workers’ unofficial slack channel. After receiving positive feedback from the
AMT community, we released the qualification test. We also included three
questions in the qualifying test asking AMT workers to annotate YouTube videos
whose annotation labels were known in advance. These videos were already
annotated previously by the authors. AMT workers who correctly labeled all
three videos (100% score) qualified for the YouTube annotation task. In
addition to getting a perfect score on the qualification test, we also
required AMT workers to have at least a 90% approval rating on the AMT
platform.

YouTube annotation task: The YouTube annotation task required AMT workers to
assign a label to the video and also provide the rationale behind selecting
the label. We enforced a minimum word limit of 10 characters for the
rationale. We released YouTube videos in a batch sizes of 10 and 15 and
obtained three annotations for each video. The majority response was selected
as the annotation label for the video. In total, we obtained annotations for
545 videos, out of which consensus was reached on 516 videos (Supporting: 26,
Opposing: 74, Neutral: 77, Irrelevant: 318, YouTube video in a language other
than English: 6, URL not accessible: 15).

###  A.3. Annotating YouTube channels for partisan bias

Our dataset of unique videos came from a large number of YouTube channels
($\sim$17.5K) devoted to both news and non-news content. We coded the leaning
of the channel on a 5-point Likert scale (far-left, center-left, neutral,
center-right, and far-right) using computational methods and several
heuristics. First, to identify news-related channels, we used several pattern-
matching techniques (e.g., finding keyword news in the channel’s name, etc.)
and discovered a total of 802 news channels. Then we used existing datasets on
media bias from mediabiasfactcheck.com and allsides.com for annotating the
channels. For channels whose annotations were not available in the datasets,
we manually went through their title, description, sample videos, and related
information from their website, Wikipedia, and/or google search to identify
their leaning or the leaning of their affiliations. Many local news channels
such as KHOU191919https://www.youtube.com/c/KHOU or
KPRC202020https://www.youtube.com/c/KPRC2Click2Houston are affiliated with
national channels. If we did not find the bias ratings for such local
channels, we assigned them the label of their affiliations. For example, KHOU
is associated with center-left CBS and thus, was also assigned a center-left
rating. We assigned channels that didn’t fall under the news category the
neutral label. We manually checked a random sample (n=50) of non-news channels
and found only one channel that had content about the news. Therefore, this
process produced channel bias annotations (to be used as a feature in our
classifier) with reasonable accuracy for our study, given that channel bias
detection is not the main focus of our work.


# Constraints on millicharged particles by neutron stars

Xi Huang1,2,3, Xiao-Ping<EMAIL_ADDRESS>(Xiao-Ping Zheng),
<EMAIL_ADDRESS>(Xi Huang), Wei-Hua Wang1, and Shao-Ze Li1 1Institute of
Astrophysics, Central China Normal University, Wuhan 430079, China
2School of Electronic and Electrical Engineering, Wuhan Textile University,
Wuhan 430073, China
3Key Laboratory of Quark and Lepton Physics (Ministry of Education), Central
China Normal University, Wuhan 430079, China

###### Abstract

We have constrained the charge-mass ($\varepsilon-m$) phase space of
millicharged particles through the simulation of the rotational evolution of
neutron stars, where an extra slow-down effect due to the accretions of
millicharged dark matter particles is considered. For a canonical neutron star
of $M=1.4~{}M_{\odot}$ and $R=10~{}{\rm km}$ with typical magnetic field
strength $B_{0}=10^{12}$ G, we have shown an upper limit of millicharged
particles, which is compatible with recently experimental and observational
bounds. Meanwhile, we have also explored the influences on the $\varepsilon-m$
phase space of millicharged particles for different magnetic fields $B_{0}$
and dark matter density $\rho_{\rm{DM}}$ in the vicinity of the neutron star.

###### pacs:

95.35.+d, 95.30.Cq, 97.60.Jd, 97.60.Gb

##  I INTRODUCTION

Since Zwicky (1933) proposed the problem of the “missing mass”, the
theoretical and experimental studies on dark matter (DM) have attracted more
and more attention. It is well-known that the total energy of the Universe
contains about 27% DM and about 5% baryon matter as well as about 68% dark
energy according to the most recent cosmological results based on Planck
measurements of the cosmic microwave background (CMB) temperature and lensing-
potential power spectra (Ade et al.,, 2014). In recent years, experimental
physicists have concentrated their attention on the direct and indirect
detections of DM particles. Direct search experiments aim to detect individual
interactions, i.e., DM particles scatter off target nuclei of detectors, such
as DAMA (Savage et al.,, 2009), CDMS II (Ahmed et al.,, 2010), CoGeNT (Aalseth
et al.,, 2011), CRESST-II (Angloher et al.,, 2012), XENON100 (Aprile et al.,,
2012), LUX (Akerib et al.,, 2014), PandaX (Xiao et al.,, 2014) and so on.
Indirect detection experiments mainly probe annihilation or decay products
(gamma-rays, neutrinos and charged cosmic rays) of DM, such as AMS-02, PAMELA,
Fermi-LAT, IACTs, IceCube, ANTARES, Super-K (Rott,, 2013; Conrad,, 2014). From
a theoretical point of view, several candidates of DM particles are supposed,
for example, neutralinos (Jungman et al.,, 1996; Bertone et al.,, 2005),
Majorana neutrinos and technibaryons (Gudnason et al.,, 2006; Kouvaris,,
2007). In this paper, we are interested in another candidate of DM in the form
of millicharged (MC) particles (Goldberg and Hall,, 1986; Kouvaris,, 2013;
Kouvaris and Perez-Garcia,, 2014).

The electric charge of all the particles in the Standard Model appears to be
an integer multiple of $e/3$, where $e$ is the charge of the electron.
However, MC particles have electric charge $e^{\prime}=\varepsilon e$, where
$\varepsilon$ is any real number and $\varepsilon<1$. They can be either
bosons or fermions. MC particles were first proposed in order to solve the DM
puzzle a long time ago (Goldberg and Hall,, 1986; De Rújula et al.,, 1990;
Dimopoulos et al.,, 1990). Hereafter millicharged dark matter particles (MCDM)
have been studied extensively (see Ref. (Kouvaris,, 2013) and references
therein). It is worth noticing that Huh et al. (Huh et al.,, 2008) presented a
possible explanation of the 511 KeV galactic $\gamma$-ray due to positron
productions from MCDM. MC particles have an obvious impact on the standard
picture of the Universe in many ways. Firstly MC particles can significantly
influence the expansion rate of the Universe and the baryon-to-photon ratio
during the epoch of big bang nucleosynthesis (BBN) in the early Universe
(Davidson et al.,, 2000; Melchiorri et al.,, 2007; Berezhiani et al., 2013a,
). Secondly, MC particles also may explain the creation of galactic magnetic
fields at the cosmological epoch of the galaxy formation (Berezhiani et al.,
2013b, ). Thirdly, the anisotropy power spectrum of the CMB is affected by MC
particles in several respects (Dubovsky and Gorbunov,, 2001; Dolgov et al.,,
2013).

At present MC particles are already constrained by experimental and
observational data (Prinz et al.,, 1998; Davidson et al.,, 2000; Langacker and
Steigman,, 2011; Jaeckel et al.,, 2013; Berezhiani et al., 2013a, ), including
the $\varepsilon-m$ ($m$ is the mass of MC particle) phase space and the
fraction of MC particles. These existing constraints on MC particles mainly
contain laboratory bounds (including the limits from experiments searching for
MC particles (Mitsui et al.,, 1993; Prinz et al.,, 1998)), cosmological bounds
(BBN, CMB anisotropy) and constraints from stellar evolution (globular
clusters, white dwarfs, red giants, and supernova 1987A). For very light MC
particles, $m<$ 1 keV, the reactor experiments show a strict limit
$\varepsilon<10^{-5}$ (Gninenko et al.,, 2007). For MC particles that are
lighter than electrons, $m<m_{e}$, the best particle physics bound
$\varepsilon<3.4\times 10^{-5}$ is given from the data that ortho-positronium
decay to dark matter pairs (Badertscher et al.,, 2007). For heavier MC
particles, $m>m_{e}$, the bounds become weaker (Prinz et al.,, 1998). For $m=$
1 MeV, the bound is $\varepsilon<4.1\times 10^{-5}$, and for $m=$ 100 MeV, the
bound is up to $\varepsilon<5.8\times 10^{-4}$. If the MC particles become
heavier than 100 MeV, the bound $\varepsilon\sim 10^{-2}$ is allowed, while
for $m>$ 1 GeV it can be as large as $\varepsilon=0.1$.

With the discovery of neutron star (NS) by Bell and Hewish (Hewish et al.,,
1968), the properties of NS have been studied by many theoretical and
experimental physicists (Glendenning,, 1997; Page and Reddy,, 2006; Lattimer
and Prakash,, 2007; Potekhin,, 2010). NS is a compact star, which is
characterized by strong gravitational field, electromagnetic field, extreme
strong and weak interaction. Thus it is usually regarded as a “natural
laboratory” with extreme physical condition. At present it seems that all
currently observed pulsar periods mainly lie between milliseconds and a few
seconds. As it can be seen from the $P-\dot{P}$ ($P$ and $\dot{P}$ are the
spin period and its time derivative, respectively) diagram (see Fig. 1), most
radio pulsars have a period of around one second and the period derivatives in
the range of $10^{-16}$ to $10^{-14}$ s s-1. However, one group of sources
having higher magnetic field ($10^{13-15}$ G), relatively long periods ($>$1
s), and high period derivatives ($10^{-13}-10^{-9}$ s s-1 ) are magnetars. The
other group of sources are millisecond pulsars with old age ($\sim 10^{9}$ yr)
and lower magnetic field ($10^{8-9}$ G). In the ATNF pulsar catalogue
(Manchester et al.,, 2005), the longest spin period for normal radio pulsars
to date is $\sim$8.51 s of PSR J2144-3933 whose characteristic age is
$2.72\times 10^{8}$ yr and surface magnetic field is $2.08\times 10^{12}$ G
(Young et al.,, 1999). Except for short-lived glitches (McCulloch et al.,,
1983), the rate of rotation of a NS drops steadily as a function of time,
especially for millisecond pulsars which have very low values of $\dot{P}$ and
very low timing noise. It is generally believed that the spin-down of the NS
is due to magnetic dipole radiation (MDR) and electromagnetic torques in the
magnetosphere (Contopoulos and Spitkovsky,, 2006).

Limits on DM from neutron stars have been deeply investigated in a large
amount of literatures so far (Bertone and Fairbairn,, 2008; Kouvaris,, 2008;
Kouvaris and Tinyakov,, 2010; de Lavallaz and Fairbairn,, 2010; Fan et al.,,
2011; Kouvaris and Tinyakov,, 2011; Kouvaris,, 2012; Huang et al.,, 2014;
Zheng et al.,, 2014). Recently, Kouvaris and Perez-Garcia (Kouvaris and Perez-
Garcia,, 2014) have discovered that the electric charges could be expelled
from the star as MCDM are accreted onto the NS. The escaping charged particles
will provide an extra current, thus the torque is produced. They find that
this mechanism yields an extra spin-down of neutron stars and the braking
indices can be substantially smaller than 3 predicted by the MDR model.

We follow the philosophy of Kouvaris and Perez-Garcia (Kouvaris and Perez-
Garcia,, 2014), but explore how to constrain the $\varepsilon-m$ of MC
particles via the currently observed pulsar periods. We consider the effect of
the additional torque by the extra currents due to the accretions of MCDM onto
the NS on the period of pulsars. We also impose a limit of pulsar’s period in
the $P-\dot{P}$ diagram on the period’s evolution equation as a cut-off. By
doing so, the $\varepsilon-m$ of MC particles will be constrained under the
given DM density.

This paper is organized as follows. Firstly, we revisit the mechanism of the
accretions of MCDM onto the NS (electromagnetic accretion), then the
rotational evolution equations of a NS considered the MDR and the torques,
which are produced by the Goldreich and Julian (GJ) current (Goldreich and
Julian,, 1969) $I_{\rm GJ}$ and the extra current $I_{\rm DM}$, are shown in
Section 2. Secondly, we numerically show the bounds of the $\varepsilon-m$ of
MC particles in Section 3. Finally, we present the conclusions in Section 4.

##  II THE MODELS

First we show how the accretions of MCDM onto the NS can lead to an excess of
currents (Kouvaris and Perez-Garcia,, 2014). If the Larmor radius $r_{\rm
L}=\frac{m\upsilon_{\perp}}{\varepsilon eB}$ of MCDM is much smaller than the
curvature radius $R_{\rm curv}=|\frac{1}{B}\frac{{\rm d}B}{{\rm d}r}|^{-1}$,
it may follow the magnetic field lines in the magnetosphere within the light
cylinder of the NS and cross the surface of the star. $\upsilon_{\perp}$ is
the particle’s velocity perpendicular to the magnetic field $B$. As the
accumulations of MCDM trapped in the NS, once the Coulomb repulsive force
becomes larger than the gravitational force, the electric charges will be
expelled from the star. The expelled charges are in the form of either
electrons or protons, which depend on the charge of the accreted MCDM. Once
the equilibrium between the accretions of MCDM and the expulsions of electric
charges has been established, stable extra currents will be formed.

As stated previously, the slow-down of the rotation powered pulsars is due to
the MDR and the braking torques provided by the outflowing plasmas (Kramer et
al.,, 2006). The rotational kinetic energy loss rate of the NS is
$\dot{E}=\mathcal{I}\Omega\dot{\Omega}=-L$, where $\mathcal{I}$ and $\Omega$
are the moment of inertia and the angular velocity of the star, respectively,
and $L$ can be expressed as two components (Harding et al.,, 1999)

$L=L_{\rm orth}\sin^{2}\theta+L_{\rm align}\cos^{2}\theta.$ (1)

In the above equation, $L_{\rm orth}$ represents the “orthogonal” case that
the angular momentum of the star is decreased by the MDR, $L_{\rm align}$
represents the “aligned” case that the torque is produced by the electric
current from escaping charged particles that follow the open magnetic-field
lines in the magnetosphere of the NS, and $\theta$ is the angle between the
rotational and the magnetic axes (magnetic inclination angle). The two
components can be written as

$L_{\rm orth}=\frac{B_{0}^{2}\Omega^{4}R^{6}}{4c^{3}},\hskip 28.45274ptL_{\rm
align}=\frac{B_{0}\Omega(\Omega-\Omega_{\rm death})R^{3}I}{2c^{2}},$ (2)

where $B_{0}$ and $R$ are the magnetic field strength on the polar cap (the
open magnetic-field lines in the light cylinder connect to the surface of the
star) and radius of the star, respectively, $c$ is the light speed, and
$\Omega_{\rm death}$ is the angular velocity below which the pulsar emission
dies. The total current $I$ can be written as the sum of $I_{\rm GJ}$ and
$I_{\rm DM}$, $I=I_{\rm GJ}+I_{\rm DM}$, where $I_{\rm GJ}$ represents the
emission of relativistic charged particles from the surface cap regions
($\sim\pi R_{C}^{2}$, $R_{C}$ is the polar cap radius) of the NS, due to a
particle acceleration gap potential $V_{\rm{gap}}$ of the order of $10^{12}$ V
(Hibschman and Arons,, 2001) that develops along open magnetic field lines in
the vicinity of the polar cap, and $I_{\rm DM}$ represents the extra current
from the expelled electric charges (electrons or protons) due to the
accretions of MCDM onto the NS.

According to the pioneering work of Goldreich and Julian (Goldreich and
Julian,, 1969), the GJ current is $I_{\rm GJ}=\pi R_{C}^{2}\rho_{\rm GJ}c$,
where $\rho_{\rm GJ}$ is the GJ charge density. The polar cap radius can be
expressed as $R_{C}=R\sin{\theta_{\rm P}}=R(R/R_{\rm L})^{1/2}$, where
$\theta_{\rm P}$ is the angle of the polar cap to the center of the star and
$R_{\rm L}=cP/2\pi$ is the light cylinder radius (Sturrock,, 1971). In the
polar cap region, the GJ charge density is given by (Goldreich and Julian,,
1969)

$\rho_{\rm GJ}=7\times
10^{10}e\left(\frac{B_{0}}{10^{12}~{}\rm{G}}\right)\left(\frac{\rm~{}s}{P}\right)~{}\rm{cm^{-3}},$
(3)

thus, we can easily deduce

$I_{\rm GJ}\simeq 1.4\times
10^{30}e\left(\frac{B_{0}}{10^{12}~{}\rm{G}}\right)\left(\frac{\rm~{}s}{P}\right)^{2}~{}\rm{s^{-1}}.$
(4)

For the extra current $I_{\rm DM}$ due to the accretions of MCDM onto the NS,
we directly adopt the results in Ref. (Kouvaris and Perez-Garcia,, 2014),

$I_{\rm DM}\simeq 1.0\times 10^{29}\varepsilon e\left(\frac{\rho_{\rm
DM}}{0.3~{}\rm{GeV}~{}\rm{cm^{-3}}}\right)\left(\frac{1~{}\rm{GeV}}{m}\right)\left(\frac{P}{\rm~{}s}\right)~{}\rm{s^{-1}}.$
(5)

Figure 1: Distribution of all known pulsars on the $P-\dot{P}$ diagram. Dots
are rotation powered pulsars (including normal pulsars and millisecond
pulsars, which dominate the pulsar population) (from ATNF:
http://www.atnf.csiro.au/research/pulsar/psrcat/). Triangles are magnetars
(from McGill magnetar catalog:
http://www.physics.mcgill.ca/$\sim$pulsar/magnetar/main.html). The red star
denotes the famous PSR J2144-3933, which has the longest spin period for
normal radio pulsars. The dash lines of pulsar characteristic age are defined
by ${P}/({2\dot{P}})$ and another dash lines of surface dipole magnetic field
are conventionally defined as $3.2\times 10^{19}(P\dot{P})^{1/2}$, which are
based on the MDR.

Finally, using the Eqs. (1) and (2), we can get the rotational evolution
equation of the NS

$\frac{{\rm d}\Omega}{{\rm
d}t}=-\frac{B_{0}^{2}\Omega^{3}R^{6}}{4\mathcal{I}c^{3}}\left[\sin^{2}\theta+\left(1-\frac{\Omega_{\rm
death}}{\Omega}\right)\left(1+\frac{I_{\rm DM}}{I_{\rm
GJ}}\right)\cos^{2}\theta\right].$ (6)

We define $\lambda=\frac{I_{\rm DM}}{I_{\rm GJ}}$ for convenience. As we can
see from Eq. (6), it easily recover the scene of the MDR for $\lambda=0$ (no
extra current). It is worth noticing that we can take $\frac{\Omega_{\rm
death}}{\Omega}\ll 1$ safely, since normal pulsars couldn’t be older than
$10^{7}$ to $10^{8}$ years. We compare the evolution curve of period from Eq.
(6) with that from the scene of pure MDR. Fig. 2 shows the differences between
them. It is obvious from Fig. 2 that the differences become larger gradually
as the increases of age. We use the 8.51 s period at $2.72\times 10^{8}$ yr
(which are the observational data of PSR J2144-3933) as the cut-off of neutron
stars, because it is the longest observed period of normal pulsars and
possible longer period should fall below the so-called “death line” on the
$P-\dot{P}$ diagram, where pulse emission can’t be observed. As shown in Fig.
1, the location of PSR J2144-3933 on the $P-\dot{P}$ diagram indicates that
the star could be a good candidate as the cut-off for the period evolution and
the characteristic age of normal radio pulsars. However, both the millisecond
pulsars and magnetars are unsuitable to constrain the $\varepsilon-m$ of MC
particles. It is generally believed that the millisecond pulsars have a
complicated and turned evolution. Recycling neutron stars to millisecond
periods may be a key process. Additionally, the extra current $I_{\rm DM}$ is
much smaller than the corresponding GJ current $I_{\rm GJ}$ according to Eq.
(4) for millisecond pulsars. The tiny differences have no imprint in Fig. 2
even if we overlook the recycled processes. For higher magnetic fields, there
are only 28 currently known magnetars and magnetar candidates (Olausen and
Kaspi,, 2014). The physics of magnetars is unclear so far. Their emission is
probably powered by the non-rotational kinetic energy or by their decay of
super-strong magnetic field (Goldreich and Reisenegger,, 1992; Heyl and
Kulkarni,, 1998). The pure MDR model may not be a good description for
magnetars. Therefore, we think that it is seriously inaccurate and unreliable
to constrain the parameters of MC particles by magnetars.

Figure 2: The period evolution of a canonical NS with $M=1.4~{}M_{\odot}$ and
$R=10~{}{\rm km}$ for $B_{0}=10^{12}$ G. The dash curve represents the spin-
down induced by pure MDR ($\lambda=0$ in Eq. (6)). The solid curve shows an
extra slow-down of the NS with magnetic inclination angle $\theta=45^{\circ}$
by the additional torque due to the accretions of MCDM with $m=1$ GeV and
$\varepsilon=5.04\times 10^{-2}$. The labels for the red dots represent the
8.51 s period at $2.72\times 10^{8}$ yr.

##  III NUMERICAL RESULTS AND DISCUSSIONS

Figure 3: The $\varepsilon-m$ phase space of MC particles constrained by a
canonical NS of $M=1.4~{}M_{\odot}$ and $R=10~{}{\rm km}$ with $B_{0}=10^{12}$
G for different magnetic inclination angles.

For convenience, we assume that the star consists of nonsuperfluid neutrons,
protons and electrons, and suppose that the density of the star is uniform. In
our calculations, we consider a canonical NS of $M=1.4~{}M_{\odot}$ and
$R=10~{}{\rm km}$, and take the initial period $P_{0}=1\hskip 2.84544pt{\rm
ms}$, the moment of inertia of the star $\mathcal{I}=\frac{2}{5}MR^{2}\approx
10^{45}~{}\rm{g~{}cm^{2}}$, and the local DM density in the vicinity of the NS
$\rho_{\rm{DM}}=~{}100\times 0.3~{}\rm{GeV/cm^{3}}$ ($0.3~{}\rm{GeV/cm^{3}}$
represents the standard DM density around the Earth).

The $\varepsilon-m$ phase space of MC particles could be constrained using the
following strategy. First, we plug a great number of groups of
($\varepsilon,m$) into Eqs. (4), (5) and (6) for various magnetic field
strength and different magnetic inclination angles under the given DM density
in the vicinity of the NS. Then, we simulate Eq. (6) up to $2.72\times 10^{8}$
yr as the cut-off for the age of the NS. If the evolution of the rotational
period of the star exceeds the longest observed period of normal radio pulsars
to date ($\sim$8.51 s), we consider this group of ($\varepsilon,m$) invalid.

The $\varepsilon-m$ phase space of MC particles constrained by a canonical NS
of $M=1.4~{}M_{\odot}$ and $R=10~{}{\rm km}$ with typical $B_{0}=10^{12}$ G
for different magnetic inclination angles is shown in Fig. 3. It can be seen
from Fig. 3(a) that the charge of MC particles can reach up to
$\varepsilon=2.52\times 10^{-2}$ for heavier particles with $m=$ 1 GeV, the
bound $\varepsilon>2.52\times 10^{-5}$ is ruled out apparently for MC
particles with $m=$ 1 MeV, and the limit $\varepsilon\leq 2.52\times 10^{-8}$
is allowed for lighter particles with $m=$ 1 KeV. Furthermore, if the mass $m$
decreases by one order of magnitude, the charge $\varepsilon$ would reduce by
the same one order of magnitude. As shown above, the lighter the mass, the
smaller the charge. It is obvious from Eq. (6) that the rotational angular
velocity of the star could decrease rapidly as $\lambda$ increases for the
same magnetic field strength $B_{0}$, which result in the rise of the spin
period even to the unreasonable value. Fig. 3(b), (c) and (d) show that the
upper limit of $\varepsilon$ is inversely proportional to $\cos^{2}\theta$ for
the same mass $m$ of MC particles (see Eq. (6)). To summarize, we would
present an upper limit of MC particles,
$(\frac{1~{}\rm{GeV}}{m})\times\varepsilon\leq 2.52\times
10^{-2}/\cos^{2}\theta$, which is consistent with experimental and
observational bounds (Gninenko et al.,, 2007; Badertscher et al.,, 2007; Prinz
et al.,, 1998).

It is worth stressing that, as shown in Eq. (6), the rate of deceleration of
the rotation $\dot{\Omega}$ also depends on the magnetic field strength
$B_{0}$. For $B_{0}=10^{11}$ G, the bounds become weaker, the charge
$\varepsilon$ of MC particles can’t be constrained for heavier particles with
$m=$ 1 GeV, however up to $\varepsilon\leq 3.26\times 10^{-3}/\cos^{2}\theta$
for $m=$ 1 MeV and $\varepsilon\leq 3.26\times 10^{-6}/\cos^{2}\theta$ for
lighter particles with $m=$ 1 KeV. On the other hand, it can be seen from Eq.
(5) that, the extra current $I_{\rm DM}$ relies on the DM density
$\rho_{\rm{DM}}$ in the vicinity of the NS. Although $\rho_{\rm{DM}}$ around
the NS is uncertain to date, we could draw a conclusion that, the larger the
DM density, the smaller the charge for the MC particles with the same mass,
e.g., if the DM density $\rho_{\rm{DM}}$ increases by one order of magnitude,
the charge $\varepsilon$ would decrease by one order of magnitude
correspondingly for the same mass $m$ of MC particles.

##  IV CONCLUSIONS

We have constructed the constraints of the $\varepsilon-m$ of MC particles by
neutron stars based on an enhanced slow-down of neutron stars due to an extra
current yield by the accretions of MCDM. For a canonical NS of
$M=1.4~{}M_{\odot}$ and $R=10~{}{\rm km}$ with typical magnetic field strength
$B_{0}=10^{12}$ G, we have shown an upper limit of MC particles,
$(\frac{1~{}\rm{GeV}}{m})\times\varepsilon\leq 2.52\times
10^{-2}/\cos^{2}\theta$, which indicates the charge would be smaller as the
mass of MC particles becomes lighter and is compatible with experimental and
observational bounds. The specific limits are as follows, the charge of MC
particles could rise to $\varepsilon=2.52\times 10^{-2}/\cos^{2}\theta$ for
heavier particles with $m=$ 1 GeV, the bounds $\varepsilon\leq 2.52\times
10^{-5}/\cos^{2}\theta$ and $\varepsilon\leq 2.52\times
10^{-8}/\cos^{2}\theta$ are allowed for the MC particles with $m=$ 1 MeV and
$m=$ 1 KeV, respectively, i.e., if the mass $m$ decreases by one order of
magnitude, the charge $\varepsilon$ would reduce by the same one order of
magnitude. However, for the NS with lower magnetic fields ($10^{10-11}$ G),
the bounds become weaker. In particular, for millisecond pulsars ($10^{8-9}$
G) and magnetars ($10^{13-15}$ G), it is not appropriate to bound the
$\varepsilon-m$ of MC particles. In addition, we have also investigated the
influence on the $\varepsilon-m$ of MC particles for the different DM density
$\rho_{\rm{DM}}$ in the vicinity of the NS. It is obvious that, the larger the
DM density, the smaller the charge for the MC particles with the same mass.

The model we adopted is uniform stellar configuration. However, it is well-
known that the neutron stars constructed by the realistic equations of states
can be approximated as the uniform case. Our work have shown an upper limit of
the $\varepsilon-m$ of MC particles based on the new mechanism, and the
results will be unchanged in the order of magnitude when considering the
realistic equations of states.

## ACKNOWLEDGMENTS

The authors would like to thank the referee very much for helpful comments and
Yunwei Yu for useful discussions, which have significantly improved our work.
We also thank Shuhua Yang for help in English writings. This work is supported
by the National Natural Science Foundation of China (Grant No. 11178001) and
CCNU-QLPL Innovation Fund (Grant No. QLPL2015P01).

## References

  * Ade et al., (2014) P. A. R. Ade et al. (Planck Collaboration), Astron. Astrophys. 571, 16 (2014).
  * Savage et al., (2009) C. Savage, K. Freese, P. Gondolo, and D. Spolyar, J. Cosmol. Astropart. Phys. 09, 036 (2009).
  * Ahmed et al., (2010) Z. Ahmed et al. (CDMS II Collaboration), Science 327, 1619 (2010).
  * Aalseth et al., (2011) C. E. Aalseth et al., Phys. Rev. Lett. 106, 131301 (2011).
  * Angloher et al., (2012) G. Angloher et al., Eur. Phys. J. C 72, 1971 (2012).
  * Aprile et al., (2012) E. Aprile et al. (XENON100 Collaboration), Phys. Rev. Lett. 109, 181301 (2012).
  * Akerib et al., (2014) D. S. Akerib et al. (LUX Collaboration), Phys. Rev. Lett. 112, 091303 (2014).
  * Xiao et al., (2014) M. J. Xiao et al. (PandaX Collaboration), Sci. China Phys. Mech. Astron. 57, 2024 (2014).
  * Rott, (2013) C. Rott, Nucl. Phys. B, Proc. Suppl. 235, 413 (2013).
  * Conrad, (2014) J. Conrad, arXiv:1411.1925.
  * Jungman et al., (1996) G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. 267, 195 (1996).
  * Bertone et al., (2005) G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279 (2005).
  * Gudnason et al., (2006) S. B. Gudnason, C. Kouvaris, and F. Sannino, Phys. Rev. D 74, 095008 (2006).
  * Kouvaris, (2007) C. Kouvaris, Phys. Rev. D 76, 015011 (2007).
  * Goldberg and Hall, (1986) H. Goldberg and L. J. Hall, Phys. Lett. B 174, 151 (1986).
  * Kouvaris, (2013) C. Kouvaris, Phys. Rev. D 88, 015001 (2013).
  * Kouvaris and Perez-Garcia, (2014) C. Kouvaris and M. A. Perez-Garcia, Phys. Rev. D 89, 103539 (2014).
  * De Rújula et al., (1990) A. De Rújula, S. L. Glashow, and U. Sarid, Nucl. Phys. B 333, 173 (1990).
  * Dimopoulos et al., (1990) S. Dimopoulos, D. Eichler, R. Esmailzadeh, and G. D. Starkman, Phys. Rev. D 41, 2388 (1990).
  * Huh et al., (2008) J.-H. Huh, J. E. Kim, J.-C. Park, and S. C. Park, Phys. Rev. D 77, 123503 (2008).
  * Davidson et al., (2000) S. Davidson, S. Hannestad, and G. Raffelt, J. High Energy Phys. 05, 003 (2000).
  * Melchiorri et al., (2007) A. Melchiorri, A. Polosa, and A. Strumia, Phys. Lett. B 650, 416 (2007).
  * (23) Z. Berezhiani, A. Dolgov, and I. Tkachev, J. Cosmol. Astropart. Phys. 02, 010 (2013).
  * (24) Z. Berezhiani, A. D. Dolgov, and I. I. Tkachev, Eur. Phys. J. C 73, 2620 (2013).
  * Dubovsky and Gorbunov, (2001) S. L. Dubovsky and D. S. Gorbunov, Phys. Rev. D 64, 123503 (2001).
  * Dolgov et al., (2013) A. D. Dolgov, S. L. Dubovsky, G. I. Rubtsov, and I. I. Tkachev, Phys. Rev. D 88, 117701 (2013).
  * Prinz et al., (1998) A. A. Prinz et al., Phys. Rev. Lett. 81, 1175 (1998).
  * Langacker and Steigman, (2011) P. Langacker and G. Steigman, Phys. Rev. D 84, 065040 (2011).
  * Jaeckel et al., (2013) J. Jaeckel, M. Jankowiak, and M. Spannowsky, Phys. Dark Univ. 2, 111 (2013).
  * Mitsui et al., (1993) T. Mitsui et al., Phys. Rev. Lett. 70, 2265 (1993).
  * Gninenko et al., (2007) S. N. Gninenko, N. V. Krasnikov, and A. Rubbia, Phys. Rev. D 75, 075014 (2007).
  * Badertscher et al., (2007) A. Badertscher et al., Phys. Rev. D 75, 032004 (2007).
  * Hewish et al., (1968) A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott, and R. A. Collins, Nature 217, 709 (1968).
  * Glendenning, (1997) N. K. Glendenning, Compact stars (Springer-Verlag, New York, 2000)
  * Page and Reddy, (2006) D. Page and S. Reddy, Annu. Rev. Nucl. Part. Sci. 56, 327 (2006).
  * Lattimer and Prakash, (2007) J. M. Lattimer and M. Prakash, Phys. Rep. 442, 109 (2007).
  * Potekhin, (2010) A. Y. Potekhin, Phys. Usp. 53, 1235 (2010).
  * Manchester et al., (2005) R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, Astron. J. 129, 1993 (2005).
  * Young et al., (1999) M. D. Young, R. N. Manchester, and S. Johnston, Nature 400, 848 (1999).
  * McCulloch et al., (1983) P. M. McCulloch, P. A. Hamilton, G. W. R. Royle, and R. N. Manchester, Nature 302, 319 (1983).
  * Contopoulos and Spitkovsky, (2006) I. Contopoulos and A. Spitkovsky, Astrophys. J. 643, 1139 (2006).
  * Bertone and Fairbairn, (2008) G. Bertone and M. Fairbairn, Phys. Rev. D 77, 043515 (2008).
  * Kouvaris, (2008) C. Kouvaris, Phys. Rev. D 77, 023006 (2008).
  * Kouvaris and Tinyakov, (2010) C. Kouvaris and P. Tinyakov, Phys. Rev. D 82, 063531 (2010).
  * de Lavallaz and Fairbairn, (2010) A. de Lavallaz and M. Fairbairn, Phys. Rev. D 81, 123521 (2010).
  * Fan et al., (2011) Y. Z. Fan, R. Z. Yang, and J. Chang, Phys. Rev. D 84, 103510 (2011).
  * Kouvaris and Tinyakov, (2011) C. Kouvaris and P. Tinyakov, Phys. Rev. D 83, 083512 (2011).
  * Kouvaris, (2012) C. Kouvaris, Phys. Rev. Lett. 108, 191301 (2012).
  * Huang et al., (2014) X. Huang, W. Wang, and X. P. Zheng, Sci. China Phys. Mech. Astron. 57, 791 (2014).
  * Zheng et al., (2014) H. Zheng, K. J. Sun, and L. W. Chen, arXiv:1408.2926.
  * Goldreich and Julian, (1969) P. Goldreich and W. H. Julian, Astrophys. J. 157, 869 (1969).
  * Kramer et al., (2006) M. Kramer et al., Science 312, 549 (2006).
  * Harding et al., (1999) A. K. Harding, I. Contopoulos, and D. Kazanas, Astrophys. J. Lett. 525, L125 (1999).
  * Hibschman and Arons, (2001) J. A. Hibschman and J. Arons, Astrophys. J. 554, 624 (2001).
  * Sturrock, (1971) P. A. Sturrock, Astrophys. J. 164, 529 (1971).
  * Olausen and Kaspi, (2014) S. A. Olausen and V. M. Kaspi, Astrophys. J. Suppl. Ser. 212, 6 (2014).
  * Goldreich and Reisenegger, (1992) P. Goldreich and A. Reisenegger, Astrophys. J. 395, 250 (1992).
  * Heyl and Kulkarni, (1998) J. S. Heyl and S. R. Kulkarni, Astrophys. J. Lett. 506, L61 (1998).