††thanks<EMAIL_ADDRESS>
# Testing unified models for the origin of ultrahigh-energy cosmic rays and
neutrinos:
Multimessenger approaches with x-ray observations

Shigeru Yoshida<EMAIL_ADDRESS>International Center for
Hadron Astrophysics, Chiba University, Chiba 263-8522 Japan  Kohta Murase
Department of Physics; Department of Astronomy & Astrophysics; Center for
Multimessenger Astrophysics, Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA
and
Center for Gravitational Physics and Quantum Information, Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan

###### Abstract

The unified models of astrophysical sources to account for ultrahigh-energy
cosmic rays (UHECRs) and high-energy cosmic neutrinos with energies greater
than 100 TeV have been discussed. Based on model-independent arguments, we
argue that if the photomeson production is the dominant mechanism, the most
probable candidate sources are x-ray transient objects, allowing for the semi-
transparency for the photomeson production. We develop a generic model of
high-energy neutrino emitters accompanied by x-ray emission, and present how
multimessenger observations can place significant constraints on the source
parameters that characterize the common sources of neutrinos and UHECRs, such
as the cosmic-ray loading factor. The requirements of UHECR acceleration,
escape, and energetics further constrain the magnetic field and the bulk
Lorentz factor of the sources. The resulting bounds provide diagnoses of the
unified models, which demonstrates the importance of current and future x-ray
facilities such as MAXI and Einstein Probe.

###### pacs:

98.70.Sa, 95.85.Ry

##  I  Introduction

The cosmic background radiation in high-energy and ultrahigh-energy (UHE) sky
at $\gtrsim 1$ PeV ($10^{15}$ eV) is formed by cosmic rays and neutrinos. The
precise measurements of ultrahigh-energy cosmic rays (UHECRs) by the 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. It has been widely recognized that neutrino detection
is a smoking gun for identifying the origins of high-energy cosmic rays, as
high-energy neutrinos can mainly be produced by the hadronic processes
involving high-energy nucleons or nuclei, while high-energy electromagnetic
(EM) emissions can naturally be realized by the leptonic processes originating
in electrons or positrons and magnetic fields. The reality is not that simple,
however. The existence of high-energy protons or nuclei does not necessarily
imply cosmic-ray emission because these charged particles must escape from
their acceleration site, which is not guaranteed to be possible. Any dense
environment formed by radiation or matter within or in the vicinity of the
acceleration region causes significant energy losses to prevent radiation of
cosmic rays. It leads to neutrino emission not accompanied by escaping cosmic
rays.

The recent discovery of neutrino emission from the Seyfert galaxy NGC 1068 [6,
7] indeed belongs to this case, as the parent protons to produce neutrinos are
unlikely to run away from the inner region close to the central supermassive
black hole [8]. Moreover, the energy range of cosmic background neutrinos
discovered by IceCube is mostly below 10 PeV ($10^{16}$ eV), which is orders
of magnitude lower than the main extragalactic UHECR energies ($\gtrsim
10^{19}$ eV), and it is not obvious that the measured neutrinos are associated
with UHECRs. The question of whether the two primary components of UHE sky,
cosmic rays, and neutrinos, are relevant to each other has not yet been
answered. There are various candidates of UHECR sources, including active
galactic nuclei (AGNs), gamma-ray bursts (GRBs), and tidal disruption events
(TDEs) [9].

As shown in Fig. 1, the observed energy flux of high-energy neutrinos above
$~{}\sim 100$ TeV is comparable to that of UHECRs at $\gtrsim 10^{19}$ eV.
This suggests that neutrino background radiation may originate from the same
class of astrophysical objects to produce UHECRs. It may be plausible that the
cosmic-ray radiation from UHECR accelerators can be understood in a common
unified scheme and even the connection with the gamma-ray background radiation
may be expected [10, 11].

Motivated by this observational fact, Yoshida and Murase (YM20 [12]) built the
generic unification model to account for the observed neutrinos at energies
greater than $\sim 100$ TeV and UHECRs, based on the photohadronic interaction
framework. The UHECR luminosity density and the optical depth to the
photomeson production are constrained by the measured UHECR and neutrino
fluxes and the other conditions required to emit UHECR nucleons or nuclei. The
generic constraints enable us to evaluate whether a given class of
astronomical objects is qualified as the common origin of UHECRs and
neutrinos, which will be useful as a probe of the origin of UHECRs with
multimessenger observations.

In this paper, we discuss which objects among the known astronomical object
classes meet the criteria for UHECR accelerators in the unified UHE particle
emission scheme. We deconstruct the generic constraints obtained by YM20 into
a set of straightforward analytical formulas, which allows us to diagnose the
likelihood of a given object being a candidate of the common sources. We argue
that the most plausible candidates are x-ray transients allowing the semi-
transparency for the photomeson production. We then construct a generic
modeling of neutrino emitters with optically thin x-ray targets by the
unification scheme of neutrino and UHECR origins based on YM20. The results
will be described in the parameterization to characterize the sources, such as
the source number density, the Lorentz bulk factor of plasma, and the cosmic-
ray loading factor, for a given source luminosity in the x-ray band. We
present how current and future multimessenger observations involving x-rays
constrain these parameters, by which we pin down the unified origin or fully
rule out the UHE unification scenario in the photohadronic framework.

This paper is organized as follows. In Sec. II, we revisit the framework of
YM20 [12], and introduce the analytical formulas to describe necessity
conditions for common UHECR and neutrino sources. In Sec. III, we evaluate
each of the known astronomical object classes that have been considered as
UHECR sources in the literature to see their applicability to the unified
source scheme. In Sec. IV, we introduce a generic x-ray – neutrino – UHECR
source model with a set of parameters observable by the present or future
x-ray missions. We present constraints in the parameter space placed by x-ray
observations and various UHECR source conditions. We summarize the
implications of the present constraints and future perspective in Sec. V. The
standard cosmology with $H_{0}\simeq 73.5$ km $\sec^{-1}$ Mpc-1,
$\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$ is assumed throughout the paper.
Primed (’) characters represent quantities in the comoving frame of plasma
outflow.

Figure 1: Energy fluxes of high-energy cosmic background radiations. The small
data points represent the UHECR fluxes measured by PAO [1] and IceTop [13].
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 called PEPE (PeV Energy Partially-
contained Events) [14]. The thick line with arrows indicates the differential
upper limit obtained by the EHE (Extremely-High Energy neutrino search [15].
The neutrino fluxes are the all-flavor-sum fluxes
$E_{\nu}^{2}\Phi_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}$.

##  II  Generic unified model

Figure 2: The allowed region in the parameter space of luminosity density,
$Q_{\rm UHECR}$, and damping factor $\displaystyle{1-e^{-\tau_{p\gamma 0}}}$
[12]. 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.

In YM20, a nonthermal photon field represented by a power-law spectrum filling
the acceleration and/or the emission region in a source provides a target for
UHECRs to yield secondary neutrinos via photohadronic ($p\gamma$)
interactions. The $p\gamma$ interaction efficiency is parameterized by the
$p\gamma$ optical depth, $\tau_{p\gamma 0}$, at the fiducial UHECR energy
$\varepsilon_{\rm UHECR}^{\rm FID}=10$ PeV. The $p\gamma$ optical depth is
generically given by

$\displaystyle\tau_{p\gamma}(\varepsilon_{p})$ $\displaystyle\approx$
$\displaystyle\tau_{p\gamma 0}{\left(\frac{\varepsilon_{p}}{{\varepsilon}_{\rm
UHECR}^{\rm FID}}\right)}^{\alpha_{\gamma}-1}$ (1) $\displaystyle\approx$
$\displaystyle\frac{B^{\prime}}{\Gamma^{2}}\sqrt{\frac{L^{\prime}_{\gamma}}{\xi_{\rm
B}}}\beta^{-1}C(\alpha_{\gamma}){\left(\frac{\varepsilon_{p}}{{\varepsilon}_{\rm
UHECR}^{\rm FID}}\right)}^{\alpha_{\gamma}-1}.$

Here $\Gamma$ is the bulk Lorentz factor of the plasma in the interaction
site, $\beta$ is the plasma velocity, $\alpha_{\gamma}$ is the power law index
of the target photon spectrum, $B^{\prime}$ is the magnetic field strength
measured in the plasma rest frame, and $L^{\prime}_{\gamma}$ is the target
photon luminosity in the plasma rest frame. The magnetic field loading factor,
$\xi_{\rm B}$, is the ratio of the magnetic energy density
$U^{{}^{\prime}}_{\rm B}$ to the photon radiation energy density and defined
by

$\displaystyle\xi_{\rm B}$ $\displaystyle\equiv$ $\displaystyle
U^{\prime}_{\rm B}\left(\frac{L_{\gamma}}{4\pi\Gamma^{2}R^{2}c}\right)^{-1}.$
(2)

The coefficient $C(\alpha_{\gamma})$ in Eq. (1) is determined by the
photomeson production cross section and the target photon spectral shape, and
given by

$\displaystyle C(\alpha_{\gamma})$ $\displaystyle\sim$ $\displaystyle
2.4\times{10}^{-24}~{}{\rm erg}^{-1}~{}{\rm cm}^{3/2}~{}{\rm s}^{1/2}$ (3)
$\displaystyle\times$ $\displaystyle\left(\frac{2}{1+\alpha_{\gamma}}\right).$

The necessity conditions to be qualified for UHECR emitters can also be
formulated using the optical depth, which we describe later.

For a given $\tau_{p\gamma 0}$, the analytical formulation to calculate the
neutrino flux has been derived in YM20. Using the analytical representation of
the UHECR flux taking into account the energy loss effect during the
propagation in the cosmic microwave background (CMB) field filling the
intergalactic space, the model has constrained the parameter space of
$\tau_{p\gamma 0}$ and the integrated luminosity density of UHECRs in the
local universe $Q_{\rm UHECR}$ by comparisons to the measured UHECR and
neutrino fluxes. Figure 2 displays the resultant constraints obtained by YM20,
for the two cases when the extragalactic UHECRs are dominated by protons, and
dominated by nuclei, considering silicon as a benchmark, respectively.

Our goal here is to interpret these constraints by a set of simple
approximated formulas to quickly investigate a given astronomical object for
qualifications as the UHECR-neutrino common source. We start with the UHECR
energetics argument. The UHECR differential luminosity density is estimated as
(e.g., Ref. [11] and references therein)

$\displaystyle E_{\rm CR}\frac{dQ_{\rm UHECR}}{dE_{\rm CR}}$
$\displaystyle\approx$ $\displaystyle 6.3\times 10^{43}~{}{\rm erg\
Mpc^{-3}yr^{-1}}$ (4) $\displaystyle\times$ $\displaystyle\left(\frac{E_{\rm
CR}}{10^{19.5}\ {\rm eV}}\right)^{2-\alpha_{\rm CR}}$

where $\alpha_{\rm CR}$ is the power-law index of UHECR spectrum. The
integrated luminosity density of UHECRs above the energy of
$\varepsilon_{p}\geq\varepsilon_{\rm UHECR}^{\rm FID}(=10~{}{\rm PeV)}$ is
obtained by the transformation from the differential luminosity density above
and given by

$\displaystyle Q_{\rm UHECR}$ $\displaystyle\approx$ $\displaystyle 2.3\times
10^{45}~{}{\rm erg\ Mpc^{-3}yr^{-1}}$ (5) $\displaystyle\approx$
$\displaystyle n_{0}^{\rm eff}\xi_{\rm UHECR}L_{\gamma}$

when we use $\alpha_{\rm CR}=2.3$ as a benchmark. Here, the UHECR loading
factor, $\xi_{\rm UHECR}$, is the ratio of the UHECR bolometric luminosity
$L_{\rm UHECR}$ and the target photon luminosity $L_{\gamma}$, and $n_{0}^{\rm
eff}$ is the effective local source number density at redshift $z=0$. Note
that this effective number density can be different from the number density
measured by UHECR observations. This is because cosmic rays are intervened by
magnetic fields and the number density in the cosmic-ray sector should be
affected by time delays [16, 17]. One can see that this is consistent with the
range of y-axis in the allowed region shown in Fig. 2. This energetics
condition above effectively sets the requirement of the UHECR loading factor
for a given $L_{\gamma}$ and $n_{0}^{\rm eff}$, such as

$\xi_{\rm UHECR}\sim 0.7\left(\frac{L_{\gamma}}{10^{46}{\rm
erg/s}}\right)^{-1}\left(\frac{n_{0}^{\rm eff}}{10^{-8}{\rm
Mpc^{-3}}}\right)^{-1}.$ (6)

Note that $L_{\gamma}$ should be evaluated at the representative energy band,
where UHECRs with $\varepsilon_{\rm UHECR}^{\rm FID}(=10~{}{\rm PeV})$ mainly
interact via the $\Delta(1232)$ resonance of photohadronic collisions (see Eq.
(12) in YM20 and Eq. (13) in Sec. IV.1), and $n_{0}^{\rm eff}$ is determined
at the local density represented by this luminosity (see Eq. 9 of Ref. [10]).
In the standard candle source approximation, it is identical to the net local
source density.

Next, we derive the requirements by the all-sky neutrino flux. The neutrino
all-flavor-sum emissivity from a source is connected to the primary UHECR
emissivity as [18]

$\varepsilon_{\nu}^{2}\frac{d\dot{N}_{\nu}}{d\varepsilon_{\nu}}\approx\xi_{\pi}\langle
y_{\nu}\rangle\langle x\rangle\tau_{p\gamma}\varepsilon_{\rm
CR}^{2}\frac{d\dot{N}_{\rm CR}}{d\varepsilon_{\rm CR}}$ (7)

for a hadronically thin (i.e.$\tau_{p\gamma}\lesssim 5$) source. Here $\langle
x\rangle\sim 0.2$ is the average inelasticity of the $p\gamma$ collision, and
$\langle x\rangle\tau_{p\gamma}$ corresponds to the effective optical depth
($f_{p\gamma}$) used in the literature. Also, $\langle y_{\nu}\rangle\sim 1/4$
is the average fraction of energy channeling into a neutrino from the
secondary produced pion, and $\xi_{\pi}\sim 3/2$ is the average multiplicity
of neutrinos from a single pion produced by the photohadronic interaction
considering the $\pi^{+}:\pi^{0}$ ratio is $\approx 1:1$. 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. (7). We get

$\displaystyle\tau_{p\gamma 0}Q_{\rm UHECR}$ $\displaystyle\approx$
$\displaystyle 9.3\times 10^{43}\left(\frac{E_{\nu}^{2}\Phi_{\nu}}{2\times
10^{-8}\ [{\rm GeV\ cm^{-2}s^{-1}sr^{-1}]}}\right)$ (8)
$\displaystyle\times\left(\frac{\xi_{z}}{2.8}\right)^{-1}\quad{\rm erg\
Mpc^{-3}yr^{-1}},$

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. (5) with this neutrino
fiducial flux condition, Eq. (8), sets the lower bound of the $p\gamma$
optical depth, which is [19]

$\displaystyle\tau_{p\gamma 0}$ $\displaystyle\gtrsim$ $\displaystyle
0.04\left(\frac{\xi_{z}}{2.8}\right)^{-1}.$ (9)

Third, we consider the conditions for the acceleration of UHECRs. The powered
required from the Hillas condition is represented as the condition of
$L^{\prime}_{\gamma}$ as a function of the maximal accelerated cosmic-ray
energy $\varepsilon_{i}^{\rm max}$ (e.g., Refs. [20, 12, 9]). This condition
can be translated into the lower bound of the magnetic field loading factor
$\xi_{B}$ defined as Eq. (2). We get

$\displaystyle\xi_{B}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}c\eta^{2}\beta^{2}L^{{}^{\prime}-1}_{\gamma}\left(\frac{\varepsilon_{i}^{\rm
max}}{Ze}\right)^{2}$ $\displaystyle\gtrsim$ $\displaystyle
1.7\eta^{2}\beta^{2}\left(\frac{L_{\gamma}}{10^{46}{\rm
erg/s}}\right)^{-1}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\left(\frac{\varepsilon_{i}^{\rm
max}}{Z10^{11}~{}{\rm GeV}}\right)^{2},$

where $\eta\gtrsim 1$ represents the efficiency of the particle acceleration,
and the limit of $\eta\rightarrow\beta^{-2}$ corresponds to the Hillas
condition in the case of diffusive shock acceleration [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 photomeson
production time scale is essentially counted with the $p\gamma$ optical depth,
$\tau_{p\gamma}$, in the present scheme, and any sources with
$\tau_{p\gamma}(\varepsilon_{i}^{\rm max})\lesssim 5$ implies that the energy
losses by the photomeson production is not a deciding factor to limit the
UHECR acceleration and escape processes. As the BH process is in general
important 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 into the upper bound
$\tau_{p\gamma 0}$ as [12]

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

Note that the diffusion motion of cosmic rays in the photohadronic interaction
zone makes this condition even more stringent. Thus, the bound placed by Eq.
(11) above can be regarded as conservative.

If the bulk of the measured UHECRs is dominated by heavier nuclei rather than
nucleons as strongly indicated by the PAO data, the further severe requirement
must be satisfied – The nuclei survival condition [21]. We require that nuclei
with $A>1$ and $Z>1$ are accelerated and survive. This is possible only if the
time scale of the photodisintegration is slower than the dynamical time scale,
leading to the condition on photodisintegration optical depth as
$\tau_{A\gamma}\lesssim A$. We find that this condition sets the upper bound
of the $p\gamma$ optical depth as [12]

$\displaystyle\tau_{p\gamma 0}$ $\displaystyle\lesssim$ $\displaystyle
A\frac{\int ds\frac{\sigma_{p\gamma}(s)}{s-m_{p}^{2}}}{\int
ds\frac{\sigma_{A\gamma}(s)}{s-m_{A}^{2}}}{\left[\left(\frac{s_{\rm
GDR}-m_{A}^{2}}{s_{\Delta}-m_{p}^{2}}\right)\left(\frac{\varepsilon_{p}^{\rm
10PeV}}{\varepsilon_{i}^{\rm max}}\right)\right]}^{\alpha_{\gamma}-1}$ (12)
$\displaystyle\lesssim$ $\displaystyle
0.4~{}{\left(\frac{A}{56}\right)}^{-0.21},$

where $s_{\Delta}$ is the Mandelstam variable at the $\Delta$ resonance of the
photo-meson interactions and $s_{\rm GDR}$ is the Mandelstam variable at the
giant dipole resonance of the photodisintegration. This is equivalent to the
effective nucleus-survival bound obtained by Ref. [21].

##  III  Diagnosis of candidate sources for the unified model

The series of equations to describe the conditions for accounting for the
UHECRs and $0.1-1$ PeV energy neutrino measurements discussed in the previous
section are generic without relying on specific model-dependent arguments, in
exchange for more conservative constraints compared to those imposed by a
model dedicated for a given object. In other words, a class of sources failing
to meet the criteria cannot be the unified source without introducing more
complicated modeling such as multizones for cosmic ray accelerations and
neutrino productions. Here we run the generic diagnosis to check if any known
class of astronomical object remains in a possible candidate to unify the
picture of UHE astroparticle radiation.

###  III.1 Steady sources

Table 1: Parameters characterizing UHECR/neutrino/EM emission and the resultant constraints on the UHECR loading factor, $\xi_{\rm UHECR}$, the magnetic field parameter $\xi_{\rm B}$, and the photohadronic optical depth at 10 PeV $\tau_{p\gamma 0}$, imposed by the conditions for UHECR acceleration, the fiducial neutrino flux, UHECR escape, and UHECR nucleus survival. The various sites/populations in the AGN family are listed. Note that parameters have significant uncertainties although only specific parameters are shown in this table. We use $\varepsilon_{i}^{\rm max}=10^{11}$ GeV.  | RL AGN (BL Lac jet) | RL AGN (FSRQ jet) | RQ AGN (jet) | RL AGN (hot disk)
---|---|---|---|---
$\Gamma\beta$ of the outflow |  $\sim$ 10 |  $\sim$ 10 | $\sim 1$ | $\sim 0.01$
target photon energy | UV/x-ray | opt/UV | opt/UV | IR/opt
$L_{\gamma}^{\rm eff}\ [{\rm erg/s}]$ | ${\rm a~{}few}\times 10^{45}$ | ${\rm a~{}few}\times 10^{47}$ | ${\rm a~{}few}\times 10^{43}$ | ${\rm a~{}few}\times 10^{41}$
$n_{0}^{\rm eff}\ [{\rm Mpc^{-3}}]$ | $\sim 10^{-9}$ | $\sim 10^{-11}$ | $\sim 10^{-6}$ | $\sim 10^{-7}$
$n_{0}^{\rm tot}\ [{\rm Mpc^{-3}}]$ | $\sim 10^{-7}-10^{-6}$ | $\sim 10^{-9}-10^{-8}$ | $\sim 10^{-4}-{10}^{-3}$ | $\sim 10^{-5}-{10}^{-4}$
$R\ [{\rm cm}]$ | ${\rm a~{}few}\times{10}^{17}$ | ${\rm a~{}few}\times{10}^{17}$ | ${\rm a~{}few}\times{10}^{18}$ | ${\rm a~{}few}\times{10}^{14}$
$B^{\prime}\ [{\rm G}]$ | $\sim 0.1$ | $\sim 1$ | $\sim 0.01$ | $\sim 100$
$\xi_{\rm B}$ | $\sim 1$ | $\sim 1$ | $\sim 1$ | $\sim 100$
$\tau_{p\gamma 0}$ by Eq. (1) | $\sim 10^{-5}$ | $\gtrsim 10^{-3}$ | $\sim 10^{-4}$ | $\sim 1$
$\xi_{\rm UHECR}$: Eq. (6) | $\sim 10-100$ | $\sim 10-100$ | $\sim 1-10$ | $\sim 1000-10000$
$\xi_{\rm B}$ by Acceleration: Eq. (LABEL:eq:hillas_condition) | $\gtrsim 0.3\eta^{2}(\frac{Z}{10})^{-2}$ | $\gtrsim 0.3\eta^{2}(\frac{Z}{1})^{-2}$ | $\gtrsim 1\eta^{2}(\frac{Z}{10})^{-2}$ | $\gtrsim 0.03\eta^{2}(\frac{Z}{10})^{-2}$
$\tau_{p\gamma 0}$ by $\nu$ Flux: Eq. (9) | $\gtrsim 0.3$ | $\gtrsim 0.01$ | $\gtrsim 0.04$ | $\gtrsim 0.3$
$\tau_{p\gamma 0}$ by Escape: Eq. (11) | $\lesssim 1{(\frac{A}{2Z})}^{4}$ | $\lesssim 1{(\frac{A}{Z})}^{4}$ | $\lesssim 1{(\frac{A}{2Z})}^{4}$ | $\lesssim 1{(\frac{A}{2Z})}^{4}$
$\tau_{p\gamma 0}$ by Nuclei survival: Eq. (12) | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$

AGNs have been proposed in the literature for many decades as a possible UHECR
origin and high-energy neutrino emitters. The UHECR emission from the radio-
loud (RL) AGN jets represents a classical scenario discussed in the past and
present [9]. The fact that blazars dominate the high-energy gamma-ray sky
maintains the AGN jets as viable UHECR accelerators [22]. The jet activity in
EM emission varies with time, suggesting that the UHE particle emission can
also be time-dependent, and neutrinos may be predominantly produced during the
flaring phase [23, 24]. However, because the flaring time scale can be the
order of months, which is far longer than the real transient class we discuss
later, we consider the average behavior of the jets here for holding the
simplicity.

Table 1 lists the parameters obtained by the unification model for the
representative populations/acceleration sites in the AGN family. Several
remarks on the general constraints should be mentioned here. First, larger
values of the magnetic field parameter, $\xi_{\rm B}$, are favored by the
UHECR acceleration condition (Eq. (LABEL:eq:hillas_condition)), but higher
$\xi_{\rm B}$ severely constrains the $p\gamma$ optical depth by the UHECR
escape condition (Eq. (11)), in return, which hardly meets the requirement to
explain the neutrino flux given by Eq. (9). This is the fundamental reason why
the parameter space allowed in the unification scenario is rather limited.
This is a solid argument independent of the details on source physics. Second,
the UHECR energetics solely determines the UHECR loading factor condition (Eq.
(6)) and any rare class of the object needs $\xi_{\rm UHECR}\gg 1$. This may
not be favored by the aspect of the energy budget of the central engine. Also,
the source number density has been observationally constrained by the search
for small-scale anisotropy in the UHECR arrival direction distribution [25]
although details depend on the magnetic field due to the time delay of cosmic
rays [17].

Jets of BL Lac objects (BL Lacs) are promising sites of UHECR acceleration,
especially if UHECRs are nuclei [22]. The acceleration condition can be
satisfied for $\xi_{\rm B}(\sim 1)$ (e.g., Ref. [26]), and small values of
$\tau_{p\gamma 0}$ imply that protons and nuclei can be accelerated to the UHE
region without suffering from the energy losses by the photohadronic
collisions. However, for this reason, it is typically difficult to consider BL
Lacs as the sources that mainly contribute to the all-sky neutrino flux (e.g.,
Refs. [27, 28]). The estimated optical depth to the photomeson production is
$\displaystyle\tau_{p\gamma 0}\sim 7\times 10^{-6}\left({B^{\prime}}/{{\rm
0.1G}}\right)\left({\Gamma}/{10}\right)^{-3}\left({L_{\gamma}}/{2\times
10^{45}{\rm erg/s}}\right)^{\frac{1}{2}}\xi_{B}^{-\frac{1}{2}}$, which
indicates that BL Lacs would be too dim to be viable for the unification.

Flat-spectrum radio quasars (FSRQs) can be interesting as the sources of UHECR
accelerators and neutrino emitters especially in the presence of external
fields [29, 30, 31]. Using Eq. (1), the photomeson optical depth due to
internal photons is expected to be small, but external photons play an
important role as targets because they are boosted in the comoving frame of
the jet. Assuming that $L_{\rm ext}\sim 0.01L_{\gamma}$, the optical depth due
to external photons is $\displaystyle\tau_{p\gamma}\sim
1\left({B^{\prime}}/{{\rm
1~{}G}}\right)\left({\Gamma}/{10}\right)\left({L_{\rm ext}}/{10^{45}{\rm
erg/s}}\right)^{\frac{1}{2}}{\xi_{B}}^{-\frac{1}{2}}\left(\varepsilon_{p}/\varepsilon_{\rm
UHECR}^{\rm FID}\right)^{0.5}$. Given that $\tau_{p\gamma 0}\sim 0.1-1$, UHECR
escape and efficient neutrino production can be concurrent in principle. Large
photon luminosities with $L_{\gamma}\gtrsim 10^{46}\ {\rm erg/s}$ meets the
acceleration condition given by Eq. (LABEL:eq:hillas_condition) even for
protons ($Z=1$), while it is challenging for nuclei to survive especially in
the presence of external photons. This implies that FSRQs could only be the
common origin for UHECR protons and neutrinos, but not for UHECR nuclei.
However, we should remark that the unification model by FSRQs would be
inconsistent with the other aspects of the neutrino data. Cosmogenic neutrinos
produced by UHECR protons emitted from strongly evolved sources such as FSRQs
would have overshot the present upper limit of neutrino flux at EeV ($10^{18}\
{\rm eV}$) range placed by IceCube and PAO [32, 33]. The present constraints
placed by the cosmogenic neutrino search become certainly weaker if UHECRs are
far dominated by heavy nuclei, but this loophole is soon to be substantially
narrower when the sensitivity of the neutrino telescopes reaches to
$E_{\nu}^{2}\Phi_{\nu}\sim 10^{-9}~{}{\rm GeV~{}cm^{-2}~{}s^{-1}~{}sr^{-1}}$
by accumulating more data or by the next-generation detectors such as IceCube-
Gen2.

UHECRs may be accelerated in weak jets of radio-quiet (RQ) AGN [34]. However,
RQ AGNs do not have powerful jets, so that only heavy nuclei can be
accelerated to ultrahigh energies. The efficiency of neutrino production is
also predicted to be very low, and they are unlikely to contribute to the all-
sky neutrino flux without violating gamma-ray data; instead, nuclei may
survive against photodisintegration [34]. Neutrino production in RQ AGN such
as NGC 1068 would preferentially occur in the vicinity of supermassive black
holes, including hot coronal regions [18, 8], in which $\tau_{p\gamma}\gtrsim
1$ is expected and the all-sky neutrino flux and neutrino emission from NGC
1068 can simultaneously be explained [35]. However, small radii prevent cosmic
rays from being accelerated to ultrahigh energies.

The hot disk of radio-loud AGN (RL AGN) is expected to be radiatively
inefficient and collisionless for protons, in which protons could be
accelerated without suffering from efficient $p\gamma$ interactions. Protons
and nuclei may be accelerated by turbulence via stochastic acceleration [36,
37], and magnetic reconnections could also lead to the UHECR production if the
disk is magnetically arrested [37, 38]. High-energy neutrinos can also be
produced especially in the PeV or higher-energy range. Here we note that the
photomeson optical depth in Table 1 should rather be regarded as upper limits,
and realistic values considering realistic spectral energy distributions can
readily be much smaller [37, 38]. We also note that large values of $\xi_{B}$
and $\xi_{\rm UHECR}$ are necessary to satisfy the UHECR energetics, but we
note that such a possibility of dark accelerators is energetically allowed
when the disk is radiatively inefficient and the accretion power is much
larger than the radiation luminosity.

###  III.2 Transient sources

Table 2: Same as Table 1 but the various transient objects are listed. Note that parameters have significant uncertainties although only specific parameters are shown in this table. We use $\varepsilon_{i}^{\rm max}=10^{11}$ GeV. Star-formation evolution is assumed for supernovae and GRBs, while no evolution is assumed for TDEs. | Jetted TDE | TDE wind | LL GRB | Engine-driven SN
---|---|---|---|---
$\Gamma\beta$ of the outflow | $\sim 10$ | $\sim 0.3$ | $\sim 5$ | $\sim 0.3$
target photon energy | x-ray | opt/UV | x-ray | opt/UV
$L_{\gamma}\ [{\rm erg/s}]$ | $\sim 10^{47}$ | $\sim 10^{44}$ | $\sim 10^{47}$ | $\sim 10^{44}$
$\rho_{0}\ [{\rm Mpc^{-3}yr^{-1}}]$ | $\sim 10^{-11}-10^{-10}$ | $\sim 10^{-7}-10^{-6}$ | $\sim 10^{-7}-10^{-6}$ | $\sim 10^{-6}-10^{-5}$
$\Delta T\ [{\rm s}]$ | $\sim 10^{6}-10^{7}$ | $\sim 10^{6}-10^{7}$ | $\sim 10^{3}-10^{4}$ | $\sim 10^{6}-10^{7}$
$R\ [{\rm cm}]$ | ${\rm a~{}few}\times{10}^{15}$ | $\sim{10}^{17}$ | ${\rm a~{}few}\times{10}^{15}$ | $\sim{10}^{17}$
$B^{\prime}\ [{\rm G}]$ | $\sim 300$ | $\sim 1$ | $\sim 100$ | $\sim 1$
$\xi_{\rm B}$ | $\sim 1$ | $\sim 1$ | $\sim 1$ | $\sim 1$
$\tau_{p\gamma 0}$ by Eq. (1) | $\sim 0.1$ | $\sim 0.03$ | $\sim 1$ | $\sim 0.03$
$\xi_{\rm UHECR}$: Eq. (6) | $\sim 100-1000$ | $\sim 1-10$ | $\sim 10-100$ | $\sim 0.1-1$
$\tau_{p\gamma 0}$ by $\nu$ Flux: Eq. (9) | $\gtrsim 0.1$ | $\gtrsim 0.1$ | $\gtrsim 0.03$ | $\gtrsim 0.03$
$\xi_{\rm B}$ by Acceleration: Eq. (LABEL:eq:hillas_condition) | $\gtrsim 10^{-2}\eta^{2}(\frac{Z}{10})^{-2}$ | $\gtrsim 1(\frac{\eta}{10})^{2}(\frac{Z}{10})^{-2}$ | $\gtrsim 0.01\eta^{2}(\frac{Z}{10})^{-2}$ | $\gtrsim 1(\frac{\eta}{10})^{2}(\frac{Z}{10})^{-2}$
$\tau_{p\gamma 0}$ by Escape: Eq. (11) | $\lesssim 1{(\frac{A}{2Z})}^{4}$ | $\lesssim 3{(\frac{A}{2Z})}^{4}$ | $\lesssim 1{(\frac{A}{2Z})}^{4}$ | $\lesssim 3{(\frac{A}{2Z})}^{4}$
$\tau_{p\gamma 0}$ by Nuclei survival: Eq. (12) | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$ | $\lesssim 0.4(\frac{A}{56})^{-0.21}$

Powerful transient objects (see Table 2) have also been discussed in the
literature as possible UHECR origins.

Jetted TDEs are so powerful that protons and nuclei could be accelerated to
ultrahigh energies [39, 40], and the nucleus survival condition can be
satisfied especially if the jet power is not sufficiently large [41]. Neutrino
production is reasonably efficient [42], and the approximate estimate gives
$\displaystyle\tau_{p\gamma}\sim 0.1\left({B^{\prime}}/{{\rm
300G}}\right)\left({\Gamma}/{10}\right)^{-3}\left({L_{\gamma}}/{10^{47}{\rm
erg/s}}\right)^{\frac{1}{2}}\xi_{B}^{-\frac{1}{2}}\left(\varepsilon_{p}/\varepsilon_{\rm
UHECR}^{\rm FID}\right)$. One of the caveats in this scenario is that it may
be difficult to meet the energetics condition for explaining both neutrinos
and UHECRs, and their contribution to the all-sky neutrino flux is expected to
be subdominant [42]. The requirement for the UHECR loading factor, $\xi_{\rm
UHECR}\gtrsim 100$, may also be challenging in light of the efficiency of
cosmic-ray acceleration. This shortcoming of the energetics may be resolved if
more generous yet-to-be-detected low-luminosity jetted TDEs exist [41] or
there may be more missing jetted TDEs.

Non-jetted TDEs are known to be more abundant objects that are of interest as
potential neutrino sources. The disk-driven wind launched from the vicinity of
the black hole may be a plausible site for cosmic-ray acceleration [41]. The
estimated optical depth to photomeson production is
$\displaystyle\tau_{p\gamma}\sim 0.05\left({B^{\prime}}/{{1\rm
G}}\right)\left({L_{\gamma}}/{10^{44}{\rm
erg/s}}\right)^{\frac{1}{2}}\xi_{B}^{-\frac{1}{2}}\left(0.3/{\beta}\right)\left(\varepsilon_{p}/\varepsilon_{\rm
UHECR}^{\rm FID}\right)^{0.8}$, and $pp$ interactions can also be efficient in
this model [43]. It is not trivial to satisfy the acceleration condition, and
only nuclei could be accelerated to ultrahigh energies [41]. Also, we note
that the energy dependence of the $p\gamma$ optical depth is important. In
non-jetted TDEs, target photons are mainly thermal, and neutrinos are expected
to be emitted at PeV or higher energies [43, 44, 45]. Because of the
nonresonant interactions, $\tau_{p\gamma}$ does not decrease even at higher
energies.

Low-luminosity (LL) GRBs and engine-driven supernovae are among the most
promising candidates for the unification model of high-energy neutrinos [46,
47, 48, 49, 50] and UHECRs [46, 51, 52, 53, 54]. For LL GRBs, the estimated
$p\gamma$ optical depth, $\displaystyle\tau_{p\gamma}\sim
0.4\left({B^{\prime}}/{{\rm
100G}}\right)\left({\Gamma}/{5}\right)^{-3}\left({L_{\gamma}}/{10^{47}{\rm
erg/s}}\right)^{\frac{1}{2}}\xi_{B}^{-\frac{1}{2}}\left(\varepsilon_{p}/\varepsilon_{\rm
UHECR}^{\rm FID}\right)^{1.2}$, meets all the conditions listed in Table 2.
Although the rate density, $\rho_{0}$, and the representative magnetic field,
$B^{\prime}$, are highly uncertain, the LL GRB hypothesis provides the viable
parameter space to simultaneously explain the UHECRs and the high-energy
neutrinos [46, 55]. UHECR acceleration may preferentially occur at external
shocks [51, 52]. Since target photons are expected in the optical and
ultraviolet range, neutrinos are mostly emitted at EeV energies, in which PeV
neutrinos may come from different emission regions such as choked jets [49,
50] (see also Fig. 17 of Ref. [54]).

Note that we do not consider classical high-luminosity (HL) GRBs as the
sources for the unification model. This is because the GRBs have already been
excluded as the main sources of IceCube neutrinos through the stacking
analysis [56, 57] and observations of the brightest GRB 2201009A [58, 59].
However, we note that HL GRBs still remain viable as the dominant sources of
UHECRs [60].

##  IV  Neutrino emission from UHECR sources with x-ray targets

We have seen that LL GRBs are the promising candidate for the UHE particle
emitters in the unification scheme. Jetted TDEs and possibly low-luminosity
TDEs could be considered. Both sources are known as x-ray emitters. This is
not an accidental coincidence. In the photohadronic framework, it is a bulk of
x-ray photons with which high-energy cosmic ray protons are colliding in the
production of neutrinos if the bulk Lorentz factor in plasma is $\Gamma\sim$
3–10, as the condition of the $\Delta$-resonance in the photomeson production
requires the photon energy being
$\displaystyle\varepsilon_{\gamma}\gtrsim(s_{\Delta}-m_{p}^{2})\Gamma^{2}/(4\varepsilon_{p})\sim
15.5(\Gamma/10)^{2}(\varepsilon_{p}/1{\rm PeV})^{-1}$ keV. Thus, it is
naturally expected that high energy neutrino emission takes place in an
environment filled with UV to soft x-ray photons. These photons, which may be
boosted by $\Gamma$, can be observed as keV-energy x-ray emissions if the
sources are optically thin. A search for neutrino and soft x-ray coincidence
emission is, therefore, a powerful approach to identifying such common sources
or constraining the parameters characterizing the emitters.

In this section, we construct a generic phenomenological model to account for
the neutrinos observed by IceCube, assuming high-energy protons colliding with
x-ray photons. Employing this model, we present how the measured neutrino data
and the searches for the coincident x-ray signals characterize the high-energy
cosmic-ray sources. The key parameters to characterize the source environment
are the PeV-energy cosmic-ray loading factor $\xi_{\rm CR}$ and the plasma
bulk Lorentz factor $\Gamma$. They are subject to be constrained by the x-ray
photon luminosity measured by x-ray observatories. We demonstrate that the
neutrino and soft x-ray coincidence searches can uniquely probe the parameter
space to constrain the condition for the origin of high-energy neutrinos and
UHECRs. The modeling and the parameterizations are entirely based on YM20
[12], but we redefine some of the critical parameters specifically for lucidly
associating the neutrino observations to x-ray luminosity observable by the
telescopes/sky monitors.

###  IV.1  Generic phenomenological model

In YM20 [12], neutrino sources are assumed to supply the nonthermal target
photons interacting with the accelerated cosmic-ray protons. The reference
UHECR proton energy $\varepsilon_{\rm UHECR}^{\rm FID}$ was fixed to be 10 PeV
in the observer frame, and the photon reference energy is given by the
$\Delta$-resonance condition as,

$\varepsilon^{\prime}_{\gamma
0}\approx\frac{(s_{\Delta}-m_{p}^{2})}{4}\frac{\Gamma}{\varepsilon_{\rm
UHECR}^{\rm FID}}.$ (13)

Thus, the representative photon energy range to calculate the bolometric
photon luminosity $L^{\prime}_{\gamma}$, which gauges the central engine
power, must vary from optical to x-rays, depending on the bulk Lorentz factor.
In the present parameterization, we instead fix the photon reference energy in
the plasma rest frame. We denote this energy as $\varepsilon^{\prime}_{X,b}$.
The target x-ray photon spectrum is represented as a broken power law (BPL),
instead of a single power law employed in YM20, and written as

$\frac{dn_{X}}{d\varepsilon^{\prime}_{X}}=\frac{K^{\prime}_{X}}{\varepsilon^{\prime}_{X,b}}\left(\frac{\varepsilon^{\prime}_{X}}{\varepsilon^{\prime}_{X,b}}\right)^{-\alpha_{X}},$
(14)

which is distributed from $\varepsilon^{\prime}_{\rm min}$ to
$\varepsilon^{\prime}_{\rm max}$. The power law indices are

$\alpha_{X}=\left\\{\begin{array}[]{l}\alpha_{X}^{\rm L}\approx 1\ \
(\varepsilon^{\prime}_{X}<\varepsilon^{\prime}_{X,b})\\\ \alpha_{X}^{\rm
H}\approx 2.2\ \ (\varepsilon^{\prime}_{X}\geq\varepsilon^{\prime}_{X,b})\\\
\end{array}\right.$ (15)

This choice of parameters incorporates a typical x-ray emission expected in LL
GRBs or TDEs.

We introduce the reference bolometric x-ray luminosity $L_{X}^{\rm REF}$ of
the source defined in the observer frame, which is subject to the measurement
(or constraints) by x-ray observatories with the energy band of
$[\varepsilon_{X,{\rm min}}^{\rm REF},\varepsilon_{X,{\rm max}}^{\rm REF}]$.
In the case of the Monitor of All-sky x-ray Image (MAXI) telescope [61],
$[\varepsilon_{X,{\rm min}}^{\rm REF},\varepsilon_{X,{\rm maz}}^{\rm
REF}]=[2{\rm keV},10{\rm keV}]$ for example. Taking into account the
differences in the energy band coverage, this reference x-ray luminosity
$L_{X}^{\rm REF}$ is connected to the source photon luminosity, $L_{X}^{\rm
L}$ and $L_{X}^{\rm H}$, respectively, which is the bolometric luminosity
below (above) the BPL break energy $\varepsilon^{\prime}_{X,b}$.

The reference x-ray luminosity $L_{X}^{\rm REF}$ then determines the target
photon energy density and the magnetic field strength $B^{\prime}$ through the
equipartition parameter $\xi_{\rm B}$ as described in YM20, requiring that the
magnetic energy density in the plasma rest frame balances the photon energy
density with the spectrum following the BPL written in Eq. (14) via $\xi_{\rm
B}$. Note that the photon energy density is given by the entire photon field
with luminosity $L^{\prime}_{X}=L_{X}^{{}^{\prime}{\rm
L}}+L_{X}^{{}^{\prime}{\rm H}}$ but not $L_{X}^{\rm REF}$ with the fixed
energy band $[\varepsilon_{X,{\rm min}}^{\rm REF},\varepsilon_{X,{\rm
max}}^{\rm REF}]$. Note that the shape of the distribution follows the energy
spectrum given by Eq. (14) while the normalization $K^{\prime}_{X}$ is solely
determined by $B^{\prime}$ and $\xi_{\rm B}$ which gives the magnetic energy
density balancing to the photon energy density.

The reference cosmic-ray (CR) proton energy is defined with the
$\Delta$-resonance condition given by Eq. (13), and obtained by

$\varepsilon_{p0}[\Gamma]=\frac{(s_{\Delta}-m_{p}^{2})}{4}\frac{\Gamma}{\varepsilon^{\prime}_{X,b}},$
(16)

and, therefore, varies with $\Gamma$. For this reference energy, the spectrum
of cosmic ray protons injected from the sources is given by

$\frac{d\dot{N}_{\rm CR}}{d\varepsilon_{p}}=\frac{K_{\rm
CR}}{\varepsilon_{p0}[\Gamma]}\left(\frac{\varepsilon_{p}}{\varepsilon_{p0}[\Gamma]}\right)^{-\alpha_{\rm
CR}}$ (17)

and the cosmic-ray bolometric luminosity $L_{\rm CR}$ is determined by the
reference x-ray luminosity $L_{X}^{\rm REF}$ via the cosmic-ray loading factor
$\xi_{\rm CR}$ as $L_{\rm CR}=\xi_{\rm CR}L_{X}^{\rm REF}$. For a more
straightforward interpretation of the parameters, we define $L_{\rm CR}$ as
the bolometric luminosity above the fixed fiducial CR energy
$\varepsilon_{p}^{{\rm FID}}=1$ PeV which is independent of $\Gamma$. The
fiducial cosmic-ray energy is different from $\varepsilon_{p0}[\Gamma]$ given
by Eq. (16). We apply the corrections below when necessary. The normalization
factor of the cosmic-ray spectrum $K_{\rm CR}$ is linked to the x-ray
luminosity as $\displaystyle K_{\rm CR}\approx(\alpha_{\rm CR}-2)\xi_{\rm
CR}L_{X}^{\rm REF}(\varepsilon_{p0}^{\rm
FID}/\varepsilon_{p0}[\Gamma])^{\alpha_{\rm CR}-1}/\varepsilon_{p0}^{\rm FID}$
for $\alpha_{\rm CR}>2$.

Following the formulation in YM20 [12], the optical depth of the photo-meson
production is given by

$\tau_{p\gamma}^{\rm L/H}\approx\tau_{0}^{{\rm
L/H}}\left(\frac{\varepsilon_{p}}{\varepsilon_{p0}[\Gamma]}\right)^{\alpha_{X}^{\rm
L/H}-1}$ (18)

where $\tau_{p\gamma}^{{\rm L}}$ ($\tau_{p\gamma}^{{\rm H}}$) is the optical
depth in the photon field below (above) the break energy
$\varepsilon^{\prime}_{X,b}$. The cosmic-ray energy independent term
$\tau_{0}^{\rm L/H}$ is written as

$\tau_{0}^{\rm L/H}=\sqrt{\frac{L_{X}^{\rm L/H}}{\xi_{\rm B}^{\rm
L/H}}}\frac{B^{\prime}}{\Gamma^{2}}\frac{2}{1+\alpha_{X}^{\rm
L/H}}\frac{\beta^{-1}}{y_{\alpha^{\rm
L/H}}(\varepsilon^{{}^{\prime}\rm{L/H}}_{X,{\rm
min}},\varepsilon^{{}^{\prime}\rm{L/H}}_{X,{\rm max}})}\mathcal{F},$ (19)

where $y_{\alpha}$ is the power-law integral function

$y_{\alpha}(\varepsilon^{\prime}_{\rm min},\varepsilon^{\prime}_{\rm
max})\equiv\int^{\frac{\varepsilon^{\prime}_{\rm
max}}{\varepsilon^{\prime}_{X,b}}}_{\frac{\varepsilon^{\prime}_{\rm
min}}{\varepsilon^{\prime}_{X,b}}}dxx^{-\alpha+1}$ (20)

and its integral covers the corresponding energy band below (above) the photon
break energy. The form factor, $\displaystyle\mathcal{F}\approx 4.3\times
10^{-25}(\varepsilon^{\prime}_{X,b}/200\ {\rm eV})^{-1}\ {\rm erg}^{-1}{\rm
cm}^{\frac{3}{2}}{\rm s}^{\frac{1}{2}}$ is given by

$\mathcal{F}=\frac{1}{4\pi\sqrt{c}\varepsilon^{\prime}_{X,b}}\int
ds\frac{\sigma_{p\gamma}(s)}{s-m_{p}^{2}}.$ (21)

$\xi_{\rm B}^{\rm L/H}$ in Eq. (19) is the effective equipartition parameter
of the B-field for the photon energy field below (above) the break energy, and
given by

$\displaystyle\xi_{\rm B}^{\rm L}$ $\displaystyle=$ $\displaystyle\xi_{\rm
B}\left(1+\frac{L_{X}^{\rm H}}{L_{X}^{\rm L}}\right)$ $\displaystyle\xi_{\rm
B}^{\rm H}$ $\displaystyle=$ $\displaystyle\xi_{\rm B}\left(1+\frac{L_{X}^{\rm
L}}{L_{X}^{\rm H}}\right)$ (22)

The differential neutrino luminosity emitted from a source is approximated to
be [19, 12]

$\displaystyle\frac{d\dot{N}_{\nu}}{d\varepsilon_{\nu}}\approx\int
d\varepsilon_{p}\frac{d\dot{N}_{\rm CR}}{d\varepsilon_{p}}(\tau_{p\gamma}^{\rm
L}(\varepsilon_{p})+\tau_{p\gamma}^{\rm
H}(\varepsilon_{p}))Y(\varepsilon_{\nu};\varepsilon_{p})f_{\rm
sup}(\varepsilon_{\nu}).\,\,\,\,\,\,\,\,\,$ (23)

Here $Y(\varepsilon_{\nu};\varepsilon_{p})$ denotes the energy distribution of
the neutrinos produced by an interaction of a cosmic-ray proton. The details
of the $Y$ formulation can be found in [19, 12]. The analytical representation
in this formulation is given in the Appendix A.

The suppression factor in Eq. (23), $\displaystyle f_{\rm
sup}(\varepsilon_{\nu})=1-\exp(-t^{\prime}_{\pi/\mu,{\rm
syn}}/t^{\prime}_{\pi/\mu,{\rm dec}})$, accounts for synchrotron cooling of
pions and muons. The ratio of the time scale can be rewritten in the ratio of
the neutrino energy to the critical energy as

$\frac{t^{\prime}_{\pi/\mu,{\rm syn}}}{t^{\prime}_{\pi/\mu,\rm
dec}}=\left(\frac{\varepsilon_{\nu,\pi/\mu}^{\rm
syn}}{\varepsilon_{\nu}}\right)^{2},$ (24)

and the critical energy is given by

$\varepsilon_{\nu,\pi/\mu}^{\rm
syn}\approx\Gamma\kappa_{\pi,\mu}\sqrt{\frac{6\pi}{\tau_{\pi,\mu}\sigma_{T}cB^{\prime
2}}\frac{(m_{\pi/\mu}c^{2})^{5}}{(m_{e}c^{2})^{2}}},$ (25)

where $\kappa_{\pi,\mu}$ is the inelasticity from pion (muon) to a neutrino in
the decay process. In this work, $\kappa_{\pi}$ is approximated by $\sim
1-r_{\pi}$ [62], where $r_{\pi}=m_{\mu}^{2}/m_{\pi}^{2}\simeq 0.57$ is the
muon-to-pion mass-squared ratio. The other fraction goes to a muon, and
$\kappa_{\mu}$ is approximated as $\sim 0.3$.

Table 3: Main parameters in the UHECR–neutrino–x-ray unified source model. Their baseline values are listed in the third column. The numbers in a bracket are the values for representing the different environments or investigating parameter dependence. See the text for the details.  Reference bolometric x-ray Luminosity | $L_{X}^{\rm REF}$ |  $5\times 10^{46}$($\lesssim 3\times 10^{45}$)[erg/s]
---|---|---
Reference Photon Energy | $\varepsilon^{\prime}_{X,b}$ | 0.5 [keV]
x-ray observation band to determine $L_{X}^{\rm REF}$ | $[\varepsilon^{\rm REF}_{X,{\rm min}},\varepsilon^{\rm REF}_{X,{\rm max}}]$ | [2, 10] [keV]
Source Magnetic Field | $B^{\prime}$ | 100 [G]
Equipartition parameter | $\xi_{\rm B}$ | 0.1
Fiducial cosmic-ray Energy | $\varepsilon_{p}^{\rm FID}$ | 1 [PeV]
Fiducial UHECR Energy | $\varepsilon_{\rm UHECR}^{\rm FID}$ | 10 [PeV]
Bolometric cosmic-ray luminosity above $\varepsilon_{p}^{\rm FID}$ | $L_{\rm CR}$ | floating
cosmic-ray Loading factor | $\xi_{\rm CR}$ | $\displaystyle\equiv L_{\rm CR}/L_{X}^{\rm REF}$
cosmic-ray spectrum power law index | $\alpha_{\rm CR}$ | 2.3 (2.1)
Luminosity Density of UHECRs above $\varepsilon_{\rm UHECR}^{\rm FID}$ | $Q_{\rm UHECR}$ |  $2.3(0.85)\times 10^{45}$ [erg Mpc-3 yr-1]

For a given $L_{X}^{\rm REF}$, the neutrino luminosity $\displaystyle
d\dot{N}_{\nu}/d\varepsilon_{\nu}$ decreases with increasing $\Gamma$. There
are two main reasons for this dependence. First, higher $\Gamma$ sources are
dimmer in the plasma rest frame as $L^{\prime}_{X}=L_{X}/\Gamma^{2}$. Second,
the interaction zone becomes more compact for higher $\Gamma$ due to the
Lorentz boost. These factors cause $\sim\Gamma^{-2}$ dependence on
$\tau_{0}^{\rm L/H}$ in Eq. (19) (There are the other minor $\Gamma$
dependence primarily brought by $\varepsilon_{p0}[\Gamma]$).

Table 3 lists the main parameters and their baseline values in the model.

###  IV.2  Contribution to the all-sky neutrino flux

A population of neutrino sources contributes to the diffuse cosmic background
flux. Assuming emission from standard candles (i.e., identical sources over
redshifts), the differential flux of diffuse neutrinos from these sources
across the universe is calculated by

$\Phi_{\nu}(E_{\nu})=\frac{c}{4\pi}\int\limits_{z_{\rm min}}^{z_{\rm
max}}dz(1+z)\left|\frac{dt}{dz}\right|\\\
\times\frac{d\dot{N}_{{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}}{d\varepsilon_{\nu}}\left|{}_{\varepsilon_{\nu}=E_{\nu}(1+z)}n_{0}^{\rm
eff}\Psi(z),\right.$ (26)

where $n_{0}^{\rm eff}\Psi(z)$ is the comoving source number density given the
local source number density $n_{0}^{\rm eff}$ and the cosmological evolution
factor $\Psi(z)$. Sources are distributed between redshift $z_{\rm min}$ and
$z_{\rm max}$.

The evolution factor $\Psi(z)$ is parameterized as $(1+z)^{m}$ such that the
parameter $m$ represents the “scale” of the cosmological evolution that is
often used in the literature. The evolution factor for the neutrino sources is
unknown, but many of the proposed sources are related to supernovae, which
approximately trace the star formation rate (SFR). Following Refs. [63, 64],
we parameterize $\Psi(z)$ as

$\displaystyle\Psi(z)\propto\left\\{\begin{array}[]{ll}(1+z)^{3.4}&(0\leq
z\leq 1)\\\ {\rm constant}&(1\leq z\leq 4)\end{array}\right.,$ (29)

to approximately trace SFR.

Figure 3: The cosmic diffuse neutrino fluxes from a population of sources with
the photon target energy density represented by BPL described as Eq. (14),
along with the neutrino and UHECR observed data shown in Fig. 1. We assume
$L_{X}^{\rm REF}=5\times 10^{46}$ erg/s, $n_{0}^{\rm eff}=3\times 10^{-11}\
{\rm Mpc}^{-3}$, $\alpha_{\rm CR}=2.3$, and the cosmic-ray loading factor
$\xi_{\rm CR}=10$. The three cases of $\Gamma$ values, $\Gamma=2$ (Top),
$\Gamma=3$ (Middle), $\Gamma=10$ (Bottom) are shown. Figure 4: The cosmic
diffuse neutrino energy fluxes
$E_{\nu}^{2}\Phi_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}|_{\rm 1PeV}$ [${\rm GeV\
cm^{-2}\ s^{-1}\ sr^{-1}}$] on the plane of the plasma bulk Lorentz factor
$\Gamma$ and the local source number density $n_{0}^{\rm eff}$. We assume
$L_{X}^{\rm REF}=5\times 10^{56}$ erg/s and the cosmic-ray loading factor
$\xi_{\rm CR}=10$.

Figure 3 shows the diffuse background neutrino fluxes produced by a population
of the sources with $L_{X}^{\rm REF}=5\times 10^{56}$ erg/s, $n_{0}^{\rm
eff}=3\times 10^{-11}\ {\rm Mpc}^{-3}$, $\alpha_{\rm CR}=2.3$, and $\xi_{\rm
CR}=10$. The IceCube data can constrain these parameters by comparison to the
calculated fluxes. As shown in Fig. 4, the predicted fluxes in the source
parameter plane of $\Gamma$–$n_{0}^{\rm eff}$ can constrain or estimate the
cosmic-ray loading factor $\xi_{\rm CR}$ for a given source x-ray luminosity
$L_{X}^{\rm REF}$. We discuss the present constraints in the next section.

###  IV.3  Constraints on source parameters

The present generic model indicates that the neutrino intensity is determined
by the several source parameters characterizing the photohadronic interaction
efficiency evaluated in the form of optical depth $\tau_{0}$, and the power of
cosmic-ray emission represented by the cosmic-ray loading factor $\xi_{\rm
CR}$. Since $\displaystyle d\dot{N_{\nu}}/d\varepsilon_{\nu}\propto\xi_{\rm
CR}(L_{X}^{\rm REF})^{3/2}B^{\prime}/\sqrt{\xi_{\rm B}}$ ($\displaystyle
d\dot{N_{\nu}}/d\varepsilon_{\nu}\propto\xi_{\rm CR}L_{X}^{\rm REF}$ when the
flux reaches the calorimetric limit) (See Eqs. (19), (46)), the neutrino data
provided by IceCube can constrain these parameters. Especially if searches for
coincident x-ray emission associated with a cosmic neutrino event determine or
constrain $L_{X}^{\rm REF}$, we can determine or place a bound on $\xi_{\rm
CR}$, a likelihood of a given astronomical object class being the origin of
the cosmic rays. In the following sections, we first present the constraints
on the source parameters with the baseline magnetic field configuration of
$B^{\prime}=100$ G and $\xi_{\rm B}=0.1$, and then expand the constraints to
more generic bounds that are valid for the other configurations.

####  IV.3.1 Constraints from all-sky flux observations

Figure 5: The constraints on the cosmic-ray loading factor $\xi_{\rm CR}$
(left) and the UHECR bolometric luminosity $L_{\rm UHECR}$ [erg/s] above
$\varepsilon_{\rm UHECR}^{\rm FID}=10$ PeV (right) on the plane of the plasma
bulk Lorentz factor $\Gamma$ and the local source number density $n_{0}^{\rm
eff}$. The upper (lower) panels display the case of $\alpha_{\rm
CR}=2.3(2.1)$. The baseline magnetic field configuration of $B^{\prime}=100$ G
and $\xi_{\rm B}=0.1$ is assumed. The case of the observed x-ray luminosity in
the energy band of $[2{\rm keV},10{\rm keV}]$ being $L_{X}^{\rm REF}=5\times
10^{46}$ erg/s is shown. The parameters to yield the number of neutrino
multiplet sources of $2.44$ per 20 years is shown by the blue curve. The phase
space below this curve is disfavored if the IceCube measurement with 20-year
data does not detect any neutrino multiplet within the observational time
window of $T_{\rm w}=10^{4}$ sec. The parameters to meet the bolometric
luminosity density of cosmic rays above 10 PeV $\mathcal{Q}_{\rm
UHECR}=2.3\times 10^{45}(\alpha_{\rm CR}=2.3),8.5\times 10^{44}(\alpha_{\rm
CR}=2.1)$ erg Mpc-3yr-1 is shown by the red line. The parameter space in the
right of the red line is unlikely to represent UHECR sources because of the
UHECR energetics limit. The region below the blue curve is disfavored by the
neutrino multiplet detection limit. See the text for details.

Figure 6: The same as Figure 5, but when the upper limit of the coincident
x-ray emission is obtained. The upper (lower) panels display the case of
$\alpha_{\rm CR}=2.3(2.1)$. The case of $L_{X}^{\rm UL}=3\times 10^{45}$ erg/s
in Eq. (32) is displayed. The left panel shows the lower bound of cosmic-ray
loading factor $\xi_{\rm CR}$ and the right panel shows the UHECR luminosity
$L_{\rm UHECR}$ [erg/s]. The parameter space above the red line is unlikely
due to the UHECR energetics constraint. The region below the blue curve is
disfavored by the neutrino multiplet detection limit. See the text for
details.

As the cosmic diffuse neutrino flux $\Phi_{\nu}(E_{\nu})\propto n_{0}^{\rm
eff}\xi_{\rm CR}(L_{X}^{\rm REF})^{3/2}B^{\prime}/\sqrt{\xi_{\rm B}}$, the
cosmic-ray loading factor $\xi_{\rm CR}$ is bounded by the comparison of the
measured diffuse flux data to the predicted flux for a given source population
characterized by $n_{0}^{\rm eff}$ and $\Gamma$, when $L_{X}^{\rm REF}$ is
determined or constrained by the neutrino – x-ray coincident search. We
consider a standard candle source with the photon target energy density given
by Eq. (14). The reference x-ray luminosity $L_{X}^{\rm REF}$ is calculated in
the band of $[\varepsilon_{X,{\rm min}}^{\rm REF},\varepsilon_{X,{\rm
max}}^{\rm REF}]=[2{\rm keV},10{\rm keV}]$. We assume that the population of
this standard candle source with the evolution tracing SFR (Eq. (29)) gives
the cosmic neutrino diffuse background flux.

In order to obtain $\xi_{\rm CR}$ consistent with the IceCube measurements, we
estimated the neutrino intensity at 100 TeV and the approximated power law
index for each of the flux realizations by our model, and compared them with
the profiled likelihood landscape by the diffuse $\nu_{\mu}$ data [5]. In
addition, we demand the predicted flux to meet the differential
$\nu_{e+\mu+\tau}$ flux limit placed by the extremely high energy (EHE)
analysis [15]. This is the same method to extract the allowed parameter space
in YM20.

The left panels of Fig. 5 show the value of cosmic-ray loading factor
$\xi_{\rm CR}$ accounting for the flux data. The case of $L_{X}^{\rm
REF}=5\times 10^{46}$ erg/s is shown. This luminosity is within the expected
range of the LL GRBs [46]. We find that lesser populated ($n_{0}^{\rm
eff}\lesssim 10^{-10}\ {\rm Mpc}^{-3}$) source class with higher plasma
Lorentz factor ($\Gamma\gtrsim 5$) requires $\xi_{\rm CR}\gtrsim 100$. While
on the contrary, highly-populated sources ($n_{0}^{\rm eff}\gtrsim 10^{-8}\
{\rm Mpc}^{-3}$) with low $\Gamma$ need $\xi_{\rm CR}\ll 1$. Because we expect
$\xi_{\rm CR}$ to be not far from unity from theoretical viewpoint, any source
classes in these parameter spaces are unlikely to be valid neutrino source
candidates. It is demonstrated that the measurement of x-ray luminosity leads
to bounding the range of the cosmic-ray loading factor, the indicator of
likelihood to be the high energy neutrino sources for a given set of the
source parameters such as $n_{0}^{\rm eff}$ and $\Gamma$.

We can extend the bounds obtained here to be universally valid for various
magnetic field strengths $B^{\prime}$, the equipartition parameter $\xi_{\rm
B}$ and the x-ray luminosity $L_{X}^{\rm REF}$. Because
$\displaystyle\Phi_{\nu}\propto n_{0}^{\rm eff}\xi_{\rm CR}(L_{X}^{\rm
REF})^{3/2}B^{\prime}/(\Gamma^{2}\sqrt{\xi_{\rm B}}\beta)$ in the most region
of the parameter space, the constraints on the cosmic-ray loading factor shown
in Fig. 5, which are essentially placed by requiring that the neutrino energy
flux does not violate the EHE limit, are approximately represented as

$\xi_{\rm CR}\lesssim 15(11)\left(\frac{L_{X}^{\rm REF}}{5\times 10^{46}\ {\rm
erg/s}}\right)^{-\frac{3}{2}}\left(\frac{n_{0}^{\rm eff}}{1\times 10^{-10}\
{\rm Mpc}^{-3}}\right)^{-1}\\\ \left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\beta,$
(30)

for $\alpha_{\rm CR}=2.3\ (2.1)$. We confirmed that this scaling law in the
relativistic plasma flow case (i.e., $\beta=1$) is valid except
$\Gamma\lesssim 1.4$ where the neutrino flux reaches the colorimetric limit
departing from this scaling behavior.

This cosmic-ray loading factor can also be translated to the characteristic
UHECR luminosity emitted by the source. Using the EHE limit at 20 PeV,
$E_{\nu}^{2}\Phi_{\nu}\approx 1.9\times{10}^{-8}~{}{\rm GeV}~{}{\rm
cm}^{-2}~{}{\rm s}^{-1}~{}{\rm sr}^{-1}$, which is also comparable to the all-
sky neutrino flux in the PeV range, the bolometric luminosity of UHECR
emission at $\varepsilon_{p}\geq\varepsilon_{\rm UHECR}^{\rm FID}$ is given by

$L_{\rm UHECR}=\xi_{\rm CR}L_{X}^{\rm REF}\left(\frac{\varepsilon_{\rm
UHECR}^{\rm FID}}{\varepsilon_{p}^{\rm FID}}\right)^{-\alpha_{\rm CR}+2}\\\
\simeq 3.6(3.8)\times 10^{47}\left(\frac{L_{X}^{\rm REF}}{5\times 10^{46}\
{\rm erg/s}}\right)^{-\frac{1}{2}}\left(\frac{n_{0}^{\rm eff}}{10^{-10}\ {\rm
Mpc}^{-3}}\right)^{-1}\\\ \left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\ {\rm
erg/s},$ (31)

where $\varepsilon_{p}^{\rm FID}(={\rm 1PeV})$ is the fiducial cosmic-ray
energy to define the bolometric cosmic-ray luminosity in the present
formulation (see Table 3) and $\varepsilon_{\rm UHECR}^{\rm FID}={\rm 10PeV}$
is the threshold cosmic-ray energy to define the bolometric UHECR luminosity.
$L_{\rm UHECR}$ on the plane of $\Gamma-n_{0}^{\rm eff}$ is shown in the right
panels of Fig. 5.

####  IV.3.2 Constraints from searches for x-ray coincidences

EM counterparts of neutrinos allow us to search for neutrino sources more
efficiently. In this regard, Ref. [10] considered limits placed by targeted
searches for coincident gamma rays associated with steady neutrino source
candidates, assuming $pp$ scenarios (see their Fig. 7). In $p\gamma$
scenarios, hadronic gamma rays are cascaded down to lower energies, and it is
better to use target photons as a proxy. The best-known example is HL GRBs,
but the gamma-ray search is not effective for many transients such as LL GRBs
and TDEs. Here we highlight the importance of all-sky x-ray monitors such as
MAXI and Einstein Probe.

If we find no coincident x-ray emission associated with cosmic neutrino
signals, we can set upper limits on $L_{X}^{\rm REF}$ as a function of
$n_{0}^{\rm eff}$. The MAXI facility [61] detected a LL GRB candidate of
$L_{X}^{\rm REF}\simeq 1.2\times 10^{46}$ erg/s at a distance of $\sim 360$
Mpc [65]. As the detector received $\sim 30$ x-ray photons with the expected
background of $\sim 10$ photons in this observation, the indicated
sensitivity, $2\times
10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$, will turn into
the upper limit on $L_{X}^{\rm REF}$ for a given $n_{0}^{\rm eff}$. We have

$L_{X}^{\rm REF}\leq L_{X}^{\rm UL}\left(\frac{n_{0}^{\rm eff}}{5.2\times
10^{-9}\ {\rm Mpc}^{-3}}\right)^{-\frac{2}{3}},$ (32)

$L_{X}^{\rm UL}=3\times 10^{45}$ erg/s corresponds to the estimated value for
the MAXI sensitivity, and the Euclidean geometry and no enhancement from the
background are assumed.

On the other hand, when we consider $p\gamma$ scenarios accounting for the
all-sky neutrino flux measured by IceCube, we can derive upper limits on the
neutrino luminosity or lower limits on $n_{0}^{\rm eff}$ [10]. This is clear
in the calorimetric limit for neutrino production, in which
$\displaystyle\Phi_{\nu}\propto n_{0}^{\rm eff}\xi_{\rm CR}L_{X}^{\rm REF}$,
and we obtain “lower limits” on the cosmic-ray loading factor. If the system
is optically thin to the photomeson production, we have
$\displaystyle\Phi_{\nu}\propto n_{0}^{\rm eff}\xi_{\rm CR}(L_{X}^{\rm
REF})^{3/2}B^{\prime}/\sqrt{\xi_{\rm B}}$, and the lower bound becomes
independent of $n_{0}^{\rm eff}$ as $L_{X}^{\rm REF}\propto{n_{0}^{\rm
eff}}^{-2/3}$. Then, assuming that x-ray transients such as LL GRBs and jetted
TDEs contribute to the all-sky neutrino flux via $p\gamma$ interactions with
the x-rays, the upper limit on the x-ray luminosity of neutrino transients
leads to the following constraint on the parameter space,

$\xi_{\rm CR}\gtrsim 19(14)\left(\frac{L_{X}^{\rm UL}}{3\times 10^{45}\ {\rm
erg/s}}\right)^{-\frac{3}{2}}\\\ \left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\beta,$
(33)

for $\alpha_{\rm CR}=2.3(2.1)$.

Figure 6 shows the resulting lower bounds on $\xi_{\rm CR}$. The obtained
lower bounds are nearly independent of $n_{0}^{\rm eff}$ as expected, which
are considered as general bounds under the present sensitivity of x-ray
detection by a wide field of view observatory such as MAXI. We note that both
optically thin and thick cases are consistently considered in numerical
calculations.

The corresponding UHECR luminosity needed for explaining the all-sky neutrino
flux is given by Eq. (31), which is now constrained as

$L_{\rm UHECR}=\xi_{\rm CR}L_{X}^{\rm REF}\left(\frac{\varepsilon_{\rm
UHECR}^{\rm FID}}{\varepsilon_{p}^{\rm FID}}\right)^{-\alpha_{\rm CR}+2}\\\
\gtrsim 4.0(4.2)\times 10^{47}\left(\frac{L_{X}^{\rm UL}}{3\times 10^{45}\
{\rm erg/s}}\right)^{-\frac{1}{2}}\left(\frac{n_{0}^{\rm eff}}{10^{-10}\ {\rm
Mpc}^{-3}}\right)^{-\frac{2}{3}}\\\ \left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\ {\rm
erg/s}.$ (34)

This “lower-limit” luminosity is shown in the right panel of Fig. 6 for the
baseline magnetic field configuration of $B^{\prime}=100$ G and $\xi_{\rm
B}=0.1$.

####  IV.3.3 Constraints from neutrino multiplet detection

A population of nearby luminous transient neutrino sources causes detectable
neutrino multiplets, two (doublet) or more neutrinos originating from the same
direction $\leq\Delta\Omega$ within a certain time frame. A null detection of
such neutrino multiplets also constrains the source neutrino energy outputs
and the flare(burst) rate density [66, 10], as they decide how often we see
bright transient neutrino emission from nearby sources. In the present
neutrino–x-ray common source modeling, these constraints translate to the
bounds on the plasma bulk Lorentz factor $\Gamma$ and the local source number
density $n_{0}^{\rm eff}$ for a given x-ray luminosity $L_{X}^{\rm REF}$.

The number of neutrino sources that could produce multiplets from a $2\pi$
solid angle sky detected in the observation time $T_{\rm obs}$ is given by
[10]

$N_{2\pi}^{\rm M}=\frac{2\pi}{4\pi}\frac{T_{\rm obs}}{T_{\rm
w}}\int\limits_{z_{\rm min}}^{z_{\rm
max}}dzd_{z}^{2}(1+z)\left|\frac{dt}{dz}\right|P_{\rm p}^{n\geq 2}[\mu^{\rm
s}]n_{0}^{\rm eff}\psi(z),$ (35)

where $T_{\rm w}\ll T_{\rm obs}$ is a time window to search for neutrino
multiplets, $P_{\rm p}^{n\geq 2}$ is the Poisson probability of producing
multiple neutrinos for the mean number of neutrinos $\mu^{\rm s}$ from a
source at redshift $z$, given by

$\mu^{\rm s}(\Omega,z)=T_{\rm w}~{}\frac{1}{3}\int
dE_{\nu}A_{\nu_{\mu}}(E_{\nu},\Omega)\phi_{\nu}^{\rm fl}(E_{\nu},z).$ (36)

Here $A_{\nu_{\mu}}$ is the detection area of cosmic muon neutrinos for a
given neutrino telescope. Note that the $1/3$ factor is applied to convert the
all-flavor-sum neutrino flux to that of per-flavor, assuming the equal
neutrino flavor ratio. $\phi_{\nu}^{\rm fl}$ is given by

$\phi_{\nu}^{\rm fl}=\frac{1}{4\pi
d_{z}^{2}}\frac{d\dot{N}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}\left|{}_{\varepsilon_{\nu}=E_{\nu}(1+z)}\right.$
(37)

where $\varepsilon_{\nu}$ and $E_{\nu}=\varepsilon_{\nu}{(1+z)}^{-1}$ are the
neutrino energies at the time of emission and arrival at Earth’s surface,
respectively. The proper distance, $d_{z}$, is calculated via

$d_{z}=\frac{c}{H_{0}}\int\limits_{0}^{z}dz^{{}^{\prime}}\frac{1}{\sqrt{\Omega_{\rm
M}(1+z^{{}^{\prime}})^{3}+\Omega_{\Lambda}}}.$ (38)

We consider the possible constraints placed by the neutrino multiplet search,
using $A_{\nu_{\mu}}$ provided by an underground neutrino telescope model with
a 1 km3 detection volume [67, 10] as a benchmark. The search time window
$T_{\rm w}$ is usually chosen to accommodate a typical flare time scale in a
transient neutrino source. Keeping LL GRBs as a baseline source candidate in
mind, we set $T_{\rm w}=10^{4}$ sec to demonstrate how a multiplet search
could constrain the source characteristics. Note that if the actual burst time
$\Delta T$ is shorter than $T_{\rm w}/(1+z)$, the total neutrino energy output
from a source $\varepsilon_{\nu}^{\rm fl}=L_{\nu}\Delta T$ is fully
observable. The energy contributing to the multiplet detection, $L_{\nu}T_{\rm
w}$, would be less than $\varepsilon_{\nu}^{\rm fl}$ otherwise. A similar
argument is also held for total cosmic-ray energy output. We also point out
that the sensitivity to $L_{X}^{\rm REF}$ by an x-ray telescope is always
conservative when the exposure time of an x-ray monitoring telescope for a
given patch of the sky, $T_{X}$, is shorter than $T_{\rm w}$. The observation
by the MAXI telescope belongs to this case as $T_{X}=40$ sec.

For $T_{\rm w}=10^{4}$ sec, the number of the expected atmospheric neutrino
background is $\ll 10^{-2}$ for $\Delta\Omega\approx 1^{\circ}\times
1^{\circ}$ in IceCube. Therefore, the search for neutrino multiplets can be
conducted under a background-free environment, and the multiplet source number
$N_{2\pi}^{\rm M}$ directly tells the sensitivity. A null detection of any
neutrino multiple events by the observation with $T_{\rm obs}=20$ year would
constrain the viable parameter space of $\Gamma-n_{0}^{\rm eff}$ by setting
$N_{2\pi}^{\rm M}\leq 2.44$ (The Feldman-Cousins 90% CL Poisson upper limit
[68]). The blue curves in Figs. 5 and 6 represent the resultant constraints.
The parameter space below the curve is ruled out by 90 % CL. The space for
rarer source population (i.e., smaller $n_{0}^{\rm eff}$) is disfavored as
expected. Note that the multiplet search cannot constrain the parameters by
the present IceCube sensitivity in the harder UHECR spectrum case
($\alpha_{\rm CR}=2.1$) because the resultant hard neutrino spectrum yields a
lower event rate.

One can notice that the bound placed in the parameter space on the $\Gamma-
n_{0}^{\rm eff}$ plane for the fixed x-ray luminosity case (i.e., Fig. 5) is
nearly identical to that for the upper limit case (i.e., Fig. 6), indicating
that the constraints on $n_{0}^{\rm eff}$ and $\Gamma$ are independent of
$L_{X}^{\rm REF}$. We demonstrate that this is indeed true, and the bound is
universal regardless of not only the x-ray luminosity but also the magnetic
field configuration of $B^{\prime}$ and $\xi_{\rm B}$ by the following
argument. $\displaystyle P_{\rm p}^{n\geq 2}\simeq(\mu^{\rm s})^{2}/2$ as
$\mu_{s}\ll 1$. Then $\displaystyle N_{2\pi}^{\rm M}\propto n_{0}^{\rm
eff}\xi_{\rm CR}^{2}(L_{X}^{\rm REF})^{3}B^{\prime 2}/(\xi_{\rm B}d_{z^{\rm
min}})$. As the proper distance of a source at $z=z_{\rm min}$, $d_{z^{\rm
min}}$, is approximately $\displaystyle\sim\left(3/4\pi n_{0}^{\rm
eff}\right)^{1/3}$, $\displaystyle N_{2\pi}^{\rm M}\propto{n_{0}^{\rm
eff}}^{4/3}\xi_{\rm CR}^{2}(L_{X}^{\rm REF})^{3}B^{\prime 2}/\xi_{\rm B}$.
Then the cosmic-ray loading factor upper limit set by $N_{2\pi}^{\rm M}\leq
2.44$ is written as $\xi_{\rm CR}^{\rm UL}\propto{n_{0}^{\rm
eff}}^{-2/3}(L_{X}^{\rm REF})^{-3/2}\sqrt{\xi_{\rm B}}/B^{\prime}$. The
cosmic-ray loading factor determined by the cosmic diffuse neutrino flux must
be lower than this upper limit, i.e., $\xi_{\rm CR}\leq\xi_{\rm CR}^{\rm UL}$.
As represented in Eq. (30), $\xi_{\rm CR}\propto{n_{0}^{\rm
eff}}^{-1}(L_{X}^{\rm REF})^{-3/2}\sqrt{\xi_{\rm B}}/B^{\prime}$ (primarily
because $\Phi_{\nu}(E_{\nu})\propto n_{0}^{\rm eff}\xi_{\rm CR}(L_{X}^{\rm
REF})^{3/2}B^{\prime}/\sqrt{\xi_{\rm B}}$) which has the same dependencies on
$L_{X}^{\rm REF},B^{\prime}$, and $\xi_{\rm B}$ as $\xi_{\rm CR}^{\rm UL}$
follows. Thus, these dependencies are canceled out in the condition of
$\xi_{\rm CR}\leq\xi_{\rm CR}^{\rm UL}$.

####  IV.3.4 Additional constraints for unification scenarios

Figure 7: Constraints on the UHECR-neutrino common source model. The colored
area represents the allowed parameter space by the UHECR energetic condition
(thick solid line), the acceleration condition (thick dashed line), the escape
condition (dotted line), and the constraint by multiplet (vertical dash-dotted
line). Left: The case of $L_{X}^{\rm REF}=5\times 10^{46}$ erg/s and
$B^{\prime}=100$ G. Middle: The case of $L_{X}^{\rm REF}=5\times 10^{46}$
erg/s and $B^{\prime}=1000$ G. Right: The case of $L_{X}^{\rm REF}=1\times
10^{45}$ erg/s and $B^{\prime}=100$ G.

If the neutrino and x-ray emitters we consider here are also emitting UHECRs,
we can place further constraints in the parameter space. We first discuss the
energetics conditions as we did in Section II. The UHECR luminosity density
condition demanded by the UHECR energetics is approximately given by

$n_{0}L_{\rm UHECR}=n_{0}\xi_{\rm CR}L_{X}^{\rm
REF}\left(\frac{\varepsilon_{\rm UHECR}^{\rm FID}}{\varepsilon_{p}^{\rm
FID}}\right)^{-\alpha_{\rm CR}+2}\lesssim Q_{\rm UHECR}.$ (39)

$\mathcal{Q}_{\rm UHECR}$ [12, 11] is given by Eq. (5) and listed in Table 3.
Because the cosmic-ray loading factor $\xi_{\rm CR}$ or equivalently the UHECR
luminosity $L_{\rm UHECR}$ are determined by the cosmic background neutrino
flux, this condition above sets bounds on the $\Gamma-n_{0}$ plane. The red
line in Fig. 5 and 6 corresponds to this boundary. Any sources found in the
parameter space beyond this line would overproduce the observed UHECR flux and
thus are unlikely to be the common UHECR and neutrino origin. The constraints
for the cases other than the baseline magnetic field parameters of
$B^{\prime}=100$ G and $\xi_{\rm B}=0.1$ are approximately obtained by Eq.
(31) or (34) with Eq. (39). We get

$\left(\frac{L_{X}^{\rm REF}}{5\times 10^{46}\ {\rm
erg/s}}\right)^{-\frac{1}{2}}\left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\\\ \left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\beta\lesssim
2.0(1.0),$ (40)

for the fixed x-ray luminosity case when the UHECR spectrum power-law index
$\alpha_{\rm CR}=2.3(2.1)$ and

$\left(\frac{n_{0}^{\rm eff}}{1\times 10^{-10}\ {\rm
Mpc}^{-3}}\right)^{\frac{1}{3}}\left(\frac{L_{X}^{\rm UL}}{3\times 10^{45}\
{\rm erg/s}}\right)^{-\frac{1}{2}}\\\ \left(\frac{B^{\prime}}{100\ {\rm
G}}\right)^{-1}\left(\frac{\xi_{\rm
B}}{0.1}\right)^{\frac{1}{2}}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\beta\lesssim
1.8(0.9),$ (41)

for the scenario when obtaining the upper limit of the associated x-ray
luminosity.

UHECR emissions also require cosmic rays to leave the sources before losing
their energies via synchrotron cooling. This “survival” condition leads to the
bound on $B^{\prime}$ and $\Gamma$ represented by [12]

$\displaystyle B^{\prime}$ $\displaystyle\leq$ $\displaystyle\frac{6\pi
m_{p}^{4}c^{9/2}}{\sigma_{T}m_{e}^{2}{(2\xi_{B}L_{X}^{\rm
REF})}^{1/2}}\frac{\Gamma^{3}}{\varepsilon_{p}^{\rm max}}$ (42)
$\displaystyle\lesssim$ $\displaystyle 7\times
10^{2}\left(\frac{\Gamma}{10^{0.5}}\right)^{3}\left(\frac{L_{X}^{\rm
REF}}{5\times 10^{46}\ {\rm erg/s}}\right)^{-\frac{1}{2}}$
$\displaystyle\left(\frac{\xi_{\rm
B}}{0.1}\right)^{-\frac{1}{2}}\left(\frac{\varepsilon_{p}^{\rm max}}{10^{10}\
{\rm GeV}}\right)^{-1}\ {\rm G}.$

We note that details depend on the diffusion process, which is particularly
relevant for nonrelativistic systems. Here, we conservatively demand protons
with energies of $\varepsilon_{p}\lesssim\varepsilon_{p}^{\rm max}=10^{10}$
GeV to meet the escape condition since cosmic-ray nuclei would then meet the
condition even at energies beyond $10^{11}$ GeV. The bound can be translated
into the upper bound of $\xi_{\rm B}$ for a given $B^{\prime}$ and we get

$\xi_{\rm B}\lesssim 6\left(\frac{L_{X}^{\rm REF}}{5\times 10^{46}\ {\rm
erg/s}}\right)\left(\frac{B^{\prime}}{100{\rm
G}}\right)^{2}\left(\frac{\Gamma}{10^{0.5}}\right)^{6}$ (43)

Note that we cannot constrain the parameters by the escape condition when only
the upper limit of the x-ray luminosity is obtained.

The UHECR acceleration condition given by Eq. (LABEL:eq:hillas_condition)
places the lower limit of $\xi_{\rm B}$ for a given x-ray luminosity as

$\xi_{\rm B}\gtrsim 3\times 10^{-3}\left(\frac{L_{X}^{\rm REF}}{5\times
10^{46}\ {\rm
erg/s}}\right)^{-1}\left(\frac{\Gamma}{10^{0.5}}\right)^{2}\left(\frac{\varepsilon_{p}^{\rm
max}}{10^{10}\ \rm{GeV}}\right)^{2}$ (44)

The UHECR energetics condition by Eq. (40), the escape condition by Eq. (43),
and the acceleration condition by Eq. (44) lead to the $n_{0}^{\rm eff}$
independent bound on the equipartition parameter $\xi_{\rm B}$ and the plasma
Lorentz bulk factor $\Gamma$. Figure 7 shows the resultant allowed parameter
space on $\xi_{\rm B}$ and $\Gamma$. The energetics and escape conditions lead
to $\xi_{\rm B}\lesssim 1$, which is consistent with Ref. [12]. Combining the
UHECR energetics condition (Eq. (40)) and the acceleration requirement (Eq.
(44)) leads to the universal upper limit of the plasma Lorentz bulk factor
$\Gamma$ as

$\Gamma\lesssim 7(6)\left(\frac{L_{X}^{\rm REF}}{5\times 10^{46}\ {\rm
erg/s}}\right)^{\frac{1}{3}}\left(\frac{B^{\prime}}{100{\rm
G}}\right)^{\frac{1}{3}},$ (45)

for $\alpha_{\rm CR}=2.3(2.1)$.

##  V Discussion

The synergy between neutrinos and X-ray observations is crucial for testing
some models such as LL GRBs. The Wide-Field Telescope (WFT) of Einstein Probe
has a sensitivity of $\sim 3\times{10}^{-11}~{}{\rm erg}~{}{\rm
cm}^{-2}~{}{\rm s}^{-1}$ at $0.5-4$ keV. For example, for a LL GRB with
$L_{\gamma}\sim{10}^{46}~{}{\rm erg}~{}{\rm s}^{-1}$ in the WFT band, the
detection horizon is $z_{\rm lim}\sim 0.3-0.4$, and the number of LL GRBs
within 1 sr could be $\sim 100$ per year, although in reality it would be
optimistic since, e.g., the peak energy will not always be in the WFT band.
Most of the all-sky neutrino flux comes from LL GRBs with redshifts of $\sim
1-2$, which are beyond the detection horizon. Thus, x-ray counterparts should
be found for only a fraction of the alerts. Nevertheless, should no coincident
neutrinos be discovered in conjunction with a larger sample of X-ray
transients identified by Einstein Probe, the surveyed range of $L_{X}^{\rm
REF}$ can be enhanced by a factor of $\sim 10$, thereby enabling constraints
on $\xi_{\rm CR}$ to be stringent as $\gtrsim 1000$.

We also note that the constraints from searches for x-ray coincidences and
multiplets are general, which are applicable even if neutrinos and UHECRs are
produced in different regions (e.g., neutrinos from choked jets and UHECRs
from the external reverse shock) but by the same population [54]. On the other
hand, it is crucial to know the nature of x-ray transients, including source
redshifts. Some of the x-ray transients can be HL GRBs at high redshifts [69,
70, 71], in which we will be able to constrain rarer and more luminous gamma-
ray transients.

In addition to real-time searches for neutrinos, stacking analyses will also
be useful. These analyses allow us to look for lower-energy neutrinos,
although the current sensitivity is largely limited by the sample size.
Performing stacking analyses with optical transients [72, 73, 74, 75] and
follow-up searches for optical counterparts [66] will also be useful,
especially in the era of the Rubin Observatory. Furthermore, it would be
beneficial to conduct real-time subthreshold analyses on neutrinos and X-ray
transients. AMON (Astrophysical Multimessenger Observatory Network) has
attempted to achieve this [76, 77, 78, 79]. Enhancing the probability of
discovering candidate sources of neutrino transients through searches for
lower-energy neutrinos and/or weaker X-ray transients is a viable strategy.

##  VI  Summary

We presented diagnoses for the unified picture of high-energy neutrinos and
UHECRs, focusing on $p\gamma$ scenarios. The framework is largely source
independent and useful for determining the parameter space required for more
sophisticated source models. Given that the UHECR energy generation rate
density is comparable to that of high-energy neutrinos, the system must be
semi-transparent to the photomeson production, particularly if the spectrum is
hard [19]. This substantially limits the allowed model parameter space.
Furthermore, the search for neutrino multiplets has excluded rare, bright
source populations [10]. As the production of neutrinos is inefficient if the
luminosity of the source is too low, the cosmic-ray loading factor must be
sufficiently high to satisfy the unification requirement for dim sources (see
Fig. 5). We investigated the impact of x-ray counterpart searches and
demonstrated that the nondetection of x-ray emission allows us to place lower
bounds on the cosmic-ray loading factor, which is a crucial quantity for
source energetics (see Fig. 6). The ranges of magnetic fields and Lorentz
factor are also limited depending on the source luminosity (see Fig. 7).

We discussed different classes of astrophysical sources that can potentially
satisfy the requirements for the unified models. Among them, LL GRBs and TDEs
are of particular interest as x-ray transients that can be tested for current
and near-future x-ray missions. It is possible that realistic sources may not
be standard candles. Therefore, it is necessary to obtain a sufficiently large
sample to account for the diversity of source parameters in order to obtain
reliable constraints. Nevertheless, our results demonstrate that real-time
neutrino-triggered searches for x-ray transients are a powerful tool for
testing the unified models for IceCube neutrinos and UHECRs. This is
encouraging for current and future x-ray satellites with wide-field coverage,
such as MAXI, Einstein Probe, HiZ-GUNDAM, and Athena. Neutrino detectors with
bigger volumes or better angular resolutions are also beneficial for more
efficient and frequent follow-up observation, for which KM3Net, Baikal-GVD,
P-ONE, and especially IceCube-Gen2 will play important roles.

###### Acknowledgements.

The authors are grateful to Wataru Iwakiri for his useful input on the x-ray
observations with MAXI and NICER. K.M. also thanks Bing Zhang for discussions
during the 3rd Nanjing conference. This work by S.Y.is supported by JSPS
KAKENHI grant Nos. 18H05206, 23H04892, and Institute for Advanced Academic
Research of Chiba University. The work of K.M. is supported by the NSF Grant
Nos. AST-2108466, AST-2108467, and AST-2308021, and KAKENHI Nos. 20H01901 and
20H05852.

##  Appendix A Analytical Formulas for calculating neutrino flux

When we focus on the leading terms for simplicity in the integration of the
formula in Eq. (23), the neutrino luminosity originated from the target photon
field below (above) the break energy is obtained by [19],

$\frac{d\dot{N}^{\rm
L/H}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}\approx(\alpha_{\rm
CR}-2)\xi_{\rm CR}\left(\frac{\varepsilon_{p}^{\rm
FID}}{\varepsilon_{p0}[\Gamma]}\right)^{\alpha_{\rm CR}}\frac{L_{X}^{\rm
REF}}{(\varepsilon_{p}^{\rm FID})^{2}}\tau_{0}^{\rm
L/H}\frac{3}{1-r_{\pi}}\frac{1}{x_{\Delta}^{+}-x_{\Delta}^{-}}f_{\rm
sup}\mathcal{I}_{\rm P}^{\rm L/H}.$ (46)

Here $x_{\Delta}^{+(-)}$ is the maximal (minimal) bound of the relative energy
of emitted pion normalized by the parent cosmic-ray energy. A kinematic
relation represents them [64] as

$x_{\Delta}^{\pm}=\frac{(s_{\Delta}+m_{\pi}^{2}-m_{p}^{2})\pm\sqrt{(s_{\Delta}+m_{\pi}^{2}-m_{p}^{2})^{2}-4s_{\Delta}m_{\pi}^{2}}}{2s_{\Delta}}.$
(47)

The neutrino energy term $\mathcal{I}_{\rm P}^{\rm L/H}$ is a power-law form
in most of the relevant neutrino energy ranges. We get

$\displaystyle\mathcal{I}_{\rm P}^{\rm
L/H}=\left\\{\begin{array}[]{ll}\frac{1}{(\alpha_{\rm
CR}+2)^{2}}\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
min}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{(\alpha_{X}^{\rm
L/H}+1)}\left(\frac{\varepsilon_{\nu}}{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm min}]}\right)^{-(\alpha_{\rm
CR}+2)},&(\varepsilon_{\nu 0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
min}]\leq\varepsilon_{\nu})\\\ \frac{1}{(\alpha_{\rm CR}+1-\alpha_{X}^{\rm
L/H})^{2}}\left[\left(\frac{\varepsilon_{\nu}}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}-\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
min}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}\right]&\\\ +\frac{1}{(\alpha_{\rm
CR}+2)^{2}}\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
min}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})},&(\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]\leq\varepsilon_{\nu}\leq\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm min}])\\\ \frac{1}{(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})^{2}}\left[\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm
L/H})}-\left(\frac{\varepsilon_{\nu}}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{-}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}\right]&\\\ +\frac{1}{(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}\ln{\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]}{\varepsilon_{\nu}}\right)},&(\varepsilon_{\nu
0}^{-}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]\leq\varepsilon_{\nu}\leq\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm max}])\\\ \frac{1}{(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm L/H})}\left(\frac{\varepsilon_{\nu
0}^{+}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm
max}]}{\varepsilon_{p0}[\Gamma]x_{\Delta}^{+}(1-r_{\pi})}\right)^{-(\alpha_{\rm
CR}+1-\alpha_{X}^{\rm
L/H})}\ln{\left(\frac{x_{\Delta}^{+}}{x_{\Delta}^{-}}\right)},&(\varepsilon_{\nu}\leq\varepsilon_{\nu
0}^{-}[\varepsilon^{{}^{\prime}\rm{L/H}}_{\rm max}]).\\\ \end{array}\right..$
(49)

Here, $\varepsilon_{\nu 0}^{\pm}[\varepsilon^{\prime}_{X}]$ is given by

$\varepsilon_{\nu
0}^{\pm}[\varepsilon^{\prime}_{X}]=\frac{s_{\Delta}-m_{p}^{2}}{4\varepsilon^{\prime}_{X}}\Gamma
x_{\Delta}^{\pm}(1-r_{\pi}).$ (50)

This parameterization is understood by the fact that the main energy range of
neutrinos produced by the photon energy $\varepsilon^{\prime}_{X}$ is
approximately represented by $\varepsilon_{\nu
0}^{-}[\varepsilon^{\prime}_{X}]\leq\varepsilon_{\nu}\leq\varepsilon_{\nu
0}^{+}[\varepsilon^{\prime}_{X}]$.

$\varepsilon_{\rm min/max}^{{}^{\prime}\rm L/H}$ specified the photon energy
range (in the plasma rest frame) in the BPL spectrum represented by Eq. (14).
They are given by

$\displaystyle\varepsilon^{{}^{\prime}\rm L}_{\rm
min}=\varepsilon^{\prime}_{\rm min}$ ,
$\displaystyle\quad\varepsilon^{{}^{\prime}\rm L}_{\rm
max}=\varepsilon^{\prime}_{X,b}$ $\displaystyle\varepsilon^{{}^{\prime}\rm
H}_{\rm min}=\varepsilon^{\prime}_{X,b}$ ,
$\displaystyle\quad\varepsilon^{{}^{\prime}\rm H}_{\rm
max}=\varepsilon^{\prime}_{\rm max}$ (51)

The total differential neutrino yield emitted from a source is then given by

$\frac{d\dot{N}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}=\frac{d\dot{N}^{\rm
L}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}+\frac{d\dot{N}^{\rm
H}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}$ (52)

Most of the parameter space relevant to the present sensitivities of x-ray
observation facilities involves optically thin sources. UHECR emitters also
must be optically thin. Nevertheless, we introduced the calorimetric limit in
the calculations of the neutrino flux by employing the limit on the optical
depth as

$\tau_{0}^{\rm L/H}\leq 1/\kappa_{p\gamma}\approx 5$ (53)

where $\kappa_{p\gamma}\sim 0.2$ is the proton inelasticity. For the parameter
set beyond this limit, we placed $\tau_{0}=5$ to calculate the neutrino yield
as the calorimetric condition.

When the target photon spectrum follows the thermal spectrum, the neutrino
yield is obtained by following Eq. (7) in Ref [64] with minor modifications
and re-parameterization, and given by

$\frac{d\dot{N}_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}}{d\varepsilon_{\nu}}\approx(\alpha_{\rm
CR}-2)\xi_{\rm CR}\left(\frac{\varepsilon_{p}^{\rm
FID}}{\varepsilon_{p0}[\Gamma]}\right)^{\alpha_{\rm CR}}\frac{L_{X}^{\rm
REF}}{(\varepsilon_{p}^{\rm
FID})^{2}}\frac{3}{1-r_{\pi}}\frac{1}{x_{\Delta}^{+}-x_{\Delta}^{-}}f_{\rm
sup}\tau_{p\gamma}(\varepsilon_{\nu}).$ (54)

where the optical depth $\tau_{p\gamma}$ is given by

$\tau_{p\gamma}(\varepsilon_{\nu})=\frac{15}{32\pi^{5}}\sqrt{\frac{L_{X}}{\xi_{\rm
B}}}\frac{B^{\prime}}{\Gamma^{2}}\frac{1}{(k_{\rm
B}T)^{3}\sqrt{2c}}\int\limits^{\infty}_{\frac{\varepsilon_{\nu}}{(1-r_{\pi})x^{+}_{\Delta}}}d\varepsilon_{p}\frac{\Gamma^{3}}{\varepsilon_{p}^{3}}\left(\frac{\varepsilon_{p}}{\varepsilon_{p0}[\Gamma]}\right)^{-\alpha_{\rm
CR}}F(s,\varepsilon_{p})$ (55) $\displaystyle F(s,\varepsilon_{p})$
$\displaystyle\equiv$ $\displaystyle\int
ds(s-m_{p}^{2})\sigma_{p\gamma}(s)\ln{\left(\frac{x_{\Delta}^{+}}{\xi_{p}}\right)}\\{-\ln{(1-e^{-\Gamma\frac{s-m_{p}^{2}}{4k_{\rm
B}T\varepsilon_{p}}})}\\}$ (56) $\displaystyle\approx$
$\displaystyle(s_{\Delta}-m_{p}^{2})\Delta_{s_{\Delta}}\sigma_{p\gamma}^{\Delta}\ln{\left(\frac{x_{\Delta}^{+}}{\xi_{p}}\right)}\\{-\ln{(1-e^{-\Gamma\frac{s_{\Delta}-m_{p}^{2}}{4k_{\rm
B}T\varepsilon_{p}}})}\\}$ (57)
$\xi_{p}=\begin{cases}x_{\Delta}^{-}&\varepsilon_{\nu}\leq(1-r_{\pi})x_{\Delta}^{-}\varepsilon_{p},\\\
{\varepsilon_{\nu}\over(1-r_{\pi})\varepsilon_{p}}&\text{otherwise}.\end{cases}$
(58)

The simplification from Eq. (56) to Eq. (57) is based on the $\Delta$
resonance approximation.

##  Appendix B UHECR-Neutrino common Sources with thermal x-ray emission

The x-rays in the cosmic-ray acceleration/neutrino production region can be
thermalized if the environment is optically thick at the initial phase of the
system. A choked-jet scenario [49, 50] is a good example of this possibility.
As long as considering the UHECR-neutrino common sources, the emission region
must still be optically thin when emitting UHECRs and neutrinos. Here, we
briefly review our modeling of the optically thin thermal x-ray source in a
generic way to see if the outcomes we presented in the earlier sections are
solid or not.

Figure 8: Same as Figure 3, but the thermal photon case. The three cases of
$\Gamma$ values ,$\Gamma=2$ (Top), $\Gamma=3$ (Middle), $\Gamma=10$ (Bottom)
are presented.

Figure 9: Same as Fig. 5, but the case when the target photon spectrum follows
the thermal distribution given by Eq. (59). The x-ray thermal energy is
$k_{\rm B}T=0.177{\rm keV}$ for corresponding to
$\varepsilon^{\prime}_{X,b}=0.5{\rm keV}$ in the nonthermal emission case.

The target x-ray photons are now assumed to follow the quasi-Planck
distribution instead of the BPL written as Eq. (14). It is given by

$\frac{dn_{X}}{d\varepsilon_{X}^{\prime}}=\frac{\epsilon_{\rm
T}}{\pi^{2}\hbar^{3}c^{3}}\frac{\varepsilon^{{}^{\prime}2}_{X}}{\exp^{\frac{\varepsilon^{\prime}_{X}}{k_{\rm
B}T}}-1}$ (59)

$\epsilon_{\rm T}=1$ corresponds to the black body distribution

The x-ray luminosity density is associated with the thermal photon energy
density,

$\frac{L^{\prime}_{X}}{4\pi R^{2}c}=\epsilon_{\rm
T}\frac{\pi^{2}}{15\hbar^{3}c^{3}}(k_{\rm B}T)^{4}$ (60)

$R$ is the distance of an acceleration and emission region from the central
engine.

The x-ray density is also associated with the magnetic energy density via
$\xi_{\rm B}$ and we get

$\frac{B^{{}^{\prime}2}}{8\pi}=\xi_{\rm B}\frac{L^{\prime}_{X}}{4\pi R^{2}c}$
(61)

We then get the relations between $\epsilon_{\rm T}$, $B^{\prime}$, and
$\xi_{\rm B}$ as

$\displaystyle\varepsilon_{\rm T}$ $\displaystyle=$
$\displaystyle\frac{15\hbar^{3}c^{3}}{8\pi^{3}}\frac{B^{{}^{\prime}2}}{(k_{\rm
B}T)^{4}}\xi_{\rm B}^{-1}$ $\displaystyle\approx$ $\displaystyle 4.6\times
10^{-10}\left(\frac{k_{\rm B}T}{0.5{\rm
keV}}\right)^{-4}\left(\frac{B^{\prime}}{100~{}{\rm
G}}\right)^{2}\left(\frac{\xi_{\rm B}}{0.1}\right)^{-1}$

It implies that the reasonable range of $\xi_{B}$ and $B^{\prime}$ requires
$\epsilon_{\rm T}\lesssim 10^{-9}\ll 1$, providing the optically thin
environment.

Hereafter we consider the case when $k_{\rm
B}T=0.355\varepsilon^{\prime}_{X,b}=0.177$ keV so that its peak energy is
close to $\varepsilon^{\prime}_{X,b}$. The analytical formulas to calculate
the neutrino yield are described in the Appendix and given by Eq. (54). Figure
8 shows the resultant neutrino background flux with the same baseline
parameters as Fig. 3. The spectrum gets rapidly steeper at energies beyond the
peak energy $\varepsilon_{\nu}^{\rm peak}$ which corresponds to the
representative energy of neutrinos produced by protons with energies of
$\varepsilon_{p}\simeq\varepsilon_{p0}[\Gamma]$. Although the introduced
analytical $\Delta$-resonance approximation artificially enhances this
feature, it implies that this scenario contributes mostly to the cosmic
background flux in the energy region below 100 TeV. We, thus, scan the range
of cosmic-ray loading factor $\xi_{\rm CR}$ to meet $\displaystyle 2\times
10^{-8}\leq
E_{\nu}^{2}\Phi_{\nu_{e}+\nu_{\mu}+\nu_{\tau}}|_{E_{\nu}=\varepsilon_{\nu}^{\rm
peak}}\leq 2\times 10^{-7}$ [GeV cm-2 sec-1 sr-1]. The resultant upper bound
of $\xi_{\rm CR}$ and the corresponding $L_{\rm UHECR}$ are displayed in Fig.
9. The constraints are similar to the nonthermal BPL case summarized in Fig.
5. The conclusion remains the same, given that the thermal x-ray field in the
neutrino production zone is optically thin, determined by $B^{\prime}$ and
$\xi_{\rm B}$ as represented in Eq. (LABEL:eq:glay_body_scale).

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