# Black Holes as Neutrino Factories

Yifan Chen<EMAIL_ADDRESS>Niels Bohr International Academy, Niels Bohr
Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark  Xiao Xue
<EMAIL_ADDRESS>II. Institute of Theoretical Physics, Universität Hamburg,
22761 Hamburg, Germany  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607, Hamburg, Germany  Vitor Cardoso<EMAIL_ADDRESS>Niels Bohr
International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen,
Denmark  CENTRA, Departamento de Física, Instituto Superior Técnico – IST,
Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

###### Abstract

Ultralight bosons can grow substantially in the vicinity of a black hole,
through superradiant energy extraction. Consequently, such bosons can
potentially reach field values close to the Planck scale, making black holes
powerful transducers of such fields. If a scalar field couples to neutrino, it
can trigger parametric production of neutrinos, and potentially quench their
superradiant growth. During this saturation phase, scalar clouds can
accelerate neutrinos to the TeV energy scale, generating fluxes that surpass
those produced by atmospheric neutrinos.

††preprint: DESY-23-109

## I Introduction.

Hypothetical bosons with ultralight masses are promising candidates for
beyond-the-standard-model particle physics. They are motivated by their
potential to explain the smallness of the neutron electric dipole moment
Peccei and Quinn (1977), and they are natural outcomes of fundamental theories
with extra dimensions Svrcek and Witten (2006); Abel  _et al._ (2008);
Arvanitaki  _et al._ (2010); Goodsell  _et al._ (2009). When these bosons
comprise dark matter Preskill  _et al._ (1983); Abbott and Sikivie (1983);
Dine and Fischler (1983); Nelson and Scholtz (2011) and possess masses lower
than $\mathcal{O}(1)$ eV, they behave as coherent waves due to their
substantial occupation numbers Hu  _et al._ (2000).

The signals associated with ultralight bosons are typically proportional to
their energy density or field value. When the Compton wavelength of these
bosons is comparable to the gravitational radius of a rapidly rotating black
hole (BH), a bound state can form through the superradiance mechanism, which
involves the extraction of the BH rotational energy Penrose and Floyd (1971);
Zel’Dovich (1971); Brito  _et al._ (2015a). The process then leads to the
condensation of a bosonic structure – a boson cloud – in the BH exterior
Detweiler (1980); Cardoso and Yoshida (2005); Dolan (2007). Superradiant
clouds can reach a total mass of approximately $\mathcal{O}(10)\%$ of the BH
mass Brito  _et al._ (2015b); East and Pretorius (2017); Herdeiro  _et al._
(2022), corresponding to a field value approaching the Planck scale Chen  _et
al._ (2023a). Thus, Kerr BHs can serve as powerful transducers for ultralight
bosons Brito  _et al._ (2015a). Detection based on superradiance does not
necessarily require ultralight bosons to constitute the majority of dark
matter. It can be achieved, for example, through the spin-down of BHs
Arvanitaki and Dubovsky (2011); Arvanitaki  _et al._ (2015); Brito  _et al._
(2015b); Baryakhtar  _et al._ (2017); Brito  _et al._ (2017a); Cardoso  _et
al._ (2018); Davoudiasl and Denton (2019); Brito  _et al._ (2020); Stott
(2020); Ünal  _et al._ (2021); Saha  _et al._ (2022), gravitational-wave
signals from boson clouds Arvanitaki and Dubovsky (2011); Yoshino and Kodama
(2012, 2014); Arvanitaki  _et al._ (2015); Yoshino and Kodama (2015);
Baryakhtar  _et al._ (2017); Brito  _et al._ (2017b, a); Isi  _et al._ (2019);
Siemonsen and East (2020); Sun  _et al._ (2020); Palomba  _et al._ (2019);
Brito  _et al._ (2020); Zhu  _et al._ (2020); Tsukada  _et al._ (2021); Yuan
_et al._ (2021a); Abbott  _et al._ (2022a); Yuan  _et al._ (2022); Chen  _et
al._ (2023a); Brito and Shah (2023), and axion cloud-induced birefringence
Chen  _et al._ (2020); Yuan  _et al._ (2021b); Chen  _et al._ (2022a, b).

The couplings between ultralight bosons and standard model neutrinos are
prevalent in various beyond-standard-model theories, which are motivated by
neutrino mass generation Gelmini and Roncadelli (1981); Chikashige  _et al._
(1981); Aulakh and Mohapatra (1982) and grand unification theories Georgi and
Glashow (1974); Pati and Salam (1974); Mohapatra and Pati (1975); Fritzsch and
Minkowski (1975); Georgi (1975). Nevertheless, investigating these couplings
presents significant challenges due to the inherent difficulties associated
with neutrino production and detection. The constraints on these couplings
primarily arise from neutrino self-interactions mediated by ultralight bosons
Kolb and Turner (1987); Huang  _et al._ (2018); Brune and Päs (2019);
Forastieri  _et al._ (2019); Berryman  _et al._ (2022); Li and Xu (2023), and
distortions in neutrino oscillations caused by ultralight dark matter Reynoso
and Sampayo (2016); Berlin (2016); Krnjaic  _et al._ (2018); Brdar  _et al._
(2018); Davoudiasl  _et al._ (2018); Liao  _et al._ (2018); Capozzi  _et al._
(2018); Huang and Nath (2018); Farzan (2019); Cline (2020); Dev  _et al._
(2021); Losada  _et al._ (2022); Huang and Nath (2022); Chun (2021); Dev  _et
al._ (2023); Losada  _et al._ (2023a); Brzeminski  _et al._ (2022); Alonso-
Álvarez  _et al._ (2023); ChoeJo  _et al._ (2023); Losada  _et al._ (2023b).

In this work, we demonstrate that scalar clouds surrounding BHs can generate
substantial neutrino emissions through parametric production Greene and Kofman
(1999, 2000). Moreover, the spatial gradient of scalar clouds can efficiently
accelerate the produced neutrinos to TeV-level energies. Given the significant
field values achievable through superradiant production, a deeper range of
coupling constants can be explored by comparing the anticipated fluxes with
those observed in diffusive atmospheric neutrinos Vitagliano  _et al._ (2020).

## II Superradiant Clouds.

An ultralight boson field can form (quasi-)bound states outside a BH due to
gravitational attraction. These bound states exhibit discrete quantum numbers
and are influenced by the gravitational fine-structure constant, denoted as
$\alpha\equiv G_{\rm N}M_{\rm BH}\mu$ Detweiler (1980); Brito  _et al._
(2015a). Here, $G_{\rm N}$ is Newton’s constant, while $M_{\rm BH}$ and $\mu$
represent the BH mass and the boson mass, respectively. Superradiance occurs
when the BH rotates at a sufficient velocity for its angular velocity to
surpass the angular phase velocity of specific boson states, leading to
exponential transfer of rotational energy into these states (Zel’Dovich, 1971;
Brito  _et al._ , 2015a).

We focus on the superradiant ground state of a massive scalar $\phi$, whose
orbital angular momentum corotates with the BH. Its wave function can be
approximated as Detweiler (1980)

$\begin{split}\phi(\vec{x},t)=&\Psi_{0}(t)R_{\alpha}(r)\sin\theta\cos(\mu
t-\varphi),\end{split}$ (1)

in the Newtonian, $\alpha\ll 1$ limit. Here, $\Psi_{0}$ represents the peak
field value of the scalar cloud, $R_{\alpha}(r)={\rm
Exp}[1-\alpha^{2}r/(2r_{g})]\,\alpha^{2}r/(2r_{g})$ denotes the normalized
radial potential. We adopt Boyer-Lindquist coordinates $(t,r,\theta,\varphi)$,
and $r_{g}\equiv G_{\rm N}M_{\rm BH}$ represents the gravitational radius. It
is worth noting that the total mass of the cloud, $M_{\rm cloud}$, should not
exceed approximately $10\%$ of the BH mass. Beyond this threshold, the BH spin
decreases to a value that no longer satisfies the superradiant condition,
unless there is an external input of angular momentum to the BH. By
considering the relationship $M_{\rm cloud}\approx
186\Psi^{2}_{0}/(\alpha^{3}\mu)$, we observe that the field value can approach
the Planck scale:

$\Psi_{0}\approx 1.1\times 10^{16}\,{\rm
GeV}\left(\frac{\alpha}{0.2}\right)^{2}\left(\frac{M_{\rm cloud}}{0.1M_{\rm
BH}}\right)^{1/2}.$ (2)

Hereafter, we adopt $\Psi_{0}^{10\%}$ as the maximum allowable field value by
setting $M_{\rm cloud}/M_{\rm BH}=0.1$ in Eq. (2).

On the other hand, exponential growth can be halted prior to the occurrence of
a strong backreaction on the BH if there exists an interaction between the
boson cloud and other fields. In such cases, the cloud reaches a quasi-
equilibrium state where the interaction induces a steady outflow of energy,
effectively balancing the superradiant gain. Examples of such interactions
include quartic self-interactions Yoshino and Kodama (2012); Fukuda and
Nakayama (2020); Baryakhtar  _et al._ (2021); Omiya  _et al._ (2022), axion-
photon coupling Rosa and Kephart (2018); Boskovic  _et al._ (2019); Ikeda  _et
al._ (2019); Spieksma  _et al._ (2023), and electron-positron pair production
Fukuda and Nakayama (2020); Blas and Witte (2020); Siemonsen  _et al._ (2023).

## III Neutrino Emission from Scalar Clouds.

We investigate a neutrino-philic coupling between a scalar field denoted as
$\phi$ and the left-handed neutrino $\nu_{L}$, which is described using two-
component spinor notation:

$g_{\phi\nu}\,\phi\,\nu_{L}\nu_{L}.$ (3)

Here, $g_{\phi\nu}$ represents the coupling constant. The scalar field
involved in this interaction is commonly referred to as the Majoron. Its
coupling has been extensively discussed in the context of mediating neutrino
self-interactions, a comprehensive summary can be found in Ref. Berryman  _et
al._ (2022). Various constraints on the value of $g_{\phi\nu}$ exist when
considering the case of an ultralight $\phi$, including supernova (SN) 1987A
Kachelriess  _et al._ (2000); Farzan (2003), double beta decay Gando  _et al._
(2012); Brune and Päs (2019), big bang nucleosynthesis Huang  _et al._ (2018),
and the cosmic microwave background Forastieri  _et al._ (2019); Li and Xu
(2023). These studies have established an approximate upper limit of
$g_{\phi\nu}<3\times 10^{-7}$. For simplicity, we assume a universal coupling
to all mass eigenstates in the following.

Given the significant field value that scalar clouds can attain and the light
neutrino masses ($<0.12$ eV Aghanim  _et al._ (2020); Singh and Mirza (2023)),
we expect that the coupling in Eq. (3) induces a coherent oscillating mass
term $g_{\phi\nu}\phi$ for neutrinos around the cloud. The amplitude of this
oscillation $g_{\phi\nu}\phi_{0}$ is more significant than the bare mass of
neutrino $m_{\nu}$, where $\phi_{0}\equiv\Psi_{0}R_{\alpha}(r)\sin\theta$
represents the local amplitude of the scalar field. When the effective mass,
$m_{\rm eff}=g_{\phi\nu}\phi+m_{\nu}$, crosses zero, neutrinos undergo
parametric excitation. This production mechanism was previously studied during
preheating Greene and Kofman (1999, 2000), and the average production rate per
unit volume is given by

$\Gamma_{\phi\nu}\approx\frac{g_{\phi\nu}^{2}\phi_{0}^{2}\mu^{2}}{48\pi^{3}}\sqrt{\frac{\mu}{g_{\phi\nu}\phi_{0}}},$
(4)

see Supplemental Material for details. Relativistic neutrinos are produced
with an initial momentum
$|\vec{p}^{\,i}_{\nu}|\sim\sqrt{g_{\phi\nu}\phi_{0}\mu}/2$. The corresponding
de Broglie wavelength of the neutrinos, $\sim 1/|\vec{p}^{\,i}_{\nu}|$, is
significantly shorter than the size of the cloud, $\sim 1/(\alpha\mu)$. This
allows us to safely ignore the finite-size effects when using Eq. (4).

Once produced, the neutrinos immediately experience an effective force arising
from the varying mass term Uzan  _et al._ (2020). The equation of motion for
their worldline is given by:

$\frac{{\rm d}p^{\alpha}_{\nu}}{{\rm
d}t}=-\frac{1}{p^{0}_{\nu}}\Gamma^{\alpha}_{\kappa\beta}\,p_{\nu}^{\kappa}\,p_{\nu}^{\beta}-\frac{1}{2p_{\nu}^{0}}\nabla^{\alpha}m_{\rm
eff}^{2}.$ (5)

Here, $p_{\nu}^{\alpha}$ represents the $4$-momentum of the neutrinos, and
$\Gamma^{\alpha}_{\kappa\beta}$ refers the Christoffel symbols of the Kerr
metric. The spatial components of Eq. (5) contain the effective force
$-\vec{\nabla}m_{\rm eff}^{2}/(2p_{\nu}^{0})$, which accelerates the neutrinos
after their production. The final accelerated energy, denoted as
$\omega^{\nu}_{\rm acc}$, can be estimated by considering the $0$-component of
Eq. (5), resulting in $\omega^{\nu}_{\rm acc}\sim g_{\phi\nu}\phi_{0}$. It is
essential to emphasize that both the spatial gradient and temporal oscillation
of the cloud wave-function are necessary for the acceleration process, as
demonstrated in Eq. (5). Notably, neutrinos produced during the previous
crossing, having been accelerated to much higher energies than
$|\vec{p}^{\,i}_{\nu}|$, would not influence subsequent production, akin to
the pair production of millicharged particles from superconducting cavities
Berlin and Hook (2020).

Figure 1: Top panel: examples of neutrino trajectories originating
simultaneously from distances of $10\,r_{g}$ (orange) and $100\,r_{g}$ (green)
with respect to the BH. The direction of the black arrow indicates the BH
spin. The scalar force lines present at the production time and the density of
$\phi_{0}^{2}$ are depicted in gray. Bottom panel: The observed neutrino
energy spectrum $\Delta\Phi_{\nu}\equiv{\rm d}\Phi_{\nu}/({\rm
d}\cos\theta_{o}\,{\rm d}{\rm log}_{10}\omega_{\rm acc}^{\nu})$ for distant
observers located at inclination angles $\theta_{o}$. $\Phi_{\nu}^{\rm tot}$
is the total flux emitted.

To obtain the angular distribution and spectrum of the outgoing neutrino
fluxes, we perform simulations of neutrino trajectories originating from
various points within scalar clouds. Details can be found in the Supplemental
Material. The top panel of Fig. 1 presents two examples of neutrino
trajectories emitted simultaneously from distances of $10\,r_{g}$ (shown in
orange) and $100\,r_{g}$ (shown in green) in a scalar cloud background with
$\alpha=0.2$. Notably, the trajectory originating from the outer region
directly leaves the cloud in the radial direction, while the other trajectory
first curls around the BH spin axis (shown in black). Additionally, we depict
the force lines from $-\vec{\nabla}\phi^{2}$ at the production time and
density of $\phi_{0}^{2}$ in gray. The bottom panel displays the outgoing
spectrum as a function of the observer’s inclination angle $\theta_{o}$. The
flux is slightly higher when observed from nearly face-on angles
($|\cos\theta_{o}|>3/4$), forming a jet-like structure of neutrinos. This
enhanced flux is primarily due to the tendency of neutrinos produced in the
inner region of the cloud to be trapped in perpendicular directions while
maintaining acceleration along the polar axis.

Based on our simulations, we have determined the average energy of the
neutrino fluxes:

$\bar{\omega}^{\nu}_{\rm acc}\approx 2.7\,{\rm
TeV}\,\left(\frac{g_{\phi\nu}}{10^{-8}}\right)\,\left(\frac{\Psi_{0}}{10^{12}\,{\rm
GeV}}\right).$ (6)

This value is generally independent of $\alpha$, as demonstrated in the
Supplemental Material. We can estimate the differential fluxes received by a
distant observer using

$\begin{split}&\frac{{\rm d}\Phi_{\nu}}{{\rm d}\omega^{\nu}_{\rm
acc}}(\bar{\omega}^{\nu}_{\rm
acc})\approx\frac{3\int\Gamma_{\phi\nu}\,d^{3}\vec{x}}{4\pi
d^{2}\,\bar{\omega}^{\nu}_{\rm acc}}\\\ \simeq&1.2\times 10^{-7}\,{\rm
cm^{-2}s^{-1}\,GeV^{-1}}\left(\frac{g_{\phi\nu}}{10^{-8}}\right)^{1/2}\left(\frac{\Psi_{0}}{10^{12}\,{\rm
GeV}}\right)^{1/2}\\\ &\times\left(\frac{10^{-12}\,{\rm
eV}}{\mu}\right)^{1/2}\left(\frac{0.25}{\alpha}\right)^{3}\left(\frac{10\,{\rm
kpc}}{d}\right)^{2}.\end{split}$ (7)

Here, the factor $3$ accounts for the number of neutrino mass eigenstates, and
$d$ represents the distance between the BH and the observer.

We will now examine the value of $\Psi_{0}$ in the presence of neutrino
production and acceleration. This condition can be solved by ensuring that the
energy carried away by neutrinos balances the energy injected by the BH
through superradiance, i.e.,

$\Gamma_{\rm SR}^{\phi}\,M_{\rm cloud}=3\,\bar{\omega}^{\nu}_{\rm
acc}\int\Gamma_{\phi\nu}\,d^{3}\vec{x}.$ (8)

Here, $\Gamma_{\rm SR}^{\phi}$ represents the scalar superradiant rate, which
is approximately $\alpha^{8}a_{J}\mu/24$ when $\alpha\ll 1$ Detweiler (1980).
The parameter $a_{J}$ denotes the dimensionless spin parameter of the BH.
Solving Eq. (8) yields the critical value for $\Psi_{0}$:

$\begin{split}{\Psi}_{0}^{c}\approx 10^{12}\,{\rm
GeV}\left(\frac{\mu}{10^{-12}\,{\rm
eV}}\right)\left(\frac{\alpha}{0.25}\right)^{16}\left(\frac{10^{-8}}{g_{\phi\nu}}\right)^{5}\left(\frac{a_{J}}{0.9}\right)^{2}.\end{split}$
(9)

For the benchmark parameters, corresponding to a BH of mass $33\,M_{\odot}$,
substituting the saturation field value from Eq. (9) into Eq. (6) and (7)
yields steady emission of neutrinos with a typical energy scale of TeV and
fluxes approximately equal to $1.2\times 10^{-7}$ cm-2s-1GeV-1. Notably, these
values are significantly higher than the flux of atmospheric neutrinos, which
is on the order of $10^{-11}$ cm-2s-1GeV-1 at TeV Vitagliano  _et al._ (2020).
Interestingly, both the neutrino fluxes and the average energy
$\bar{\omega}^{\nu}_{\rm acc}$ during the saturation phase increase as
$g_{\phi\nu}$ decreases. This behavior arises from the fact that a lower value
of $g_{\phi\nu}$ leads to a higher critical amplitude $\Psi_{0}^{c}$.

## IV Spin Measurements and Neutrino Detection.

The key ingredients here are rapidly rotating BHs. Recent observations have
confirmed the existence of both stellar-mass BHs and supermassive BHs, with
masses ranging from a few $M_{\odot}$ to $10^{10}\,M_{\odot}$. Despite the
lower scalar mass $\mu$, which reduces the saturated neutrino fluxes and
average energy, scalar clouds surrounding supermassive BHs can still generate
and accelerate neutrinos, as long as $g_{\phi\nu}\Psi_{0}\gg m_{\nu}$ Greene
and Kofman (2000). Hence, we will consider both types of BHs in our analysis.

There are different methods for identifying stellar-mass BHs of interest, many
of which rely on electromagnetic observations. One such method involves
utilizing X-ray emissions to detect the motion of binary systems located at
substantial distances Lewin  _et al._ (1997). Spectroscopic observations, such
as iron lines, can provide further information about the spins of these BHs
Brenneman and Reynolds (2006). Another technique for identifying isolated BHs
involves microlensing Mao  _et al._ (2002); Bennett  _et al._ (2002).
Typically, these potential BH candidates are distributed throughout the Milky
Way, with a distance of $\mathcal{O}(1)$ kpc from Earth. Therefore, we adopt a
benchmark value of $d=1$ kpc for stellar-mass BHs ranging from $3\,M_{\odot}$
to $100\,M_{\odot}$.

A different approach involves gravitational-wave observations Barack  _et al._
(2019). When two BHs merge, they produce a BH remnant whose spin can be
inferred from the gravitational wave signal. These events happen at distances
typically exceeding $\mathcal{O}(10)$ Mpc. Nevertheless, for a critical field
value $\Psi_{0}^{c}$ close to $\Psi_{0}^{10\%}$, the neutrino fluxes continue
to exceed the atmospheric background even at a distance of approximately Gpc.
Consequently, we anticipate significant opportunities for neutrino detection
following such merger events. Gravitational wave observations can serve as
triggers for detecting neutrino signals, which experience delayed arrival
times due to superradiant timescales. The correlation between these two types
of observations introduces novel prospects in the field of multi-messenger
astronomy.

The remarkable advancements in Very-Long-Baseline Interferometry (VLBI)
technology have enabled the observation of supermassive BHs. The Event Horizon
Telescope, with unprecedented angular resolution, has captured horizon-scale
images of two nearby supermassive BHs: M87⋆ ($M_{\rm BH}\simeq 6.5\times
10^{9}\,M_{\odot}$, $d\simeq 16.4$ Mpc) Akiyama  _et al._ (2019), and Sgr A⋆
($M_{\rm BH}\simeq 4.3\times 10^{6}\,M_{\odot}$, $d\simeq 8.2$ kpc) Akiyama
_et al._ (2022). These results are also in favor of nearly face-on
observations and high spins of the two BHs. Moreover, there is promising
potential to employ correlations between VLBI observations and neutrino
experiments to investigate additional supermassive BHs located at greater
distances Kovalev  _et al._ (2023). An intriguing feature that can be
anticipated from supermassive BHs is the periodic modulation of the neutrino
flux. For instance, the cloud surrounding M87⋆ oscillates with a period of
$\mathcal{O}(10)$ days, a time span that is sufficiently long to be resolved.

Figure 2: Prospects for constraining the coupling $g_{\phi\nu}$ of a neutrino-
philic scalar. The stellar-mass BHs considered have masses spanning from
$3\,M_{\odot}$ to $100\,M_{\odot}$ and are assumed to be located at a distance
of $1$ kpc. The spin parameter is held constant at $a_{J}=0.8$. Previous
exclusion from SN1987A Kachelriess  _et al._ (2000); Farzan (2003) is shown
for comparison.

When the interaction between $\phi$ and neutrinos is extremely weak, a steady
state is not reached before the BH spins down. Consequently, detecting a BH
with a high spin can rule out the corresponding superradiant window mass,
similar to the constraints imposed on minimally coupled bosons or axions with
weak self-interaction Arvanitaki and Dubovsky (2011); Arvanitaki  _et al._
(2015); Brito  _et al._ (2015b); Baryakhtar  _et al._ (2017); Brito  _et al._
(2017a); Cardoso  _et al._ (2018); Davoudiasl and Denton (2019); Brito  _et
al._ (2020); Stott (2020); Ünal  _et al._ (2021); Saha  _et al._ (2022). A
quantitative criterion for this exclusion is that
${\Psi}_{0}^{c}>\Psi_{0}^{10\%}$, which implies that the cloud extracts $10\%$
of the BH energy without reaching the saturation phase. The blue region in
Fig. 2 represents this condition. In our analysis, we consider a fixed spin
parameter $a_{J}=0.8$ with the highest possible value of $\alpha=0.25$. The
lower bound of $\alpha$ is set at $0.05$. For stellar-mass BHs, the
corresponding scalar mass $\mu\equiv\alpha/(G_{N}M_{\rm BH})$, is divided into
two regions. In the low-mass region, we fix $M_{\rm BH}=100\,M_{\odot}$ and
vary $\alpha$ from $0.05$ to $0.25$. In the high-mass region, we fix
$\alpha=0.25$ and vary $M_{\rm BH}$ from $100\,M_{\odot}$ to $3\,M_{\odot}$.
We also include the constraints from the two nearby supermassive BHs.

The parameter space within the red region in Fig. 2 represents the range in
which observable neutrino fluxes from a saturating cloud are expected. In this
region, the cloud no longer exponentially extracts BH spin, thus enabling the
coexistence of a high spin BH and the saturated cloud. Therefore, we maintain
the value of $a_{J}=0.8$, while $\mu$ follows the same division as the one
used for spin measurement. The upper boundary of the blue region is determined
by requiring the neutrino fluxes in Eq. (7) to be higher than $1\%$ of the
diffusive atmospheric neutrino background Vitagliano  _et al._ (2020) at
$\bar{\omega}^{\nu}_{\rm acc}$. The atmospheric neutrino has already been well
measured by IceCube Aartsen  _et al._ (2015), Super-Kamiokande Richard  _et
al._ (2016), ANTARES Albert  _et al._ (2018), and Baikal-GVD Allakhverdyan
_et al._ (2023). A target source with neutrino fluxes at $1\%$ of the
diffusive background can be easily distinguished with an angular resolution of
$<10^{\circ}$ Aartsen  _et al._ (2014); Aiello  _et al._ (2019); Aartsen  _et
al._ (2019, 2021); Fiorillo  _et al._ (2023); Abbasi  _et al._ (2023).

## V Discussion.

Various intriguing phenomena can arise in the region of strong fields,
particularly related to particle production. These phenomena have been
previously examined in the preheating stage of the early universe Zeldovich
and Starobinsky (1971); Kofman  _et al._ (1997); Greene and Kofman (1999,
2000) and in the context of strong field quantum electrodynamics Schwinger
(1951); Grib  _et al._ (1994). Rotating BHs provide an excellent platform for
studying particle production, as superradiant clouds can reach field values
close to the Planck scale.

In this study, we propose a novel approach to investigate the interaction
between ultralight bosonic fields and neutrinos through BH superradiance.
Superradiant scalar clouds play dual roles in the generation and acceleration
of neutrinos, necessitating both periodic oscillation and spatial gradient of
the cloud wave-functions. The observation prospects can be examined from two
perspectives. First, BH spin measurements can exclude negligible interactions
with low values of $g_{\phi\nu}$. Second, neutrino fluxes from a target BH
with high spin can constrain the region of strong interaction. Our findings
demonstrate that the emitted neutrino fluxes can significantly surpass the
diffusive background, both for stellar-mass BHs and supermassive BHs.

Apart from interactions involving a hidden scalar, it is also natural to
consider interactions with a vector field in the presence of new gauge groups
Georgi and Glashow (1974); Pati and Salam (1974); Mohapatra and Pati (1975);
Fritzsch and Minkowski (1975); Georgi (1975). The vector cloud can generate
neutrinos through Schwinger pair production Schwinger (1951); Grib  _et al._
(1994) and further accelerate them using the electric fields of the vector.
However, the couplings between vector fields and neutrinos are subject to
strict constraints due to their coupling to other standard model fermions
Graham  _et al._ (2016); Pierce  _et al._ (2018); Abbott  _et al._ (2022b);
Shaw  _et al._ (2022); Xue  _et al._ (2022); Ge and Pasquini (2022); Yu  _et
al._ (2023) or the non-conservation of currents Laha  _et al._ (2014); Bakhti
and Farzan (2017); Escudero  _et al._ (2019); Bahraminasr  _et al._ (2021);
Dror (2020); Ekhterachian  _et al._ (2021); Singh  _et al._ (2023). Thus
sophisticated model building is necessary in the neutrino sector to yield
observable consequences.

Notice that the neutrino production from a bosonic cloud can also apply to
other coupled fermions in the hidden sector. In addition, dark matter
particles around the cloud can be directly accelerated to higher energies.
Consequently the superradiant cloud can serve as a source for boosted dark
matter, offering potential for detection through direct detection experiments
and neutrino detectors D’Eramo and Thaler (2010); Huang and Zhao (2014);
Agashe  _et al._ (2014); Kopp  _et al._ (2015); Bhattacharya  _et al._ (2017);
Kamada  _et al._ (2018a); Kachulis  _et al._ (2018); Chatterjee  _et al._
(2018); Kamada  _et al._ (2018b); McKeen and Raj (2019); Argüelles  _et al._
(2020); Kamada and Kim (2020); Berger  _et al._ (2021); Abi  _et al._ (2020).
The specific incoming directions of these particles can induce a daily
modulation of signals due to scattering with Earth materials Ge  _et al._
(2021); Fornal  _et al._ (2020); Chen  _et al._ (2023b); Xia  _et al._ (2022);
Cui  _et al._ (2022); Qiao  _et al._ (2023) or directional detectors Vahsen
_et al._ (2021).

###### Acknowledgements.

We are grateful to Mauricio Bustamante, Yuxin Liu, and Xin Wang for useful
discussions. This work is supported by VILLUM FONDEN (grant no. 37766), by the
Danish Research Foundation, and under the European Union’s H2020 ERC Advanced
Grant “Black holes: gravitational engines of discovery” grant agreement no.
Gravitas–101052587. X.X. is supported by Deutsche Forschungsgemeinschaft under
Germany’s Excellence Strategy EXC2121 “Quantum Universe” - 390833306. Views
and opinions expressed are however those of the author only and do not
necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them. This work was supported by FCT (Fundação para a Ciência
e Tecnologia I.P, Portugal) under project No. 2022.01324.PTDC. This project
has received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
101007855.

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Supplemental Material: Black Holes as Neutrino Factories

###  .1 Particle Production from Time-Varying Backgrounds

We begin by considering a bosonic field $\chi$ in a time-varying background
Zeldovich and Starobinsky (1971); Kofman  _et al._ (1997). The equation of
motion in the frequency domain can be expressed as

$\ddot{\chi}_{k}+\omega_{k}(t)^{2}\chi_{k}=0.$ (S1)

Here, $\omega_{k}(t)$ represents the time-varying energy, which is modulated
by the background fields to which $\chi$ couples. The solutions to Eq. (S1) in
the adiabatic representation are given by Kofman  _et al._ (1997)

$\chi_{k}(t)=\frac{\alpha_{k}(t)}{\sqrt{2\omega_{k}}}e^{-i\int^{t}_{0}\omega_{k}dt}+\frac{\beta_{k}(t)}{\sqrt{2\omega_{k}}}e^{+i\int^{t}_{0}\omega_{k}dt}.$
(S2)

The coefficients $\alpha_{k}$ and $\beta_{k}$ satisfy the following equations:

$\dot{\alpha}_{k}=\frac{\dot{\omega}_{k}}{2\omega_{k}}e^{+2i\int^{t}_{0}\omega_{k}dt}\beta_{k},\quad\dot{\beta}_{k}=\frac{\dot{\omega}_{k}}{2\omega_{k}}e^{-2i\int^{t}_{0}\omega_{k}dt}\alpha_{k},$
(S3)

and they adhere to the normalization condition
$|\alpha_{k}|^{2}-|\beta_{k}|^{2}=1$. Initially, at $t=0$, the vacuum state is
defined as $\alpha_{k}=1$ and $\beta_{k}=0$. The quantities $\alpha_{k}(t)$
and $\beta_{k}(t)$ correspond to the coefficients of the Bogoliubov
transformation of the creation and annihilation operators, respectively, which
diagonalize the Hamiltonian at each moment in time $t$. The particle
occupation number for a momentum mode $k$ is given by $n_{k}=|\beta_{k}|^{2}$,
which yields the particle number density per unit volume as

$n_{\chi}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}dkk^{2}\left|\beta_{k}\right|^{2}.$
(S4)

When $|\beta_{k}|\ll 1$ holds true, we can approximate the solution for Eq.
(S3) as

$\beta_{k}\simeq\int_{0}^{t}dt^{\prime}\frac{\dot{\omega}}{2\omega}\exp\left(-2i\int_{0}^{t^{\prime}}dt^{\prime\prime}\omega\left(t^{\prime\prime}\right)\right).$
(S5)

Consequently, efficient particle production is expected when the non-adiabatic
condition is satisfied:

$\left|\frac{\dot{\omega}}{\omega^{2}}\right|\gg 1.$ (S6)

Using the interaction term $g_{\phi\chi}^{2}\phi^{2}\chi^{2}/2$ as an example,
we consider the following scenario: $\phi=\phi_{0}\cos(\mu t)$ represents the
coherently oscillating background scalar with amplitude $\phi_{0}$ and
frequency $\mu$, and $g_{\phi\chi}$ represents the coupling constant. The
frequency relation for $\chi$ is Kofman  _et al._ (1997)

$\omega_{\chi}^{2}=k^{2}+m_{\chi}^{2}+g_{\phi\chi}^{2}\phi_{0}^{2}\cos^{2}(\mu
t).$ (S7)

Here, $m_{\chi}$ denotes the bare mass term of $\chi$. In this context, our
focus lies on the region where $g_{\phi\chi}^{2}\phi_{0}^{2}/\mu^{2}$ is
substantially greater than $1$, while $m_{\chi}$ can be considered negligible.
Within a narrow region of $\phi$ during a single oscillation period, momentum
values below the typical scale $k_{\star}=\sqrt{g_{\phi\chi}\phi_{0}\mu}/2$
satisfy the non-adiabatic condition (S6). Consequently, the occupation number
$n_{k}$ for $k<k_{\star}$ is exponentially produced with a time dependence of
approximately $\exp(2\mu\tau_{k}t)$, where $\tau_{k}\approx\mathcal{O}(0.1)$
Kofman  _et al._ (1997).

Fermionic fields with a Yukawa coupling $g_{\phi\nu}\phi\nu_{L}\nu_{L}$ and a
bare mass $m_{\nu}$ exhibit a frequency relation for the mode function given
by Greene and Kofman (1999, 2000)

$\omega_{\nu}^{2}=k^{2}+\left(m_{\nu}+g_{\phi\nu}\phi_{0}\cos(\mu
t)\right)^{2}.$ (S8)

In the near black hole region where the resonance parameter $q\equiv
g_{\phi\nu}^{2}\phi_{0}^{2}/\mu^{2}$ is significantly high and
$g_{\phi\nu}\phi_{0}\gg m_{\nu}$, the non-adiabatic condition is met when the
effective mass term, $m_{\rm eff}\equiv m_{\nu}+g_{\phi\nu}\phi_{0}\cos(\mu
t)$, crosses zero. During each crossing, the background ground instantly
contributes to the $k-$mode with a factor of $|\beta_{k}|^{2}=e^{-\pi
k^{2}/(g_{\phi\nu}\phi_{0}\mu)}+\cdots$ Greene and Kofman (1999, 2000), where
$\cdots$ contains the terms related to previously produced fermions.
Exponential growth is excluded due to Pauli blocking. Instead, a fermion
sphere with a radius of approximately
$k_{\star}=\sqrt{g_{\phi\nu}\phi_{0}\mu}/2$ is formed immediately after a
single kick. In cases where neutrinos are produced from a superradiant scalar
background, neutrinos produced during the previous kick are accelerated to
much higher energy scales and will not obstruct subsequent production.
Consequently, the average production rate per unit volume is estimated as:

$\Gamma_{\phi\nu}\approx\frac{1}{2\pi^{2}}\frac{k_{\star}^{3}}{3}\frac{\mu}{\pi}=\frac{g_{\phi\nu}^{2}\phi_{0}^{2}\mu^{2}}{48\pi^{3}}\sqrt{\frac{\mu}{g_{\phi\nu}\phi_{0}}}.$
(S9)

Here, the factor $\pi/\mu$ accounts for the time between two crossings.
Furthermore, the de Broglie wavelength of the initially produced fermions,
approximately $2\pi/k_{\star}$, is considerably shorter than the cloud size.
Thus, the finite size of the cloud can be disregarded when considering the
production rate.

The scalar cloud displays an oscillatory component in its wavefunction, which
exhibits a dependency on the azimuthal angle, i.e., $\phi\propto\cos(\mu
t-\varphi)$. This dependence arises from the orbital angular momentum of the
cloud. Notably, at a specific time $t$, the non-adiabatic condition is
triggered on a plane that is approximately aligned with $\varphi=\mu
t\pm\pi/2$. As a result, the cloud oscillation induces a periodic rotation of
the production plane around the spin axis of the black hole.

One key difference between boson and fermion production is the requirement for
the bare mass term. In Eq. (S8), the effective mass is the linear sum of the
bare mass and the oscillating term, so only $g_{\phi\nu}\phi_{0}\gg m_{\nu}$
is necessary to trigger the parametric excitation Greene and Kofman (2000). On
the other hand, for bosons, $m_{\chi}$ needs to be smaller than $k_{\star}$
due to their similar roles in Eq. (S8). In our cases, due to the smallness of
the neutrino mass and the significant field value of the scalar cloud,
neutrino production can be achieved for both stellar-mass black holes and
supermassive black holes with different ranges of $\mu$.

###  .2 Neutrino Trajectory

Immediately after production, neutrinos start propagating under the scalar
background outside the black hole. This is effectively equivalent to
considering geodesics with a varying mass term denoted by $m_{\rm
eff}=g_{\phi\nu}\phi+m_{\nu}$. The worldline action for the neutrino is given
by:

$S_{\nu}=-\int{\rm d}\tau\,m_{\rm
eff}\sqrt{-g_{\alpha\beta}u^{\alpha}_{\nu}u^{\beta}_{\nu}},$ (S10)

where $g_{\alpha\beta}$ is the Kerr metric, and $u^{\alpha}_{\nu}$ is the
$4$-velocity of the neutrino particle. From Eq. (S10), we obtain the
corresponding Euler-Lagrange equation:

$\frac{{\rm d}u_{\nu}^{\alpha}}{{\rm
d}\tau}=-\Gamma^{\alpha}_{\kappa\beta}u_{\nu}^{\kappa}u_{\nu}^{\beta}-(g^{\alpha\beta}+u_{\nu}^{\alpha}u_{\nu}^{\beta})\frac{\nabla_{\beta}m_{\rm
eff}}{m_{\rm eff}},$ (S11)

where $\tau$ is the proper time, and $\Gamma^{\alpha}_{\kappa\beta}$ are the
Christoffel symbols of the Kerr metric. By using the relations
$p^{\alpha}_{\nu}=m_{\rm eff}u^{\alpha}_{\nu}$ and ${\rm d}t=u^{0}_{\nu}{\rm
d}\tau$, we can rewrite Eq. (S11) as:

$\frac{{\rm d}p^{\alpha}_{\nu}}{{\rm
d}t}=-\frac{1}{p^{0}_{\nu}}\Gamma^{\alpha}_{\kappa\beta}\,p_{\nu}^{\kappa}\,p_{\nu}^{\beta}-\frac{1}{2p_{\nu}^{0}}\nabla^{\alpha}m_{\rm
eff}^{2}.$ (S12)

Here, $p^{0}_{\nu}$ satisfies the on-shell condition $p^{0}_{\nu}=\sqrt{m_{\rm
eff}^{2}+|\vec{p}_{\nu}|^{2}}$. The time-component of Eq. (S12) shows that
$p_{\nu}^{0}$ will be accelerated to the same order as $m_{\rm eff}$ after a
time-scale of approximately $1/\mu$.

The two terms on the right-hand side of Eq. (S12) correspond to gravitational
lensing and scalar force Uzan  _et al._ (2020), respectively. We compare their
relative contributions using Cartesian coordinates $(t,x,y,z)$:

$\displaystyle-\frac{\Gamma^{\alpha}_{\kappa\beta}\,p_{\nu}^{\kappa}\,p_{\nu}^{\beta}}{m^{2}_{\rm
eff}/r_{g}}$ $\displaystyle\approx$
$\displaystyle-\frac{r_{g}^{2}}{r^{2}}\,\left(\mathcal{O}(1)+\mathcal{O}(\frac{r_{g}}{r})\right),$
(S13) $\displaystyle-\frac{\vec{\nabla}m^{2}_{\rm eff}}{m^{2}_{\rm
eff}/r_{g}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\,\hat{r}-\frac{2\,r_{g}}{r\cos(\alpha
t-\phi)\sin\theta}\,\hat{n}_{\perp}+\cdots.$ (S14)

Here, $\cdots$ represents the influence of the neutrino bare term $m_{\nu}$,
which is negligible in the region where $g_{\phi\nu}\phi_{0}\gg m_{\nu}$.
$\hat{n}_{\perp}\equiv(\cos(\alpha t),\sin(\alpha t),0)$ is a unit directional
vector rotating on the $x-y$ plane. It is clear that the scalar force
dominates over gravitational lensing in the majority part of the cloud. Thus,
in the simulation, we neglect the effect of the latter.

When neutrinos are produced at larger radii, the first term on the right-hand
side of Eq. (S14) dominates, accelerating neutrinos along the radial
direction. This causes the nearly isotropic distribution of fluxes in Fig. 1
of the main text for $|\cos\theta_{o}|<3/4$. On the other hand, the second
term on the right-hand side of Eq. (S14) is non-negligible in the inner region
of the cloud. It traps the trajectories in regions with small inclination
angles ($|\cos\theta_{o}|>3/4$), causing an excess of neutrino flux.

###  .3 Trajectory and Flux Simulation

In this section, we will discuss the simulation of neutrino trajectories and
the recording of the flux at infinity. For convenience, we introduce
dimensionless quantities:

$(\tilde{t},\tilde{x},\tilde{y},\tilde{z})\equiv(t/r_{g},x/r_{g},y/r_{g},z/r_{g}),\qquad\tilde{p}^{\alpha}_{\nu}\equiv\frac{p^{\alpha}_{\nu}}{g_{\phi\nu}\Psi_{0}},\quad\tilde{m}_{\rm
eff}\equiv\frac{m_{\rm eff}}{g_{\phi\nu}\Psi_{0}}.$ (S15)

The trajectories with these new definitions become:

$\frac{{\rm d}\tilde{x}^{\alpha}}{{\rm
d}\tilde{t}}=\frac{\tilde{p}_{\nu}^{\alpha}}{\tilde{p}^{0}_{\nu}},\qquad\frac{{\rm
d}\tilde{p}_{\nu}^{\alpha}}{{\rm
d}\tilde{t}}=-\frac{\nabla^{\alpha}\tilde{m}^{2}_{\rm
eff}}{2\tilde{p}^{0}_{\nu}},\qquad\tilde{p}^{0}_{\nu}=\sqrt{\tilde{m}^{2}_{\rm
eff}+|\vec{\tilde{p}}_{\nu}|^{2}}.$ (S16)

We employ a Monte-Carlo simulation to generate neutrino events, where the
initial positions of events are distributed within $2<r/r_{g}<40/\alpha^{2}$.
The upper range is chosen to be much higher than the cloud size. We use the
generalized acceptance-rejection method Casella  _et al._ (2004) for the
distribution weighted by the production rate
$\Gamma_{\phi\nu}\propto\phi_{0}^{3/2}$. Without loss of generality, we set
the initial time of each event at $\tilde{t}=0$. We relocate the azimuthal
angle $\varphi$ to either $\pi/2$ or $-\pi/2$, depending on the initial sign
of $\varphi$.

To solve the trajectory of each event using Eq. (S16), we need to specify the
initial momentum and the neutrino bare mass. The oscillatory part of the mass
takes a value to cancel the bare mass, resulting in a vanishing
$\tilde{m}_{\rm eff}$. Both the initial momentum with a value below
$\tilde{k}_{\star}=\sqrt{\phi_{0}\mu/(g_{\phi\nu}\Psi_{0}^{2})}/2$ and the
dimensionless neutrino mass $m_{\nu}/(g_{\phi\nu}\Psi_{0})$ are several orders
of magnitude lower than $1$ in the relevant parameter space of this work. We
test the convergence of the trajectories, which demonstrates independence of
the initial momentum value and directions. This is because the scalar
background immediately contributes to $\mathcal{O}(1)$ values for both
$\tilde{m}_{\rm eff}$ and $|\vec{\tilde{p}}_{\nu}|$. Hence, in practice, we
choose a small enough value of momentum with random directions and vanishing
$\tilde{m}_{\rm eff}$ as the initial condition.

The end of each trajectory is set to be at $\tilde{t}=400/\alpha^{2}$, where
both the momentum and outgoing direction $(\theta,\varphi)$ converge to nearly
constant values. We assume that the observation takes a time longer than the
oscillation period $2\pi/\mu$, allowing us to neglect the azimuthal angle
$\varphi$ dependence of the event. The final momentum and polar angle
information are then recorded as

$\omega_{\rm
acc}^{\nu}/(g_{\phi\nu}\Psi_{0})=\tilde{p}^{0}_{\nu},\qquad\cos\theta_{o}=\tilde{p}^{z}_{\nu}/\tilde{p}^{0}_{\nu},$
(S17)

respectively. In Fig. 1 of the main text, we show the results for
$\alpha=0.2$, while in Fig. S1, we present the results for $\alpha=0.05$ and
$0.4$ for comparison. The average momentum turns out to be universal for
different $\alpha$.

Figure S1:  Same as Fig. 1 of the maintext with $\alpha=0.05$ and $0.4$.