Resonant SIMP dark matter

1.3 MB·

by  Soo-Min Choi

Resonant SIMP dark matter
Soo-Min Choi and Hyun Min Lee∗
Department of Physics, Chung-Ang University, Seoul 156-756, Korea.
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Abstract
2

] We consider a resonant SIMP dark matter in models with two singlet complex scalar
h
fields charged under a local dark U(1) . After the U(1) is broken down to a Z
p D D 5
- discrete subgroup, the lighter scalar field becomes a SIMP dark matter which has the
p
enhanced 3 2 annihilation cross section near the resonance of the heavier scalar
e
→
h field. Bounds on the SIMP self-scattering cross section and the relic density can be
[
fulfilled at the same time for perturbative couplings of SIMP. A small gauge kinetic

2 mixing between the SM hypercharge and dark gauge bosons can be used to make
v
SIMP dark matter in kinetic equilibrium with the SM during freeze-out.
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∗Email: [email protected]
1 Introduction
Indirectevidencesfordarkmatterareincreasinginbothdiversityandprecision,asobserved
in Cosmic Microwave Background anisotropies and missing masses of galaxies and galaxy
clusters, etc. Thus, dark matter has been one of the driving forces for going beyond the
Standard Model (SM), mostly, under the name of the Weakly Interacting Massive Particles
(WIMP). WIMP of weak-scale mass could be a natural outcome of the solution for the
hierarchy problem in the SM, but there has been no conclusive hint for WIMP or new
particles of weak scale yet in many indirect and direct searches on Earth or in satellites.
On the other hand, light dark matter of sub-GeV scale mass might have been elusive and
less explored in previous searches, so it is important to devote more efforts to building
consistent scenarios for that possibility and testing them by experiments.
Strongly Interacting Massive Particles (SIMP) [1] have been recently suggested as an
alternativethermaldarkmatter,therelicabundanceofwhichisdeterminedfromthefreeze-
out of the 3 2 annihilation of dark matter, instead of the 2 2 annihilation. The SIMP
→ →
mechanism is based on the assumption that the 2 2 annihilation is suppressed and dark
→
matterisinkineticequilibriumwiththethermalplasmaatthetimeoffreeze-out[1,2,3,4].
Concrete models for SIMP dark matter have been proposed in the literature [5, 2, 3, 6, 12]
andareviewonSIMPdarkmattercanbefoundinRef.[8]. SIMPdarkmattertypicallyhas
asub-GeVmassandalargeself-scatteringcrosssectionaboutσ /m 1cm2/g, unlike
DM DM
∼
the WIMP case. Then, although such a large self-scattering cross section is constrained
by Bullet cluster [9] and spherical halo shapes [10], it can lead to distinct signatures in
galaxies and galaxy clusters, such as the off-set of the dark matter subhalo from the galaxy
center, as hinted in Abell 3827 [11].
We briefly review on the production mechanism of SIMP dark matter. First, the
Boltzmann equation for the SIMP number density n includes the additional terms from
DM
the 3 2 annihilation processes as follows,
→
dn
DM +3Hn = σv2 (n3 n2 neq )
dt DM −(cid:104) (cid:105)3→2 DM − DM DM
σv (n2 (neq )2) (1)
−(cid:104) (cid:105)2→2 DM − DM
where σv2 α3eff is the effective 3 2 annihilation cross section and σv is the
(cid:104) (cid:105)3→2 ≡ m5DM → (cid:104) (cid:105)2→2
2 2 annihilation cross section into a pair of SM particles. When 3 2 annihilation
→ →
is dominant, the Boltzmann equation can be rewritten in terms of the DM abundance,
Y = n /s, as
DM
dY
= λ σv2 x−5(Y3 Y2Y ) (2)
3→2 eq
dx − (cid:104) (cid:105) −
(cid:113)
with λ s2(m )/H(m ) where s(m ) = 2π2g m3 and H(m ) = π2g m2DM.
≡ DM DM DM 45 ∗s DM DM 90 ∗ MP
Then, the approximate solution to the Boltzmann equation leads to the DM relic density
1
as
1.05 10−10GeV−2
Ω h2 = × . (3)
DM (cid:16)g3/2m2DM (cid:82)∞dxx−5 σv2 (cid:17)1/2
∗ MP xF (cid:104) (cid:105)3→2
Therefore, assuming that the 3 2 annihilation cross section is assumed to be s-wave and
→
taking H(T ) = n2 σv2 at freeze-out, the SIMP relic density condition is satisfied for
F DM(cid:104) (cid:105)
m = α [0.17g2 /(x4g1/2)T2M ]1/3 where T is the temperature at matter-radiation
DM eff ∗s F ∗ eq P eq
equality given by T = 0.8eV and g ,g are the effective numbers of relativistic species
eq ∗ ∗s
in radiation and entropy densities, respectively. Then, for x mDM 20 and g = 10.75,
F ≡ TF ≈ ∗
we get m (35α )MeV. Thus, we need to choose α = 1 30 for the SIMP mass
DM eff eff
≈ −
being in the range between 35MeV and 900MeV. Consequently, the fact that the correct
relic density for SIMP requires such a large effective DM coupling could be in a tension
with the validity of perturbativity and unitarity [6, 3, 13]. Furthermore, since the DM self-
α2
scattering cross section behaves as σ eff , the resultant large effective DM coupling
self ∼ m2DM
is also constrained by the Bullet cluster or spherical halo shapes.
In this article, we consider a novel possibility to generate the tree-level 5-point interac-
tion for dark matter by exchanges of a heavy field in a local dark U(1) model with two
D
complex singlet scalar fields in the dark sector. The resonant enhancement of the 3 2
→
annihilation can tolerate the necessity of introducting large couplings and avoid the strong
constraints from the Bullet cluster and halo shapes as well as unitarity bounds. We show
how the parameter space of scalar interactions is constrained by perturbativity/unitarity
and bounds on self-interactions and discuss how the resonant SIMP dark matter can be
searched for.
2 A gauged Z symmetry
5
We introduce a U(1) gauge symmetry which is broken down to a Z discrete subgroup
D 5
due to the VEV of a complex singlet scalar φ carrying a charge q = +5 under U(1) . On
φ D
the other hand, assumed that a complex scalar χ carries a charge q = +1 under U(1)
χ D
and it does not get a VEV, it can be a candidate for stable dark matter. In order for χ
to be a SIMP dark matter, we need to induce a 5-point interaction for χ but there is no
such interaction at tree level due to the remaining Z symmery. Therefore, we introduce
5
an additional singlet scalar S carrying a charge q = +3 under U(1) . U(1) charges are
S D D
given in Table 1. Then, after integrating out the scalar field S, we can obtain an effective
5-point self-interaction, χ5, respecting a Z discrete symmetry. If the additional scalar field
5
S is not decoupled, there is a possibility that the resulting 3 2 annihilation for dark
→
matter can be enhanced due to the resonance of the scalar field S. Moreover, the scalar
field S, if lighter than χ, can be a SIMP dark matter too and its 3 2 annihilation can
→
be enhanced in a similar matter at the resonance of the heavier scalar field χ.
2
φ S χ
U(1) +5 +3 +1
D
Table 1: U(1) charges.
D
The Lagrangian for two singlet complex scalars, χ and S, dark Higgs φ and dark gauge
boson V , in our model, is given by
µ
1
= V Vµν + D φ 2 + D χ 2 + D S 2 V
hid µν µ µ µ hid
L −4 | | | | | | −
where the field strength tensor for dark photon is V = ∂ V ∂ V , and covariant
µν µ ν ν µ
−
derivativesareD φ = (∂ iq g V )φ,D χ = (∂ iq g V )χ,andD S = (∂ iq g V )S
µ µ φ D µ µ µ χ D µ µ µ S D µ
− − −
with q = +5,q = +1, q = +3 and g being dark gauge coupling. The scalar potential
φ χ S D
in the hidden sector is V is given by
hid
V = m2 φ 2 +m2 χ 2 +m2 S 2
hid − φ| | χ| | S| |
+λ φ 4 +λ χ 4 +λ S 4
φ χ S
| | | | | |
+λ φ 2 χ 2 +λ S 2 χ 2 +λ φ 2 S 2 (4)
φχ Sχ φS
| | | | | | | | | | | |
1 1 1
+ λ φ†S2χ† + λ φ†Sχ2 + λ S†χ3 +h.c..
1 2 3
√2 √2 6
We note that there can be extra quartic couplings between the SM Higgs doublet H and
the singlet scalars, e.g. H 2 χ 2, but we assume that they are suppressed enough to satisfy
| | | |
the bounds on the invisible decay of Higgs boson [14, 3]. Thus, we don’t consider extra
quartic couplings any more in the following discussion.
After expanding the dark Higgs φ around a nonzero VEV as φ = √1 (v + h ), the
D D
2
renormalizable interactions between χ and S in eq. (5) become
1 1 1
= λ v(cid:48)S2χ† λ v(cid:48)Sχ2 λ S†χ3 +h.c.. (5)
S,χ 1 2 3
L −2 − 2 − 6
Therefore, the resulting cubic and quartic couplings respect the Z discrete symmetry
5
and are responsible for generating the 5-point interactions for χ or S. Moreover, λ in
1,2
the cubic interactions are responsible for the decay of the heavier scalar to a pair of the
lighter ones, if kinematically allowed. On the other hand, the dark gauge boson gets mass,
m = 5g v , due to the U(1) breaking, and it can couple to the charged particles in
V D D D
the SM through the gauge kinetic mixing and thus play a role of messenger between dark
matter and the SM.
3 Resonant enhancement of the 3 2 annihilation
→
We assume that the dark Higgs and the dark gauge boson are heavier than dark matter
such that their contributions to the annihilation of dark matter are suppressed. However,
3
χ χ χ χ∗ χ
χ∗
χ χ∗ χ S χ S χ χ∗
S
χ∗ χ∗ S
χ S χ∗ χ χ χ χ∗
χ χ∗ χ χ∗ χ χ χ∗
S S χ∗ S
χ S χ χ χ S χ∗
S
S
χ χ∗ χ χ∗ χ χ∗ χ
Figure 1: Feynman diagrams for χχχ χ∗χ∗.
→
Z(cid:48) gaugebosoncontributesdominantlytothekineticscatteringbetweenSIMPdarkmatter
and the SM charged leptons [2, 3, 4].
First, taking m > 2m , the singlet scalar S decays into a pair of dark matter χ. In
S χ
this case, while the 2 2 (semi-) annihilation processes in hidden sector are kinematically
→
forbidden, the 3 2 process, χχχ χ∗χ∗, is a dominant annihilation process. But, the
→ →
dark Higgs or the dark gauge boson does not contribute to the processes even through
intermediate states, unlike the Z case [3]. Moreover, the 3 2 process for χ is made
3
→
possible due to the exchanges of the scalar S as shown in Fig. 1.
In the non-relativistic limit for dark matter, the squared amplitude for the χχχ χ∗χ∗
→
process is
25m2R2(cid:12) λ (37m4 21m2m2 +2m4)
2 = χ 2(cid:12) 3 χ − χ S S
|Mχχχ→χ∗χ∗| 3 (cid:12)(m2 +m2)(4m2 m2 +iΓ m )(9m2 m2 +iΓ m )
χ S χ − S S S χ − S S S
6m2R R (11m4 8m2m2 +m4) (cid:12)2
χ 1 2 χ − χ S S (cid:12) (6)
−(m2 +m2)2(4m2 m2 +iΓ m )(9m2 m2 +iΓ m )(cid:12)
χ S χ − S S S χ − S S S
where R λ v(cid:48)/(√2m ) and the decay width for S is given by
1,2 1,2 χ
≡
m2R2 (cid:16) 4m2(cid:17)1/2
χ 2 χ
Γ = 1 . (7)
S 16πm − m2
S S
Then, for CP conservation, the DM number density is given by n = n + n , for
DM χ χ∗
n = n , and the effective 3 2 annihilation cross section is obtained as
χ χ∗
→
√5 α3
σv2 = 2 eff. (8)
(cid:104) (cid:105)χ,3→2 1536πm3 |Mχχχ→χ∗χ∗| ≡ m5
χ χ
We note that all the Z -invariant quartic couplings between χ and S participate in the
5
χχχ χ∗χ∗ process.
→
Assuming that the dark Higgs and dark gauge boson are heavy enough and ignoring
the mixing quartic coupling between χ and dark Higgs field, we also obtain the 2 2
→
4
self-scattering processes for χ as
1
σ = ( 2 + 2). (9)
χ,self 64πm2 |Mχχ| |Mχχ∗|
χ
with
(cid:12) (cid:18) 2g2 m2(cid:19) m2R2 (cid:12)2
2 = 2(cid:12)(cid:12)2 λ + D χ + χ 2 (cid:12)(cid:12) ,
|Mχχ| (cid:12) χ m2 4m2 m2 +iΓ m (cid:12)
V χ − S S S
(cid:12) (cid:18) g2 m2(cid:19) m2R2(cid:12)2
|Mχχ∗|2 = 4(cid:12)(cid:12)(cid:12)2 λχ − Dm2 χ − mχ2 2(cid:12)(cid:12)(cid:12) .
V S
Here, we note that unitarity bounds on self-scattering are , < 8π.
χχ χχ∗
|M | |M |
Remarkably, the annihilation cross section for χχχ χ∗χ∗ is then enhanced near
→
the resonance with m = 3m , as can be seen in eq.(8) with (6). Including a nonzero
S χ
velocity of dark matter in the center of mass energy s = 9m2(1+v2 /4) in the propagators
χ rel
in eq. (6), the annihilation cross section for χχχ χ∗χ∗ before thermal average has a
→
temperature-dependent pole as follows,
c γ2
(σv2) χ S (10)
χ ≡ m5 ((cid:15) v2 /4)2 +γ2
χ S − rel S
where γ m Γ /(9m2), (cid:15) (m2 9m2)/(9m2) parametrizes the off-set from the
S ≡ S S χ S ≡ S − χ χ
resonance pole, and c is a constant parameter. Then, following the similar steps as for
χ
WIMP in Ref. [16, 17, 18], we obtain the general result for the thermal-averaged SIMP
annihilation cross section as follows,
x3/2 (cid:90) ∞
(σv2) = dvv2(σv2) e−xv2/4
χ χ
(cid:104) (cid:105) 2√π
0
2c (cid:16) (cid:17)
= χ x3/2√πγ Re z1/2e−xzSErfc( ix1/2z1/2) (11)
m5 S S − S
χ
where x m /T, z (cid:15) +iγ and
χ S S R
≡ ≡
2 (cid:90) ∞
Erfc(x) e−t2dt. (12)
≡ √π
x
In particular, for γ 1, by using the formula,
S
(cid:28)
(cid:18) (cid:19)
γ (cid:12)
S (cid:12) = πδ(x), (13)
x2 +γS2 (cid:12)γS(cid:28)1
we get the approximate form for the thermal-averaged SIMP annihilation cross section,
2c
(σv2) χ x3/2√πγ (cid:15)1/2e−x(cid:15)S θ((cid:15) ) (14)
(cid:104) χ(cid:105) ≈ m5 S S S
χ
5
where θ(x) is the Heaviside step function with θ(x) = 1, x 0, and θ(x) = 0, x < 0.
≥
Thus, for (cid:15) > 0, i.e. m > 3m , the tail of the Maxwell-Boltzmann distribution at large
S S χ
velocities allows for a resonant enhancement of the annihilation cross section. On the other
hand, for (cid:15) < 0, i.e. m < 3m , the annihilation cross section almost vanishes, because
S S χ
the center of mass with nonzero velocity is always above the resonance.
Consequently, the SIMP relic density can be determined by
m s /ρ0
Ω = χ 0 c (15)
χ (cid:16) (cid:17)1/2
2λJ(x )
f
where s and ρ0 are the entropy and critical densities at present, λ s(m )2/H(m ) and
0 c ≡ χ χ
(cid:90) ∞ (σv2)
χ
J(x ) (cid:104) (cid:105) dx. (16)
F ≡ x5
xF
with x = m /T 10 20 at freeze-out temperature. For γ 1, the J factor is
F χ F S
(cid:39) − (cid:28)
approximated as
2c
J(x ) χ √πγ (cid:15)1/2θ((cid:15) )F((cid:15) ). (17)
F ≈ m5 S S S S
χ
with
(cid:90) ∞
F((cid:15) ) dxx−7/2e−x(cid:15)S (18)
S
≡
xF
1 (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
= 32√π(cid:15)5/2Erfc x1/2√(cid:15) +x−5/2e−xF(cid:15)S 24+8x (cid:15) ( 2+4x (cid:15) ) .
60 − S F S F F S − F S
For instance, (cid:15) (cid:46) x−1, we get F((cid:15) ) 2x−5/2e−xF(cid:15)S. Then, we get Ω /Ω0 = (cid:112)J /J
S F S ≈ 5 F χ χ 0 ∼
(γ /(cid:15) )1/2(x (cid:15) )−3/2 where Ω0 and J are computed for v = 0 and γ (cid:15) is assumed.
S S F S χ 0 rel S (cid:28) S
Therefore, as the resonant enhancement is improved with nonzero temperature taken into
account, a smaller SIMP coupling is favored for a correct relic density, being consistent
with perturbativity for SIMP dark matter.
For simplicity, we choose λ = 0 and compare the relic density of the χ SIMP dark
3
matter as a function of the resonance mass m in Fig. 2. Here, the temperature of dark
S
matter is taken to zero or nonzero near the resonance in dashed or solid lines. For nonzero
λ , the 3 2 annihilation cross section gets smaller or larger, depending on whether the
3
→
sign of λ is the same as λ λ or not, but the results are not different qualitatively. In
3 1 2
Fig. 3, we also show the relic density of the χ SIMP dark matter, depending on the SIMP
mass and the SIMP coupling, R of order one, in solid lines. Consequently, we find that
1
the relic density changes significantly, depending on the mass and width of the resonance1.
1See Ref. [16, 17, 18] for the temperature effect on the resonant enhancement of the relic density for
WIMP dark matter. More general discussion on the temperature effects on SIMP dark matter will be
published elsewhere [19].
6
mχ=30MeV,γS=0.1 mχ=30MeV,γS=0.01
0.20 10 0.20 10
R1=2
0.15 0.15
g] g]
2m/ 2m/
2hΩχ 0.10 1 mc[χ 2hΩχ 0.10 1 cm[χ
σ/self σ/self
0.05 0.05
R1=2
0.00 0.1 0.00 0.1
0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.06 0.07 0.08 0.09 0.10 0.11 0.12
mS[GeV] mS[GeV]
Figure 2: Temperature effect near the resonance. The relic density of χ SIMP is given as
a function of m for zero or nonzero temperature of dark matter in dashed or solid lines.
S
We took R = 2. Planck 3σ band on the relic density is imposed in horizontal light-blue
1
region. For m = 500MeV, g = 0.1 and λ = 1, self-scattering cross section (σ /m )
V D χ self χ
is shown in units of 1cm2/g in dot-dashed line.
In both Fig. 2 and Fig. 3 , we depict the self-scattering cross section per SIMP mass,
σ /m , in dot-dashed lines, and impose the Planck 3σ values [15] on the relic density
self χ
in light-blue region. As a result, the correct relic density can be obtained for the SIMP
coupling of order one near the resonance, while the bounds on the self-scattering cross
section, σ /m < 1cm2/g, obtained from Bullet cluster [9] and halo shapes [10], as well
self χ
as unitartity and perturbativity bounds, are satisfied. The smaller the width of the scalar
field S, the smaller the self-scattering cross section and the larger SIMP masses are allowed
to satisfy the correct relic density and the bounds on self-scattering. We note that another
resonance at m = 2m appears in both 2 2 and 3 2 processes, so the region near
S χ
→ →
those additional resonances is disfavored by unitarity or bounds on self-scattering.
Now we consider the case with 2m < m for which χ decays into a pair of S and S
S χ
is stable. While the semi-annnihilation channels for S are closed kinematically, the 3 2
→
annihilation channel, SSS S∗S∗ is possible as shown in Fig. 4. In the non-relativistic
→
limit for dark matter, the squared amplitude for the SSS S∗S∗ process is
→
300R4R2(m4 8m2m2 +11m4)2m6
2 = 1 2 χ − χ S S χ (19)
|MSSS→S∗S∗| (m2 +m2)4[(9m2 m2)2 +Γ2m2][(4m2 m2)2 +Γ2m2]
χ S S − χ χ χ S − χ χ χ
where the decay width for χ is given by
m R2 (cid:16) 4m2 (cid:17)1/2
Γ = χ 1 1 S . (20)
χ 16π − m2
χ
7
mχ=30MeV,γS=0.1 mχ=100MeV,γS=0.1
0.20 10 0.20 10
0.15 0.15 R1=4
1
g] g]
2m/ 2m/
2hΩχ 0.10 R1=1 1 mcσ/[selfχ 2hΩχ 0.10 mcσ/[selfχ
R1=6 0.1
0.05 0.05
R1=3
0.00 0.1 0.00 10-2
0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.20 0.25 0.30 0.35 0.40
mS[GeV] mS[GeV]
mχ=30MeV,γS=0.01 mχ=100MeV,γS=0.01
0.20 10 0.20 10
R1=2
R1=7
0.15 R1=3 0.15 1
g] g]
2m/ 2m/
2hΩχ 0.10 1 cm[χ 2hΩχ 0.10 0.1 mc[χ
σ/self σ/self
R1=10
0.05 0.05 10-2
0.00 0.1 0.00 10-3
0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.20 0.25 0.30 0.35 0.40
mS[GeV] mS[GeV]
Figure 3: Relic density of χ SIMP as a function of the mediator mass m . We took R
S 1
of order one in solid lines. Planck 3σ band on the relic density is imposed in horizontal
light-blue region. For m = 500MeV, g = 0.1 and λ = 1, self-scattering cross section
V D χ
(σ /m ) is shown in units of 1cm2/g in dot-dashed line.
eff χ
We note that the above squared amplitude is the same form as eq. (6 with λ = 0 in
3
the case of χχχ χ∗χ∗ process, but with R and m being interchanged by R and
1 χ 2
→
m , respectively. Then, for CP conservation, the DM number density is given by n =
S DM
n +n , for n = n , and the effective 3 2 annihilation cross section is obtained as
S S∗ S S∗
→
√5 α3
σv2 = 2 eff. (21)
(cid:104) (cid:105)S,3→2 1536πm3 |MSSS→S∗S∗| ≡ m2
S S
We note that only λ quartic couplings between χ and S participate in the SSS S∗S∗
1,2
→
8
S χ S χ S∗ S S∗ S S∗
S∗ χ χ
S χ S χ S∗ S χ S
S∗ χ
S S∗ S S S S∗
Figure 4: Feynman diagrams for SSS S∗S∗.
→
process.
Similarly to the χ SIMP case, when dark Higgs and dark gauge boson are heavy enough
and ignoring the mixing quartic coupling between S and dark Higgs field, the 2 2 self-
→
scattering processes for S is given by
1
σ = ( 2 + 2). (22)
S,self 64πm2 |MSS| |MSS∗|
S
with
2 = 2(cid:12)(cid:12)(cid:12)2(cid:16)λ + 18gD2 m2S(cid:17)+ R12m2χ (cid:12)(cid:12)(cid:12)2,
|MSS| (cid:12) S m2 4m2 m2 +iΓ m (cid:12)
V S − χ χ χ
(cid:12)(cid:12) (cid:16) 9g2 m2 (cid:17) (cid:12)(cid:12)2
|MSS∗|2 = 4(cid:12)(cid:12)2 λS − mD2 S −R12(cid:12)(cid:12) .
V
Here, we note that unitarity bounds on self-scattering are , < 8π.
SS SS∗
|M | |M |
Like the case with m > 2m , the annihilation cross section for SSS S∗S∗ is en-
S χ
→
hanced near the resonance with m = 3m . Including a nonzero velocity of dark matter in
χ S
the center of mass energy s = 9m2(1+v2 /4) in the propagators in eq. (6), the annihilation
S rel
cross section for SSS S∗S∗ before thermal average has a temperature-dependent pole
→
as follows,
c γ2
(σv2) S χ (23)
S ≡ m5 ((cid:15) v2 /4)2 +γ2
S χ − rel χ
where γ m Γ /(9m2), (cid:15) (m2 9m2)/(9m2) parametrizes the off-set from the
χ ≡ χ χ S χ ≡ χ − S S
resonance pole, and c is a constant parameter. Then, we can apply the results obtained
S
for χ SIMP dark matter.
In Fig. 5, we show the relic density of the S SIMP dark matter, depending on the
SIMP mass and scalar coupling, R , of order one, in solid lines, and also depict the self-
2
scattering cross section per SIMP mass, σ /m , in dot-dashed lines. Similarly to the
self S
χ SIMP case, we find that the relic density changes significantly, depending on the mass
and width of the resonance. We note that the correct relic density for S SIMP can be
obtained for a relatively small SIMP coupling R than in the case of χ SIMP, because the
2
9