#### Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry

Eur. Phys. J. C
Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry
Zhi-Long Han 2
Weijian Wang 1
Ran Ding 0
0 Center for High Energy Physics, Peking University , Beijing 100871 , China
1 Department of Physics, North China Electric Power University , Baoding 071003 , China
2 School of Physics and Technology, University of Jinan , Jinan 250022, Shandong , China
In the light of the e+ + e− excess observed by DAMPE experiment, we propose an anomaly-free radiative seesaw model with an alternative leptophilic U (1)X gauge symmetry. In the model, only right-handed leptons are charged under U (1)X symmetry. The tiny Dirac neutrino masses are generated at one-loop level and charged leptons acquire masses though the type-I seesaw-like mechanism with heavy intermediate fermions. In order to cancel the anomaly, irrational U (1)X charge numbers are assigned to some new particles. After the spontaneous breaking of U (1)X symmetry, the dark Z2 symmetry could appear as a residual symmetry such that the stability of inert particles with irrational charge numbers are guaranteed, naturally leading to stable DM candidates. We show that the Dirac fermion DM contained in the model can explain the DAMPE excess. Meanwhile, experimental constraints from DM relic density, direct detection, LEP and anomalous magnetic moments are satisfied.
1 Introduction
It is well known that new physics beyond the Standard
Model (SM) is needed to accommodate two open
questions: the tiny neutrino masses and the cosmological dark
matter (DM) candidates. The scotogenic model, proposed
by Ma [
1,2
], is one of the attractive candidate, which
attributes the tiny neutrino masses to the radiative
generation and the DM is naturally contained as intermediate
messengers inside the loop. In the original models, an ad hoc
Z2 or Z3 symmetry serves to guarantee the stability of DM,
whereas such discrete symmetry would be broken at
highscale [3]. Perhaps a more reasonable scenario is regarding
the discrete symmetry as the residual symmetry originated
from the breaking of a continuous U (1) symmetry at high
scale. Along this line, several radiative neutrino mass
models [
4–23
] were proposed based on gauged U (1)B−L
theory, which is simplest and well-studied gauge extension of
SM.
Very recently, a sharp excess in the e+ +e− flux is reported
by Dark Matter Particle Explorer (DAMPE), which can be
interpreted as the annihilation signal of DM with the mass
around 1.5 TeV [
24
]. For this mass scale, the DM
annihilation cross section times velocity to explain the excess is in
the range of 10−26–10−24 cm3/s if one further assumes the
existence of a nearby dark subhalo about 0.1–0.3 kpc
distant from the solar system [
25
]. Inspired by this assumption,
various models [
26–58
] have been proposed. On the other
hand, the associated proton–antiproton pair excess has not
been observed. Therefore a leptophilic gauge theory rather
than the U (1)B−L ones seems more attractive.
In this work, we present a radiative neutrino mass model
based on an alternative leptophilic U (1)X gauge symmetry.
In the model only the right-handed SM leptons are charged
under the U (1)X symmetry, resulting in the direct Yukawa
couplings forbidden in the lepton sector. We will show that
the Dirac neutrino masses are generated radiatively and the
charged leptons, acquire masses via seesaw-like mechanism.
The heavy fermions we added for anomaly-free cancellation
play as the intermediate fermions in lepton mass generation.
After the spontaneous symmetry breaking(SSB) of U (1)X ,
the dark Z2 symmetry could appear as a residual symmetry
[
59
] such that the stability of a classes of inert particles are
protected by the irrational U (1)X charge assignments from
decaying into SM particles, naturally leading to stable DM
candidates.
The rest of this paper is organised as follows. In Sect. 2,
the model is set up. In Sect. 3, we focus on DM phenomenon
and its implication on DAMPE results. Conclusions are given
in Sect. 4.
2 Model setup
2.1 Particle content and anomaly cancellation
All the field contents and their charge assignments under
SU (2)L × U (1)Y × U (1)X gauge symmetry are summarized
in Table 1. First of all, the U (1)X symmetry proposed here
is a right-handed leptophilic gauge symmetry since, in SM
sector, only right-handed leptons carry U (1)X charges. As a
result, the SM Yukawa coupling L¯ E R for charged lepton
mass generation is strictly forbidden. In order to generate the
minimal Dirac neutrino mass, two right-handed neutrino field
νRi (i = 1, 2) are added being coupled with U (1)X , leading
to the zero mass for the lightest neutrino. The direct Yukawa
coupling L¯ νR is also forbidden. Instead we have introduce
several Dirac fermions as the intermediated fields for lepton
mass generation with their corresponding chiral components
Ri/Li (i = 1–9) and FRi/Li (i = 1–4) respectively. In the
scalar sector, we further add inert doublet scalars η1, η2 and
one inert singlet scalar χ . An SM singlet scalar σ is added
being responsible for U (1)X breaking.
First and foremost, we check the anomaly cancellations for
the new gauge symmetry in the model. The [SU (3)C ]2U (1)X
and [SU (2)L ]2U (1)X anomalies are zero because quark and
left-handed leptons are not assumed coupled to U (1)X . We
then find all other anomalies are also zero because
[U (1)Y ]2U (1)X : −3 × (−1)2 × 3n + 9 × (−1)2
×2n − 9 × (−1)2 × n = 0
[U (1)X ]2U (1)Y : −3 × (3n)2 × (−1) + 9 × (2n)2
×(−1) − 9 × n2 × (−1) = 0
[Gravity]2U (1)Y : Str [QY ]SM + (−1) − (−1) = 0
U (1)3Y : Str [Q3Y ]SM + (−1)3 − (−1)3 = 0
[Gravity]2U (1)X : −3 × 3n − 2 × 2n + 9 × 2n
−9 × n + 4 × Q FL − 4 × Q FR = 0
[U (1)X ]3 : −3 × (3n)3 − 2 × (2n)3 + 9 × (2n)3
−9 × n3 + 4 × Q3FL − 4 × Q3FR = 0
(1)
μ2σ + λσ vσ2 + 21 λσ vφ2 + 21 λσ η1 u
Here, Q FL = (√11 + 1)n/2 and Q FR = (√11 − 1)n/2
are the U (1)X charge of FL and FR as shown in Table 1
respectively. In order to cancel anomaly, the FLi/Ri fermions
acquire irrational U (1)X charge numbers. Similar scenarios
also appeared in radiative inverse or linear seesaw models
[
8,9
] where other solutions to anomaly free conditions with
irrational B − L charges of mirror fermions were found.
2.2 Scalar sector
The scalar potential in our model is given by
V =μ2 †
+ μ2η1 η1†η1 + μ2η2 η2†η2 + μ2χ |χ |2 + μ2σ |σ |2
+ λ ( † )2 + λη1 (η1†η1)2
+ λη2 (η2†η2)2 + λχ |χ |4 + λσ |σ |4
+ λη1 ( † )(η1†η1) + λη1 (η1† )( †η1)
+ λη2 ( † )(η2†η2) + λη2 (η2† )( †η2)
+ λχ |χ |2( † ) + λσ |σ |2( † )
+ λη1η2 (η1†η1)(η2†η2) + λη1η2 (η1†η2)(η2†η1)
+ λχη1 |χ |2(η1†η1) + λσ η1 |σ |2(η1†η1)
+ λχη2 |χ |2(η2†η2)+λσ η2 |σ |2(η2†η2)+λχσ |χ |2|σ |2
+ [μ(η1† )σ +λ(η2† )χ σ + h.c.]
The scalars and σ and η1 with their vevs after SSB of
U (1)X can be parameterized as
G+
= vφ+φ√0φ+i Gφ , η1 =
2
σ = vσ + σ√02+ i Gσ .
Then the minimum of V is determined by
η+
u 1 0 ,
√2 + η1
+ λ vφ2 + 21 λσ vσ2 + 21
u μ2η1 + λη1 u2 + 21 λσ η1 vσ2 + 21
For a large and negative μ2σ , there exists a solution with u2
vφ2 vσ2 as
vσ2
,
where vφ 246 GeV is the vev of the SM Higgs doublet
scalar and vσ is responsible for the SSB of U (1)X
symmetry. We have set a positive μ2η1 , hence the vev of η1 scalar is
not directly acquired as that of and σ but induced from
μ(η1† )σ term. Note that η2 and χ do not acquire vevs
because of positive μ2η2 , μ2χ and the absence of linear terms,
like χ σ k .
The mass spectrum of scalar σ, and η1 can be obtained
with the their vevs and the cross terms in Eq. (2). In the
condition of u2 vφ2 vσ2 , the contributions from v and
vσ to scalar masses are dominant, and mixings between η1
and other CP-even scalars are negligible small. Then the two
CP-even scalars h and H with mass eigenvalues are given by
m2h,H
2 2
λ vφ + λσ vσ ∓
(λ vφ − λσ vσ2 )2 + λ2σ v2 v2 ,
2
φ σ
with the mixing angle
tan 2α = λσλvσσ2 −v λvσvφ2 ,
where we take scalar h as the SM-like Higgs boson and H
the heavy Higgs boson. A small mixing angle sin α ∼ 0.1 is
assumed to satisfy Higgs measurement [
60
]. Note that due to
the lack of ( †η1,2)2 term, we actually have nearly
degen0
erate masses for the real and imaginary part of η1,2 [
61
],
and they are assumed to be degenerate for simplicity in the
following discussion. The masses of scalar doublet η1 are
μ2η1 + 21 (λφη1 + λφη1 )vφ2 + 21 λσ η1 vσ2 ,
2 1 2 1 2
μη1 + 2 λφη1 vφ + 2 λσ η1 vσ .
Hereafter, we take degenerate η1 scalars and m H,η1 ∼
500 GeV for illustration. For a complete detail of mass
spectrum of , σ, η1 scalars, one can refer some models which
shares part of the scalar potential, e.g. Ref. [
61
]. On the other
hand, we pay more attention to inert scalars η2 and χ which
are closely related to neutrino mass generation and DM. Note
that scalars η2, χ do not mix with , σ and η1. The two mass
(4)
(5)
(6)
(7)
(8)
(9)
eigenstate of neutral complex scalars η20 and χ are obtained
by
S1
S2
=
cos θ − sin θ
sin θ cos θ
λvφ vσ
, sin 2θ = m2S1 − m2S2 ,
Mη22 + Mχ2 ±
(Mη22 − Mχ )2 + λ2vφ2 vσ2 ,
with mass eigenvalues
1
m2S1,2 = 2
where
Mη22
Mχ2
μ2η2 + 21 (λφη2 + λφη2 )vφ2 + 21 λσ η2 vσ2 ,
μ2χ + 21 λχφ vφ2 + 21 λχσ vσ2 .
Meanwhile, the mass of inert charged scalar η2± is
Mη2±
2
2 1 2 1 2
μη2 + 2 λφη2 vφ + 2 λσ η2 vσ
m S1,S2,η2± ∼ 10 TeV, are assumed.
As will shown in Sect. 3, the DAMPE excess favors fermion
DM m F1 ∼ 1.5 TeV. Therefore, heavier inert scalars, e.g.,
2.3 Lepton masses The Yukawa interactions related to charged lepton mass generation is given by
L1 ⊃ y1 L¯ η1 R + y2 E¯ R L σ + y ¯ L R σ + h.c.
(14)
the charged lepton masses are generated though the diagram
in Fig. 1. In the basis of (l¯L , ¯ L ) and (E R , R ), we obtain
the 12 × 12 effective mass matrix
(10)
(11)
(12)
(13)
Φ
σ
Then the charged lepton mass is obtained as Ml
y1 y2u/(√2y). Correct charged lepton mass can be acquired
e μ τ
with y1,2 = 8.5 × 10−4, y1,2 = 1.2 × 10−2 and y1,2 =
5.0 × 10−2 for u = 10 GeV and y = 0.01.
The Yukawa sector for Dirac neutrino mass generation is
given by
L2 ⊃ h1 L¯ FR η2 + h2ν¯ R FL χ † + f F¯L FR σ + h.c.
(16)
The effective mass matrix for active neutrinos depicted in
Fig. 2 is expressed as
(15)
(mν )αβ =
sin 2θ
8π 2
6
k=1
m2S2
− m2Fk − m2S2 log
h1αk m Fk hk2β
m2S2
m2Fk
m2S1
m2Fk − m2S1 log
m2S1
m2
k
where m Fk (k = 1 − 4) denote the masses inert Dirac
fermions. Typically, mν ∼ 0.1 eV can be realised with
θ ∼ 10−3, h1 h2 ∼ 10−4, m F ∼ 1.5 TeV and
m S1,S2 ∼ 10 TeV. From Eqs. (2), (14) and (16) , one can
confirm that after the symmetry breaking with vφ and vσ there
exists a residual Z2 symmetry for which the irrational U (1)X
charged particles (FRi/Li , η1 and χ ) are odd while other are
even. Therefore the lightest particles with irrational charges
can not decay into SM particles and thus can be regarded as
DM candidate.
2.4 Lepton flavor violation
The new Yukawa interactions of charged lepton will induce
lepton flavor violation processes at one-loop level. Taking
tan 2θZ =
the radiative decay α → β γ for an illustration, the
corresponding branching ratio is calculated as [
62
]
⎨⎪⎧ 9 ⎛ ⎞
yiβ∗ yiα
1 1 F1 ⎝
h1 ∼ 10−4 with m F ∼ 1.5 TeV, mη+ ∼ 10 TeV are taken
to reproduce lepton masses. Due to s2mall Yukawa coupling
h and heavy mass of η2±, contribution of charged scalar η2±
is suppressed. The predicted branching ratios are BR(μ →
eγ ) 1.6 × 10−15, BR(τ → eγ ) 4.6 × 10−15 and
BR(τ → μγ ) 9.4 × 10−13, which are clearly below
current experimental limits [
63–65
].
2.5 Mixing in the gauge sector (17)
Since η1 is charged under both U (1)Y and U (1)X , its vev
u will induce mixing between Z0 and Z0 at tree level. The
resulting mass matrix in the (Z0, Z0) basis is given by [
66
]
⎫
2⎬⎪
⎪⎭
(18)
(19)
(20)
(21)
(22)
(23)
M 2 =
41 g2nZ2 g(vZφ2g+u2u )
2
n2 gZ g u2
g 2n2(vσ2 + u2)
.
The eigenvalues of M 2 are
1
m2Z,Z = 2
with mixing angle given by
2M122
M222 − M121 .
M121 + M222 ∓
(M121 − M222)2 + 4M142 ,
As u2 vφ2 vσ2 in this model, we have m2Z g2Z vφ2 /4,
m2Z g 2n2vσ2 , and the mixing angle θZ ∼ u2/vσ2 is
naturally suppressed. Typically, for u ∼ 10 GeV and vσ ∼
10 TeV, we have θZ ∼ 10−6. Therefore, the dilepton
signature pp → Z → + − at LHC is dramatically suppressed
by the tiny mixing angle θZ . For light Z around EW-scale,
the four lepton signature pp → + − Z → + − + −
is promising at LHC [
67
]. As shown in next section, the
DAMPE excess favors heavy Z 3 TeV. In this case, the
Z can hardly be detected at LHC, but are within the reach of
the 3 TeV CLIC in the e+e− → Z → μ+μ− channel [
68
].
2.6 LHC signature
In this subsection, we qualitatively discuss possible
signatures of new particles at LHC. Since decays of η1 scalars and
depend on their masses, the resulting signatures would be
different. Considering the mass spectrum m < mη1 , the
decay mode of η1 scalars are η10 → − +, η± → ν ±,
and decay modes of are ± → ± Z , ±h, νW ±. The
promising signature would be pp → + − → + − Z Z ,
leading to same signature as charged fermion in type-III
seesaw [
69
]. In the opposite case m > mη1 , the decay mode
of η1 scalars are η10 → + −, η1± → ±ν, and new decay
modes of exotic charged fermion ± → ±η10, νη1± are also
possible. Note that η1 scalars are responsible for charged
fermion mass, hence η10 → τ +τ − and η1± → τ ±ν are the
dominant decay mode. The promising signature would be
pp → η10η10∗ → τ +τ −τ +τ −, pp → η1±η10 → τ ±ντ +τ −,
similar as the lepton-specific 2HDM [
70
].
For the mass of scalar singlet m H ∼ 500 GeV with not too
small mixing angle α ∼ 0.1, the promising signature would
be gg → H → W +W −, Z Z , hh at LHC [
71
]. Provided
m < m H,η1 , the new decay channel H → ± ∓ is also
allowed. Then, the new signature gg → H → ± ∓ with
± further decaying into ± Z , νW ± is a good way to probe
the corresponding Yukawa coupling y2 E¯ R L σ introduced
in this model.
As for the inert scalars, the most promising signature in
principle would be pp → η2+η2− → + F1 + − F¯1, i.e.,
+ − + E T , for fermion DM at LHC [
72
]. But actually, this
dilepton signature is suppressed dramatically by heavy mass
of the inert charge scalar mη± ∼ 10 TeV in our consideration
[
73
], thus it is hard to probe2 at LHC. Similarly, the mono- j
signature pp → η20η20∗ j → νν¯ F1 F¯1 j , i.e., j + E T , is also
challenging at LHC.
3 DAMPE dark matter
Motivated by recent DAMPE excess around 1.5 TeV, we
focus on DM phenomenon in this section. Here, we consider
the lightest Dirac fermion F1 as DM candidate. The relevant
interactions mediated by the new gauge boson Z for DM
and leptons are
LZ ⊃ g Zμ Q ER E¯ R γ μ E R + QνR ν¯ R γ μνR
+Q FL F¯L1γ μ FL1 + Q FR F¯R1γ μ FR1 ,
with mass of gauge boson Z given by m Z g nvσ . In
the following numerical calculation, we will take n = 1/3
for illustration. Therefore, we have √Q ER = 1, QνR = 2/3,
Q FL = (√11 + 1)/6, and Q FR = ( 11 − 1)/6.
The dominant annihilation channels for DM F1 are
F¯1 F1 → ¯ , ν¯ ν, Z Z .
Provided m Z > m F1 , then the annihilation channel F¯1 F1 →
Z Z is not allowed kinematically. Hence, F¯1 F1 → ¯ , ν¯ ν
become dominant, which would be able to interpret the
DAMPE e+ + e− excess when m F1 ∼ 1.5 TeV.
3.1 Constraints
In this part, we summarize some relevant constraints for
DAMPE DM. To research the DM phenomenon, we
implement this model into FeynRules [
74
] package. Then,
for DM relic density, we require the results calculated by
micrOMEGAs4.3.5 [
75
] in 1σ range of Planck
measurements: h2 = 0.1199 ± 0.0027 [
76
].
As for direct detection, the leptophilic Z will mediate
DM-electron scattering at tree level, with the
corresponding cross section constrained by XENON100, i.e., σe <
10−34cm2[
77
]. Because of XENON100 sensitive to
axialvector couplings, the analytical expression for axial-vector
DM-electron scattering is given by [
78
]
σe = 3(gaF ga )2 me2
π m4 ≈ 3(gaF ga )2
Z
×3.1 × 10−39cm2,
m Z
10 GeV
−4
where gaF = g (Q FR − Q FL )/2 = −g /6 and ga =
g Q ER /2 = g /2. For g ∼ 0.1, m Z ∼ 3 TeV, the predicted
value is far below current experimental bound. Instead, we
consider the loop induced DM-nucleus scattering with the
cross section calculated as [
78
]
α2 Z 2μ2N
σN = 9π 3 A2m4
Z
=e,μ,τ
gvF gv log
where μN = m N m F1 /(m N + m F1 ) is the reduced
DMnucleus mass, gvF = g (Q FR + Q FL )/2 = g √11/6, gv =
g Q ER /2 = g /2 and μ = m Z / gv gv is the cut-off scale.
F
Since current most strict direct detection constraint is
performed by PandaX [
79
], we take Z = 54, A = 131 and
(24)
(25)
(26)
(27)
Fig. 3 Left: allowed region for DAMPE DM in the g –m Z plane.
The green line delimit the relic density in the 1σ range: h2 =
0.1199±0.0027, while blue and red line correspond to LEP and PandaX
bound respectively. Right: predicted value of current σ v in the halo
as a function of m Z . The green line satisfy the observed relic density,
while the red curves are excluded by PandaX
’
g
m N = 131 GeV for the target nucleus charge, mass number
and mass respectively.
The leptophilic Z will contribute to anomalous magnetic
moments of leptons [
80
]
(28)
a
g 2
m2
12π 2 m2
Z
For an universal gauge–lepton coupling, the precise
measurement of aμ = (27.8 ± 8.8) × 10−10 [
81
] set a
stringent bound, i.e., g 5 × 10−3m Z /1 GeV.
Meanwhile, searches for leptophilic Z at LEP in terms of
fourfermion operators provide a much more stringent bound:
g 2 × 10−4m Z /1 GeV [
82
].
3.2 Fitting the DAMPE excess
To determine the allowed parameter space under above
constraints from relic density, direct detection and collider
searches, we scan over the g –m Z plane while fix m F1 =
1500 GeV. The results are depicted in Fig. 3. Since the
dominant annihilation channels into leptons are via s-channel,
the resonance production of Z will diminish the required
g coupling for correct relic density. And currently, the most
stringent bound is from direct detection, which constrains Z
around the resonance region. In Fig. 3, the predicted value of
current σ v in the halo is also shown. Slightly below the
resonance, the Breit–Wigner mechanism [
83
] greatly enhances
the annihilation cross section. In contrast, we see a strong dip
10 23
10 24
10 25
3
s
just above the resonance. Considering the fact that DAMPE
excess favor σ v > 10−26cm3/s as well as PandaX has
excluded the region m Z < 2810 GeV ∪ m Z > 3380 GeV,
the possible region to interpret DAMPE excess falls in the
range m Z ∈ [2810, 3000] GeV.
Based on the above analysis, we select a benchmark point
(see Table 2) to fit the sharp DAMPE excess by taking into
account contributions from both nearby subhalo and
Galactic halo. In our numerical calculation, we respectively use
GALPROP [87,88] and micrOMEGAs packages [
75
] to
evaluate the background flux coming from various astrophysical
sources and the flux due to DM annihilation in Galactic halo.
While for subhalo contribution, we numerically solve
following steady-state diffusion equation [
84
]
∂
−∇ · K (E )∇ f (x , E ) − ∂ E
!
b(E ) f (x , E ) = Q(x , E ) ,
with the source term
"
σ v
Q(x , E ) = 2m2DM
ρ2(r )d V δ3(x − xsub) ,
by using Green function method. In Eqs. (29) and (30),
K (E ) = K0(E /E0)δ is the diffusion coefficient, b(E ) =
E 2/(E0τE ) is the positron loss rate due to the synchrotron
radiation and inverse Compton scattering, σ v the thermal
averaged cross section at present, ρ(r ) and xsub the density
profile and location of nearby subhalo, respectively. Here we
(29)
(30)
adopt propagation parameters as [
25
]: K0 = 0.1093 kpc2
Myr−1, δ = 1/3, L = 4 kpc (the half height of the Galactic
diffusion cylinder), τE = 1016 s (the typical loss time) and
E0 = 1 GeV. In addition, we assume both Galactic halo and
subhalo are follow NEW density profile [
85,86
]:
ρs
ρ(r ) = (r/rs )(1 + r/rs )2
.
The Galactic halo is normalized by the local density ρ at
Sun orbit R , which are respectively fixed as ρ = 0.4
GeVcm−3 and R = 8.5 kpc. While for nearby subhalo,
the parameters ρs and rs can be determined by its viral mass
Mvir. The fitting result for our benchmark point is presented
in Fig. 4 together with DAMPE data points. From which, we
find that a nearby subhalo with a distance of 0.1 (0.3) kpc
and the viral mass 3 × 107 M (3 × 108 M ) can account
for the DAMPE excess for our model.
(31)
In this paper, we propose an anomaly-free radiative seesaw
model with an alternative leptophilic U (1)X gauge
symmetry. Under the U (1)X symmetry, only right-handed leptons
are charged. Charged leptons acquire mass via the type-I
seesaw-like mechanism with heavy intermediate fermions
added also for anomaly-free cancellation. Meanwhile, tiny
neutrino masses are generated at one-loop level with DM
candidate in the loop.
Provided all other particles are heavy enough, the
dominant annihilation channel for DM F1 is F¯1 F1 → ¯ , ν¯ ν
mediated by the new leptophilic gauge boson Z . Motivated
by the observed DAMPE e+ + e− excess around 1.5 TeV, we
fix m F1 = 1.5 TeV while consider possible constraints from
relic density, direct detection and collider searches. Under all
these constraints, a benchmark points, i.e., m Z = 2950 GeV,
is chosen from the viable region m Z ∈ [2810, 3000] GeV.
After fitting to the observed spectrum, we find that the
DAMPE excess can be explained by a nearby subhalo with
a distance of 0.1 (0.3) kpc and the viral mass 3 × 107 M
(3 × 108 M ).
Acknowledgements The work of Weijian Wang is supported by
National Natural Science Foundation of China under Grant number
11505062, Special Fund of Theoretical Physics under Grant number
11447117 and Fundamental Research Funds for the Central
Universities under Grant number 2014ZD42. We thank Qiang Yuan for help on
DAMPE spectrum fitting.
Open Access This article is distributed under the terms of the Creative
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