aainstitutetext: Tsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, Chinabbinstitutetext: Key Laboratory for
Particle Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai
200240, Chinaccinstitutetext: Department of Physics, Yantai University, Yantai
264005, China

# Boosted Dark Matter From Centaurus A
and Its Detection

Chen Xia a,b Chuan-Yang Xing c Yan-Hao Xu<EMAIL_ADDRESS>chuan-yang-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Dark matter can be boosted by high energy particles in astrophysical
environments through elastic scattering. We study the production of boosted
dark matter via scattering with electrons in the relativistic jet of the
closest active galactic nucleus, Centaurus A, and its detection in the Super-
Kamiokande experiment. Since there are a huge number of electrons in the jet
and dark matter is extremely dense around the supermassive black hole that
powers the jet, the number of boosted dark matter is tremendously large.
Compared to boosted dark matter from blazars, the dark matter flux from
Centaurus A is enhanced due to the proximity of Centaurus A. The constraint on
dark matter-electron scattering cross section set by Super-Kamiokande is more
stringent, down to $\sim 10^{-36}\,\mathrm{cm}^{2}$ for $\mathrm{MeV}$ dark
matter.

##  1 Introduction

With enormous cosmological and astrophysical observations suggesting the
existence of dark matter (DM), the nature of DM remains a mystery
Bertone:2004pz ; Bauer:2017qwy ; Cooley:2021rws . DM in the form of exotic
particles is well motivated by theories, notably the Weakly Interacting
Massive Particles (WIMPs) paradigm, which predicts DM particles with mass and
coupling strength similar to the electroweak sector of the Standard Model
(SM). Inspired by WIMPs, numerous DM direct detection experiments
SENSEI:2019ibb ; PandaX-4T:2021bab ; CDEX:2022fig ; LZ:2022lsv ;
SuperCDMS:2020aus ; DarkSide-50:2022qzh ; XENON:2023cxc  have been established
and improved in the past decades. However, still no conclusive DM signal has
been detected. The constraints imposed by direct detection experiments on
WIMPs have become increasingly stringent.

In the past years, sub-GeV mass scale DM has attracted increasing attention
SENSEI:2019ibb ; XENON:2019gfn ; SuperCDMS:2020aus ; PROSPECT:2021awi ;
COHERENT:2021pvd ; CDEX:2022kcd ; Super-Kamiokande:2022ncz ; CRESST:2022lqw ;
DAMIC-M:2023gxo . Inside conventional direct detection experiments, sub-GeV
halo DM particles are too light to trigger detectable signals and are much
less constrained. To enhance the detectability of sub-GeV DM in the direct
detection experiments, various kinds of boosted DM scenarios have been
proposed, such as DM boosted by annihilation of heavy DM Agashe:2014yua ,
solar reflection An:2017ojc ; Emken:2021lgc ; An:2021qdl , and high-energy
cosmic rays in the Milky Way Bringmann:2018cvk ; Ema:2018bih ;
Cappiello:2019qsw ; Ge:2020yuf ; Guo:2020drq ; Xia:2020apm ; Xia:2021vbz ;
Wang:2023wrx . The boosted DM carries enough energy to deposit recoil energy
above the threshold in DM direct detection experiments PandaX-II:2021kai ;
CDEX:2022fig  or neutrino experiments Super-Kamiokande:2022ncz ;
PROSPECT:2021awi .

Recently, the scenario of blazar boosted DM has shown the potential to derive
more stringent constraints Wang:2021jic ; Calabrese:2021src ; Bhowmick:2022zkj
. Blazar is a type of active galactic nucleus (AGN) with a relativistic jet
directed very nearly towards the Earth. AGN is theorized to contain a
supermassive black hole in the center which produces the jet. The relativistic
particles in the jet can boost DM particles around the supermassive black hole
through leptonic or hadronic scattering. The boosted DM propagating to the
Earth can be detected in underground experiments.

In this work, we propose that the boosted DM from the closest AGN, the AGN in
the galaxy Centaurus A (Cen A), can dominate over that from blazars studied in
Refs. Wang:2021jic ; Calabrese:2021src  assuming DM is coupled to electrons.
Note that Cen A is not a blazar since its jets do not point towards the Earth.
Instead, the viewing angle between the jet axis and our line-of-sight to Cen A
is quite large, which is measured to be $\theta_{\mathrm{LOS}}\sim
12^{\circ}\mathrm{-}45^{\circ}$ Hardcastle:2003ye . Inside the jet of Cen A,
the majority of charged particles propagate along the jet axis. Since the
particles in the jet are highly relativistic, the upscattered light DM is
primarily forward. Therefore, the boosted DM particles mostly travel along the
jet as well. The number of boosted DM particles with a large collimation
angle, $\sim\theta_{\mathrm{LOS}}$ with respect to the jet axis, so that they
can travel to the Earth, is suppressed. That is, the fraction of boosted DM
from Cen A to the Earth is lower than the fraction from blazars. However, we
note that Cen A is much closer to us than blazars. The luminosity distance of
Cen A is $d_{L}\simeq 3.7\,\mathrm{Mpc}$ Harris:2009wj ; Zhang:2023ewt . For
comparison, the luminosity distance of the blazar BL Lacertae (BL Lac) studied
in Refs. Wang:2021jic ; Calabrese:2021src  is $\simeq 322.7\,\mathrm{Mpc}$,
around two orders of magnitude further than Cen A. Therefore, we can expect a
four-order-of-magnitude enhancement of DM flux from Cen A at the Earth than
that from BL Lac. This can compensate for the suppression of boosted DM moving
at large angles within the jet. The astrophysical environments of Cen A and BL
Lac are similar Boettcher:2013wxa ; Banik:2020ffo , so we can expect the total
amounts of boosted DM produced in these two AGNs to be of the same order.
Consequently, with the suppression from viewing angles and the enhancement
from luminosity distances taken into account, the flux of boosted DM from Cen
A is about two orders of magnitude higher than that from blazars. The
constraints on the DM-electron scattering cross section from the Super-
Kamiokande (Super-K) experiment can be improved by around one order of
magnitude. In addition, the Earth attenuation effect on DM particles before
reaching the underground detector is considered in this work. This effect
obstructs DM particles with large cross sections, leading to an upper boundary
of the exclusion region.

This paper is organized as follows: In section 2, we introduce the
distribution of electrons inside the jet of Cen A. In section 3, we calculate
the flux of boosted DM. The attenuation of boosted DM inside the Earth is
carried out in section 4. Then, we analyze the detection of the boosted DM in
Super-K detector and derive constraints in section 5. We summarize our results
in section 6.

##  2 The Relativistic Jet of Centaurus A

The distribution of high-energy electrons in the relativistic jet is an
essential ingredient in the production of boosted DM. Our understanding of the
distribution of electrons is based on electromagnetic observations of the core
region of Centaurus A (Cen A). The observations span a range from microwaves
to $\gamma$-rays, and are typically plotted in energy versus wavelength, known
as the spectral energy distribution (SED). Similar to other AGNs, the SED of
Cen A exhibits two peaks (see in e.g. Ref. Banik:2020ffo ): the lower-energy
one is located in the infrared band $(\simeq 1\,\mathrm{eV})$, and the higher-
energy peak is in the $\gamma$-ray region $(\simeq 1\,\mathrm{MeV})$. These
two peaks are theorized to have different physical origins Ghisellini:1998it ;
Moderksi:2005jw ; Boettcher:2006pd ; Celotti:2007rb ; Finke:2008pe ;
Boettcher:2013wxa ; Blandford:2018iot ; Cerruti:2020lfj . The first peak is
dominated by the synchrotron radiation of relativistic charged particles
within the jet, where strong tangled magnetic fields exist. On the other hand,
the second peak is due to the inverse Compton scattering of high-energy
particles with background photons in the jet.

Based on these two interactions, there can be various models to reproduce the
two-peak SEDs. Firstly, the dominant types of charged particles can be
different. In a leptonic model, the electromagnetic radiation is mainly
produced by electrons, whereas in a hadronic model, it results dominantly from
protons. In addition, the number of emission regions within the jet can vary.

Here we employ the simple two-zone model from Ref. Banik:2020ffo  for the
relativistic jet of Cen A. In this model, one of the emission zones locates in
the inner jet that is close to the supermassive black hole at the center of
Cen A. The other emission region is considerably distant $(\sim\mathrm{kpc})$
from the inner jet. Since DM is dense in the region near the supermassive
black hole (see Sec. 3), we can concentrate on the distributions of charged
particles in the core emission region and neglect the $\mathrm{kpc}$ emission
region. In the core emission region, the dominant charged particles are
electrons.

The core emission region is modeled as a spherical blob with radius
$R^{\prime}_{b}$. Note that in this paper, we use primed symbols to denote
quantities in the blob’s comoving frame and use non-primed ones for those in
the observer’s frame. The blob propagates outwards along the jet
relativistically with a boost factor $\Gamma_{B}$ and velocity
$\beta_{B}=\sqrt{1-\Gamma_{B}^{-2}}$. In the blob’s comoving frame, electrons
are homogeneously distributed within the blob and move isotropically. The
spectrum of the electrons is assumed to follow a broken power-law
Banik:2020ffo ,

$\frac{dn^{\prime}_{e}}{dE^{\prime}_{e}}=\begin{cases}k_{e}\left(\frac{E^{\prime}_{e}}{m_{e}}\right)^{-\alpha_{1}},&\gamma^{\prime}_{e,{\min}}\leq
E^{\prime}_{e}/m_{e}\leq\gamma^{\prime}_{b}\\\
k_{e}(\gamma^{\prime}_{b})^{\alpha_{2}-\alpha_{1}}\left(\frac{E^{\prime}_{e}}{m_{e}}\right)^{-\alpha_{2}},&\gamma^{\prime}_{b}\leq
E^{\prime}_{e}/m_{e}\leq\gamma^{\prime}_{e,{\max}}\end{cases},$ (1)

where $\gamma^{\prime}_{e,{\min}}$ and $\gamma^{\prime}_{e,{\max}}$ denote the
minimal and maximal boost factors of electrons in the blob, respectively, and
$\gamma^{\prime}_{b}$ represents the break boost factor of the spectrum. The
normalization constant $k_{e}$ can be determined by the comoving luminosity of
electrons, $L^{\prime}_{e}$, in the blob’s comoving frame Banik:2019jlm ,

$L^{\prime}_{e}=\pi R^{\prime
2}_{B}\beta_{B}\int_{\gamma^{\prime}_{e,{\min}}m_{e}}^{\gamma^{\prime}_{e,{\max}}m_{e}}E^{\prime}_{e}\frac{dn^{\prime}_{e}}{dE^{\prime}_{e}}dE^{\prime}_{e}.$
(2)

The distribution of electrons in the observer’s frame is connected with Eq.
(1) by a Lorentz boost,

$\frac{d^{2}n_{e}}{dE_{e}d\Omega_{e}}=\Gamma_{B}\left|\begin{array}[2]{cc}\frac{\partial
E^{\prime}_{e}}{\partial E_{e}}&\frac{\partial E^{\prime}_{e}}{\partial u}\\\
\frac{\partial u^{\prime}}{\partial E_{e}}&\frac{\partial u^{\prime}}{\partial
u}\end{array}\right|\frac{d^{2}n^{\prime}_{e}}{dE^{\prime}_{e}d\Omega^{\prime}_{e}}.$
(3)

Here $u\equiv\cos\theta_{e}$ ($u^{\prime}\equiv\cos\theta^{\prime}_{e}$)
represents the cosine of the angle between the electron momentum and the jet
axis in the observer’s (blob’s comoving) frame. The factor $\Gamma_{B}$ arises
from the length contraction effect of relativistically moving objects. Using
the assumption that the electrons are isotropically distributed in the blob
frame,
$\frac{d^{2}n^{\prime}_{e}}{dE^{\prime}_{e}d\Omega^{\prime}_{e}}=\frac{1}{4\pi}\frac{dn^{\prime}_{e}}{dE^{\prime}_{e}}$,
we can calculate the explicit expression of the electron distribution in the
observer’s frame as follows,

$\frac{d^{2}n_{e}}{dE_{e}d\Omega_{e}}=\frac{1}{4\pi}\frac{\beta_{e}}{\sqrt{(1-\beta_{B}\beta_{e}u)^{2}-(1-\beta_{B}^{2})(1-\beta_{e}^{2})}}\frac{dn^{\prime}_{e}}{dE^{\prime}_{e}}(E^{\prime}_{e}),$
(4)

where $\beta_{e}=\sqrt{1-\left(\frac{E_{e}}{m_{e}}\right)^{-2}}$ is the
velocity of the electron and
$E^{\prime}_{e}=\Gamma_{B}(1-\beta_{B}\beta_{e}u)E_{e}$.

Note that the radius of the blob is smaller than the length of the inner jet.
In this simple blob model, electrons are predominantly distributed inside the
blob, and the number of electrons outside the blob is negligible. Therefore,
in addition to the distribution of electrons in Eq. (4), we also need to know
the position of the blob to calculate the boosted DM flux, since the number
density of DM particles varies at different locations. We assume the complete
picture for the jet is that the blob is propagating outwards within the inner
jet. When a blob reaches the boundary of the inner jet, it has lost most of
its energy and a new blob is born near the supermassive black hole. In this
picture, there is only one blob at a time. The inner jet of Cen A has been
imaged by the Event Horizon Telescope EventHorizonTelescope:2021iqj  with high
resolution. According to the imaged jet structure, the length of the core
emission region is approximately $L_{\mathrm{core}}\simeq 500R_{\mathrm{S}}$,
where $R_{\mathrm{S}}$ denotes the Schwarzschild radius of the supermassive
black hole. Since the speed of the blob is close to the speed of light, the
lifetime of a blob can be estimated as $\sim 10^{5}\,\mathrm{s}$. Therefore,
to calculate the number of boosted DM over a period as long as years, we can
effectively use the average number density within the whole inner jet, which
is given by

$\frac{d^{2}n_{e,\mathrm{eff}}}{dE_{e}d\Omega_{e}}=\frac{R_{B}}{L_{\mathrm{core}}}\frac{d^{2}n_{e}}{dE_{e}d\Omega_{e}}=\frac{R^{\prime}_{B}/\Gamma_{B}}{L_{\mathrm{core}}}\frac{d^{2}n_{e}}{dE_{e}d\Omega_{e}},$
(5)

Note that the averaging factor $R_{B}/L_{\text{core}}$ is not considered in
the previous work on blazar boosted DM Wang:2023wrx , so our calculation is
more conservative.

Figure 1: Effective distribution of electrons concerning the incident angle
$\theta_{e}$ with energy larger than $E_{0}$. The minimum $E_{0}$ is taken to
be $0.01\,\mathrm{GeV}$, $0.1\,\mathrm{GeV}$ and $1\,\mathrm{GeV}$ for the
red, yellow and green curve.

Note again that the relativistic jet of Cen A is not directed towards the
Earth. To produce boosted DM propagating to the Earth, the oblique angle
between the momentum of boosted DM and the jet axis should be
$\theta_{\chi}=\theta_{\mathrm{LOS}}$, where $\theta_{\mathrm{LOS}}$ is the
viewing angle between the jet axis and our line-of-sight to Cen A. This angle
$\theta_{\chi}$ is determined by the incident angle $\theta_{e}$ of the
initial state electron and the scattering angle. To get a large
$\theta_{\chi}$, we need a large incident angle $\theta_{e}$ or a large
scattering angle. Here we firstly investigate the distribution of electrons
with respect to the incident angle $\theta_{e}$ between the electron momentum
and the jet axis. For this purpose, we define the effective distribution of
electrons with energy exceeding a minimum $E_{0}$ as,

$\frac{dn_{e,\mathrm{eff}}}{d\Omega_{e}}(E_{e}>E_{0})=\int_{E_{0}}\frac{d^{2}n_{e,\mathrm{eff}}}{dE_{e}d\Omega_{e}}dE_{e}.$
(6)

According to Eq. (5) and the parameters about the jet of Cen A adopted from
Ref. Banik:2020ffo  (see Tab. 1 below), in Fig. 1, we show the effective
distribution versus the incident angle $\theta_{e}$ with three different
minima $E_{0}$. We can see that most electrons are propagating along the jet
axis. This is because the Lorentz boost from the blob’s comoving frame to the
observer’s frame is in this direction. We note that the portion of electrons
traveling at a relatively large angle against the jet axis is non-negligible.
For example, more than one-third of electrons have an incident angle
$\theta_{e}>15^{\circ}$, and around one percent of electrons propagate with
$\theta_{e}>42^{\circ}$. These electrons with large incident angles can boost
DM particles with large oblique angles, allowing them to travel to the Earth.
Therefore, compared to the scenario where DM is boosted in blazars
Wang:2021jic ; Granelli:2022ysi , the fraction of boosted DM propagating
towards the Earth from Cen A due to large viewing angle is suppressed by one
to two orders of magnitude at most.

##  3 Boosted Dark Matter Flux

Another essential ingredient for calculating boosted DM flux is the
distribution of DM particles in Cen A. Around supermassive black holes that
grow adiabatically, DM is accreted to form a dense spike Gondolo:1999ef .
Consider a DM halo model with an inner cusp before the black hole formation,
and the initial density profile follows a simple power-law, $\rho(r)\propto
r^{-\gamma}$, where $\gamma$ is the power-law index. As the formation of the
supermassive black hole, the DM profile will evolve to be Gondolo:1999ef ,

$\rho^{\prime}(r)\propto\left(1-\frac{4R_{\mathrm{S}}}{r}\right)^{3}r^{-\gamma_{\mathrm{sp}}},\qquad\mathrm{with}\qquad\gamma_{\mathrm{sp}}=\frac{9-2\gamma}{4-\gamma}.$
(7)

For DM halos initially following the Navarro–Frenk–White (NFW) profile
Navarro:1995iw , $\gamma=1$, we have $\gamma_{\mathrm{sp}}=\frac{7}{3}$. That
is, the final DM profile is much steeper than the initial one. If DM particles
are too close to the supermassive black hole, they will fall into the black
hole. The DM number density vanishes at a radius $r<4R_{\mathrm{S}}$
Gondolo:1999ef . The normalization constant for the profile $\rho^{\prime}(r)$
can be determined by the DM mass enclosed within a large radius. Following
Ref. Ullio:2001fb ; Gorchtein:2010xa ; Wang:2021jic ; Granelli:2022ysi , the
mass of enclosed DM can be of the same order as the black hole mass,

$\int_{4R_{\mathrm{S}}}^{{10^{5}R_{\mathrm{S}}}}4\pi
r^{2}\rho^{\prime}(r)dr\simeq M_{\mathrm{BH}}.$ (8)

In this normalization relation, the upper limit is conventionally taken to be
$10^{5}R_{\mathrm{S}}$ Gorchtein:2010xa ; Wang:2021jic ; Granelli:2022ysi .

If DM can annihilate with each other, the annihilation channel will reduce the
DM density in the spike, setting a maximal DM energy density to be
$\rho_{\mathrm{core}}=m_{\chi}/\left\langle\sigma
v_{\mathrm{rel}}\right\rangle t_{\mathrm{BH}}$ Gondolo:1999ef . Here
$m_{\chi}$ denotes the DM particle mass; $\left\langle\sigma
v_{\mathrm{rel}}\right\rangle$ is the thermal averaging of its annihilation
cross section times relative velocity; $t_{\mathrm{BH}}$ is the lifetime of
the black hole. The final DM spike profile becomes Gondolo:1999ef

$\rho_{\mathrm{sp}}(r)=\frac{\rho^{\prime}(r)\rho_{\mathrm{core}}}{\rho^{\prime}(r)+\rho_{\mathrm{core}}}.$
(9)

With the electron distribution in Eq. (5) and the DM profile in Eq. (9), we
can calculate the production rate of boosted DM inside the jet. The number of
boosted DM particles produced per unit volume per unit time can be written as

$\frac{dn_{\chi}}{dt}=\int\frac{d^{2}n_{e,\mathrm{eff}}}{dE_{e}d\Omega_{e}}n_{\chi}\frac{d\sigma_{\chi
e}}{du_{\mathrm{sc}}}\beta_{e}dE_{e}d\Omega_{e}du_{\mathrm{sc}}.$ (10)

In this equation, $n_{\chi}=\frac{\rho_{\mathrm{sp}}}{m_{\chi}}$ is the DM
number density. $\frac{d\sigma_{\chi e}}{du_{\mathrm{sc}}}$ denotes the
differential cross section with respect to the cosine of the scattering angle
in the observer’s frame, $u_{\mathrm{sc}}\equiv\cos\theta_{\mathrm{sc}}$. The
non-relativistic motion of DM in the spike is neglected, and the relative
velocity of DM and electrons is equal to the electron velocity
$\beta_{e}\approx 1$. The total number of boosted DM particles is obtained by
integrating $\frac{dn_{\chi}}{dt}$ over the jet volume,
$\frac{dN_{\chi}}{dt}=\pi R_{B}^{\prime 2}\int\frac{dn_{\chi}}{dt}dr$. We are
interested in the spectrum of boosted DM particles travelling towards the
Earth, rather than the total number of boosted DM particles. To achieve this,
we can introduce two $\delta$-functions into the expression for
$\frac{dN_{\chi}}{dt}$,

$\frac{dN_{\chi}}{dtdE_{\chi}d\Omega_{\chi}}=\frac{R_{B}^{\prime
2}}{2}\int\frac{d^{2}n_{e,\mathrm{eff}}}{dE_{e}d\Omega_{e}}n_{\chi}\frac{d\sigma_{\chi
e}}{du_{\mathrm{sc}}}\beta_{e}\delta(E_{\chi}-E_{\chi}(E_{e},u_{\mathrm{sc}}))\delta^{2}(\Omega_{\chi}-\Omega_{\chi}(\Omega_{e},\Omega_{\mathrm{sc}}))dE_{e}d\Omega_{e}d\Omega_{\mathrm{sc}}dr.$
(11)

Here $E_{\chi}$ is the energy of the boosted DM, and $\Omega_{\chi}$ is the
solid angle of the boosted DM momentum. $\Omega_{\mathrm{sc}}$ is the
scattering solid angle of the outgoing DM momentum $\mathbf{p}_{\chi}$
relative to the injecting electron momentum $\mathbf{p}_{e}$. The function
$\Omega_{\chi}(\Omega_{e},\Omega_{\mathrm{sc}})$ indicates that
$\Omega_{\chi}$ can be fully determined by the solid angle of the injecting
electron $\Omega_{e}$ and the scattering solid angle $\Omega_{\mathrm{sc}}$.
Furthermore, the function $E_{\chi}(E_{e},u_{\mathrm{sc}})$ describes the
relation between the energy of the boosted DM and the injecting electron
energy and the scattering angle. Its form can be determined by the energy-
momentum conservation condition of the process,

$E_{\chi}(E_{e},u_{\mathrm{sc}})=\frac{\gamma_{\mathrm{CM}}^{2}+(\gamma_{\mathrm{CM}}^{2}-1)u_{\mathrm{sc}}^{2}}{\gamma_{\mathrm{CM}}^{2}-(\gamma_{\mathrm{CM}}^{2}-1)u_{\mathrm{sc}}^{2}}m_{\chi},$
(12)

where
$\gamma_{\mathrm{CM}}=\frac{E_{e}+m_{\chi}}{\sqrt{(E_{e}+m_{\chi})^{2}-|\mathbf{p}_{e}|^{2}}}$
is the boost factor from the center-of-mass frame to the observer’s frame.

Figure 2: Differential cross section versus scattering angles in the
observer’s frame. The injecting electron energy is set to be
$E_{e}=1\,\mathrm{GeV}$. The red, yellow and green curves correspond to
different DM masses, $m_{\chi}=100\,\mathrm{MeV},\,1\,\mathrm{MeV}$ and
$10\,\mathrm{keV}$.

As discussed in Sec. 2, to produce a boosted DM particle with a large oblique
angle against the jet axis, we need a large incident angle $\theta_{e}$ or a
large scattering angle $\theta_{\mathrm{sc}}$. The distribution of scattering
angle is described by the differential cross section $\frac{d\sigma_{\chi
e}}{du_{\mathrm{sc}}}$. We assume that the scattering of DM and electrons is
isotropic in the center-of-mass frame. The differential cross section in the
observer’s frame is then,

$\frac{d\sigma_{\chi
e}}{du_{\mathrm{sc}}}=\frac{4\gamma_{\mathrm{CM}}^{2}u_{\mathrm{sc}}}{\left(\gamma_{\mathrm{CM}}^{2}-(\gamma_{\mathrm{CM}}^{2}-1)u_{\mathrm{sc}}^{2}\right)^{2}}\frac{\sigma_{\chi
e}}{2}.$ (13)

Note that the $\theta_{\mathrm{sc}}$ in $u_{\mathrm{sc}}$ is the scattering
angle of the upscattered DM. In Fig. 2, we show the differential cross section
for different DM masses with $E_{e}=1\,\mathrm{GeV}$. We can see that the
scattering is mainly forward in the observer’s frame. Especially for light DM
with $m_{\chi}\lesssim 1\,\mathrm{MeV}$, the possibility of scattering
processes with a scattering angle $\theta_{\mathrm{sc}}>15^{\circ}$ is
significantly suppressed. This means that to produce boosted light DM
particles travelling towards the Earth, we need the incident angle of
electrons to be large in the first place,
$\theta_{e}\approx\theta_{\mathrm{LOS}}$. For heavier DM with $m_{\chi}\simeq
100\,\mathrm{MeV}$, the suppression from large scattering angle is modest,
even smaller than the suppression arising from the electron number densities
with large incident angles as shown in Fig. 1. In this case, the dominant
contribution to boosted DM comes from the scattering of the electrons moving
along the jet with large scattering angles.

In the case of isotropic scattering of DM and electrons, we can integrate out
the $\delta$-functions in Eq. (11) to obtain,

$\frac{dN_{\chi}}{dtdE_{\chi}d\Omega_{\chi}}=\pi R_{B}^{\prime 2}\sigma_{\chi
e}\int\frac{\rho_{\mathrm{sp}}}{m_{\chi}}dr\times\int_{E_{e}^{\min}}\frac{dE_{e}}{\Delta
E_{\chi}^{\max}}\frac{d\varphi_{\mathrm{sc}}}{2\pi}\frac{dn_{e,\mathrm{eff}}}{dE_{e}d\Omega_{e}}\beta_{e},$
(14)

where $\Delta E_{\chi}^{\max}=2(\gamma_{\mathrm{CM}}^{2}-1)m_{\chi}$ is the
maximal energy transfer from the electron to the DM, and the integral lower
limit $E_{e}^{\min}$ is the minimal electron energy producing boosted DM with
energy $E_{\chi}$,

$E_{e}^{\min}=\frac{E_{\chi}-m_{\chi}}{2}\left(1+\sqrt{\frac{(E_{\chi}+m_{\chi})(m_{\chi}(E_{\chi}-m_{\chi})+2m_{e}^{2})}{m_{\chi}(E_{\chi}-m_{\chi})^{2}}}\right).$
(15)

The DM flux arriving at the Earth from Cen A can be derived from this rate as

$\frac{d\Phi_{\oplus}}{dE_{\chi}}=\frac{1}{d_{L}^{2}}\frac{dN_{\chi}}{dtdE_{\chi}d\Omega_{\chi}},$
(16)

where $d_{L}=3.7~{}\mathrm{Mpc}$ is the luminosity distance of Cen A.

Parameter | Symbol | Model $A$ | Model $A^{\prime}$ | Model $B$
---|---|---|---|---
Luminosity Distance | $d_{L}(\mathrm{Mpc})$ | 3.7 | 3.7 | 3.7
Black hole mass | $M_{\mathrm{BH}}(M_{\odot})$ | $5\times 10^{7}$ | $5\times 10^{7}$ | $5\times 10^{7}$
Boost factor of the blob | $\Gamma_{B}$ | 3 | 3 | 3
Viewing angle of the jet | $\theta_{\mathrm{LOS}}$ | $\mathbf{42^{\circ}}$ | $\mathbf{42^{\circ}}$ | $\mathbf{15^{\circ}}$
Minimal boost factor of electron | $\gamma^{\prime}_{\mathrm{e},\min}$ | $1$ | $1$ | $1$
Break boost factor of electron | $\gamma^{\prime}_{b}$ | $1.2\times 10^{3}$ | $1.2\times 10^{3}$ | $1.2\times 10^{3}$
Maximal boost factor of electron | $\gamma^{\prime}_{\mathrm{e},\max}$ | $10^{6}$ | $10^{6}$ | $10^{6}$
Electron spectral power-law index | $\alpha_{1}$ | 1.7 | 1.7 | 1.7
Electron spectral power-law index | $\alpha_{2}$ | 4 | 4 | 4
Radius of the blob | $R^{\prime}_{B}(\mathrm{cm})$ | $2.2\times 10^{15}$ | $2.2\times 10^{15}$ | $2.2\times 10^{15}$
Electron luminosity | $L^{\prime}_{\mathrm{e}}(\mathrm{erg}/\mathrm{s})$ | $10^{42}$ | $10^{42}$ | $10^{42}$
DM profile power-law index | $\gamma_{\mathrm{sp}}$ | $\frac{7}{3}$ | $\frac{7}{3}$ | $\frac{7}{3}$
DM annihilation cross section | $\left\langle\sigma v_{\mathrm{rel}}\right\rangle(\mathrm{cm}^{3}/\mathrm{s})$ | $\mathbf{0}$ | $\mathbf{10^{-28}}$ | $\mathbf{0}$
Black hole lifetime | $t_{\mathrm{BH}}(\mathrm{yr})$ | $10^{9}$ | $10^{9}$ | $10^{9}$
Table 1: Parameters of three models for boosted DM from Centaurus A, including
electron distribution parameters in the jet and DM density profile parameters.
The differences between models are marked in bold.

In this work, the parameters for the jet of Cen A used to calculate the
boosted DM flux are adopted from Ref. Banik:2020ffo  and summarized in Tab. 1
as model $A$. For DM profile, we take the DM annihilation cross section to be
zero and the black hole lifetime to be $10^{9}$ years in model $A$ following
Ref. Wang:2021jic . We also introduce the model $A^{\prime}$ that adjusts the
annihilation cross section to be $10^{-28}~{}\text{cm}^{3}/\text{s}$ based on
model $A$. Additionally, the uncertainty of the viewing angle between the jet
axis and our line-of-sight to Cen A is quite large, measured to be
$\theta_{\mathrm{LOS}}\sim 12^{\circ}\mathrm{-}45^{\circ}$ by the TANAMI
program Muller:2014wja . To illustrate the variations in boosted DM flux for
different viewing angles, we define model $B$ based on model $A$ by replacing
the value of $\theta_{\mathrm{LOS}}$ from $42^{\circ}$ to $15^{\circ}$.

In Fig. 3, we show the spectra of boosted DM flux from Cen A calculated with
the three models and different DM particle masses. We can see that the DM flux
is larger in model $B$ than that in model $A$, since the viewing angle
$\theta_{\mathrm{LOS}}$ in model $B$ is smaller. DM flux is significantly
suppressed in model $A^{\prime}$, since the DM number density in the spike is
much lower in this model due to DM annihilation. Comparing the two cases with
different DM particle masses in Fig. 3, the flux of the lighter one is larger
due to the enhanced DM number density in the spike.

In addition, we can compare the boosted DM flux from Cen A to that from the
blazar BL Lacertae (BL Lac), whose jet points towards the Earth. The electron
luminosities, the masses of the central supermassive black holes, and hence
the number densities of DM particles in the spike are similar in these two
AGNs Boettcher:2013wxa ; Banik:2020ffo . Therefore, the total numbers of
boosted DM particles from Cen A and BL Lacertae are comparable. On one hand,
our scenario experiences a suppression from the fraction of boosted DM
propagating towards the Earth, which is around two orders of magnitude. On the
other hand, Cen A is around $10^{2}$ times closer than BL Lacertae, leading to
a four-order-of-magnitude enhancement in DM flux. Consequently, the boosted DM
flux from Cen A is expected to be about two orders of magnitude larger than
that from BL Lacertae.

Note that we do not include the averaging factor in Eq. (5) for boosted DM
flux from BL Lacertae to reproduce results in Ref. Wang:2021jic . The
parameters for the DM spike of BL Lac is set to be the same as model $A$. Due
to our more conservative treatment about the averaging for Cen A, the DM flux
from model $A$ is about one order of magnitude higher than that from BL
Lacertae, as shown in Fig. 3.

Figure 3: The spectrum of boosted DM Flux from Centaurus A with DM-electron
scattering cross section $\sigma_{\chi e}=10^{-30}\,\mathrm{cm}^{2}$. The DM
mass is $1\,\mathrm{MeV}$ for the solid curves and $10\,\mathrm{keV}$ for the
dashed curves. The red, yellow and green curves corresponds to model $A$,
$A^{\prime}$ and $B$, respectively.

##  4 Dark Matter Attenuation

The boosted DM from Cen A can be detected in the Super-Kamiokande (Super-K)
experiment. The Super-K experiment is located at a depth of $\sim 1$ km
underground Super-Kamiokande:2002weg . Before arriving at the underground
detector, the boosted DM particles may lose energy and deflect due to elastic
scattering off the electrons within the Earth, resulting in a distorted
underground DM flux. This phenomenon is known as the Earth attenuation effect.
For sufficiently large cross sections, this effect can be strong enough to
completely shield the DM signal from the underground detector.

The Earth attenuation effect can be estimated by solving the one-dimensional
energy loss equation Starkman:1990nj ; Bringmann:2018cvk ; Ema:2018bih , which
is appropriate when particles travel in straight lines. Other analytic methods
taking into account the deflection of DM particles based on single scattering
Kavanagh:2016pyr  and the summation of multiple scatterings Cappiello:2023hza
are also developed. A more sophisticated method is the numerical Monte Carlo
simulation Collar:1992qc ; Mahdawi:2017cxz ; Emken:2017qmp ; Chen:2021ifo ,
which simulates the trajectory of DM particles and statistically reconstructs
the underground DM flux. This method accounts for the three-dimensional
propagation and scattering of DM particles, as well as the spherical geometry
of the Earth. In this work, the underground boosted DM flux is simulated using
the DarkProp code DarkProp . This computational tool has been used to analyze
the attenuation of relativistic cosmic ray boosted DM for both DM-nucleus
Xia:2021vbz  and DM-electron Xia:2022tid  scatterings, and to study the
attenuation of non-relativistic halo DM Qiao:2023pbw .

The lower boundary of the exclusion region on the DM-electron scattering cross
section in this work turns out to be small enough that the Earth is almost
transparent to DM particles. So the simulation of the Earth attenuation
focuses on determining the upper boundary of the cross section exclusion
region, where DM particles lose significant energies that they cannot trigger
the detector even if the zenith direction of the detector points to Cen A. In
this case the core and mantle structures of the Earth is not important.
Therefore, we adopt a simplified homogeneous Earth model with chemical
composition and density the same as the outer crust, in which the number
density of electrons is $n_{e}\approx 8\times 10^{23}~{}\text{cm}^{-3}$. Since
the energy of DM particles considered in this work must be higher than the
threshold of electron recoil energy in Super-K detector, which is $0.1$ GeV,
the binding effect of the electrons within the atoms is neglected. In our
simulation, DM particles are injected from the same direction with energies
sampled from the surface flux, and the simulation of each trajectory is
terminated either when the particle leaves the Earth or its energy becomes
lower than $0.1$ GeV. After simulation, the underground flux is reconstructed
as a global average by collecting all the crossing events at the spherical
surface at the depth of 1 km, as described in Ref. Xia:2021vbz .

Figure 4:  The underground boosted DM fluxes corresponding to model $A$ with
$m_{\chi}=1$ MeV for cross sections $\sigma_{\chi e}=10^{-33}$, $10^{-30}$,
and $3\times 10^{-28}~{}\text{cm}^{2}$. The surface fluxes without attenuation
are also shown in dashed lines for comparison.

The reconstructed underground boosted DM fluxes corresponding to model $A$ are
shown in Fig. 4 with DM particle mass $m_{\chi}=1$ MeV for different cross
sections. For the small cross section $\sigma_{\chi
e}=10^{-33}~{}\text{cm}^{2}$, the underground flux nearly coincides with the
surface flux. This can be elucidated by estimating the mean-free-path of DM
particle traveling in the Earth, which is $\ell\equiv(n_{e}\sigma_{\chi
e})^{-1}\approx 1.25\times 10^{4}$ km for $\sigma_{\chi
e}=10^{-33}~{}\text{cm}^{2}$. This length is comparable to the diameter of the
Earth, so the Earth attenuation effect is negligible for $\sigma_{\chi
e}\lesssim 10^{-33}~{}\text{cm}^{2}$. For the larger cross section
$\sigma_{\chi e}=10^{-30}~{}\text{cm}^{2}$, $\ell$ reduces to $12.5$ km. DM
particles from the front side of the detector can easily cross the 1 km depth
overburden, while those coming from the other side of the Earth are blocked.
Therefore, the Earth blocks half of the flux resulting in a factor 2 reduction
compared to the surface flux, as shown in Fig. 4. For $\sigma_{\chi e}=3\times
10^{-28}~{}\text{cm}^{2}$, the mean-free-path is only about 41.7 m. Thus, the
1 km overburden causes strong attenuation and the underground flux is reduced
more than 10 orders of magnitude. At such large cross sections, although the
surface flux is proportional to $\sigma_{\chi e}$, the underground flux
decreases exponentially with $\sigma_{\chi e}$ increasing.

##  5 Detection and Constraints

After passing through the Earth, DM particles can scatter on target electrons
in Super-K detectors and produce detectable Cherenkov light, which can be used
to constrain the region of DM-electron scattering cross section. Since the
signal energy $(>0.1~{}\rm GeV)$ under consideration is much higher than the
energy of target electrons, we assume reasonably that target electrons in the
detector are at rest. The differential event rate per unit target mass can be
written as

$\frac{d\Gamma}{dE_{e}}=\mathcal{N}_{e}\int_{E_{\chi}^{\min}}dE_{\chi}\frac{\sigma_{\chi
e}}{\Delta E^{\text{max}}_{e}}\frac{d\Phi_{\oplus}}{dE_{\chi}},$ (17)

where $\mathcal{N}_{e}$ is the number of electrons per unit target mass, taken
as $\mathcal{N}_{e}\approx 3.3\times 10^{26}~{}\rm{kg^{-1}}$ for water
Cherenkov detectors, and $E_{\chi}^{\min}$ and $\Delta E_{e}^{\max}$ are given
by the expressions of $E_{e}^{\min}$ and $\Delta E_{\chi}^{\max}$ in Eq. (14)
with $\chi$ and $e$ interchanged, respectively.

The Super-K Collaboration has reported a search for boosted DM using
$161.9~{}\rm{kt\cdot yr}$ data in Ref. Super-Kamiokande:2017dch , which gives
data for three energy bins, including the coordinates of events points,
predicted backgrounds and signal detection efficiencies. In this work, the
analysis is based on the first energy bin in the range of $0.1\sim
1.33~{}\mathrm{GeV}$ of this data set. Since the boosted DM particles come
from the point source of Cen A and the scattering is almost forward, the
direction of recoil electrons are nearly the same. We therefore define a
proper searching cone around the Cen A that include most of the recoil
directions. The half-opening angle of the cone is determined by making the
probability of the electron recoil direction in the cone equal 0.95, for the
minimal incoming DM energy reproducing electron recoil energy that exceeds the
threshold $0.1~{}\mathrm{GeV}$,

$\left.\int_{\cos{\delta}}^{1}du^{e}_{\rm sc}\frac{1}{\sigma_{\chi
e}}\frac{d\sigma_{\chi e}}{du^{e}_{\rm
sc}}\right|_{E_{\chi}=E_{\chi}^{\mathrm{th}}}\simeq
0.95\,,\quad\text{with}\quad E_{\chi}^{\mathrm{th}}\equiv
E_{\chi}^{\min}(E_{e}=0.1~{}\mathrm{GeV})\,,$ (18)

where $\frac{d\sigma_{\chi e}}{du^{e}_{\rm sc}}$ has the same form as Eq. (13)
with $\chi$ and $e$ interchanged, and $u^{e}_{\rm sc}$ is the scattering angle
of the electron. The numerical solution to Eq. (18) gives $\delta\approx
23.7^{\circ}$.

We use the standard Poisson method to constrain the DM cross section for
different DM masses. Given an expectation value of $\lambda=s+b$ events with
$s$ the theoretical prediction and $b$ the expected background, the
probability of observing $N_{\mathrm{obs}}$ events is given by the Poisson
distribution $P(N_{\mathrm{obs}}|\lambda)$. The 95$\%$ confidence level (C.L.)
upper limit $\lambda_{p}$ is determined by requiring $P(N<N_{\rm
obs}|\lambda_{p})=0.05$. Then the exclusion region of the cross section
$\sigma_{\chi e}$ is obtained from $\lambda<\lambda_{p}$.

The theoretical prediction $s$ is given by

$s=0.95\times\epsilon\times E_{\rm X}\int dE_{e}\frac{d\Gamma}{dE_{e}},$ (19)

where the coefficient 0.95 comes from the fact that the search cone contains
95% of the total events, as implied by Eq. (18), $\epsilon$ is the signal
efficiency taken as 0.93 Super-Kamiokande:2017dch , $E_{\rm
X}=161.9~{}\rm{kt\cdot yr}$ is the exposure of the Super-K data, and the
integral is performed from 0.1 GeV to 1.33 GeV following the first bin of the
data. For the background event number $b$, we use an isotropic distribution
from the Super-K Monte Carlo simulation Super-Kamiokande:2017dch , which
implies $b\approx 168.4$ for $\delta=23.7^{\circ}$. The observed events
$N_{\mathrm{obs}}$ within the search cone around Cen A direction,
$(201.4^{\circ},-43.0^{\circ})$ in equatorial coordinate system, can be
selected from the data, which gives $N_{\mathrm{obs}}=177$.

Figure 5: Exclusion regions in the $(m_{\chi},\sigma_{\chi e})$ plane at 95%
C.L. derived from the first bin of Super-K data (shadow area) for model $A$,
model $A^{\prime}$, model $B$ of boosted DM from Cen A and the blazar BL Lac.
A selection of other constraints from XENON-1T XENON:2019gfn , PandaX-4T
PandaX:2022xqx , SENSEI SENSEI:2023zdf , cosmic ray Cappiello:2018hsu , and
CDEX with solar reflection Zhang:2023xwv  are shown for comparison. The
vertical brown dashed line stands for the BBN constraint on thermalized Dirac
fermionic DM from Ref. Sabti:2019mhn .

In Fig. 5, we show the exclusion regions at 95% C.L. calculated from the three
models studied in this work, as well as that from the blazar BL Lac for
comparison. For model $A$, the lower boundary of the exclusion region is
derived to be $5\times 10^{-35}~{}\text{cm}^{2}$ for $m_{\chi}=10^{-3}$ GeV,
and the constraint is more stringent for lighter DM. If the DM annihilation
channel is open, as in model $A^{\prime}$, the constraint becomes weaker,
since the DM number density in the spike is lower. In model $B$, the viewing
angle of Cen A is taken as $\theta_{\rm LOS}=15^{\circ}$. The constraint can
be one order of magnitude stronger than that from model $A$ with $\theta_{\rm
LOS}=42^{\circ}$. Comparing model $A$ to BL Lac, we have estimated that the
boosted DM flux from the former is about 10 times higher than the latter.
However, the constraint on the cross section from Cen A is not as low as
around $\sqrt{10}$ times lower than that for BL Lac, as expected from that the
number of DM events is proportional to $\sigma_{\chi e}^{2}$. This is because
the number of observed events in the searching cone around Cen A is larger
than that of BL Lac in Super-K data, and the limit for Cen A is moderately
weakened. The upper boundary of the exclusion region due to the Earth
attenuation is derived to be $\sim 5\times 10^{-28}~{}\mathrm{cm}^{2}$. Larger
cross sections may be probed by astrophysical observations such as cosmic ray
scattering Cappiello:2018hsu . Other constraints from selected DM direct
detection experiments, SENSEI SENSEI:2023zdf , XENON-1T XENON:2019gfn ,
PandaX-4T PandaX:2022xqx , and CDEX with solar reflection Zhang:2023xwv , as
well as big bang nucleosynthesis (BBN) constraints on Dirac fermionic DM
Sabti:2019mhn  are also shown in Fig. 5.

##  6 Conclusion

There are various high energy processes in astrophysics, and dark matter can
be boosted in these environments. The recoil energy of boosted dark matter can
exceed the detection threshold in underground experiments. We study the
production and detection of boosted dark matter from the jet of Centaurus A,
the closest active galactic nucleus. Since the jet of Centaurus A is not
directed towards the Earth, the angle of the momentum of boosted dark matter
against the jet axis should be quite large so that they can propagate to the
Earth. Although most of the boosted dark matte particles travel along the jet,
our research shows that the fraction of boosted dark matter with large oblique
angles is still sizable. Due to the proximity of Centaurus A, compared to that
from blazars, boosted dark matter flux from Centaurus A is enhanced by around
two orders of magnitude. These boosted dark matter particles are detectable in
the Super-Kamiokande experiment, which results in an exclusion limit on sub-
GeV dark matter-electron scattering cross section as low as $\sim
10^{-36}\,\mathrm{cm}^{2}$. Considering the Earth attenuation effect, the
upper boundary of the exclusion region is set to be $\sim 5\times
10^{-28}~{}\mathrm{cm}^{2}$.

###### Acknowledgements.

We thank Jin-Wei Wang and Shang-Ming Chen for helpful discussions. This work
is supported by the National Natural Science Foundation of China (NSFC) No.
12247141 and No. 12247148, by the Natural Science Foundation of Shandong
Province under the grants No. ZR2023QA149, by the Project of Shandong Province
Higher Educational Science and Technology Program under grants No. 2022KJ271,
and by NSFC No.12375101, 12090060 and 12090064. The Monte Carlo simulations in
this paper were run on the Siyuan-1 cluster supported by the Center for High
Performance Computing at Shanghai Jiao Tong University.

## References

  * (1) G. Bertone, D. Hooper and J. Silk, _Particle dark matter: Evidence, candidates and constraints_ , _Phys. Rept._ 405 (2005) 279 [hep-ph/0404175].
  * (2) M. Bauer and T. Plehn, _Yet Another Introduction to Dark Matter: The Particle Physics Approach_ , vol. 959 of _Lecture Notes in Physics_ , Springer (2019), 10.1007/978-3-030-16234-4, [1705.01987].
  * (3) J. Cooley, _Dark Matter direct detection of classical WIMPs_ , _SciPost Phys. Lect. Notes_ 55 (2022) 1 [2110.02359].
  * (4) SENSEI collaboration, _SENSEI: Direct-Detection Constraints on Sub-GeV Dark Matter from a Shallow Underground Run Using a Prototype Skipper-CCD_ , _Phys. Rev. Lett._ 122 (2019) 161801 [1901.10478].
  * (5) PandaX-4T collaboration, _Dark Matter Search Results from the PandaX-4T Commissioning Run_ , _Phys. Rev. Lett._ 127 (2021) 261802 [2107.13438].
  * (6) CDEX collaboration, _Constraints on sub-GeV dark matter boosted by cosmic rays from the CDEX-10 experiment at the China Jinping Underground Laboratory_ , _Phys. Rev. D_ 106 (2022) 052008 [2201.01704].
  * (7) LZ collaboration, _First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041002 [2207.03764].
  * (8) SuperCDMS collaboration, _Light Dark Matter Search with a High-Resolution Athermal Phonon Detector Operated Above Ground_ , _Phys. Rev. Lett._ 127 (2021) 061801 [2007.14289].
  * (9) DarkSide-50 collaboration, _Search for low-mass dark matter WIMPs with 12 ton-day exposure of DarkSide-50_ , _Phys. Rev. D_ 107 (2023) 063001 [2207.11966].
  * (10) XENON collaboration, _First Dark Matter Search with Nuclear Recoils from the XENONnT Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041003 [2303.14729].
  * (11) XENON collaboration, _Light Dark Matter Search with Ionization Signals in XENON1T_ , _Phys. Rev. Lett._ 123 (2019) 251801 [1907.11485].
  * (12) PROSPECT, (PROSPECT Collaboration)* collaboration, _Limits on sub-GeV dark matter from the PROSPECT reactor antineutrino experiment_ , _Phys. Rev. D_ 104 (2021) 012009 [2104.11219].
  * (13) COHERENT collaboration, _First Probe of Sub-GeV Dark Matter beyond the Cosmological Expectation with the COHERENT CsI Detector at the SNS_ , _Phys. Rev. Lett._ 130 (2023) 051803 [2110.11453].
  * (14) CDEX collaboration, _Constraints on Sub-GeV Dark Matter–Electron Scattering from the CDEX-10 Experiment_ , _Phys. Rev. Lett._ 129 (2022) 221301 [2206.04128].
  * (15) Super-Kamiokande collaboration, _Search for Cosmic-Ray Boosted Sub-GeV Dark Matter Using Recoil Protons at Super-Kamiokande_ , _Phys. Rev. Lett._ 130 (2023) 031802 [2209.14968].
  * (16) CRESST collaboration, _Results on sub-GeV dark matter from a 10 eV threshold CRESST-III silicon detector_ , _Phys. Rev. D_ 107 (2023) 122003 [2212.12513].
  * (17) DAMIC-M collaboration, _First Constraints from DAMIC-M on Sub-GeV Dark-Matter Particles Interacting with Electrons_ , _Phys. Rev. Lett._ 130 (2023) 171003 [2302.02372].
  * (18) K. Agashe, Y. Cui, L. Necib and J. Thaler, _(In)direct Detection of Boosted Dark Matter_ , _JCAP_ 10 (2014) 062 [1405.7370].
  * (19) H. An, M. Pospelov, J. Pradler and A. Ritz, _Directly Detecting MeV-scale Dark Matter via Solar Reflection_ , _Phys. Rev. Lett._ 120 (2018) 141801 [1708.03642].
  * (20) T. Emken, _Solar reflection of light dark matter with heavy mediators_ , _Phys. Rev. D_ 105 (2022) 063020 [2102.12483].
  * (21) H. An, H. Nie, M. Pospelov, J. Pradler and A. Ritz, _Solar reflection of dark matter_ , _Phys. Rev. D_ 104 (2021) 103026 [2108.10332].
  * (22) T. Bringmann and M. Pospelov, _Novel direct detection constraints on light dark matter_ , _Phys. Rev. Lett._ 122 (2019) 171801 [1810.10543].
  * (23) Y. Ema, F. Sala and R. Sato, _Light Dark Matter at Neutrino Experiments_ , _Phys. Rev. Lett._ 122 (2019) 181802 [1811.00520].
  * (24) C.V. Cappiello and J.F. Beacom, _Strong New Limits on Light Dark Matter from Neutrino Experiments_ , _Phys. Rev. D_ 100 (2019) 103011 [1906.11283].
  * (25) S.-F. Ge, J. Liu, Q. Yuan and N. Zhou, _Diurnal Effect of Sub-GeV Dark Matter Boosted by Cosmic Rays_ , _Phys. Rev. Lett._ 126 (2021) 091804 [2005.09480].
  * (26) G. Guo, Y.-L.S. Tsai and M.-R. Wu, _Probing cosmic-ray accelerated light dark matter with IceCube_ , _JCAP_ 10 (2020) 049 [2004.03161].
  * (27) C. Xia, Y.-H. Xu and Y.-F. Zhou, _Constraining light dark matter upscattered by ultrahigh-energy cosmic rays_ , _Nucl. Phys. B_ 969 (2021) 115470 [2009.00353].
  * (28) C. Xia, Y.-H. Xu and Y.-F. Zhou, _Production and attenuation of cosmic-ray boosted dark matter_ , _JCAP_ 02 (2022) 028 [2111.05559].
  * (29) W. Wang, W.-L. Xu, J.M. Yang and R. Zhu, _Direct detection of cosmic ray-boosted puffy dark matter_ , _Nucl. Phys. B_ 995 (2023) 116348 [2305.12668].
  * (30) PandaX-II collaboration, _Search for Cosmic-Ray Boosted Sub-GeV Dark Matter at the PandaX-II Experiment_ , _Phys. Rev. Lett._ 128 (2022) 171801 [2112.08957].
  * (31) J.-W. Wang, A. Granelli and P. Ullio, _Direct Detection Constraints on Blazar-Boosted Dark Matter_ , _Phys. Rev. Lett._ 128 (2022) 221104 [2111.13644].
  * (32) R. Calabrese, M. Chianese, D.F.G. Fiorillo and N. Saviano, _Direct detection of light dark matter from evaporating primordial black holes_ , _Phys. Rev. D_ 105 (2022) L021302 [2107.13001].
  * (33) S. Bhowmick, D. Ghosh and D. Sachdeva, _Blazar boosted dark matter — direct detection constraints on $\sigma$e$\chi$: role of energy dependent cross sections_, _JCAP_ 07 (2023) 039 [2301.00209].
  * (34) M.J. Hardcastle, D.M. Worrall, R.P. Kraft, W.R. Forman, C. Jones and S.S. Murray, _Radio and x-ray observations of the jet in centaurus a_ , _Astrophys. J._ 593 (2003) 169 [astro-ph/0304443].
  * (35) G.L.H. Harris, M. Rejkuba and W.E. Harris, _The Distance to NGC 5128 (Centaurus A)_ , _Publ. Astron. Soc. Austral._ 27 (2010) 457 [0911.3180].
  * (36) B.T. Zhang and K. Murase, _Nuclear and electromagnetic cascades induced by ultra-high-energy cosmic rays in radio galaxies: implications for Centaurus A_ , _Mon. Not. Roy. Astron. Soc._ 524 (2023) 76 [2302.14048].
  * (37) M. Boettcher, A. Reimer, K. Sweeney and A. Prakash, _Leptonic and Hadronic Modeling of Fermi-Detected Blazars_ , _Astrophys. J._ 768 (2013) 54 [1304.0605].
  * (38) P. Banik, A. Bhadra and A. Bhattacharyya, _Interpreting correlated observations of cosmic rays and gamma-rays from Centaurus A with a proton blazar inspired model_ , _Mon. Not. Roy. Astron. Soc._ 500 (2020) 1087 [2102.00213].
  * (39) G. Ghisellini, A. Celotti, G. Fossati, L. Maraschi and A. Comastri, _A Theoretical unifying scheme for gamma-ray bright blazars_ , _Mon. Not. Roy. Astron. Soc._ 301 (1998) 451 [astro-ph/9807317].
  * (40) R. Moderksi, M. Sikora, P.S. Coppi and F.A. Aharonian, _Klein-Nishina effects in the spectra of non-thermal sources immersed in external radiation fields_ , _Mon. Not. Roy. Astron. Soc._ 363 (2005) 954 [astro-ph/0504388].
  * (41) M. Boettcher, _Modeling the Emission Processes in Blazars_ , _Astrophys. Space Sci._ 309 (2007) 95 [astro-ph/0608713].
  * (42) A. Celotti and G. Ghisellini, _The power of blazar jets_ , _Mon. Not. Roy. Astron. Soc._ 385 (2008) 283 [0711.4112].
  * (43) J.D. Finke, C.D. Dermer and M. Böttcher, _Synchrotron Self-Compton Analysis of TeV X-ray Selected BL Lacertae Objects_ , _Astrophys. J._ 686 (2008) 181 [0802.1529].
  * (44) R. Blandford, D. Meier and A. Readhead, _Relativistic Jets from Active Galactic Nuclei_ , _Ann. Rev. Astron. Astrophys._ 57 (2019) 467 [1812.06025].
  * (45) M. Cerruti, _Leptonic and Hadronic Radiative Processes in Supermassive-Black-Hole Jets_ , _Galaxies_ 8 (2020) 72 [2012.13302].
  * (46) P. Banik and A. Bhadra, _Describing correlated observations of neutrinos and gamma-ray flares from the blazar TXS 0506+056 with a proton blazar model_ , _Phys. Rev. D_ 99 (2019) 103006 [1908.11849].
  * (47) Event Horizon Telescope collaboration, _Event Horizon Telescope observations of the jet launching and collimation in Centaurus A_ , _Nature Astron._ 5 (2021) 1017 [2111.03356].
  * (48) A. Granelli, P. Ullio and J.-W. Wang, _Blazar-boosted dark matter at Super-Kamiokande_ , _JCAP_ 07 (2022) 013 [2202.07598].
  * (49) P. Gondolo and J. Silk, _Dark matter annihilation at the galactic center_ , _Phys. Rev. Lett._ 83 (1999) 1719 [astro-ph/9906391].
  * (50) J.F. Navarro, C.S. Frenk and S.D.M. White, _The Structure of cold dark matter halos_ , _Astrophys. J._ 462 (1996) 563 [astro-ph/9508025].
  * (51) P. Ullio, H. Zhao and M. Kamionkowski, _A Dark matter spike at the galactic center?_ , _Phys. Rev. D_ 64 (2001) 043504 [astro-ph/0101481].
  * (52) M. Gorchtein, S. Profumo and L. Ubaldi, _Probing Dark Matter with AGN Jets_ , _Phys. Rev. D_ 82 (2010) 083514 [1008.2230].
  * (53) C. Müller et al., _TANAMI monitoring of Centaurus A: The complex dynamics in the inner parsec of an extragalactic jet_ , _Astron. Astrophys._ 569 (2014) A115 [1407.0162].
  * (54) Super-Kamiokande collaboration, _The Super-Kamiokande detector_ , _Nucl. Instrum. Meth. A_ 501 (2003) 418.
  * (55) G.D. Starkman, A. Gould, R. Esmailzadeh and S. Dimopoulos, _Opening the Window on Strongly Interacting Dark Matter_ , _Phys. Rev. D_ 41 (1990) 3594.
  * (56) B.J. Kavanagh, R. Catena and C. Kouvaris, _Signatures of Earth-scattering in the direct detection of Dark Matter_ , _JCAP_ 01 (2017) 012 [1611.05453].
  * (57) C.V. Cappiello, _Analytic Approach to Light Dark Matter Propagation_ , _Phys. Rev. Lett._ 130 (2023) 221001 [2301.07728].
  * (58) J.I. Collar and F.T. Avignone, _Diurnal modulation effects in cold dark matter experiments_ , _Phys. Lett. B_ 275 (1992) 181.
  * (59) M.S. Mahdawi and G.R. Farrar, _Closing the window on $\sim$GeV Dark Matter with moderate ($\sim$ $\mu$b) interaction with nucleons_, _JCAP_ 12 (2017) 004 [1709.00430].
  * (60) T. Emken and C. Kouvaris, _DaMaSCUS: The Impact of Underground Scatterings on Direct Detection of Light Dark Matter_ , _JCAP_ 10 (2017) 031 [1706.02249].
  * (61) Y. Chen, B. Fornal, P. Sandick, J. Shu, X. Xue, Y. Zhao et al., _Earth Shielding and Daily Modulation from Electrophilic Boosted Dark Matter_ , 2110.09685.
  * (62) C. Xia, “DarkProp.” http://yfzhou.itp.ac.cn/darkprop, 2021.
  * (63) C. Xia, Y.-H. Xu and Y.-F. Zhou, _Azimuthal asymmetry in cosmic-ray boosted dark matter flux_ , _Phys. Rev. D_ 107 (2023) 055012 [2206.11454].
  * (64) M. Qiao, C. Xia and Y.-F. Zhou, _Diurnal modulation of electron recoils from DM-nucleon scattering through the Migdal effect_ , _JCAP_ 11 (2023) 079 [2307.12820].
  * (65) Super-Kamiokande collaboration, _Search for Boosted Dark Matter Interacting With Electrons in Super-Kamiokande_ , _Phys. Rev. Lett._ 120 (2018) 221301 [1711.05278].
  * (66) PandaX collaboration, _Search for Light Dark Matter with Ionization Signals in the PandaX-4T Experiment_ , _Phys. Rev. Lett._ 130 (2023) 261001 [2212.10067].
  * (67) SENSEI collaboration, _SENSEI: First Direct-Detection Results on sub-GeV Dark Matter from SENSEI at SNOLAB_ , 2312.13342.
  * (68) C.V. Cappiello, K.C.Y. Ng and J.F. Beacom, _Reverse Direct Detection: Cosmic Ray Scattering With Light Dark Matter_ , _Phys. Rev. D_ 99 (2019) 063004 [1810.07705].
  * (69) Z.Y. Zhang et al., _Experimental Limits on Solar Reflected Dark Matter with a New Approach on Accelerated Dark Matter-Electron Analysis in Semiconductors_ , 2309.14982.
  * (70) N. Sabti, J. Alvey, M. Escudero, M. Fairbairn and D. Blas, _Refined Bounds on MeV-scale Thermal Dark Sectors from BBN and the CMB_ , _JCAP_ 01 (2020) 004 [1910.01649].


# Bound state eigenvalues from transmission coefficients

Zafar Ahmed1,2 and Koushik Bhattacharya3<EMAIL_ADDRESS><EMAIL_ADDRESS>$~{}^{1}$Nuclear Physics Division, Bhabha Atomic Research
Cntre,Mumbai 400 085, India;
$~{}^{2}$Homi Bhabha National Institute, Anushaktinagar, Mumbai, 400094, India
$~{}^{3}$Human Resource Development Division, Bhabha Atomic Research
Centre,Mumbai 400 094, India

###### Abstract

Experts know that bound state energy eigenvalues of a potential well are poles
of its transmission amplitude, $t(E)$. The textbooks do well by solving the
bound states and scattering states separately, but the connection goes
unheeded. Here, we present interesting and instructive illustrations, where we
extract bound state eigenvalues from the known transmission coefficients
$(T(E)=|t(E)|^{2})$ for six analytically solvable potential wells.

A one dimensional potential well that vanishes asymptotically can possess both
bound and scattering states. In the former case, energy is negative and
discrete:$E_{n}<0$, in the latter, the energy is positive and continuous,
where one can define a probability of transmission which is called the
transmission coefficient: $T(E)$. A square well and a Dirac delta well are
simple examples [1-4]. Usually, textbooks discuss these two types of states
separately. Here we present the connection between negative energy bound
states with the negative energy poles of $T(E)$. This has been slightly
touched upon in the book by Merzbacher [1] but not without an error wherein
the transmission coefficient $T(E)$ at negative energy eigenvalues is
mistakenly shown to become 1 in a Fig. 6.9, there. A potential barrier is a
positive potential with a maximum that has only scattering states. If
$V(x)=-V_{0}f(x/a)$ represents a well of depth $V_{0}$, its inverted form
$V(x)=U_{0}f(x/a)$ gives a barrier of height $U_{0}$, here $V_{0},U_{0}>0.$ In
simple cases $f(\pm\infty)=0$, in more advanced cases $f(x/a)$ may diverge on
one or both sides [2,3].

In textbooks, students are often asked to find the discrete set of energies at
which a square well or a barrier transmits fully [1,4]. This infinite set of
discrete energies also correspond to the bound state eigenvalues of an
infinitely deep well but for an additive constant ($-V_{0}$). These discrete
eigenvalues are different from the finite number of eigenvalues of a finite
square well.

For simplicity, assume a single minimum potential well that vanishes
asymptotically. For scattering of a particle with the energy $E>0$ from the
left of this potential, we require asymptotic form of the solution $\psi(x)$
of Schrödinger equation

$\frac{d^{2}\psi(x)}{dx^{2}}+\frac{2m}{\hbar^{2}}(E-V(x))\psi(x)=0.$ (1)

as

$\psi_{<}(x)\sim A(k)e^{ikx}+B(k)e^{-ikx},\psi_{>}(x)\sim C(k)e^{ikx}.$ (2)

The reflection and transmission amplitudes are defined as

$t(k)=C(k)/A(k),\quad r(k)=B(k)/A(k).$ (3)

where the wave number $k=\sqrt{2mE}/\hbar$ and the energy $E>0$. The
transmission and reflection coefficients are defined as $T(E)=|t(k)|^{2}$ and
$R(E)=|r(k)|^{2}$, respectively, such that $R(E)+T(E)=1$. For transition of
$\psi(x)$ to bound states at negative energies $(E<0)$, let
$\kappa=\sqrt{-2mE}/\hbar$ such that $k=i\kappa$. We get

$\psi_{<}(x)\sim A(i\kappa)e^{-\kappa x}+B(i\kappa)e^{\kappa
x},\psi_{>}(x)\sim C(i\kappa)e^{-\kappa x}.$ (4)

For bound states, to attain asymptotic convergence of $\psi(x)$ on both sides,
we set $A(\kappa)=0$ and this is how both $t(i\kappa)$ and $r(i\kappa)$ become
infinity and we get discrete values of $\kappa(=\kappa_{n})$.

Experts know that physical poles of $t(i\kappa)$ yield bound state
eigenvalues. However, as mostly scattering coefficients and not the amplitudes
are remembered, we propose to extract bound states $E_{n}$ from $T(E)$ or
$R(E)$. Also if $\hat{T}(E)$ is known for a barrier of height $U_{0}>0$, we
may change $U_{0}\rightarrow-V_{0}$ to get $T(E)$ for the well. Even if the
well diverges to $\pm\infty$, the obtained $T(E)$ will suffice for obtaining
the eigenvalues of bound states. One may see Illustrations 3 and 4.

It will be well to point out the possible trivial/un-physical pole. In Eq.
(4), $\kappa$ needs to be positive, otherwise $\psi(x)$ will diverge
asymptotically. This is despite the fact that
$E=-\frac{\kappa^{2}\hbar^{2}}{2m}$ is insensitive to the sign of $\kappa$.
Therefore, while extracting the poles, care must be taken to choose
$\kappa_{n}>0$. Moreover, when the discrete spectrum is finite, this physical
restriction defines the upper bound of the quantum number $n$. In this regard,
see Illustrations 4 and 5.

One may get trivial/u-physical poles of complex amplitudes $t(k)$ or $r(k)$
having energy dependent uni-modular phase factor like $(1-ik)/(1+ik)$. Then on
changing $k\rightarrow i\kappa$, one will get a pole $\kappa=1\implies E=-1$
(taking $\hbar^{2}=1=2m$), which being removable and independent of the
potential parameters in $T(E)=|t|^{2}$ and $R(E)=|r|^{2}$ is inconsequential
and hence un-physical. The reflection amplitude $r(k)$ of the exponential well
$V(x)=-V_{0}e^{-2|x|}$ [5] has a factor of $\Gamma(ik)/\Gamma(-ik)$ whose
modulus is 1, but by changing $k\rightarrow i\kappa$ and taking $\kappa=n$, we
encounter an un-physical pole as $\Gamma(-n)=\infty,n\in I^{+}$ [6], This
gives rise to fake energy eigenvalues $E=-n^{2}$. Thus finding poles of $T(E)$
or $R(E)$ is more robust and meaningful. Also the complex transmission
amplitude $t(k)$ may involve more advanced functions like gamma functions
$\Gamma(z)$ of complex variables but they get eliminated and give rise to
simple form for $T(E)$, see Illustration 6 below in this regard.

Illustration 1: Dirac Delta $V(x){=}-V_{0}\delta(x),V_{0}{>}0$

The transmission amplitude is given as $t(k)=\hbar^{2}k/(\hbar^{2}k-imV_{0})$
[4], whose pole is at $k=imV_{0}/\hbar^{2}$ giving the commonly known single
bound state eigenvalue of this potential well at
$E=k^{2}=-mV_{0}^{2}/(2\hbar^{2})$. Similarly,
$T(E)=2\hbar^{2}E/(2\hbar^{2}E+mV_{0}^{2})$, has a pole at the same energy.

Illustration 2: Square well potential

For this well $V(|x|\leq a/2)=-V_{0},V(|x|>a/2)=0$, whose depth is $V_{0}$ and
width is $a$, the commonly known eigenvalue equations for the even and odd
parity states are given by

$\displaystyle Ka\tan(Ka/2)=\kappa a~{}~{}\text{and}~{}~{}-Ka\cot(Ka/2)=\kappa
a,$ (5) $\displaystyle
K=\sqrt{2m(V_{0}+E)}/\hbar,~{}\kappa=\sqrt{-2mE}/\hbar,E<0.$

Let us convert (5) to $\sin$ and $\cos$ form respectively as

$|\sin(Ka/2)|=\frac{\kappa}{k_{0}}~{}\text{and}~{}|\cos(Ka/2)|=\frac{\kappa}{k_{0}}~{}~{},k_{0}=\frac{\sqrt{2mV_{0}}}{\hbar},$
(6)

$\tan(Ka/2)$ should be positive and negative, respectively. This will be
useful in sequel and such equations for eigenvalues of a square well have been
set up earlier for simplified graphical calculations in [7,8]. For a square
well of depth $V_{0}$ and width $a$, the transmission amplitude is given by
[1-4]

$t(k)=\frac{e^{2ika}}{\cos(Ka)-i\frac{k^{2}+K^{2}}{2kK}\sin(Ka)},E>0.$ (7)

For $E<0$, change $k\rightarrow i\kappa$, the poles of (7) are given by

$\tan(Ka)=\frac{2\kappa
K}{K^{2}-\kappa^{2}}=\frac{2\tan(Ka/2)}{1-\tan^{2}(Ka/2)}.$ (8)

Solving it as an equation in $\tan(Ka/2)$, we retrieve the energy eigenvalue
equations (5). The transmission coefficient of a square barrier of width $a$
and height $U_{0}>0$ is commonly known [1-4] as

$\displaystyle\hat{T}(E)=\frac{4\epsilon(1-\epsilon)}{4\epsilon(1-\epsilon)+\sinh^{2}(\alpha\sqrt{1-\epsilon})},$
(9)
$\displaystyle\epsilon=\frac{E}{U_{0}},\alpha=\frac{\sqrt{2mU_{0}}}{\hbar}a,E,V_{0}>0.$

By changing $U_{0}\to-V_{0},~{}\alpha\to ik_{0}a,~{}\epsilon=-\varepsilon$, we
get $T(E)$ for the square well as

$\displaystyle
T(E)=\frac{4\varepsilon(1+\varepsilon)}{4\epsilon(1+\varepsilon)+\sin^{2}(k_{0}\sqrt{1+\varepsilon)}},$
(10) $\displaystyle\varepsilon=\frac{E}{V_{0}},E,V_{0}>0.$

This also follows from (7) as $T(E)=|t(k)|^{2}$. Now we look for the poles of
$T(E)$ in (10), when $E<0$. The denominator gets factored in $\varepsilon$,
the poles are given by

$\displaystyle-\varepsilon{=}\frac{1}{2}(1\pm\cos(Ka)){\implies}|\sin(Ka/2)|=\sqrt{-\varepsilon}~{}\text{and}$
(11) $\displaystyle~{}~{}|\cos(Ka/2)|=\sqrt{-\varepsilon},$

which is nothing but the energy eigenvalue equations given in (6) and
equivalently in (5). To re-emphasize, the poles given by two parts in (11)
will give correct eigenvalues of the square well only if their roots meet the
criterion that $\tan(Ka/2)$ is positive and negative, respectively. This is a
slightly inconvenient aspect of square well for extracting eigenvalues from
$T(E)$ rather than $t(E)$ as done from Eq. (7). The following illustrations
are free from this hassle.

A potential barrier that diverges to $-\infty$ on one side or both, upon
inversion ($U_{0}\to-V_{0}$) gives a well which diverges to $+\infty$ on
respective sides, both $T$ and $R$ become un-physical yet they remain useful
to extract discrete energies. The following two illustrations are for two such
cases.

Illustration 3: Harmonic oscillator: $V(x)=\frac{1}{2}m\omega^{2}x^{2}$:

The transmission coefficient of the parabolic barrier
$V(x)=V_{0}-\frac{1}{2}m\Omega^{2}x^{2}$ is well known as Hill-Wheeler formula
[3,9]

$\hat{T}(E)=\frac{1}{1+\exp\left(\frac{2\pi(V_{0}-E)}{\hbar\Omega}\right)}.$
(12)

If we put $\Omega=i\omega$ and $V_{0}=0$, the parabolic barrier becomes a well
for which we have

$T(E)=\frac{1}{1+\exp\left(\frac{2i\pi E}{\hbar\omega}\right)}.$ (13)

This being imaginary, it is un-physical and a warning that a harmonic
oscillator which is an inverted parabolic barrier does not admit scattering
states, as it diverges to $+\infty$ on both sides of the well. However, (13)
yields poles which are the commonly known energy eigenvalues [1-4] of the
harmonic oscillator.

$\displaystyle e^{2i\pi E/(\hbar\omega)}=-1=e^{(2n+1)i\pi},~{}n=0,1,2,3,..$
(14) $\displaystyle\implies E_{n}=(n+1/2)\hbar\omega.$

Illustration 4: Morse oscillator, $V(x)=V_{0}[e^{2x/a}-2e^{x/a}]$

This potential well [2,3] diverges to $+\infty$ on the right side, so it
reflects a particle coming from left completely without transmission. But its
corresponding barrier $V(x)=-U_{0}[e^{2x/a}-2e^{x/a}],U_{0}>0$ admits simple
exact transmission coefficient [10] which is given for $E>0$ as

$\displaystyle\hat{T}(E)=\frac{1-e^{-4\pi\alpha}}{1+e^{2\pi(\beta-\alpha)}},~{}\alpha=\sqrt{\frac{E}{\Delta}},~{}\beta=\sqrt{\frac{U_{0}}{\Delta}},$
(15) $\displaystyle~{}\Delta=\frac{\hbar^{2}}{2ma^{2}}.$

By changing $U_{0}\to-
V_{0}(\beta=i\eta(\eta=\sqrt{V_{0}/\Delta}))$,$\alpha=i\gamma(\gamma=\sqrt{-E/\Delta}$),
we get $T(k)$ which becomes non-real and un-physical, nevertheless we get the
poles as

$\displaystyle
e^{2i\pi(\eta-\gamma_{n})}{=}{-}1{=}e^{\pm(2n+1)i\pi}\implies\gamma_{n}{=}[\eta-(n+1/2)]$
(16) $\displaystyle>0,\implies
E_{n}=\gamma_{n}^{2}=-[\sqrt{V_{0}}-(n+1/2)\sqrt{\Delta}]^{2},$

which are the commonly known eigenvalues of the Morse potential. Since Morse
potential well [2,3] admits only a finite number of eigenvalues , then
$n=0,1,2,...<[\sqrt{V_{0}/\Delta}-1/2]$, here $[z]$ means the integer-part of
$z$ .

Illustration 5: Soliton (Eckart) potential well,
$V(x)=-V_{0}\text{sech}^{2}(x/a),V_{0}>0$

The transmission coefficient [2,3] for this barrier
$V(x)=U_{0}\text{sech}^{2}(x/a)$ is given as

$\displaystyle T(E)=\frac{\cosh(2\pi\alpha)-1}{\cosh(2\pi\alpha)+\cos
2\pi\beta},~{}\alpha=\sqrt{\frac{E}{\Delta}},$ (17)
$\displaystyle\beta=\sqrt{1/4-\frac{U_{0}}{\Delta}}.$

Let us change $U_{0}\to-V_{0},~{}\alpha\to
i\gamma(\gamma=\sqrt{-E/\Delta}),~{}\beta=\eta=\sqrt{1/4+V_{0}/\Delta}$.We get

$\displaystyle\cos(2\pi\gamma_{n})=-\cos(2\pi\eta)=\cos(2\pi\eta-\pi)\implies
2\pi\gamma_{n}$ (18) $\displaystyle=\pm
2n\pi\pm(2\pi\eta-\pi)\implies\gamma_{n}=\eta-(n+1/2)]>0$
$\displaystyle\implies E_{n}=\gamma_{n}^{2}=-\Delta[\eta-(n+1/2)]^{2}.$

In Eq. (16) and (18) the signs $\pm$ have to be chosen such that
$\gamma_{n}>0$ $(\psi(x)\sim e^{-\gamma_{n}|x|})$ can decay when $x\to\infty$
and $n$ gets restricted rightly as these potentials have a finite number of
bound states.

Illustration 6: Scarf II potential

This exactly solvable potential is given by [11,12]

$V(x){=}\frac{\hbar^{2}}{2m}[(B^{2}{-}A^{2}{-}A)\text{sech}^{2}x{+}B(2A{+}1)\text{tanh}x{~{}}\text{sech}x].~{}(19)$

Here $A$ and $B$ are essentially real and positive and $V(x)$ is Hermitian
asymmetric potential well. The very interesting transmission amplitude for
(19) is given as [11]

$t(k){=}\frac{\Gamma({-}A{-}ik)\Gamma(1{+}A{}-ik)\Gamma(\frac{1}{2}{+}iB{-}ik)\Gamma(\frac{1}{2}{-}iB{-}ik)}{\Gamma(-ik)\Gamma(1+ik)\Gamma^{2}(1/2-ik)}~{}(20)$

Noting that $\Gamma(-N)=\infty,N=0,1,2,3,...$, changing $k\rightarrow i\kappa$
the poles of $t(k)$ can be obtained as $-A-ik=-n\implies\kappa^{(1)}=(A-n)$.
The second $\Gamma(z)$, gives $\kappa^{(2)}=-n-A-1<0$, Third and fourth
$\Gamma(z)$ gives $\kappa^{(3,4)}=(n+1/2)\pm iB$. Among these poles, only
$\kappa^{(1)}$ is physical. We get

$\kappa^{2}_{n}=(A-n)^{2}{\implies}E_{n}=-\frac{\hbar^{2}}{2m}(A-n)^{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(21)$

These are exactly the eigenvalues obtained by the usual method of bound states
[12], Interestingly, we can find $T(E)=t(k)t^{*}(k)$ using (20) by writing
$T(E)={\cal N}(E)/{\cal D}(E)$, where

$\displaystyle{\cal
N}(E){=}\Gamma(-z)\Gamma(-z^{*})\Gamma(1{+}z)\Gamma(1{+}z^{*})\Gamma(\frac{1}{2}{+}ix_{1})\Gamma(\frac{1}{2}{-}ix_{1})~{}(22)$
$\displaystyle\Gamma(\frac{1}{2}{+}ix_{2})\Gamma(\frac{1}{2}{-}ix_{2}),z=(A+ik),x_{1}=(B+k),x_{2}=(B-k)$

and

${\cal
D}(E){=}\Gamma(-ik)\Gamma(ik)\Gamma(1{+}ik)\Gamma(1-ik)\Gamma^{2}(\frac{1}{2}-ik)\Gamma^{2}(\frac{1}{2}+ik)~{}(23)$

Using the properties of $\Gamma$ functions [6]:
$\Gamma(1+z)=z\Gamma(z),\Gamma(-z)\Gamma(z)=-\pi/z\sin(\pi z)$,
$\Gamma(1/2+u)\Gamma(1/2-u)=\pi/\cos(\pi u)$. Further, after interesting
manipulations with trigonometric and hyperbolic functions, we get

$T(E)=\frac{\sinh^{2}2\pi k}{(\cosh 2\pi k-\cos 2\pi A)(\cosh 2\pi k+\cosh
2\pi B)}~{}~{}~{}(24)$

One can easily see that when $k\rightarrow i\kappa$ in above, we get poles of
$T(k)$ as $\cos 2\pi\kappa=\cos 2\pi A\implies\kappa_{n}=(n\pm A)$ Choosing
the $-$ sign and letting $0\leq n<A$ ensure a finite number of bound state
eigenvalues which are identically the same as in (21).

At research levels, non-orthodox complex (Parity-Time) potentials (invariant
under the joint action of $i\to-i,x\to-x$) [13] and pseudo-Hermitian
potentials [14] are known to have real discrete spectrum. The extraction of
eigenvalues from the transmission coefficient has revealed a surprising and
unnoticed presence of real discrete spectrum in a non-Hermitian potential,
accidentally [15]. Interestingly, this model is neither PT-symmetric nor
apparently pseudo-Hermitian. However, the correctness of the claimed spectrum
has been tested by the other usual methods of finding real energy bound
states.

Lastly, we hope that students will feel self-persuaded to extract eigenvalues
of bound states of more advanced and involved potential wells which have
interesting analytic forms for transmission coefficients. These potentials
are: Rosen-Morse [16], Ginocchio [17] and one more potential [18] which
interpolates between Eckart and Morse potentials. Even potential models which
have $T(E)$ in terms of higher order functions (see [19]) are worth studying
in this regard. The poles of reflection coefficient $R(E)$ can also be
considered which are usually the same as that of $T(E)$. In more involved
cases, if they are not the same, then the common poles of the two will need to
be considered.

## Acknowledgement

We thank the physics trainees of 64th batch of Orientation Course of
Engineering and Science (OCES) of the Human Resource Development Division
(HRDD) of Bhabha Atomic Research Center for their keen interest in the course
of Quantum Mechanics.

## References

## References

  * (1) Merzbacher E. Quantum Mechanics (John Wiley and Sons, Inc.: New York) $2^{nd}$edition (1970).
  * (2) Landau L. D. and Lifshitz E. M. Quantum Mechanics (Pergamon Press, N.Y) $3^{rd}$ edition (1977).
  * (3) Rapp D. Quantum Mechanics (Holt, Rinehart and Winston, Inc.: N.Y.) (1971).
  * (4) Griffith D. J. Introduction to Quantum Mechanics $2^{nd}$ edition (New-Delhi: Pearson) (2011).
  * (5) Ahmed Z., Sharma V., Sharma M., Singhal A, Kaiwart R. and Priyadarshini P. Eur. J. Phys. 38 025401 (2017).
  * (6) Abromowitz M., Stegun I.A, Handbook of Mathematical Functions (Dover: New York) (1070).
  * (7) Yoav P.et al Quantum Mechanics, Schaum’s outlines, 2nd Ed. (Tata McGraw Hill: New Delhi) (2010).
  * (8) Lima F.M.S., Am. J. Phys. 88, 1019 (2020).
  * (9) Bates D. R. Quantum Theory (I Elements) (Academic Press: N.Y.) pp. 238-242 (1961)
  * (10) Ahmed Z., Phys. Lett. A 157 1 (1991).
  * (11) Khare A. and Sukhatme U.P. J. Phys. A: Math. Gen. 21 L501 (1988).
  * (12) Gendenshtein L. Zh. Eksp. Teor. Fiz. 38 299 1983 (JETP Lett. 38 356)
  * (13) Bender C.M., Brody D.C., and Jones H. F. Am . J. Phys. 71, 1095 (2003).
  * (14) Mostafazadeh A., J. Math. Phys. 43 205; 2814 (2002).
  * (15) Ahmed Z. and Joseph A.N., Phys. Lett. A 379, 865 (2015).
  * (16) Morse P. M. and Feshbach H. Methods of Theoretical Physics, Vol. 2(McGraw-Hill Book Co. Inc.: N.Y) pp. 1651-1660 (1953).
  * (17) Ginocchio J.N. Ann. Phys. N.Y. 152 203 (1984).
  * (18) Ahmed Z, Phys. Rev. A 47 4761 (1993).
  * (19) Ahmed Z., Ghosh D., Kumar S. , Turumella N. , Eur. J. Phys. 39 025404 (2018).