A Fresh Look at keV Sterile Neutrino Dark Matter from Frozen-In Scalars

Sterile neutrinos with a mass of a few keV can serve as cosmological warm dark matter. We study the production of keV sterile neutrinos in the early universe from the decay of a frozen-in scalar. Previous studies focused on heavy frozen-in scalars with masses above the Higgs mass leading to a hot spectrum for sterile neutrinos with masses below 8-10 keV. Motivated by the recent hints for an X-ray line at 3.55 keV, we extend the analysis to lighter frozen-in scalars, which allow for a cooler spectrum. Below the electroweak phase transition, several qualitatively new channels start contributing. The most important ones are annihilation into electroweak vector bosons, particularly W-bosons as well as Higgs decay into pairs of frozen-in scalars when kinematically allowed.


Introduction
dominant. Neutrino masses can be generated in different ways. In this work we use the type-II seesaw mechanism [78][79][80][81][82][83]. The paper is organised as follows. In section 2, we introduce the model. In section 3, the dark matter production is explained. In section 4, we discuss the free-streaming horizon of the dark matter in order to determine whether the keV sterile neutrino constitutes HDM, WDM, or CDM. Finally we conclude in section 5. Technical details are collected in the appendices. Cross section and decay widths can be found in appendix A. Their thermal averages are given in appendix B. A short summary of the treatment of effective degrees of freedom is given in appendix C.

Model
Apart from the SM particle fermion content, we introduce one light SM singlet fermion N , one scalar singlet φ and one scalar Higgs triplet ∆. For simplicity we introduce a discrete Z 4 version of lepton number 3 with the following transformation properties of the fields The most general Yukawa interactions in the lepton sector are given by After all scalars obtain a vacuum expectation value (vev) the neutral lepton mass matrix in the basis (ν, N ) reads where m L = y L v ∆ , m LN = y LN v, and m N = y N v φ . In the limit of m N m LN , m L , there is one heavier mass eigenstate with a mass of m N and the active neutrino mass matrix is given by 3 The breaking of the discrete symmetry Z 4 could lead to the formation of domain walls [84], which alter the history of the universe. There are several different ways to avoid this problem. See for example ref. [85][86][87][88][89]. The mechanism even works without imposing a Z 4 symmetry, but the following additional terms are introduced Obviously the tadpole term of φ can be removed by shifting φ. The term (ρ 5 H T iσ 2 ∆ † H + h.c.) is the usual term in the type-II seesaw mechanism. If the explicit mass term of N is of the correct size and the couplings ρ 3 H † Hφ and ρ 4 φTr(∆ † ∆) are sufficiently small, the production of a keV sterile neutrino from a frozen-in scalar works similarly without the Z 4 symmetry.
We are particularly interested in the case of small couplings y LN , which results in a negligibly small seesaw contribution to the light neutrino mass, i.e. m 2 LN m L m N and therefore m ν ≈ m L . The most general scalar potential is given by We use the freedom to re-phase the fields H, ∆ to make κ real and positive as well as the vev of H, v. Minimising the potential, we see that there is no spontaneous breaking of CP and the vevs are given by in the limit of a weakly coupled field φ and a small vev v ∆ . We checked that there are no charge-breaking minima. At leading order there is no mixing between the different scalar fields. The mixing is suppressed by the small Higgs portal couplings, λ Hφ , λ ∆φ , κ and the assumption that the vev of ∆ is generated via its coupling to H and φ. The scalar masses at leading order are given by There are 3 Goldstone bosons G ± ≈ h ± , G 0 ≈ a H , which are absorbed into the SM gauge bosons to become the massive gauge bosons, W ± and Z 0 . The mixing between the different states is suppressed by the small couplings between the singlet φ and the scalars with a SM charge, H and ∆. We will assume in the rest of the paper that the triplet ∆ is sufficiently heavy and its couplings are sufficiently small 4 such that it does not contribute to the production of the scalar singlet σ via triplet Higgs annihilations (∆∆ ↔ σσ). The only relevant mixing for the production via freeze-in is the mixing between the Higgs doublet and the singlet φ, which is given by The relevant couplings for the production of dark matter via freeze-in are the Higgs portal interactions The decay of the sterile neutrino to an active neutrino and a photon is determined by the mixing with active neutrino flavour α via the small mixing angle According to [25; 26], if we want to explain the 3.55 keV line from the decay of the sterile neutrino DM, the active-sterile mixing is constrained to be α sin 2 (2θ α ) 7 × 10 −11 .

Dark Matter Production
We consider the sterile neutrino production in the early universe. If the keV sterile neutrino gets into thermal equilibrium with SM particles, it will be overproduced and will overclose the universe after freeze-out. Therefore usually a keV sterile neutrino is very weakly coupled to the thermal bath and it is assumed that the initial abundance is zero. We concentrate on the freeze-in mechanism as main production mechanism, which has been first discussed in [23]. In the freeze-in mechanism, keV sterile neutrinos are produced in two steps. First a feebly-coupled scalar field, σ, is produced via a tiny Higgs portal coupling, λ Hφ , which has to be small enough such that σ is always out of thermal equilibrium. This scalar subsequently decays into keV sterile neutrinos. The authors of Ref. [23] consider scalars with a mass heavier than the Higgs mass. In this region of parameter space the main process is Higgs annihilation (hh ↔ σσ). See Fig. 1a for a typical evolution of the abundances of the keV sterile neutrino (red) and the scalar σ (blue) for a frozen-in scalar with a mass larger than the mass of the Higgs. As it can be seen in [23] and we explain in more detail in the next section, it turns out that heavy scalars lead to larger free-streaming scales for light keV sterile neutrinos, like a 7.1 keV sterile neutrino. Thus it is interesting to consider scalars lighter than the Higgs and consequently below the EW phase transition. In addition to Higgs annihilation, there are the following additional processes: • annihilation of vector bosons: • annihilation of SM fermions:f f ↔ σσ • Higgs decay to pairs of the scalar σ as well as pairs of keV sterile neutrinos directly.
A typical evolution for a frozen-in scalar with mass m σ = 60 GeV is shown in Fig. 1b. Note that the production of the keV sterile neutrino is dominated by Higgs decay and very weakly depends on the physics above the EW phase transition. The abundances Y σ,N ≡ n σ,N /s normalised to the entropy density s are described by the following Boltzmann equations:  where the superscripts A, D, and HD denote annihilation, decay and Higgs decay terms, respectively. Before the SM particles get out of thermal equilibrium, the different terms are given by Y eq X (T ) = n eq X (T )/s(T ) denotes the equilibrium abundance, G N Newtons constant, g ef f (T ) the effective degrees of freedom at temperature T , and g * (T ) is defined in the usual way. The precise definitions of the thermally averaged cross sections and decay widths as well as g ef f and g * are collected in the appendices. We show the different contributions for six different benchmark points. We fix the keV sterile neutrino mass to m N = 7.1 keV and the scalar self-coupling λ φ = 0.5, which fixes the vev of the scalar singlet σ. We choose the Higgs portal coupling λ Hφ such that the observed DM abundance of Ω DM = 0.1199 ± 0.0027 [90] is obtained at 2σ. We also include the contribution from the DW mechanism using the approximate formula [91] Table 1: Benchmark points. We fix λ φ = 0.5 and the keV sterile neutrino mass m N = 7.1 keV. T in denotes the temperature when 80% of the keV sterile neutrino DM are produced and r F S denotes the free-streaming horizon scale, which is discussed in sec. 4.
fixing sin 2 (2θ) = 7 × 10 −11 . All parameter choices are collected in Tab [39; 92]). Note that the contribution from the Higgs decay to pairs of sterile neutrinos is negligible for our chosen parameters. We do not show its contribution in the plots. Higgs annihilation is the only process above the EW phase transition, because the other processes are proportional to the EW vev and are absent above the EW phase transition. In our numerical analysis we simply set the corresponding annihilation cross section to zero above the EW phase transition. A proper treatment requires the inclusion of thermal corrections to properly treat the temperature dependence during the EW phase transition. Below the EW phase transition we have to distinguish between frozen-in scalars with masses m σ > m h /2 and m σ < m h /2: for m σ > m h /2 there is a sizeable contribution from EW gauge boson annihilations, particularly annihilation of W -bosons, besides Higgs annihilation, as it can be seen in the plots for m σ = 65, 100 GeV, but for m σ < m h /2, the most important process is Higgs decay (See the plots for m σ = 30 GeV and m σ = 60 GeV.). As the coupling λ Hφ is extremely small, the contribution to the invisible decay width of the Higgs is negligibly small.

Free-Streaming Horizon
The free-streaming horizon can be used as the indicator whether the keV sterile neutrinos are HDM, WDM, or CDM. It is the comoving mean distance which a collision-less gravitationally unbound particle travels where t 0 denotes time today, t in is its production time, v(t) its mean velocity at time t, and a(t) the scale factor at time t.
In our discussion we will neglect the DW contribution and concentrate on the main production mechanism via decays of frozen-in scalars. The DW contribution generally leads to a larger free-streaming horizon scale, since it is hotter than the contribution from the frozen-in scalar. Numerically the DW contribution of a keV sterile neutrino with mass m N = 7.1 keV and sin 2 (2θ) = 7×10 −11 constitutes less than 5% of the cosmological DM abundance and is generally strongly constrained [14; 15]. A detailed study of the free-streaming horizon scale is beyond the scope of this paper. Guided by our numerical results in Fig. 2, we estimate the production time t in 5 of the keV sterile neutrinos to be the time, when 80% have been produced. 6 We follow the discussion of the free-streaming horizon in [23]. We assume an instantaneous transition between the relativistic and the non-relativistic regimes of the keV sterile neutrino, i.e. v(t) 1 for t < t nr and v(t) where t nr is the time when the particle becomes non-relativistic, that is p(t nr ) = m N . Our numerical results in Fig. 2 show that the frozen-in scalar with masses m σ 30 GeV mostly decays when it is non-relativistic. If the frozen-in scalar is lighter, a significant fraction will already decay when they are relativistic. We will focus on frozen-in scalar with masses m σ 30 GeV and thus can safely assume that it decays non-relativistically. The rest of the discussion follows ref. [23]. The distribution of the DM produced from a non-relativistic parent σ is given by [93][94][95][96] f (p, where β is a normalisation factor and T DM = p cm a(t d )/a(t) is the DM temperature. Using the DM momentum in the centre-of-mass frame, The average momentum p(t) can be calculated to be In the radiation dominated era, the scale factor a ∝ t 1/2 . Thus from Eq. (21) and p(t nr ) = m N , the time when N becomes non-relativistic is given by The relation between the production time t in and the production temperature T in is given by t in In our case DM becomes non-relativistic before the time of matter-radiation equality, t eq = 1.9 × 10 11 s. Thus the free-streaming horizon can be calculated as 2 √ t eq t nr a eq + √ t eq t nr a eq ln t eq t nr + 3 √ t eq t nr a eq (24) √ t eq t nr a eq 5 + ln t eq t nr .
where a eq is the scale factor at matter-radiation equality. Including entropy dilution the final expression is [23] r FS √ t eq t nr a eq 5 + ln t eq t nr /ξ 1/3 .
where the entropy dilution factor is given by We have taken into account the scalar field σ and the sterile neutrino N . DM with a freestreaming scale larger than the size of a dwarf galaxy (r F S 0.1 Mpc) constitutes HDM. The distinction between WDM and CDM is arbitrary. We follow ref. [23] and consider DM with a The free-streaming horizon depends both on the mass of the scalar field and the mass of sterile neutrino. In Fig. 3 we plot the free-streaming horizon vs. the keV sterile neutrinos mass for different values of the scalar mass m σ = 30, 60, 65, 100, 170, 500 GeV. The HDM (CDM) regions are coloured red (blue). Note that keV sterile neutrinos with m N = 7.1 keV are WDM for m σ 170 GeV and become too hot for larger frozen-in scalar masses. Demanding that the keV sterile neutrino accounts for the cosmologically observed DM abundance at 1σ and fixing λ s = 0.5, we can plot the required value of the Higgs portal coupling λ Hφ as a function of the keV sterile neutrino mass m N and the frozen-in scalar mass m σ . This is shown in Fig. 4. The blue (red) coloured region indicates the CDM (HDM) region. The black lines are contour lines of equal λ Hφ assuming a vanishing active-sterile mixing angle, θ = 0. The jaggedness of the black contour lines appears because it is not possible to fix the DM abundance to a number, but we only demand it to lie within the 1σ allowed region. The grey shading in the background indicates the size of λ Hφ . Darker regions correspond to larger values of λ Hφ . The magenta line marks the Higgs mass m h and the orange line marks m h /2. To the left of the orange line, Higgs decays will dominate the production of the frozen-in scalar σ and the required value of λ Hφ is generically smaller. We find λ Hφ 10 −8 for m σ < m h /2 and λ Hφ 10 −8 for m σ > m h /2.

Conclusion
We studied sterile neutrino dark matter production from the decay of a frozen-in scalar. The previous study of this production mechanism [23] focused on heavy frozen-in scalars with masses above the Higgs mass and light keV sterile neutrinos with masses below 10 keV turned out to have a free-streaming scale larger than the size of dwarf galaxies and are thus too hot and excluded. Motivated by the hints for an X-ray line at 3.55 keV, we are considering lighter frozen-in scalars. This leads to a smaller free-streaming horizon and thus lighter sterile neutrinos are allowed. We find that Higgs decay is the dominant production channel when kinematically allowed. For m σ > m h /2 Higgs decay becomes kinematically forbidden and the main production channels are annihilation into Higgs pairs as well as pairs of EW gauge bosons, particularly W -bosons. Above the EW phase transition the frozen-in scalar and thus the keV sterile neutrino is only produced by Higgs annihilations. Furthermore we calculated the free-streaming horizon to show the viable region in parameter space, where sterile neutrinos are WDM or CDM. A 7.1 keV sterile neutrino requires the frozenin scalar to be lighter than approximately 170 GeV. Demanding that the keV sterile neutrino accounts for the cosmological DM abundance, we determined the necessary value of the Higgs portal coupling λ Hφ . Above m h /2 the coupling is λ Hφ 10 −8 , below m h /2 the coupling is generically smaller, λ Hφ 10 −8 . Neutrino masses are naturally obtained using the type-II seesaw mechanism [78][79][80][81][82][83], as we discussed in section 2. After the frozen-in scalar and the SM Higgs obtain a vev, a small vacuum expectation value is generated for the Higgs triplet. We discussed the most minimal model of a keV sterile neutrino production from a frozen-in scalar. It might be interesting to promote the discrete Z 4 symmetry to a continuous U (1). This introduces a (pseudo) Goldstone boson (pGB) which substantially modifies the production. As we expect the pGB to have strong couplings to the scalar σ via its quartic interaction, it will be efficiently produced and the cosmological DM abundance might be explained by a mixture of keV sterile neutrinos and the pGB.

A Cross sections and Decay Widths
The relevant cross sections W ab = 4E a E b σv for annihilation into fermions W f f , vector bosons W V V and the Higgs W hh 7 are given by [97] W hh = λ 2 The annihilation into Z (W )-bosons is The partial Higgs decay width to scalars is given by and the partial decay widths of σ as well as the Higgs to two keV neutrinos N are described by where y N is chosen to be real without loss of generality and sin γ denotes the mixing between the Higgs and the scalar σ, which is defined in Eq. (12).

B Thermal Averages
The thermal average of a partial decay width Γ(X → ii) of a particle X with mass m X can be calculated as follows with x = m X /T . Following ref. [97; 98], we write the thermally averaged annihilation cross section, σv , as where x = m σ /T and n eq denotes the equilibrium number density of σ. It is given by where g σ = 2s σ + 1 are the spin degrees of freedom of σ and K 2 (x) denotes the modified Bessel function of second kind. W ab is defined as 4E a E b σv. The relevant cross sections W ab are given in App. A.

C Effective Degree of Freedom
We follow the discussion in [99]. The energy density and the entropy density are defined as In order to define the total effective number of degrees of freedom, we first have to define the effective number of degrees of freedom for energy and entropy for each individual particle where ρ i (T ) (s i (T )) is the energy (entropy) density of each particle species and x i = m i /T with m i being the mass of the particle species. The total energy effective degrees of freedom is where g c (T ) is the energy effective degree of freedom including all coupled species at temperature T . The second contribution includes all species that are already decoupled at temperature T . Like in [99], we also choose T d i = m i /20 for the decoupling temperature. The total entropy effective degrees of freedom can be calculated as where h c (T ) =