Effects of Bound States on Dark Matter Annihilation

17 Pages·2016·1.71 MB·English

CALT-TH-2016-005
Effects of Bound States on Dark Matter Annihilation
Haipeng An ∗, Mark B. Wise †, Yue Zhang ‡
6
1
0 Walter Burke Institute for Theoretical Physics,
2 California Institute of Technology, Pasadena, CA 91125

r
p
A

6 Abstract: We study the impact of bound state formation on dark matter annihilation rates in

models where dark matter interacts via a light mediator, the dark photon. We derive the general
]
h
cross section for radiative capture into all possible bound states, and point out its non-trivial
p
- dependenceonthedarkmattervelocityandthedarkphotonmass. Forindirectdetection,ourresult
p
shows that dark matter annihilation inside bound states can play an important role in enhancing
e
h signal rates over the rate for direct dark matter annihilation with Sommerfeld enhancement. The
[ effects are strongest for large dark gauge coupling and when the dark photon mass is smaller than

1 the typical momentum of dark matter in the galaxy. As an example, we show that for thermal
v dark matter the Fermi gamma ray constraint is substantially increased once bound state effects
6
are taken into account. We also find that bound state effects are not important for dark matter
7
7 annihilation during the freeze out and recombination epochs.
1
0
.
4
0
6
1
:
v
i
X
r
a
∗[email protected]
†[email protected]
‡[email protected]
1
1 Introduction
It is very likely that dark matter (DM) requires degrees of freedom, that are not in the standard model
(SM) for its explanation. As its name implies the electromagnetic interactions of DM must be small. A
convenient way to realize this is to suppose that the DM is not charged under the SM gauge group. A
simplemodelofthistypethatdoesn’thaveanyfinetunings,beyondtheusualonestokeepthecosmological
constant and Higgs mass small, is to have the DM be a Dirac fermion coupled to a new massive U(1)
D
gauge boson (the dark photon). Since the DM couples to a conserved current the new gauge boson can
have a mass term in the Lagrange density. This model adds to the SM seven new dark degrees of freedom:
two spin components for both the DM and anti-DM particles, and the three spin states for the massive
dark photon. The model has three additional parameters, a dark fine structure constant, α = g2 /(4π),
D D
the DM mass m and the dark photon mass m . In addition there is one dimensionless renormalizable
D V
coupling κ that characterizes the kinetic mixing of the hypercharge U(1) and U(1) kinetic terms. It is
Y D
only through gravity and this kinetic mixing that SM degrees of freedom communicate with those in the
dark sector.
In this paper we will assume thermal DM i.e., at early times when the universe is at a very high
temperature the DM sector is in thermal equilibrium, and moreover has the same temperature as the SM.
As the universe evolves it cools and when the temperature drops below the DM mass, DM and anti-DM
particles start to annihilate, eventually freezing out at T ∼ m /30. In this scenario achieving the correct
D
the DM density relates the dark fine structure constant to the DM mass, roughly α ∼ 0.02(m /1TeV).
D D
For DM much heavier than TeV, one needs to take into account of the Sommerfeld effect during freeze
out and α is somewhat lower than the value predicted by the above relation. For example, α = 0.2 for
D D
m = 16.7TeV.
D
The light mediator scenario with m (cid:29) m has been studied for a variety of reasons. In the same
D V
region of parameter space there are indirect detection signals that are the topic of study in this paper.
For α m /(2m ) > 0.84, two body DM-anti-DM bound states exist. They are the analog of positronium
D D V
bound states in electromagnetism. It is the impact of these darkonium bound states on the rate for
DM-anti-DM annihilation in the galaxy today that we focus on.
The region of parameter space in the m −m plane where bound states exist and are potentially
V D
important for DM cosmology is shown in Fig. 1. We assume these darkonium states are non-relativistic
and so we restrict our attention to the region of parameter space where α < 0.3 or equivalently (for
D
thermal DM m < 30TeV). Very small dark photon masses, m < 30MeV are inconsistent with direct
D V
detection, the supernova constraints and the requirement that dark photons decay away before big bang
nucleosynthesis (BBN). The region of Fig. 1 below the green line does not have darkonium states and the
region between the brown and green lines has darkonium states but the mass of the dark photon is too
large for these bound states to be produced with the low kinetic energy for the DM during recombination
or today. Thus, the region of parameter space for our study of the impact of bound states on indirect
detection signals is the triangular region marked as the “Focus of this study” in Fig. 1.
The main finding of our work is that the bound state effects are important for DM indirect detection
when m /m (cid:46) 10−3 and α (cid:38) 0.1. The goal of this paper is not to provide a comprehensive list of all
V D D
the indirect detection constraints but rather to highlight the important role bound state formation plays
in this region of parameter space. To this end we focus on the photon spectrum resulting from DM near
the center of our galaxy annihilating either directly or through bound states.
In section 2, we derive the general cross section for dark matter bound state formation via radiation
of an on-shell dark photon and its dependence on the dark matter velocity and the dark photon mass. In
section 3, we apply our results to calculate the indirect detection constraints on dark matter annihilation
anddiscusstheimportanceofboundstateeffects. Wediscusstheboundstateeffectsondarkmatterfreeze
out and on the cosmic microwave background (CMB) in sections 4 and 5, and conclude in section 6.
2
100.
50. CMB ΑΑDD(cid:62)(cid:62)00..33
FFooccuuss ooff tthhiissssttuuddyy
10.
mm
(cid:72)(cid:76)mTeVD 15.. IIwwnnSSccNNiiootthh&&nnssLLBBiiUUssBBtteeXXNNnn,,tt BB oouunnddoorrssttddaattuueerriissnnccggaarrnneennccoooottffmm oobbrriinnaattiioonn
0.5 ttooddaayy BBoouunnddssttaatteess
ddoonnootteexxiisstt
0.1
0.001 0.01 0.1 1 10 100 1000
m (cid:72)GeV(cid:76)
V
Figure 1: The parameter space relevant to this study (marked by “Focus of this study”), where m is the dark
D
matter mass and m is the dark photon mass. The value of the dark fine structure constant α = g2 /(4π)
V D D
is chosen to give the correct relic abundance for dark matter. We do not consider α < 0.3 (above the black
D
curve) where the next-to-leading order corrections would be large. Dark matter bound states do not exist below
the green curve. Between the brown and green curves, dark matter bound states exist but the dark photon
is too heavy for the bound states to be produced with the low kinetic energy such as in the galaxy today or
during recombination. The region to the left of the blue curve is excluded by direct detection, the supernova
constraints and demanding that dark photons decay before BBN, see also Fig. 4. The yellow region is excluded
by CMB constraints on dark matter annihilation during recombination, see section 5.
2 Bound State Formation Cross Section
The Lagrangian for the dark sector is
1 1 κ
L = L +χ¯iγµ(∂ −ig V )χ−m χ¯χ− V Vµν + m2 V Vµ− B Vµν , (1)
SM µ D µ D 4 µν 2 V µ 2cosθ µν
w
whereB ishyper-chargefieldstrengthtensorandθ istheWeinbergangle. Henceκisthekineticmixing
µν w
between the photon and the vector field V.
The interaction Hamiltonian for radiating one dark photon V can be separated into transverse and
longitudinal mode parts. In the center-of-mass frame, the Hamiltonian for radiating transverse V’s is
(cid:18) (cid:19)
g k (cid:104) (cid:16)r(cid:17) (cid:16) r(cid:17)(cid:105)
HT = D A +A − , (2)
int µ T 2 T 2
where r is the relative coordinate of χ and χ¯, k is the relative DM momentum and µ = m /2 is the
D
DM reduced mass. For the transverse modes, the polarization vectors satisfy (cid:15)+(q)(cid:15)+∗(q)+(cid:15)−(q)(cid:15)−∗(q) =
i j i j
δ −q q /|q|2, and q is the three momentum of the radiated dark photon. For radiating a longitudinal V,
ij i j
using current conservation qµJ = 0, the interaction Hamiltonian can be written as
µ
(cid:18) (cid:19)
g m (cid:104) (cid:16)r(cid:17) (cid:16) r(cid:17)(cid:105)
HL = D V φ −φ − . (3)
int |q| L 2 L 2
3
Effectively it is equivalent to radiating a scalar particle.
For bound state formation, |q| ∼ α2 µ and |r| ∼ 1/(α µ), where α = g2 /(4π). Thus we use the
D D D D
dipole approximation and only keep the leading terms in q · r. Then the matrix elements for the free-
bound transitions with transverse and longitudinal V radiation are
(cid:90) (cid:90)
r·q
M = g d3r Ψ∗(r)(E −E )r·(cid:15) Ψ (r) , M = −ig d3r Ψ∗(r)m Ψ (r) . (4)
T D f i f T i L D f V |q| i
The total cross section for bound state formation from a scattering state with momentum k at infinity
is
(σv)B = α3πD (cid:88)(cid:18)ωn2(cid:96)+ 12m2V(cid:19)(cid:113)ωn2(cid:96)−m2V (cid:34)(cid:96)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) drr3Rn(cid:96)(r)Rk(cid:96)−1(cid:12)(cid:12)(cid:12)(cid:12)2+((cid:96)+1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) drr3Rn(cid:96)(r)Rk(cid:96)+1(cid:12)(cid:12)(cid:12)(cid:12)2(cid:35) , (5)
n,(cid:96)
where v is the relative velocity and ω = E +k2/(2µ) is the sum of the binding energy of the (n(cid:96))’th
n(cid:96) n(cid:96)
bound state level and the kinetic energy of incoming state. For a Yukawa potential, the binding energy in
general depends on both n and (cid:96).
A couple of remarks relevent for the evaluation of Eq. (5) are:
• The sum over n,(cid:96) includes all the energy level satisfying ω > m . For low velocity DM with
n(cid:96) V
k2/(2µ) (cid:28) m , this amounts to E > m . As a rough estimate we can use E ∼ α2 µ/(2n2) for
V n(cid:96) V n(cid:96) D
the binding energy. Then if E > m , the ratio of the bound state size to the screening length of
n(cid:96) V
the Yukawa potential, a m ∼ nm /(α µ) is less than α /(2n) (cid:28) 1. In other words those bound
n V V D D
states, that are deep enough to emit an on-shell V, in their formation, are all much smaller than
1/m . Therefore, from now on, we will approximate the relevant bound states as Coulomb bound
V
states, with energies E = E , ω = ω , that are (cid:96) independent.
n(cid:96) n n(cid:96) n
• The bound and scattering wavefuctions that solve the Schr¨odinger equation are written as
(cid:88) (cid:88)
ψ (r) = R (r)Y (rˆ), ψ (r) = R (r)Y (rˆ)Y∗ (kˆ) . (6)
n n(cid:96) (cid:96)m k k(cid:96) (cid:96)m (cid:96)m
(cid:96)m (cid:96)m
Using the above approximation, the radial Coulomb wavefuctions for bound states are
(cid:115)
2 (n+(cid:96))! (2r)(cid:96) (cid:18) 2r (cid:19)
R (r) = e−(r/na0)F 1+(cid:96)−n,2+2(cid:96), , (7)
n(cid:96) n(cid:96)+2(2(cid:96)+1)! (n−(cid:96)−1)! a(cid:96)0+3 1 na0
where a = 1/(α µ) is the Bohr radius.
0 D
For the scattering state radial wave functions, we numerically solve the Schr¨odinger equation with
a Yukawa potential and energy eigenvalue E = k2/(2µ) using the “shooting method”. For the (cid:96)’th
partial wave, define R (r) = r(cid:96)−1φ(r), the Schr¨odinger equation for φ(r) is
k(cid:96)
2(cid:96) (cid:18) 2αµe−mVr 2(cid:96)(cid:19)
φ(cid:48)(cid:48)(r)+ φ(cid:48)(r)+ k2+ − φ(r) = 0 . (8)
r r r2
The boundary condition at r = 0 is φ(0) = 0 and φ(cid:48)(0) = c. Because Eq. (8) is a linear equation,
the overall normalization of R is proportional to the parameter c, and we fix it by requiring that R
k(cid:96)
matches to the (cid:96)-th partial of a plane wave as r → ∞, i.e.,
(4π) (cid:16) π (cid:17)
rR (r → ∞) ∼ i(cid:96)cos kr−((cid:96)+1) +δ . (9)
k(cid:96) k(cid:96)
k 2
4
Σ(cid:72)GeV(cid:45)2(cid:76)
0.1
0.01
0.001
10(cid:45)4
10(cid:45)5
mV(cid:72)GeV(cid:76)
5 10 20 50 100
Figure 2: Dark photon mass dependence in the bound state formation cross section in today’s galaxy (thick red
curve), calculatedfromthegeneralformula, Eq.(5). Wehavefixedtheotherparameterstobem = 16.7TeV,
D
α = 0.2 and the velocity v = 10−3. The solid black curve is obtained from the modified Kramers formula
D
Eq. (13), which does not capture the resonance effects. The horizontal dotted line is the original Kramers
formula in the Coulomb limit (m → 0), Eq. (11), and the dashed line corresponds to only ground state
V
formation with massless dark photon with n = 1.
In the m → 0 limit, R is given by the Coulomb scattering wave function,
V k(cid:96)
π (cid:12) (cid:12)
R (r) = 4πe2ka0 (cid:12)(cid:12)Γ(1+(cid:96)− kai0)(cid:12)(cid:12)(2kr)(cid:96)e−ikr F (cid:18)1+(cid:96)+ i ,2+2(cid:96),2ikr(cid:19) . (10)
k(cid:96) 1 1
(2(cid:96)+1)! ka
0
We first discuss the m dependence of the bound state formation cross section in Eq. (5). In the Coulomb
V
limit m → 0, it is the Kramers formula for the recombination cross section [1]. For α (cid:29) k/µ ≡ v, the
V D
leading terms of the Kramers formula for the cross section are [2]
32π α3 (cid:104) (cid:16)α (cid:17) (cid:105)
σ = √ D ln D +0.16+O(v/α) . (11)
B 3 3µ2v2 v
The logarithmic factor arises from the sum over n in Eq. (5). For given level n (cid:29) 1 [3],
32π α E2
(σ ) (cid:39) √ D 0 . (12)
B n 3 3 µ2 (µv2/2)(µv2/2+E0/n2)n3
Recall E is the binding energy of the Coulomb bound state, E = α2 µ/2. The condition α > v implies
0 0 D D
E > µv2/2. It is important to note that σ ∼ 1/n for small n and σ ∼ 1/n3 for large n. The transition
0 n n
between these two behavious occurs around n ∼ α /v. The sum of σ from n = 1 to n results in
trans D n trans
the logarithmic factor ln(α /v) in Eq. (11). §
D
For m (cid:54)= 0 the situation is more complicated. For non-zero mediator mass m , we solved for the cross
V V
section numerically using the approach described above. As m grows, the first effect it has is to change
V
§We have verified the Kramers formulae (11) and (12) numerically. The calculations taking into account only the capture
into ground state [4,5] would miss the logarithmic factor and underestimate the cross section, for the cases α (cid:29) v and/or
D
(cid:112)
α (cid:29) m /m . WealsonotethatRefs.[6,7]triedtosumoveralltheenergylevels(n≥1)butconcludedityieldsafactor
D V D
of π2/6 compared to the ground state case. However, they used the recombination cross section Eq. (75.6) in [3] that only
applies for n = 1 and also underestimated the enhancement. Moreover, these previous works have assumed massless dark
photon limit which would neglect the nontrivial m dependence as shown in Fig. 2.
V
5
Σv(cid:72)cm3(cid:144)sec(cid:76)
DirectAnnihilation Bound State
10(cid:45)22
10(cid:45)24
10(cid:45)26 Born approximation
Freeze
Recombination Galaxy
out
10(cid:45)28 v
10(cid:45)10 10(cid:45)8 10(cid:45)6 10(cid:45)4 0.01 1
Figure 3: Dark matter bound state formation cross section for various dark matter velocities (thick red curve),
calculated from the general formula, Eq. (5). We have fixed the other parameters, m = 16.7TeV, α = 0.2
D D
andm = 10GeV. Forcomparison,wealsoshowthedirectannihilationcrosssectionintheBornapproximation
V
(dashed black) and the one with Sommerfeld enhancement (thick blue curve). The bound state formation cross
section plays the most important role for dark matter indirect detection, if m /m < v < α .
V D D
the upper limit of the sum over n. The largest n corresponds to the highest energy level that has a
max
large enough binding energy that allows V to be produced on-shell. When we are still within the Coulomb
(cid:113)
limit, n = α µ . When n < n , the Kramers cross section is modified to
max D 2mV max trans
32π α3 (cid:18) (cid:114) µ (cid:19)
σ (cid:39) √ D ln α . (13)
B 3 3µ2v2 D 2mV
The condition that n < n is equivalent to µv2/2 < m and we find numerically that Eq. (13) is
max trans V
valid in the range 1µv2 < m < µv. In the region m > µv, the cross section is resonantly enhanced
2 V V
when the kinetic energy of the incoming state is nearly degenerate with a resonance state of the Yukawa
potential. These features are shown in Fig. 2.
Next, we discuss the velocity dependence of the bound state formation cross section, and highlight the
comparison with the Sommerfeld enhanced cross section of direct annihilation, often used for computing
DM annihilation in the literature [4,8–10]. This comparison is shown in Fig. 3 for m = 16.7TeV,
D
(cid:112)
α = 0.2 and m = 10GeV. We find that in the region m /µ < v < 2m /µ the bound state formation
D V V V
cross section is consistently larger than the direct annihilation cross section with Sommerfeld enhancement
(labelledwithsubscriptA).Forthisrangeofrelativevelocities, thetwocrosssectionscanbeapproximated
as
32πα3 (cid:18) (cid:114) µ (cid:19) π2α3
(σv) ∼ √ D ln α , (σv) ∼ D . (14)
B 3 3µ2v D 2mV A 2µ2v
The direct annihilation cross section (σv) is obtained by enhancing the Born level cross section by the
A
s-wave Sommerfeld factor, defined as |R |2/(4π)2 in our convention.
k,(cid:96)=0
The ratio of the two above cross sections is
(cid:18) (cid:114) (cid:19)
Bound state formation rate (σv) 64 µ
= B = √ ln α . (15)
D
Direct Annihilation rate (σv)A 3 3π 2mV
6
With the parameters used in Fig. 3, the bound state formation cross section can be larger by more than
one order of magnitude over the Sommerfeld enhanced annihilation cross section.
(cid:112)
Forv > 2m /µ, theboundstateproductioncrosssectionisgivenbytheKramersformulainEq.(11)
V
and the the logarithmic enhancement factor is suppressed compared with the region of velocity we have
just discussed. This is the region to the right in Fig. 3. Eventually, at v ∼ α , the argument of the log
D
factor is ∼ 1 and the Kramers and Sommerfeld cross sections become comparable to each other.
For v < m /µ, the kinetic energy of the incoming state 1µv2 becomes smaller than the height of
V 2
bump of the Yukawa potential barrier V ∼ (cid:96)((cid:96)+1)m2 /µ for the (cid:96) (cid:54)= 0 partial wave, located near
barrier V
r ∼ 1/m . As v decreases, it becomes increasingly more difficult for these partial wave states to penetrate
V
through the barrier to find the bound state wavefunction. In this region, the contributions to bound state
formation cross section from all partial waves with l (cid:54)= 0 are suppressed. Eventually, at very tiny v, only
the ks → np,(n ≥ 2) transitions can happen. In this region the bound state formation cross section is
smaller than the Sommerfeld enhanced annihilation cross section.
The above velocity dependence can have important impact on indirect detection of DM annihilation
in the Milky way galaxy, where the DM velocity is ∼ 10−3. We find for m /µ < 10−3 and α > 0.1
V D
(corresponding to multi-TeV scale thermal DM), it is much more likely for two DM particles to form a
bound state than directly annihilate.
After a bound state is formed, it could either annihilate decaying to dark photon V’s or de-excite to a
lower state. The annihilation decay rate for the n(cid:96) bound state goes as,
(cid:16)α (cid:17)2(cid:96)+3
D (5−C)/2
Γ ∼ α µ , (16)
n,s,(cid:96)→V(cid:48)s n D
where s = 0,1 is the total spin angular momentum of the bound state, n is the principal quantum number,
(cid:96) is the orbital angular momentum, and C = (−1)(cid:96)+S is the charge conjugation. For C = 1, the bound
state decays into 2V’s, while for C = −1, it decays into 3V’s due to the Furry’s theorem in the dark sector.
The (cid:96) dependence arises because the annihilation decay amplitude is proportional to the (cid:96)-th derivative
of the zero point wavefunction at the origin, (d(cid:96)/dr(cid:96))R (0). Each derivative yields a power of α . For
n(cid:96) D
smaller n (and hence (cid:96)) the time scale for darkonium annihilation to dark gauge bosons is extremely short
compared with the the age of our galaxy which in units of inverse GeV is, τ ∼ 0.62×1042GeV−1. But
g
for larger principal quantum numbers n and values of (cid:96) that is not the case. As an explicit example, we
consider the parameters m = 16.7TeV, α = 0.2 and m = 1GeV. For these parameters n = 12
D D V max
and for this value of the principal quantum number, (cid:96) = n − 1 = 11 and C = −1 the bound state
max
annihilation decay has a lifetime associated with it that is about one order of magnitude larger than the
age of our galaxy.
Darkonium states with larger n and (cid:96) produced in our galaxy do disappear but the route is through
de-excitation to lower values of n and (cid:96) and then annihilation to energetic V’s with E (cid:39) m /2. For
V D
simplicity we consider the case where the transition is dark electric dipole to either a real or if that is
kinematically not allowed virtual V. When this can occur via a real dark V the rate is very rapid. For
de-excitation through a virtual V we estimate the rate for the bound state transition n,(cid:96) → n−1,(cid:96)±1 to
be
κ2αα13 µ5
Γ ∼ D . (17)
n→n−1 n19 4π2m4
V
It is convenient to introduce the “partial lifetimes”, τ = 1/Γ . Assuming the transition to the
n→n−1 n→n−1
ground state occurs changing n by one unit at a time the total rate is
(cid:32) (cid:44)(cid:88)n (cid:33) κ2αα13 µ5
Γ ∼ 1 τ ∼ 20 D . (18)
n→1 i→i−1 n20 4π2m4
i=2 V
7
Figure 4: Constraints on the m − κ parameter space for fixed m = 16.7TeV and α = 0.2. The blue
V D D
region is excluded by the LUX result in dark matter direct detection [13]. The gray region is excluded from
the supernova cooling argument [11,12]. In the green region, the dark photon lives longer than a second and
would threaten the success of BBN assuming the dark and SM sectors had similar temperatures [14,15]. In the
allowed (white) region, all possible dark matter bound states have lifetime shorter than the age of the Milky
Way galaxy.
We find in this case for the allowed values of κ, m and m that this de-excitation rate is shorter than
D V
theageofthegalaxy. ThisisillustratedinFig.4usingthesameparametersasbefore(i.e.,m = 16.7TeV,
D
α = 0.2, m = 1GeV and n = 12). All the darkonium bound states produced in our galaxy do decay
D V max
converting eventually into two or three very energetic V’s and some softer V’s which all decay to standard
model particles.
Without loss of generality, we imagine there is just one type of bound state. Then its number density
n satisfies the rate equation
B
dn 1
B = −Γ n + n2 (σv) , (19)
dt B B 4 D B
where n is the unbound DM particle density ¶, Γ is the DM decay rate and (σv) is the DM bound
D B B
state production cross section multiplied by the relative velocity of the DM particles. In the galaxy, if only
a small fraction of DM have formed bound states or annihilated today, the free DM density n is constant
D
in time. Solving the rate equation for the number density of dark bound states using this approximation
gives
n (t) = 1 n2 (σv) (cid:0)1−e−ΓBt(cid:1) . (20)
B 4Γ D B
B
Taking the time t to be the age of our galaxy τ ∼ 1018 seconds, for Γ t (cid:29) 1, the value of n approaches
g B B
an equilibrium value given by, 4n Γ = n (σv) .
B B D B
(tot)
The DM today includes both free DM and the ones inside (unstable) bound states n = n +2n .
D D B
ThetotalannihilationrateRrelevantforindirectdetectionsignal, includingDMannihilationbothdirectly
(∝ (σv) ) and inside the bound states (∝ (σv) ), is
A B
1 1 (cid:104) (cid:105)
R = 2n Γ + n2 (σv) = n2 (σv) +(σv) , (21)
B B 2 D A 2 D A B
¶n is the sum of DM and anti-DM particle number densities.
D
8
where in the second step, we have used the above equilibrium value for n . Hence the indirect detection
B
signal discussed in the next section is fully determined by the sum of DM direct annihilation and bound
state formation rates.
3 Indirect Detection
In this section, we quantify the importance of bound state formation on DM indirect detection. The V
particles from the annihilation of DM will further decay into SM charged particles via the kinetic mixing
with the photon and Z boson, and contribute to the cosmic gamma ray spectrum. We will consider the
Fermi constraint on the photon spectrum from DM annihilation at the galactic center [16]. The goal of
this paper is not to provide a comprehensive list of all the constraints but rather to highlight the important
role bound state formation plays in some regions of parameter space.
The gamma ray flux at the earth is obtained from the DM annihilation rate averaged over the galactic
center region via,
dΦγ = (cid:88)3 1 dNγ(n) (cid:90) dΩ(cid:90) dsρ(r(s,θ))2(σv) , (22)
dE 16πm2 dE nV
γ n=2 D γ l.o.s.
where ρ is the DM density profile, s is the distance of the annihilation point to the earth, θ is the
angle between the line of sight and the direction of the galactic center in view of the earth, r(s,θ) =
(cid:113)
r2 +s2−2r scosθ, and r = 8.4kpc is the distance between the earth (or the sun) and the galactic
(cid:12) (cid:12) (cid:12)
center. We use the NFW profile for dark matte mass density distribution in the galaxy,
ρ
0
ρ(r) = , (23)
(r/R)(1+r/R)2
where G is the Newton’s constant, R = 20kpc, and we choose ρ = 0.34GeV/cm3 such that near the
N 0
solar system ρ(r ) = 0.4GeV/cm3. For the gamma ray observation by Fermi-LAT, the Ω integral covers
(cid:12)
the 15◦×15◦ region around the galactic center [16].
As discussed in the previous section, the annihilation of DM could happen in two ways
χχ¯ → 2V ,
χχ¯ → B → nV (n = 2,3) .
The first line is the usual direct annihilation with Sommerfeld enhancement, while the second line corre-
spondstohavingboundstateformationandthenDMannihilatingwithintheboundstates. Thenumberof
dark photons resulting from this annihilation, n = 2,3, depends on theC-parity C = (−1)(cid:96)+S of the bound
(n)
state B that decays via DM annihilation. In Eq. (22), the function dN /dE is the photon spectrum at
γ γ
the source, depending on the number of V’s in the final state (n = 2,3). We describe the details of our
(n=2,3)
calculation of dN /dE in appendix A.
γ γ
The cross sections for n = 2,3 are related to the ones discussed in the previous section as
(σv) = (σv) +(σv) f , (24)
2V A B 2
(σv) = (σv) (1−f ) . (25)
3V B 2
The factor f is the fraction of bound states annihilating into 2V from the state with C = +1. The rest of
2
the bound states will annihilate to 3V with C = −1. As shown by Fig. 5, we find the 3V channel yields
only a slightly larger (and slightly softer) gamma ray flux than the 2V channel. Thus our numerical results
9
1019
5(cid:76)m 3V
(cid:144)Vc 3(cid:180)1018 2V
e
M
(cid:72)
(cid:77)
Γ
E 1018
d
(cid:145) v
(cid:70)Γ Σ
d
(cid:73)
2EΓ
3(cid:180)1017
1017
103 104 105 106 107
EΓ (cid:72)MeV(cid:76)
Figure 5: An example of gamma ray fluxes per unit cross section as a function of photon energy for the 2V and
3V channels. Here the dark matter is m = 16.7TeV and dark photon mass is m = 10GeV.
D D
are insensitive to the value of f . For convenience, we use f = 1 in the following calculations, which yields
2 2
the most conservative limits.
In general, one has to calculate the cross section (σv) within the line of sight integral because the
nV
bound state formation cross section is velocity dependent as discussed in Fig. 3, and the DM velocity near
the galactic center depends on the position r. For simplicity, we neglect the r dependence and assume
Maxwell-BoltzmannvelocitydistributionofDMwithareasonableroot-mean-squarevaluev = 150km/s
rms
throughout the signal region at the galactic center as suggested in [17]. This approximation does not affect
the main point of our work.
WecomparethegammarayspectrumwiththeonefromthegalacticcenterobservedbyFermi-LAT[16].
We find that for the multi TeV DM in this study, the resulting gamma ray spectrum is peaked around a
few hundred GeV to a TeV, while in the Fermi data, a spectrum decreasing with energy is provided only in
the window 1-100 GeV. Therefore, the last bin with E ∼ 80GeV provides the strongest upper limit. As
γ
discussed above, we assume that all DM bound states annihilate decay into 2V. The relevant cross section
is just
(σv) = (σv) +(σv) . (26)
2V A B
This quantity as a function of m is shown in Fig. 6, for m = 16.7TeV, α = 0.2. For m less than the
V D D V
typical DM momentum at the galactic center, (σv) has similar dependence on m as Fig. 2, because the
2V V
DM annihilation via bound state formation gives the dominant effect for indirect detection. In contrast,
we also show the direct annihilation cross section with Sommerfeld enhancement (often considered in
the literature) in the blue curve, which can be lower than the total effective cross section by an order
of magnitude. The solid black curve is the most conservative upper limit on (σv) by assuming zero
eff
background and requiring the signal from annihilation itself does not exceed the Fermi observation. The
dashedblackcurvecorrespondstotakingintoaccountthebackgroundusingtheinterstellaremissionmodels
discussed in [16], which sets a more stringent upper limit than the conservative one by a factor of ∼4.
Clearly, the effect of bound state formation at the galactic center can make a large difference. Given the
other parameters, the upper limit on the dashed curve already rules out the region m < 20GeV, while it
V
would still be allowed if we only considered the direct annihilation channel with Sommerfeld enhancement.
In Fig. 7, we show the impact of bound state formation on indirect detection in the m versus m
V D
parameter space plane. The value of α is chosen to give the correct thermal relic density for DM (see
D
10