Colder Freeze-in Axinos Decaying into Photons

We point out that 7 keV axino dark matter (DM) in the R-parity violating (RPV) supersymmetric (SUSY) Dine-Fischler-Srednicki-Zhitnitsky model can simultaneously reproduce the 3.5keV X-ray excess, and evade stringent constraints from the Ly-alpha forest data. Peccei-Quinn symmetry breaking naturally generates both axino interactions with minimal SUSY standard model particles and RPV interactions. The RPV interaction introduces an axino-neutrino mixing and provides axino DM as a variant of sterile neutrino DM, whose decay into a monochromatic photon can be detected by X-ray observations. Axinos, on the other hand, are produced by freeze-in processes of thermal particles in addition to the Dodelson-Widrow mechanism of sterile neutrinos. The resultant phase space distribution tends to be colder than the Fermi-Dirac distribution. The inherent entropy production from late-time saxion decay makes axinos even colder. The linear matter power spectrum satisfies even the latest and strongest constraints from the Ly-alpha forest data.

Introduction -Supersymmetry (SUSY) and Peccei-Quinn (PQ) symmetry are two of the most promising extensions of the standard model (SM).While SUSY resolves the hierarchy between the electroweak scale and the quantum gravity or grand unification scale [1], PQ symmetry explains why quantum chromodynamics (QCD) preserves CP symmetry accurately [2].The extensions of the SM with these symmetries introduce natural dark matter (DM) candidates: neutralino and axion, respectively.While a promising parameter region of axion DM is still under investigation [3], neutralino DM is already tightly constrained by direct and indirect searches [4].On the other hand, combining the two symmetric extensions introduces another attractive DM candidate: axino (ã), which is the fermion SUSY partner of QCD axion (a) [5].The mass of axino is generated by SUSY breaking, so naively is of order the gravitino mass.In some models, however, the axino mass can be of order keV [6], where axino is the lightest SUSY particle and thus a warm dark matter (WDM) candidate.
In this letter, we consider the R-parity violating (RPV) SUSY Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model [7].The SUSY µ-term is generated by PQ symmetry breaking à la the Kim-Nilles mechanism [8].The interaction responsible for the µ-term mediates axino and minimal supersymmetric standard model (MSSM) particles.The production of axinos follows the freeze-in nature of feebly interacting massive particles (FIMPs) [9], since the interactions with MSSM particles are apparently renormalizable but are feeble due to the suppression by the PQ symmetry breaking scale, v PQ 10 9 GeV.In addition, bilinear R-parity violating (bRPV) terms are generated in a similar way [10].Such interactions trigger an axino-neutrino mixing like sterile neutrino [11], and the resultant axino DM decay into a monochromatic photon can be detected by X-ray observations.In light of the recent evidence of an anomalous 3.5 keV X-ray line in the Andromeda galaxy and galaxy clusters, it is timely to consider models of keV-mass decaying DM.The line excess in the XMM-Newton and Chandra data was first reported by two independent groups in February 2014 [12,13].Subsequent studies showed that similar excesses are also found in the Galactic Center [14] and in the Suzaku data [15].While there are reports of a null detection (e.g., in observations of dwarf spheroidal galaxies [16]), the decaying DM explanation of the 3.5 keV line excess is yet to be excluded (see [17] for a thorough review).
Constraints from Ly-α forest data are rather relevant, in general, when decaying 7 keV DM is considered as an origin of the 3.5 keV line excess.For example, 7 keV sterile neutrino DM from scalar particle decay satisfies the less stringent constraints, m WDM > 2.0 [19] and 3.3 keV [20], but is in tension with the recently updated ones, m WDM > 4.09 [21] and 5.3 keV [22], as shown by direct comparisons of linear matter power spectra [18].One may wonder why m WDM > 5.3 keV disfavors 7 keV DM.Note that such bounds are derived under the assumption that WDM particles follow the Fermi-Dirac distribution with two spin degrees of freedom, and they reproduce the observed DM mass density by tuning the temperature (T WDM ) for a given mass (m WDM ): where T ν is the neutrino temperature, and g * (T dec ) is the effective massless degrees of freedom when the WDM particles are decoupled.This relation clearly shows that we need g * (T dec ) ∼ 7000, which implies a large entropy arXiv:1707.02077v1[hep-ph] 7 Jul 2017 dilution factor, ∆ ∼ 70, in addition to the full SM degrees of freedom, g SM = 106.75,even if the WDM particles decouple before the electroweak phase transition.The phase space distribution of freeze-in axinos varies depending on its production processes, and thus it is affected by the mass spectrum of MSSM particles involved in freeze-in processes.We obtain the resultant phase space distribution by integrating the Boltzmann equation, and find that it is typically colder than the Fermi-Dirac one. 1 Saxion (s), which is the scalar partner of axion, also makes axinos DM colder, since its late-time decay injects a certain amount of entropy to the thermal bath after axino decoupling.We calculate the resultant linear matter power spectra of freeze-in 7 keV axino DM, and show that they are concordant with the current constraints from the Ly-α forest data.
Model -The DFSZ solution to the strong CP problem invokes a coupling between a PQ symmetry breaking field (X) and the up-and down-type Higgs doublets (H u,d ).Its SUSY realization is given by the following superpotential: where y 0 is a dimensionless constant and M * is a cutoff scale.The PQ charges of X, H u , and H d are respectively −1, 1, and 1.Once the field X develops its vacuum expectation value (VEV), i.e., A is the axion superfield, the µ-term and an axino interaction are generated as where µ = y 0 v 2 PQ /(2M * ).The approximate equality is valid when one considers the axino interaction.If M * ∼ 10 16 GeV, y 0 ∼ 0.1, and v PQ ∼ 10 10 GeV, one finds µ ∼ 500 GeV.This is a well-known solution to µ-term generation by the Kim-Nilles mechanism [8].From this renormalizable interaction, freeze-in production of axinos occurs dominantly when the cosmic temperature (T ) is of order the mass of the other SUSY particle involved in the process [25][26][27].The contributions from dimensionfive anomaly operators (e.g., axino-gluino-gluon) are suppressed [26].
The bRPV term is also generated as [10] If M * ∼ 10 16 GeV, y i ∼ 1, and v PQ ∼ 10 10 GeV, one 1 Different realizations of 7 keV axino DM decay were considered in Refs.[23].Nevertheless, none of them discussed a phase space distribution of axino DM or Ly-α forest constraints.
FIG. 1. Axino phase space distributions from respective production processes.The red, blue, and yellow solid lines show q 2 f (q) respectively from Higgsino 2-body decay and s-and t-channel scatterings, while the purple solid line shows that from wino 3-body decay.For comparison, the Fermi-Dirac distribution is shown by the dashed line.Each distribution is normalized such that dqq 2 f (q) = 1.
finds µ i ∼ MeV.This term generates mixing between active neutrinos and axino.The mixing angle is given by where v u is the VEV of H u and m ã is the axino mass.One finds that the mixing parameter of sin 2 2θ ∼ 10 −10 is easily obtained, so axino DM decay can be an origin of the 3.5 keV X-ray line excess like sterile neutrino DM.From this mixing, axinos are produced by the Dodelson-Widrow mechanism [28], but they account for only a few % of the total DM density [29].Therefore, there must exist a more efficient production mechanism of axinos: freeze-in production via the µ-term interaction.
Freeze-in Production -The production of axino is governed by the following Boltzmann equation: where f ã(t, p) is the axino phase space distribution as a function of the cosmic time (t) and the axino momentum (p), R(t) is the cosmic scale factor, E is the axino energy, and C(t, p) is the collision term.Due to feeble interactions of axino, one can safely neglect f ã in the collision term.Then by integrating the both sides from t = t i to t = t f , one finds Once one collects all the relevant contributions to the collision term, it is easy to obtain the axino phase space  Higgsino NLSP is shown.The SM-like Higgs mass and soft masses at Q = mtc are calculated by SUSY-HIT v1.5a [31].
The masses of all the other SUSY particles are taken to be 10 TeV.
distribution.We do not provide details here, but refer readers to Ref. [30].
For the freeze-in production of axinos, the contributions of 2-body and 3-body decays, and s-and t-channel scatterings are taken into account.In Fig. 1, phase space distributions, in form of q 2 f ã(q) (q = p ã/T ã), are shown for Higgsino 2-body decay ( H → H + ã), s-channel scattering (t+ t → H+ã), t-channel scattering ( H+t → ã+t), and wino 3-body decay ( W → H + H + ã). 2 Here we define the axino temperature by T ã = (g * (T )/g * (T th )) 1/3 T , where T th is set to the mass of the other SUSY particle involved in the freeze-in process.While all the freezein processes shown in Fig. 1 have a colder phase space distribution than the Fermi-Dirac distribution, a 3-body decay case has the coldest distribution.The reason is that 3-body decay leads to a smaller kinetic energy of the final-state axino than the other processes at a given temperature.However, when one considers a realistic example, such a 3-body decay rarely dominates over other processes, so the resulting axino phase space distribution follows those of 2-body decay or s-and t-channel scatterings.
For a realistic analysis, we consider a benchmark point where the Higgsino-like neutralino is the next-to-lightest SUSY particle (NLSP).The mass spectrum is shown in Table I.In this benchmark scenario, the dominant process is Higgs decay into Higgsino and axino, while Higgsino 3-body decay and s-and t-channel scatterings also contribute. Figure 2 shows the resultant axino phase space distribution accompanied by the contributions of the respective processes. 3It is clearly shown that the freeze-in production of axinos leads to a colder phase 2 In Fig. 1, we take tops (t and t) and Higgses to be massless, while introducing the thermal mass of intermediate Higgs in tchannel scattering.In the realistic analysis with the benchmark point below, however, we take into account the Higgs soft masses while tops are still massless. 3We add the s-channel scattering contribution to that of the 2- space distribution than the Fermi-Dirac one.
Ly-α Constraints -In order to examine whether 7 keV freeze-in axino DM with the phase space distribution obtained above is concordant with the constraints from the Ly-α forest data, we calculate linear matter power spectra by using a Boltzmann solver, CLASS [32].We define the squared transfer function by the ratio of the body decay.This is because we define the s-channel scattering contribution by subtracting the Higgs pole from the matrix element to avoid the double counting of the 2-body decay.See Ref. [30] for details.WDM linear matter power spectrum to the cold dark matter one, which is denoted by T 2 (k) as a function of the wave number, k. Figure 3 compares T 2 (k) in the benchmark point with those for the Ly-α forest lower bounds of m WDM = 2.0, 3.3, and 4.09 keV.For comparison, we also show T 2 (k) for 7 keV axino DM from UV production via non-renormalizable operators (more specifically, W + H → H + ã), where the produced axinos follow the Fermi-Dirac distribution. 4It is clearly shown that 7 keV axino DM from UV production is disfavored by the Ly-α forest data, when one incorporates the m WDM > 3.3 keV or stronger constraint.On the contrary, 7 keV axino DM from freeze-in production in our benchmark scenario shows larger T 2 (k) so that it is allowed even by the constraint of m WDM > 3.3 keV.It is, however, still in tension with the stronger constraint, m WDM > 4.09 keV.
In this regard, one can conclude that a certain amount of entropy production is still necessary, when the stronger Ly-α forest constraints, m WDM > 4.09 and 5.3 keV, are taken into account.In Fig. 4, we find that 7 keV axino DM with ∆ = 4.7 fits the strongest lower bound from the Ly-α forest data, m WDM = 5.3 keV, very well. 5It means that we need only a mild entropy dilution factor, ∆ > 4.7, to evade the Ly-α forest constraints.
In the SUSY DFSZ model, such an entropy dilution factor is easily achieved by late-time saxion decay.Saxions are abundantly produced in the form of coherent 4 In this case, the axino phase space distribution is slightly different from the Fermi-Dirac one, since axinos are not thermalized [30]. 5We can infer the entropy dilution factor by comparing the second moments of the phase space distribution of 7 keV axino DM and of the Fermi-Dirac one with m WDM = 5.3 keV [30,33].In this case, the total axino density is dominated by the freeze-in contribution.Consequently, we find that the total mass density of 7 keV axinos also meets the observed DM one, i.e., Conclusions -We have examined 7 keV axino DM in the RPV SUSY DFSZ model by incorporating the 3.5 keV X-ray line excess and the Ly-α forest constraints.The model naturally introduces two key ingredients: 1) the µ-term interaction, which is responsible for freeze-in production of axinos and 2) the bRPV term, which is responsible for axino-neutrino mixings.While the 3.5 keV line excess is easily explained by 7 keV axino DM decay via an axino-neutrino mixing, the constraints from the Ly-α forest data impose a colder phase space distribution on axino DM.Freeze-in production of axinos via the µ-term interaction indeed leads to a colder phase space distribution.As a result, the axino phase space distribution meets the most stringent limit from the Ly-α forest data with the mild entropy production from late-time saxion decay, which is inherent in the model.We stress that, even with entropy production, the whole DM density is explained and dominated by the freeze-in axinos.
The result shown in this letter implies that X-ray observations determine the axino mass and its mixing parameter with active neutrinos, while Ly-α forest data and the observed DM mass density narrow down the saxion mass as well as the PQ breaking scale.Once the observational aspects of freeze-in axinos become evident, we can constrain and probe the underlying PQ breaking sector and its communication with the SUSY breaking sector.We also emphasize that our analysis of the resultant axino phase space distribution and linear matter power spectrum can be easily applied to other freeze-in

Total 2 -FIG. 2 .FIG. 3 .
FIG.2.Axino phase space distributions for the benchmark point.The red solid line shows the total axino phase space distribution normalized such that dqq 2 f (q) = 1.The blue solid line is sum of the contributions from Higgs 2-body decay and Higgsino s-channel scattering, and the yellow solid line is the contribution from Higgsino t-channel scattering (multiplied by 10 for visualization).The normalized Fermi-Dirac distribution is shown by the dashed line.

7 5FIG. 4 .
FIG. 4. Squared transfer functions with the entropy production from late-time saxion decay.Red crossed points show T 2 (k) for the benchmark points with ∆ = 4.7.Blue solid line shows T 2 (k) for mWDM = 5.3 keV corresponding to the most stringent lower bound from the Ly-α forest data.

× 10 2 7 s 0 10 16 2 . ( 9 )
where m s is the saxion mass, T R is the reheat temperature, s 0 is the saxion initial amplitude, and T s is determined by(3/R) (dR/dt)| T =Ts = m s .Such saxions dominates the energy density of the Universe at the temperature, GeV min[T R , T s ] 10 GeV GeV For ∆ = 4.7, it is required that saxion decay occurs at T = T s D 53 GeV, since the entropy dilution factor from saxion decay is determined by the temperature ratio: ∆ = T s e /T s D [34].The decay temperature of T s D 53 GeV is realized when saxion with m s = 110 GeV decays dominantly into b b, and v PQ = 2.5×10 10 GeV [35].

TABLE I .
MSSM parameters of the benchmark point with