[a]Shigeru Yoshida

# Multimessenger astronomy driven by high-energy neutrinos

###### Abstract

The possible connection between high energy neutrinos in the energy region
above 100 TeV and ultrahigh energy cosmic rays (UHECRs) at energies above
$10^{19}$ eV motivates multi-messenger observation approaches involving
neutrinos and the multi-wavelength electro-magnetic (EMG) signals. We have
constructed a generic unification scheme to model the neutrino and UHECR
common sources. Finding the allowed space of the parameters on the source
characteristics allows a case study to evaluate the likelihood of each of the
known source classes being such unified sources. The likely source candidates
are transient or flaring objects mainly in optical and X-ray bands. We propose
the two feasible strategies to identify these sources. One is to introduce a
sub-threshold triggering in a wide field of view X-ray observatory for
following up neutrino detections, and the other is to search for EMG
counterparts associated with detections of multiple neutrino events coming
from the same direction within a time scale of $\lesssim 30$ days. Sources
with a total neutrino emission energy greater than $\sim 10^{51}$ erg are
accessible with the present or near-future high energy neutrino observation
facilities collaborating with X-rays and optical telescopes currently in
operation. The neutrino-driven multi-messenger observations provide a smoking
gun to probe the hadronic emission sources we would not be able to find
otherwise.

##  1 Introduction

The cosmic background radiation in ultra-high energy (UHE) sky at $\gg$ TeV is
formed by cosmic rays and neutrinos. The precise measurements of ultrahigh-
energy cosmic rays (UHECRs) by Pierre Auger Observatory (PAO) with high
statistics have now revealed the detailed structure of their energy spectrum
[1]. The IceCube Neutrino Observatory has discovered [2, 3] and measured the
high energy neutrino radiation in the UHE sky [4, 5], realizing the
observation window of the penetrating messengers to study the UHE emissions.
As shown in Fig. 1, we have found that the observed energy flux of high-energy
neutrinos above $~{}\sim 100$ TeV is comparable to that of UHECRs at $\gtrsim
10^{19}$ eV. It suggests that the neutrino radiation may originate in the same
astronomical objects to radiate UHECRs we have been detecting. It may be
plausible that the UHE cosmic radiation can be understood in a common unified
scheme.

Figure 1: The energy fluxes of UHE cosmic background radiations. The small
data points represent the UHECR fluxes measured by PAO [1]. The rest of the
data shows the neutrino energy fluxes and the upper limits measured with
IceCube. The thick black data points were obtained by the neutrino-induced
cascade measurements [4]. The shaded region indicates the flux space
consistent with the $\nu_{\mu}$ induced track measurements [5]. The blue data
point shows the flux of the 6 PeV-energy $\bar{\nu_{e}}$ event estimated by
the dedicated search channel (PEPE) [6]. The thick line with arrows indicates
the differential upper limit obtained by the Extremely-high Energy (EHE)
neutrino search [7]. The neutrino fluxes are the all-flavor-sum fluxes
$E_{\nu}^{2}\Phi_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}$.

We have built the unification generic model to account for the observed
neutrinos at energies greater than $\sim 100$ TeV and UHECRs, based on the
photo-hadronic interaction framework [8]. This modeling enables us to evaluate
if a given class of astronomical objects are qualified as the possible common
origin of UHECRs and neutrinos. It is a viable tool to probe the UHECR origin
with the multi-messenger observations.

In this article, we discuss the possibilities of the UHECR-neutrino common
sources for a broad spectrum of the astronomical objects classes with the
generic unification model. We then discuss how we can identify these sources.
As the viable source candidates are transients, we argue that neutrino follow-
up observations in optical and X-ray bands are feasible methods to find the
sources of hadronic emissions. We propose practical strategies to pin down
neutrino sources that could not be identified by neutrino observations alone.
Finally, we conclude that multi-messenger observations with neutrinos,
optical, and X-ray photons pioneer in the field of high energy astrophysics
with the presently and soon-to-be available facilities.

##  2 A generic unification model

We constructed the generic framework to describe a unified origin of UHECRs
and high energy neutrinos with the parameterization less dependent on the
details of the source environment and the model-dependent micro-physics
processes [8]. Our goal here is to derive generic and model-independent
constraints on characteristics of the possible UHECR-neutrino common emitters.
The resultant constraints are considered to be conservative and weaker than
those obtained by the model-dependent arguments applied to each of the
specific classes of astronomical objects, but the results based on our generic
framework are universal and robust as long as the UHE particle emission
mechanism is mostly determined in a simple setup with physics processes well
approximated by a first leading order effect. We introduce the Occam’s razor
principle here to judge whether a given astronomical object class can be the
unified origin. In this sense, our arguments provide a guidance for
astronomers to conduct the multi-messenger observations.

We made the following assumptions to build our generic unification framework.

One zone

The cosmic ray accelerations and their interactions to produce secondary
neutrinos occur in the same place.

Escape from sources

The energy spectrum of accelerated UHECRs running away from the acceleration
zone is not drastically distorted in their escape process. This assumption may
affect the UHECR energetics condition we will discuss below. Considering the
uncertainties of the UHECR escape process, we use the measured UHECR intensity
as the upper limit to constrain the source luminosity rather than to fit it
with the spectrum calculated by the generic model.

Optically thin environment

The sources are optically thin for UHE protons interacting with photons, and
their emission is directly observed without absorption. This assumption is
valid for photo-hadronic scenarios with one zone modelling.

Photon spectrum

The spectrum of photons interacting with UHECRs to produce neutrinos is
described by a power-law. However we note that the thermal photon yield can
also be reasonably approximated by a power-law form in the energy range
relevant to the high energy neutrino emission within a factor of two
allowance.

Cosmological evolution

The UHECR-neutrino common sources follow the cosmological evolution tracing
the star formation rate (SFR) or any other similar evolutions. However, we
parameterize the evolutions to estimate the conditions for the different
evolution cases when necessary.

###  2.1 The source modelling

The power of the unified sources is gauged by the bolometric photon luminosity

$L^{\prime}_{\gamma}=4\pi R^{2}c\int\limits_{\varepsilon_{\gamma}^{\rm
min}}^{\varepsilon_{\gamma}^{\rm
max}}\frac{dn_{\gamma}}{d\varepsilon^{\prime}_{\gamma}}\varepsilon^{\prime}_{\gamma}d\varepsilon^{\prime}_{\gamma},$
(1)

where $R$ is the distance of the UHECR acceleration and emission site from the
source center. Primed (’) characters represent quantities measured in the rest
frame of plasma with the Lorentz bulk factor $\Gamma$. The photon density
spectrum follows a power-law form as

$\frac{dn_{\gamma}}{d\varepsilon^{\prime}_{\gamma}}=\frac{K^{\prime}_{\gamma}}{\varepsilon^{\prime}_{\gamma
0}}\left(\frac{\varepsilon^{\prime}_{\gamma}}{\varepsilon^{\prime}_{\gamma
0}}\right)^{-\alpha_{\gamma}},$ (2)

where $\varepsilon^{\prime}_{\gamma 0}$ is the reference energy in the engine
frame, and it is associated with the representative energy of UHECRs
$\varepsilon_{p0}^{\Delta}$ by

$\varepsilon_{\gamma
0}=\frac{(s_{R}-m_{p}^{2})}{4}\frac{\Gamma^{2}}{\varepsilon_{p0}^{\Delta}}$
(3)

where $s_{R}\approx 1.47$ GeV2 is the Mandelstam variable at the $\Delta$
resonance in the photopion production. The representative CR energy
$\varepsilon_{p0}^{\Delta}$ is set to be 10 PeV in the present formulation, as
this energy range of cosmic ray protons should produce the PeV-energy
neutrinos IceCube has detected. The spectrum of UHECRs emitted from the
sources is assumed to follow a power-law with index of $\alpha_{\rm CR}$.

The bolometric luminosity of UHECRs at energies above
$\varepsilon_{p0}^{\Delta}=10$ PeV is connected to $L_{\gamma}$ via the CR
loading factor $\xi_{\rm CR}$ as $L_{\rm CR}\approx\xi_{\rm
CR}L_{\gamma}=\xi_{\rm CR}L^{\prime}_{\gamma}\Gamma^{2}$.

The neutrino luminosity with respect to a given $L_{\rm CR}$ is determined by
the $p\gamma$ interaction optical depth, an average number of the interaction
times before cosmic ray protons escape from the interaction site. It is
approximately given by [8]

$\displaystyle\tau_{p\gamma}(\varepsilon_{i})$ $\displaystyle\approx$
$\displaystyle\tau_{p\gamma
0}{\left(\frac{\varepsilon_{i}}{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)}^{\alpha_{\gamma}-1}$
(4) $\displaystyle\approx$
$\displaystyle\left[\frac{2}{1+\alpha_{\gamma}}\frac{K^{\prime}_{\gamma}R}{\Gamma}\int
ds\frac{\sigma_{p\gamma}(s)}{s-m_{p}^{2}}\right]{\left(\frac{\varepsilon_{i}}{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)}^{\alpha_{\gamma}-1}$
$\displaystyle\approx$
$\displaystyle\frac{B^{\prime}}{\Gamma^{2}}\sqrt{\frac{L^{\prime}_{\gamma}}{\xi_{B}}}C(\alpha_{\gamma},\tilde{\varepsilon}_{p0}^{\Delta}){\left(\frac{\varepsilon_{i}}{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)}^{\alpha_{\gamma}-1}.$
(5)

Proceeding from Eq. (4) to (5), the explicit dependence on $R$ is eliminated
by considering the energy density balance between the photon radiation
$L^{\prime}_{\gamma}$ and the magnetic energy with the B-field strength
$B^{\prime}$ via the equipartition parameter $\xi_{B}$. The constant $C$
depends only on the photon spectrum power-law index $\alpha_{\gamma}$ and the
representative CR energy $\varepsilon_{p0}^{\Delta}$ and is approximately
given by

$\displaystyle C(\alpha_{\gamma},\tilde{\varepsilon}_{p0}^{\Delta})$
$\displaystyle\sim$ $\displaystyle 2.4\times{10}^{-24}~{}{\rm erg}^{-1}~{}{\rm
cm}^{3/2}~{}{\rm s}^{1/2}$ (6) $\displaystyle\times$
$\displaystyle\left(\frac{2}{1+\alpha_{\gamma}}\right)\left(\frac{\tilde{\varepsilon}_{p0}^{\Delta}}{10~{}{\rm
PeV}}\right).$

###  2.2 The required source conditions

A UHECR source must meet the following necessity conditions : The
acceleration, the escape, and the survival requirements.

In order to accelerate cosmic rays to UHE range, the acceleration time must be
faster than the dynamical time scale. It sets the lower bound for the magnetic
field as

$B^{\prime}\gtrsim\frac{\varepsilon_{i}^{\rm max}}{eZ}\frac{\eta}{R}\approx
1.1\times 10^{5}\eta\left(\frac{R}{3\times 10^{12}\ {\rm
cm}}\right)^{-1}\left(\frac{\varepsilon_{i}^{\rm max}}{Z10^{11}~{}{\rm
GeV}}\right)\ {\rm G},$ (7)

where $\eta\gtrsim 1$ is the particle acceleration efficiency term. This
condition, also known as the Hillas condition when
$\eta\rightarrow\beta^{-2}$, can be transformed to the constraint on the
target photon luminosity $L^{\prime}_{\gamma}$, the gauge of the source engine
power in the present generic modelling scheme,

$\displaystyle L^{\prime}_{\gamma}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\xi_{B}^{-1}c\eta^{2}\beta^{2}\left(\frac{\varepsilon_{i}^{\rm
max}}{Ze}\right)^{2}$ (8) $\displaystyle\simeq$ $\displaystyle 1.7\times
10^{45}~{}\xi_{B}^{-1}\eta^{2}\beta^{2}\left(\frac{\varepsilon_{i}^{\rm
max}}{Z10^{11}~{}{\rm GeV}}\right)^{2}\quad{\rm erg/s}$ (9)

In addition, to ensure that UHECRs can leave the sources before losing their
energies, the escape time scale must be faster than the cosmic ray energy loss
time scale. The energy loss processes consist of the $p\gamma$ photo-meson
production, Bethe-Heitler (BH) interactions, and the synchrotron cooling. The
photo-meson production time scale is essentially counted with the $p\gamma$
optical depth, $\tau_{p\gamma}$, in the present scheme, and any sources with
$\tau_{p\gamma}(\varepsilon_{i}^{\rm max})\lesssim 1$ implies that the energy
loss by the $p\gamma$ photo-meson production is not a deciding factor to limit
the UHECR acceleration and escape processes. We examine this $\tau_{p\gamma}$
condition by estimating $\tau_{p\gamma}$ using Eq. (5) in Section 2.4. As the
BH process is in general important only if the photon spectrum is softer as
$\alpha_{\gamma}\gtrsim 2$, the UHECR escape condition is formulated as a
necessity condition by requiring the dynamical time scale faster than the
synchrotron cooling time scale. It has been found that this condition is
transformed to the upper bound of the $p\gamma$ optical depth at the cosmic
ray reference energy $\varepsilon_{p0}^{\Delta}=10$ PeV, $\tau_{p\gamma 0}$,
as

$\displaystyle\tau_{p\gamma 0}$ $\displaystyle\lesssim$ $\displaystyle 6\times
10^{-1}\frac{2}{1+\alpha_{\gamma}}\left(\frac{\xi_{B}}{0.1}\right)^{-1}{\left(\frac{A}{Z}\right)}^{4}{\left(\frac{\varepsilon_{i}^{\rm
max}}{10^{11}\ {\rm GeV}}\right)}^{-1}.$ (10)

If the measured bulk of UHECRs is dominated by heavier nuclear rather than
nucleon as strongly indicated by the data obtained by PAO, the further severe
requirement must be satisfied – The nuclei survival condition [9]. That is, we
require that nuclei with $A>1$ and $Z>1$ are accelerated and survive. This is
possible only if the time scale of the photo-disintegration is slower than the
dynamical time scale which leads to the condition on photo-disintegration
optical depth as $\tau_{A\gamma}\lesssim A$. We found that this condition sets
the upper bound of the $p\gamma$ optical depth as [8]

$\displaystyle\tau_{p\gamma 0}$ $\displaystyle\lesssim$ $\displaystyle
A\frac{\int ds\frac{\sigma_{p\gamma}(s)}{s-m_{p}^{2}}}{\int
ds\frac{\sigma_{A\gamma}(s)}{s-m_{A}^{2}}}{\left[\left(\frac{s_{\rm
GDR}-m_{A}^{2}}{s_{R}-m_{p}^{2}}\right)\left(\frac{\tilde{\varepsilon}^{\Delta}_{p0}}{\varepsilon_{i}^{\rm
max}}\right)\right]}^{\alpha_{\gamma}-1}$ (11) $\displaystyle\lesssim$
$\displaystyle
0.4~{}{\left(\frac{A}{56}\right)}^{0.79}=0.2\left(\frac{A}{24}\right)^{0.79}.$

###  2.3 The constraints due to the UHECR and and neutrino fluxes

Figure 2: The allowed region in the parameter space of luminosity per unit
volume, $L^{\prime}_{\gamma}\mathcal{N}_{\Gamma}$, and damping factor
$\displaystyle{1-e^{-\tau_{p\gamma 0}}}$ [8]. The region enclosed by the solid
lines displays the allowed space by the UHECR and the neutrino flux
requirements. The shaded region represents the parameter space allowed also by
considering the UHECR proton escape condition or the nuclear survival
condition. The left panel shows the proton model while the right panel shows
the case of primary silicon nuclei.

UHECR sources in the unification scenario must provide both the UHECR flux and
the high energy neutrino flux that are consistent with the measurements. The
analytical formulation to calculate the spectrum of UHECRs on the Earth and
that of secondary produced neutrinos have been derived in Ref. [10, 8] to
place the constraints on the source parameters of $L^{\prime}_{\gamma}$,
$\tau_{p\gamma 0}$, and the boosted source number density $N_{\Gamma}\equiv
n_{0}\xi_{\rm CR}\Gamma^{2}$ where $n_{0}$ is the comoving number density in
the present epoch. Note that $L^{\prime}_{\gamma}N_{\Gamma}=\xi_{\rm
CR}L_{\gamma}n_{0}=L_{\rm CR}n_{0}$ is the bolometric luminosity density of
UHECRs above $\varepsilon_{p0}^{\Delta}=10$ PeV. Figure 2 shows the resultant
constraints. The region enclosed in the solid lines is the allowed parameter
space by the flux conditions. The conditions were set by the criteria that
ensure the consistency to the neutrino measurements with IceCube including the
upper limit of flux at $\varepsilon_{\nu}\gtrsim 100$ PeV, and that the UHECR
flux on the earth would not exceed the integral flux above $10^{19}$ eV
measured by PAO.

We can interpret the allowed parameter space shown in Fig. 2 from the context
of the UHECR energetics and the analytical estimate of the fiducial neutrino
flux. The UHECR differential luminosity density is estimated as [11]

$\displaystyle E_{\rm CR}\frac{dQ_{\rm CR}}{dE_{\rm CR}}$
$\displaystyle\approx$ $\displaystyle 6.3\times 10^{43}[{\rm erg\
Mpc^{-3}yr^{-1}}]\ \left(\frac{E_{\rm CR}}{10^{19.5\ {\rm
eV}}}\right)^{2-\alpha_{\rm CR}}$ (14) $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{lr}1.8\times 10^{44}\ [{\rm erg\
Mpc^{-3}yr^{-1}}]&\alpha_{\rm CR}=2.3,E_{\rm CR}=10^{18}\ {\rm eV}\\\
3.4\times 10^{44}\ [{\rm erg\ Mpc^{-3}yr^{-1}}]&\alpha_{\rm CR}=2.5,E_{\rm
CR}=10^{18}\ {\rm eV}\end{array}\right.$

As a representative case, we consider $\alpha_{\rm CR}=2.3$ hereafter. From
Eq. (14), the resultant bolometric UHECR energy density above the reference
energy above $\varepsilon_{p0}^{\Delta}=10$ PeV is given by

$\displaystyle n_{0}\xi_{\rm CR}L_{\gamma}$ $\displaystyle\approx$
$\displaystyle 13E_{\rm CR}\frac{dQ_{\rm CR}}{dE_{\rm CR}}|_{E_{\rm
CR}=10^{18}\ {\rm eV}}$ (15) $\displaystyle\approx$ $\displaystyle 2.3\times
10^{45}\quad{\rm erg\ Mpc^{-3}yr^{-1}},$

which is consistent with the allowed region of the parameter space. This
energetics condition above effectively sets the requirement of the CR loading
factor for a given $L_{\Gamma}$ and $n_{0}$ such as

$\xi_{\rm CR}\approx 0.7\left(\frac{L_{\gamma}}{10^{46}{\rm
erg/s}}\right)^{-1}\left(\frac{n_{0}}{10^{-8}{\rm Mpc^{-3}}}\right)^{-1}.$
(16)

The neutrino emissivity from a source is connected to the primary UHECR
emissivity as [12]

$\varepsilon_{\nu}^{2}\frac{d\dot{N}_{\nu}}{d\varepsilon_{\nu}}\approx\xi_{\pi}<x><y_{\nu}>\tau_{p\gamma}\varepsilon_{\rm
CR}^{2}\frac{d\dot{N}_{\rm CR}}{d\varepsilon_{\rm CR}}A^{2-\alpha_{\rm CR}}$
(17)

for a hadronically thin (i.e.$\tau_{p\gamma}\lesssim 5$) source. Here
$\xi_{\pi}\sim 3/2$ is the average multiplicity of neutrinos from a single
pion produced by the photo-hadronic interaction, and $<y_{\nu}>\sim 1/4$ is
the average fraction of energy channeling into a neutrino from the secondary
produced pion and $<x>\sim 0.2$ is the average ineasticity of the $p\gamma$
collision. Since the energy flux of the high energy cosmic background
neutrinos can be approximately written using the source emissivity
$\displaystyle\varepsilon_{\nu}^{2}d\dot{N}_{\nu}/d\varepsilon_{\nu}$, we can
relate the $p\gamma$ optical depth to the required bolometric UHECR luminosity
density for a given energy flux of neutrinos via Eq. (17). We get

$\displaystyle\tau_{p\gamma
0}L^{\prime}_{\gamma}\mathcal{N}_{\Gamma}=\tau_{p\gamma 0}n_{0}\xi_{\rm
CR}L_{\gamma}$ $\displaystyle\approx$ $\displaystyle 9.3\times
10^{43}\left(\frac{E_{\nu}^{2}\Phi_{\nu}}{2\times 10^{-8}\ [{\rm GeV\ cm^{-2}\
sec^{-1}sr^{-1}]}}\right)$ (18)
$\displaystyle\times\left(\frac{\xi_{z}}{2.8}\right)^{-1}A^{0.3}\quad{\rm erg\
Mpc^{-3}yr^{-1}},$

for $\alpha_{\rm CR}=2.3$. Here $\displaystyle\xi_{z}\equiv(1/t_{\rm H})\int
dt\Psi(z)/(1+z)$ is a dimensionless parameter that depends on the redshift
evolution function $\Psi(z)$ of the sources. This relation well represents the
allowed parameter space shown in Fig. 2.

Combining the UHECR luminosity density given by Eq. (15) with this neutrino
fiducial flux condition, Eq. (18), sets the lower bound of the $p\gamma$
optical depth, which is

$\displaystyle\tau_{p\gamma 0}$ $\displaystyle\gtrsim$ $\displaystyle
0.04A^{0.3}\left(\frac{\xi_{z}}{2.8}\right)^{-1}$ (19) $\displaystyle\gtrsim$
$\displaystyle
0.1\left(\frac{A}{28}\right)^{0.3}\left(\frac{\xi_{z}}{2.8}\right)^{-1}.$

As the UHECR escape condition sets the upper bound of the optical depth (see
Eq. (10)), this fiducial neutrino flux requirements leads to a necessity
condition of the B-field equipartition parameter.

$\xi_{\rm B}\lesssim
1.5\left(\frac{A}{Z}\right)^{4}\left(\frac{\xi_{z}}{2.8}\right){\left(\frac{\varepsilon_{i}^{\rm
max}}{10^{11}\ {\rm GeV}}\right)}^{-1}A^{-0.3}$ (20)

###  2.4 The case study

| AGN corona | BL Lac | FSRQ | Radio Gal. MAD
---|---|---|---|---
$\Gamma$ of the target photons |  $\approx$ 1 |  $\approx$ 10 |  $\approx$ 1 |  $\approx$ 1
target photon energy: Eq. (3) | opt/UV/X-ray | X-ray | UV/Opt | UV/Opt
$L_{\gamma}\ [{\rm erg/s}]$ | $10^{44}$ | $2\times 10^{44}$ | $4\times 10^{46}$ | $10^{41}$
$n_{0}\ [{\rm Mpc^{-3}}]$ | $5\times 10^{-6}$ | $3\times 10^{-7}$ | $3\times 10^{-10}$ | $2\times 10^{-6}$
$B^{\prime}\ [{\rm G}]$ | 300 | 1 | 1 | 100
$\xi_{\rm B}$ | 13 | 75 | 0.1 | $4\times 10^{5}$
Acceleration: Eq. (9) | $\xi_{\rm B}\gtrsim 0.087(\frac{Z}{14})^{-2}$ | $\xi_{\rm B}\gtrsim 4.3(\frac{Z}{14})^{-2}$ | $\xi_{\rm B}\gtrsim 0.04(\frac{Z}{1})^{-2}$ | $\xi_{\rm B}\gtrsim 87(\frac{Z}{14})^{-2}$
$\tau_{p\gamma 0}$ by Escape: Eq. (10) | $\lesssim 0.005(\frac{\xi_{\rm B}}{13})^{-1}$ | $\lesssim 7.5\times 10^{-4}(\frac{\xi_{B}}{80})^{-1}$ | $\lesssim 0.6(\frac{\xi_{B}}{0.1})^{-1}$ | $\lesssim 8\times 10^{-8}(\frac{\xi_{B}}{4\times 10^{5}})^{-1}$
$\tau_{p\gamma 0}$ by Nuclei survival: Eq. (11) | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$
$\tau_{p\gamma 0}$ by $\nu$ Flux: Eq. (19) | $\gtrsim 0.1(\frac{\xi_{z}}{2.8})^{-1}(\frac{A}{28})^{0.3}$ | $\gtrsim 0.1(\frac{\xi_{z}}{2.8})^{-1}(\frac{A}{28})^{0.3}$ | $\gtrsim 0.01(\frac{\xi_{z}}{8.4})^{-1}$ | $\gtrsim 0.5(\frac{\xi_{z}}{0.6})^{-1}(\frac{A}{28})^{0.3}$
$\xi_{\rm CR}$: Eq. (16) | $\approx 1.2$ | $\approx 1.2$ | $\approx 5.8$ | $\approx 350$
Table 1: The parameters of the neutrino emission characteristics in the
unification model and the constraints on $\xi_{B}$, the B-field equipartition
parameter, and $\tau_{p\gamma 0}$, the photo-hadronic optical depth at the
cosmic ray reference energy of $\varepsilon^{\Delta}_{p0}=10$ PeV imposed by
the conditions for UHECR-neutrino common sources. The various sites/population
in AGN family are listed.

Tables 1 and 2 list the various source classes together with their
characteristic parameters and the constraints on $\xi_{B}$ and $\tau_{p\gamma
0}$ imposed by the conditions we discussed earlier.

The AGN corona [13] is already disfavored as a common UHECR neutrino source
regardless of any model-dependent arguments. The conditions on
$\tau_{p\gamma}$ from the UHECR escape and the fiducial neutrino flux
requirements cannot hold concurrently. This is primarily because the corona is
expected to be strongly magnetized ($\xi_{\rm B}\gg 1$). The approximate
estimate using Eq. (5) gives $\displaystyle\tau_{p\gamma}\approx
100\left({B^{\prime}}/{\rm 0.3kG}\right)\left({L_{\gamma}}/{10^{44}{\rm
erg/s}}\right)^{\frac{1}{2}}$
$\left({\xi_{B}}/{13}\right)^{-\frac{1}{2}}\left({\beta}/{0.02}\right)^{-1}\left({\varepsilon_{i}}/{\tilde{\varepsilon}_{p0}^{\Delta}}\right)^{0.8}$,
indicating that AGN corona may be a TeV-PeV neutrino source candidate, but
certainly not a UHECR source since $\tau_{p\gamma}\gg 1$ prevents hadrons from
being accelerated to UHE range.

The BL Lac [14] should also meet the contradicting $\tau_{p\gamma}$ conditions
demanded by the UHECR escape and neutrino fiducial flux requirements, which
makes it difficult to be considered as a common source. The estimated optical
depth is $\displaystyle\tau_{p\gamma 0}\approx 4\times
10^{-6}\left({B^{\prime}}/{{\rm 1G}}\right)\left({\Gamma}/{10}\right)^{-3}$
$\left({L_{\gamma}}/{2\times 10^{44}{\rm
erg/s}}\right)^{\frac{1}{2}}\left({\xi_{B}}/{80}\right)^{-\frac{1}{2}}$, which
indeed suggests that BL Lacs may be UHECR sources but too dark in neutrinos.

UHE emission from FSRQ [15] is an interesting possibility. The UHECR escape
and the neutrino flux conditions can be concurrent in principle, providing
that $\tau_{p\gamma 0}\sim 0.1-1$ is in the plausible range at the FSRQ
system. Using Eq. (5), we indeed estimate $\displaystyle\tau_{p\gamma}\approx
1.4\left({B^{\prime}}/{{\rm 1G}}\right)\left({L_{\gamma}}/{4\times 10^{46}{\rm
erg/s}}\right)^{\frac{1}{2}}\left({\xi_{B}}/{0.1}\right)^{-\frac{1}{2}}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)^{0.5}$.
The high photon luminosity ($L_{\gamma}\gtrsim 10^{46}\ {\rm erg/s}$) meets
the acceleration condition given by Eq. (9) for protons ($Z=1$) even if
$\xi_{\rm B}\ll 1$ while the nuclei survival condition could be only barely
satisfied. It implies that FSRQs may be the common origin for UHECR protons
and neutrinos, though not UHECR nuclei. However, this hypothesis is
disfavored, since the strongly evolved sources such as FSRQs are not likely to
be UHECR origins if the proton component is not negligible at the UHE range
[16, 17]. This is because the GZK cosmogenic neutrinos would have overshot the
present upper limit of neutrino flux at EeV ($10^{18}\ {\rm eV}$) range placed
by IceCube and PAO.

A scenario of hadronic emissions from magnetically arrested disks (MAD) in a
subclass of radio galaxies has been proposed [18], motivated by the GeV-TeV
gamma-ray observations from nearby radio galaxies. The cosmic ray
accelerations can be plausible with the framework of radiatively inefficient
accretion flows (RIAFs). Because of the highly magnetized environment
($\xi_{\rm B}\gg 100$), MAD may be producing UHECR nuclei though UHECR protons
cannot escape. It certainly meets the acceleration condition given by Eq. (9)
for silicons ($Z=14$). The energetics condition, Eq. (16), requires $\xi_{\rm
CR}\gg 10$ but this constraints can be relaxed if we expand this model to the
entire Fanaroff-Riley I galaxies by including high-excitation radio galaxies,
bringing $n_{0}\approx 10^{-4}\ {\rm Mpc}^{-3}$. However, it is unlikely to
emit neutrinos with the intensity measured by IceCube. The approximate
estimate using Eq. (5) gives $\displaystyle\tau_{p\gamma}\approx 2.5\times
10^{-3}\left({B^{\prime}}/{{\rm
100G}}\right)\Gamma^{-3}\left({L_{\gamma}}/{10^{41}{\rm
erg/s}}\right)^{\frac{1}{2}}\left({\xi_{B}}/{3.7\times
10^{5}}\right)^{-\frac{1}{2}}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)^{1.2}$,
which is far below the fiducial neutrino flux condition, Eq. (19). This is
primarily because the MAD is optically too thin, given that the luminosity of
the target photons (optical/UV) $L_{\gamma}\sim 10^{41}\ {\rm erg/s}$. Note
that the high magnetic field strength would cause the synchrotron cooling of
the secondary produced muons by the photo-hadronic collisions, which
suppresses neutrinos with energies higher than $\sim 10$ PeV.

| jetted TDE | TDE corona | Low Luminosity GRB | Engine-driven SNe
---|---|---|---|---
|  |  |  | afterglow
$\Gamma$ of the target photons |  $\approx$ 10 |  $\approx$ 1 |  $\approx$ 2–10 |  $\approx$ 1
target photon energy: Eq. (3) | X-ray | opt/UV/X-ray | X-ray | UV/Opt
$L_{\gamma}\ [{\rm erg/s}]$ | $10^{47}$ | $3\times 10^{43}$ | $10^{47}$ | $3\times 10^{45}$
$n_{0}\ [{\rm Mpc^{-3}}]$ | $3\times 10^{-12}(\frac{\Delta T}{10^{6}{\rm s}})$ | $4\times 10^{-7}(\frac{\Delta T}{10^{7}{\rm s}})$ | $3\times 10^{-11}(\frac{\Delta T}{3\times 10^{3}{\rm s}})$ | $10^{-9}(\frac{\Delta T}{10^{4}{\rm s}})$
$B^{\prime}\ [{\rm G}]$ | 500 | $10^{3}$ | 80 | 1.6
$\xi_{\rm B}$ | 1 | 45 | 0.1 | 4
Acceleration: Eq. (9) | $\xi_{\rm B}\gtrsim 0.009(\frac{Z}{14})^{-2}$ | $\xi_{\rm B}\gtrsim 0.3(\frac{Z}{14})^{-2}$ | $\xi_{\rm B}\gtrsim 0.009(\frac{Z}{14})^{-2}$ | $\xi_{\rm B}\gtrsim 0.3(\frac{Z}{14})^{-2}$
$\tau_{p\gamma 0}$ by Escape: Eq. (10) | $\lesssim 0.06(\frac{\xi_{\rm B}}{1})^{-1}$ | $\lesssim 1.3\times 10^{-3}(\frac{\xi_{B}}{45})^{-1}$ | $\lesssim 0.6(\frac{\xi_{B}}{0.1})^{-1}$ | $\lesssim 0.015(\frac{\xi_{B}}{4})^{-1}$
$\tau_{p\gamma 0}$ by Nuclei survival: Eq. (11) | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$ | $\lesssim 0.2(\frac{A}{28})^{0.79}$
$\tau_{p\gamma 0}$ by $\nu$ Flux: Eq. (19) | $\gtrsim 0.5(\frac{\xi_{z}}{0.5})^{-1}(\frac{A}{28})^{0.3}$ | $\gtrsim 0.5(\frac{\xi_{z}}{0.5})^{-1}(\frac{A}{28})^{0.3}$ | $\gtrsim 0.1(\frac{\xi_{z}}{2.8})^{-1}(\frac{A}{28})^{0.3}$ | $\gtrsim 0.1(\frac{\xi_{z}}{2.8})^{-1}(\frac{A}{28})^{0.3}$
$\xi_{\rm CR}$: Eq. (16) | $\approx 220$ | $\approx 5.6$ | $\approx 24$ | $\approx 24$
Table 2: Same as Table 1 but the various transient objects are listed.

Powerful transient objects (table 2) have also been discussed in the
literature for considering UHECR origins. Jetted TDEs cannot be UHECR proton
sources because of the escape requirement, but potentially UHE nuclei sources
[19]. However, the tight margin between the nuclei survival condition and the
fiducial neutrino flux requirement needs a fine parameter tuning for
qualifying this object to be the UHECR-neutrino common sources. The
approximate estimate indeed gives $\displaystyle\tau_{p\gamma}\approx
0.34\left({B^{\prime}}/{{\rm
500G}}\right)\left({\Gamma}/{10}\right)^{-3}\left({L_{\gamma}}/{10^{47}{\rm
erg/s}}\right)^{\frac{1}{2}}\xi_{B}^{-\frac{1}{2}}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)$
which could barely meet the both conditions, considering the unavoidable
uncertainties of the parameter values. The more serious issue is that it is
more difficult to meet the energetics condition. The resultant CR loading
factor $\xi_{\rm CR}\gg 10$ raises questions about credibility of the TDE
scenario.

Non-jetted TDEs are more generous objects and may alleviate the energetics
issue. The wind driven by TDE or the possible corona may be a plausible site
for cosmic-ray acceleration [20]. The demerit is their possible optically
thick environment. The TDE corona scenario expects
$\displaystyle\tau_{p\gamma}\approx 20\left({B^{\prime}}/{{\rm
1kG}}\right)\Gamma^{-3}\left({L_{\gamma}}/{3\times 10^{43}{\rm
erg/s}}\right)^{\frac{1}{2}}\left({\xi_{B}}/{45}\right)^{-\frac{1}{2}}\left({\beta}/{0.1}\right)^{-1}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)^{0.8}$
and the UHE nuclei cannot survive. The high magnetic field ($\sim\ {\rm kG}$)
naturally sets such dense photon environment that break down nuclei via photo-
disintegration. UHECR protons cannot escape either.

Low Luminosity GRBs [21] are among the most promising candidates for the
unification scenario. The estimated optical depth,
$\displaystyle\tau_{p\gamma}\approx 0.1\left({B^{\prime}}/{{\rm
80G}}\right)\left({\Gamma}/{10}\right)^{-3}\left({L_{\gamma}}/{10^{47}{\rm
erg/s}}\right)^{\frac{1}{2}}$
$\left({\xi_{B}}/{0.1}\right)^{-\frac{1}{2}}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)^{1.2}$
exactly meets all the conditions as listed in table 2. The CR lading factor
required for the UHECR energetics may be high, but it can be relaxed, given
that the rate density $\rho_{0}\sim 3\times 10^{-7}\ {\rm Mpc}^{-3}{\rm
yr}^{-1}$ is still quite uncertain.

The external acceleration during the afterglow of engine-driven SNe has also
been among the UHECR nuclei emission models [22]. As seen in Table 2, the
UHECR escape condition prevents the afterglow from being the common UHECR
protons and neutrino sources regardless of model-dependent arguments. Just as
in the other transients, there is the (tight) margin to meet both the nuclei
survival condition and the fiducial neutrino flux requirement. However, the
estimate using Eq. (5) gives $\displaystyle\tau_{p\gamma}\approx 1.1\times
10^{-4}\left({B^{\prime}}/{{\rm 1.6G}}\right)\left({\Gamma}/{10}\right)^{-3}$
$\left({L_{\gamma}}/{3\times 10^{45}{\rm
erg/s}}\right)^{\frac{1}{2}}\left({\xi_{B}}/{4}\right)^{-\frac{1}{2}}\left({\varepsilon_{i}}/{{\tilde{\varepsilon}_{p0}^{\Delta}}}\right)^{0.7}$,
which does not satisfy the neutrino flux condition. This source is
hadronically too thin to explain the 100 TeV-PeV energy neutrinos measured by
IceCube.

##  3 Identification of UHECR sources with neutrino follow-up observations

Many of the potential high energy neutrino/UHECR sources are transient
emitters. Low Luminosity GRBs, or jetted-TDEs are among the representative
examples. SNe with strong interaction with circumstellar material and other
non-relativistic transients may contribute to the TeV neutrino sky [23, 24]
although they are unlikely to be the sources of UHECRs. Moreover, the neutrino
emissions from jets in AGNs (e.g., FSRQs) are expected to occur in flare
rather than a steady manner [25]. Thus it is a powerful method to search for
electromagnetic (EMG) counterparts by follow-up observations triggered by a
neutrino detection in order to identify neutrino or UHECR sources.

It is straightforward to find the neutrino-EMG association with a rare type of
the objects. The GeV-energy gamma-ray blazars detected by Fermi-LAT belong to
this category. However more abundant classes of objects such as SNe or low-
luminosity GRBs would yield more frequent chance coincidences between neutrino
and EMG detections, which makes it challenging to claim robust associations.
Moreover, the optical sky is filled with many SNe irrelevant from neutrino
emissions (e.g., type Ia) and they cause significant contamination in optical
follow-ups. The longer duration of neutrino transient emissions reaching to
weeks expected from circumstellar SNe or TDEs would yield even more severe
contamination by the unrelated SNe. The simple multi-messenger strategy faces
the difficulty here.

There are two feasible solutions. The first approach is to conduct follow-up
observations in X-ray band. The X-ray sky is quieter as SNe are not luminous
X-ray transients. The drawback is that many of the neutrino source candidates
we discussed above may not be bright enough for the existing X-ray telescopes
with the rapid follow-up capability. The low luminosity GRBs are good
examples. Their dim luminosity of $L_{\rm X}\sim 10^{47}\ {\rm erg/s}$ is
below the regular detection threshold unless a progenitor happens to be
located in the neighborhood of our galaxy. It is, therefore, necessary to
implement a sub-threshold detection trigger on a X-ray observatory with wide
field of view. The Monitor of All-sky X-ray Image (MAXI) telescope [26] can
meet this demand as it is regularly monitoring all sky at a single photon
counting level. The MAXI-IceCube coincidence search with the sub-threshold
detection algorithms will be scheduled in near future.

The second approach is to search for neutrino multiplet, two (doublet) or more
neutrinos originating from the same direction within a certain time frame
[27]. Since only nearby sources can yield detectable neutrino multiplets, we
can rule out any distant EMG transient counterpart candidates found in a
follow-up observation triggered by a multiplet detection. Therefore, searches
for the neutrino-optical association can be performed under less contaminated
environment. It has been found that $\sim 90$ % of sources to produce
detectable neutrino multiplets by a $1\ {\rm km}^{3}$ neutrino telescope are
localized within $z\lesssim 0.15$ while the distribution of sources to yield a
singlet neutrino detection extends up to $z\gtrsim 2$ [27]. Confining neutrino
sources within the local universe realizes the following strategy of optical
follow-up observations for claiming robust neutrino-optical associations.

Figure 3: Probability distribution of the redshift of the closest counterpart
with bin size $\Delta z=0.005$ [27]. The bin size is chosen for illustrative
purposes. The blue curve represents the case of the signal hypothesis that the
object is the neutrino source to yield the detected neutrino multiplet, and
the red curve shows the case of the coincident background hypothesis, that is
the chance coincident detection of an unassociated SN.
${\mathcal{E}}_{\nu}^{\rm fl}=1\times 10^{49}$ erg and $R_{0}=3\times 10^{-6}$
Mpc-3 yr-1 are assumed for the multiplet source.

We anticipate $\gtrsim$ 100 SNe in an optical follow-up observation to search
for any optical counterpart. They are mostly not associated with the neutrino
detection as mentioned earlier, though it is always possible that one of them
can be the real counterpart. Using the fact that sources producing neutrino
multiplet are localized in the low redshift universe, we can distinguish
unassociated SNe from the source to emit neutrino multiplet. Among the optical
transients found in a follow-up observation triggered by a neutrino multiplet
detection, the closest object is the most likely neutrino source. Because the
expected redshift distribution of the source to yield neutrino doublet is
quite different from that of unassociated SNe, we can judge if the closest
counterpart is indeed associated with the neutrino doublet detection in a
statistical way. Figure 3 shows the probability distributions of the redshift
of the closest object for the two possibilities. The pronounced difference
between them can construct a test statistic to examine which hypothesis is
favored. For example, finding an SN-like transient at $z=0.04$ ($\approx 170\
{\rm Mpc}$) in an optical follow-up observation leads to $\sim 2.7\sigma$
significance against the hypothesis of the incorrect coincident SN detection.

Figure 4: Number of sources to yield neutrino multiplet, $N^{\rm
M}_{\Delta\Omega}$, during time $\Delta T=$ 30 days in $\Delta\Omega=1\ {\rm
deg}^{2}$ of sky on the parameter space of $({\mathcal{E}}_{\nu}^{\rm
fl},R_{0})$, the output neutrino energy from a source and the burst density
rate [27]. A criteria to suppress the annual false alarm rate below $\sim
0.25$ for 2$\pi$ sky is applied.

Identifying the closest object from the numerous transients found in an
optical survey triggered by the neutrino detection requires extensive
spectroscopic measurements, which may not always be feasible. One of the
practical solutions is to rely on the photometric redshift of the host
galaxies whose information is available in the data taken by the survey
facilities with the wide field of view such as the Vera C. Rubin Observatory.
Another way to perform the intensive spectroscopy will be brought by the prime
focus spectrograph on Subaru [28]. It has a remarkable capability of a wide-
field simultaneous spectrography with high multiplicity. It is of great
importance for neutrino and optical astronomers to closely collaborate for
discoveries of yet unknown transient neutrino sources.

Searches for neutrino multiplets are powerful not just for reliably
identifying the EMG counterparts, but also for revealing/constraining the
source characteristics. Figure 4 shows the number of sources to yield
detectable neutrino multiples, $N^{\rm M}_{\Delta\Omega}$, from the sky patch
of $\Delta\Omega=1\ {\rm deg}^{2}$ [27]. The search time scale is assumed to
be $\Delta T=30$ days, considering the typical time scale of the neutrino
transient emission from TDEs and core-collapse SNe and being long enough to
cover faster transients such as low luminosity GRBs and GRB afterglow. Only
the diagonal region is displayed where it is consistent with the cosmic
neutrino diffuse background flux,
$E_{\nu}^{2}\Phi_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}\approx 10^{-8}\ {\rm GeV\
cm^{-2}\ sec^{-1}sr^{-1}}$. Because a neutrino telescope looks for $\sim 2\pi$
sky, the parameter space of $N^{\rm M}_{\Delta\Omega}\gtrsim 10^{-6}$ is
accessible by a $1\ {\rm km}^{3}$ neutrino telescope with 5 year observation.
Null detection of the neutrino multiplets with the criteria of the false alarm
rate ${\rm FAR}\lesssim 0.25\ {\rm yr}^{-1}$ leads to the allowed parameter
space of

${\mathcal{E}}_{\nu}^{\rm fl}\lesssim 5\times 10^{51}\ {\rm erg},\quad
R_{0}\gtrsim 2\times 10^{-8}~{}{\rm Mpc}^{-3}~{}{\rm yr}^{-1},$ (21)

if the transients of $\Delta T\lesssim 30$ days are the major sources to
contribute the high energy cosmic neutrino diffuse background radiation [27].
It will constrain the models involving jetted TDEs and super-luminous SNe in
future.

##  4 Summary

The observational fact that the neutrino energy flux at $100\ {\rm TeV}\sim 1\
{\rm PeV}$ is comparable with that of UHECRs is understandable by the UHECR-
neutrino multi-messenger approach. Based upon the framework of photo-hadronic
interactions for producing the secondary neutrinos, we have constructed the
generic scheme to describe the common sources of UHECRs and high energy
neutrinos with less model-dependent way and obtained the viable parameter
space required to explain the diffuse high-energy neutrino flux above 100 TeV
energies and the UHECR flux above 10 EeV, simultaneously. The five conditions
have essentially constrained the allowed parameter space which is rather
narrow – requirements for the UHECR acceleration, the UHECR escape, the UHECR
nuclei survival, the UHECR energetics, and the fiducial neutrino flux. For an
astronomical object with a given photon luminosity, a number density, magnetic
field strength, and its equipartition parameter $\xi_{\rm B}$, the basic
requirements to qualify as the common sources are represented in the form of
conditions of $\xi_{\rm B}$, $p\gamma$ optical depth at UHECR energy of 10
PeV, and the CR loading factor $\xi_{\rm CR}$. Among the known astronomical
object classes, we have found that the low luminosity GRBs are the most likely
candidate, followed by the jetted TDEs and FSRQs with the extreme parameter
tuning. We would like to emphasize, however, that our framework to provide the
generic constraints is applicable to as-yet-unknown source populations which
may be revealed in future neutrino-driven multi-messenger observations.

These common source candidates are transients in optical and X-ray bands. In
addition, many other neutrino source candidates at TeV sky such as the
circumstellar SNe are also transients. Thus it is a key to conduct the multi-
messenger follow-up observations in oder to identify the neutrino sources.
Overcoming the difficulty that the optical transient sky is filled with
numerous SNe, the two approaches have been proposed with the presently
operating facilities – introducing a sub-threshold detection channel to a wide
field of view X-ray observatory responding to high energy neutrino detection,
and the neutrino multiplet detection to trigger ToO observations with optical
telescopes. We are living in a new era to utilize neutrino, optical, and X-ray
messengers to reveal the origin of cosmic rays, and study the hadronic
emissions.

Acknowledgments

The studies written in this article have benefited from the extensive
discussions with Kohta Murase. The author is also deeply grateful to Masaomi
Tanaka, Nobuhiro Shimizu, and Aya Ishihara for their valuable inputs. Special
thanks go to IceCube Collaboration. This work is supported partly by JSPS
KAKENHI grant No.18H05206 and 23H04892.

## References

  * [1] Alexander Aab et al.  Combined fit of spectrum and composition data as measured by the Pierre Auger Observatory.  JCAP, 04:038, 2017.  [Erratum: JCAP 03, E02 (2018)].
  * [2] M.G. Aartsen et al.  First observation of PeV-energy neutrinos with IceCube.  Phys. Rev. Lett., 111:021103, 2013.
  * [3] M.G. Aartsen et al.  Evidence for High-Energy Extraterrestrial Neutrinos at the IceCube Detector.  Science, 342:1242856, 2013.
  * [4] M.G. Aartsen et al.  Characteristics of the diffuse astrophysical electron and tau neutrino flux with six years of IceCube high energy cascade data.  Phys. Rev. Lett., 125(12):121104, 2020.
  * [5] R. Abbasi et al.  Improved Characterization of the Astrophysical Muon–neutrino Flux with 9.5 Years of IceCube Data.  Astrophys. J., 928(1):50, 2022.
  * [6] M. G. Aartsen et al.  Detection of a particle shower at the Glashow resonance with IceCube.  Nature, 591(7849):220–224, 2021.
  * [7] M.G. Aartsen et al.  Differential limit on the extremely-high-energy cosmic neutrino flux in the presence of astrophysical background from nine years of IceCube data.  Phys. Rev. D, 98(6):062003, 2018.
  * [8] Shigeru Yoshida and Kohta Murase.  Constraining photohadronic scenarios for the unified origin of IceCube neutrinos and ultrahigh-energy cosmic rays.  Phys. Rev. D, 102(8):083023, 2020.
  * [9] Kohta Murase and John F. Beacom.  Neutrino Background Flux from Sources of Ultrahigh-Energy Cosmic-Ray Nuclei.  Phys. Rev. D, 81:123001, 2010.
  * [10] Shigeru Yoshida and Hajime Takami.  Bounds on the origin of extragalactic ultrahigh energy cosmic rays from the IceCube neutrino observations.  Phys. Rev. D, 90(12):123012, 2014.
  * [11] Kohta Murase and Masataka Fukugita.  Energetics of High-Energy Cosmic Radiations.  Phys. Rev. D, 99(6):063012, 2019.
  * [12] Kohta Murase, Dafne Guetta, and Markus Ahlers.  Hidden Cosmic-Ray Accelerators as an Origin of TeV-PeV Cosmic Neutrinos.  Phys. Rev. Lett., 116(7):071101, 2016.
  * [13] Kohta Murase, Shigeo S. Kimura, and Peter Meszaros.  Hidden Cores of Active Galactic Nuclei as the Origin of Medium-Energy Neutrinos: Critical Tests with the MeV Gamma-Ray Connection.  Phys. Rev. Lett., 125(1):011101, 2020.
  * [14] G. Ghisellini, F. Tavecchio, L. Foschini, G. Ghirlanda, L. Maraschi, and A. Celotti.  General physical properties of bright Fermi blazars.  MNRAS, 402(1):497–518, February 2010.
  * [15] Kohta Murase, Yoshiyuki Inoue, and Charles D. Dermer.  Diffuse Neutrino Intensity from the Inner Jets of Active Galactic Nuclei: Impacts of External Photon Fields and the Blazar Sequence.  Phys. Rev. D, 90(2):023007, 2014.
  * [16] M.G. Aartsen et al.  Constraints on Ultrahigh-Energy Cosmic-Ray Sources from a Search for Neutrinos above 10 PeV with IceCube.  Phys. Rev. Lett., 117(24):241101, 2016.  [Erratum: Phys.Rev.Lett. 119, 259902 (2017)].
  * [17] Alexander Aab et al.  Probing the origin of ultra-high-energy cosmic rays with neutrinos in the EeV energy range using the Pierre Auger Observatory.  JCAP, 10:022, 2019.
  * [18] Shigeo S. Kimura and Kenji Toma.  Hadronic High-energy Emission from Magnetically Arrested Disks in Radio Galaxies.  Astrophys. J., 905(2):178, 2020.
  * [19] Daniel Biehl, Denise Boncioli, Cecilia Lunardini, and Walter Winter.  Tidally disrupted stars as a possible origin of both cosmic rays and neutrinos at the highest energies.  Sci. Rep., 8(1):10828, 2018.
  * [20] Kohta Murase, Shigeo S. Kimura, B. Theodore Zhang, Foteini Oikonomou, and Maria Petropoulou.  High-Energy Neutrino and Gamma-Ray Emission from Tidal Disruption Events.  Astrophys. J., 902(2):108, 2020.
  * [21] Kohta Murase, Kunihito Ioka, Shigehiro Nagataki, and Takashi Nakamura.  High Energy Neutrinos and Cosmic-Rays from Low-Luminosity Gamma-Ray Bursts?  Astrophys. J. Lett., 651:L5–L8, 2006.
  * [22] B. Theodore Zhang and Kohta Murase.  Ultrahigh-energy cosmic-ray nuclei and neutrinos from engine-driven supernovae.  Phys. Rev. D, 100(10):103004, 2019.
  * [23] Ke Fang, Brian D. Metzger, Indrek Vurm, Elias Aydi, and Laura Chomiuk.  High-energy Neutrinos and Gamma Rays from Nonrelativistic Shock-powered Transients.  Astrophys. J., 904(1):4, 2020.
  * [24] Kohta Murase.  New Prospects for Detecting High-Energy Neutrinos from Nearby Supernovae.  Phys. Rev. D, 97(8):081301, 2018.
  * [25] Kenji Yoshida, Maria Petropoulou, Kohta Murase, and Foteini Oikonomou.  Flare Duty Cycle of Gamma-Ray Blazars and Implications for High-energy Neutrino Emission.  Astrophys. J., 954(2):194, 2023.
  * [26] M. Matsuoka et al.  The MAXI Mission on the ISS: Science and Instruments for Monitoring All-Sky X-Ray Images.  PASJ, 61:999, October 2009.
  * [27] Shigeru Yoshida, Kohta Murase, Masaomi Tanaka, Nobuhiro Shimizu, and Aya Ishihara.  Identifying High-energy Neutrino Transients by Neutrino Multiplet-triggered Follow-ups.  Astrophys. J., 937(2):108, 2022.
  * [28] N. Tamura et al.  Prime Focus Spectrograph for the Subaru telescope: massively multiplexed optical and near-infrared fiber spectrograph.  Journal of Astronomical Telescopes, Instruments, and Systems, 1:035001, July 2015.