Exploiting a future galactic supernova to probe
neutrino magnetic moments

Sudip Jana,1*** E-mail<EMAIL_ADDRESS>Yago P Porto-Silva,2†††
E-mail<EMAIL_ADDRESS>Manibrata Sen1‡‡‡ E-mail: manibrata@mpi-
hd.mpg.de

1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany
2Instituto de Física Gleb Wataghin - UNICAMP, 13083-859,
Campinas, São Paulo, Brazil

###### Abstract

A core-collapse supernova (SN) offers an excellent astrophysical laboratory to
test non-zero neutrino magnetic moments. In particular, the neutronization
burst phase, which lasts for few tens of milliseconds post-bounce, is
dominated by electron neutrinos and can offer exceptional discovery potential
for transition magnetic moments. We simulate the neutrino spectra from the
burst phase in forthcoming neutrino experiments like the Deep Underground
Neutrino Experiment (DUNE), and the Hyper-Kamiokande (HK), by taking into
account spin-flavour conversions of supernova neutrinos caused by interactions
with ambient magnetic fields. We find that the neutrino transition magnetic
moments which can be explored by these experiments for a galactic SN are an
order to several orders of magnitude better than the current terrestrial and
astrophysical limits. Additionally, we also discuss how this realization might
provide light on three important neutrino properties: (a) the Dirac/Majorana
nature, (b) the neutrino mass ordering, and (c) the neutrino mass-generation
mechanism.

###### Contents

  1. 1 Introduction
  2. 2 Supernova neutronization burst
  3. 3 Neutrino flavour Conversion in the presence of magnetic moments
    1. 3.1 Evolution equation
    2. 3.2 Level crossing scheme
      1. 3.2.1 Normal mass ordering (NO)
      2. 3.2.2 Inverted mass ordering (IO)
  4. 4 Sensitivity in upcoming experiments
  5. 5 Implications on neutrino properties
  6. 6 Conclusions

##  1 Introduction

The discovery of neutrino oscillations, driven by non-zero neutrino masses and
mixing, provided the first robust evidence of physics beyond the Standard
Model (SM) [1]. In order to account for the tiny neutrino masses and mixing,
the SM must be extended. In these extensions, a generic outcome of mechanisms
of neutrino mass generation is the existence of a non-zero neutrino magnetic
moment $(\mu_{\nu})$ through quantum loop corrections. Thereby, a careful
theoretical and experimental study of neutrino electromagnetic interactions
may help shed some light on the underlying theory. Within the SM minimally
extended to contain Dirac neutrino masses, neutrino magnetic moments are
$\mu_{\nu}\lesssim 10^{-19}$ $\mu_{B}$ [2], where $\mu_{B}=e/(2m_{e})$ is the
Bohr magneton, $e$ is the electron charge, and $m_{e}$ is the electron mass.
On the other hand, the strength of transition neutrino magnetic moments, in
case of Majorana scenario in the standard seesaw mechanism, is even smaller
and of the order of $\sim 10^{-23}\leavevmode\nobreak\ \mu_{B}$ [3]. Such a
small values $\mu_{\nu}$ is far beyond the reach of current experimental
capabilities.

New symmetries and interactions [4, 5, 6, 7, 8] can change the above picture,
and allow for larger values of $\mu_{\nu}$, without getting into unrealistic
values of neutrino masses (for a review, see Ref. [9]). Neutrino magnetic
moments can either be flavour-diagonal and/or flavour off-diagonal, depending
on the nature of the neutrino. Dirac neutrinos can have an intrinsic magnetic
moment that gives rise to spin precession for a given flavour $\alpha$, i.e.
$\nu_{\alpha L}\rightarrow\nu_{\alpha R}$ (where $L,R$ refer to left-handed
helicity and right-handed helicity respectively) in the presence of
sufficiently strong magnetic fields [2], as well as a transition magnetic
moment (TMM) that induces flavour precession in addition to rotation of the
spin, i.e., $\nu_{\alpha L}\rightarrow\nu_{\beta R}$ [3, 10, 11]. For Dirac
neutrinos, the latter processes would cause the resultant neutrinos to be
invisible since they will have the wrong helicity to be detected through weak
interactions. Within the minimally extended SM, such TMMs of Dirac neutrinos
are small due to a cancellation as a result of the Glashow-Iliopolus-Maiani
(GIM) mechanism. On the other hand, Majorana neutrinos can have only
transition moments, and the resultant spin-flavour precession would convert a
neutrino into an antineutrino of a different flavour, and vice-versa [10].
Furthermore, coherent forward scattering of neutrinos off the surrounding
matter background can resonantly enhance these spin-flavour conversions,
thereby termed as the resonant spin-flavour precession (RSFP) [12, 13, 14].
This is very similar to matter enhancement of neutrino oscillations due to the
Mikheyev-Smirnov-Wolfenstein (MSW) effect [15, 16].

Bounds on neutrino magnetic moments can be placed using neutrinos from a
variety of terrestrial as well as astrophysical sources. The search for the
neutrino magnetic moments began seven decades ago [17], even before the
neutrino was discovered. However, it became a very popular topic three decades
ago when the Chlorine experiment found an apparent time variation of solar
neutrino flux in anti-correlation with the Sun-spot activity [18, 19]. After
that, several neutrino experiments (reactor-based experiments like KRASNOYARSK
[20], ROVNO [21], MUNU [22], TEXONO [23], GEMMA [24] and CONUS [25]; solar
neutrino experiment like Borexino [26]; accelerator-based experiments like
LAPMF [27] and LSND experiment [28]) investigated neutrino magnetic moments by
looking at $\nu_{e}-e$ scattering in general. The best constraint from these
lab-based experiments, coming from Borexino, sets $\mu_{\nu}<2.8\times
10^{-11}\leavevmode\nobreak\ \mu_{B}$ [26]. The GEMMA [24] and CONUS [25]
collaborations placed somewhat less stringent constraints on neutrino magnetic
moments of the order of $\mathcal{O}(10^{-11})\leavevmode\nobreak\ \mu_{B}.$
Other terrestrial experiments mentioned above have shown sensitivity to
neutrino magnetic moments of the order of
$\mathcal{O}(10^{-10})\leavevmode\nobreak\ \mu_{B}$. The topic of neutrino
magnetic moments started receiving renewed attention when the XENON1T
experiment [29] observed an excess of low energy electron recoil events, which
can be explained by a neutrino transition magnetic moment
$\mu_{\nu}\in(1.4,2.9)\times 10^{-11}\mu_{B}$ at $90\%$ C.L.

Strong bounds on neutrino magnetic moment can arise from astrophysical setups
as well [30, 31, 32]. The presence of a non-zero neutrino magnetic moment
allows for a direct coupling between neutrinos and photons, thereby allowing
for neutrino radiative decays, as well as plasmon decays to neutrino-
antineutrino pairs. The strongest bounds usually arise from globular cluster
stars, where plasmon decay can delay helium ignition, leading to anomalous
cooling of stars. Absence of any such observational evidence leads to
$\mu_{\nu}\leq 3\times 10^{-12}\mu_{B}$ [33]. For an updated bound, check
[34]. Similar cooling bounds also exist from observations of neutrinos from
SN1987A. The presence of a non-zero neutrino magnetic moment allows for
$\nu_{L}\rightarrow\nu_{R}$ conversions, leading to further cooling, and hence
a shorter duration of neutrino emission than observed. Constraining the amount
of $\nu_{R}$ produced, one arrives at $\mu_{\nu}\leq 10^{-11}\,\mu_{B}$ [35].
Cosmological constraints on neutrino magnetic moments arise from observation
of the primordial abundances of elements during Big Bang Nucleosynthesis
(BBN), which sets constraints of the order of
$\mathcal{O}(10^{-10})\leavevmode\nobreak\ \mu_{B}$ [36].

The neutrino flux from a core-collapse supernova (SN) can be an excellent tool
to constrain neutrino magnetic moments. Neutrino flavour evolution inside a SN
is currently an unsolved problem in the realm of particle physics. The dense
matter environment, coupled with strong magnetic fields, makes a SN ideal for
studying flavour transitions due to RSFP [13, 14]. This is specially relevant
for Majorana neutrinos, where non-zero TMMs can lead to
$\nu_{e}\rightarrow\bar{\nu}_{x}$, where $\nu_{x}$ is linear combination of
the non-electron flavours, $\nu_{\mu}$ and $\nu_{\tau}$. Related studies were
performed in [37, 38, 39, 40], where this mechanism was shown to mix different
neutrino flavours, causing spectral changes. A simple, effective two-flavour
analysis of neutrino flavour evolution in the presence of non-zero TMMs
identifies, in addition to the usual MSW resonances, two extra resonances: (i)
the RSFP-H, associated with the atmospheric mass-squared difference $(\Delta
m^{2}_{31})$, occurring at high densities, and (ii) the RSFP-L, associated
with the solar mass-squared difference $(\Delta m^{2}_{21})$, occurring at
lower densities. Ref. [40] studied the impact of such RSFP in a three-flavour
setup and discussed an additional type of resonance, the RSFP-E (electron-
type), which is present in a pure three-flavour analysis. The RSFP-E can
result in converting $\nu_{e}\rightarrow\bar{\nu}_{x}$ in the normal mass
ordering (NO), while in the inverted mass ordering (IO), it would result in
$\bar{\nu}_{e}\rightarrow\nu_{x}$. In all these cases, it was found that all
the resonances associated with spin-flavour precession depend very sensitively
on the electron number fraction per nucleon, denoted by $Y_{e}$. These results
can be further complicated due to the presence of collective flavour
oscillations, arising out of the non-linearity of the equations of motion
associated with neutrino flavour evolution within a SN [41, 42].

However, flavour transitions due to RSFP will not be very effective if the
spectra of different neutrino flavours are similar. This is mostly true of the
cooling phase – a period existing for $\mathcal{O}(1-10)$s post-bounce.
However, the first $\mathcal{O}(30)$ms post-bounce is dominated by the
deleptonization epoch, also known as the neutronization burst (described in
detail in the next section). This epoch is characterized by a pure $\nu_{e}$
flux, with very little contamination of $\bar{\nu}_{e}$ and $\nu_{\mu,\tau}$.
This phase is a robust feature of all hydrodynamic simulations, and hence acts
as an excellent laboratory to test new neutrino properties. As a result, any
new physics mechanism that ends up converting $\nu_{e}$s would cause a
dramatic change in the $\nu_{e}$ spectra. In particular, it was discussed in
Ref [40, 39] that combination of MSW and RSFP transitions can result in
$\nu_{e}\rightarrow\bar{\nu}_{e}$ conversions during this epoch. A smoking-gun
signal of such conversion would be a suppression of the $\nu_{e}$ flux while
leading to an enhancement in the $\bar{\nu}_{e}$ channel, which can be
detected in experiments sensitive to neutrinos from a SN.

In this work, we expand on this idea to utilize the neutronization burst from
a future galactic SN to put some of the strongest constraints on neutrino
TMMs. We simulate the neutrino spectra from the burst phase in upcoming
neutrino experiments such as the Deep Underground Neutrino Experiment (DUNE)
[43, 44], and the Hyper-Kamiokande (HK) [45]. Since DUNE has maximum
sensitivity to the $\nu_{e}$ spectra, while HK can measure the
$\overline{\nu}_{e}$, both these experiments can be used to constrain TMMs of
neutrinos using the neutronization burst spectra. The large sizes of these
detectors and their detection channels make them ideal for probing the effect
of TMM on the burst spectra. We find that the values of neutrino TMM that can
be probed by these experiments for a SN occurring at 10 kpc are two or three
orders of magnitude better than the current terrestrial as well as
astrophysical constraints. We perform a parameterized study of the effect of
neutrino TMM in this paper. In order to relate it to the fundamental
parameters of a given model, we consider a simple model based on $SU(2)_{H}$
horizontal symmetry [6], which can give rise to large neutrino magnetic
moments. We translate the bounds obtained from our analysis onto the model
parameters.

The paper is structured as follows: in the next section, we briefly discuss
the supernova neutronization burst. Then we analyze in detail the neutrino
flavour conversion in the presence of neutrino magnetic moments. Subsequently,
we analyze the discovery potential of neutrino magnetic moments in forthcoming
experiments, and finally, we discuss the implications on neutrino properties
before we conclude.

##  2 Supernova neutronization burst

The supernova neutronization burst is a period of deleptonization of the SN
core, which lasts for about 30 ms after core-bounce [46]. During collapse, as
the core stiffens and reaches nuclear densities, a shockwave is launched,
which travels outwards dissociating the surrounding nuclei on its way into
corresponding nucleons. Electron capture on protons leads to a large burst of
$\nu_{e}$, which dominates the spectral content during this period, with
subdominant contributions from $\bar{\nu}_{e}$ and $\nu_{\mu,\tau}$.

Figure 1: Time-integrated neutrino number fluxes (upto $50\,$ms) for the
neutronization burst epoch. Note that $\nu_{e}$ dominates over other flavours
upto $E\lesssim 25\,$MeV.

The neutrino spectra from a SN can be parameterized using the “alpha-fit”
spectra [47]:

$F^{0}_{\nu}(E)=\frac{L_{\nu}}{\langle
E_{\nu}\rangle^{2}}\frac{(\alpha+1)^{(\alpha+1)}}{\Gamma(\alpha+1)}\left(\frac{E}{\langle
E_{\nu}\rangle}\right)^{\alpha}{\rm exp}\left[-(\alpha+1)\frac{E}{\langle
E_{\nu}\rangle}\right]\,,$ (1)

where $L_{\nu}$ is the neutrino luminosity, $\langle E_{\nu}\rangle$ is the
average energy of the neutrino, $\Gamma(z)$ denotes the Euler gamma function,
and the pinching parameter, $\alpha$, defined as,

$\frac{1}{1+\alpha}=\frac{\langle E^{2}\rangle-\langle E\rangle^{2}}{\langle
E\rangle^{2}}\,.$ (2)

is related to the spectral width. Fig. 1 shows the time-integrated (upto
$t=50\,$ms) neutrino spectra during this period, obtained from a spherically
symmetric simulation of a $15\,M_{\odot}$ progenitor [48]. Evidently, this
period is dominated by the $\nu_{e}$ spectra for $E\leq 25\,$MeV, while the
non-electron flavours dominate above that energy.

Deep inside the SN, the matter densities are high enough that the $\nu_{e}$ is
produced as the instantaneous Hamiltonian eigenstate associated with the
largest eigenvalue. During evolution, the flavour eigenstates decohere, and
hence can be treated as an incoherent superposition of mass eigenstates.
Assuming adiabatic evolution inside the SN, the $\nu_{e}$ flux at the Earth
can be written as [49]

$F_{\nu_{e}}(E)=\frac{1}{4\pi R^{2}}\left[|{\rm
U}_{eh}|^{2}\,F^{0}_{\nu_{e}}(E)+(1-|{\rm
U}_{eh}|^{2})\,F^{0}_{\nu_{\mu,\tau}}(E)\right]$ (3)

where $U_{eh}$ is the corresponding flavour element of the PMNS matrix, and
$h$ denotes the heaviest mass eigenstate. In the NO, the $\nu_{e}$ produced
inside the SN comes out of it as a $\nu_{3}$, whereas in the IO, it comes out
as a $\nu_{2}$. Since the other neutrino flavours are sub-dominant during this
epoch, the flux in the NO is suppressed with respect to that in the IO by a
factor of $\sim|{\rm U}_{e3}|^{2}/|{\rm U}_{e2}|^{2}\simeq 0.1$. In principle,
this suppression of the flux in the NO as compared to the IO can be utilized
to probe the neutrino mass ordering [49]. For antineutrinos, since the sign of
the matter potential inverts, a $\bar{\nu}_{e}$ produced will exit the SN as
the lightest mass eigenstate. However, this entire picture changes as soon as
neutrinos acquire a non-trivial magnetic moment, as we will discuss in the
next section.

##  3 Neutrino flavour Conversion in the presence of magnetic moments

Neutrinos can have Dirac and/or Majorana magnetic moments, which can be
parameterized by the following interaction Lagrangians,

$\displaystyle\mathcal{L}_{D}\supset\mu_{D}\overline{\nu_{L}}\sigma_{\eta\delta}\nu_{R}F^{\eta\delta}$
(4) $\displaystyle\mathcal{L}_{M}\supset\mu_{\alpha\beta}\nu_{\alpha
L}^{T}C\sigma_{\eta\delta}\nu_{\beta L}F^{\eta\delta}$ (5)

where $F^{\eta\delta}$ is the electromagnetic field-strength tensor, and
$\mu_{D}$ and $\mu_{\alpha\beta}$ denote the Dirac (or intrinsic), and
Majorana (or transition) magnetic moment operators. Here $\nu_{\alpha}$ and
$\nu_{\beta}$ denote neutrino flavours $\alpha$ and $\beta$ respectively,
since transition magentic moments (TMM) connect neutrinos of different
flavours. In the presence of a magnetic field in the exploding star, such TMM
can lead to $\nu_{\alpha}\rightarrow\bar{\nu}_{\beta}$ transition, which
violates lepton number by 2 units. On the other hand, the intrinsic magnetic
moment (IMM) can only allow for $\nu_{\alpha L}\rightarrow\nu_{\alpha R}$ to
occur, without changing the neutrino flavour. For Majorana neutrinos, only the
TMMs $(\alpha\neq\beta)$ are non-zero, while the IMMs $\mu_{\alpha\alpha}$ are
zero111Due to $CPT$ invariance, if $\alpha=\beta$, the Majorana moment will be
exactly zero. A particle’s magnetic moment should be equal and opposite to
that of its anti-particle, but since a Majorana particle is its own anti-
particle, its diagonal moments must disappear. This does not preclude off-
diagonal transition magnetic moments..

In this section we focus on TMMs of Majorana neutrinos, and study their
flavour evolution as they propagate radially out in the magnetic fields of the
exploding star. In such situations, neutrinos undergo several resonances
driven by the matter enhanced MSW, as well as spin-flavour precession
transitions. The interplay between RSFP and subsequent MSW conversions along
the SN profile brings an interesting possibility where the $\nu_{e}$ flux
during the neutronization burst can be efficiently converted into
$\bar{\nu}_{e}$ giving a very distinctive signal in experiments sensitive to
SN neutrinos.

###  3.1 Evolution equation

We consider a three-flavour setup, and study the evolution of neutrinos and
antineutrinos in the presence of a finite transverse magnetic field intensity
$B_{\perp}$. Ignoring temporal variations, the Equation of Motion (EoM) in
matrix form is given by,

$i\frac{d}{dr}\left(\begin{array}[]{c}\nu\\\
\bar{\nu}\end{array}\right)=\left(\begin{array}[]{cc}H_{\nu}&B_{\perp}M\\\
-B_{\perp}M&H_{\bar{\nu}}\end{array}\right)\left(\begin{array}[]{c}\nu\\\
\bar{\nu}\end{array}\right),$ (6)

where $r$ is the radial coordinate in the SN. The neutrino flavour dynamics is
governed by the Hamiltonian,

$H_{\nu}=\frac{1}{2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+\left(\begin{array}[]{ccc}V_{\nu_{e}}&0&0\\\
0&V_{\nu_{\mu}}&0\\\ 0&0&V_{\nu_{\tau}}\end{array}\right),$ (7)

where $U$ is the PMNS matrix, $E$ is neutrino energy and $\Delta m^{2}_{ij}$
gives the mass square difference between vacuum eigenstates $\nu_{i}$ and
$\nu_{j}$, and $V$ denotes the forward scattering potential due to coherent
scattering of neutrinos on electrons and nucleons. Assuming equal number
densities of electron and protons ($n_{e}=n_{p}$), the potentials, at leading
order, have the following form,

$V_{\nu_{e}}=\sqrt{2}G_{F}(n_{e}-\frac{1}{2}n_{n})\hskip
14.22636pt\text{and}\hskip
14.22636ptV_{\nu_{\mu,\tau}}=-\frac{1}{2}\sqrt{2}G_{F}n_{n},$ (8)

where $n_{e}$ and $n_{n}$ represent the electron and neutron number densities,
respectively. The corresponding Hamiltonian for the antineutrinos,
$H_{\bar{\nu}}$, only differs in the sign of the potential, such that
$V_{\bar{\nu}_{e}}=-V_{\nu_{e}}$ and
$V_{\bar{\nu}_{\mu,\tau}}=-V_{\nu_{\mu,\tau}}$. Note that the relevant
quantity for the flavour evolution is the difference of the $\nu_{e}$ and
$\nu_{\mu,\tau}$ potentials, since an overall phase term can always be rotated
away.

The matrix $M$ contains the TMMs:

$M=\left(\begin{array}[]{ccc}0&\mu_{e\mu}&\mu_{e\tau}\\\
-\mu_{e\mu}&0&\mu_{\mu\tau}\\\
-\mu_{e\tau}&-\mu_{\mu\tau}&0\end{array}\right).$ (9)

The three flavours of neutrinos and anti-neutrinos are grouped together in a
column given by

$\nu=\left(\begin{array}[]{c}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{array}\right),\quad\bar{\nu}=\left(\begin{array}[]{c}\bar{\nu}_{e}\\\
\bar{\nu}_{\mu}\\\ \bar{\nu}_{\tau}\end{array}\right).$ (10)

It is the $B_{\perp}M$ term that couples the neutrinos to the antineutrinos,
and gives rise to RSFP conversions. Note that in this analysis, we neglect the
neutrino self-interaction terms, which are known to cause collective
oscillations among different flavours [50, 51]. This is primarily because the
negligible proportions of $\bar{\nu}_{e}$ and $\nu_{\mu,\tau}$ suppresses
collective oscillations during the neutronization epoch.

Figure 2: The quantities, $\rho(1-2Y_{e})$ and $\rho Y_{e}$, relevant for RSFP
and MSW transitions respectively. These are taken from the static profile of a
$15M_{\odot}$ star with solar metallicity [52]. Here we consider
$R_{\odot}=696340$ Km.

We rewrite the EoMs in terms of the electron number fraction per nucleon
$Y_{e}=n_{e}/(n_{p}+n_{n})$, such that the electron number density
$n_{e}=Y_{e}\rho/m_{N}$, where $\rho$ is the matter density and $m_{N}$ is the
nucleon mass. For our discussions with regards to resonant flavour
conversions, the relevant potential difference for the MSW flavour conversion
is

$V_{\nu_{e}}-V_{\nu_{\mu,\tau}}=\sqrt{2}G_{F}\frac{\rho}{m_{N}}Y_{e}=-(V_{\bar{\nu}_{e}}-V_{\bar{\nu}_{\mu,\tau}}).$
(11)

whereas for the RSFP flavour conversions, it is

$V_{\nu_{e}}-V_{\bar{\nu}_{\mu,\tau}}=-\sqrt{2}G_{F}\frac{\rho}{m_{N}}(1-2Y_{e})=-(V_{\bar{\nu}_{e}}-V_{\nu_{\mu,\tau}}).$
(12)

Clearly, in order to proceed with the neutrino flavour evolution, it is
important to know the matter profile $\rho$, as well as the electron number
fraction, $Y_{e}$. We use the static profile model for a progenitor mass
$15M_{\odot}$ and solar metallicity as shown in Fig. 2 [52]. We focus on the
almost isotopically neutral region where $(1-2Y_{e})$ is small.

The magnetic field within a SN can be parameterized as a dipole,

$B_{\perp}=B_{0}\left(\frac{r_{0}}{r}\right)^{3},$ (13)

where $r_{0}$ is the radius of the iron core, which is set to $r_{0}\approx
0.0024R_{\odot}$ for a progenitor mass $15M_{\odot}$ [53]. The magnitude of
the magnetic field at the equator of the iron surface is given by $B_{0}$, and
its value can be deduced from the observation of white dwarfs to be in the
range $10^{7}$ G $\lesssim B_{0}\lesssim 10^{10}$ G [53].

###  3.2 Level crossing scheme

To understand the neutrino flavour evolution, it is useful to work in the
basis in which the $\nu_{\mu}-\nu_{\tau}$ and
$\bar{\nu}_{\mu}-\bar{\nu}_{\tau}$ subspaces are simultaneously diagonal. The
flavour states in the new basis are primed: $\nu_{\mu}^{\prime}$,
$\nu_{\tau}^{\prime}$, $\bar{\nu}_{\mu}^{\prime}$ and
$\bar{\nu}_{\tau}^{\prime}$. In addition, the magnetic moment elements in Eq.
9 alter accordingly to the change of basis and are called
$\mu_{e\mu^{\prime}}$ and $\mu_{e\tau^{\prime}}$. For simplicity, we set
$\mu_{\mu\tau}=0$ as all our results are insensitive to it222For a detailed
discussion of the bounds on $\mu_{\mu\tau}$, see [54].

We compute the energy levels of the instantaneous matter eigenstates of the
Hamiltonian in Eq. 6 for both mass orderings and show them in Fig. 3. The
values of the mixing parameters were artificially chosen for illustration
purpose in Fig. 3, but the arrangements of the resonances is in good agreement
with calculation using current mixing best fit values. For very high
densities, $\rho\gtrsim 10^{6}$ $\text{g/cm}^{3}$, the matter eigenstates
coincide with the flavour states in the primed basis. Both diagrams display
five resonances: the usual MSW-H (converts
$\nu_{e}\rightleftharpoons\nu_{\tau}^{\prime}$ for NO and
$\bar{\nu}_{e}\rightleftharpoons\bar{\nu}_{\tau}^{\prime}$ for IO), and MSW-L
(converts $\nu_{e}\rightleftharpoons\nu_{\mu}^{\prime}$ for both orderings),
and three additional spin-flavour resonances, RSFP-H
($\bar{\nu}_{e}\rightleftharpoons\nu_{\tau}^{\prime}$ for NO and
$\nu_{e}\rightleftharpoons\bar{\nu}_{\tau}^{\prime}$ for IO), RSFP-L
($\bar{\nu}_{e}\rightleftharpoons\nu_{\mu}^{\prime}$ for NO and IO) and RSFP-E
($\nu_{e}\rightleftharpoons\bar{\nu}_{\mu}^{\prime}$ for NO and
$\bar{\nu}_{e}\rightleftharpoons\nu_{\mu}^{\prime}$ for IO). Out of these, the
MSW-H, MSW-L, RSFP-H, and RSFP-L can be estimated using the two-flavour
approximation. The condition for the MSW-H(L) is given by

$\sqrt{2}G_{F}\frac{\rho}{m_{N}}Y_{e}\approx\frac{|\Delta
m^{2}_{j1}|}{2E}\cos\theta_{1j}$ (14)

where $j=3$ for H and $j=2$ for L. For RSFP-H(L), the corresponding resonance
condition is

$\sqrt{2}G_{F}\frac{\rho}{m_{N}}(1-2Y_{e})\approx\frac{|\Delta
m^{2}_{j1}|}{2E}\cos\theta_{1j}.$ (15)

On the other hand, the RSFP-E only appears in a full three-flavour analysis
[40]. Furthermore, note that the RSFP-H(L) happen at higher densities than
MSW-H(L), because the former is driven by $(1-2Y_{e})\lesssim 10^{-3}$,
whereas the latter is governed only by $Y_{e}\approx 0.5$, hence the densities
need to be higher in RSFP-H(L).

The corresponding adiabaticity parameter for the MSW and RSFP resonances are

$\gamma_{\text{MSW-H(L)}}=\frac{\sin^{2}2\theta_{1j}}{\cos
2\theta_{1j}}\frac{\Delta m_{j1}^{2}}{2E}\left|\frac{1}{\rho
Y_{e}}\frac{d\left(\rho Y_{e}\right)}{dr}\right|^{-1},$ (16)

and

$\gamma_{\text{RSFP-H(L)}}\simeq\frac{8E}{\Delta
m_{j1}^{2}}\left(\mu_{e\beta^{\prime}}B_{\perp}\right)^{2}\left|\frac{1}{\rho(1-2Y_{e})}\frac{d\left(\rho(1-2Y_{e})\right)}{dr}\right|^{-1},$
(17)

where $\beta=\mu$ for L and $\beta=\tau$ for H. Eqs. (16) and (17) must be
evaluated at the resonance point and adiabaticity is guaranteed for $\gamma\gg
1$. When neutrinos encounter an adiabatic resonance, the transitions between
different energy levels in Fig. 6 are negligible. If adiabaticity is not met,
a instantaneous matter eigenstate follows the dashed lines in Fig. 6 and hops
from its energy level to a close one with probability,

$P_{\rm res}\approx e^{-\frac{\pi}{2}\gamma_{\rm res}},$ (18)

where $\gamma_{\rm res}$ assumes the form in Eqs. (16) or (17) with the
subscript ${\rm res}$ referring to the specific resonance. In the non-
adiabatic limit, $\gamma_{\rm res}\ll 1$, the hopping probability in (18)
approaches unity and the matter eigenstates corresponding the two energy
levels involved in the resonance exchange their previous fluxes. For lack of
precise analytical expression for the adiabaticity of RSFP-E [40], we rely on
numerical numerical calculations. In what follows, we discuss these resonances
in detail for both mass orderings.

####  3.2.1 Normal mass ordering (NO)

The analysis is simplified because of the adiabaticity of the MSW-H and MSW-L
resonances. We find, numerically, for

$\mu_{\nu}B_{0}\lesssim 10^{-2}\mu_{B}\text{G}\,,$ (19)

the RSFP-L and RSFP-E are completely non-adiabatic. On the other hand, the
RSFP-H can transit from completely non-adiabatic to adiabatic in the interval

$0.5\times 10^{-3}\mu_{B}\text{G}\lesssim\mu_{\nu}B_{0}\lesssim
10^{-2}\mu_{B}\text{G}\,.$ (20)

The closer $\mu_{\nu}B_{0}$ is to $10^{-2}\mu_{B}\text{G}$, the more adiabatic
is the RSFP-H, whereas for values $\gtrsim 10^{-2}\mu_{B}\text{G}$,
adiabaticity at RSFP-H saturates to a maximum. The probability that
transitions happen between energy levels of neutrinos due to non-adiabaticity
of the RSFP-H is given by $p_{H}$ (i.e., $p_{H}=P_{\text{RSFP-H}}$, see Eq.
(18)).

Figure 3: Energy level crossing diagrams for neutrinos and antineutrinos for
the Hamiltonian considered in Eq. (6). The top panel shows the crossings for
NO, while the bottom panel shows the same for IO. The axes are arbitrarily
scaled to highlight the resonances.

The energy levels of matter eigenstates in case of NO are given in the top
panel of Fig. 3. The coloured lines indicate the evolution of the matter
eigenstates (solid for neutrinos, dashed for antineutrinos), whereas the
dashed black lines track the flavour content. We find that in the presence of
non-zero magnetic moment deep inside the star, the $\nu_{e}$ state coincides
with initial $\bar{\nu}_{2}$, and $\bar{\nu}_{\mu}^{\prime}$ with $\nu_{3}$,
therefore $F_{\bar{\nu}_{2}}=F_{\nu_{e}}^{0}$, whereas
$F_{\nu_{3}}=F_{\bar{\nu}_{\mu}^{\prime}}^{0}$. As the non-adiabatic RSFP-E
resonance is hit, the flux associated with the matter eigenstates $\nu_{2}$
and $\nu_{3}$ switch, thereby $F_{\nu_{e}}^{0}=F_{\nu_{3}}$ and
$F_{\bar{\nu}_{\mu}^{\prime}}^{0}=F_{\bar{\nu}_{2}}$. The next resonance
encountered by $\nu_{3}$ is the MSW-H, which is adiabatic, while
$\bar{\nu}_{2}$ finds no other resonances, and hence the neutrino states exit
the SN as

$\displaystyle F_{\nu_{3}}$ $\displaystyle=$ $\displaystyle
F_{\nu_{e}}^{0},\,$ (21) $\displaystyle F_{\bar{\nu}_{2}}$ $\displaystyle=$
$\displaystyle F_{\bar{\nu}_{\mu}^{\prime}}^{0}.$ (22)

The $\bar{\nu}_{e}$ is produced deep inside as a $\nu_{2}$, whereas the
$\nu_{\tau}^{\prime}$ is produced mainly as a $\nu_{1}$. As these states
evolve, they encounter the RSFP-H, where the $\nu_{2}$ can flip to the
$\nu_{1}$ and vice versa with a probability $p_{H}$. At this stage, the
$\nu_{2}$ and $\nu_{1}$ fluxes can be written as an admixture of the initial
$\bar{\nu}_{e}$ flux and the $\nu_{\tau}^{\prime}$ flux. As these states
propagate, the $\nu_{2}$ crosses adiabatic MSW-H and MSW-L resonances,
therefore $F_{\nu_{2}}$ remains unchanged, and finally exits the SN as

$F_{\nu_{2}}=(1-p_{H})F_{\bar{\nu}_{e}}^{0}+p_{H}F_{\nu_{\tau}^{\prime}}^{0}\,.$
(23)

Meanwhile, after the RSFP-H, the $\nu_{1}$ flux can be written as
$F_{\nu_{1}}=p_{H}F_{\bar{\nu}_{e}}^{0}+(1-p_{H})F_{\nu_{\tau}^{\prime}}^{0}$.
As it propagates outwards, it encounters two resonances as well, the non-
adiabatic RSFP-L, and the adiabatic MSW-L (check inset plot). At the RSFP-L
point, transition happens between $\nu_{1}$ and $\bar{\nu}_{1}$. Since initial
$\bar{\nu}_{1}$ flux deep inside the SN is
$F_{\bar{\nu}_{1}}=F_{\nu_{\mu}^{\prime}}^{0}$, after encountering the RSFP-L
resonance, one has

$\displaystyle F_{\nu_{1}}$ $\displaystyle=$ $\displaystyle
F_{\nu_{\mu}^{\prime}}^{0}\,,$ (24) $\displaystyle F_{\bar{\nu}_{1}}$
$\displaystyle=$ $\displaystyle
p_{H}F_{\bar{\nu}_{e}}^{0}+(1-p_{H})F_{\nu_{\tau}^{\prime}}^{0}.$ (25)

Finally, as the $\bar{\nu}_{3}$ is not affected by any of the resonances, we
have

$F_{\bar{\nu}_{3}}=F_{\bar{\nu}_{\tau}^{\prime}}^{0}.$ (26)

Therefore, the $\nu_{e}$ and the $\bar{\nu}_{e}$ fluxes at the Earth are

$\displaystyle F_{\nu_{e}}$ $\displaystyle=$
$\displaystyle|U_{e1}|^{2}F_{\nu_{\mu}^{\prime}}^{0}+|U_{e2}|^{2}\left[(1-p_{H})F_{\bar{\nu}_{e}}^{0}+p_{H}F_{\nu_{\tau}^{\prime}}^{0}\right]+|U_{e3}|^{2}F_{\nu_{e}}^{0},$
(27) $\displaystyle F_{\bar{\nu}_{e}}$ $\displaystyle=$
$\displaystyle|U_{e1}|^{2}\left[p_{H}F_{\bar{\nu}_{e}}^{0}+(1-p_{H})F_{\nu_{\tau}^{\prime}}^{0}\right]+|U_{e2}|^{2}F_{\bar{\nu}_{\mu}^{\prime}}^{0}+|U_{e3}|^{2}F_{\bar{\nu}_{\tau}^{\prime}}^{0}.$
(28)

Now, if the neutrino magnetic moment were zero, there would be no RSFP, and
the flux arriving at the Earth can be obtained by setting $p_{H}=1$ in the
above equations. Thus, in the limit where all the non-electron neutrino fluxes
are set equal
$(F_{\nu_{\mu}^{\prime}}=F_{\nu_{\tau}^{\prime}}=F_{\bar{\nu}_{\mu}^{\prime}}=F_{\bar{\nu}_{\tau}^{\prime}}=F_{\nu_{x}})$,
one can write Eq. 28 as

$\displaystyle F_{\nu_{e}}$ $\displaystyle=$ $\displaystyle
F_{\nu_{e}}^{MSW}-|U_{e2}|^{2}(1-p_{H})(F_{\nu_{x}}^{0}-F_{\bar{\nu}_{e}}^{0})$
(29) $\displaystyle F_{\bar{\nu}_{e}}$ $\displaystyle=$ $\displaystyle
F_{\bar{\nu}_{e}}^{MSW}+|U_{e1}|^{2}(1-p_{H})(F_{\nu_{x}}^{0}-F_{\bar{\nu}_{e}}^{0})$
(30)

where $F_{\nu}^{MSW}=F_{\nu}(p_{H}=1)$.

With this information, we can calculate the changes in the neutronization
fluence due to the presence of a non-zero TMM. From Fig. 1, note that for
$E\lesssim 20$ MeV, $F_{\bar{\nu}_{e}}^{0}\approx F_{\nu_{x}}^{0}$, and hence
the effect due to a non-zero TMM cancels out. On the other hand, for $E\gtrsim
20$ MeV, we have $F_{\nu_{x}}^{0}>F_{\nu_{e}}^{0}$. This ends up reducing the
$\nu_{e}$ flux, and increasing the $\bar{\nu}_{e}$ flux received at the Earth.
We estimate that in case of complete adiabaticity of the RSFP-H resonance
($p_{H}=0$), for $E=25$ MeV, one has

$\frac{F_{\nu_{e}}}{F_{\nu_{e}}^{MSW}}\sim 0.85\hskip
14.22636pt\text{and}\hskip
14.22636pt\frac{F_{\bar{\nu}_{e}}}{F_{\bar{\nu}_{e}}^{MSW}}\sim 1.5\hskip
14.22636pt$ (31)

Therefore, fingerprints of RSFP conversion can be visible in upcoming
experiments sensitive to the neutronization burst flux of a galactic SN.

####  3.2.2 Inverted mass ordering (IO)

A similar argument can be applied if the mass ordering is inverted (see bottom
panel of Fig. 3). At very high densities, we have
$F_{\nu_{1}}=F_{\nu_{e}}^{0}$ and
$F_{\nu_{2}}=F_{\bar{\nu}_{\tau}^{\prime}}^{0}$. As the neutrinos propagate, a
RSFP-H causes a flip between a $\nu_{1}$ and $\nu_{2}$ with probability
$p_{H}$. Since the MSW-L resonance is adiabatic, the $\nu_{2}$ finally emerges
as

$F_{\nu_{2}}=p_{H}F_{\nu_{e}}^{0}+(1-p_{H})F_{\bar{\nu}_{\tau}^{\prime}}^{0}.$
(32)

On the other hand, the $\nu_{1}$, with flux
$F_{\nu_{1}}=(1-p_{H})F_{\nu_{e}}^{0}+p_{H}F_{\bar{\nu}_{\tau}^{\prime}}^{0}$
just after RSFP-H, goes through the adiabatic MSW-H and the non-adiabatic
RSFP-E, where it flips completely to $\bar{\nu}_{1}$, and vice-versa. To get
the final $\nu_{1}$ flux, we need to track the flavour evolution of
$\bar{\nu}_{1}$. The $\bar{\nu}_{1}$ is produced deep inside as a
$\bar{\nu}_{e}$, while $\bar{\nu}_{3}$ is produced as a $\nu_{\mu}^{\prime}$.
At RSFP-L, $\bar{\nu}_{1}$ and $\bar{\nu}_{3}$ exchange fluxes so that
$F_{\bar{\nu}_{1}}=F_{\nu_{\mu}^{\prime}}^{0}$ and
$F_{\bar{\nu}_{3}}=F_{\bar{\nu}_{e}}^{0}$. The $\bar{\nu}_{3}$ flux remains
unchanged and exits the SN as

$F_{\bar{\nu}_{3}}=F_{\bar{\nu}_{e}}^{0}.$ (33)

On the other hand, the $\bar{\nu}_{1}$ exchanges flux with $\nu_{1}$ at RSFP-E
and reaches the surface of the star as

$F_{\bar{\nu}_{1}}=(1-p_{H})F_{\nu_{e}}^{0}+p_{H}F_{\bar{\nu}_{\tau}^{\prime}}^{0}\,,$
(34)

whereas the $\nu_{1}$ is emitted as

$F_{\nu_{1}}=F_{\nu_{\mu}^{\prime}}^{0}\,.$ (35)

The $\bar{\nu}_{2}$ and $\nu_{3}$ have relatively trivial dynamics, and they
are emitted as

$\displaystyle F_{\nu_{3}}$ $\displaystyle=$ $\displaystyle
F_{\nu_{\tau}^{\prime}}^{0}\,,$ (36) $\displaystyle F_{\bar{\nu}_{2}}$
$\displaystyle=$ $\displaystyle F_{\bar{\nu}_{\mu}^{\prime}}^{0}.$ (37)

Collecting these fluxes, the $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes at the Earth
are given by

$\displaystyle F_{\nu_{e}}$ $\displaystyle=$
$\displaystyle|U_{e1}|^{2}F_{\nu_{\mu}^{\prime}}^{0}+|U_{e2}|^{2}\left[p_{H}F_{\nu_{e}}^{0}+(1-p_{H})F_{\bar{\nu}_{\tau}^{\prime}}^{0}\right]+|U_{e3}|^{2}F_{\nu_{\tau}^{\prime}}^{0}\,,$
(38) $\displaystyle F_{\bar{\nu}_{e}}$ $\displaystyle=$
$\displaystyle|U_{e1}|^{2}\left[(1-p_{H})F_{\nu_{e}}^{0}+p_{H}F_{\bar{\nu}_{\tau}^{\prime}}^{0}\right]+|U_{e2}|^{2}F_{\bar{\nu}_{\mu}^{\prime}}^{0}+|U_{e3}|^{2}F_{\bar{\nu}_{e}}^{0}\,.$
(39)

In terms of the pure MSW fluxes (which is obtained by setting $p_{H}=1$ in
Eqs. (38-39)), we get

$\displaystyle F_{\nu_{e}}$ $\displaystyle=$ $\displaystyle
F_{\nu_{e}}^{MSW}-|U_{e2}|^{2}(1-p_{H})(F_{\nu_{e}}^{0}-F_{\nu_{x}}^{0})\,$
(40) $\displaystyle F_{\bar{\nu}_{e}}$ $\displaystyle=$ $\displaystyle
F_{\bar{\nu}_{e}}^{MSW}+|U_{e1}|^{2}(1-p_{H})(F_{\nu_{e}}^{0}-F_{\nu_{x}}^{0})\,,$
(41)

where all the non-electron neutrino fluxes are set equal
$(F_{\nu_{\mu}^{\prime}}=F_{\nu_{\tau}^{\prime}}=F_{\bar{\nu}_{\mu}^{\prime}}=F_{\bar{\nu}_{\tau}^{\prime}}=F_{\nu_{x}})$.
Thus, the RSFP signal for inverted ordering will be more relevant at
$E\lesssim 20$ MeV once $F_{\nu_{e}}^{0}\gg F_{\nu_{x}}^{0}$. At $E=10$ MeV we
estimate

$\frac{F_{\nu_{e}}}{F_{\nu_{e}}^{MSW}}\sim 0.3\hskip
14.22636pt\text{and}\hskip
14.22636pt\frac{F_{\bar{\nu}_{e}}}{F_{\bar{\nu}_{e}}^{MSW}}\sim 7.$ (42)

From Eqs. (31) and (42), the RSFP signal will be more dominant in the IO
compared to NO, due to a large enhancement in the $\bar{\nu}_{e}$ flux, which
is quite uncharacteristic of the neutronization epoch. As a result,
experiments sensitive to $\bar{\nu}_{e}$ from a SN will suddenly see a large
flux during the shock breakout phase, which can be a smoking-gun signal of the
presence of neutrino TMM.

##  4 Sensitivity in upcoming experiments

In this section, we study the impact of RSFP in the neutronization burst
signal at upcoming neutrino detectors. As discussed in the previous sections,
the effect of the RSFP is to reduce the $\nu_{e}$ flux, and increase the
$\bar{\nu}_{e}$. Depending on the mass ordering, the effect can be quite
large. As a result, sensitivity can be maximised using a combination of
detectors sensitive to the $\nu_{e}$ and the $\bar{\nu}_{e}$ flux. We focus on
the Deep Underground Neutrino Experiment (DUNE), which can detect $\nu_{e}$,
and the Hyper-Kamiokande (HK) which is mainly sensitive to the $\bar{\nu}_{e}$
flux. The number of events of species $\nu_{\alpha}$ detected per unit energy
is

$\frac{dN_{\nu_{\alpha}}}{dE_{r}}=\frac{N_{\rm tar}}{4\pi R^{2}}\int
dE_{t}F_{\nu_{\alpha}}(E_{t})\sigma_{\alpha}(E_{t})W(E_{r},E_{t}),$ (43)

with $N_{\rm tar}$ the number of targets in the detector material, $R$ the
distance of the supernova from the Earth, $F_{\nu_{\alpha}}$ is the
$\nu_{\alpha}$ flux that reaches the detector, $\sigma_{\alpha}$ is the
relevant neutrino interaction cross section and $W$ is the Gaussian energy
resolution function with width $\sigma_{E}$ that depends on that experimental
setup. $E_{t}$ is the actual value of neutrino energy, while $E_{r}$ is the
experimentally reconstructed energy. For the analysis presented in this paper,
we choose a fiducial distance of $R=10\,$kpc. For the neutrino oscillation
parameters, we assume the best fit values in [55].

The TMM matrix $M$ in Eq. 9, with $\mu_{\mu\tau}=0$, has two independent
elements $\mu_{e\mu}$, $\mu_{e\tau}$. We consider the following cases:

  1. 1.

both $\mu_{e\mu}$ and $\mu_{e\tau}$ are equal, set to $\mu_{\nu}$, and
considered free parameters.

  2. 2.

$\mu_{e\mu}=0$ and $\mu_{e\tau}=\mu_{\nu}$ is a free parameter.

  3. 3.

$\mu_{e\mu}=\mu_{\nu}$ is considered a free parameter, while $\mu_{e\tau}=0$.

Because the TMMs always appear together with $B_{\perp}(r)$ in the equations,
all our results are sensitive to the quantity $\mu_{\nu}B_{0}$, and not
independently to $\mu_{\nu}$ or $B_{0}$. Hence, to derive constraints on
$\mu_{\nu}$, one needs prior knowledge of $B_{0}$ from simulations and/or
astrophysical observations.

The schemes of level crossing and resonances are the same for all cases. For
the specific interval in eq. (20), MSW-H and MSW-L are always adiabatic while
RSFP-L and RSFP-E are non-adiabatic. However, the degree of adiabaticity of
RSFP-H, given by the hoping probability $p_{H}$, is strongly dependent on the
configuration of the matrix $M$, specifically on the magnitude of
$\mu_{e\tau^{\prime}}$ (see Eq. (17)). For the set of mixing parameters and
magnetic field assumed in this work,
$(\mu_{e\tau^{\prime}})_{1}>(\mu_{e\tau^{\prime}})_{3}>(\mu_{e\tau^{\prime}})_{2}$
for the entire energy range close to RSFP-H, where the subscript points to one
of the cases above. Therefore, the strength of conversions at RSFP-H has the
following hierarchy: case 1$>$ case 3 $>$ case 2 for the same value of
$\mu_{\nu}B_{0}$. As a consequence, the fluxes on Earth will change
accordingly and that translates into different experimental sensitivities.

The DUNE underground facility in South Dakota uses $40$-kton of liquid argon
in time projection chambers. The experiment will be able to detect neutrinos
from few MeV up to GeV. The relevant interaction for SN neutrinos that reach
the detector is the charged current scattering with argon nuclei,
$\nu_{e}+{}^{40}\mathrm{Ar}\rightarrow{}^{40}\mathrm{K}^{*}+e^{-}$. The final
state consists of electrons, and excited states of the potassium nuclei, which
de-excite producing a cascade of photons in the energy range between 5 and 50
MeV. For our analysis, we use the Monte Carlo event simulator MARLEY to
generate the $\nu_{e}-\mathrm{Ar}$ interaction cross section. The energy
resolution is taken to be
$\sigma_{E}/\mathrm{MeV}=0.11\sqrt{E_{r}/\mathrm{MeV}}+0.02E_{r}/\mathrm{MeV}$,
which implies that $\sigma_{E}/E_{r}\sim 5\%$ at $10$ MeV. In addition, we use
bins of $5\,$MeV, with the energy reconstruction threshold set at $4\,$MeV.

HK, an upgrade of the Super-Kamiokande (SK) detector, is a next generation
water-Cherenkov detector, with a fiducial volume of $187$ kt in each of its
two tanks. It is expected to record a huge number of events in case of a
galactic SN. HK primarily detects $\bar{\nu}_{e}$ through the Inverse Beta
Decay (IBD) channel, $\bar{\nu}_{e}+p\rightarrow e^{+}+n$. The IBD cross
section for the MeV range is extracted from [56] and the energy resolution is
the same of SK, namely $\sigma_{E}/\mathrm{MeV}=0.6\sqrt{E_{r}/\mathrm{MeV}}$
with $\sigma_{E}/E_{r}\sim 20\%$ at $10$ MeV. We simulate events at HK
detector using bins of $8$ MeV, and a detector threshold of $3$ MeV.

We compute the number of events at each detector for values of the parameter
$\mu_{\nu}B_{0}$ lying within the range given in (20), such that the RSFP-L/E
are non-adiabatic, whereas the RSFP-H can transition from non-adiabatic to
adiabatic. We focus on the most general Case 1. In Fig. 4, we show how the
neutrino event spectra is altered at DUNE and HK, assuming normal mass
ordering, in the presence of a non-zero neutrino TMM. The figure nicely
illustrates the features we discussed in Sec. 3.2 using the level crossing
diagram: a spectral distortion, resulting in suppression of events for DUNE,
while simultaneously enhancing the events in HK, for bins higher than $10$
MeV. We find that our results are sensitive to values of $\mu_{\nu}B_{0}$ as
small as $10^{-3}\mu_{B}\,$G; the spectral distortion becomes more prominent
as these values are increased. Case 2 and 3 also demonstrate similar spectral
distortions, albeit smaller, and hence we do not show them here.

In fig. 5, we show a similar analysis for the IO, where this effect is
expected to be more drastic. We find that for energy bins $E_{r}\lesssim 25$
MeV, there is a sharp decrease in events for DUNE, while HK records an almost
three-fold increase in number of events. This is consistent with our
understanding in Sec. 3.2 (see the estimations in eq. (42)). The decrease in
events in DUNE, combined with the sharp increase in HK, illustrates that the
combination of RSFP-H with MSW-H is very efficient in converting $\nu_{e}$ to
$\bar{\nu}_{e}$. This can be considered as a smoking gun signal of non-zero
TMMs of neutrinos.

Figure 4: Expected event spectrum for a supernova explosion $10$ kpc away in
DUNE (left, $\nu_{e}$ channel) and HK (right, $\bar{\nu}_{e}$ channel) for
distinct values of $\mu_{\nu}B_{0}$. We assume NO for neutrino masses.

Figure 5: Expected event spectrum for a supernova explosion $10$ kpc away in
DUNE (left, $\nu_{e}$ channel) and HK (right, $\bar{\nu}_{e}$ channel) for
distinct values of $\mu_{\nu}B_{0}$. We assume IO for neutrino masses.

We perform a $\chi^{2}$ analysis to study the sensitivity of DUNE and HK to
the presence of neutrino TMM, using the signals from a future galactic SN. We
use the Poissonian $\chi^{2}$ estimator,

$\chi^{2}=\sum_{i=1}^{n}2\left[F_{i}-D_{i}+D_{i}\ln\left(D_{i}/F_{i}\right)\right]\,,$
(44)

where $F_{i}$ is the number of events predicted for a specific non zero value
of $\mu_{\nu}B_{0}$ and $D_{i}$ is the expected number of events assuming
$\mu_{\nu}B_{0}=0$. The subscript $i$ refers to the $i$-th bin and $n$ is the
number of bins of the experiment.

Figs. 6 and 7 show $\chi^{2}$ as a function of $\mu_{\nu}B_{0}$ for NO and IO,
respectively. The null hypothesis is that neutrinos undergo pure MSW flavour
conversions, and there is no RSFP involved. As discussed before, the results
are sensitive to the magnitude of the product $\mu_{\nu}B_{0}$, and it is not
possible to disentangle one effect from the other. DUNE shows very little
sensitivity to RSFP for NO, as we show in Fig. 6. On the other hand, for IO
(see Fig. 7), DUNE can exclude at $95\%$ C.L. ,

$\displaystyle(\mu_{\nu}B_{0})_{1}$ $\displaystyle\gtrsim$ $\displaystyle
3\times 10^{-3}\mu_{B}\text{G}\,,$ (45) $\displaystyle(\mu_{\nu}B_{0})_{2}$
$\displaystyle\gtrsim$ $\displaystyle 7\times 10^{-3}\mu_{B}\text{G}\,,$ (46)
$\displaystyle(\mu_{\nu}B_{0})_{3}$ $\displaystyle\gtrsim$ $\displaystyle
6\times 10^{-3}\mu_{B}\text{G}\,,$ (47)

where the subscripts refers to specific cases 1, 2 and 3 respectively.

Figure 6: $\chi^{2}$ in Eq. (44) as a function of $\mu_{\nu}B_{0}$ assuming NO
for DUNE (left) and HK (right). The analysis compares the RSFP hypothesis with
the pure MSW one for the different cases 1, 2 and 3 defined in the text. As
expected, the strength of the signal and hence the sensitivity is always
greater for case 1 for both mass orderings.

Figure 7: $\chi^{2}$ in Eq. (44) as a function of $\mu_{\nu}B_{0}$ assuming NO
for DUNE (left) and HK (right) that compares the RSFP hypothesis with the pure
MSW one for the different cases 1, 2 and 3 defined in the text.

Due to its large size, HK will be sensitive to both the mass orderings. For
the NO, HK can exclude, at $95\%$ C.L.,

$\displaystyle(\mu_{\nu}B_{0})_{1}$ $\displaystyle\gtrsim$ $\displaystyle
4.5\times 10^{-3}\mu_{B}\text{G}\,,$ (48) $\displaystyle(\mu_{\nu}B_{0})_{2}$
$\displaystyle\gtrsim$ $\displaystyle 9\times 10^{-3}\mu_{B}\text{G}\,,$ (49)
$\displaystyle(\mu_{\nu}B_{0})_{3}$ $\displaystyle\gtrsim$ $\displaystyle
8\times 10^{-3}\mu_{B}\text{G}\,.$ (50)

The best sensitivity to TMM comes from HK in case of IO, and it can exclude,
at $95\%$ C.L.,

$\displaystyle(\mu_{\nu}B_{0})_{1}$ $\displaystyle\gtrsim$ $\displaystyle
0.6\times 10^{-3}\mu_{B}\text{G}\,,$ (51) $\displaystyle(\mu_{\nu}B_{0})_{2}$
$\displaystyle\gtrsim$ $\displaystyle 1.4\times 10^{-3}\mu_{B}\text{G}\,,$
(52) $\displaystyle(\mu_{\nu}B_{0})_{3}$ $\displaystyle\gtrsim$ $\displaystyle
1.2\times 10^{-3}\mu_{B}\text{G}.$ (53)

As discussed, for a given $B_{0}$, these bounds above can be translated into
bounds on the TMM of neutrinos. Table I summarizes the derived sensitivities
to $\mu_{\nu}$ assuming $B_{0}=10^{10}\,$G.

Experiments | Sensitivities on Neutrino Magnetic Moments (in $\mu_{B}$)
---|---
CASE 1 | CASE 2 | CASE 3
NO | IO | NO | IO | NO | IO
HK | $4.5\times 10^{-13}$ | $6\times 10^{-14}$ | $9\times 10^{-13}$ | $1.4\times 10^{-13}$ | $8\times 10^{-13}$ | $1.2\times 10^{-13}$
DUNE | $-$ | $3\times 10^{-13}$ | $-$ | $7\times 10^{-13}$ | $-$ | $6\times 10^{-13}$

Table I: Experimental sensitivities on neutrino magnetic moments for different
benchmark scenarios for a fixed value of magnetic field strength
$B_{0}=10^{10}$ G.

Uncertainties may affect the derived sensitivity. However, such impact need
not be drastic as the neutronization burst is a robust feature of all SN
simulations with a relatively weak dependence on progenitor mass and/or
specific traits of the models [57, 58, 59, 60]. Indeed, other studies show
that different progenitor models and neutrino emission parameters yield
changes that are smaller than the statistical uncertainties for current water
Cherenkov detectors [57]. For this reason, we do not perform a dedicated study
about the influence of uncertainty in the fluence parameters on the expected
sensitivities to $\mu_{\nu}B_{0}$.

Systematic uncertainties can also impact the above sensitivities, but owing to
energy dependent spectral distortion of the neutrino signal in case of nonzero
$\mu_{\nu}B_{0}$, only mild changes are expected. To see how systematics
affect the $\chi^{2}$, we modify Eq. (44) as

$\chi^{2}=\sum_{i=1}^{n}2\left[(1+\xi)F_{i}-D_{i}+D_{i}\ln\left(\frac{D_{i}}{(1+\xi)F_{i}}\right)\right]+\frac{\xi^{2}}{\sigma^{2}}\,,$
(54)

where $\xi$ is nuisance parameter and $\sigma$ an overall normalization error
that we assume to be $50\%$. $\xi$ is allowed to vary in the $3\sigma$ range
and we minimize the $\chi^{2}$ over it. We focus on case 1 and summarize new
exclusion limits at $95\%$ C.L. below

$\displaystyle\text{DUNE(IO):}\hskip 5.69046pt(\mu_{\nu}B_{0})_{1}$
$\displaystyle\gtrsim$ $\displaystyle 3.5\times 10^{-3}\mu_{B}\text{G}\,,$
(55) $\displaystyle\text{HK(NO):}\hskip 5.69046pt(\mu_{\nu}B_{0})_{1}$
$\displaystyle\gtrsim$ $\displaystyle 6\times 10^{-3}\mu_{B}\text{G}\,,$ (56)
$\displaystyle\text{HK(IO):}\hskip 5.69046pt(\mu_{\nu}B_{0})_{1}$
$\displaystyle\gtrsim$ $\displaystyle 0.7\times 10^{-3}\mu_{B}\text{G},$ (57)

with no sensitivity from DUNE in case of normal ordering.

A final comment on the adiabaticity of the RSFP-E resonance is in order. For
$\mu_{\nu}B_{0}>10^{-2}$ $\mu_{B}\text{G}$, the RSFP-E starts to be partially
adiabatic. This can end up affecting the energy spectra in both the mass
orderings. We checked that it is possible for the event spectra in the NO even
to mimic that of the IO in certain cases. We do not comment on these cases
further and leave the analysis to a future study.

##  5 Implications on neutrino properties

In this section, we mull over the possible impact of our results on the
uncharted territories of neutrino physics. To do so, we first briefly
summarize the experimental constraints and theoretical predictions on neutrino
magnetic moments, along with our results, in Fig. 8. Existing limits on
neutrino magnetic moments are shown in blue, while the red lines indicate the
future sensitivities on neutrino magnetic moments at DUNE and HK, based on our
analysis, by setting magnetic field strength $B_{0}=10^{10}$ G. Note that as
compared to existing limits, the sensitivity can be upgraded by two or three
orders of magnitude (from $\mathcal{O}[10^{-11}]\leavevmode\nobreak\ \mu_{B}$
to $\mathcal{O}[10^{-14}]\leavevmode\nobreak\ \mu_{B}$) by utilizing the
neutrino spectra from the SN neutronization burst phase in forthcoming
neutrino experiments like the DUNE, and HK. This is more stringent than the
expected sensitivity when considering the scattering at the DUNE near detector
[61]. This will have consequences for three of the most important unanswered
questions in neutrino physics:

Figure 8: Summary of experimental constraints and theoretical predictions on
neutrino magnetic moments. The red lines indicate the future sensitivities on
neutrino magnetic moments at DUNE and HK experiments based on our analysis.
Here we set magnetic field strength $B_{0}=10^{10}$ G. Theoretical predictions
for maximum possible strengths of neutrino magnetic moments are summarized
from Ref. [6] and references therein. See text for details.

  * •

Dirac/Majorana nature of neutrinos: Based on an Effective Field Theory (EFT)
analysis, it has been shown that Dirac neutrino magnetic moments over
$10^{-14}\mu_{B}$ would not be natural, since this would generate unacceptably
large neutrino masses at higher loops [62]. Weak interaction corrections to
neutrino mass originating from the magnetic moment operator are excessive in
the case of a Dirac neutrino magnetic moment. However, such a correction is
small for Majorana neutrinos since it is proportional to the mass differences
of charged leptons. Because of EFT naturalness considerations, the transition
magnetic moments for Majorana neutrinos are permitted to be significantly
greater than the Dirac scenario [63, 64]. For instance, if the new physics
scale is roughly a TeV, EFT would allow the transition magnetic moment as
large as $\mu_{\nu}\sim 10^{-7}\mu_{B}$. Therefore, if DUNE or Hyper-
Kamiokande experiment measure neutrino magnetic moments at the level of
$10^{-13}\leavevmode\nobreak\ \mu_{B}$, it is more likely that neutrinos are
Majorana in nature.

  * •

Neutrino mass ordering: The SN neutronization burst has long been advertised
as one of the cleanest means to identify the neutrino mass ordering. The
presence (absence) of a distinct peak during the burst phase indicates
inverted (normal) mass ordering. However, this is true only in the absence of
any new physics in the neutrino sector. If neutrinos have a finite magnetic
moment, it can result in the suppression of the peak in the IO due to
$\nu_{e}\rightarrow\bar{\nu}_{e}$ conversion. As a result, this might make
identification of the mass ordering from the burst phase ambiguous.

  * •

Neutrino mass generation mechanism: Due to similar chiral structure of the
neutrino magnetic moment and neutrino mass operator, while removing the photon
line from the loop diagram that yields the neutrino magnetic moment, a
neutrino mass term is generated, and $\mu_{\nu}$ generally becomes
proportional to $m_{\nu}$. One can estimate $m_{\nu}$ originating from such
diagrams as

$m_{\nu}\sim\frac{\mu_{\nu}}{\mu_{B}}\,\frac{\Lambda^{2}}{2m_{e}},$ (58)

where $\Lambda$ is the mass scale of the heavy particle inside the loop
diagram. Experimental searches generally prefer charged particles with masses
heavier than 100 GeV [65]. Without extra symmetries (and in the absence of
severe fine-tuning), this implies that neutrino mass $m_{\nu}$ must be $\sim
0.1$ MeV in order to generate neutrino magnetic moments in the order of
$10^{-11}\leavevmode\nobreak\ \mu_{B}$ for a new physics scale $\sim 100$ GeV.
This prediction of active neutrino mass contradicts the observed neutrino
masses by six orders of magnitude. Therefore, one requires leptonic symmetries
[4, 7, 6] in order to get large neutrino magnetic moments while being
consistent with tiny neutrino masses and mixings utilizing their different
Lorentz structure. There are several BSM extensions to generate neutrino
masses and mixings. However, most of these extensions cannot accommodate large
neutrino magnetic moments ($\gtrsim 10^{-14}\mu_{B}$). For instance, if right-
handed neutrinos are introduced in the SM to generate tiny Dirac neutrino
mass, neutrino magnetic moment can be expressed as [66]

$\mu_{\nu}=\frac{eG_{F}m_{\nu}}{8\sqrt{2}\pi^{2}}=3\times
10^{-20}\mu_{B}\,\left(\frac{m_{\nu}}{0.1\leavevmode\nobreak\ {\rm
eV}}\right)\leavevmode\nobreak\ ,$ (59)

and it can be as large as $\sim 10^{-20}\leavevmode\nobreak\ \mu_{B}$. In the
standard seesaw scenario, where neutrinos are Majorana particles, the
transition magnetic moments as a result of Standard Model interactions are
given by [3]

$\mu_{ij}=-\frac{3eG_{F}}{32\sqrt{2}\pi^{2}}(m_{i}\pm
m_{j})\sum_{\ell=e,\mu,\tau}U_{\ell i}^{*}U_{\ell
j}\frac{m_{\ell}^{2}}{m_{W}^{2}}.$ (60)

The strength of transition neutrino magnetic moment is even smaller and of the
order of $\sim 10^{-23}\leavevmode\nobreak\ \mu_{B}$ in this scenario. In the
minimal Left-Right symmetric model, neutrino magnetic moment can be
comparatively enhanced due to non-standard interactions via $W_{R}^{\pm}$
gauge boson [9]. However, the current experimental limit on the mixing angle
between $W_{R}^{\pm}$ and $W^{\pm}$ does not permit the strength to be greater
than $\sim 10^{-14}\leavevmode\nobreak\ \mu_{B}$. Similar strength of neutrino
magnetic moment can be achievable in $R$-parity-violating supersymmetric
extensions [67]. It was recently systematically analyzed and demonstrated in
Ref. [6] that if neutrino magnetic moments are measured at the current
experimental sensitivity $\mathcal{O}(10^{-11})\leavevmode\nobreak\ \mu_{B}$
level, neutrinos are most likely Majorana particles, and the models based on
$SU(2)_{H}$ symmetry[6, 5, 68] fit well within this category. Neutrino mass
models [69, 70, 65] based on spin-symmetry mechanism [7] can accommodate
neutrino transition magnetic moments as large as
$\mathcal{O}(10^{-12})\leavevmode\nobreak\ \mu_{B}$ [6]. All these predictions
on neutrino magnetic moments in different neutrino mass models [6, 71, 9] are
shown in green coloured lines in Fig. 8. It is quite interesting to see that
the investigation of neutrino magnetic moments can be an excellent tool in
searching for the theory underlying the neutrino mass generating mechanism.

Figure 9: Different theoretical and experimental limits and future
sensitivities on $SU(2)_{H}$ horizontal symmetric model. See text for details.

Any UV complete model of large neutrino magnetic moments is expected to
provide a more extensive phenomenology. For completeness of our study, we
consider the model based on $SU(2)_{H}$ horizontal symmetry [6] which
generates naturally large neutrino transition magnetic moments consistent with
light neutrino masses and other existing experimental constraints. Several
phenomenological consequences of this model have been recently studied in a
different context [6, 72]. Following similar conventions and the same
Lagrangian structure from Ref. [6], we summarize the different
phenomenological implications in the $SU(2)_{H}$ model parameter space in Fig.
9. The transition magnetic moments of neutrino arise through charged Higgs
induced quantum loop corrections. Due to SU(2)H symmetric limit, the
contributions from two loop diagrams add up for neutrino magnetic moments,
whereas it subtracts for neutrino mass contributions. In Fig. 9, $h^{+}$
denotes the responsible charged scalar field, and $f$ is the relevant Yukawa
term contributing to the neutrino magnetic moments. The colour shaded regions
are excluded from current experimental searches: the light blue shaded region
is excluded from charged Higgs searches at LEP experiment; the red shaded
region is excluded from Borexino NSI searches [73]; NSI from the global fit to
neutrino oscillation data [74] constrains dark green shaded region, IceCube
atmospheric data [75] on NSI imposes bound as indicated by light green shaded
region; the yellow shaded region is bounded from $\tau$ lifetime constraints
[6]; the gray shaded region is excluded from neutrino magnetic moment searches
at Borexino experiment [26]. There is an LEP mono-photon search limit [76] as
well; however, we find that it is comparatively less constrained than the
above-mentioned scenarios. We also project the future DUNE sensitivity [77]
from NSI searches by the blue dashed line. The brown dashed line represents
the future sensitivity from IceCube by looking at Glashow-like resonance
features [78, 79] induced by charged scalars in the ultrahigh-energy neutrino
event spectrum. The gray dashed line is the astrophysical constraint
originating from red giant and horizontal branch stars $|\mu_{\nu}|\leq
1.5\times 10^{-12}\mu_{B}$ (95% CL) [80, 81, 34]. However, it has been
recently pointed out that this astrophysical limit can be relaxed by
considering “neutrino trapping mechanism" [6, 72]. The red dashed lines
indicate the future sensitivities on neutrino magnetic moments at DUNE and HK
experiments based on our analysis and setting magnetic field strength
$B_{0}=10^{10}\,$G. We can see that a large region of the $SU(2)_{H}$ model
parameter space can be explored by looking at imprints of neutrino magnetic
moments on supernova neutrino signal at DUNE and HK.

##  6 Conclusions

The neutronization burst from a future galactic SN has the potential to
provide a wealth of information on neutrino transition magnetic moments (TMM).
The neutrino spectra during this phase is usually dominated by a large
fraction of $\nu_{e}$ with sub-dominant $\bar{\nu}_{e}$ and $\nu_{\mu,\tau}$.
However, the presence of a non-zero neutrino TMM can change this picture
completely. The combination of spin-flavour conversions due to a TMM, and
resonant flavour conversions due to mass-mixings, can lead to a suppression of
the $\nu_{e}$, while simultaneously enhancing the $\bar{\nu}_{e}$ spectra in
this epoch. This tell-tale signature can be used to put strong bounds on
neutrino TMMs using the spectra from a future galactic SN. With immense
experimental effort underway in the detection of neutrinos from a future
galactic SN, it is timely to analyze the impact of neutrino TMM on the signal.

In this work, we have studied the neutrino flavour evolution inside a SN in
the presence of a finite TMM, and considered the effect it has on the spectra
from the burst phase. By simulating the event spectra in upcoming neutrino
experiments like as DUNE and Hyper-Kamiokande, we have found that the neutrino
TMMs that may be probed by these experiments for a SN happening at 10 kpc are
two or three orders of magnitude (from
$\mathcal{O}[10^{-11}]\leavevmode\nobreak\ \mu_{B}$ to
$\mathcal{O}[10^{-14}]\leavevmode\nobreak\ \mu_{B}$ for the allowed range of
magnetic field strengths) better than the current terrestrial and
astrophysical bounds. We have discussed the uncertainties present in such an
analysis, and the kind of impact it can have on our results. Furthermore, we
have analyzed how this realization can shed light on three essential neutrino
properties: (a) the Dirac/Majorana character of the neutrino, (b) mass
ordering, and (c) the neutrino mass generation mechanisms. Finally, for
completeness of our study, we have considered a model based on a $SU(2)_{H}$
horizontal symmetry, which generates large neutrino TMMs, and extrapolated our
bounds onto the model parameter space. We have found that in such models, a
large section of the parameter space can be explored using the neutrino signal
from a future galactic SN.

## Acknowledgments

We thank Evgeny Akhmedov, Manfred Lindner, Orlando Peres and Suprabh Prakash
for useful discussions. MS would like to thank Ivan Martinez-Soler for help
with MARLEY simulations. YPPS thanks the Max-Planck-Institut für Kernphysik
Particle and Astroparticle Physics division for warm hospitality during the
completion of this work. The work of YPPS is supported in part by the FAPESP
funding Grants No. 2014/19164-453 6, No. 2017/05515-0 and No. 2019/22961-9.

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