal{O}(10)G_{F}$. Clearly
NSSI with an interaction strength greater or equal in magnitude to the Fermi
interaction is still allowed from these results.

Recently, it has been understood that the extremely high densities of core
collapse supernova, in conjunction with the relatively large neutrino flux,
could be an ideal environment to look for NSSI Raffelt:2013rqa ;
Blennow:2008er ; Nakazato:2012qf ; Dighe:2017sur ; Das:2017iuj ; Duan:2006an ;
Duan:2008eb . With the construction of the next generation of large scale
neutrino oscillation experiments (DUNE, Hyper-K), the next occurence of a
galactic supernova could give access to NSSI parameters
$\leq\mathcal{O}(10^{-1})G_{F}$ if it is less than $\sim 10$ kpc away.
Analytic as well as large scale computational techniques have been developed
both to study the effects of standard neutrino oscillations Duan:2006an ;
Duan:2008eb ; Fogli:2004ff ; Dighe:1999bi ; Doring:2019axc ; Shalgar:2019kzy ,
and to look for features of NSSI at the flavor evolution and spectra level
Dighe:2017sur ; Kharlanov:2019cpk . However, most of the previous works use
schematic neutrino flux data that have relatively large spectra difference
between flavors, while realistic supernova neutrino data during the relevant
emission periods are often far more degenerate, suppressing oscillation
features. Moreover, although there has been significant progress in next-
generation neutrino detectors since the first detection of SN1987A, the actual
number of neutrino events that can be detected from a future SN is still
limited due to the extremely weak interactions of neutrinos. Thus, some
features of NSSI at the supernova emission level might not be preserved at the
detector data level, a concern we wish to explore.

In this paper we build on previous work by applying computational techniques
to neutrino flux from a realistic numerical supernova simulation
Nakazato:2012qf , and look for NSSI signatures by simulating detection data at
the Hyper-K detector Abe:2018uyc . We develop simple physical observables to
examine in the event of future galactic supernova and estimate the sensitivity
of Hyper-K to NSSI after applying these more realistic simulations of neutrino
flux and detector response.

##  2 Neutrino Production in Core Collapse Supernova

We begin with a brief description of the current picture of neutrino
production in a core collapse supernova. When a sufficiently massive star
($M>8M_{\odot}$) can no longer sustain itself through nuclear fusion, its core
collapses to a proto-neutron star, ejecting its stellar envelope and setting
off the supernova explosion. Neutrinos are produced in three distinct stages:

  1. 1.

Initial collapse and neutronization burst of neutrinos (0 $<t<$ 100 ms)
As the star runs out of nuclear fuel, its core is supported mainly by
degenerate electrons. As the core contracts, these electrons are captured by
nuclei and free protons, producing a large flux of $\nu_{e}$. The reduction in
degeneracy pressure further collapses the core, leading to a runaway
production of $\nu_{e}$ which stream out of the star. When the density of the
inner core approaches $3\times 10^{11}\ \text{g}/\text{cm}^{3}$ the neutrinos
become trapped in the dense matter Nbook . The core stops collapsing and
rebounds, sending a shockwave outwards, dissipating energy by
photodissociation of nuclei leaving behind a plethora of free nucleons. These
nucleons capture electrons, producing a large number of $\nu_{e}$ which pile
up behind the shock wave, until they reach a zone of low enough density and
are released in a few milliseconds. This is known as the neutronization burst.

  2. 2.

Shockwave stall and thermal neutrino flux (100 ms $<t\lesssim$ 300 ms)
In the next phase of neutrino production, the shockwave stalls and matter
accretes on the proto-neutron star and heats up, leading to the thermal
production of neutrinos and anti-neutrinos of all flavors. The low matter
density outside the core means this thermal flux of neutrinos can freely
stream out of the star, corresponding to a period of neutrino emission dubbed
the accretion phase.

  3. 3.

Helmholtz cooling phase and degenerate neutrino flux (300 ms $<t$)
After the shockwave is revived the explosion sets in, with accretion by the
proto-neutron star stopping shortly after. The star enters a Kelvin-Helmholtz
cooling phase, during which it emits neutrinos and antineutrinos of all
flavors on a time scale of $\sim 10$ seconds.

The above serves as a qualitative picture of neutrino production and emission
which depends on the supernova model used. For a more detailed overview of
neutrino production in supernova, see Mirizzi et al. Mirizzi:2015eza . In Fig.
1 we show the luminosity of supernova neutrinos vs. time from a 13 solar mass
star using simulations by Nakazato et al. Nakazato:2012qf . These simulations
neither include NSSI nor take into account subsequent neutrino oscillations
(implementation of which we make in this paper), but instead give the
approximate time evolution of the neutrino flux as the core collapse proceeds.
The left plot of Fig. 1 covers the whole time profile from infall to 20
seconds where the initial spike of $\nu_{e}$ from the neutronization burst is
prominent, followed by an increase in flux of $\nu_{x}$ ($x$ is any other
(anti)lepton flavor) over the accretion phase. The right plot shows the
cooling phase from $\sim 1$ second on, showing the near degeneracy of the flux
between all flavors.

Figure 1: Plots of luminosity over time from reproducing results from Nakazato
et al. Nakazato:2012qf  for a model with initial stellar mass $13m_{\odot}$,
metallicity 0.02, and shock revival time 300 ms. The left panel corresponds to
the whole time profile from infall to 20 seconds after core bounce. The right
panel gives a zoomed in look at the luminosity evolution during the later
cooling stage. Since the luminosity of $\nu_{\mu},\bar{\nu}_{\mu},\nu_{\tau}$,
and $\bar{\nu}_{\tau}$ are degenerate, they are denoted collectively as
$\nu_{x}$.

##  3 NSSI and Neutrino Transport in Supernova Simulations

The dynamics of neutrino transport in supernova is a non-linear problem
involving three distinct oscillation contributions:

  1. 1.

Vacuum oscillations, which depend on the splittings between mass eigenstates
and their relative ordering,

  2. 2.

The MSW effect, which can cause fast flavor conversion in the presence of
extremely large matter densities,

  3. 3.

Forward elastic scattering between neutrinos. This leads to non-linear
‘collective oscillations’ that couple neutrinos of different energies, and
produce bipolar oscillations that split and/or swap the energy spectrum of
neutrinos of different flavors.

The evolution of the neutrino flavor ensemble in the presence of self
interactions was first described by Sigl and Raffelt Sigl:1992fn . The
equation of motion for each neutrino and antineutrino momentum mode is given
by

$\displaystyle i\dot{\rho}_{p}$
$\displaystyle=[(\Omega^{0}_{p}+V+\Omega^{S}_{p}),\rho_{p}],$
$\displaystyle-i\dot{\bar{\rho}}_{p}$
$\displaystyle=[(\Omega^{0}_{p}-V-\Omega^{S}_{p}),\bar{\rho}_{p}].$ (3)

Here $\rho_{p}(\bar{\rho}_{p})$ is the density matrix for neutrinos(anti-
neutrinos) of momentum $p$ in the flavor basis. The diagonal elements give the
occupation numbers of each neutrino flavor and the off-diagonal elements
contain correlations between the mixing flavors Sigl:1992fn . For ultra-
relativistic neutrinos,
$\Omega^{0}_{p}=U\text{diag}(m^{2}_{1},m^{2}_{2},m^{2}_{3})U^{\dagger}/2p$ is
the free Hamiltonian ($U$ is the PMNS matrix),
$V=\sqrt{2}G_{F}n_{e}\text{diag}(1,0,0)$ in the weak interaction basis gives
the contribution to forward scattering off of electrons, and the last term,
$\Omega^{S}_{p}$, is the additional contribution to scattering from self
interactions:

$\Omega^{S}_{p}=\sqrt{2}G_{F}\int
d^{3}q(1-\vec{v}_{q}\cdot\vec{v}_{p})\\{G(\rho_{q}-\bar{\rho}_{q})G+GTr[(\rho_{q}-\bar{\rho}_{q})G]\\}.$
(4)

From here on we work in the 2 x 2 flavor space of $\nu_{e}$ and $\nu_{x}$,
where $x=\mu,\tau$. In the SM, the $G$ matrix is the identity. In the presence
of NSSI, after parameter redefinitions discussed in Dighe:2017sur , $G$ takes
the form

$G=\begin{bmatrix}1+g_{3}&g_{1}\\\ g_{1}&1-g_{3}\end{bmatrix},$ (5)

where $g_{1}$ contibutes to flavor-violating NSSI (FV-NSSI), and $g_{3}$
contributes to flavor-preserving NSSI (FP-NSSI). Evaluating what possible
constraints can be placed on these two couplings with Hyper-K is a central
goal of this paper.

In the supernova environment, there are two broad regimes of neutrino
transport and oscillation: the early high matter density regime where the
neutrino mean free paths are less than the neutrino oscillation lengths, and
the late coherent regime where neutrinos stream out freely from the proto-
neutron star. In the first regime, the high matter density suppresses
collective neutrino oscillations Chakraborty:2011gd , which only become
relevant when the neutrino number density is comparable to or greater than the
nucleon number density. Since NSSI modify collective neutrino interactions,
this first regime is less discerning of NSSI, and so we turn to the second
regime. For the numerical supernova simulations used in this paper
Nakazato:2012qf , we identify the second regime as roughly 1 second after core
bounce, when the inner core of the progenitor star has settled into a proto-
neutron star of radius $\sim$ 10km, from which neutrinos stream out freely,
until 20 seconds, when the flux has significantly dropped off. This period,
from $1$ to $20$ seconds, is the most sensitive to NSSI, so it will be the
focus of this paper.

To simulate neutrino flavor evolution in the coherent regime ($t>1\text{ s}$),
we follow the computational approach developed by Duan et al. Duan:2008eb .
Several approximations are made in this approach. First, neutrino emission
during this regime follows the “neutrino bulb model,” in which neutrinos are
emitted isotropically outwards from the surface of the proto-neutron star. A
simplified supernova matter profile for the late time is assumed, which only
depends on the distance from the center of the neutron star. Finally, in this
paper, to allow the calculation of neutrino emission and evolution in the
entire late time period from 1 to 20 seconds after core bounce, we use the
single angle approximation instead of the multi-angle approximation used in
Duan:2008eb . In this approximation, all neutrinos evolve in the same way as a
radially-propagating neutrino. In the real supernova environment, there are a
variety of physical processes that can affect neutrino transport and
evolution, complicating the simplified procedure adopted here. However, we
justify our approximations by reasoning that the effect of coupled neutrino
oscillations and their modification by NSSI will always be present in any
supernova environment, and the results from Duan et al. Duan:2006an ;
Duan:2008eb  as well as our own calculations have given us confidence that
most of the qualitative and even the quantitative results are likely to
survive in a more advanced supernova simulation. If more sophisticated
supernova models become available, such as the complete evolution of the
matter density profile, they can always be incorporated into the same approach
to modify our results, with qualitative features and approximate sensitivity
estimates expected to be only lightly touched.

##  4 Computational Approach and Simulation Results

Our computational approach is organized as follows. First, we apply the
neutrino flavor transformation calculation procedure by Duan et al.
Duan:2008eb  to neutrino emission at a particular point during the late time
regime of the numerical supernova simulation, chosen to be 5 seconds after
core bounce, and compare the standard oscillation results with those obtained
by Duan et al. using schematic supernova neutrino flux. The supernova model
used has initial mass $M_{\text{init}}=13M_{\odot}$ and metallicity $Z=0.02$,
and we use the same mixing angles and matter density profile (appropriately
scaled to a star with $M_{\text{init}}=13M_{\odot}$), as those found in
Duan:2008eb . The choice of 5 seconds in our initial calculation is simply to
confirm our simulation matches those found in the literature (it does) before
extending it to the whole late time coherent regime. Then we incorporate the
NSSI modification to collective oscillation and observe how different values
of flavor-preserving ($g_{3}$) and flavor-violating ($g_{1}$) parameters alter
the flavor evolution and neutrino spectra. Finally, we extend these
calculations to the entire late time coherent regime from 1 to 20 seconds
after core bounce, and feed the resulting post-oscillation neutrino flux to a
simulation of the Hyper-K detector sntools  to look for signatures of NSSI in
the detector data. The results of these calculations are presented in the
following sections.

###  4.1 Standard Oscillation - No NSSI

Figure 2: Plots of energy-averaged neutrino survival probability evolution in
the core-collapsed supernova environment. The upper panels correspond to the
NH, lower panels to the IH, left panel $\nu_{e}$, and right panels
$\bar{\nu}_{e}$. Three oscillation regimes can be identified from the plots.
Near the proto-neutron star is the synchronized regime, where collective
oscillations are suppressed by the large matter density, and neutrinos
experience $\nu$-enhanced MSW flavor transformations. Far away from the star
the neutrino flux is negligible and neutrinos undergo conventional vacuum/MSW
oscillations. In between these two regions is the bipolar regime where
neutrinos experience collective oscillations and spectral swaps/splits
develop.

Fig. 2 shows energy-averaged survival probability plots for the normal (NH)
and inverted hierarchy (IH) for $\nu_{e}$ and $\bar{\nu_{e}}$. Three distinct
oscillation regimes can be identified in the plots as a function of the radial
distance from the neutrinosphere. Our results agree with Duan:2008eb , and
show that collective oscillation phenomena only become important when the
neutrino density becomes equal to or greater than the matter density
($n_{\nu}\geq n_{e}$), corresponding to the intermediate region of $50\text{
km}<r<200\text{ km}$. At the energy spectra level, a spectral swap/split
agreeing with Duan:2008eb  is observed. This swap/split is on the order of a
couple of MeV. As can be seen in Fig. 3, the energy reconstruction limitations
of Hyper-K Abe:2018uyc  make this behavior, which is only slightly modified by
the addition of NSSI, largely unobservable.

Figure 3: Plots of visible lepton energy in Hyper-K from ES and IBD
interactions. On the left is the NH and on the right is the IH. Figures
include standard neutrino oscillations as well as contributions from FP-NSSI
($g_{1}\neq 0$) and FV-NSSI ($g_{3}\neq 0$). As can be seen, no
splitting/swapping in the energy spectrum is visible.

###  4.2 NSSI Added

Figure 4: Plots of energy-averaged survival probabilities with different
values of FP-NSSI parameter $g_{3}$, the FV-NSSI parameter $g_{1}$ is set to
zero for all cases. The panels are ordered as in Fig. 2 but with the x axis
extended to 1000km, and put in log scale to focus in on the region where NSSI
is most important. The x axis has also bFrom the plots we can see that larger
values of $g_{3}$ correspond to a delay of the onset time of the bipolar
regime in the NH case, as indicated by the vertical lines. The effect is less
apparent in the IH case. Small differences in the final survival probability
develop between different values of $g_{3}$ for $\nu_{e}$ in the NH case, and
for $\bar{\nu}_{e}$ in the IH case.

Now we study the effects of NSSI by scanning $g_{1}$ and $g_{3}$ over the
values {0.01, 0.1, 0.2}. We turn on only one NSSI parameter at a time to more
accurately understand their individual effects on the neutrino spectra. Figs.
4 ($g_{3}$ turned on) and 5 ($g_{1}$ turned on) show the energy-averaged
survival probability as a function of distance from the neutrinosphere for the
NH and IH and for $\nu_{e}$ and $\bar{\nu}_{e}$.

In Fig. 4 we see that in the NH (top), larger values of $g_{3}$ tend to delay
the onset of collective oscillation effects, while for the IH (bottom) the
delay disappears. The final survival probability is slightly modified for
$\nu_{e}$ in the NH (top left) and $\bar{\nu}_{e}$ in the IH (bottom right),
with no change in final survival probability for the opposite cases.

In Fig. 5 we see the flavor-violating coupling, $g_{1}$, has a much more
dramatic impact on the survival probabilities. Larger values of $g_{1}$ tend
to advance the onset of collective oscillations, as can be seen in all four
plots, as well as substantially modify the final survival probability for
$\nu_{e}$ in the NH (top left) and $\bar{\nu}_{e}$ in the IH (bottom right),
with little to no change in final survival probability for the opposite cases.

An analytical and qualitative explanation for this behavior can be found in
Dighe:2017sur , where the authors obtain a linearized, analytic formula for
the rate of flavor conversions, represented by $S_{\omega,u}$, where $\omega$
is the neutrino energy and $u=sin^{2}(\theta_{R})$ represents the emission
angle. For only one non-zero NSSI parameter these equations of motion are
eigenvalue equations with solutions that grow like $e^{\kappa t}$ for $\kappa$
real. For FP-NSSI (FV-NSSI) one finds that $\kappa$ is reduced (enhanced) by a
factor of $1-g_{3}^{2}$ ($1+g_{1}^{2}$) relative to the no-NSSI case, which
leads to the delay (advancement) of collective oscillation effects. Thus, our
results agree with Dighe:2017sur .

As our goal is to understand the effects of NSSI on the observed spectrum at
Hyper-K of a future galactic supernova, we are most interested in the
differences between the energy-averaged final survival probabilities for
different NSSI parameter values. The final flux of electron neutrinos
($\nu^{\prime}s$ per $4\pi$ steradian per second), $F^{f}_{e}$, is given in
terms of the initial fluxes $F_{e},F_{x}$ and the survival probability
$P_{\nu_{e}\nu_{e}}$ (the survival probability at the surface of the star) by

$F^{f}_{e}=P_{\nu_{e}\nu_{e}}F_{e}+(1-P_{\nu_{e}\nu_{e}})F_{x}=P_{\nu_{e}\nu_{e}}(F_{e}-F_{x})+F_{x},$
(6)

with an analogous formula for anti-electron neutrinos. During the late-time
cooling phase, where NSSI have the largest impact, the flux hierarchy is
$F_{x}>F_{e},\bar{F}_{e}$ Nakazato:2012qf . Therefore, larger values of
$P_{\nu_{e}\nu_{e}}$ correspond to a smaller final flux. Examining Figs. 4 and
5, we can predict that for $\nu_{e}(\bar{\nu}_{e})$ in the NH(IH), the final
flux in the flavor-violating NSSI scenario will be larger(smaller) than the
final flux in both the flavor-preserving NSSI and standard scenarios. As the
cooling phase continues, the fluxes of different flavors become more and more
degenerate and this difference approaches zero.

Figure 5: Plots of energy-averaged survival probabilities with different
values of flavor-violating NSSI parameter $g_{1}$, the flavor-preserving NSSI
parameter $g_{3}$ is set to zero for all cases. The panels are ordered as in
Fig. 2 but with the x axis extended to 1000km, and put in log scale to focus
in on the region where NSSI is most important.Here, larger values of $g_{1}$
correspond to an advance of the onset time of the bipolar regime in the NH
case. Large differences can be observed in the final survival probability
between different values of $g_{3}$ for $\nu_{e}$ in the NH case, and for
$\bar{\nu}_{e}$ in the IH case.

###  4.3 Hyper-K Sensitivity to NSSI

In this section we discuss the detection capabilities of Hyper-K. The peak of
the supernova neutrino energy spectrum is around $10-15$ MeV, with an
exponentially falling tail. Only three interaction channels have low enough
thresholds at these energies: inverse beta decay (IBD)
$\bar{\nu}_{e}+\,p\rightarrow e^{+}+\,n$, neutrino-electron elastic scattering
(ES) $\nu+e^{-}\rightarrow\nu+e^{-}$, and charged current scattering off of
oxygen-16 nuclei (OCC) $\nu_{e}(\bar{\nu}_{e})+^{16}O\rightarrow
e^{-}(e^{+})+X$. The initial burst of neutrinos from the neutronization phase
of the supernova will mostly be $\nu_{e}$, which will allow for an unambiguous
identification of the supernova onset time to within 1 ms Abe:2018uyc . At
Hyper-K, we expect the event distribution to be dominated by IBD, resulting in
50,000 to 75,000 events, along with 3,000 to 4,000 ES and 1,000 to 14,000 OCC
events detected over a 20 second period for a core collapse supernova at 10
kpc. Roughly 70 - 80% of these events occur in the 1 to 20 second time period
of interest, as can be gleaned from Abe:2018uyc . Looking at Fig. 5, this is
optimal for constraining flavor-violating NSSI in the IH, as the
$\bar{\nu}_{e}$ final survival probability depends strongly on the value of
$g_{1}$. In the NH, the final $\nu_{e}$ survival probability also depends
strongly on $g_{1}$ but poor statistics in the ES channel worsen the ability
of Hyper-K to distinguish between NSSI and no NSSI.

In Fig. 6 we show the number of IBD (left) and ES (right) events vs. time for
both the NH (top) and IH (bottom), for no NSSI,
$\\{g_{1},g_{3}\\}=\\{0,0.2\\}$ (FP-NSSI), and $\\{g_{1},g_{3}\\}=\\{0.2,0\\}$
(FV-NSSI). No corresponding figure is shown for the OCC events as their
contribution to the overall event rate in the 1 to 20 second time window is at
least an order of magnitude smaller than the ES and IBD event rates. It is
easy to see that the IBD channel cannot easily constrain NSSI in the NH
because the final survival probability for $\bar{\nu}_{e}$ (Figs. 4 and 5)
does not depend strongly enough on $g_{1}$ or $g_{3}$. Similarly, ES at
Hyper-K cannot well constrain NSSI in either hierarchy, but this is mainly due
to poor statistics in this channel. The largest difference can be seen in the
bottom left plot of Fig. 6 in the IH, IBD channel, where there is a
significant reduction in the event rate for FV-NSSI interactions with strength
$g_{1}=0.2$.

Figure 6: Plot of number of detected events over time, starting at $t=1$
second. $t=0$ is set to the time of the observed onset of the neutrino flux.
The upper panels correspond to NH, lower panels IH, left panels the inverse
beta decay channel, and the right panels the elastic scattering channel. Error
bars are statistical only. Since the IBD channel correspond to $\bar{\nu_{e}}$
detection and ES channel mainly correspond to $\nu_{e}$ detection, as
predicted by the flavor evolution plots in Figs. 4 and 5, we should expect to
see significant differences between flavor-violating NSSI and flavor-
preserving NSSI/standard for IBD in the IH case, and for ES in the NH case,
and the difference should decrease as neutrino fluxes become more degenerate
over time. This is clearly the case for the IBD channel. However, it is not
observed in the ES channel, due to poor $\nu_{e}$ statistics from water
cherenkov detectors such as Hyper-K.

Based on numerical supernova simulations Nakazato:2012qf , neutrino flux and
luminosity during the cooling phase are largely independent of supernova
models, so we should restrict any observable to this phase of the supernova (1
s $<$ $t$ $<$ 20 s), which fortuitously is the most sensitive to NSSI.
Additionally, to remove the dependence on the distance of the supernova from
the detector (which affects the absolute neutrino flux), and minimize the
effect of systematic errors, one can take ratios of events in different
interaction channels or in different time bins.

In the standard Hyper-K setup with only water, it is difficult to distinguish
between IBD and ES interactions on an event by event basis. This is because
electrons and positrons are indistinguishable in a water cherenkov detector,
but two solutions are possible. ES and IBD events may be separated on a
statistical basis by the angular distributions of the produced (anti-)lepton,
as electrons from ES will be approximately forward going and positrons
produced from IBD will be essentially isotropic. The second option is neutron
tagging which the EGADS collaboration is currently investigating by doping
water with Gadolinium Xu:2016cfv . If Gadolinium (Gd) is added to the water in
Hyper-K then IBD interactions are identifiable via the resultant neutron
capture. Gadolinium has a substantial thermal neutron capture cross section of
49,000 barns and emits an 8 MeV gamma cascade which can be detected via its
cherenkov light. At a concentration of 0.1-0.2% the efficiency for neutron
capture is roughly 90%.

A combination of statistical, cut based methods as well as the addition of
gadolinium could allow for IBD identification with an efficiency between
$90-100\%$. In this case we identify an observable, $R_{\text{IBD}}$(7), the
ratio of total ES + OCC events to total IBD events from $1\text{ s}<t<20\text{
s}$ after the onset of the neutrino flux, to be an indicator of NSSI.

$R_{\text{IBD}}=\frac{\text{ES and OCC events from 1 s $<$ $t$ $<$ 20 s
}}{\text{IBD events from 1 s $<$ $t$ $<$ 20 s}},$ (7)

where $t=0$ is the standardly defined onset time of the neutrino flux.

Assuming $100\%$ IBD identification efficiency, we show results for
$R_{\text{IBD}}$ in Table 1 where the errors on each ratio are statistical.
FV-NSSI in the IH with $g_{1}=0.2$ or greater will lead to an increase in this
ratio of more than $10\%$, which we estimate to be discernible from the no
NSSI standard scenario. For IBD identification below $100\%$ efficiency, the
discerning power of this observable rapidly deteriorates. This is because the
number of IBD events is an order of magnitude larger than ES and OCC events,
so any migration of IBD events due to misidentification will quickly change
$R_{\text{IBD}}$. Below in Fig. 7 we show the sensitivity of $R_{\text{IBD}}$
to the IBD identification efficiency by plotting this observable for FV-NSSI
in the IH as a function of the IBD identification efficiency. As the IBD
identification efficiency drops below $99\%$ $R_{\text{IBD}}$ rapidly deviates
from its nominal value at $100\%$ efficiency making NSSI vs no NSSI
distinctions difficult. Thus, this observable becomes less reliable as the IBD
identification efficiency deteriorates.

$R_{\text{IBD}}\times 100\%$

| Standard | FP-NSSI | FV-NSSI
---|---|---|---
NH | 12.95$\pm$0.37% | 12.75$\pm$0.36% | 12.72$\pm$0.36%
IH | 10.14$\pm$0.26% | 10.21$\pm$0.26% | 11.52$\pm$0.30%

Table 1: $R_{\text{IBD}}$(7), the ratio of total events detected during the
late time regime ($1\text{ s}<t<20\text{ s}$) in the ES and OCC channels to
those in the IBD channel for both NH and IH cases, assuming $100\%$ IBD
identification efficiency. The ratio provides a largely model-independent
signature for NSSI modifications. In this case, there is a significant
difference between flavor-violating NSSI and flavor-preserving NSSI/standard
in the IH case. With better ES channel statistics, such as from a liquid-Ar
detector like DUNE, we expect to see much smaller statistical variation in the
ratios, as well as a difference in the NH case, as predicted by the flavor
history plots in Fig. 4 & 5. Figure 7: Plot of $R_{\text{IBD}}$ for FV-NSSI in
the IH vs IBD identification efficiency. As the IBD identification efficiency
drops below $99\%$ $R_{\text{IBD}}$ rapidly deviates from its value at $100\%$
efficiency making NSSI vs no NSSI distinctions difficult. Thus, this
observable becomes less reliable as the IBD identification efficiency
deteriorates.

In the event that the IBD identification efficiency is not adequate for
looking for the effects of NSSI, one should construct an observable that is
independent of the ability of Hyper-K to identify specific neutrino
interaction types. We take this observable, $R_{\text{10/20}}$, to be the
ratio of the total number of events between 1 and 11 seconds to the total
number of events between 11 and 20 seconds.

$R_{\text{10/20}}=\frac{\text{total events from 1 s $<$ $t$ $<$ 11 s
}}{\text{total events from 11 s $<$ $t$ $<$ 20 s}},$ (8)

where $t=0$ is the standardly defined onset time of the neutrino flux.

The behavior of this observable can be seen from Fig. 6. For IBD in the
inverted hierarchy, one can see that the Standard, FP-NSSI, and FV-NSSI event
rates are nearly degenerate for $t>11\text{ s}$, but the FV-NSSI event rate is
suppressed in the region from $1\text{ s}<t<11\text{ s}$. In the NH this
suppression is not present because the final survival probability is only very
weakly dependent on $g_{1}$ and $g_{3}$ as can be seen in Figs. 4 and 5. We
show results for $R_{\text{10/20}}$ in Table 2 where the errors on each ratio
are statistical. FV-NSSI in the IH with $g_{1}=0.2$ or greater will lead to an
increase in this ratio of $10\%$, which again we estimate to be discernible
from the no NSSI standard scenario. We note that this observable is not
sensitive to the 20 second endpoint because statistics rapidly fall for
$t>20\text{ s}$.

$R_{\text{10/20}}\times 100\%$

| Standard | FP-NSSI | FV-NSSI
---|---|---|---
NH | 20.76$\pm$0.50% | 20.16$\pm$0.49% | 20.38$\pm$0.40%
IH | 16.54$\pm$0.35% | 17.14$\pm$0.36% | 18.32$\pm$0.40%

Table 2: $R_{\text{10/20}}$(8), the ratio of total events detected during
$1\text{ s}<t<11\text{ s}$ in to those detected during $11\text{ s}<t<20\text{
s}$ for both NH and IH cases. This ratio provides a largely model-independent
signature for NSSI modifications that is independent of Hyper-K’s ability to
identify the neutrino interaction type. In this case, there is a significant
difference between flavor-violating NSSI and flavor-preserving NSSI/standard
in the IH case.

##  5 Conclusion

Probing NSSI with supernova neutrinos presents a unique opportunity to search
for new physics in the neutrino sector. By using a realistic supernova
simulation under a suite of reasonable assumptions and combining it with
advanced detector simulations, we have evaluated the ability of Hyper-K to
constrain NSSI. We have shown that a single galactic supernova combined with
Hyper-K’s massive detection volume will allow one to constrain flavor-
violating NSSI in the IH to two orders of magnitude smaller than current
bounds ($g<0.2$ vs. $g<10$ previously) even if Hyper-K is unable to identify
the neutrino interaction type. Additionally, a complementary experiment like
DUNE, which uses liquid argon, will give a much better measurement of the
$\nu_{e}$ flux from a supernova, allowing us to probe NSSI in the NH as well.
Future work will focus on higher order improvements to the supernova
simulation, such as implementing more realistic density profiles, full three
flavor neutrino framework, and multi-angle instead of single angle
approximation. We also wish to extend these results to DUNE after detector
response characterization is more mature.

###### Acknowledgements.

We thank Advanced Research Computing at the University of Michigan, Ann Arbor
for their computational resources, as well as M. Buschmann, Z. Zhang, and K.
Nakazato for helpful comments and discussion. The authors are supported by DoE
grant de-sc0007859. The work of N. Steinberg was supported by a Leinweber
Graduate Fellowship.

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