# Using Bayesian Inference to Distinguish Neutrino Flavor Conversion Scenarios
via a Prospective Supernova Neutrino Signal

Sajad Abbar  0000-0001-8276-997X Max-Planck-Institut für Physik (Werner-
Heisenberg-Institut), Boltzmannstr. 8, 85748 Garching, Germany Maria Cristina
Volpe  Astro-Particule et Cosmologie (APC), CNRS UMR 7164, Universite Denis
Diderot, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France

###### Abstract

The upcoming galactic core-collapse supernova is expected to produce a
considerable number of neutrino events within terrestrial detectors. By using
Bayesian inference techniques, we address the feasibility of distinguishing
among various neutrino flavor conversion scenarios in the supernova
environment, using such a neutrino signal. In addition to the conventional
MSW, we explore several more sophisticated flavor conversion scenarios, such
as spectral swapping, fast flavor conversions, flavor equipartition caused by
non-standard neutrino interactions, magnetically-induced flavor equilibration,
and flavor equilibrium resulting from slow flavor conversions. Our analysis
demonstrates that with a sufficiently large number of neutrino events during
the supernova accretion phase (exceeding several hundreds), there exists a
good probability of distinguishing among feasible neutrino flavor conversion
scenarios in the supernova environment.

††preprint: MPP-2024-11

##  I Introduction

Neutrinos originating from core-collapse supernovae (CCSNe) present an
unprecedented opportunity for exploring the intricacies of astroparticle
physics. Understanding the behavior and characteristics of these neutrinos can
significantly contribute to the broader comprehension of neutrino physics in
extreme conditions and offers invaluable insights into the physics governing
the CCSNe Horiuchi:2018ofe ; Raffelt:2007nv ; Janka:2017vcp ;
volpe2023neutrinos .

Since the detection of Supernova 1987A (SN 1987A) and its corresponding
neutrino signal, there has been a substantial body of research focusing on
studying a galactic SN neutrino signals for various purposes. These studies
have focused on a number of issues, such as developing insight into the SN
explosion mechanism Loredo:2001rx ; Pagliaroli:2008ur , understanding neutrino
spectral parameters Harada:2023elm ; GalloRosso:2018ugl ; GalloRosso:2017mdz ,
analyzing SN remnant features Suwa:2022vhp ; GalloRosso:2017hbp , and
exploring non-standard physics in extreme conditions Kolb:1987qy ;
Shalgar:2019rqe ; deGouvea:2022dtw . Specifically, Refs. Hyper-
Kamiokande:2021frf ; Olsen:2022pkn ; Saez:2024ayk  have recently employed
Bayesian techniques to differentiate between various SN models by analyzing a
future SN neutrino signal.

One of the most captivating aspects of SN neutrinos is that they are thought
to undergo a phenomenon known as _collective_ neutrino oscillations
pantaleone:1992eq ; sigl1993general ; Pastor:2002we ; duan:2006an ;
duan:2006jv ; duan:2010bg ; Mirizzi:2015eza ; volpe2023neutrinos . This
intriguing behavior arises from an intricate interplay between propagating
neutrinos and the dense background neutrino gas, where the coherent forward
scatterings of neutrinos play a major role. What makes this phenomenon
particularly fascinating is its nonlinear and collective nature.

The phenomenon of neutrino flavor conversion (FC) within the SN environment
displays a remarkable richness. It has been established that varying the
physics conditions in the SN environment can give rise to different FC
scenarios. While the conventional Mikheev-Smirnov-Wolfenstein (MSW) effect
arises from a resonant behavior resulting from the interplay between matter
and vacuum Hamiltonian terms, the SN environment introduces more intricate FC
scenarios for neutrinos.

Initially, the understanding of neutrino FCs in the SN environment was mostly
centered around the so-called phenomenon of slow modes (driven by the neutrino
vacuum frequency), and specifically the well-established phenomenon of
spectral swapping duan:2006an ; duan:2006jv ; duan:2010bg ; dasgupta:2009mg ;
Galais:2011gh ; Dasgupta:2021gfs . This phenomenon involves the interchange of
spectra between $\nu_{e}$($\bar{\nu}_{e}$) and $\nu_{x}$($\bar{\nu}_{x}$)
across a range of neutrino energies.

However, further exploration revealed that the actual FC scenarios within more
realistic supernova models can be considerably diverse. For example, studies
have demonstrated the existence of fast flavor conversions (FFCs) when the
angular distributions of $\nu_{e}$ and $\bar{\nu}_{e}$ exhibit significant
enough differences, so that they can cross each other sawyer2005speed ;
sawyer2016neutrino ; Morinaga:2021vmc  (see volpe2023neutrinos  for a review
on flavor mechanisms).

Additionally, in the presence of strong magnetic fields, (partial) flavor
equilibration between neutrinos and antineutrinos may occur due to the
presence of the neutrino magnetic moment Abbar:2020ggq ; Sasaki:2021bvu ;
Kharlanov:2020cti . Another interesting FC scenario occurs when neutrinos can
experience non-standard self interactions (NSSI). Such NSSI introduce yet
another facet of flavor equilibration among various neutrino (antineutrino)
flavors Abbar:2022jdm ; Stapleford:2016jgz .

In this paper, we utilize Bayesian inference techniques to discern between
various neutrino FC scenarios by analyzing a future SN neutrino signal. Our
specific focus lies within the SN accretion phase, characterized by
distinctive energy spectra among different neutrino species. We demonstrate
that a sufficiently large number of neutrino events significantly enhances the
likelihood of distinguishing between different FC scenarios in the SN
environment.

The paper is structured as follows. In Sec. II, we describe the SN neutrino
spectra. We then discuss the neutrino FC scenarios in Sec. III. Sec. IV
introduces the neutrino detection channels we consider in this study. Before
concluding in Sec. VI, we present the results of our Bayesian analysis of a
prospective SN neutrino signal in Sec. V.

##  II SN neutrino signal

To ensure the maximum impact of the FC on neutrino energy spectra, we
strategically select the SN accretion phase, anticipating the greatest
disparity between different neutrino spectra. As a first step, we make the
simplifying hypothesis that the neutrino spectral parameters are stationary
during the emission of the neutrino signal. This assumption can be justified
on the basis of two key factors. Firstly, during the SN accretion phase, the
neutrino spectral parameters can reasonably be considered constant, at least
for several tens of milliseconds, probably allowing for the accumulation of a
sufficient number of events. Additionally, our methodology allows us to
determinine the minimum time interval for which the spectral parameters should
remain constant to obtain any conclusive result.

In scenarios where neutrino FC is not present, the time-integrated neutrino
energy-differential number flux of a specific neutrino species, $\nu_{\beta}$,
at a distance $D$ from the SN can be written as Mirizzi:2015eza ,

$\mathcal{F_{\nu_{\beta}}}(E_{\nu})=\frac{\mathcal{E}_{\nu_{\beta}}}{4\pi
D^{2}\langle E_{\nu_{\beta}}\rangle}f_{\nu_{\beta}}(E_{\nu})$ (1)

with

$f_{\nu_{\beta}}(E_{\nu})=\frac{1}{T_{\nu_{\beta}}\Gamma(1+\alpha_{\nu_{\beta}})}\bigg{(}\frac{E_{\nu}}{T_{\nu_{\beta}}}\bigg{)}^{\alpha_{\nu_{\beta}}}\exp(-E_{\nu}/T_{\nu_{\beta}})$
(2)

being the normalized $\nu_{\beta}$ spectrum, where $E_{\nu}$ is the neutrino
energy. Here, $\alpha_{\nu_{\beta}}$ and $\langle E_{\nu_{\beta}}\rangle$ are
the pinching parameter and the average neutrino energy, the parameters which
describe the normalized spectrum and $T_{\nu_{\beta}}=\langle
E_{\nu_{\beta}}\rangle/(1+\alpha_{\nu_{\beta}})$. In addition,
$\mathcal{E}_{\nu_{\beta}}$ is the time-integrated neutrino luminosity for the
time interval of interest, i.e., $\mathcal{E}_{\nu_{\beta}}=\int_{t}^{t+\Delta
t}\mathrm{d}tL_{\nu_{\beta}}(t)$, with $L_{\nu_{\beta}}$ being the neutrino
luminosity.

Table 1: The neutrino energy spectra we consider in this study, relevant to
the SN accretion phase. While the LSD spectrum shows significant variations
among neutrino species, in the SSD one the spectral variations are less
prominent Serpico:2011ir ; Fischer:2009af . LSD
---
$\mathcal{E}_{\nu_{e}}:\mathcal{E}_{\bar{\nu}_{e}}:\mathcal{E}_{\nu_{x}}=1:1:0.33$
$\langle E_{\nu_{e}}\rangle:\langle E_{\bar{\nu}_{e}}\rangle:\langle
E_{\nu_{x}}\rangle=9:12:16.5$
${\alpha}_{\nu_{e}}:{\alpha}_{\bar{\nu}_{e}}:{\alpha}_{\nu_{x}}=3.2:4.5:2.3$
SSD
$\mathcal{E}_{\nu_{e}}:\mathcal{E}_{\bar{\nu}_{e}}:\mathcal{E}_{\nu_{x}}=1:1:0.75$
$\langle E_{\nu_{e}}\rangle:\langle E_{\bar{\nu}_{e}}\rangle:\langle
E_{\nu_{x}}\rangle=9:12:14$
${\alpha}_{\nu_{e}}:{\alpha}_{\bar{\nu}_{e}}:{\alpha}_{\nu_{x}}=3.26:4.76:2.26$

In the following, we analyze two distinctive neutrino energy spectra relevant
to the SN accretion phase, as described in Table. 1: one with large spectral
difference (LSD), showing significant variations among neutrino species, and
another with small spectral difference (SSD), where spectral variations are
less prominent. Here, we make the assumption that $\nu_{x}$ and
$\bar{\nu}_{x}$ have identical spectral shapes. Also note that the average
energies are in MeV, and that there is no unit in front of $\mathcal{E}$,
given that we here consider the scenario where the number of observed neutrino
event are fixed (see the next section for more details). While the latter
profile is more suitable for SN models with lighter progenitors, the former is
more appropriate for models involving heavier ones Serpico:2011ir ;
Fischer:2009af .

##  III Neutrino flavor conversion scenarios

In this section, we briefly describe the FC mechanisms under consideration. It
is important to highlight that apart from the conventional MSW mechanism, all
the FC scenarios we explore stem from coherent neutrino-neutrino scatterings
(mean-field approximation). As a result, the total number of (anti)neutrinos
within a designated energy/momentum range remains conserved.

Figure 1:  Initial spectra of $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ (derived
from the LSD profile), as well as the resulting $\bar{\nu}_{e}$ spectrum post
FC for different FC scenarios. Here, the initial spectra (LSD) are assumed to
exist at the neutrinosphere. Then, we have applied the FC scenarios (defined
in Sec. III) given in the title of each panel.

To provide the reader with an impression of the various FC scenarios, Fig. 1
displays the initial spectra of $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ derived
from the LSD profile, alongside the resulting $\bar{\nu}_{e}$ spectrum post
the given FC scenario.

###  III.1 No flavor conversions (NFC)

In our first example, we examine the FC scenario where the emitted neutrino
fluxes within the SN core precisely translates to larger distances without
alteration. This scenario serves as a benchmark for comparison. Surprisingly,
while seemingly idealistic, this scenario is not entirely implausible, given
the fact we do not yet know to which extent FC in the SN core influences the
SN neutrino fluxes.

###  III.2 MSW effect

The established MSW mechanism stands as one of the most extensively explored
neutrino FC mechanisms. It occurs due to a resonant behavior generated by a
cancellation in the diagonal term of the Hamiltonian, between the matter and
vacuum components. This effect is highly sensitive to the neutrino mass
hierarchy. Vacuum neutrino oscillation experiments provide accurate knowledge
of mass-squared differences. Due to matter effects in the Sun, we also know
that $\Delta m_{21}^{2}>0$. However, the sign of the atmospheric mass
splitting, $\Delta m_{31}^{2}$, is currently unknown since current measurement
solely revealing its absolute value. Hence, we can have either a normal
hierarchy (NH) with $\Delta m_{31}^{2}>0$, or an inverted hierarchy (IH) with
$\Delta m_{31}^{2}<0$. Assuming a perfect adiabatic MSW, in NH one has
Dighe:1999bi ,

$\begin{split}\mathcal{F}_{\nu_{e}}&=\sin^{2}\theta_{13}\
\mathcal{F}^{0}_{\nu_{e}}+\cos^{2}\theta_{13}\ \mathcal{F}^{0}_{\nu_{x}},\\\
\mathcal{F}_{\bar{\nu}_{e}}&=\cos^{2}\theta_{12}\cos^{2}\theta_{13}\
\mathcal{F}^{0}_{\bar{\nu}_{e}}+(1-\cos^{2}\theta_{12}\cos^{2}\theta_{13})\
\mathcal{F}^{0}_{\bar{\nu}_{x}},\\\
2\mathcal{F}_{\nu_{x}}&=\cos^{2}\theta_{13}\
\mathcal{F}^{0}_{\nu_{e}}+(1+\sin^{2}\theta_{13})\
\mathcal{F}^{0}_{\nu_{x}},\\\
2\mathcal{F}_{\bar{\nu}_{x}}&=(1-\cos^{2}\theta_{13}\cos^{2}\theta_{12})\
\mathcal{F}^{0}_{\bar{\nu}_{e}}+(1+\cos^{2}\theta_{13}\cos^{2}\theta_{12})\
\mathcal{F}^{0}_{\bar{\nu}_{x}},\\\ \end{split}$ (3)

while for IH, it is given by,

$\begin{split}\mathcal{F}_{\nu_{e}}&=\cos^{2}\theta_{13}\sin^{2}\theta_{12}\
\mathcal{F}^{0}_{\nu_{e}}+(1-\sin^{2}\theta_{12}\cos^{2}\theta_{13})\
\mathcal{F}^{0}_{\nu_{x}},\\\
\mathcal{F}_{\bar{\nu}_{e}}&=\sin^{2}\theta_{13}\
\mathcal{F}^{0}_{\bar{\nu}_{e}}+\cos^{2}\theta_{13}\
\mathcal{F}^{0}_{\bar{\nu}_{x}},\\\
2\mathcal{F}_{\nu_{x}}&=(1-\cos^{2}\theta_{13}\sin^{2}\theta_{12})\
\mathcal{F}^{0}_{\nu_{e}}+(1+\sin^{2}\theta_{12}\cos^{2}\theta_{13})\
\mathcal{F}^{0}_{\nu_{x}},\\\
2\mathcal{F}_{\bar{\nu}_{x}}&=\cos^{2}\theta_{13}\
\mathcal{F}^{0}_{\bar{\nu}_{e}}+(1+\sin^{2}\theta_{13})\
\mathcal{F}^{0}_{\bar{\nu}_{x}}.\\\ \end{split}$ (4)

Here $\mathcal{F}^{0}_{\nu}$ denotes the neutrino fluxes at the neutrinosphere
before flavor conversion mechanisms produce spectral distortions.

###  III.3 Spectral swapping (SS)

The initial investigations into collective neutrino oscillations within the SN
environment focused on simplified symmetric models, such as the stationary
spherically symmetric neutrino bulb model. These studies revealed a
significant phenomenon: spectral swapping phenomenon which involves the
swapping of spectra between $\nu_{e}$($\bar{\nu}_{e}$) and
$\nu_{x}$($\bar{\nu}_{x}$) over a range of neutrino energies, due to the
collective oscillations of SN neutrinos duan:2006an ; duan:2006jv ;
duan:2010bg ; dasgupta:2009mg ; Dasgupta:2021gfs ; volpe2023neutrinos . Though
this was first observed in stationary spherically symmetric models, in Ref.
Martin:2019dof  the authors showed that SS can also exist in multi-dimensional
models provided that FCs do not occur very deep inside the SN core.

The spectral swap can occur around the neutrino energies where there is a zero
crossing in the energy spectrum defined as,

$g_{\omega}=\frac{|\Delta
m^{2}|}{2\omega^{2}}\begin{cases}&\mathcal{F}^{0}_{\nu_{e}}-\mathcal{F}^{0}_{\nu_{x}}\quad(\omega>0)\\\
&\mathcal{F}^{0}_{\bar{\nu}_{x}}-\mathcal{F}^{0}_{\bar{\nu}_{e}}\quad(\omega<0)\\\
\end{cases}$ (5)

where $\omega=\frac{|\Delta m^{2}|}{2E_{\nu}}$ and $|\Delta m^{2}|$ is the
neutrino mass squared difference. Note that there can be a crossing (by
default) when $\omega\rightarrow 0$, corresponding to
$E_{\nu}\rightarrow\infty$, which if unstable leads to an SS for the tail of
the energy spectra.

The spectral crossings exhibit instability in the IH (NH) scenarios when the
crossing slope is positive (negative), resulting in a swap across a finite
range of $\omega$. While zero crossings in $g_{\omega}$ serve as a criterion
for instability concerning SS, determining the swapping width is not
straightforward. In practice, there might exist even crossings that do not
induce any spectral swapping.

In our computations, we prioritize practicality over ambition when estimating
the width of the SS. Specifically, if the crossing at $\omega=0$ is unstable,
we approximate $\Delta\omega\simeq 0.7$ km-1. Alternatively, for each
crossing, we assume that SS occurs within $\Delta E_{\nu}=\pm 7.5$ MeV. These
assumptions align with the simulation results duan:2006an ; dasgupta:2009mg .
Furthermore, we have verified that the width of the crossings, if sufficiently
large, does not impact the results presented in the subsequent section.

###  III.4 Slow flavor conversions (SFC)

In the context of slow modes, in some symmetric models like the bulb model, it
has been shown that some flavor equilibration can be reached Raffelt:2007yz .
Such equilibrium leads to neutrino flavor equipartition up to the constraint
of neutrino electron lepton number conservation. The resulted neutrino spectra
can then be found as,

$\begin{split}\mathcal{F}_{\nu_{e}}&=\mathcal{F}_{\rm{eq}}+\max(0,\mathcal{L}),\\\
\mathcal{F}_{\bar{\nu}_{e}}&=\mathcal{F}_{\rm{eq}}+\max(0,-\mathcal{L}),\\\
\mathcal{F}_{\nu_{x}}&=\mathcal{F}_{\bar{\nu}_{x}}=\mathcal{F}_{\rm{eq}},\end{split}$
(6)

where $\mathcal{F}_{\rm{eq}}$ is the equilibrium $\mathcal{F}$, defined by,

$\mathcal{F}_{\rm{eq}}=\begin{cases}\frac{1}{3}(\mathcal{F}^{0}_{\bar{\nu}_{e}}+2\mathcal{F}^{0}_{\bar{\nu}_{x}}),&\text{if
}\mathcal{L}>0,\\\
\frac{1}{3}(\mathcal{F}^{0}_{\nu_{e}}+2\mathcal{F}^{0}_{\nu_{x}}),&\text{if
}\mathcal{L}\leq 0.\end{cases}$ (7)

where $\mathcal{L}=\mathcal{F}^{0}_{\nu_{e}}-\mathcal{F}^{0}_{\bar{\nu}_{e}}$,
provides information on the neutrino electron lepton number. For more details
on the SFC prescription, we refer an interested reader to Ref. Ehring:2023abs
; Ehring:2023lcd .

###  III.5 Fast flavor conversions (FFC)

One of the most recent developments in the field of neutrino FC in the SN
environment has been the discovery of FFCs Sawyer:2005jk ; Sawyer:2015dsa ;
Chakraborty:2016lct . The assessment of the outcomes of FFCs has been
extensively explored through localized dynamical simulations conducted within
confined spaces using periodic boundary conditions Bhattacharyya:2020dhu ;
Bhattacharyya:2020jpj ; Wu:2021uvt ; Richers:2021nbx ; Zaizen:2021wwl ;
Richers:2021xtf ; Bhattacharyya:2022eed ; Grohs:2022fyq ; Abbar:2021lmm ;
Richers:2022bkd ; Zaizen:2022cik ; Xiong:2023vcm . Insights obtained from
these investigations reveal a tendency toward kinematic decoherence during
flavor conversions, leading to the emergence of quasistationary states. These
stable states can be characterized by survival probabilities, governed by the
conservation of neutrino lepton number, and have shown potential for accurate
analytical modeling Xiong:2023vcm .

In this study, we take the analytical survival probability developed in Ref.
Xiong:2023vcm , assuming an axisymmetric neutrino angular distributions. In
addition, we assumed the maximum entropy angular distribution
Cernohorsky:1994yg , defined as,

$\mathcal{G}_{\nu}(\mu)=\exp(\eta+a\mu),$ (8)

where $\eta$ and $a$ are some arbitrary parameters and $\mu=\cos\theta_{\nu}$,
with $\theta_{\nu}$ being the zenith angle of the neutrino velocity, and,

$\mathcal{G}_{\nu}(\mu)=\int_{0}^{\infty}\int_{0}^{2\pi}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}\
\mathcal{G}_{\nu}(\mathbf{p}),$ (9)

where $\mathcal{G}_{\nu}(\mathbf{p})$’s are the neutrino occupation numbers of
different flavors. Here, $\phi_{\nu}$ is the azimuthal angle of the neutrino
velocity. If one is then provided with the flux factors of the energy-
integrated as well as those of each energy bin, one can find the outcome of
FFC for each energy bin. Here, the flux factor is defined as,

$\mathbf{F}_{\nu}=\frac{I_{1}}{I_{0}}$ (10)

where $I$’s are the radial moments of the neutrino angular distributions,
defined as,

$I_{n}=\int_{-1}^{1}\mathrm{d}\mu\ \mu^{n}\
\int_{0}^{\infty}\int_{0}^{2\pi}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}\
\mathcal{G}_{\nu}(\mathbf{p}),$ (11)

for the energy-integrated moments, and as,

$I_{n,i}=\frac{E_{\nu,i}^{2}\Delta
E_{\nu,i}}{(2\pi)^{3}}\int_{-1}^{1}\mathrm{d}\mu\ \mu^{n}\
\int_{0}^{2\pi}\mathrm{d}\phi_{\nu}\ \mathcal{G}_{\nu}(\mathbf{p}),$ (12)

when the corresponding quantities for each energy bin are required. Here,
$E_{\nu,i}$ and $\Delta E_{\nu,i}$ are the mean energy and the width of the
i-th energy bin. For the the energy-integrated flux factors, we take:
$\mathbf{F}_{\nu_{e}}=0.5$, $\mathbf{F}_{\bar{\nu}_{e}}=0.7$, and
$\mathbf{F}_{\bar{\nu}_{e}}=0.8$. Then for each energy bin we adopted an
assumption describing the relationship as
$\mathbf{F}_{\nu,i}=\mathbf{F}_{\nu}\ (70-E_{\nu,i})^{2}/60^{2}$. Here, we
have assumed that the spectra approach zero for $E_{\nu,i}\gtrsim 60$ MeV.
Note that it is anticipated that the flux factor will decrease with increasing
neutrino energy, and this reduction is expected to be nonlinear, due to the
nonlinear scaling of the neutrino scattering cross-section (with the matter)
with neutrino energy in the SN environment. While our assumption regarding the
dependence of the flux factor on energy is speculative, it aligns with both of
these anticipated conditions. Otherwise, we regard this assumption as an
initial exploration, considering it as the first stage of such a study.
Further investigations may involve exploring alternative formulations for the
energy dependence of the flux factor.

###  III.6 Neutrino flavor equipartition (EQP)

The existence of neutrino non-standard self-interactions (NSSI) has been
established as a mechanism capable of inducing equipartition between neutrinos
and antineutrinos separately Abbar:2022jdm , given by,

$\begin{split}\mathcal{F}_{\nu_{e}}&=\mathcal{F}_{\nu_{x}}=\frac{1}{3}(\mathcal{F}^{0}_{\nu_{e}}+2\mathcal{F}^{0}_{\nu_{x}}),\\\
\mathcal{F}_{\bar{\nu}_{e}}&=\mathcal{F}_{\bar{\nu}_{x}}=\frac{1}{3}(\mathcal{F}^{0}_{\bar{\nu}_{e}}+2\mathcal{F}^{0}_{\bar{\nu}_{x}}).\\\
\end{split}$ (13)

This phenomenon occurs due to the introduction of non-standard off-diagonal
terms in the neutrino-neutrino interaction Hamiltonian, resulting in the
emergence of unavoidable neutrino flavor instabilities. These instabilities
subsequently trigger flavor decoherence, ultimately leading to a state of
flavor equipartition within both the neutrino and antineutrino sectors.

In several beyond Standard Model (BSM) theories of particle physics
Bialynicka-Birula:1964ddi ; Bardin:1970wq , such neutrino NSSIs are permitted.
These interactions are facilitated through a vector mediator, which modifies
the effective Lagrangian to $\mathcal{L}_{\mathrm{eff}}\supset
G_{\mathrm{F}}[\mathsf{G}^{\alpha\beta}\bar{\nu}_{\alpha}\gamma^{\mu}\nu_{\beta}][\mathsf{G}^{\xi\eta}\bar{\nu}_{\xi}\gamma_{\mu}\nu_{\eta}]$.
This revised Lagrangian closely resembles the neutrino-neutrino interaction
Lagrangian of the Standard Model (SM), but now introduces new interaction
terms that couple neutrinos of different flavors via
$\mathsf{G}^{\alpha\beta}$’s.

###  III.7 Magnetically-induced flavor conversion (MFC)

Neutrinos, which possess exceedingly small yet non-zero magnetic moments (see
Refs. Giunti:2008ve ; Broggini:2012df ; Studenikin:2016ykv  for an in-depth
review), can exhibit modified flavor evolution when exposed to external
magnetic fields. This effect becomes particularly pronounced in environments
with ultra-strong magnetic fields (exceeding $10^{15}$ Gauss), such as neutron
star mergers and magneto-rotational CCSNe Mosta:2015ucs . These extreme
conditions present ideal scenarios for investigating the interplay between
neutrinos and magnetic fields and their impact on collective neutrino
oscillations. For example for Majorana neutrinos, the introduction of a
magnetic term can induce neutrino-antineutrino oscillations.

Recent studies have highlighted the potential significance of the magnetic
term when it reaches a comparable magnitude to other terms in the Hamiltonian.
Here we follow Refs. Abbar:2020ggq ; Sasaki:2021bvu ; Kharlanov:2020cti  that
have found an equipartition among $\nu_{e}$ ($\bar{\nu}_{e}$) and
$\bar{\nu}_{x}$ ($\nu_{x}$) when the contribution coming from the neutrino
magnetic moment to the neutrino Hamiltonian becomes as sizeable as the other
terms, described by:

$\begin{split}\mathcal{F}_{\nu_{e}}&=\mathcal{F}_{\bar{\nu}_{x}}=\frac{1}{3}(\mathcal{F}^{0}_{\nu_{e}}+2\mathcal{F}^{0}_{\bar{\nu}_{x}}),\\\
\mathcal{F}_{\bar{\nu}_{e}}&=\mathcal{F}_{\nu_{x}}=\frac{1}{3}(\mathcal{F}^{0}_{\bar{\nu}_{e}}+2\mathcal{F}^{0}_{\nu_{x}}).\\\
\end{split}$ (14)

##  IV Neutrino cross sections

Before discussing the outcomes of our statistical analysis, let us first
briefly outline the neutrino detection channels under consideration in our
study. In our statistical analysis, we consider neutrino events as can be seen
by water Cherenkov detectors, namely Super-Kamiokande and Hyper-Kamiokande
Hyper-Kamiokande:2018ofw ; Super-Kamiokande:2002weg . In particular, we take
inverse beta decay (IBD) and neutrino elastic scattering on electrons (ES) as
the relevant processes 111Note that there are a range of other possibilities
for neutrino detection which are beyond the scope of this work and we leave
them for a future study. .

In this study, we build upon our previous works outlined in Refs.
GalloRosso:2018ugl ; GalloRosso:2017mdz ; GalloRosso:2017hbp  (see these
References for further details and for the neutrino cross sections). In
particular, in our calculations, we work under the assumptions of a 100%
detector efficiency and the absence of background events. We also ignore the
possible existence of a time offset between the start of neutrino emission and
the detection of the first event. These considerations are made to determine
the very minimum number of events necessary to derive conclusive results. The
energy-differential rate for the reaction $j$ generated by the neutrino
species $\nu_{\beta}$ can be calculated as,

$\frac{\mathrm{d}N_{\nu_{\beta},j}}{\mathrm{d}E}=n_{\rm{T},j}\int_{E_{\mathrm{th}}}^{\infty}\mathrm{d}E_{\nu}\
\mathcal{F_{\nu_{\beta}}}(E_{\nu})\sigma_{\nu_{\beta},j}(E_{\nu})$ (15)

where $E_{\mathrm{th}}$ is the neutrino energy threshold for each process,
$n_{\rm{T},j}$ is the number of targets for the process $j$, and
$\sigma_{\nu_{\beta},j}(E_{\nu})$ is the energy dependent cross section for a
given reaction and species.

For the IBD events, we use the analytical formula developed in Ref.
Strumia:2003zx ,

$\sigma_{\rm{IBD}}\simeq 10^{-43}\mathrm{cm}^{2}p_{e}E_{e}E_{\nu}^{x},$ (16)

with $x=-0.07056+0.02018\ \mathrm{ln}E_{\nu}-0.001953\ \mathrm{ln}^{3}E_{\nu}$
and $E_{e}\simeq E_{\nu}-\Delta$, with $\Delta\simeq 1.293$ MeV. For the ES
channel, we use tree level expression of the cross section Formaggio:2012cpf ;
Tomalak:2019ibg :

$\frac{\mathrm{d}\sigma_{\nu_{\beta},\rm{ES}}}{\mathrm{d}y}=\frac{2G_{\rm{F}}^{2}m_{e}E_{\nu}}{\pi}\bigg{[}g^{2}_{\nu_{\beta}}+g^{\prime
2}_{\nu_{\beta}}(1-y)^{2}-g_{\nu_{\beta}}g^{\prime}_{\nu_{\beta}}\frac{m_{e}}{E_{\nu}}y\bigg{]}$
(17)

where $g$ and $g^{\prime}$ can be obtained using the Weinberg weak-mixing
angle and are different for different neutrino species (see Table. 3 of Ref.
GalloRosso:2017mdz ), $G_{\rm{F}}$ is the Fermi constant, and $m_{e}$ is the
electron mass. Here, $y$ is the electron kinetic energy, $K_{e}=E_{e}-m_{e}$,
divided by the neutrino energy, for which
$0<y=K_{e}/E_{\nu}<(1+m_{e}/2E_{\nu})^{-1}$.

Figure 2:  Event rate distributions for the IBD and ES detection channels, as
a function of the electron total and kinetic energies, $E_{e}$ and $K_{e}$,
respectively. Here, the EQP FC scenario is employed.

In Fig. 2, we present an illustrative event rate distribution given our
equipartition FC prescription, provided both for the IBD and ES detection
channels. It is important to note that the anticipated total number of ES
events is expected not to exceed a few percent of the IBD events.

##  V Bayesian analysis of the supernova neutrino events

Figure 3:  Bayes factor comparison among various FC scenarios across different
observed neutrino events, $N_{\rm{obs}}=100,300,$ and 1000. The x-axis denotes
the null hypothesis, representing the model by which the neutrino events were
generated, while the y-axis corresponds to the alternative hypothesis. Here,
the diagonal values are by default set to zero. The upper panels show the
results using the LSD spectrum, incorporating both the IBD and ES detection
channels. In the middle panels, we present the Bayes factor calculations
similar to the upper panels, but excluding the ES channel. The lower panels
exhibit the Bayes factor values derived from the SSD spectrum calculations
(including both IBD and ES channels).

In this section, we provide the results of our Bayesian analysis to
differentiate neutrino FC scenarios via a future SN neutrino signal. For a
specific FC scenario, $\rm{M}_{a}$, an event with an energy $E_{i}$ has the
probability distribution:

$p(E_{i}|\mathrm{M}_{a})=\frac{1}{\langle
N\rangle}\frac{\mathrm{d}N}{\mathrm{d}E},$ (18)

where $\mathrm{d}N/\mathrm{d}E$ can be found using Eq. (15) for each case of
FC considered, and $\langle
N\rangle=\int_{E_{\mathrm{th}}}^{\infty}\mathrm{d}E(\mathrm{d}N/\mathrm{d}E)$
is the expected total number of events.

One can then find the likelihood of observing a set of events with energies
$\\{E_{i}\\}$ for $i$ in $(1,2,..,N_{\rm{obs}})$ as,

$P(\\{E_{i}\\}|\mathrm{M}_{a})=\prod_{i=1}^{N_{\rm{obs}}}p(E_{i}|\mathrm{M}_{a}),$
(19)

where $N_{\rm{obs}}$ is the observed number of neutrino events. In our
calculation, we derive $N_{\rm{obs}}$ events using Eq. (15). We then calculate
the likelihood of observing a set of events with energies $\\{E_{i}\\}$ using
the above equation.

Assuming that the different FC scenarios have the same priors, one can then
find the Bayes factor,

$\mathrm{BF}_{10}=\frac{P(\\{E_{i}\\}|\mathrm{M}_{1})}{P(\\{E_{i}\\}|\mathrm{M}_{0})},$
(20)

where $\mathrm{M}_{0}$ represents the null hypothesis, which is the model the
events are generated with, and $\mathrm{M}_{1}$ is the alternative hypothesis
to be compared with the null hypothesis. The guidelines for interpreting the
natural logarithm of $\mathrm{BF}_{10}$ are as follows: i) $\mathrm{ln\
BF}_{10}\in(0–1)$: Not significant, ii) $\mathrm{ln\ BF}_{10}\in(1–3)$:
Indicates a positive evidence, iii) $\mathrm{ln\ BF}_{10}\in(3–5)$: Reflects
strong evidence, and iv) $\mathrm{ln\ BF}_{10}>5$: Demonstrates very strong
evidence.

In line with Ref. Hyper-Kamiokande:2021frf , our analysis revolves around a
fixed number of observed neutrino events, irrespective of the absolute SN
neutrino luminosities and its distance from the detector. In this context, the
absolute neutrino luminosities of the SN and its distance from the detector
become irrelevant in our evaluations, i.e., they can not be considered as free
parameters. This is because, in any case, specific values for these parameters
must be assumed to obtain the given number of events for each FC scenario
under consideration. What is important in our analysis are the spectral
parameters and the relationship between the luminosities of various neutrino
species.

We consider calculations in which the number of observed events are 100, 300,
and 1000. In addition, in our calculations, we assume a constant ratio between
the number of observed events in the IBD and ES channels, namely
$N_{\rm{ES}}=0.05\ N_{\rm{IBD}}$, consistent with the expectations in a water
Cherenkov detector.

###  V.1 Analysis assuming known spectral parameters

In our first evaluation of the differentiating between various FC scenarios,
we work under the assumption that the spectral parameters are known, essential
for calculating the likelihoods outlined in Eqs. (19) and (20). This
assumption stands on two grounds. Firstly, it offers an initial estimation of
our capability of differentiating between various FC scenarios. Should the FC
models remain indistinguishable given the spectral parameters, discerning them
in situations requiring optimisation for obtaining spectral parameters (as
discussed in the following section) becomes a considerable challenge.
Moreover, it is plausible to consider the availability of such spectral
information, particularly in models tied to specific progenitors once multi-
dimensional simulations by different groups will reach convergence.
Nevertheless, these preliminary calculations yield crucial initial insights
that prove invaluable later in the analysis.

In Fig. 3, we present the Bayes factor obtained from Eq.(20) across various FC
scenarios, given the spectral parameters. The results are shown for
$N_{\rm{obs}}=100,300,$ and 1000. The upper panels illustrate outcomes derived
from events generated using the LSD spectrum, with more pronounced differences
among neutrino spectra, where we have considered both the IBD and ES detection
channels. Our approach involves running 1000 Monte Carlo (MC) simulations for
each scenario, subsequently averaging the results to obtain the Bayes factor’s
average.

Let us make a few remarks. Firstly, even with $N_{\rm{obs}}=100$, discerning
certain FC models becomes feasible. However, a larger event count
significantly enhances our ability to differentiate between the FC scenarios.
Then upon reaching $N_{\rm{obs}}=1000$, perfect differentiation among all the
FC models becomes achievable. It is also illuminating to note that in these
calculations, we have not consider the MFC scenario, since $\nu_{x}$ and
$\bar{\nu}_{x}$ are indistinguishable observationally, MFC behaves closely to
that of the EQP scenario.

To provide the reader with some insight into the Bayes factor values, Table 2
presents $\mathrm{ln\ BF}_{10}$ corresponding to the LSD calculation with
$N{\rm{obs}}=300$. It is important to note that several computations exhibit
clear distinguishability among FC scenarios. However, certain entries in the
table display values below 10, suggesting a potential lack of strong
differentiation when considering the whole range of MC calculations. In
specific cases, the samples with the most adverse impact on the Bayes factor
might reduce $\mathrm{ln\ BF}_{10}$ value to less than 5.

Table 2:  $\mathrm{ln\ BF}_{10}$ values corresponding to the LSD calculation
with $N_{\rm{obs}}=300$ (the upper middle panel of Fig. 3). The rows/columns
indicate the null/alternative hypothesises. The red cells show the values
below 5. Note that several cells show clear distinguishability among FC
models. However, certain entries in the table display values below 10,
suggesting a potential lack of strong differentiation when considering the
whole range of MC calculations.

In the middle panels of Fig. 3, we display the Bayes factor for the
calculations closely resembling those in the upper panel with the LSD
spectrum, except for the exclusion of the ES channel. It is important to note
the minimal discrepancy between the upper and middle panels, indicating that
the primary influential factor here is the number of observed neutrino events.
The ES channel, while included in the calculations of the upper panel, appears
to play a subdominant role and does not significantly impact the outcomes.

In the lower panels of Fig. 3, the Bayes factor for the SSD spectrum
calculations is displayed, revealing more subtle variations across the
neutrino spectra. Comparatively, the distribution of the Bayes factor appears
less prominent than in the middle and upper panels. However, large values of
$\mathrm{ln\ BF}_{10}$ can still be observed and most of the FC scenarios can
be distinguished well from each other. This suggests that even in less
favorable scenarios, there remains strong potential for discerning between
different FC scenarios.

###  V.2 FC scenarios, including the MSW effect

Figure 4:  The Bayes factor regarding FC scenarios with MSW. Note that here we
have considered the LSD spectrum and known spectral parameters. The x-axis
denotes the null hypothesis, representing the model by which the neutrino
events were generated, while the y-axis corresponds to the alternative
hypothesis. The diagonal values are by default set to zero.

In the preceding part, we focused on a single FC scenario. However and in
practice, neutrinos are expected to undergo the MSW effect after encountering
FC mechanisms in the deeper SN zones. Hence, it follows that each FC scenario
should encompass an MSW effect on top, either within the NH or IH paradigm. In
our computations, we consider various FC scenarios, and the final neutrino
fluxes which are detailed in Sec.III. Given that the MSW effect is expected to
occur far from other FC scenarios, we proceed by assuming that the MSW effect
can be superimposed. In this context, we consider the final
$\mathcal{F}_{\nu}$ post FC scenario as an input ($\mathcal{F}^{0}_{\nu}$) to
Eqs.(3) and (4), accounting for the MSW effect.

In Fig. 4, we illustrate the Bayes factor regarding FC scenarios with MSW. In
this context, we have considered the LSD spectrum assuming known spectral
parameters, as above, and with $N_{\rm{obs}}=300$ and $1000$ . It is evident
that the introduction of FC scenarios with MSW on top amplifies the
degeneracy, thereby complicating the distinction between different FC
scenarios. Note that the EQP scenario does not include any MSW on top, since
the existing equipartition is already blind to MSW effect.

###  V.3 Unknown spectral parameters

Figure 5:  The Bayes factor regarding the case where the spectral parameters
are not known beforehand. The x-axis denotes the null hypothesis (obtained
from the LSD spectrum), representing the model by which the neutrino events
were generated, while the y-axis corresponds to the alternative hypothesis.
The diagonal values are by default set to zero.  Figure 6:  The Bayes factor
regarding the case where the spectral parameters are unknown, with
$n_{\rm{IBD}}=10000$. The x-axis/y-axis denote the null/alternative
hypothesises.

Up to this point, our approach involved assuming that the spectral parameters
were known once the Bayes factor was calculated. However, in reality, the
Bayes factor remains unknown at the time of detection. As a result, one should
first optimize over the likelihood distribution to determine the most probable
spectral parameters. In Fig. 5, we show our findings in such a scenario. In
our computations, we initially optimized the likelihood for each model,
obtaining the optimal spectral parameters and subsequently, we calculated the
likelihood for each FC scenario. During the optimization process, we assume
the spectral parameters to be confined in the following intervals:

$\begin{split}&L_{\bar{\nu}_{e}}/L_{\nu_{e}},\
L_{\nu_{x}}/L_{\nu_{e}}\in(0.1,1.5),\\\ &\langle E_{\nu_{e}}\rangle\in(5,15),\
\langle E_{\bar{\nu}_{e}}\rangle\in(8,18),\ \langle
E_{\nu_{x}}\rangle\in(10,20)\ \mathrm{MeV},\\\ &\alpha_{\nu_{e}}\in(2.5,4),\
\alpha_{\bar{\nu}_{e}}\in(3.5,5),\ \alpha_{\nu_{x}}\in(1.5,3),\\\ \end{split}$
(21)

which could be considered conservative enough for the SN accretion phase. To
generate the neutrino signal for each model, we utilized the LSD spectrum in
these calculations. In addition, being guided by the results in the previous
section, we only consider the IBD detection channel to enhance the computation
efficiency.

When spectral parameters remain unknown, distinguishing between various FC
scenarios becomes notably challenging, necessitating a larger number of
observed neutrino events. However and despite this complexity, there remains
optimism regarding the possibility of distinguishing between different FC
scenarios.

In our current study, we have focused on $N_{\rm{obs}}$ values up to 1000.
However, it is plausible to extend this analysis to encompass a larger number
of neutrino events during the SN accretion phase, especially when
incorporating the Hyper-Kamiokande or/and the inclusion of multiple detectors
is considered. In Fig. 6, we present calculations where $N_{\rm{obs}}=10000$.
This expanded dataset distinctly separates a larger number of FC scenarios,
showing their discernibility from one another.

##  VI DISCUSSION AND OUTLOOK

Neutrino FC within the SN environment is an intricate and multifaceted
phenomenon. Previous studies have shown that varying physics within the SNe
can yield diverse FC scenarios affecting neutrino flavor evolution in these
extreme settings. In this study we have employed Bayesian techniques to
discern and differentiate between various FC scenarios in the SN environment
using the neutrino signal from a future galactic CCSN. Our findings
demonstrate a promising prospect: with a sufficiently large number of observed
events (exceeding a few hundred), there exists a high probability of
distinguishing between different FC scenarios based on the neutrino signals
emitted by the next galactic CCSN.

In our analysis, we examine two distinct neutrino energy spectra designed to
characterize the neutrino spectra throughout the SN accretion phase. One
spectrum exhibits more pronounced variations among neutrino species, while the
other displays comparatively minor fluctuations. Our findings illustrate that
while distinguishing among the FC scenarios may be more discernible in the
former spectrum, there remains a considerable probability of distinguishing
between different FC scenarios even in cases where spectral differences are
less pronounced.

In addition, in our calculations, we incorporate two distinct neutrino
detection channels: IBD and ES. While the ES channel exhibits sensitivity to
all neutrino species and possesses the potential to distinguish among
different neutrino flavors, our findings illustrate that the crucial factor
here lies in the events number. Surprisingly, our results indicate that
incorporating the ES channel does not significantly enhance our analysis due
to its relatively lower number of events.

In realistic scenarios it is crucial to consider two important factors. The
first is that neutrinos may encounter FC scenarios with MSW effect on top.
This means they could undergo flavor conversions deep within the SN core while
also experiencing the MSW effect at larger radii. Moreover, the spectral
parameters of neutrinos might remain unknown at the time of detection, which
necessitates an optimization to finding the spectral parameters. Our findings
demonstrate that both these effects unfavorably impact our ability to
distinguish among different FC scenarios and increase the minimum number of
events required for efficient FC scenario differentiation.

In summary, our findings highlight the potential to differentiate various FC
scenarios within the SN environment by analyzing a future SN neutrino signal.
However, there remain crucial paths for further investigation. Primarily, our
calculations excluded any information regarding the SN models. One can, in
principle, link various FC scenarios to SN models derived from CCSN
simulations. This connection has been explored to some extent in the studies
detailed in Refs. Hyper-Kamiokande:2021frf ; Olsen:2022pkn ; Saez:2024ayk ,
though only focusing on the MSW effect. The incorporation of SN models has the
potential to improve the ability to differentiate between FC scenarios in
calculations where spectral parameters are unknown. However, it is crucial to
note that introducing the complexity of various SN models might simultaneously
hinder the distinguishability of different FC scenarios by introducing a
potential degeneracy. Additionally, our FC scenarios presumed perfect
conversions, which may not hold true in reality. For example, the MSW effect
might not achieve complete adiabaticity, resulting in final flux values
deviating from those of an ideal adiabatic MSW scenario. Similar uncertainties
could exist in other FC scenarios, where actual equipartition might only be
partial, differing from our assumed perfect conditions. Assessing the
distinguishability of various FC scenarios under partial conversions would be
a valuable exploration. Furthermore, our focus was solely on the IBD and ES
detection channels. While our results suggest that incorporating the ES
channel does not significantly impact our analysis, this might differ for
other available neutrino detection channels. Considering the imperative need
to comprehend neutrino FC mechanisms within the SN environment, addressing
these crucial aspects significantly enhance our ability to extract insights
from the neutrino signal of a future galactic CCSN.

## Acknowledgments

S.A. is deeply grateful to Georg Raffelt for insightful conversations. S.A.
was supported by the German Research Foundation (DFG) through the
Collaborative Research Centre “Neutrinos and Dark Matter in Astro- and
Particle Physics (NDM),” Grant SFB-1258, and under Germany’s Excellence
Strategy through the Cluster of Excellence ORIGINS EXC-2094-390783311. We
would also like to acknowledge the use of the following softwares: Scikit-
learn pedregosa2011scikit , Keras chollet2015keras , Matplotlib Matplotlib ,
Numpy Numpy , SciPy SciPy , and IPython IPython .

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# Fabrication of Ultra-High Q Silica Microdisk Using Chemo-Mechanical
Polishing

S. Honari  S. Haque  T.Lu<EMAIL_ADDRESS>Electrical and Computer
Engineering Department, University of Victoria,
EOW 448, 3800 Finnerty Rd., Victoria, BC V8P 5C2, Canada

###### Abstract

Here we demonstrate that adding a chemo-mechanical polishing (CMP) procedure
to conventional photolithography, a silica microdisk with ultra-high quality
factors ($>10^{8}$) can be fabricated. By comparing with the intrinsic optical
quality factor (Q) measured at 970 nm, we observe that due to the
significantly reduced surface roughness, at 1550 nm wavelength the water
molecule absorption at the cavity surface supersedes Rayleigh scattering as
the dominant factor for Q degradation.

††preprint: AIP/123-QED

The study of whispering gallery mode (WGM) micro resonators has dominated many
fields of science over past decades. Microcavities have become one of the most
attractive optical components to study in a broad range of scientific
disciplines, including but not limited to, nanoparticle detection, biosensing,
quantum information, comb generation, and optomechanics Vahala (2003); Lu  _et
al._ (2011); Baaske, Foreman, and Vollmer (2014); Kippenberg, Holzwarth, and
Diddams (2011); Papp  _et al._ (2014); Kippenberg and Vahala (2008). The power
of microcavities to confine the light is mostly measured by optical quality
factor (Q) as the higher the Q goes the more intense the light inside the
cavity becomes. Campillo, Eversole, and Lin (1991) High Q microcavities are
essential for better detection in sensing applications, lower threshold power
for comb generation, and parametric oscillation Kippenberg, Spillane, and
Vahala (2004); Ilchenko  _et al._ (2004) and narrower linewidth for laser
oscillators. Li  _et al._ (2012)

One of the most important limiting factors for achieving high Q is Rayleigh
scattering induced by surface roughness Gorodetsky, Pryamikov, and Ilchenko
(2000). Hence, efforts to increase the optical Q has mainly relied on using
the expensive fabrication techniques such as stepper lithography Lee  _et al._
(2012) or CO_2 laser reflow Armani  _et al._ (2003) to decrease the surface
roughness of microcavities and increase the Q. However, these techniques come
with either the need for expensive machinery or incompatibility with
monolithic integration. Silica microdisks with Q exceeding 1 billion have been
fabricated recently by carving out the Silica and making trenches on grown
silica film rather that making stand alone disks. Wu  _et al._ (2020) High Q
microresonators and low loss waveguides from other nonlinear materials have
also been demonstrated with deposition on silica platforms to take advantage
of smooth surface and low loss nature of the oxide. Kim  _et al._ (2020)

Chemo-mechanical polishing (CMP) has been used to increase the optical Q for
LiNbO_3 Wu  _et al._ ; Wang  _et al._ (2019) and Si_3N_4 Ji  _et al._ (2017)
microdisks, but to the best of our knowledge, this technique has not been used
to fabricate SiO_2 microdisks. In this paper we incorporate CMP to
conventional photolithography to fabricate ultra-high Q microdisks. Through
scanning electron microscopy (SEM) and numerical modelling, we further analyze
limiting factors to further Q improvement.

Figure 1: a) Photolithograhpy of the resist to transfer the pattern, b) HF wet
etch to make the disk, c) CMP step to increase the smoothness of the surface,
and d) etching Si substrate to make the pillar.

Our fabrication process consists of four steps: a) conventional
photolithography using deep UV light, b) wet etching of SiO_2 using buffered
HF, c) chemomechanical polishing of the SiO_2 disks with silica slurry, and d)
XeF_2 dry etch of the Silicon layer to form the pillars (Fig 1).

The fabrication details are as follows. After applying a standard wafer
cleaning process, we spin coat a silica-on-silicon wafer (SoS, University
Wafers) with positive resist (S1813). A UV lithography is then applied to
transfer the microdisk pattern from the photomask onto the wafer followed by a
buffered HF (Transene) wet etch to form disks on the silica thin film. To
further improve the surface smoothness of the disks, we proceed to the
polishing step using slurry of 70-nm-diameter silica nanoparticles. Before the
polishing step can be done, it is important to change the surface chemistry of
the sample. Hydrogen passivation surface of the sample after HF wet etching
makes the surface extremely hydrophobic, Trucks  _et al._ (1990) which can
hinder the flow of slurry and reduce the effectiveness of the CMP step
significantly. In order to remedy this problem, we used a 10 minutes Piranha
solution (H_2SO_4/H_2O_2) to clean the surface and make it hydrophylic. Hand
polishing of the sample was then performed using the silica nanoparticle
slurry to reduce the surface roughness. RCA cleaning steps has also been
performed after polishing to wash off the slurry particles stick to the
surface of the disk. In the last step of fabrication, XeF_2 gas was used
inside an etching chamber to undercut the disk and form the pillars by etching
the silicon underneath the disks. This step is crucial for achieving high Q
micro resonators, since the light silicon interaction inside the cavity should
be minimized. The maximum amount of undercut is limited by the buckling effect
that happens at larger undercuts, and can degrade the Q or even make the disk
crack. However, the problem can be mitigated by high temperature annealing
(1,000 ${}^{\circ}{\rm C}$). Wu  _et al._ (2020)

Figure 2: Transmission spectra of a 1 mm disk when the 970 nm probe laser
wavelengths scanned toward a) red (blue traces) and b) blue detune (blue
trace). The intrinsic Qs of $1.8\times 10^{8}$ and $2.0\times 10^{8}$ were
obtained through least square fitting to a Lorentzian function (red traces)
respectively, indicating the intrinsic Q to be $(1.9\pm 0.1)\times 10^{8}$. c)
Intrinsic Q as a function of disk diameter at 970 nm (blue squares), 1550 nm
(red squares) wavelength bands with the lengths of the error bars
corresponding to the differences of the measured intrinsic Q at red and blue
detune. In comparison, the recorded values of 7.5 ${\rm\mu}$m and 10
${\rm\mu}$m thick microdisks by Lee et. al. Lee  _et al._ (2012) are displayed
as green squares. Note that, after polishing, the silica thin film thickness
of our disks reduced from 10 ${\rm\mu}$m to 7.5 ${\rm\mu}$m. d) The side view
SEM micrograph of a polished disk at its edge shows the side wall vanishes
after polishing. e) the topview micrograph further confirms the sharp corner
at the edge of the upper surface were polished away as oppose to the
unpolished disk displayed in f). The comparison also demonstrates the reduced
surface roughness through CMP.

In the next step, we measure the intrinsic Qs at 970 nm and 1550 nm wavelength
bands. Here, external cavity tunable lasers (Samtec TSL-550 for 1550 nm, and
Newfocus TLB 6718-D for 970 nm) were used to probe the microdisks through a
tapered optical fiber. As shown in Fig. 2a, by linearly scanning the laser
around the cavity resonance wavelength, a Lorentzian shaped signal will appear
at the output transmission of the tapered fiber, from which intrinsic Q can be
calculated. Here, to accurately readout the optical frequency, we placed a
fiber Mach-Zehnder interferometer in parallel to the microdisk cavity as a
reference. Lu  _et al._ (2011) Note that,the intrinsic Q values measured by
red detuning and blue detuning of the laser wavelengths are slightly
different, due to the thermal effects. Therefore, we take the average of both
Qs as the estimated value of intrinsic Q and their difference as the
uncertainty. For example, as shown on Fig. 2a, the Q measured from blue
detuning on a 10 ${\rm\mu}$m (7.5 ${\rm\mu}$m after polishing) thick, 1mm
diameter microdisk using a 970 nm laser was $2.0{\times}10^{8}$ while Fig. 2b
shows that from a red detuned measurement the Q of $1.8{\times}10^{8}$ was
obtained. Consequently we determine the intrinsic Q to be $(1.9\pm
0.1){\times}10^{8}$ as shown in Fig. 2c (blue square with error bars
representing the uncertainty). We also used our fabrication process to make
500 ${\rm\mu}$m disks, and the average intrinsic Q of $(1.8\pm
0.4)\times{10^{8}}$ was obtained. In order to better compare our work with
previously reported results (green squares), Lee  _et al._ (2012) we used 1550
nm laser to measure the Q as well. We observed the optical Q drops to $(1.2\pm
0.2){\times}10^{8}$ for 1 mm disk and $(6.9\pm 0.2){\times}10^{7}$ for 500
${\rm\mu}$m disk (red squares in Fig. 2b). Note that the Q of our 1 mm disk is
slightly better than the equivalent 7.5 ${\rm\mu}$m thick disks reported in.
Lee  _et al._ (2012) The side view and top view SEM micrographs in Fig. 2d-e
show reduced roughness on the polished surface compared to the unpolished one
shown in Fig. 2f and confirms our process make the disk edges more smooth,
which will reduce the Rayleigh scattering. The edge shape has altered after
the CMP step and the sharp angle between the top surface and the sidewall has
changed to a smooth gradual change in angle, so that top surface and the
sidewall are almost indistinguishable. This change in turn would affect the
mode position inside the cavity as shown in the inset of Fig. 2d.

Further, the observation of lower Q at longer wavelength suggests Rayleigh
scattering is not the dominant factor for Q degradation. Instead, water
absorption at the disk surface dominates the Q at 1550 nm since otherwise
surface roughness induced Rayleigh scattering should make the Q at shorter
wavelength (970 nm) even lower. Borselli  _et al._ (2004) This contradicts the
previous observations that Q at 1550 nm is limited by surface roughness.
Vernooy  _et al._ (1998); Rokhsari, Spillane, and Vahala (2004); Ganta, Dale,
and Rosenberger (2014) To confirm, we numerically simulate the modes and
compare the intrinsic Q at both wavelengths. In simulation, the side view SEM
image of the disk in Fig. 2d was converted to a contour curve for COMSOL
simulation as shown on its inset (black trace). In the absence of water layer,
the Q of $2.12{\times}10^{10}$ at 970 nm and $2.69\times 10^{10}$ at 1550 nm
were obtained through simulations. Since our SEM does not provide sufficient
resolution for surface roughness, the simulated Q were mainly silica material
absorption limited. We then added a layer of water at the disk surface and
obtained a Q of $1.56{\times}10^{8}$ at 1550 nm when the water layer is 1 nm
thick. This is in close agreement to our experiment observation around this
wavelength. On the other hand, the simulated Q at 970 nm was $4.14\times
10^{9}$ due to the significantly lower water absorption at this wavelength
band. As the experimental measurement (Q=(1.9$\pm$0.1$)\times 10^{8}$) is much
lower than the simulation value, we confirm that at 970 nm Q is still limited
by Rayleigh scattering. In fact, assuming the Rayleigh scattering induced Q
(${\rm Q}_{ss}$) has $\lambda^{3}$ dependence Borselli  _et al._ (2004) and
${\rm Q}_{ss}\sim 1.9\times 10^{8}$ at 970 nm, one would expect ${\rm
Q}_{ss}\sim{7.8\times 10^{8}}$ at 1550 nm, which is much higher than the
measured Q. From these observations, we concluded that at 1550 nm, Q is
limited by a 1-nm water layer absorption while at 970 nm surface roughness is
still the limiting factor for Q.

In summary, we made microdisks with ultra high optical Q using conventional
photolithography in conjunction with CMP. As a result, surface roughness is no
longer a limiting factor for Q at 1550 nm. The Q at shorter wavelength can be
further improved by using smaller nanoparticles as slurry to achieve lower
surface roughness. Through simulation, we further find that by removing the 1
nm water layer at the microdisk surface, a Q value close to $10^{9}$ can be
reached at 1550 nm.

This work was supported in part by the Nature Science and Engineering Research
Council of Canada (NSERC) Discovery (Grant No. RGPIN-2020-05938), and Threat
Reduction Agency (DTRA) Thrust Area 7, Topic G18 (Grant No.GRANT12500317). We
would like to acknowledge CMC Microsystems for the provision of products and
services that facilitated this research, including the use of COMSOL for the
numerical analysis. The authors would like to thank Dr. Elaine Humphrey, Mr.
Jon Rudge, and the staff of Advanced Microscopy Facility (AMF) at University
of Victoria for their help and constructive and valuable discussions. The data
that support the findings of this study are available from the corresponding
author upon reasonable request.

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