Sommerfeld effect in freeze-in dark matter Fucheng ZhongaXinyu Wangb aSchool of Physics and Astronomy Sun Yat-sen University Zhuhai Campus 2 Daxue Road Xi-

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Sommerfeld eﬀect in freeze-in dark matter

Fucheng ZhongaXinyu Wangb

aSchool of Physics and Astronomy, Sun Yat-sen University, Zhuhai Campus, 2 Daxue Road, Xi-

angzhou District, Zhuhai, P. R. China

bCenter for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Bei-

jing 100875, China

E-mail: zhongfch@mail2.sysu.edu.cn,wangxy525@mail2.sysu.edu.cn

Abstract: If two annihilation products of dark matter (DM) particles are non-relativistic

and coupled to a light force mediator, their plane wave functions are modiﬁed due to

multiple exchanges of the force mediators. This gives rise to the Sommerfeld eﬀect (SE).

We consider the attractive and repulsive force SE on the relic density in diﬀerent phases of

freeze-in DM. We ﬁnd that in the pure freeze-in region, the attractive/repulsive force SE

slightly increases/decreases DM relic density by less than 20% for TeV-scale DM. In the

reannihilation region, if the portal coupling κis suﬃciently large (by comparing the portal

reaction rate to the Hubble rate), DM density will reach its equilibrium, and subsequently

freeze out. Compared to the case without the SE, the presence of the attractive SE leads to

an enlarged cross-section. As a result, a higher equilibrium value of DM density is reached,

and a lower relic density is obtained after the subsequent freeze-out. However, the repulsive

SE has the opposite inﬂuence. In the dark sector (DS) freeze-out region, also known as

the middle ﬂat plateau or “mesa” in the phase diagram, the SE has a signiﬁcant impact

on DM relic abundance. In this region, the attractive SE suppresses DM relic density by

simultaneously enlarging the cross-section of the portal and DS internal interaction. In

contrast, the repulsive SE will have the opposite eﬀect. Finally, in the usual freeze-out

region, DM relic density is suppressed or enhanced by an enlarged or reduced cross-section

of the portal, respectively, due to the presence of the attractive or repulsive SE. In summary,

when considering the constraint of producing correct DM relic abundance, the inclusion of

SE in the portal reaction or DS internal reaction will modify the model parameters, resulting

in a band-like possible parameter space.

Keywords: dark matter, freeze-in, Sommerfeld eﬀect, relic abundance

arXiv:2210.12505v2 [hep-ph] 8 Oct 2023

Contents

1 Introduction 1

2 Sommerfeld eﬀect and its approximations 3

2.1 Approximations of Sommerfeld eﬀects 3

2.2 The thermal average of the SE-modiﬁed cross-section 5

3 Freeze-in with the SE and upper bound 5

3.1 The SE on pure freeze-in relic density 6

3.1.1 Initial State SE 7

3.1.2 Final State SE 7

3.1.3 Simple 2→2cross-section 7

3.2 SE in DM reannihilation 8

4 The SE in hidden dark sector 11

4.1 Thermalization of the dark sector with SE 12

4.2 Freeze-out of dark sector 14

5 The Model 15

5.1 U(1) dark photon 15

5.2 2→2point contact FIMP 16

6 Conclusion 19

A Kinematics 20

B The upper bound of pure freeze-in 21

C The αthreshold on pure freeze-in 21

D The Boltzmann equation with DS 22

E Balance condition 24

1 Introduction

The Sommerfeld eﬀect [1], as depicted in Fig 1, has been well-considered in freeze-out

scenario [2,3] but not in the freeze-in scenario [4–8]. We suggest the existence of the SE in

the freeze-in mechanism will modify the abundance of dark matter (DM) and constraints.

In the non-relativistic (low-velocity) region, if there is some long-range interaction between

– 1 –

the annihilation particles or their annihilation products, the non-perturbative eﬀects like

the SE or bound state eﬀects [9], need to be considered.

In recent times, the freeze-in dark matter mechanism has gained popularity as several

novel models have proposed it as a viable explanation for the existence of dark matter.

These models may include axion-like particles (ALPs), dark photons, dark Z’ bosons, Feebly

Interacting Massive Particles (FIMPs), or sterile neutrinos [10]... DM was generated with

an initial zero density during freeze-in. The dark sector (DS) will undergo a heating and

then a subsequent cooling process. After decoupling, the DM relic density is proportional to

⟨σv⟩. Hence, the mechanism requires a small coupling to match the observed relic density.

The cross-section correction eﬀect will change the relic density. If there are various particles

in the DS, there will be diﬀerent “phases” according to the various couplings between the

Standard Model (SM) particles and the DS, as well as the coupling within DS.

The relic density of WIMPs is commonly determined through the freeze-out mechanism

[2,3], in which DM particles are initially in thermal equilibrium with the SM and then

decouple as their interaction rate becomes smaller than the Hubble expansion rate. Since

the DM relic density is proportional to 1/⟨σv⟩, a relatively large cross-section is needed to

get a proper DM density. The cross-section enhancement eﬀect will suppress the ﬁnal relic

density of DM, so it requires a larger mass or smaller coupling with the SM.

Non-perturbative eﬀects, such as the SE, bound-state formation, resonance eﬀects,

and co-annihilation, have been well-studied in the context of freeze-out [9,11–16], but

they are rare in the context of freeze-in. One reason for this is that freeze-out typically

occurs at a low temperature compared to the mass of the DM particles (x=m/T ∼25),

resulting in a low velocity for the DM particles before or after annihilation, which can lead

to rich non-perturbative phenomena. In contrast, at the end of freeze-in, the temperature

is approximately equal to the DM mass, resulting in a relatively high temperature during

DM production. At high temperatures, the annihilation cross-section becomes simpler and

is roughly proportional to 1/s, with negligible non-perturbative eﬀects. However, we found

that the SE still makes a signiﬁcant contribution to the cross-section in diﬀerent phases of

the DM freeze-in. In this paper, we focus on the possible SE during the freeze-in process

and we analyze its correction to the DM abundance. Furthermore, we provide a preliminary

investigation of the SE inﬂuence on the boundary and parameter space via speciﬁc models.

We concentrate on the SE in the infrared (IR) freeze-in, rather than ultraviolet (UV)

freeze-in. DM is assumed to have a negligible initial abundance, and its interaction with

the particles in the bath can be so feeble1that it was never in thermal equilibrium with

the SM plasma. The feeble interaction leads to the continuous production of DM until

the reaction rate becomes smaller than the Hubble expansion rate, then DM abundance is

gradually ﬁxed. The typical freeze-in temperature is about x=mr/T ∼2−5[5], where

mrrepresents the relevant mass for the Boltzmann suppression. The relevant particles

are moving at nearly non-relativistic velocities. Furthermore, it is natural for the relevant

particles to exchange light force mediators. In the Standard Model, particle-antiparticle

pairs exchange gauge bosons, and the same may happen in the dark sectors. This is the

1The feeble coupling between DM and standard model particles is about 10−7or less [6,17,18].

– 2 –

··· α

DS V S

· · ·

DS V S

Figure 1. The initial SE (left panel) and the ﬁnal SE (right panel). The middle circle stands for

tree-level interaction.

source of the SE. We suggest that the SE in freeze-in has a correction on DM abundance,

which is well beyond the percent level accuracy of the observational value [19].

Our work is organized as follows: In Sec.2, we make an approximation of the SE in

the low-velocity limit, including the SE resonance situation, to analyze the SE for general

models. Next, in Sec.3, we calculate the SE-corrected DM relic density and the enhancement

ratio in a pure freeze-in region. We also consider DM reannihilate when the SE correction

is large enough. There is a competition between the DM freeze-in and freeze-out. The SE

in the DS is considered in Sec.4, including its eﬀect on DS thermalization and freeze-out. It

gives a similar result as the ordinary WIMP freeze-out. In Sec.5, we provide speciﬁc models

to show the SE inﬂuence on parameter space. Finally, we give some conclusions in Sec.6.

2 Sommerfeld eﬀect and its approximations

In this section, we focus on the contribution of the SE in diﬀerent velocity regions. The

low velocity-dependent part of SE often provides the most signiﬁcant modiﬁcations to DM

production. Identifying the contribution of diﬀerent parts helps us understand how the SE

modiﬁes DM production and the physics involved.

2.1 Approximations of Sommerfeld eﬀects

The s-wave Sommerfeld factor (SF) with massless mediators is given by

So=πα/v

1−e−πα/v ,(2.1)

where vis the particle velocity in the center of mass (COM) frame and αis the coupling

constant with the force mediators. With massive mediators, the analytic SF using the

Hulthén potential approximation [11,20–22] is given by:

Sϕ=π

ϵv

sinh( 2πϵv

π2ϵϕ/6)

cosh( 2πϵv

π2ϵϕ/6)−cos(2πq1

π2ϵϕ/6−ϵ2

(π2ϵϕ/6)2)

(2.2)

for the cases of DM singlet or zero mass gap multiplet, where ϵv=v/α, ϵϕ=mϕ/(αmχ).

Fig. 2indicates that the majority contribution of SF comes from the low-velocity region.

An intuitive explanation is that annihilating particles have suﬃcient time to exchange many

mediators, and the contribution of high-order loop diagrams cannot be neglected.

The behavior of SF under diﬀerent velocities can be simpliﬁed into three parts: low-

velocity resonance region where S∼v−2, low velocity without resonance region where

– 3 –

Sϕ

Sϕ_reso

10-40.001 0.010 0.100 1

100

1000

104

105

106

α=0.1, mχ=1, mϕ=0.1

Sϕ

Sϕ_reso

10-40.001 0.010 0.100 1

1000

105

107

α=1, mχ=1, mϕ=0.1

Figure 2. The Sommerfeld factor and its parameters are shown in the panel’s title. The red,

green, and blue lines correspond to the massless mediator, massive mediator, and massive mediator

resonance situations, respectively. Note that for So,mϕ= 0; for Sϕ_reso,mϕ= 6αmχ/π2. The

values of αand mχremain the same in each panel.

S∼v−1, and the constant region. The SF can be simpliﬁed as:

S≃a+bv−1+cv−2,(2.3)

where astands for the constant term, brepresents the part without resonance, and cis

for the part with resonance. Including the saturation case for the very low velocity (a

constant term), the asymptotic behavior of SF under diﬀerent velocity regions, DM mass,

and coupling constant can be expressed as follows:

S∼









6 csc2(q6

ϵϕ)/ϵϕv < α4ϵϕ

α2/v2ϵv<< ϵϕ,

πα/v ϵv>> ϵϕ, v < vlim

1vlim < v

(2.4)

In the saturation region, SF saturates at v∼α4ϵϕ. The saturation is caused by the ﬁnite

lifetime of bound states [13,23,24]. In the relativistic region, S∼1[21,25], where vlim

is the velocity beyond which the SE becomes negligible. Resonance occurs when there are

zero-energy bound states in two body systems [12]. The situations ϵϕ>1or ϵv>1will

lead to SF ∼ O(1) [21]. The ﬁrst situation represents a mediator with a large mass and

short force range that cannot provide enough SE enhancement, and the second situation

indicates that the SE is insigniﬁcant in the relativistic region. The resonance condition can

be written as:

mϕ≃6αmχ

π2n2, n = 1,2,3. . . (2.5)

We take n= 1 to achieve maximum enhancement.

The p-wave SF can be obtained through the s-wave SF using the following formula [25]:

Sp=(c−1)2+ 4a2c2

1+4a2c2×S(2.6)

– 4 –

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Sommerfeldeffectinfreeze-indarkmatterFuchengZhongaXinyuWangbaSchoolofPhysicsandAstronomy,SunYat-senUniversity,ZhuhaiCampus,2DaxueRoad,Xi-angzhouDistrict,Zhuhai,P.R.ChinabCenterforAdvancedQuantumStudies,DepartmentofPhysics,BeijingNormalUniversity,Bei-jing100875,ChinaE-mail:zhongfch@mail2.sysu.edu.cn,...

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Sommerfeld effect in freeze-in dark matter Fucheng ZhongaXinyu Wangb aSchool of Physics and Astronomy Sun Yat-sen University Zhuhai Campus 2 Daxue Road Xi-

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