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PHYSICAL REVIEW D 96, 031702(R) (2017) Sterile neutrino dark matter with supersymmetry
Bibhushan Shakya1,2 and James D. Wells2,3 1Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA 2Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 3Deutsches-Elektronen Synchrotron (DESY), Notkestraße 85, Hamburg D-22607, Germany (Received 5 January 2017; revised manuscript received 29 June 2017; published 18 August 2017) Sterile neutrino dark matter, a popular alternative to the WIMP paradigm, has generally been studied in non-supersymmetric setups. If the underlying theory is supersymmetric, we find that several interesting and novel dark matter features can arise. In particular, in scenarios of freeze-in production of sterile neutrino dark matter, its superpartner, the sterile sneutrino, can play a crucial role in early Universe cosmology as the dominant source of cold, warm, or hot dark matter, or of a subdominant relativistic population of sterile
neutrinos that can contribute to the effective number of relativistic degrees of freedom Neff during big bang nucleosynthesis.
DOI: 10.1103/PhysRevD.96.031702
I. MOTIVATION independent of dark matter considerations, there are several compelling reasons to expect the underlying theory of A sterile neutrino is a well-motivated and widely studied nature to be supersymmetric. The purpose of this paper dark matter (DM) candidate. In the neutrino minimal is to study a supersymmetric extension of the sterile Standard Model (νMSM) [1–3], its relic abundance is neutrino dark matter framework with properties (i) and produced through its mixing with the active neutrinos (ii) above, which are generic, model-independent features via the Dodelson-Widrow mechanism [4] for keV-scale of the freeze-in mechanism. In this framework, N1 is part of masses; however, this possibility has now been ruled out by a supermultiplet that also contains a scalar, the sterile a combination of x-ray and Lyman-alpha measurements ~ sneutrino N1. The aforementioned Z2 symmetry neces- [3,5–12]. The Shi-Fuller mechanism [13] employs resonant ~ sarily requires N1 to decay into N1; furthermore, as production, but requires fine-tuned parameters and faces we will see, this decay involves the “feeble" coupling constraints from structure formation [14,15]. Thermal ~ freeze-out with additional interactions followed by appro- from (ii) above; hence, N1 can potentially be long lived. priate entropy dilution can also result in the correct relic These features allow for interesting modifications of early abundance [16–19], but it is strongly constrained by big Universe cosmology and dark matter properties. bang nucleosynthesis [20]. An alternate production mechanism that is compatible II. FRAMEWORK with all constraints is the freeze-in mechanism [21,22], The sterile neutrino DM freeze-in framework requires where the relic abundance is built up through a feeble the following Lagrangian terms [23–39] (we only list terms coupling to some particle beyond the Standard Model that will be relevant for our study): (BSM) present in the early Universe. This possibility has L ⊃ þ ϕ ¯ c þ λð † Þϕ2 ð Þ been studied by several groups in several motivated frame- yijLihNj xi Ni Ni H H : 1 works [23–37] (see Ref. [38] for a recent review). While the details differ, all of these frameworks share two common In addition to three Standard Model (SM)-singlet, sterile salient features: neutrinos Ni (the heavier two are required to generate (i) A vanishing mixing between the sterile neutrino DM active neutrino masses via the seesaw mechanism), this candidate N1 and the active neutrinos, necessary to setup also features a neutral scalar ϕ. x, y are dimen- make N1 stable or very long lived and to alleviate sionless numbers. The aforementioned requirement of tension with observations. vanishing mixing for N1 translates to yi1 → 0,corre- (ii) A feeble coupling between N1 and a BSM particle sponding to a Z2 symmetry for N1. The second term present in the early Universe, which facilitates DM leads to freeze-in production of N1 via ϕ → N1N1 “ ” 2 production. decays if the coupling x1 is feeble, x1
2470-0010=2017=96(3)=031702(6) 031702-1 © 2017 American Physical Society RAPID COMMUNICATIONS
BIBHUSHAN SHAKYA and JAMES D. WELLS PHYSICAL REVIEW D 96, 031702(R) (2017) In a supersymmetric theory, each of the above fields is part of a supermultiplet; we denote the supermultiplets as Φ and N i, with their spin ð0; 1=2Þ components being (ϕ, ψ) ~ and (Ni, Ni), respectively. The Lagrangian terms in Eq. (1) can then be generated from the following superpotential: pffiffiffi W ⊃ yijLiHuN j þ xiΦN iN i þ λΦHuHd: ð2Þ
This superpotential further generates the following addi- tional terms (we only list the ones that will be relevant for our study): pffiffiffi pffiffiffi ~ ~ ~ ~ ~ L ⊃ xiψNiNi þ λϕHuHd þ λðψhdHu þ ψhuHdÞ: ð3Þ FIG. 1. Particle masses and relevant decays. Supersymmetric particles are shown in green to highlight how the non-super- symmetric sterile neutrino freeze-in framework gets extended. In addition, the following soft terms are also generated after Particles that make up dark matter are denoted by thick lines. SUSY breaking: pffiffiffi L ⊃ ~ ~ þ ϕ ~ ~ þ λ ϕ ð Þ III. FORMALISM soft yijAyijLihuNj xiAxi N1N1 Aλ huhd: 4 The goal of this paper is to highlight new qualitative Note, in particular, that the second term can give rise to the features arising in the supersymmetric framework. We ~ ~ focus on scenarios where ϕ is in equilibrium at high decay ϕ → N1N1. In keeping with previous work on freeze-in of sterile temperatures T>mϕ, and its decays during this period ~ neutrino dark matter [23–34,36,38], we take N1 to be light result in the freeze-in production of N1 and N1.No ~ (sub-GeV scale). N2, N3 are taken to be above the GeV significant production of N1 or N1 occurs after ϕ freezes scale to ensure they decay before BBN and remain out, as it decays rapidly into lighter SM or SUSY particles. ~ ≫ ϕ compatible with cosmological constraints. The heavier For cases where mN1
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STERILE NEUTRINO DARK MATTER WITH SUPERSYMMETRY PHYSICAL REVIEW D 96, 031702(R) (2017) ∼1–30 ’ Here, SN2 3 ( for GeV-scale N2, N3 [16,40,41]) Such energetic N1 s contribute to the effective number ; Δ accounts for entropy dilution from the late freeze-out of relativistic degrees of freedom Neff during big and out-of-equilibrium decay of the other two sterile bang nucleosynthesis (BBN) (which we take to be at ¼ 4 neutrinos N2, N3. TBBN MeV). This contribution can be estimated as ~ Since N1 decays produce Higgsinos, we must ensure that ~ ρ N1 decays before Higgsino freeze-out in order for N1 to Δ ¼ N1 ð Þ Neff ; 9 ρν form the dominant DM component. Using the radiation- T¼TBBN 2 dominated time-temperature relation HðTÞ¼T =M0 with 45 2 1=2 MPl which compares the sterile neutrino energy density with the M0 ¼ð 3 Þ , where g is the number of degrees of 4π g energy density of a neutrino species in equilibrium at the Δ freedom in the bath, the temperature of the SM bath when same temperature. Current bounds on Neff at BBN are at ~ 1=2 ≈ ðΓ ~ Þ ∼0 3 1σ N1 decays is approximately Tdecay N1 M0 , where the level of . at [47]. With the simplifying ~ ~ Γ ~ assumption that all of the N1 population decays at T N1 is the decay width of N1. In our calculations, we ensure decay and N1 is produced with typical energy m ~ =2 (m ~ =3)ina that Tdecay is higher than the Higgsino freeze-out temper- N1 N1 ∼ 20 two- (three-) body decay process, which gets redshifted by ature mH~ = . 1=3 1=3 Sterile neutrino dark matter can be cold, warm, or hot, as a factor S ðg =g Þ due to subsequent entropy N2;3 SM BBN characterized by its free-streaming length Λ , defined as Δ FS dilution, Neff can be approximated as (for the three-body the distance travelled by a dark matter particle from its decay case) production at time tp to the present time t0: −8 10 ~ Z 2 mN1 GeV t0 h ð Þi ΔN ≈ Ωh v t eff 1=3 1=3 Λ ¼ dt: ð8Þ S ðg =g Þ Tdecay mN1 FS N2;3 SM BBN t aðtÞ p 2 Ωh m ~ MeV 10 1=3 ≈ 0.2 10−8 N1 : Here vðtÞ and aðtÞ are the DM velocity and the scale factor, 0 0012 . Tdecay mN1 SN2;3 respectively, at a given time t. As a rough guide, we take Λ ≲ 0 01 0 01 ≲ Λ ≲ 0 1 0 1 ≲ ð10Þ FS . Mpc, . FS . Mpc, and . Mpc Λ as corresponding to cold, warm, and hot dark matter, FS Here, Ωh2 represents the present relic abundance that respectively [39]. We note that there are several subtleties ~ related to using the free-streaming length as a measure of originated from N1 decay, as this is the only component disruption to structure formation for nonthermal distribu- that is relativistic at BBN. Δ tions [42]; however, we adhere to this simplistic approach While there are stronger constraints on Neff from the for the purposes of our paper, since we are only interested later era of cosmic microwave background (CMB) decou- in an approximate, qualitative understanding of the pos- pling, the N1 particles generally redshift and become sibilities of cold, warm, and hot dark matter. nonrelativistic by this time [39], resulting in weaker ~ Δ ~ ≫ constraints, hence we only focus on Neff during BBN. If mN1 mN1 and N1 decays extremely late, the population of N1 produced from such decays can be However, we do note that light- (sub-eV) mass sterile relativistic and act as dark radiation (see also Ref. [43] neutrinos produced in this manner could contribute to Δ for a similar setup). It is well known that a species that Neff at CMB decoupling, and might be relevant for forms all of dark matter cannot account for any measurable alleviating the recent tension between the local and dark radiation in the Universe [39,44,45]. However, this CMB-inferred measurements of the Hubble rate [48]. constraint can be circumvented in our framework, since the ~ hot N1 population produced from N1 decays does not mix IV. RESULTS ϕ with the cold N1 population from decays. The latter In this section, we investigate modifications to dark population can thus be the dominant dark matter compo- matter properties in the supersymmetric framework. ~ nent, while a subdominant, hot population from N1 decays forms dark radiation; we conservatively take this fraction to A. Abundance and composition be ≲1% (as in Ref. [46]), which should leave structure ~ The N1 population acts as multicomponent dark matter, formation unaffected. We note that heavy, long-lived N1 ϕ ~ can grow to dominate the energy density of the Universe, as the fractions produced from and N1 decays do not interact with each other. The two abundances differ by a introducing an intermediate phase of matter domination, 2 subsequently releasing entropy that reheats the thermal bath factor of ðAϕ=mϕÞ [see Eqs. (6) and (7)]. Since we expect ∼ ∼ and dilutes the dark matter abundance. This indeed occurs Aϕ mϕ mSUSY, the two abundances are generally of in parts of our parameter space, and we correct for these comparable magnitude. For given values of mϕ and mN1 , effects appropriately. the desired relic abundance can be obtained by selecting
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BIBHUSHAN SHAKYA and JAMES D. WELLS PHYSICAL REVIEW D 96, 031702(R) (2017)
FIG. 2. Parameter space with cold, warm, and hot dark matter FIG. 3. Cold, warm, and hot dark matter (black, blue, and red ¼ 1 ¼ 106 (black, blue, and red regions, respectively). For all points in the regions respectively) for mN1 MeV and m ~ GeV. 2 11 N1 plot, Ωh ¼ 0.12, mϕ ¼ 10 GeV, Aϕ=mϕ ¼ 10, S ¼ 10. ¼ 10 N2;3 We set SN2;3 .
~ ~ appropriate values of x and Aϕ as long as Eq. (5) remains through the three-body channel N1 → N1Hh with a long satisfied. Due to the presence of an additional dark matter lifetime. As discussed in the previous section, this N1 ~ production mechanism in N1 decays, the supersymmetric population can only comprise a subdominant component of framework opens up more parameter space where sterile dark matter, and we fix its abundance to 1% of the total DM neutrino dark matter can be realized. abundance by choosing Aϕ ¼ 0.1mϕ. Δ In Fig. 4, we plot Neff at BBN as a function of N1 and ~ B. Free-streaming length N1 masses from a scan over parameter space, where we ¼ 1–30 ~ scanned over SN2;3 . Red, green, blue, and black N1 decays can produce dark matter that is cold, warm, or Δ 0 5 – – hot. This is illustrated in Fig. 2, where we delineate points represent Neff in the ranges > . , 0.1 0.5, 0.01 0.1, and < 0.01, respectively; we see that large contribu- combinations of sterile neutrino and sterile sneutrino Δ masses that give rise to cold, warm, or hot dark matter tions to Neff comparable to current bounds are possible while satisfying all the enforced constraints. The largest (regions where the full dark matter relic density can be 9 12 values correspond to m ∼ MeV and m ~ ∼10 –10 GeV: achieved extend beyond the boundaries of this plot). In N1 N1 ~ ~ ~ this plot, m ~ >mψ , so that N1 decays as N1 → ψN1; for lighter N1 or heavier N1, the DM particles are not N1 ~ 11 ~ ~ sufficiently relativistic at BBN, whereas heavier N1 (which mϕ ¼ 10 GeV, so that ϕ → N1N1 is allowed at all points; ~ forces ϕ to be heavier) or lighter N1 both require larger x to Aϕ ¼ 10mϕ, so that N1 decays account for essentially all of dark matter; and x is chosen to produce the desired relic 2 ~ density Ωh ¼ 0.12. As expected, heavier N1 or lighter N1 cause dark matter particles to become more energetic, resulting in larger free streaming lengths. Note, however, that the demarcation of cold, warm, and hot regions ~ depends not only on mN1 and mN1 but also on other ~ parameters (in particular, the ones that determine the N1 lifetime). This point is illustrated in Fig. 3, where we show that all three possibilities can be realized for fixed choices 6 ~ ¼ 10 ¼ 1 of mN1 GeV and mN1 MeV by varying mϕ and Aϕ: from Eq. (7), increasing Aϕ=mϕ or decreasing mϕ requires decreasing x to maintain the correct relic density; a ~ smaller x, in turn, leads to a longer lifetime for N1, making the decay product hotter, as seen in the figure. ~ FIG. 4. ΔNeff (BBN) for different N1 and N1 masses. Red, Δ 0 5 C. Dark radiation green, blue, and black points denote Neff in the ranges > . , 0.1–0.5,0.01–0.1, and < 0.01, respectively. For all points, the Next, we consider scenarios where extremely energetic ~ ΔN contribution comes from N1 decays, which account for 1% ~ Δ eff N1 from late N1 decays contribute significantly to Neff of the dark matter abundance, while ϕ decays produce the rest of ~ ~ during BBN. Here we choose mN1 031702-4 RAPID COMMUNICATIONS STERILE NEUTRINO DARK MATTER WITH SUPERSYMMETRY PHYSICAL REVIEW D 96, 031702(R) (2017) maintain the correct dark matter abundance [see Eq. (6)], appealing given the recent claims of an x-ray line from ~ which reduces the N1 lifetime. galaxy clusters at 3.5 keV [52,53] compatible with decays of a 7 keV sterile neutrino; this direction would warrant V. DISCUSSION further study should the signal persist. We emphasize that the results in this paper continue to hold even in the absence In this paper, we have demonstrated that a supersym- of this Z2 symmetry, provided the Yukawa coupling y 1 metric extension of the widely studied sterile neutrino dark i remains much smaller than the feeble coupling x1 (so that matter framework with the basic features of dark matter none of the dominant production/decay modes change); freeze-in, namely an underlying symmetry that stabilizes this turns out to be the case for any phenomenologically the dark matter candidate and a feeble coupling that viable model of sub-GeV sterile neutrino dark matter, facilitates dark matter production, can introduce several including the candidate that explains the 3.5 keV x-ray line. qualitatively new cosmological features and dark matter The scalar ϕ mixes with the Higgs boson, and thus can properties that are not possible in the non-supersymmetric be produced via “Higgs portal” interactions or cause scenario. The presence of the superpartner, the sterile ~ deviations in Higgs couplings measurements at the LHC sneutrino N1, offers an additional production mechanism or at future colliders. The LSP, required to be lighter than for dark matter. This not only extends the allowed param- what is traditionally needed for a thermal abundance, can eter space for sterile neutrino dark matter, but it also enables also be probed through traditional LSP search channels: the scenario of multiple-component dark matter with a note that we took it to be a Higgsino, which is difficult to single constituent N1, as the fractions produced via differ- probe at colliders, but e.g. a sub-TeV wino would be within ϕ ~ ent processes ( and N1 decays) do not mix, effectively reach of the high-luminosity LHC. acting as different components. This possibility is unique to Given the phenomenological nature of this letter, we did freeze-in production, as the two fractions would thermalize not address several interesting model-building aspects. The in the standard thermal freeze-out histories if such pro- phenomenologically most interesting regions of parameter ~ ~ duction occurred before freeze-out. N1 decays can be the space require a large hierarchy between N1 and N1 masses; dominant source of dark matter production, and dark matter these could, for instance, emerge naturally from symmetry produced via its decay can be cold, warm, or hot. For considerations in the supersymmetric neutrino sector, ~ extremely long-lived N1 producing ∼1% of dark matter, which could also explain the feeble nature of the coupling, Oð0 1Þ Δ . contributions to Neff during BBN are possible, see e.g. Refs. [31,32]. For further model-building aspects which can be probed by near-future measurements (see related to sterile neutrino dark matter, the interested reader also Ref. [49]). is referred to Refs. [38,54,55]. Mixed (cold þ warm) dark matter scenarios are attractive Given the tremendous appeal of supersymmetry as part as potential solutions to issues such as the core vs cusp of the underlying theory of nature, the cosmological aspects problem and the “too big to fail" problem [50,51]. While discussed in this paper are relevant for any study on sterile such scenarios generally involve complicated frameworks neutrino dark matter. Moreover, in the absence of clear where multiple dark matter components must be motivated, observational signals of weak scale supersymmetry or and their comparable relic densities must be further WIMP dark matter, such lines of inquiry might provide explained, these issues are trivially resolved in our setup. hints on the nature and scale of supersymmetry and reveal Likewise, any general freeze-in setup with (i) an additional promising future avenues of research. production mechanism for dark matter, (ii) comparable relic abundance as the primary mechanism, and (iii) a ACKNOWLEDGMENTS longer decay lifetime to produce a distinct momentum distribution can reproduce the phenomenology discussed in We acknowledge helpful discussions with Samuel this manuscript; while these can be realized in generic Roland. The authors are supported in part by the frameworks with some model building, supersymmetry Department of Energy under Grants No. DE-SC0007859 provides a simple setting where these features are auto- and No. DE-SC0011719. B. S. also acknowledges support matically realized in a straightforward manner via the from the University of Cincinnati. J. W. wishes to acknowl- supersymmetric counterparts of existing fields and inter- edge support from the Humboldt Foundation. This work actions with fixed related properties. was performed in part at the Aspen Center for Physics, The Z2 symmetry that protects N1 need not be exact, in which is supported by National Science Foundation Grant which case N1 can decay; this prospect is especially No. PHY-1066293. 031702-5 RAPID COMMUNICATIONS BIBHUSHAN SHAKYA and JAMES D. WELLS PHYSICAL REVIEW D 96, 031702(R) (2017) [1] T. Asaka, S. Blanchet, and M. Shaposhnikov, Phys. Lett. B [28] A. Merle, V. Niro, and D. Schmidt, J. Cosmol. 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