Two-component millicharged dark matter and the EDGES 21cm signal

We propose a two-component dark matter explanation to the EDGES 21 cm anomalous signal. The heavier dark matter component is long-lived whose decay is primarily responsible for the relic abundance of the lighter dark matter which is millicharged. To evade the constraints from CMB, underground dark matter direct detection, and XQC experiments, the lifetime of the heavier dark matter has to be larger than $0.1\, \tau_U$, where $\tau_U$ is the age of the universe. Our model provides a viable realization of the millicharged dark matter model to explain the EDGES 21 cm, since the minimal model in which the relic density is generated via thermal freeze-out is ruled out by various constraints.

If the EDGES measurement is interpreted as caused by a colder gas, the question is how to cool the hydrogen gas? One simple solution is that if there exits some sort of particle interactions between dark matter (DM) and hydrogen atom, the primordial hydrogen gas can be cooled by DM, which is colder than gas. However, there exist very strong constraints from the CMB measurements as well as from other early universe measurements on DM interactions to standard model particles. Millicharged DM is one of the leading DM candidates to explain the EDGES anomaly because its interaction with baryons is proportional to v −4 , where v is the relative velocity, and thus leads to a much smaller interaction cross section in the early universe than that needed at z 17 for the EDGES interpretation so that the early universe constraints can be significantly alleviated. Millicharged DM have been extensively investigated for the interpretation of the EDGES anomaly [10][11][12][13][14][15][19][20][21][22]. The parameter space of the millicharged DM is constrained by various experiments, including accelerator experiments [24], the CMB anisotropy [25] [26] [27] [19] [20], the SN1987A [28], and the Big Bang Nucleosynthesis (BBN) [29] [12] [30]; the allowed parameter space is that the millicharged DM mass is 0.1 MeV m χ 10 MeV, the millicharge is 10 −6 Q/e 10 −4 , and the mass fraction of the millicharged DM is 0.0115% f 0.4%. However, as pointed out by Ref. [21], such a parameter space is ruled out by the N eff limit with the Planck 2018 data if the relic abundance of the millicharged DM is set by thermal freeze-out. Recently, Ref. [22] proposed a new millicharged DM model in which the sub-component millicharged DM has a sizable interaction cross section with the other DM components so that the millicharged DM can be cooled by the other DM components; this reopens the parameter space that was previously excluded by various experimental constraints.
In this paper we propose a new DM model that consists of two DM components: the lighter DM component is the millicharged DM, and the heavier DM component is unstable, which decays into the lighter DM component. In our model, the millicharged DM are primarily produced after the recombination so that the stringent constraints from CMB can be alleviated. We show that such a model can explain the 21 cm anomaly observed by EDGES and satisfy various experimental constraints. The rest of the paper is organized as follows. We present our model in section 2. We compute the number density of the two DM components as a function of redshift in section 3. The temperature change due to the heavier DM decays is derived in section 4. We provide the time evolution equations for four different physics quantities in section 5. The results of our numerical analysis is given in section 6. We summarize our findings in section 7.

The model
We extend the standard model (SM) by introducing a hidden sector that consists of three U (1) gauge bosons, X i µ (i = 1, 2, 3), and one Dirac fermion χ that is charged under both X 2 µ and X 3 µ gauge bosons. We use the Stueckelberg mechanism [31][32][33][34][35][36] to provide mass to the three U (1) gauge bosons; the new Lagrangian is given by where σ 1 , σ 2 , and σ 3 are the axion fields in the Stueckelberg mechanism, B µ is the SM hypercharge boson, m χ is the dark fermion mass, g χ 2 and g χ 3 are the gauge couplings, and m 1 , m 1 , m 2 , m 3 , and m 4 are the Stueckelberg mass terms.
After the spontaneous symmetry breaking in the SM, the mass matrix of the neutral gauge bosons in the basis (X 1 , X 2 , X 3 , B, A 3 ), where A 3 is the third component of the SU (2) L gauge bosons in the SM, is given by where v is the vacuum expectation value of the SM Higgs, and g 2 and g Y are the SU (2) L and U (1) Y gauge couplings in the SM respectively. The mass matrix has a vanishing determinant such that there exists a massless mode to be identified as the SM photon.
Because the mass matrix is block-diagonal, one can diagonalize the first two gauge bosons and the last three gauge bosons separately.
The mass matrix of the first two gauge bosons (the upper-left two by two block matrix in Eq. (2.2)) can be diagonalized via a rotation matrix R which is parameterized by a single angle θ The mass eigenstates, Z 1 and Z 2 , are related to the gauge states via Z i = R ji X j . The rotation matrix R leads to an interaction between χ and Z 1 such that We are interested in the parameter space where θ 1 so that Z 1 ∼ X 1 and Z 2 ∼ X 2 . In our analysis, we take m 1 ∼ 2m χ , m 2 < m χ , and m 1 m 2 so that the two mass eigenstates have masses m Z 1 m 1 and m Z 2 m 2 , and the mixing angle θ is given by θ m 1 m 2 /(m 2 1 − m 2 2 ). The mass matrix of the last three gauge bosons (the bottom-right three by three block matrix in Eq. (2.2)) can be diagonalized by an orthogonal matrix O such that are the gauge states, and E i = (Z , Z, γ) are the mass eigenstates. Here γ is the photon, Z is the neutral gauge boson in the weak interaction, and Z is the new massive vector boson.
is the bottom-right three by three block matrix in Eq. (2.2). Such a matrix diagonalization also leads to interactions between matter fields (both hidden sector fermion χ and SM fermions f ) and the three mass eigenstates (γ, Z, and Z ). The interaction Lagrangian can be parameterized as followsf where the vector and axial-vector couplings are given by Here Q f is the electric charge of the SM fermion, and T 3 f is the quantum number of the left-hand chiral component under SU(2) L .
Thus, the hidden sector fermion χ has a vector current interaction with the SM photon, where we have defined an electric charge for the χ particle. In our analysis, we adopt the following model parameters: m 3 = 100 TeV, and m 4 /m 3 1. In this case, the electric charge is given by where θ W is the weak rotation angle in the SM. Since m 4 /m 3 1 in our analysis, we have 1, which is often referred to as millicharge. χ is then the millicharged particle.

Two DM components
There are two DM particles in the hidden sector, the Z 1 boson and the hidden Dirac fermion χ. In the very early universe, Z 1 is the dominant DM component, which is assumed to be nonthermally produced. The Z 1 DM component is long-lived and decays intoχχ. The decay width of the Z 1 boson is given by where v χ Z 1 = g χ 2 sin θ g χ 2 θ, ∆m ≡ m Z 1 − 2m χ , and we have assumed ∆m m Z 1 . In our analysis we have θ 1 and ∆m m Z 1 so that Z 1 is long-lived with a lifetime where τ 17 ∼ 7 × 10 15 second is the time between the early universe and z = 17.
The value of ∆m cannot be very large, otherwise the DM χ is significantly heated by the decay process Z 1 →χχ so that χ is unable to cool the baryons. The velocity of the χ particle is v χ ∆m/m χ in the rest frame of Z 1 , under the assumption of ∆m m χ . For the case where m χ ∼ 100 MeV and ∆m ∼ O(meV), one has v χ ∼ 3.2 × 10 −6 . Thus, in our analysis, we assume a sufficiently small mass difference, ∆m ∼ O(meV), such that the Z 1 decay does not heat the χ DM significantly.
The χ DM component is mainly produced via the decay process Z 1 →χχ in the universe. We assume that the initial number density of χ is negligible. The relic density of χ can also be produced via thermal freeze-out. There are two processes that contribute to the χ DM annihilation cross section:χχ → γ →f f andχχ → Z 2 Z 2 . In our analysis, σ(χχ →f f ). Theχχ annihilation cross section into on-shell Z 2 bosons is given by [37] σv where r = m Z 2 /m χ . For the case where g χ 2 = 1, r = 1/2, and m χ = 5 MeV, one has σv (χχ → Z 2 Z 2 ) 0.3 barn, leading to a mass fraction as f χ ∼ 10 −11 . Thus, the contribution to the relic density of χ from thermal freeze-out is negligible, and the constraints imposed on thermal freeze-out millicharged DM (e.g. Ref. [21]) are not directly applicable to our model.
The total number of Z 1 particles in a comoving volume at time t is given by where τ is the lifetime of the Z 1 particle, and N Z 1 (0) is the total number of Z 1 particles at time t = 0. In our analysis, we set t = 0 at redshift z 0 = 10 6 . The number of χ particles at time t in a comoving volume is N Thus, the number density of χ is related to the number density of Z 1 via where n Z 1 (n χ ) is the number density of the Z 1 (χ) particle. In our analysis m Z 1 2m χ , so the mass fraction of the millicharged DM χ at redshift z in the total DM is given by where t(z) is the time between early universe (which we take to be z 0 = 10 6 ) and redshift z. 1 Fig. (1) shows the mass fraction of χ as a function of the redshift z for different lifetimes τ . CMB observations provide strong constraints on millicharged DM; only 0.4% DM can be millicharged unless the millicharge is negligible [27] [19] [20]. This leads to an lower bound on the τ , which is ∼ 3.6 × 10 15 s. Because ∆m m χ , the total DM density ρ Z 1 + ρ χ at redshift z is given by ρ Z 1 + ρ χ = ρ DM,0 (1 + z) 3 . Thus the number density of χ particles at redshift z is given by where ρ DM,0 = Ω DM ρ cr,0 is the current DM density where ρ cr,0 = 1.054h 2 × 10 4 eV cm −3 is the critical density [38]. In our analysis, we use Ω DM h 2 = 0.1186 [39].

DM temperature increase generated by decays
The DM χ is heated by the Z 1 → χχ decay process because of the difference between the Z 1 mass and twice of the χ mass. To compute this effect, consider the kinetic energy ∆q that goes into the χχ final state for the decay process Z 1 → χχ 1 The formulas to compute t(z) are given in Appendix A.  where 3k B T Z 1 /2 is the averaged kinetic energy of Z 1 with T Z 1 being the temperature of the Z 1 particle and k B being the Boltzmann constant. Here we have assumed that Z 1 is non-relativistic and ∆m is sufficiently small such that χ is also non-relativistic. The change of the particle numbers in the comoving volume per unit time due to decay are given bẏ where the dot denotes the derivative with respect to time, N χ (N Z 1 ) is the particle number of χ (Z 1 ), and Γ Z 1 is the decay width for the process Z 1 → χχ. The total kinetic energy transfer to the χ particles per unit time from Z 1 decays is given by ∆q|Ṅ Z 1 |, which is equal to the change of the kinetic energy of the χ particles per unit time Thus, the χ temperature change per unit time due to Z 1 decays is given bẏ

Time evolution equations
To compute the baryon temperature at redshift z = 17, we numerically solve the temperature evolutions for various quantities. The time evolution equation of the baryon temperature T b is given by (see e.g. [10,40]) where H is the Hubble parameter, 2 T γ is the CMB temperature, Q b is the energy transfer term due to DM-baryon scatterings, and Γ C is the Compton scattering rate which describes the effects due to CMB-baryon interactions. The Compton scattering rate is given by [10] where σ T is the Thomson cross section, f He is the Helium fraction, and U is the energy density. In our analysis, we use σ T = 6.65 × 10 −25 cm 2 [38], f He = 0.08 [10], and U = (π 2 /15) T 4 γ = 0.26(1 + z) 4 eV/cm 3 [41] [42]. The time evolution equation of the temperature T χ of the lighter DM component χ is given by where Q χ is the energy transfer term due the interaction between DM χ and baryons. The first two terms on the right-hand side of Eq. (5.3) represent the effects due to universe expansion and the DM-baryon scattering respectively, which are similar to the first two terms in Eq. (5.1). The third term on the right-hand side of Eq. (5.3) is new and is due to the decay of the Z 1 particle in our model, as discussed in section 4.
In addition, we also solve the time evolution equation of the relative bulk velocity between baryon and DM, We follow Ref. [43] to obtain the various coefficients in Eq. (5.5). The formulas of Q b , Q χ , and D(V χb ) for millicharged DM used in our analysis are given in Appendix B.

Results
We solve simultaneously the four time evolution equations for T b , T χ , V χb and x e from redshift z = 1010 to z = 10. The baryon temperature T b at z = 1010 is assumed to be equal to the CMB temperature T γ = T 0 (1 + z) where T 0 = 2.7 K, since these two components are tightly coupled in the early universe. The temperatures for both DM components are assumed to be negligible in the early universe, so we set T Z 1 = 0 and T χ = 0 at z = 1010. This is due to the fact that in our model Z 1 does not interact with the SM particles and ∆m m χ . We also set x e = 0.05 [44] and V χb = 29 km/s [10] at redshift z = 1010.  Table 1. Benchmark model points. All the masses are in unit of MeV. We take g χ 2 = g χ 3 = 1 in our analysis.
We scan the parameter space spanned by and m χ for three different decay lifetimes of Z 1 : τ = 2 × 10 16 s, 3 × 10 17 s, and 8 × 10 17 s. In our analysis, we fix ∆m = 1 meV. In order , with the mass fraction of the millicharged DM component being 0.06%, 0.004%, and 0.001% at z = 1100, and 100%, 77%, and 42% today respectively. The green shaded region is excluded by various accelerator experiments, including SLAC electron beam dump [24], CMS [45], MiniBooNE and LSND [46], ArgoNeuT [47], milliQan demonstrator [48], and others [49] [50]. The gray shaded region indicates the parameter region excluded by the dark matter direct detection (DMDD) experiments; above the DMDD region, millicharged DM is absorbed by the rocks on top of underground labs [51] [22]. The magenta region is ruled out by the rocket experiment XQC for mass fractions: f = 100% (solid) [52], f = 1% (solid) [52], and f = 0.4% (solid) [51]. The brown shaded region is excluded by the SN1987A data [53]. The black dashed vertical line indicates the upper bound on DM mass due to ∆N eff from CMB [29] [12]. The black dotted line indicates the parameter space of the minimal millicharged DM model with a mass fraction of 0.4% to explain EDGES data [22].

Conclusions
We construct a new millicharged DM model to explain the recent 21 cm anomaly. In our model, the millicharged DM χ is a subcomponent in the early universe and is mainly produced via decays of the other DM component Z 1 . The DM annihilation cross section χχ → Z 2 Z 2 is so strong that the relic abundance due to thermal freeze-out is negligible. We compute the heating term due to the decay process Z 1 →χχ and include it in our numerical calculations of the time evolution equation of the DM temperature. We find that the model can explain the EDGES 21 cm anomaly while satisfying various experimental constraints, including those from colliders, XQC, underground DMDD, and CMB.

Acknowledgement
We thank Ran Ding and Mingxuan Du for helpful discussions. The work is supported in part by the National Natural Science Foundation of China under Grant No. 11775109.

A Time
The time t(z) at redshift z is given by where H is the Hubble parameter. Here we use z 0 = 10 6 . We compute the Hubble parameter at redshift z via where H 0 ≡ 100h km s −1 Mpc −1 is the present Hubble parameter, Ω R , Ω m , and Ω Λ are the density of radiation, matter, and dark energy respectively. In our analysis, we adopt the following values: Ω R = 2.47 × 10 −5 /h 2 [54], Ω m = 0.308, Ω Λ = 0.692, and h = 0.678 [39].

B Millicharged DM formulas
We provide the formulas of σ 0,t , Q b , Q χ , and D(V χb ) for millicharged DM used in our analysis.
The scattering cross section between millicharged DM and baryons can be parameterized as σ t = σ 0,t v −4 where v is the relative velocity between DM and baryons, and σ 0,t = 2πα 2 2 ξ/µ 2 χt where α is the fine structure constant, is the millicharge, µ χt is the reduced mass of DM χ and the target particle t, and ξ is the Debye logarithm [55] [10] ξ = ln 9T 3 b /(4π 2 α 3 x e n H ) . The baryon heating term due to interactions with millicharged DM is given by [40] [10] where u 2 th ≡ T b /m b + T χ /m χ , r ≡ V χb /u th , and F (r) = erf r/ √ 2 − √ 2πre −r 2 /2 . Here we assume that electron and proton share a common temperature T b with the hydrogen atom. The DM heating term due to interactions with baryons is given by where n e = n p = n H x e is assumed. The D term in Eq. (5.4) is given by [40] where we consider both electron and proton as the target baryons.