KUNS-2675

TUM-HEP/1082/17

A Radiative Neutrino Mass Model with SIMP Dark Matter

Shu-Yu Ho,1,∗ Takashi Toma,2,† and Koji Tsumura3,‡

1Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA2Physik-Department T30d, Technische Universitat Munchen,

James-Franck-Straße, D-85748 Garching, Germany3Department of Physics, Kyoto University, Kyoto 606-8502, Japan

AbstractWe propose the first viable radiative seesaw model, in which the neutrino masses are induced radiatively

via the two-loop Feynman diagram involving Strongly Interacting Massive Particles (SIMP). The stability

of SIMP dark matter (DM) is ensured by a Z5 discrete symmetry, through which the DM annihilation rate

is dominated by the 3 → 2 self-annihilating processes. The right amount of thermal relic abundance can

be obtained with perturbative couplings in the resonant SIMP scenario, while the astrophysical bounds

inferred from the Bullet cluster and spherical halo shapes can be satisfied. We show that SIMP DM is

able to maintain kinetic equilibrium with thermal plasma until the freeze-out temperature via the Yukawa

interactions associated with neutrino mass generation.

∗ sho3@caltech.edu† takashi.toma@tum.de‡ ko2@gauge.scphys.kyoto-u.ac.jp

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I. INTRODUCTION

The standard model (SM) of particle physics is an enormously successful theory describingthe nature of the universe. Nevertheless, the origin of the non-zero neutrino mass [1–4] and theidentification of dark matter (DM) in the universe [5–8] are the lack of explanations in the SM.

As it is well known, the easiest way to account for tiny neutrino masses is the canonical seesawmechanism [9–11], in which heavy right-handed singlet neutrinos are added to the SM. However,such heavy fermions are very hard to probe by current colliders. Alternatively, people focus onradiative seesaw models [12–15], where neutrino masses are generated at loop level and the massscales of the new particles involving in the Feynman diagram can be lighter than the canonicalseesaw mechanism.

On the other hand, a number of well-motivated DM candidates have been suggested, the mostpopular among which is the Weakly Interacting Massive Particles (WIMP) with the mass rangespanning from sub-GeV to TeV scale. The WIMP DM is thermally produced in the early universe,and its relic density is usually determined by the strength of the 2 → 2 annihilation cross sectionof DM into the SM particles. The experimental investigations for the WIMP DM have null resultsso far, this motivates physicists to come up with the new perspectives for the DM nature. Recently,a novel idea of DM, Strongly Interacting Massive Particles (SIMP) [16] has gotten attention andhas been explored in the literature [17–43]. In comparison with WIMP, the relic abundance ofSIMP is determined by the strength of the 3 → 2 annihilation cross section of DM into itself, whileits mass scale spreads from MeV to sub-GeV, which may be insensitive to present direct searches.The annihilation rate for the 3 → 2 process should be larger than the 2 → 2 annihilation rate toconsider SIMP DM instead WIMP. In addition, SIMP DM has to be in kinetic equilibrium with theSM sector until the freeze-out so that the temperature of the dark sector is the same with that inthe SM sector, known as the SIMP condition. An advantage of SIMP DM opposite to the WIMPDM is that the SIMP candidate can address some astrophysical issues such as small-scale structureproblems [44] and the DM halo separation in Abell 3827 cluster [45, 46].

In the economic point of view, any realistic model beyond the SM should incorporate the abovecrucial ingredients. The most renowned one possessing these necessary components is Ma’s scoto-genic model [47], in which the WIMP DM is running in the loop diagram to produce the neutrinomasses. There are a bunch of studies along this direction [48–51].

In this article, we propose a brand-new scheme of the scotogenic model, where the role of WIMPDM is replaced by the SIMP DM. To accomplish our thought, we refer to the resonant SIMP modelconstructed in Ref. [24] and extend it by introducing more scalars and fermions for neutrino massgeneration. Hereafter, we call it νSIMP model. In this model, the complex scalar is selected as aSIMP DM candidate and is stabilized by a Z5 symmetry. The resonant effect can reduce the sizeof the quartic couplings associated with the 3 → 2 annihilation processes so that the perturbativebound and the constraints from the Bullet cluster and spherical halo shapes can be satisfied. TheSIMP condition can also be fulfilled via the new Yukawa interactions, which connects the darksector and the SM sector.

The plan of the paper is as follows. In the next section, we introduce the νSIMP model andgive a description of the relevant interactions and masses for the new particles. In Sec. III, wewrite down the neutrino mass formula. In Sec. IV, we take into account several experimental andtheoretical constraints on the model. In Sec. V, we evaluate the relic density of the resonant SIMPDM and briefly mention the restrictions from the astrophysical sources. In Sec. VI, we demonstrate

2

the allowed parameter space to make the SIMP condition work. We conclude and summarize ourstudy in Sec. VII. Some lengthy formulas, diagrams, and the benchmark points of the model areput in the appendices.

II. νSIMP MODEL

To achieve the νSIMP scenario, we add three vector-like fermions, N1,2,3 , one scalar doublet, η,and two complex singlet scalars, χ and S to the SM, all of which have charges under a conservedZ5 symmetry,1 while all of the SM particles are Z5 neutral. The particle contents and the chargeassignments are summarized in Tab. I. It follows that the lightest mass eigenstate (denoted by X)of the linear combination of χ and the neutral component of η is stable and can serve as a validSIMP DM candidate.2

The renormalizable Lagrangian for the interactions of the scalar particles in this model with oneanother and with the SM gauge bosons is

L = (DρΦ)†DρΦ + (Dρη)†Dρη + ∂ρχ∗∂ρχ + ∂ρS∗∂ρS − V , (1)

where Dρ is the SM covariant derivative, and the scalar potential V is

V = µ2ΦΦ†Φ + µ2

ηη†η + µ2

χχ∗χ + µ2

SS∗S

+ 14λΦ(Φ†Φ)2 + 1

4λη(η

†η)2 + 14λχ(χ∗χ)2 + 1

4λS(S∗S)2

+λΦη(Φ†Φ)(η†η) + λ′Φη(Φ

†η)(η†Φ) + λΦχ(Φ†Φ)(χ∗χ) + λΦS(Φ†Φ)(S∗S)

+ληχ(η†η)(χ∗χ) + ληS(η†η)(S∗S) + λχS(χ∗χ)(S∗S)

+[

12µ1χ

∗S2 + 12µ2χ

2S + 16λ3χ

3S∗ + 1√2κυ(Φ†η)χ∗ + H.c.

], (2)

with υ ' 246.22 GeV being the vacuum expectation value (VEV) of Φ. The Hermiticity of Vimplies that the parameters in the scalar potential µ2

Φ,η,χ,S ,λΦ,η,χ,S,Φη,Φχ,ΦS,ηχ,ηS,χS, and λ′Φη mustbe real. In the later sections, we will choose µ1,2 , λ3 , and κ to be real and assume λη,Φη,Φχ,ΦS,ηχ,ηSand λ′Φη are negligible since these quartic couplings are irrelevant to our numerical analysis.

E Φ N1,2,3 η χ S

SU(2) 2 2 1 2 1 1

U(1)Y −1/2 1/2 0 1/2 0 0

Z5 1 1 ω2 ω2 ω2 ω

TABLE I: Charge assignments of the fermions and scalars in the νSIMP model, where E =(ν `−

)T is

the SM lepton doublet, Φ is the SM Higgs doublet, and ω = exp(2πi/5

)is the quintic root of unity.

1 This discrete symmetry can be realized as a remnant of the U(1) gauge symmetry as discussed in Ref. [24, 52]. A

concrete example is given in Appendix A.2 In the simplest Z3 SIMP model [16], the quartic coupling in the scalar potential is too large to satisfy the bound

from perturbativity.

3

After spontaneously symmetry breaking, the scalar bosons can be parametrized by

Φ =

(0

1√2

(h+ υ

)) , η =

(η+

η0

), (3)

with h being the physical Higgs boson. The masses of h, S and η+ are then given by

m2h = 1

2λΦυ

2 , m2S = µ2

S + 12λΦSυ

2 , m2η+ = µ2

η + 12λΦηυ

2 . (4)

The κυ term in the scalar potential causes the mixing between the neutral scalars η0 and χ. Inthe basis ( η0 χ )T, the corresponding mass matrix is written as

M2ηχ ≡

(m2η m2

ηχ

m2ηχ m2

χ

)=

(µ2η + 1

2

(λΦη + λ′Φη

)υ2 1

2κυ2

12κυ2 µ2

χ + 12λΦχυ

2

). (5)

Upon diagonalizing M2ηχ, we get the mass eigenstates H and X and their respective masses mH

and mX given by(η0

χ

)=

(cξ sξ−sξ cξ

)(H

X

)≡ Oηχ

(H

X

), OT

ηχM2ηχOηχ = diag

(m2H ,m

2X

),

2m2H,X = m2

η +m2χ ±

√(m2η −m2

χ

)2+ 4m4

ηχ , sin(2ξ) ≡ s2ξ =κυ2

m2H −m2

X

, (6)

where cξ = cos ξ, sξ = sin ξ , and mH > mX . Plugging χ = − sξH + cξX into Eq.(2), one canextract the relevant interactions for the 3 → 2 annihilation processes as

L ⊃ − 12µ1cξ

[X∗S2 +X

(S∗)2 ]− 1

2µ2c

2ξ

[X2S +

(X∗)2S∗]− 1

6λ3c

3ξ

[X3S∗ +

(X∗)3S]. (7)

These couplings manifest the Z5 discrete symmetry and can produce the 5-point interactions ofX by integrating out the complex scalar field S. To generate the neutrino masses, the additionalcouplings L ⊃ − 1

2µ2

(s2ξH

2 − 2cξsξXH)S + H.c. are also required. The neutrino masses will be

calculated in the next section. From Eqs.(1), (2) and (6), the Lagrangian describing the invisibledecay channels of the Z boson and the Higgs boson is

L ⊃igws

2ξ

2cw

(X∗∂ρX −X∂ρX∗

)Zρ −

(λΦX |X|2 + λΦS |S|2

)υh ,

λΦX ≡ λΦχc2ξ + κcξsξ +

(λΦη + λ′Φη

)s2ξ , (8)

where gw is the SU(2) gauge coupling constant, and cw = cos θw with the weak mixing angle θw.There are also the gauge interactions of the exotic scalars with the photon and the weak bosons,which are related to the electroweak precision tests. We collect them in Appendix B.

The Lagrangian responsible for the masses and interactions of the vector-like fermions N1,2,3 is

LN = −MkNkPLNk + Yjk[`−j η

− − νj(cξH

∗ + sξX∗)]PRNk

− 12YLjkNjPLN

ckS∗ − 1

2YRjkNjPRN

ckS∗ + H.c. , (9)

4

where Mk represent the Dirac masses, the summation over j, k = 1, 2, 3 is implicit, the superscriptc refers to the charge conjugation, PR,L = 1

2(1 ± γ5), and `1,2,3 = e, µ, τ . Explicitly, the Yukawa

couplings Yrk and YL,Rrk are of the forms as

Y =

Ye1 Ye2 Ye3Yµ1 Yµ2 Yµ3

Yτ1 Yτ2 Yτ3

, YL,R =

YL,R11 YL,R12 YL,R13

YL,R21 YL,R22 YL,R23

YL,R31 YL,R32 YL,R33

, (10)

where Y`jk = Yjk .

III. RADIATIVE NEUTRINO MASS

In the νSIMP model, the neutrinos acquire mass radiatively through two-loop diagrams withinternal H,X, S, and Nk as shown in Fig. 1. The resulting neutrino mass matrix defined byLν = −1

2(Mν)rsνrν

cs + H.c. is given as [53, 54]

(Mν

)rs

=µ2YrjYsks2

2ξ

4(4π)4

(YLjkCLjk + YRjkCRjk

), (11)

where the loop functions are

CLjk =

∫ 1

0

dudvdwδ(u+ v + w − 1

)1− w

[IL(m2X

M2k

,m2XjS

M2k

)− IL

(m2X

M2k

,m2HjS

M2k

)

−IL(m2H

M2k

,m2XjS

M2k

)+ IL

(m2H

M2k

,m2HjS

M2k

)],

CRjk =Mj

Mk

∫ 1

0

dudvdwδ(u+ v + w − 1

)w(1− w)

[IR(m2X

M2k

,m2XjS

M2k

)− IR

(m2X

M2k

,m2HjS

M2k

)

−IR(m2H

M2k

,m2XjS

M2k

)+ IR

(m2H

M2k

,m2HjS

M2k

)], (12)

with

IL(a, b) =a2 ln a

(1− a)(a− b)+

b2 ln b

(1− b)(b− a), IR(a, b) =

a ln a

(1− a)(a− b)+

b ln b

(1− b)(b− a),

m2XjS =

um2X + vM2

j + wm2S

w(1− w), m2

HjS =um2

H + vM2j + wm2

S

w(1− w). (13)

The mass matrix in Eq. (11) is diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)matrix UPMNS as U†PMNSMνU

∗PMNS = diag

(mν1 ,mν2 ,mν3

). The mixing angles in the PMNS matrix

and neutrino mass eigenvalues are given by the global fitting to the neutrino oscillation data [55].

5

ν ν

FIG. 1: Feynman diagrams for neutrino mass generation at the two-loop level.

As we will discuss in Sec. VI, the order of magnitude of the Yukawa couplings is Yjk ∼ O(0.01−1)with 0.1 GeV .Mk . 1 GeV when the SIMP condition is imposed. In the next section, we will alsoshow that the size of the mixing angle should be sξ . 0.06 due to the constraints from the invisibledecays of the Z boson and the Higgs boson. Moreover, in order to satisfy perturbative boundson the quartic couplings and the observed relic density of DM, we find that the cubic couplingµ2 ∼ O(100 MeV). Accordingly, if one takes Yjk ∼ 0.1, sξ ∼ 0.05, µ2 ∼ 100 MeV, YL,Rjk ∼ 0.1, and

CL,Rjk ∼ 1, the correct neutrino mass scale mν ∼ 0.1 eV can be arrived. To make this model morereliable, we display the benchmark points in Appendix C.

IV. CONSTRAINTS

There are various experimental and theoretical restrictions on the masses and couplings of thenew particles in the νSIMP scenario. Experimentally, the flavor-changing radiative decay `r → `sγconstrains the Yukawa couplings Yrk . The Feynman diagram depicted such decay process is shownin Fig. 2. The branching fraction of the decay process is

B(`r → `sγ

)=

3αB(`r → `sνrνs

)64πG2

Fm4η+

∣∣∣∣∣3∑

k= 1

Y∗rkYsk F(M2

k

m2η+

)∣∣∣∣∣2

, (14)

where the fine structure constant α, the Fermi constant GF and the loop function F(z) are

α =e2

4π, GF =

1√2υ2

, F(z) =1− 6z + 3z2 + 2z3 − 6z2 ln z

6(1− z)4, (15)

with e the electromagnetic charge. The most stringent experimental limit on the µ → eγ processcomes from the MEG Collaboration [56]. The up-to-date upper bound on its branching ratio isB(µ → eγ) < 4.2 × 10−13. If we take mη+ ∼ 300 GeV and F(M2

k/m2η+) = 1/6 with mη+ Mk ,3

the Yukawa couplings are limited in Yrk . 0.02 which is in conflict with the range mentioned inthe previous section. The simplest solution to evade this severe constraint is to assume a diagonalYukawa matrix Y . In this solution, the pattern of neutrino mixing is pinned down by the structuresof the other Yukawa matrices YL,R and the mass hierarchy of the vector-like fermions.4

3 If η± decays dominantly into electron or muon, mη+ & 270 GeV is required in order to avoid the constraint from

the left-handed slepton search [57].4 If mτ > me,µ + 2Mk, the new decay modes τ → (e, µ)NN ′ → (e, µ)νν′XX open and would contribute to

τ → (e, µ) + missing energy. However, this constraint is not so stringent.

6

η- η-

γ

ℓ-

ℓ-

FIG. 2: Feynman diagram of flavor-changing radiative decay `r → `sγ .

At one-loop level, the presence of η± and Nk also induces a contribution to the anomalousmagnetic moment a`j of charged lepton `j given by

∆a`j = −m2`j

16π2m2η+

3∑k= 1

∣∣Yjk∣∣2F(M2k

m2η+

). (16)

In particular, the current experimental value for the muon anomalous magnetic moment has morethan 3σ deviation from the SM prediction : aexp

µ − aSMµ = (288 ± 80) × 10−11 [58]. Since the new

contribution given by Eq.(16) is negative, we then require |∆aµ| < 8 × 10−10 , this gives an upperbound for the Yukawa coupling Yrk . For example, by taking mη+ ∼ 300 GeV and mη+ Mk , theYukawa couplings are limited in Yrk . O(1) which is less stringent compared to the constraintsfrom the flavor-changing radiative decay.

In our study, we suggest that the lightest complex scalar X is the SIMP DM candidate. Since themass scale of SIMP DM is MeV to sub-GeV, there is then a new physics contribution to the invisibledecay width of the Z boson and the Higgs boson, namely Γnew

Z, h→ inv = Γ(Z, h→ XX). The presentexperimental bounds on these invisible decay widths are Γnew

Z→ inv < 2 MeV (at the 95% C.L.) [59] andΓnewh→ inv . 0.78 MeV. To derive the latter one, we interpret Bnew

h→ inv = BexpBSM < 0.16 [60] reported by

the ATLAS and CMS combined measurements and adopt the SM Higgs width ΓSMh = 4.08 MeV [61]

at mh = 125.1 GeV [62]. From Eqs.(6) and (8), these upper limits consequently are translated into|sξ| . 0.4 and |sξ| . 0.165 (mH/100 GeV)−1, respectively. It turns out that the constraint fromthe Higgs invisible decay width is much stronger than the Z boson one. For instance, by choosingmH ∼ 300 GeV, we then reach the upper bound |sξ| . 0.06.

The scalar masses are constrained by the oblique parameters due to their modifications to theSM gauge boson propagators [63]. From Eq.(B1) and Fig. 8 in Appendix B, those parameters arecalculated as

∆S =1

12π

[c2ξ ln(m2H/m

2η+

)+ s2

ξ ln(m2X/m

2η+

)+ c2

ξs2ξG(m2X ,m

2H

)],

∆T =1

8απ2υ2

[c2ξF(m2η+ ,m

2H

)+ s2

ξF(m2η+ ,m

2X

)− c2

ξs2ξF(m2X ,m

2H

)],

∆U =1

12π

[c2ξG(m2η+ ,m

2H

)+ s2

ξG(m2η+ ,m

2X

)− c2

ξs2ξG(m2X ,m

2H

)], (17)

7

where the loop functions are given by

F (a, b) =a+ b

2− ab

a− bln

(a

b

),

G(a, b) =22ab− 5a2 − 5b2

3(a− b)2+

(a+ b)(a2 − 4ab+ b2)(a− b)3

ln

(a

b

). (18)

Since we are interested in the scale that the mass mX is below electroweak scale, one may thinkthat more general definitions of the oblique parameters may be used [64]. However, we have checkedthe difference is not important because the most stringent constraint comes from the T -parameterwhose definition does not change even for below electroweak scale. The current constraints are givenin Ref. [65, 66] as ∆S = 0.05 ± 0.11, ∆T = 0.09 ± 0.13, ∆U = 0.01 ± 0.11 with the correlationcoefficients 0.90 (between ∆S and ∆T ), − 0.59 (between ∆S and ∆U ), and − 0.83 (between ∆Tand ∆U ). These limits imply that the heavier neutral component H and the charged componentη+ should be nearly degenerate

(mH ≈ mη+

)in the case of sξ 1 and mX mH , mη+ .

Theoretically, the quartic parameters λj are subject to the conditions of vacuum stability andperturbativity. To ensure the vacuum to be stabilized at large field values, we demand [24]

λX,S > 0 , λXS > − 12

√λXλS , |λ3| <

√18λXλSλXS − 8λ3

XS +(4λ2

XS + 3λXλS)3/2

3λX, (19)

where λX ≡ λχc4ξ ≈ λχ , and λXS ≡ λχSc

2ξ ≈ λχS because of the smallness of the mixing angle ξ .

In the previous work [24], a condition of perturbativity on the quartic couplings has been taken,which corresponds to λX,S < 16π in our convention in Eq.(2). However, this upper bound seems tobe overly optimistic when the RG running is considered. Instead, we force the relatively conservedconditions λX,S < 4π in our numerical work. Furthermore, since X plays the role of DM, it shouldnot develop the VEV. The sufficient conditions to guarantee 〈X〉 = 0 (as well as 〈S〉 = 0) are givenby

λX >µ2

2

m2S

, λS >µ2

1

m2X

, λXS > 0 , (20)

here we have assumed λ3 = 0 for simplicity.5

V. RESONANT SIMP DM AND RELIC ABUNDANCE

In order to estimate the thermal relic abundance of SIMP DM, we have to solve the Boltzmannequation of the DM number density nDM = nX + nX = 2nX (we assume there is no asymmetrybetween particles X and X ) as follows

dnDM

dt+ 3HnDM = −

⟨σ3→2υ

2rel

⟩(n3

DM − n2DMn

eqDM

), (21)

5 One can find the necessary conditions to ensure 〈X〉 = 0 by using the method in the literature [67]. However, the

analytical result is too long to read.

8

FIG. 3: Feynman diagrams for the 3 → 2 annihilation process XXX → XX, where the similar diagrams

obtained by crossing the contraction in the initial and the final states are not shown.

with H being the Hubble parameter, neqDM the DM number density at the chemical equilibrium, and

〈σ3→2υ2rel〉 ≡ 1

24〈σXXX→XXυ2

rel〉 the thermal averaged effective 3 → 2 annihilation cross section.6

By applying the standard derivation [68], the approximate solution to the Boltzmann equation forthe current relic density ΩDM is given by

ΩDMh2 ' 5.7× 108 GeV−1

mXg3/4?,f m

1/2pl J

1/2, J =

∫ ∞xf

dx

⟨σ3→2υ

2rel

⟩x5

, xf ' 20 , (22)

where x = mX/T , h denotes the normalized Hubble constant, g?,f is the number of relativisticdegrees of freedom at the freeze-out temperature, Tf = mX/xf , mpl = 1.22 × 1019 GeV is thePlanck mass. To evaluate the thermal average of the 3 → 2 annihilation cross section, we employthe formula in Ref. [26]

⟨σ3→2υ

2rel

⟩=

x3

2

∫ ∞0

dβ(σ3→2υ

2rel

)β2 exp

(−xβ

), (23)

where β = 12

(υ2

1 +υ22 +υ2

3

)with υi the velocities of three initial DM particles. In the νSIMP model,

the Feynman diagrams of the 3→ 2 process XXX → XX are shown in Fig. 3. From Eq.(7), theeffective 3→ 2 annihilation cross section under CP invariance is calculated as

σ3→2υ2rel =

25√

5µ22c

5ξ

9216πm3X

∣∣∣∣∣ 3µ1µ2

(11m4

X − 8m2Xm

2S +m4

S

)(m2X +m2

S

)2(4m2

X −m2S + imSΓS

)(s−m2

S + imSΓS)

−λ3

(37m4

X − 21m2Xm

2S + 2m4

S

)(m2X +m2

S

)(4m2

X −m2S + imSΓS

)(s−m2

S + imSΓS)∣∣∣∣∣

2

, (24)

where s = (p1 + p2 + p3)2 ' 9m2X

(1 + 2β/3

)and the momenta of DM are neglected except around

the resonance s ≈ m2S. By taking the mass spectrum 2Mk > mS > 2mX , the decay width of the

6 The definition of the effective 3 → 2 annihilation cross section depends on the model. For example, in the Z3

SIMP model [23],⟨σ3→2υ

2rel

⟩≡ 1

24

⟨σXXX→XXυ

2rel

⟩+ 1

8

⟨σXXX→XXυ

2rel

⟩.

9

particle S is computed as

ΓS = Γ(S → XX

)=

µ22c

2ξ

32πmS

√1− 4m2

X

m2S

. (25)

To enhance the 3 → 2 annihilation cross section, we pick the resonant pole mS '√s ' 3mX in

Eq.(24), and it is convenient to adjust the resonant behavior by defining the following dimensionlessparameters as

εS =m2S − 9m2

X

9m2X

, γS =mSΓS9m2

X

, (26)

where εS indicates the degeneracy between mS and 3mX , and γS is the width of the resonance.7

With these variables, the 3 → 2 annihilation cross section can be expressed in the Breit-Wignerresonant form similar to the one in Ref.[69]

σ3→2υ2rel =

cXm5X

γ2S(

εS − 2β/3)2

+ γ2S

, (27)

where the coefficient cX is

cX =25√

5πcξm2S(

m2S − 4m2

X

)[(m2S − 4m2

X

)2+m2

SΓ2S

](m2S +m2

X

)2×

[R1m

2X

(m4S − 8m2

Sm2X + 11m4

X

)m2S +m2

X

−λ3

(2m4

S − 21m2Sm

2X + 37m4

X

)3R2

]2, (28)

with R1,2 = µ1,2/mX . Utilizing Eq.(22), we present the plots of the DM relic density ΩDM versusmS with different values of mX and R1,2 in Fig. 4, where the solid lines (light dashed lines) are thepredicted values by using the thermal (non-thermal) averaged effective 3 → 2 annihilation crosssection. The orange region is the latest relic density data ΩDMh

2 = 0.1197 ± 0.0022 given by thePlanck Collaboration [70]. In these plots, we do not vary the mixing angle ξ since dependence ofthe mixing angle is extremely small as long as sξ 1. Also, in order to examine the conditions of〈X〉 = 0 in Eq.(20) easily, we again assume λ3 = 0. For nonzero λ3 , the numerical results are similaras pointed out in Ref. [24]. We have checked our choices of the values of R1,2 can accommodate therequirements of perturbativity, R2

1 < λS < 4π and R22/9 . λX < 4π with mS ' 3mX . According

to the plots, one can see that the low values of R1,2 are disfavored if the DM mass mX is heavier.

Besides fitting the relic abundance of DM, there are the other astrophysical observations from theBullet cluster [71–73] and spherical halo shapes [74], which impose the bound σself/mX . 1 cm2/gwith σself = 1

4(σXX→XX + σXX→XX + σXX→XX) the effective self-interacting cross section. We

depict in Fig. 5 the Feynman diagrams of the DM self-interacting processes in our SIMP model,

7 From Eqs.(20), (25) and (26) with ξ 1 and mS ' 3mX , one can easily show that γS ' 10−3R22 . 10−2λX .

Thus one obtains γS . 0.1 1 with the perturbative bound λX < 4π.

10

ℛ1 = 2

λ3 = 0

ξ = 0.05

mX = 30 MeVℛ2 = 5

ℛ2 = 8

ℛ2 = 10

γS ≃ 10-3ℛ22

60 70 80 90 100 110 120

0.05

0.10

0.15

0.20

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

mS (MeV)

ΩDMh2

ϵS

ℛ1 = 3

λ3 = 0

ξ = 0.05

mX = 40 MeVℛ2 = 5

ℛ2 = 8

ℛ2 = 10

γS ≃ 10-3ℛ22

80 90 100 110 120 130 140 150 160

0.05

0.10

0.15

0.20

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

mS (MeV)

ΩDMh2

ϵS

FIG. 4: The predicted relic density versus mS for nonzero (zero) temperature of DM in solid lines (light

dashed lines). The orange band is the observed value 0.1153 ≤ ΩDMh2 ≤ 0.1241 at the 95% C.L..

and their cross sections are calculated as

σXX→XX =1

64πm2X

(λX −

m2X

m2S

R22c

2ξ

)2,

σXX→XX = σXX→XX =1

128πm2X

(λX +

m2X

4m2X −m2

S

R22c

2ξ

)2, (29)

here we have neglected the contributions from the h and Z-mediated diagrams due to their smallcouplings and mass suppression. By choosing an appropriate value of λX

(R2

2m2X/m

2S < λX < 4π

),

the bounds from the Bullet cluster and spherical halo shapes can be satisfied. For instance, if wetake mX (mS) = 30 (93) MeV,R1,2 = 2, 5 and ξ = 0.05 with λX = 7, we find σself/mX ' 0.26 cm2/g.More examples and discussions can be found in Ref. [24].

FIG. 5: Feynman diagrams for the DM self-interacting processes. The upper (lower) diagrams correspond

to the process XX → XX(XX → XX

). For the process XX → XX, the relevant diagrams can be

obtained by flipping the arrows in the lower ones.

11

VI. SIMP CONDITION

In the SIMP paradigm, DM is thermally produced through the 3→ 2 annihilation process intothe particles in the dark sector rather than the 2 → 2 annihilation process into the SM particles.On the other hand, in order to keep the temperature of the dark sector as the same with the SMsector, the SIMP candidate needs to be in kinetic equilibrium with the SM sector. Thus the naivecriteria that DM can be a SIMP candidate is given by [16]

Γ2→2 < Γ3→2 < Γkin , (30)

which should be held during the freeze-out temperature. In this inequality, each reaction rateis defined by Γ2→2 = nX〈σ2→2υrel〉, Γ3→2 = n2

X〈σ3→2υ2rel〉, and Γkin = nSM〈σkinυrel〉,8 where the

number densities of DM and the SM particles are given as [39]

nX = nX '2.04× 10−9 GeV

mX

T 3 , nSM =g

2π2T 3

∫ ∞

0

z2dz

exp[√

z2 + (mSM/T )2]± 1

, (31)

with g counts the internal degrees of freedom, mSM being the mass of the SM particle, (+) appliesto fermions, and (−) pertains to bosons. It is also pointed out in Ref. [37] that a slightly strongerSIMP condition may be derived by considering the rate of energy transfer (rather than the rateof reaction) between the SM and dark sectors. This rigorous SIMP condition can be written as|E3→2| < |Ekin|, where E3→2 is the rate of the DM mass transferring to the kinetic energy in the DMbath, and Ekin is the rate of the kinetic energy of DM transfers to the thermal plasma. Quoting thedetailed calculations of the SIMP condition in Ref. [34], we then impose Γ3→2 < Γ3→2 < 10−2Γkin

in our numerical study.

From Eq.(9), the particle X can interact with the active neutrinos via the Yukawa couplings Yrk .9

The Feynman diagrams of the elastic scattering between X and the SM neutrinos are displayed inFig. 6(a), and its reaction rate can be computed as

Γkin = nν∑r,s

⟨σXνr→Xνsυrel

⟩, (32)

where the neutrino number density nν and the thermally averaged effective scattering cross sectionare given by

nν =3 ζ(3)

2π2T 3 ,

⟨σXνr→Xνsυrel

⟩=

3m2Xs

4ξ

16π

∑k,l

Re(Y∗rkYrlYskY∗sl

)(M2

k −m2X

)(M2

l −m2X

)( T

mX

), (33)

8 In the WIMP paradigm, the Boltzmann equation of the DM number density is given by

nDM + 3HnDM = −〈σ2→2υrel〉[n2

DM − (neqDM

)2],

where 〈σ2→2υrel〉 is the thermal averaged effective 2 → 2 annihilation cross section. Due to this definition, an

extra factor 1/2 is multiplied to the DM cross sections (Eq.(33) and (35)). This comes from the fact that DM and

anti-DM particles are not identical in our case [69].9 The particle S can also have the 3→ 2 annihilation processes, but it can only interact with the SM sector through

the Higgs portal. In this case, the reaction rate would be governed by the Higgs mass, and may be too small to

keep kinetic equilibrium with the SM sector.

12

ν ν

ν ν

ν

ν

ν

ν

(a) (b)

FIG. 6: (a) Feynman diagrams of the elastic scattering between X and the SM neutrinos. (b) Feynman

diagrams for the 2→ 2 annihilation process of a DM pair into a pair of the SM neutrino.

with ζ(3) ' 1.202 the Riemann zeta function of 3.

By the crossing symmetry, the Feynman diagrams for the 2 → 2 annihilation process of a DMpair into a pair of the SM neutrino are shown in Fig. 6(b),10 and the reaction rate is calculated as

Γ2→2 = nX∑r,s

⟨σXX→νrνsυrel

⟩, (34)

where the thermally averaged effective 2→ 2 annihilation cross section is given by

⟨σXX→νrνsυrel

⟩=

m2Xs

4ξ

16π

∑k,l

Re(Y∗rkYrlYskY∗sl

)(M2

k +m2X

)(M2

l +m2X

)( T

mX

), (35)

with nX given by Eq.(31). For simplification of numerical treatment, here we assume the massesof the vector-like fermions are degenerate (M1 = M2 = M3 = M). The reaction rates of the 2→ 2annihilation process and the elastic scattering are then reduced to the form as

Γ2→2 =nXm

2Xs

4ξ

16πx(M2 +m2

X

)2 Y4 , Γkin =3nνm

2Xs

4ξ

16πx(M2 −m2

X

)2 Y4 , (36)

where Y ≡[∑

r,s,k,l Re(Y∗rkYrlYskY∗sl

)]1/4. Using the SIMP condition at Tf , we illustrate the plots

of the magnitude of |Y| as a function of M in Fig. 7 with different numerical inputs based onFig. 4. As indicated in the plots, the order of the Yukawa coupling is |Y| ∼ O(0.01 − 1) with0.1 GeV .M . 1 GeV.

VII. CONCLUSION

In this paper, we have built a SIMP version of the scotogenic model, where the SIMP DMhas the responsibility to generate the neutrino masses and its stability is guaranteed by the Z5

discrete symmetry. We have considered the experimental and theoretical constraints on the massesand the couplings in the model including the neutrino masses and mixings, lepton flavor violating

10 Here we have neglected the scattering processes X`± → X`± and the annihilation channels XX → `+`− due to

the mass suppression of the Z boson and the Higgs boson.

13

Γ2→2 > Γ3→2

Γ3→2 > Γkin

10-2Γkin <

Γ3→2 <Γkin

Γ2→2 < Γ3→2 < 10-2Γkin

ℛ1,2 = 2,5

λ3 = 0

ξ = 0.05

0.2 0.4 0.6 0.8 1.00.01

0.05

0.10

0.50

1

5

M (GeV)

||

mX = 30 MeV, mS = 93 MeV

Γ2→2 > Γ3→2

Γ3→2 > Γkin

10-2Γkin <

Γ3→2 <Γkin

Γ2→2 < Γ3→2 < 10-2Γkin

ℛ1,2 = 3,8

λ3 = 0

ξ = 0.05

0.2 0.4 0.6 0.8 1.00.01

0.05

0.10

0.50

1

5

M (GeV)

||

mX = 40 MeV, mS = 128 MeV

FIG. 7: Magnitude of |Y| versus M for some choices of numerical sets. The white (red) region is the SIMP

(WIMP) paradigm, the green region is the allowed parameter space of |Y| by using the weaker SIMP

condition, and the blue area is the failure of SIMP mechanism.

processes, anomalous magnetic moment, the invisible decay modes of the Z boson and the Higgsboson, the electroweak precision data, perturbativity of the couplings and vacuum stability. In themodels of SIMP DM, a large coupling is generally required in order to reproduce the correct DMrelic abundance measured by experiments through 3 → 2 annihilating processes. This may give atension with perturbativity and potential stability. By employing the resonant mechanism in ourmodel, the correct relic abundance of DM has been reproduced, and the bounds on the quarticcouplings and the self-scattering cross section have been fulfilled at the same time. We found theparameter space of the new Yukawa interactions such that the SIMP condition is achieved. Sinceour model faces to the stringent constraints from the Higgs invisible decay and the direct search ofnew charged scalars, it will be tested in near future.

Acknowledgments

T. T. acknowledges support from JSPS Fellowships for Research Abroad. The work of K. T issupported by JSPS Grant-in-Aid for Young Scientists (B) (Grants No. 16K17697), by the MEXTGrant-in-Aid for Scientific Research on Innovation Areas (Grants No. 16H00868), and by KyotoUniversity: Supporting Program for Interaction-based Initiative Team Studies (SPIRITS).

Appendix A: Gauged U(1)B−L extension of the νSIMP model

It is believed that there is no global symmetry can exist in a theory of quantum gravity [75, 76].Under this context, the discrete symmetry we introduced in our νSIMP model may originate from agauge symmetry (gauge redundancy). At certain energy scale, this gauge symmetry is broken downto the Z5 discrete symmetry by a nonzero VEV of a scalar field. In the following, we demonstratean extension of the νSIMP model to the gauged U(1)B−L version by adding one more SM singletcomplex scalar ζ . The particle contents and the charge assignments are summarized in Tab. II.

14

In this extended model, the Lagrangian associated with the 3→ 2 processes is given by

Lζ = 1√2λ1ζ

∗χ∗S2 + 1√2λ2ζ

∗χ2S + 16λ3χ

3S∗ + H.c. . (A1)

After spontaneous symmetry breaking, the complex scalar ζ can be expanded around its VEV asζ = 1√

2

(ς + υ′

), where υ′ ≡

√2〈ζ〉. The scalar interactions between χ and S are then extracted as

Lζ ⊃ 12λ1υ

′χ∗S2 + 12λ2υ

′χ2S + 16λ3χ

3S∗ + H.c. , (A2)

which corresponding to the first three terms in the last line of Eq.(2), respectively, with µ1,2 = λ1,2υ′.

The Yukawa couplings contributed to the neutrino mass diagrams are the same with in Eq.(9), andthe lightest scalar particle involving in the diagrams can be a SIMP DM candidate. On the otherhand, the SIMP condition can be achieved by Z ′-portal instead of the Yukawa interactions due tothe new gauge boson in this model. We leave the detailed study of the model to future work.

E Φ N1,2,3 η χ S ζ

SU(2) 2 2 1 2 1 1 1

U(1)Y −1/2 1/2 0 1/2 0 0 0

U(1)B−L −1 0 −3/5 2/5 2/5 6/5 2

TABLE II: Charge assignments of the particles in the gauged U(1)B−L extension of the νSIMP model.

Appendix B: Gauge interactions

The kinetic part of the Lagrangian in Eq.(1) contains the interactions of the new scalars withthe photon and the weak bosons,

L ⊃ iη+↔∂ρη−

(eAρ + gLZρ

)+igw

2cw

[c2ξH∗↔∂ρH + s2

ξX∗↔∂ρX + cξsξ

(H∗

↔∂ρX +X∗

↔∂ρH

)]Zρ

+igw√

2

[(cξH

↔∂ρη− + sξX

↔∂ρη−

)W+ρ +

(cξη

+↔∂ρH∗ + sξη

+↔∂ρX∗

)W−ρ

]+ η+η−

(eAρ + gLZρ

)2+

g2w

4c2w

[c2ξ |H|2 + s2

ξ |X|2 + cξsξ(H∗X +HX∗

)]ZρZρ

+g2

w

2

η+η− +

[c2ξ |H|2 + s2

ξ |X|2 + cξsξ(H∗X +HX∗

)]W+ρW−

ρ , (B1)

where

W↔∂ρX = W∂ρX − X∂ρW , gL =

gw

2cw

(1− 2s2

w

), sw =

√1− c2

w . (B2)

With these gauge interactions, we draw the Feynman diagrams of the contributions to the SMgauge boson propagators in Fig. 8.

15

γ γ

η+

η+ γ γ

η+

η+

η+

η+

+ +

η+

+

+

η+

FIG. 8: Feynman diagrams for the contributions of the new scalars to the oblique parameters ∆S,∆Tand ∆U .

Appendix C: Benchmark points

Assuming YL YR and the other parameter set

mH = mη+ = 300 GeV , ξ = 0.05 ,

M1 = 0.4 GeV , M2 = 0.6 GeV , M3 = 1 GeV ,

two benchmark Yukawa couplings are given as

Y =

0.1 0 0

0 0.3 0

0 0 0.5

, YL =

2.26 1.61 0.333

1.61 1.82 0.989

0.333 0.989 0.879

× 10−3, (C1)

for mX = 30 MeV, mS = 93 MeV, µ2 = 150 MeV, and

Y =

0.1 0 0

0 0.2 0

0 0 0.3

, YL =

1.07 1.14 0.261

1.14 1.93 1.16

0.261 1.16 1.15

× 10−3, (C2)

for mX = 40 MeV, mS = 128 MeV, µ2 = 320 MeV. One can check that Eq. (C1) and (C2) satisfythe SIMP condition (Γ3→2/Γ2→2 ' 104 and Γkin/Γ3→2 ' 103) as shown in the left and right panelsof Fig. 7, respectively. These benchmark points give normal ordering neutrino mass eigenvaluesand mixing angles consistent with neutrino oscillation data. It is also possible to take benchmarkparameter sets in the cases for YL YR and inverted hierarchy, though these are not shown here.

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