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Advances in High Energy Physics
Volume 2014, Article ID 869425, 5 pages
http://dx.doi.org/10.1155/2014/869425
Research Article
Dark Atoms and the Positron-Annihilation-Line Excess
in the Galactic Bulge
J.-R. Cudell,1 M. Yu. Khlopov,2,3,4 and Q. Wallemacq1
1
IFPA, “D´epartement d’AGO”, Universit´e de Li`ege, Sart Tilman, 4000 Li`ege, Belgium
National Research Nuclear University “Moscow Engineering Physics Institute,” Moscow 115409, Russia
3
Centre for Cosmoparticle Physics “Cosmion,” 115409 Moscow, Russia
4
APC Laboratory 10, rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France
2
Correspondence should be addressed to Q. Wallemacq; [email protected]
Received 25 November 2013; Accepted 13 January 2014; Published 25 February 2014
Academic Editor: Chris Kouvaris
Copyright © 2014 J.-R. Cudell et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The
publication of this article was funded by SCOAP3 .
It was recently proposed that stable particles of charge −2, O−− , can exist and constitute dark matter after they bind with primordial
helium in O-helium (OHe) atoms. We study here in detail the possibility that this model provides an explanation for the excess
of gamma radiation in the positron-annihilation line from the galactic bulge observed by INTEGRAL. This explanation assumes
that OHe, excited to a 2s state through collisions in the central part of the Galaxy, deexcites to its ground state via an 0 transition,
emitting an electron-positron pair. The cross-section for OHe collisions with excitation to 2s level is calculated and it is shown
that the rate of such excitations in the galactic bulge strongly depends not only on the mass of O-helium, which is determined by
the mass of O−− , but also on the density and velocity distribution of dark matter. Given the astrophysical uncertainties on these
distributions, this mechanism constrains the O−− mass to lie in two possible regions. One of these is reachable in the experimental
searches for stable multicharged particles at the LHC.
1. Introduction
According to modern cosmology, dark matter corresponds
to 25% of the total cosmological density, is nonbaryonic,
and consists of new stable particles. Such particles (see [1–
6] for reviews and references) should be stable, provide
the measured dark-matter density, and be decoupled from
plasma and radiation at least before the beginning of the
matter-dominated era. It was recently shown that heavy stable
particles of charge −2, O−− , bound to primordial helium
in OHe atoms, can provide an interesting explanation for
cosmological dark matter [6, 7]. It should also be noted
that the nuclear cross-section of the O-helium interaction
with matter escapes the severe constraints [8–10] on strongly
interacting dark-matter particles (SIMPs) [8–16] imposed by
the XQC experiment [17, 18].
The hypothesis of composite O-helium dark matter, first
considered to provide a solution to the puzzles of direct darkmatter searches, can offer an explanation for another puzzle
of modern astrophysics [6, 7, 19]: this composite dark-matter
model can explain the excess of gamma radiation in the
electron-positron-annihilation line, observed by INTEGRAL
in the galactic bulge (see [20] for a review and references).
The explanation assumes that OHe provides all the galactic
dark matter and that its collisions in the central part of
the Galaxy result in 2s-level excitations of OHe which are
deexcited to the ground state by an 0 transition, in which an
electron-positron pair is emitted. If the 2s level is excited, pair
production dominates over the two-photon channel in the
deexcitation, because electrons are much lighter than helium
nuclei, and positron production is not accompanied by a
strong gamma-ray signal.
According to [21] the rate of positron production 3 ⋅
1042 s−1 is sufficient to explain the excess in the positronannihilation line from the bulge measured by INTEGRAL. In
the present paper we study the process of 2s-level excitation
of OHe from collisions in the galactic bulge and determine
the conditions under which such collisions can provide
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the observed excess. Inelastic interactions of O-helium with
matter in interstellar space and subsequent deexcitation can
give rise to radiation in the range from a few keV to a
few MeV. In the galactic bulge with radius  ∼ 1 kpc
the number density of O-helium can be of the order of
 ≈ 3 ⋅ 10−3 /3 cm−3 or larger, and the collision rate of Ohelium in this central region was estimated in [19]: / =
2 Vℎ 43 /3 ≈ 3 ⋅ 1042 3−2 s−1 , with 3 = OHe /1 TeV. At the
velocity of Vℎ ∼ 3 ⋅ 107 cm/s energy transfer in such collisions
is Δ ∼ 1 MeV3 . These collisions can lead to excitation of
O-helium. If OHe levels with nonzero angular momentum
are excited, gamma lines should be observed from transitions
( > )  = 1.598 MeV(1/2 − 1/2 ) (or from similar
transitions corresponding to the case  = 1.287 MeV) at the
level 3 ⋅ 10−4 3−2 (cm2 s MeV ster)−1 .
2. Collisional Excitation Cross-Section
The studied reaction is the collision between two OHe atoms,
both being initially in their ground state |1s⟩, giving rise to
the excitation of one of them to a |s⟩ state while the other
remains in its ground state:
OHe (1s) + OHe (1s) → OHe (1s) + OHe (s) .
(1)
If we work in the rest frame of the OHe that gets excited
and if we neglect its recoil after the collision, the differential
cross-section of the process is given by

2
⃗ 
 (1s → s) = 2⟨s, ⃗ ||1s, ⟩

× (
2
2
3 



,
+ s −
− 1s )
2
2
(2)3
(2)
where  is the mass of OHe, ⃗ and ⃗ are the momenta of
the incident OHe before and after the collision, 1s and s
are the ground-state and excited-state energies of the target
OHe, and  is the interaction potential between the incident
and the target OHe’s.
We will neglect the internal structure of the incident OHe,
so that its wave functions are plane waves. ⃗ is normalized to
obtain a unit incident current density and the normalisation
of ⃗ is chosen for it to be pointlike, that is, the Fourier
transform of (3) ()⃗ [22]:
⃗ = √
 ⋅⃗ ⃗
 ,

⃗ ⋅ ⃗
⃗ = 
(3)
,
where  ⃗ is the position vector of the incident OHe and  = ||.⃗
In the following, we will be led to consider O−− masses
which are much larger than the mass of helium or the boundstate energies. Therefore, the origin of the rest frame of the
target OHe coincides with the position of its O−− component
and its reduced mass  can be taken as the mass of helium
He .
The OHe that gets excited is described as a hydrogen-like
atom, with energy levels s = −0.5He (He O )2 /2 and
initial and final bound-state wave functions 1s and s of a
hydrogenoid atom with a Bohr radius 0 = (He He O )−1 .
The incident OHe interacts with the O−− and helium
components in the target OHe, so that the interaction
potential  is the sum of the two contributions O and He :
⃗ ),
 ()⃗ = O ()⃗ + He ( ⃗ − He
(4)
⃗ is the position vector of the helium component.
where He
The first term O gives a zero contribution to the integral
of expression (2) since the states 1s and s are orthogonal.
For the second term, we treat the incident OHe as a heavy
neutron colliding on a helium nucleus through short-range
nuclear forces. The interaction potential can then be written
in the form of a contact term:
2
⃗ ),
⃗ )=−
  ( ⃗ − He
He ( ⃗ − He
(5)
He 0
where we have normalised the delta function to obtain an
OHe-helium elastic cross-section equal to 402 .
Going to spherical coordinates for ⃗ and integrating over

 = |⃗ | in the differential cross-section (2), together with
the previous expressions (3), (4), and (5), we get
 (1s → s) = (
 2 2 
) 0 ( )
He

2

⃗
∗
s
1s 3 He  Ω,
× ∫ −.⃗ He


(6)
where  ⃗ = ⃗ − ⃗ is the transferred momentum and Ω is the
solid angle. From the integration over the delta function in
(2), we have obtained the conservation of energy during the
process:
2 = 2 + 2 (1s − s ) .
(7)
It leads to the threshold energy corresponding to 2 = 0 and
to a minimum incident velocity Vmin = √2(s − 1s )/. The
previous expression for  allows us to express the squared
modulus of  ⃗ as
2 = 2 (2 +  (1s − s )
(8)
−√2
+ 2 (1s − s ) cos ) ,
where  is the deviation angle of the incident OHe with
respect to the collision axis in the rest frame of the target OHe.
+ − pairs will be dominantly produced if OHe is excited
to a 2s state, since the only deexcitation channel is in this case
from 2s to 1s. As + − pair production is the only possible
channel, the differential pair-production cross-section  is
equal to the differential collisional excitation cross-section.
By particularizing expression (6) to the case  = 2, one finally
gets

4

22
.
= 5122 ( 2 ) 06 ( )
 cos 
 2(42 2 + 9)6
He
0
(9)
Advances in High Energy Physics
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3. The + − Pair-Production Rate in the
Galactic Bulge
The total + − pair-production rate in the galactic bulge is
given by
2
⃗
DM
()
 
⃗ ,⃗
⟨ V⟩ ()
 = ∫
2
 


(10)
where  is the volume of the galactic bulge, which is a sphere
of radius  = 1.5 kpc, DM is the energy density distribution
of dark matter in the galactic halo, and ⟨ V⟩ is the pairproduction cross-section  times relative velocity V averaged over the velocity distribution of dark-matter particles.
The total pair-production cross-section  is obtained by
integrating (9) over the diffusion angle. Its dependence on the
relative velocity V is contained in ,  , and  through  = V
and the expressions (7) and (8) of  and  in terms of .
We use a Burkert [23, 24] flat, cored, dark-matter density
profile known to reproduce well the kinematics of disk
systems in massive spiral galaxies and supported by recent
simulations including supernova feedback and radiation
pressure of massive stars [25] in response to the cuspy halo
problem:
DM () = 0
03
,
( + 0 ) (2 + 02 )
(11)
where  is the distance from the galactic center. The central
dark-matter density 0 is left as a free parameter and 0 is
determined by requiring that the local dark-matter density at
 = ⊙ = 8 kpc is ⊙ = 0.3 GeV/cm3 . The dark-matter mass
enclosed in a sphere of radius  is therefore given by
DM () = 0 03 {log (
2 + 02
)
02
+2 log (
 + 0

) − 2 arctan ( ) } .
0
0
(12)
For the baryons in the bulge, we use an exponential profile
[26] of the form
 () =
bulge
83
−/ ,
(13)
where bulge = 1010 ⊙ [27] is the mass of the bulge. This
gives the baryonic mass distribution in the galactic bulge
 () = bulge {1 − −/ (1 +
 2
+ )} .
 2
(14)
We assume a Maxwell-Boltzmann velocity distribution
for the dark-matter particles of the galactic halo, with a
velocity dispersion () and a cutoff at the galactic escape
velocity Vesc ():
 (, V⃗ℎ ) =
1 −Vℎ2 /2 ()
,

 ()
(15)
where V⃗ℎ is the velocity of the dark-matter particles in the
frame of the halo and () = 2 (√ erf(Vesc /) −
2
2
2Vesc −Vesc / ) is a normalization constant such that
V ()
∫0 esc (, V⃗ℎ )V⃗ℎ = 1.
The radial dependence of the velocity dispersion is
obtained via the virial theorem:
 () = √
tot ()
,

(16)
where tot = DM +  , while Vesc = √2.
Using the velocity distribution (15), going to center-ofmass and relative velocities V⃗CM and V,⃗ and performing the
integrals over V⃗CM , we obtain for the mean pair-production
cross-section times relative velocity
−2V2 /2
1 √2 erf (√2Vesc /) − 4Vesc  esc
⟨ V⟩ = 2
2 /2 2
 (√ erf (V /) − 2V −Vesc
)
esc
×∫
2Vesc
0
 (V) V3 −V
esc
2
/22
(17)
V,
which is also a function of  through  and Vesc . Putting (9),
(11), (12), (14), (16), and (17) together allows us to compute the
pair-production rate in the galactic bulge defined in (10) as a
function of 0 and .
4. Results
The rate of excessive + − pairs to be generated in the galactic
bulge was estimated in [21] to be /|obs = 3 × 1042 s−1 .
We computed /| for a large range of central darkmatter densities, going from 0.3 GeV/cm3 to an ultimate
upper limit of 104 GeV/cm3 [28]. For each value of 0 , we
searched for the mass  of OHe that reproduces the observed
rate. The results are shown in Figure 1.
The observed rate can be reproduced from a value of
0 ≃ 115 GeV/cm3 , corresponding to an OHe mass of  ≃
1.25 TeV. As 0 gets larger, two values of  are possible,
with the lower one going from 1.25 TeV to 130 GeV and the
upper one going from 1.25 to 130 TeV as 0 goes from 115 to
104 GeV/cm3 .
5. Conclusion
The existence of heavy stable particles is one of the most
popular solutions for the dark- matter problem. Usually they
are considered to be electrically neutral. But dark matter
can potentially be made of stable heavy charged particles
bound in neutral atom-like states by Coulomb attraction.
An analysis of the cosmological data and of the atomic
composition of the Universe forces the particle to have charge
−2. O−− is then trapped by primordial helium in neutral Ohelium states and this avoids the problem of overproduction
of anomalous isotopes, which are severely constrained by
observations. Here we have shown that the cosmological
model of O-helium dark matter can explain the puzzle of
positron line emission from the center of our Galaxy.
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Advances in High Energy Physics
10000
0 (GeV/cm3 )
[3]
[4]
1000
[5]
[6]
100
0.01
0.1
1
10
100
1000
[7]
M (TeV)
Figure 1: Values of the central dark-matter density 0 (GeV/cm3 )
and of the OHe mass  (TeV) reproducing the excess of + − pairs
production in the galactic bulge. Below the red curve, the predicted
rate is too low.
The proposed explanation is based on the assumption
that OHe dominates the dark-matter sector. Its collisions
can lead to 0 deexcitations of the 2s states excited by the
collisions. The estimated luminosity in the electron-positronannihilation line strongly depends not only on the mass of
O−− but also on the density profile and velocity distribution of
dark matter in the galactic bulge. Note that the density profile
we considered is used only to obtain a reasonable estimate
for the uncertainties on the density in the bulge. It indeed
underestimates the mass of the Galaxy, but it shows that
the uncertainties on the astrophysical parameters are large
enough to reproduce the observed excess for a rather wide
range of masses of O−− . For a fixed density profile and a fixed
velocity distribution, only two values of the O−− mass lead
to the necessary rate of positron production. The lower value
of this mass, which does not exceed 1.25 TeV, is within the
reach of experimental searches for multicharged stable heavy
particles at the LHC.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgment
The authors express their gratitude to A. S. Romaniouk for
discussions.
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