# Using supernova neutrinos to probe strange spin of proton
with JUNO and THEIA

Bhavesh Chauhan ID<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Iowa, Iowa City, IA 52242, USA

###### Abstract

The strange quark contribution to proton’s spin ($\Delta s$) is a fundamental
quantity that is poorly determined from current experiments. Neutrino-proton
elastic scattering (pES) is a promising channel to measure this quantity, and
requires an intense source of low-energy neutrinos and a low-threshold
detector with excellent resolution. In this paper, we propose that neutrinos
from a galactic supernova and their interactions with protons in large-volume
scintillation detectors can be utilized to determine $\Delta s$. The spectra
of all flavors of supernova neutrinos can be independently determined using a
combination of DUNE and Super-(Hyper-)Kamiokande. This allows us to predict
pES event rates in JUNO and THEIA, and estimate $\Delta s$ by comparing with
detected events. We find that the projected sensitivity for a supernova at 1
kpc (10 kpc), is approximately $\pm 0.01$ ($\pm 0.15$). Interestingly, the
limits from a nearby supernova would be comparable to the results from lattice
QCD, and better than polarized deep-inelastic scattering experiments. Using
supernova neutrinos provides a true $Q^{2}\rightarrow 0$ measurement, and thus
an axial-mass independent determination of $\Delta s$.

##  I Introduction

One of the very first measurement of the contribution of strange quarks to the
spin of proton was performed by the European Muon Collaboration (EMC). The
strange spin of proton ($\Delta s$) is related to the axial form-factor in the
limit of vanishing momentum-transfer ($Q^{2}$). Usually, the experiments
measure $\Delta s$ at $Q^{2}\sim\rm GeV^{2}$, and extrapolate the results to
$Q^{2}=0$ by assuming some parametrization for the axial form-factor. The EMC
collaboration obtained,

$\Delta s\,{\rm(EMC)}=-0.095\pm 0.016\pm 0.023$

through polarized deep-inelastic scattering (pDIS) Ashman  _et al._ (1989).
The contribution is not only non-zero, which is in violation of the Ellis-
Jaffe sum rule Ellis and Jaffe (1974), but also negative. Subsequent
measurements by COMPASS Alexakhin  _et al._ (2007) and HERMES Airapetian  _et
al._ (2007) have independently determined $\Delta s$, and their results are
consistent with EMC. A recent global analysis can be found in Ref. Leader  _et
al._ (2015).

On the theory front, heavy-quark contribution to the axial form-factor has
been estimated in Refs. Collins  _et al._ (1978); Mohapatra and Senjanovic
(1979); Wolfenstein (1979). In recent times, precise lattice QCD calculations
of the nuclear structure suggest that

$\Delta s\,{\rm(Lattice)}=-0.018\pm 0.006$

which is closer to zero, but negative Chambers  _et al._ (2015). Other groups
have also obtained similar estimates (see Refs. Dong  _et al._ (1995); Bali
_et al._ (2012); Engelhardt (2012)).

Since neutrinos only interact via the weak force, neutrino proton elastic
scattering (pES) is sensitive to the axial form-factor and a possible channel
for measuring $\Delta s$. A clear advantage of using neutrinos is that one
does not need to rely on flavor symmetries or fragmentation functions. With
this motivation, the E734 experiment at Brookhaven National Lab used neutrino
beams on a liquid scintillator target to determine the axial form-factor for
$Q^{2}\in 0.45$–1.05 $\rm{GeV}^{2}$ Ahrens  _et al._ (1987). Upon
extrapolating their results to $Q^{2}=0$, they obtained

$\Delta s\,{\rm(E734)}=-0.15\pm 0.09$

for fixed axial-mass parameter ($M_{A}$). If $M_{A}$ is not fixed, then there
is large uncertainty in the extracted value of $\Delta s$ as they are strongly
correlated at these energies. The Fermilab experiment MiniBooNE has also
measured $\Delta s$ through pES and obtained $\Delta s=0.08\pm 0.26$ by
measuring the ratio of neutrino-proton and neutrino-nucleon interactions. In
Ref. Sufian  _et al._ (2020), the authors reanalyze MiniBooNE data using
$z$-expansion parametrization of the form factors and obtain

$\Delta s\,{\rm(MiniBooNE)}=-0.102\pm 0.178\pm 0.080$

where the uncertainty is smaller due to fixing some parameters with results
from lattice QCD. For a measurement of $\Delta s$ that is practically
independent of $M_{A}$, one requires an intense source of low-energy neutrinos
and a detector that can efficiently measure the small energies of the
recoiling proton. In Ref. Pagliaroli  _et al._ (2013), the authors propose
using neutrinos from pion deay-at-rest to measure $\Delta s$ in a kton-scale
scintillation detector. In this paper, we look at the possibility of measuring
$\Delta s$ using supernova neutrinos and large-volume scintillation detectors.

In a core collapse supernova, nearly 99% of the binding energy of the
progenitor is released in the form of neutrinos. These neutrinos are emitted
with energies $\mathcal{O}(10)$ MeV, and can generate nuclear recoils with
$Q^{2}\sim 0.01\,{\rm GeV}^{2}$. The elastic scattering of these neutrinos
with protons has been identified as a promising channel to study the non-
electron flavors, i.e.,
$\nu_{x}=\nu_{\mu},\,\bar{\nu}_{\mu},\,\nu_{\tau},\,\bar{\nu}_{\tau}$, using
scintillation detectors Beacom  _et al._ (2002). Even though the scintillation
from recoil proton is quenched, one can get enough statistics to reconstruct
the spectrum of $\nu_{x}$ Dasgupta and Beacom (2011); Lu  _et al._ (2016); Li
_et al._ (2018, 2019); Nagakura (2020). In these reconstruction techniques,
$\Delta s$ has been regarded as a source of uncertainty in the cross-section.
In this paper, we invert the question and ask, _“how well can we constrain
$\Delta s$ using neutrinos from a galactic supernova?”_. The uncertainty in
reconstruction of $\Delta s$ can be approximated by,

$\delta{\Delta
s}\approx\frac{1.27}{2}\left[\frac{1}{N}+\left(\frac{\delta\Phi}{\Phi}\right)^{2}\right]^{1/2}$
(1)

where $N$ is the total number of events and $\delta\Phi/\Phi$ is the
fractional uncertainty in the _total_ supernova neutrino flux. A multi-kton
scintillation detector would observe few thousand pES events from a supernova
at 10 kpc, resulting in a small contribution from the first term. Hence, the
reconstruction of $\Delta s$ mainly depends on how well we can determine the
supernova neutrino spectra.

Of all currently operational neutrino telescopes, the most promising candidate
to measure supernova neutrino spectra is the water Cherenkov detector Super-
Kamiokande Ikeda  _et al._ (2007). The addition of Gadolinium (Gd) salts has
greatly enhanced the neutron tagging efficiency Beacom and Vagins (2004); Abe
_et al._ (2022) and allows for a clean and large-statistics measurement of
$\bar{\nu}_{e}$ using inverse beta decay. In order to detect other flavors,
one has to rely on future detectors such as DUNE Abi  _et al._ (2020a, b),
JUNO An  _et al._ (2016a); Djurcic  _et al._ (2015); Abusleme  _et al._
(2021), Hyper-Kamiokande Abe  _et al._ (2011), and THEIA Askins  _et al._
(2020). There are other proposed detectors such as HALO Duba  _et al._ (2008),
deuterated liquid scintillator Chauhan  _et al._ (2021), and RES-NOVA
Pattavina  _et al._ (2021) that have relatively smaller volume, but will play
an important role in detection of supernova neutrinos.

For our analysis, we find that DUNE is the most promising candidate for
measurement of $\nu_{e}$ spectrum via charged-current interactions with argon
Nikrant  _et al._ (2018). The water Cherenkov detector Super-Kamiokande (SK)
or its upgrade Hyper-Kamiokande (HK) (both assumed to have Gd) would measure
$\bar{\nu}_{e}$ spectrum via inverse-beta decay. The non-electron flavors,
i.e., $\nu_{x}$ can only be detected through neutral-current interactions. In
SK and HK, $\nu_{x}$ will be detected using elastic scattering with electrons,
which also gets contribution from $\nu_{e}$ and $\bar{\nu}_{e}$ Gallo Rosso
_et al._ (2018). Thus, a combination of DUNE and SK/HK, can detect all three
components of supernova neutrinos independent of $\Delta s$. The measured
neutrino spectra from these detectors can be used to predict the pES event
rates in scintillation detectors, and comparison with measured rates can yield
limits on $\Delta s$. This possibility is explored in detail in this paper.

So far, only linear alkyl benzene (LAB) based scintillator detectors such as
Borexino, KamLAND, SNO+, and JUNO have been considered as candidates to detect
pES events from a galactic supernova Beacom  _et al._ (2002); Dasgupta and
Beacom (2011); Lu  _et al._ (2016); Li  _et al._ (2018, 2019). In this paper,
we show that the water-based liquid scintillator (WbLS) detector, THEIA, would
be an excellent candidate to detect pES events. The higher light yield and
smaller $\,{}^{14}\rm{C}$ concentration allows for a lower threshold as
compared to JUNO, and larger events rates despite similar volume. Using a
benchmark value for fluence parameters, and uncertainties from DUNE and SK/HK
determined from Ref. Nikrant  _et al._ (2018) and Gallo Rosso  _et al._
(2018), we determine the sensitivity to $\Delta s$ using a simple Monte Carlo
simulation for uncertainty quantification. The main results of this paper are
summarized in Fig. 1, where we show our projected sensitivity from a galactic
supernova along with results from pDIS, beam-neutrino experiments, and lattice
QCD.

Figure 1:  The $1\sigma$ estimate of $\Delta s$ from the polarized deep
inelastic scattering (pDIS) experiments, $\nu$p scattering experiments, as
well as Lattice QCD are shown along with our projected sensitivity from a
supernova at 10, 5, and 1 kpc. The darker blue and orange limits are for JUNO
and THEIA respectively where the fluence parameters are determined from
DUNE+HK. The weaker limits obtained using reconstructed fluence parameters
DUNE+SK are shown with a lighter shade.

##  II Forecast for ${\rm pES}$ in THEIA

THEIA is a proposed hybrid detector that uses a cocktail-like water-based
liquid scintillator (WbLS) as target Askins  _et al._ (2020). There are two
possible configurations, THEIA25 with 25 kton WbLS (20 kton fiducial volume)
and THEIA100 with 100 kton WbLS (80 kton fiducial volume). In this work, we
provide estimate for THEIA25, with an understanding that the results for
THEIA100 can be appropriately scaled. The unique advantage of WbLS is the
ability to simultaneously detect both scintillation and Cherenkov signals of
charged particles. A large-scale WbLS detector will have excellent capacity to
study diffuse supernova neutrino background, reactor neutrinos, and new
physics scenarios such as proton decay Askins  _et al._ (2020). If operational
at the time, THEIA can also detect neutrinos from the next galactic supernova
through inverse beta decay, electron elastic scattering, and interactions with
oxygen (similar to a Cherenkov detector like SK/HK). As THEIA can also operate
as scintillation detector, it will be sensitive to pES similar to JUNO.

To estimate the spectrum of pES events in THEIA, we follow the outline of Ref.
Dasgupta and Beacom (2011). We assume a fiducial galactic supernova that emits
a total energy of $3\times 10^{53}$ erg over a duration of $\sim 15\,$s. The
fluence of neutrinos from such a supernova is given by

$\dfrac{dF_{\nu}}{dE}=\frac{1}{4\pi d^{2}}\frac{\mathcal{E_{\nu}}}{\langle
E_{\nu}\rangle}\times\frac{d\varphi_{\nu}}{dE}$ (2)

where $\nu=\nu_{e},\bar{\nu}_{e},\nu_{x}$ represents the neutrino flavors, $d$
is the distance to the supernova, $\mathcal{E}_{\nu}$ is the luminosity,
$\langle E_{\nu}\rangle$ is the average energy, and $d\varphi_{\nu}/dE$ is the
unit-normalized spectra which can be parameterized as,

$\dfrac{d\varphi_{\nu}}{dE}=\frac{(1+\alpha_{\nu})^{1+\alpha_{\nu}}}{\Gamma(1+\alpha_{\nu})}\frac{E^{\alpha_{\nu}}}{\langle
E_{\nu}\rangle^{1+\alpha_{\nu}}}\exp\left(-(1+\alpha_{\nu})\frac{E}{\langle
E_{\nu}\rangle}\right)$ (3)

where $\alpha_{\nu}$ is called the _pinching_ parameter Keil  _et al._ (2003).
For our estimates, we use $d=10$ kpc, $\alpha_{\nu}$ = 3, $\langle
E_{\nu_{e}}\rangle$=12 MeV, $\langle E_{\bar{\nu}_{e}}\rangle$=14 MeV, and
$\langle E_{\nu_{x}}\rangle$=16 MeV. We assume equipartition of among all six
flavors.

The differential cross section for pES in the Llewellyn-Smith formalism is

$\frac{d\sigma}{dQ^{2}}=\frac{G_{F}^{2}M_{p}^{2}}{8\pi E_{\nu}^{2}}\left[A\pm
B\frac{s-u}{M_{p}^{2}}+C\frac{(s-u)^{2}}{M_{p}^{4}}\right]\,,$ (4)

where $s-u=4M_{p}E_{\nu}-Q^{2}$, $Q^{2}=2M_{p}T_{p}$, and $T_{p}$ is the
kinetic energy of the recoiling proton Llewellyn Smith (1972). In terms of
$\tau=Q^{2}/4M_{p}^{2}$,

$\displaystyle A$
$\displaystyle=4\tau\left[(1+\tau)G_{A}^{2}-(1-\tau)F_{2}^{2}+\tau(1-\tau)F_{2}^{2}+4\tau
F_{1}F_{2}\right],$ $\displaystyle B$
$\displaystyle=4\tau\left[G_{A}(F_{1}+F_{2})\right],$ $\displaystyle C$
$\displaystyle=\frac{1}{4}\left[G_{A}^{2}+F_{1}^{2}+F_{2}^{2}\right],$

where $F_{1,2}$ are the vector form-factors111We use $\sin^{2}(\theta_{W})$ =
0.238, which is the appropriate low-$Q^{2}$ limit, resulting in a slightly
weaker dependence on vector form-factors as compared to Ref. Beacom  _et al._
(2002)., and $G_{A}$ is the axial form-factor Ahrens  _et al._ (1987). For
supernova neutrino interactions, the maximum kinetic energy of recoiling
protons is only a few MeV and a simpler expression for the cross section can
be derived in the limit $Q^{2}\rightarrow 0$ Beacom  _et al._ (2002). The
axial form-factor in this limit can be written as,

$\lim_{Q^{2}\rightarrow 0}G_{A}=\frac{1}{2}\left(\mathtt{g}_{A}-\Delta
s\right)$ (5)

where $\mathtt{g}_{A}=1.2755(11)$ Czarnecki  _et al._ (2018). We note that,
for $\Delta s\in\pm 0.15$, the total cross section can change by $\mp\,25\%$.
To estimate the event rates, we use $\Delta s=-0.018$, which is the central
value reported in Ref. Chambers  _et al._ (2015).

Figure 2:  The visible energy ($E_{vis}$) as a function of the recoil proton
energy in JUNO (from von Krosigk et. al. von Krosigk  _et al._ (2013)) and in
THEIA (from Callagan et. al. Callaghan  _et al._ (2022)) is shown in blue and
orange respectively. The shaded region shows the $1\sigma$ uncertainty arising
from $k_{B}$ and $C$ given in eq. (7). For comparison, we also show in gray
the quenching for $k_{B}$ = 0.015 cm/MeV which is used in Ref. Beacom  _et
al._ (2002).

Only a fraction of proton’s recoil energy is converted into detectable
scintillation signal, a phenomenon known as _quenching_. The quenched proton
energy, also called the visible energy, $E_{vis}$, is given by,

$E_{vis}=\int_{0}^{T_{p}}\frac{dE}{1+k_{B}\langle dE/dx\rangle+C\langle
dE/dx\rangle^{2}},$ (6)

where $T_{p}$ is the kinetic energy of the recoiling proton, $\langle
dE/dx\rangle$ is the average energy loss of proton in the medium, $k_{B}$ is
the Birks’ constant Birks (1951), and $C$ is the bimolecular correction Chao
_et al._ (2021). The proton light-yield of WbLS (formulated from 5% LAB) has
been recently measured in Ref. Callaghan  _et al._ (2022) where the quenching
parameters have been determined as,

$\displaystyle k_{B}^{\rm WbLS}$ $\displaystyle=(1.65\pm 0.81)\times
10^{-3}\,\text{(cm/MeV)},$ (7) $\displaystyle C^{\rm WbLS}$
$\displaystyle=(13.30\pm 2.70)\times 10^{-6}\,\text{(cm/MeV)}^{2}.$ (8)

We approximate $\langle dE/dx\rangle$ as a weighted average of $\rm H_{2}O$
and LAB using the
PSTAR222www.physics.nist.gov/PhysRefData/Star/Text/PSTAR.html database. The
dissolved scintillator concentration in THEIA would be between 0.5%–5%
depending on the performance. In this paper, we only consider the benchmark
value of 5% for which quenching measurements are available. For other
concentrations, the results can be scaled accordingly. In Fig. 2, we show
$E_{vis}(T_{p})$ for WbLS (i.e, THEIA) along with the uncertainty arising from
the measured quenching parameters. We also show $E_{vis}(T_{p})$ for LAB (i.e,
JUNO) based on the measurements of von Krosigk et. al. von Krosigk  _et al._
(2013) where the quenching parameters have been measured with 2-3% precision.
We will assume that the uncertainty in quenching parameters of WbLS can also
be significantly reduced with future measurements, and only use the central
values in our Monte Carlo.

Figure 3:  The binned event spectrum for pES from a galactic supernova in JUNO
(top) and THEIA (bottom) is shown. The expected number of events from beta-
decay of intrinsic $\,{}^{14}\rm{C}$ over 15 s interval is also shown for a
$\,{}^{14}\rm{C}$ contamination of $10^{-18}$ g/g. The gray shaded region
represents that $E_{vis}<0.1$ MeV is dominated by the dark noise of the PMTs.
As insets, we have shown the number of events in the first bin, i.e.,
$E_{vis}\in$ 0.1 – 0.2 MeV. We also indicate the number of events with $E_{\rm
vis}>$ 0.2 MeV. For THEIA, we also note down the number of events with
$E_{vis}>$ 0.6 MeV, where it would be possible to distinguish between
electrons and protons using Cherenkov to scintillation ratio.

It is well understood that the pES event rate is highly sensitive to the
$E_{vis}$ threshold. A natural limit for $E_{vis}$ arises from the dark noise
of photo multiplies tubes (PMTs). In JUNO, this _dark rate_ will overwhelm any
signal below 0.1 MeV despite sophisticated trigger schemes Fang  _et al._
(2020). THEIA plans to use a combination of ultrafast PMTs and LAPPDs (Large
Area Picosecond Photo-Detectors) Askins  _et al._ (2020); Lyashenko  _et al._
(2020), which would be different from JUNO. Based on the trigger schemes and
dark noise of individual PMTs and LAPPDs, one can determine the low-energy
threshold for THEIA. However, this is beyond the scope of this work and we
adopt 0.1 MeV as our threshold, similar to JUNO.

Another major background arises from beta-decay of intrinsic $\,{}^{14}\rm{C}$
Beacom  _et al._ (2002). For JUNO, the estimate for $\,{}^{14}\rm{C}$ levels
is around $10^{-17}$ g/g, which results in a background activity of around 42
kHz/(20 kton)An  _et al._ (2016b). The scintillation signal of these electrons
cannot be distinguished from the scintillation of recoiling protons, which
results in a wall-like background below 0.2 MeV Beacom  _et al._ (2002). As a
result, the threshold for pES events in JUNO is often considered to be 0.2
MeV. Ideally, the contamination level of $\,{}^{14}\rm{C}$ in JUNO would be
lowered to $10^{-18}$ g/g, something that is already achieved in Borexino
Alimonti  _et al._ (1998). The $\,{}^{14}\rm{C}$ levels in THEIA is not yet
determined. To approximate the $\,{}^{14}\rm{C}$ background in THEIA, we will
assume that the scintillator component is extremely radiopure and Borexino-
like $\,{}^{14}\rm{C}$ levels can be achieved in WbLS. As there are no long-
lived radio-isotopes of oxygen, only the scintillator component of WbLS gives
rise to large low-energy backgrounds.

Given the fluence, cross section, and quenching parameters, the computation of
event rates is straightforward. In Fig. 3, we show the binned pES event
spectrum ($\Delta E_{\rm bin}=0.1$ MeV) for JUNO and THEIA. Our estimates for
JUNO agree with the ones reported in Refs. An  _et al._ (2016a); Lu  _et al._
(2016); Li  _et al._ (2018, 2019), barring a small enhancement due to weaker
quenching. Considering smearing due to energy resolution, we estimate that
only 15% of $\,{}^{14}\rm{C}$ beta-decays would have $E_{vis}\in 0.1$–0.2 MeV,
and none above that. Over the 15 s duration of the supernova explosion and for
$E_{vis}\in 0.1$–0.2 MeV, we estimate that JUNO would detect $\sim$9450 events
from $\,{}^{14}\rm{C}$ beta decays, and $\sim$1400 events from pES. On the
other hand, THEIA would detect only $\sim$475 events from $\,{}^{14}\rm{C}$
beta decays, and $\sim$1150 events from pES. The duration of the supernova
burst could be $\sim 10$ s and the $\,{}^{14}\rm{C}$ backgrounds can be scaled
appropriately. The improved performance of THEIA can be attributed to the fact
that it has a larger proton-to-$\,{}^{14}\rm{C}$ ratio and weaker quenching
than JUNO. Lastly, we also note that THEIA would observe $\sim$570 events for
$E_{vis}\geq$ 0.6 MeV where it would be theoretically possible to distinguish
between electron and proton recoils using Cherenkov-to-scintillation ratio.
However, we do not anticipate any large electron backgrounds at these
energies.

Table 1:  The expected pES event rates in JUNO and THEIA from the neutronization burst phase of a supernova at 10 kpc are tabulated below. We consider two benchmark progenitor masses, i.e., 15 $M_{\odot}$ and 27 $M_{\odot}$. The event rates depend on the nuclear equation of state (LS220 Lattimer and Swesty (1991) and Shen Shen  _et al._ (1998)). We show the pES and $\,{}^{14}\rm{C}$ event rates for $E_{vis}\in 0.1$–0.2 MeV and $E_{vis}>0.2$ MeV. The $\,{}^{14}\rm{C}$ concentration is optimistically assumed to be $10^{-18}$ g/g.  Model | $E_{vis}\left[\rm MeV\right]$ | Event | JUNO | THEIA
---|---|---|---|---
15 M⊙ (LS220) | 0.1 – 0.2 | pES | 9.34 | 8.84
$\,{}^{14}\rm{C}$ | 12.60 | 0.63
$>$ 0.2 | pES | 4.71 | 5.96
$\,{}^{14}\rm{C}$ | – | –
15 M⊙ (Shen) | 0.1 – 0.2 | pES | 9.42 | 8.90
$\,{}^{14}\rm{C}$ | 12.60 | 0.63
$>$ 0.2 | pES | 4.98 | 6.24
$\,{}^{14}\rm{C}$ | – | –
27 M⊙ (LS220) | 0.1 – 0.2 | pES | 9.77 | 9.26
$\,{}^{14}\rm{C}$ | 12.60 | 0.63
$>$ 0.2 | pES | 4.95 | 6.26
$\,{}^{14}\rm{C}$ | – | –
27 M⊙ (Shen) | 0.1 – 0.2 | pES | 9.98 | 9.40
$\,{}^{14}\rm{C}$ | 12.60 | 0.63
$>$ 0.2 | pES | 5.31 | 6.65
$\,{}^{14}\rm{C}$ | – | –

The neutrinos from a core-collapse supernova can be temporally separated in
three phases: the _neutronization burst, accretion_ , and _cooling_. The
neutronization burst phase is the first $\sim$20 ms interval where a prompt
burst of $\nu_{e}$ occurs due to electron capture by free protons in the
stellar nucleus. Unlike other phases, the neutronization burst phase has a
very weak dependence on the progenitor properties, and is almost a _standard
candle_. It has been proposed that the neutronization burst can be used to
determine the supernova distance Kachelriess  _et al._ (2005); Segerlund  _et
al._ (2021), absolute neutrino mass Zatsepin (1968); Nardi and Zuluaga (2004,
2005); Pagliaroli  _et al._ (2010); Lu  _et al._ (2015); Rossi-Torres  _et
al._ (2015); Pompa  _et al._ (2022), neutrino mass ordering Brdar and Xu
(2022), as well as new physics scenarios Huang  _et al._ (2021); Tang  _et
al._ (2020). In Tab. 1, we provide our estimated event rates from pES in JUNO
and THEIA for a supernova at 10 kpc. We use the time-dependent flux parameters
provided in EstrellaNueva González-Reina  _et al._ (2022) which relies on the
simulations of Garching group 333https://wwwmpa.mpa-
garching.mpg.de/ccsnarchive/ Mirizzi  _et al._ (2016). We choose two benchmark
progenitor masses, i.e., 15 $M_{\odot}$ and 27 $M_{\odot}$ and provide
estimates using two equation of state, i.e., LS220 Lattimer and Swesty (1991)
and Shen Shen  _et al._ (1998). We find that for $E_{vis}>0.2$ MeV, where
there are no $\,{}^{14}\rm{C}$ backgrounds, we get $\sim 5$ pES events in JUNO
and $\sim 6$ pES events in THEIA. We also note that for $E_{vis}\in 0.1$–0.2
MeV, we get $\sim 9$ pES events in JUNO and THEIA, and the $\,{}^{14}\rm{C}$
backgrounds444assuming $\,{}^{14}\rm{C}$ concentration of $10^{-18}$ g/g are
$\sim 12$ and $\sim 5$ in JUNO and THEIA respectively. A key message from this
estimate is that for the first $\sim 20$ ms of a supernova, the pES event
rates in THEIA will overwhelm the $\,{}^{14}\rm{C}$ backgrounds, whereas for
JUNO, the event rates are comparable. This would have interesting applications
for supernova early warning system (SNEWS) Al Kharusi  _et al._ (2021) where
pES events may be used as secondary trigger.

##  III Extracting $\Delta s$ from $\rm{pES}$

In order to reconstruct $\Delta s$ from pES events in JUNO and THEIA, one
needs to independently determine the flux of all flavors of supernova
neutrinos. As mentioned earlier, we will rely on other neutrino detectors to
provide information on fluence parameters using interactions that do not
depend on $\Delta s$. It has been demonstrated in Ref. Nikrant  _et al._
(2018) that DUNE can measure $\nu_{e}$ fluence parameters through charged-
current interactions with argon. On the other hand, $\bar{\nu}_{e}$ and
$\nu_{x}$ parameters can be determined using water Cherenkov detectors through
inverse beta decay, elastic scattering with electrons, and interactions with
oxygen Gallo Rosso  _et al._ (2018). It must be noted that, Gadolinium loaded
Hyper-Kamiokande can attain sensitivity to $\nu_{e}$ similar to DUNE, and all
fluence parameters can be determined from a single experiment Laha and Beacom
(2014). Similarly, in Ref. Laha  _et al._ (2014) it is proposed to detect
$\nu_{e}$ using interactions on carbon in scintillator detectors. However,
both methods rely on statistical subtraction of $\nu_{x}$ events that would be
measured using pES in JUNO-like detectors, and hence indirectly depend on
$\Delta s$. Thus, we argue that DUNE is essential for this analysis, and
assume that it would be operational before the next galactic supernova. We
consider both possibilities where Super-Kamiokande (SK) may or may-not be
upgraded to Hyper-Kamiokande (HK). The uncertainties in reconstruction of
fluence parameters using these detectors is summarized in Tab. 2. We use the
DUNE sensitivity to $\nu_{e}$ from Ref. Nikrant  _et al._ (2018) and SK/HK
sensitivity to $\bar{\nu}_{e}$ and $\nu_{x}$ from Ref. Gallo Rosso  _et al._
(2018).

Table 2:  The fractional symmetric uncertainties ($1\sigma$) on the flux parameters for a galactic supernova at 10 kpc are tabulated below. The uncertainties in $\nu_{e}$ parameters measured with DUNE are adopted from Ref. Nikrant  _et al._ (2018). The uncertainties in $\bar{\nu}_{e}$ and $\nu_{x}$ parameters measured by Super-Kamiokande (SK) or Hyper-Kamiokande (HK) are taken from Ref. Gallo Rosso  _et al._ (2018). The second column gives the central value of these parameters. Parameter | Cen. | DUNE+SK | DUNE+HK
---|---|---|---
$\mathcal{E}_{\nu_{e}}$ [$10^{53}$ erg] | 0.5 | 13% | 13%
$\mathcal{E}_{\bar{\nu}_{e}}$ [$10^{53}$ erg] | 0.5 | 10% | 4%
$\mathcal{E}_{\nu_{x}}$ [$10^{53}$ erg] | 0.5 | 18% | 8%
$\langle E_{\nu_{e}}\rangle$ [MeV] | 12 | 7% | 7%
$\langle E_{\bar{\nu}_{e}}\rangle$ [MeV] | 14 | 6% | 3%
$\langle E_{\nu_{x}}\rangle$ [MeV] | 16 | 11% | 7%
$\alpha_{\nu_{e}}$ | 3.0 | 18% | 18%
$\alpha_{\bar{\nu}_{e}}$ | 3.0 | 20% | 7%
$\alpha_{\nu_{x}}$ | 3.0 | 21% | 17%

We use the technique of uncertainty quantification using Monte Carlo
simulations to estimate the sensitivity of pES in JUNO and THEIA to $\Delta
s$. Although we can, in principle, look at the spectrum of events, instead we
only use the total number of pES events to estimate $\Delta s$. For JUNO, we
use the $E_{vis}$ threshold at 0.2 MeV, and for THEIA we use $E_{vis}$
threshold at 0.1 MeV. We generate a collection of $10\times 10^{4}$ sample
observations for the four scenarios. For each sample, the fluence parameters
are randomly chosen from a Gaussian distribution with mean and variance from
Tab. 2. The true value of $\Delta s$ in the simulation is chosen to be
$0.018$, which is the central value in Ref. Chambers  _et al._ (2015). This
choice has negligible impact on our results. The supernova distance is
strongly correlated with the total neutrino luminosity. Since we are using the
reconstructed parameters, one can consider that the uncertainty in
$\mathcal{E}_{\nu}$ in Tab. 2 is in fact the uncertainty in
$\mathcal{E}_{\nu}/d^{2}$. The distribution of estimated number of events in
each scenario is asymmetric with variance that is consistent with our
expectation of a Gaussian distribution. For each sample, we determine the
expected $\Delta s$ using a simple fit. The collection of $\Delta s$ thus
obtained is used to determine the uncertainty which is reported in Tab. 3.
Note that the distribution of $\Delta s$ obtained this way is also asymmetric,
but the asymmetry is small and can be ignored. This can be clearly seen in
Fig. 4 where we show the result of one of the realizations.

Figure 4:  The histogram of reconstructed $\Delta s$ in one of the MC simulations JUNO (blue bars) and THEIA (orange bars) using uncertainties from DUNE+HK scenario and for a supernova at 10 kpc is shown. The vertical maroon line denotes the assumed true-value of $\Delta s$ in the simulation taken from Ref. Chambers  _et al._ (2015). The overlaid solid curve is a Gaussian fit with symmetric errors, and dashed curve is a fit with asymmetric errors. As the difference is very small, we only report the symmetric uncertainty in this paper.  Table 3:  The symmetrized $1\sigma$ uncertainties on the $\Delta s$ obtained from a fiducial galactic supernova are tabulated for three choices of distance. The $\nu_{e}$ spectrum is determined from DUNE. The two columns refer to the two scenario where $\bar{\nu}_{e}$ and $\nu_{x}$ are determined from gadolinium doped Super-Kamiokande (SK) or Hyper-Kamiokande (HK). The pES events can be measured using either JUNO or THEIA. The higher sensitivity of THEIA is due to weaker quenching and lower backgrounds than JUNO. Distance | Detector | DUNE+SK | DUNE+HK
---|---|---|---
10 kpc | JUNO | $\pm 0.223$ | $\pm 0.146$
THEIA | $\pm 0.160$ | $\pm 0.099$
5 kpc | JUNO | $\pm 0.117$ | $\pm 0.078$
THEIA | $\pm 0.085$ | $\pm 0.053$
1 kpc | JUNO | $\pm 0.024$ | $\pm 0.015$
THEIA | $\pm 0.017$ | $\pm 0.010$

For a supernova at 10 kpc, the reconstructed spectra of neutrinos has large
uncertainty and $\delta\Delta s$ is around $\pm 0.2$. These limits are
slightly better than the beam neutrino experiments, but weaker than pDIS
experiments. An interesting possibility, where the uncertainties in
reconstructed fluence parameters would be smaller, is to consider a nearby
supernova. It is estimated that there is a $\sim$10% chance that the next
galactic supernova is within 5 kpc of the earth and $\sim$0.25% chance that is
within 1 kpc Adams  _et al._ (2013). For more promising estimates, one can
refer to Refs Schmidt  _et al._ (2014); The  _et al._ (2006). Notably, there
are $\sim$30 core-collapse supernova candidates within 1 kpc Mukhopadhyay  _et
al._ (2020). To approximate the reconstruction efficiency of DUNE and SK/HK
for a nearby supernova, we will assume that the systematic uncertainty (mainly
arising from cross-sections) is small and the errors mentioned in Tab. 2 are
mostly statistical. For a supernova at a distance $d$, we scale the
uncertainties by a factor of $N^{-1/2}\sim d$ to obtain our estimates. One can
consider our projected sensitivity for these supernovae to be optimistic. A
careful re-analysis of reconstruction for nearby supernova is required, but
beyond the scope of this work. The $\delta\Delta s$ obtained have been
tabulated in Tab. 3 for three benchmark distances of 10, 5, and 1 kpc. We
present our results for two scenarios, with and without Hyper-Kamiokande. The
uncertainties reported in Tab. 3 for various cases are shown in Fig. 1 along
with results from other experiments for comparison.

In the method described above, one is limited by the ability to properly
reconstruct the fluence parameters using other detectors. A promising
possibility is to exploit the neutronization burst phase of the supernova. The
neutrino flux from the neutronization burst is very well predicted through
simulations, and does not depend heavily on the properties of the progenitor.
However, we find that there are three challenges in reconstructing $\Delta s$
using neutronization burst. First, despite being a _standard candle_ , there
are large (5-10%) systematic uncertainties in the flux estimates. The
differences arise partly from various nuclear physics parameters, and partly
from the unknown mass of the progenitor Kachelriess  _et al._ (2005). This is
the source of differences in event rates given in Tab. 1. Second, if the
supernova is obscured by galactic dust, the distance to the supernova will be
poorly known. As $\Delta s$ and $d$ both have similar effects on pES event
rates, they are strongly correlated and the reconstructed $\Delta s$ would
have large uncertainties as well. Lastly, only a small fraction of the total
luminosity is emitted during the neutronization burst. The pES event rates in
JUNO and THEIA are small (cf. Tab. 1), and result in large statistical
uncertainty. Our preliminary estimates suggest that the reconstructed $\Delta
s$ is not significantly better than the one obtained by the aggregate pES
events for a supernova at 10 kpc. For a nearby supernova, the uncertainty in
the distance would be much smaller. A dedicated study of reconstruction of
spectra of neutrinos from neutronization burst in DUNE and SK/HK would be
required for a more refined estimate the $\Delta s$-sensitivity. We look
forward to future work in this direction.

##  IV Summary

A core collapse supernova in our galaxy would provide an excellent opportunity
to study low-energy physics. In this paper, we looked at the possibility
whether the strange spin of proton, $\Delta s$, could be measured using
neutrinos from the next galactic supernova, and the results look promising. We
utilize the ability of near-future detectors DUNE and Hyper-Kamiokande to
reliably estimate the spectra of _all_ flavors of supernova neutrinos using
interactions that do not depend on $\Delta s$. The reconstructed spectra can
be used to estimate the pES event rate in large-volume scintillation detectors
such as JUNO and/or THEIA, and compared with the measured event rates.
Assuming that the variation only arises because of an unknown $\Delta s$, it
is straightforward to reconstruct the allowed values of $\Delta s$.

To estimate the sensitivity of JUNO and THEIA, we perform a simple Monte Carlo
simulation for uncertainty quantification. For a supernova at 10 kpc, we
estimate that JUNO would be able to constrain $\Delta s$ within $\pm 0.15$
($\pm 0.22$) using the reconstructed fluence parameters from DUNE and
Hyper-(Super-)Kamiokande; whereas THEIA would be able to constrain $\Delta s$
within $\pm 0.10$ ($\pm 0.16$) using DUNE and Hyper-(Super-)Kamiokande. The
better performance of THEIA is due to larger light yield of water-based liquid
scintillator as compared to linear alkyl benzenes, and relatively larger
proton-to-$\,{}^{14}\rm{C}$ ratio that allows for a lower threshold. We also
provide estimates for the neutronization burst phase, and note that for the
brief $\sim$20 ms window, the pES event rates in THEIA would be larger than
the $\,{}^{14}\rm{C}$ background. Moreover, the pES event rates from
neutronization burst in JUNO are comparable to $\,{}^{14}\rm{C}$ background.
We propose that pES events could also be used in supernova early warning
systems as secondary or fail-safe triggers.

If we are lucky and the next galactic supernova is within 1 kpc, then the
projected sensitivity improves to approximately $\pm 0.01$ which is comparable
to results from lattice QCD, and better than polarized deep-inelastic
scattering experiments. Such a nearby supernova would provide us with a once-
in-a-lifetime opportunity to measure the true $Q^{2}\rightarrow 0$ limit of
neutrino-proton scattering, without a dedicated experiment. The rare
circumstance and the reliance on multiple large-volume detectors is an
attestation to the difficulty of experimental determination of $\Delta s$.

## Acknowledgments

The author would like to thank Ranjan Laha, Basudeb Dasgupta, Mary Hall Reno,
Vedran Brdar, and Xunjie Xu for their useful comments on the manuscript. This
work is supported in part by US Department of Energy grant DE-SC-0010113. The
author also acknowledges the support of TIFR (India) where a part of this work
was completed.

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