Anomalous X-Ray Galactic signal from 7 keV spin- 3/2 dark matter decay
PPrepared for submission to JCAP
Anomalous X-Ray Galactic signalfrom . keV spin- / dark matterdecay Sukanta Dutta, a, † Ashok Goyal, b, (cid:63)
Sanjeev Kumar b, †† a SGTB Khalsa College, University of Delhi, Delhi, India. b Department of Physics & Astrophysics, University of Delhi, Delhi, India.E-mail: † [email protected], (cid:63) [email protected], †† [email protected] Abstract.
In order to explain the recently reported peak at . keV in the galactic x-ray spectrum, we propose a simple model. In this model, the Standard Model is extendedby including a neutral spin- / vector-like fermion that transforms like a singlet under SMgauge group. This . keV spin- / fermion is considered to comprise a portion of the observeddark matter. Its decay into a neutrino and a photon with decay life commensurate with theobserved data, fits the relic dark matter density and obeys the astrophysical constraints fromthe supernova cooling. a r X i v : . [ h e p - ph ] D ec ontents / Model 23 Galactic X-ray spectrum 34 Relic Abundance of Spin- /
45 Supernova Energy Loss 86 Results and Discussion 9
Recently X-Ray emission at ∼ . keV has been observed in the XMM- Newton X-Rayobservatory [1, 2] in many Galaxy clusters and in the Andromeda Galaxy spectra. Theobserved flux and any X-Ray line energy measured in the MOS spectra is given by Φ MOS γ = 4 . +0 . − . × − photons cm − sec − E MOS γ = 3 . ± .
02 keV (1.1)The source of this line is yet to be identified. An attractive possibility, considered in theliterature to explain the observed flux and energy, is to attribute it to the decay/ annihilationof some dark matter particle which is stable over cosmological time scale and can accountfor at least a significant fraction of dark matter relic density with mass and decay life timecommensurate with the observed data. Sterile Neutrino of mass . keV capable of produc-ing warm Dark Matter (WDM) density through resonant or non-resonant production withparameters required to produce the observed signal is an attractive proposition discussed inthe literature [3–7]. R-parity violating decays of the lightest super-symmetric particle (LSP),the decay of gravitons and axions, into neutrino photon pair and the decay of scaler field φ oraxion-like pseudoscalar fields a into photon pairs as possible explanation of the signals havebeen considered in the literature [7–16] with varying success. The scalar case is of particularinterest because the scale of the new physics may involve super-symmetry (SUSY) whichconforms to the expectations from the physics of moduli.Several new physics models beyond the standard model (SM) predict the existence ofspin- / particles. In models of super-gravity, the graviton is accompanied by spin- / grav-itino super partner. In models of composites [17], the top quark has an associated spin- / resonance. New physics models may include exotic fermions and gauge bosons which arenot present in the SM. Spin- / fermions also exist as Kaluza -Klien modes in string theory[18, 19] if one or more of compactification radii are of the scale lower than the Planck scale.The prediction of spin- / particle as a cold dark matter has been made by severalauthors in SUSY models [20, 21]. Gravitinos with mass in the keV range have been studiedas the probable WDM candidate in various SUSY models [22–25] even before the observation– 1 –f . keV X ray emission. Recently, authors of the reference [26, 27] have studied theimplication of the effective four fermion interactions involving the DM spin- / particle onrelic density, the antiproton to proton flux ratio in cosmic rays, and the elastic scatteringoff nuclei ( direct detection) in the effective field theory approach. Constraints from directdetection of dark matter exist in literature on spin- / WIMP candidates [28].A recent comprehensive analysis by the authors of reference [29] demonstrated thatthe measured flux of the . keV line can be accounted for, by the conventionally knownplasma lines without invoking the dark matter decay as its origin. This explanation, however,requires the fixing of the abundances of different elements which are still uncertain to a certaindegree. We, thus, feel that it is worthwhile to investigate alternative interpretations that areconsistent with the other astrophysical and cosmological data.In this paper, we consider a new neutral spin- / fermion assumed to be a vector-likeSM singlet. We will consider the decay of this . keV DM particles into a neutrino-photonpair ( χ → νγ ) with decay life commensurate with the observed galactic X-ray spectrum.This spin- / particle could exist as fundamental particle or could be a bound state of SMneutrino and U (1) gauge bosons. We will explore the possibility of such an exotic spin- / particle to constitute the relic dark matter for a reasonable choice of parameters and confrontthe model from cosmological and astrophysical constraints.In section 2, we describe the spin- / fermion model. In section 3, we discuss theimplication of the model to explain the observed galactic X Ray spectrum data. In section4, we obtain the relic abundance and the resulting constraints on the model parameters. Insection 5, we discuss the bounds obtained from supernova energy loss. Section 6 is devotedto results and discussion. / Model
The standard model is extended by including a spin- / , vector like particle χ , whose right-handed (RH) as well as left-handed (LH) projections transform the same way under SU (2) × SU (1) . We further let χ to be a SM singlet. Spin- / free Lagrangian is given by L = χ µ ( p χ ) Λ µν χ ν ( p χ ) whereΛ µν = ( i (cid:54) ∂ − m χ ) g µν − i ( γ µ γ ν + γ ν γ µ ) + iγ µ (cid:54) ∂γ ν + m χ γ µ γ ν . (2.1)Here χ µ satisfies Λ µν χ ν = 0 . For on mass-shell χ , we have γ µ χ µ ( p χ ) = 0 = ∂ µ χ µ ( p χ ) = ( (cid:54) p − m χ ) χ µ ( p χ ) . (2.2)The spin-sum for spin- / fermions S + µν ( p ) = / (cid:88) i = − / u iµ ( p ) u iν ( p ) and S − µν ( p ) = / (cid:88) i = − / v iµ ( p ) v iν ( p ) (2.3)are given by S ± µν ( p ) = − ( (cid:54) p ± m χ ) (cid:20) g µν − γ µ γ ν − m χ p µ p ν ∓ m χ ( γ µ p ν − γ ν p µ ) (cid:21) , (2.4)respectively. – 2 –he most general leading order standard model gauge invariant interaction betweenthe spin- / SM singlet χ and SM spin- / fermions is given by the effective dimension sixoperators: L Eff . Int . = (cid:88) i =1 C i Λ O i = C Λ l kL γ α [ γ µ , γ ν ] χ α ˜ φ B µν + C Λ l kL γ α D µ χ α D µ ˜ φ + C Λ l kL γ µ χ ν ˜ φ B µν , (2.5)where D µ ≡ i ∂ µ − i ( g s / λ a G aµ − i ( g/ τ I W Iµ − i g (cid:48) Y B µ , B µν = (cos θF µν − sin θZ µν ) , ˜ φ = iτ φ and l kL is the SM lepton doublet. The weak U (1) hyper-charge Y for φ and χ are / and , respectively.In view of the on-mass shell conditions as given in Eq. (2.2), the second operator O vanishes and the third operator O becomes identical to the first operator O . Therefore weare left with only one coupling constant C , which can be simplified to give after symmetrybreaking: L I = Cv Λ ν kL ( p ν e ) γ µ χ ρ ( p χ ) (cos θ F µρ − sin θ Z µρ ) . (2.6)Here v is the SM Higgs vacuum expectation value and Λ is the new cut-off scale. The decay width for χ → ν e γ is given by Γ χ → ν e γ = (cid:18) Cv cos θ Λ (cid:19) π m χ = 4 . × − (cid:20) C − (cid:21) (cid:20) Λ100 TeV (cid:21) − (cid:104) m χ (cid:105) sec − . (3.1)Here we have taken the coupling of spin- / particle with only one generation (say for thefirst generation only) of SM neutrino.The expected X-ray flux is proportional to the density of the decaying dark matter χ .The WDM which in the case considered here constitutes of spin- / SM singlet, is believed tocomprise a portion of the observed DM relic abundance with CDM as the dominant component[30, 31]. If the X-ray galactic signal is interpreted as coming from the spin- / WDM χ decaying into a neutrino and a photon pair, the required value of the life time of χ should begiven by τ χ ∼ . f × seconds, where f ( < f ≤ ) is the fraction of the relic dark matterdensity contributed by the WDM χ . At f = 1 . the WDM χ would account for the entire darkmatter relic density with a choice of new physics scale Λ of the order of (cid:39) TeV along withthe coupling constant C (cid:39) − . The small value of C should not be surprising as it can beconsidered to be a measure of trilinear lepton number violating coupling and hence naturallysmall. Similar situation occurs in super-symmetric models of R-parity violating interactionsconsidered in the literature [9–16] as possible explanation of the observed galactic X-ray flux.In realistic model, the mixing between photino and neutrino for example, is suppressed by asmall parameter ∼ − characterising lepton number violation [14, 32–34].– 3 –f, the . keV signal, on the other hand, is interpreted as coming from pair annihilationof . keV spin- / DM into two photons, the annihilation cross-section (cid:104) σ v (cid:105) ann . has tomatch with the best-fit decay-width of . keV DM i.e. (cid:104) σ v (cid:105) ann . ≈ χ → ν e γ n χ , (3.2)where n χ = ρ χ /m χ ≈ (cid:0) − (cid:1) cm − is the number density of spin- / DM. This trans-lates into (cid:104) σ v (cid:105) ann . fit (cid:39) × − GeV − .The spin- / particles can couple to two photons through U (1) gauge invariant dimensionseven effective Lagrangian L int . = C γγ Λ χ µ g µν χ ν F α β F α β . (3.3)This gives an annihilation cross-section σ ( χχ → γγ ) ≈ C γγ m χ / ( π Λ ) and the desired annihi-lation rate is achieved for Λ (cid:46) O (100) MeV (for C γγ (cid:46) ), which is clearly unphysical. Thus,it is unlikely that the observed galactic X-ray signal can be explained by DM χ ’s annihilationinto photons. / Since the χ ’s couple weakly to the SM particles and are nearly stable with a lifetime com-parable to the age of the Universe if they have to account for the observed X-ray flux, theywill decouple early when they are relativistic. They will, therefore, contribute to the presentmass density of the Universe as DM. Their abundance at decoupling is nearly equal to thephoton density at that time. During the adiabatic expansion of the Universe, their num-ber densities remain comparable. A rough estimate of the bound on χ mass can be ob-tained just like the bound on the neutrino mass [35] by requiring that the ratio of DM χ density to the critical density remain less than one. This gives m χ ≤ . g (cid:63) ( T D ) /g eff eV. The effective number of degrees of freedom g (cid:63) ( T D ) at decoupling time of electroweaksymmetry breaking transition is found to be about 113.75. In the computation of g (cid:63) ( T D ) ,we have included the effective degrees of freedom from all SM particles and χ , χ spin- / DM particles. However, in the MSSM, g (cid:63) ( T D ) is much larger ∼ . and thus it is notreasonable for the spin- / DM particle χ to have a mass of the order of about . keV. Figure 1 . Relevant diagrams for decay and pro-duction of the DM candidate χ . The relic abundance of dark matter χ de-pends on the sources of production of χ in the early Universe. The leading orderprocesses (shown in Figure 1) that main-tain the DM χ in equilibrium with the restof the SM plasma are the decay rate of Z → χν e + χν e and the → pair anni-hilation rates, namely, Γ( Z → χν e + χν e ) , Σ f i σ ( f i f i → χν e + χν e ) and σ ( W + W − → χν e + χν e ) where Σ f i means summation overall SM fermions (quarks and leptons). Us-ing the interaction Lagrangian given in Eq.(2.6), the decay and spin averaged annihi-lation cross-sections can be computed in a– 4 –traightforward manner. We obtain: Γ( Z → χν e + χν e ) ≈ C v sin θ Λ π m Z m χ m Z , (4.1) σ (cid:32)(cid:88) i f i f i → χν e + χν e (cid:33) ≈ παC v Λ π (cid:88) i (cid:113) − m i s (cid:18) sm χ (cid:19) s × (cid:34) cos θ Q f i s (cid:18) m i s (cid:19) + 1cos θ s − m Z ) + Γ m z (cid:26) ( g iV + g iA ) (cid:18) sm Z (cid:19) + 2 m i s (cid:32) ( g iV − g iA ) + (cid:40) (cid:18) sm Z (cid:19) − sm Z (cid:41) ( g iV + g iA ) (cid:33)(cid:41) + Q f i ( s − m Z ) s { ( s − m Z ) + Γ m Z }× (cid:26) g iv (cid:18)
38 + m Z s + 2 m Z m i s (cid:19) + 38 (cid:0) g iV + g iA (cid:1) m i s (cid:18) m i s − (cid:19)(cid:27)(cid:21) , (4.2)and σ ( W + W − → χν e + χν e ) ≈ παC v Λ π (cid:32) − m f s (cid:33) − sm χ (cid:18) sm W (cid:19) s (cid:20) m Z s s − m Z ) + Γ m Z (cid:18) − m W s (cid:19) (cid:18) m W s + 12 m W s (cid:19) + 1 s (cid:18) m W s − m W s − m W s (cid:19)(cid:21) , (4.3)where g iV and g iA are the vector and axial vector couplings of respective fermions in SM.If the decay and annihilation rates are much smaller than the Hubble expansion rate atthe temperature of the order of Elctro-Weak (EW) symmetry-breaking scale, the spin- / DMparticle χ will never be in thermal equilibrium. The Hubble expansion rate at a temperature T is given by H ( t ) ≈ . g ∗ T /m Pl where g (cid:63) is the effective number of relativistic degrees offreedom at the temperature T . The decay Z → χν e + χν e comes into play only below EWsymmetry breaking phase transition temperature T EW ∼
150 GeV . The Z bosons go out ofthe equilibrium below roughly 5 GeV, the other SM fermions remain in equilibrium muchbelow this temperature.The decay and annihilation rates can be estimated from Eqs. (4.1)-(4.3). The Z decayrate is given by Γ( Z → χν e + χν e ) ≈ . × C (cid:20) Λ1 GeV (cid:21) − GeV . (4.4)The leading terms in the cross-section corresponding to χ production through f f annihilationand W fusion processes are given by (cid:88) i σ ( f i f i → χν e + χν e ) ≈ . × C (cid:104) s (cid:105) (cid:20) Λ1 GeV (cid:21) − GeV − (4.5)– 5 –nd σ ( W + W − → χν e + χν e ) ≈ . × C (cid:104) s (cid:105) (cid:20) Λ1 GeV (cid:21) − GeV − , (4.6)respectively. One can obtain the constraint on the effective coupling C/ Λ by demanding thethermal average Γ( Z → χν e + χν e ) , (cid:68) σ (cid:16)(cid:80) f f f → χν e + χν e (cid:17) | nv (cid:69) and (cid:104) σ ( W + W − → χν e + χν e ) | nv (cid:105) to be less than the H ( T ) for T ∼ GeV ( i.e. at the EW phase transition temperature).Therefore, using g (cid:63) = 113 . , we obtain C (cid:20) Λ1 GeV (cid:21) − ≤ . × − from Z decay , (4.7) C (cid:20) Λ1 GeV (cid:21) − ≤ . × − from (cid:0) Σ f f f → χν e + χν e (cid:1) , (4.8)and C (cid:20) Λ1 GeV (cid:21) − ≤ . × − from W + W − → χν e + χν e . (4.9)The thermal averaged cross-sections (cid:104) σ | nv (cid:105) are estimated using the relation s = 4 (cid:104) E (cid:105) where (cid:104) E (cid:105) = 3 .
15 T and 2 . , and n = 34 ζ (3) π gT and ζ (3) π gT for Fermi-Dirac and Bose-Einsteinparticles, respectively.The relic density of the spin- / DM χ can be evaluated by solving the Boltzmannequation for the evolution of the number density n χ of DM χ and is given by ˙ n χ + 3 Hn χ = −(cid:104) Γ( Z → χν e ) (cid:105) ( n χ − n ) − Σ f (cid:42) σ (cid:88) f f f → χν e | v (cid:43) ( n χ n ν e − n f ) − (cid:10) σ ( W + W − → χν e ) | v (cid:11) ( n χ n ν e − n W ) . (4.10)Here, n i is the equilibrium number density of species i . The region of validity of the equationis when all the SM particles are in thermal equilibrium unlike the DM candidate χ whichis realized for T (cid:46) (15 − m i . We can than put n χ = 0 in the R.H.S. of this equation.Changing the variable from time to temperature, the equation can be put in the form: df χ dz = Γ( Z → χν e ) Km Z zf Z + (cid:88) i (cid:104) σ ( f i f i → χν e ) | v (cid:105) KZ ( f i ) , (4.11)where z = m Z /T , f χ = n χ /T , f i = n i /T , and K = 1 . √ g (cid:63) /m P l . We use Boltzmanndistribution functions for both the fermions and bosons, i.e. f i ( m i /T ) = f i (cid:18) m i m Z m Z T (cid:19) = f i ( x i z ) = g i π (cid:90) ∞ p e − √ p + x i z dp. (4.12)The thermal averaged decay rate and annihilation cross-sections can be expressed, followingRef. [36, 37], as (cid:104) Γ( Z → χν e ) (cid:105) = Γ( Z → χν e ) K ( z ) K ( z ) (4.13)– 6 –nd (cid:104) σ ( f i f i → χν e ) (cid:105) = 18 m i T K (cid:0) m i T (cid:1) (cid:90) ∞ m i σ ( f i f i → χν )( s − m i ) √ sK (cid:18) √ sT (cid:19) ds. (4.14)Here, K , ( x ) are the modified Bessel functions. In terms of scaled number density definedas N χ = f χ Km Z and by using the expressions for thermal averaged decay width and theannihilation cross-sections given in Eqs. (4.13) and (4.14), the Boltzmann equation can bewritten as dN χ dz = Γ( Z → χν e ) K ( z ) K ( z ) 1 m Z zf z ( z )+ (cid:88) i zx i m Z K ( x i z ) (cid:90) ∞ x i ( y − x i ) √ yK ( √ yz ) σ ( f i f i → χν e ) 1 z (cid:0) f i ( x i z ) (cid:1) dy. (4.15)We solve the above Boltzmann equation for the scaled number density N χ of spin- / darkmatter particle χ for . < z < corresponding to 5 GeV < T < 150 GeV.The contribution of the proposed . keV spin- / fermion χ to the relic dark matterdensity is obtained by numerically solving the Boltzmann Eq. (4.15) from the electroweakphase transition temperature to the freeze-out temperature of W (cid:48) s and Z (cid:48) s . The scalednumber density N χ ’s for the leading processes that maintain the dark matter χ in equilibriumwith the rest of SM plasma are obtained to be N χ (Γ( Z )) (cid:39) C × (cid:20) Λ1 GeV (cid:21) − ; N χ (cid:88) f σ ( f f ) (cid:39) C × (cid:20) Λ1 GeV (cid:21) − ;and N χ (cid:0) σ ( W + W − ) (cid:1) (cid:39) C × (cid:20) Λ1 GeV (cid:21) − ; (4.16)for the Z -decay, fermion-antifermion annihilation and W ± fusion processes, respectively.We find that the contribution of spin- / DM fermion to the relic density from W ± boson fusion process is about three order of magnitudes greater than the contribution fromthe rest of the processes. For our estimate of the dark matter density, we use the N χ ≈ C × [Λ / − . Thus, the number density of χ at the electroweak phase transitiontemperature is given by n χ / T (cid:12)(cid:12) T = T EW ≈ N χ / ( K M z ) ∼ / ( K M z ) . The number densityof χ ’s as the Universe cools to the present day is estimated to be n χ | T ∼ T × / ( ζ K m Z ) where T is the present day temperature ( T = 2 . K ) and ζ = g ∗ ( T EW ) / g ∗ ( T ) ∼ . .The present day dark matter relic density ρ χ | T ≈ C × − [Λ / − GeV is thenobtained by multiplying the number density n χ with its mass m χ . Since, the critical darkmatter density ρ χc ∼ . h − GeV , the Ω χ = ρ χ /ρ χc is computed as Ω χ h ≈ . × C (cid:20) Λ1 GeV (cid:21) − . (4.17)However, the desired value of Ω χ h ∼ Ω DM = 0 . will be obtained for C [Λ / − ≈ − . – 7 – Supernova Energy Loss
The . keV spin- / dark matter χ can be a source of significant energy loss in the supernovacore. The emission rates for SN 1987 A have been extensively studied for weakly interactingDM candidate particles like axions, gravitinos, right handed neutrinos, majorons, low massneutralinos, Goldstone bosons etc. in new physics models. Constraints have been put on theproperties and interactions of these particles [38–44]. The SN bound on neutrino magneticdipole moment have been one of the tightest [45]. In our estimate of the constraints on theparameters of our model, we would use the Raffelt criterion [46] that new source of coolingshould not exceed the emissivity ˙ (cid:15)/ρ = 10 ergs per gm per sec. The main source of χ pairproduction in the core of SN is through the χ ν e and/ or χ ν e production processes. Theemissivity i.e. the energy emitted per unit time and volume, is ˙ (cid:15) = (cid:90) (cid:89) i =1 d p i (2 π ) E i (2 π ) δ ( p e + + p e − − p ν e − p χ ) f f (1 − f ) (1 − f ) E χ | M | , (5.1)where | M | is the matrix element squared, summed over the initial and final states and f i ≡ [exp( E i − µ i ) /T + 1] − is the Fermi-Dirac distribution for the i th particle.In the supernova core immediately after the collapse, the temperature is high being ofthe order of tens of MeV. Even though the nucleons are nearly non-degenerate, the electronsare degenerate and the neutrinos are trapped. The core has a fixed value of the lepton number.Thus, there also exists a sub-dominant energy loss process via the neutrino pair annihilation νν → χν e + χν e . Since, the coupling of the dark matter particle χ to SM fermions is extremelyweak, the χ ’s once produced freely stream out of the SN core, their mean free path beinggreater than the core radius. We thus have µ χ ≈ . Carrying out the phase space integralsand making a change in variables from E , E , θ to E + = E + E , E − = E − E and s = 2 m e + 2 E E − (cid:126)p . (cid:126)p cos θ , we get ˙ (cid:15) = 12 π (cid:90) ∞ m e ds (cid:90) ∞√ s dE + (cid:90) √ E + s −√ E + s dE − s E + + E − f f σ ( e + e − → χν e ) . (5.2)In deriving the above expression, we have neglected the Pauli blocking terms for thefinal state particles χ and ν e which is an excellent approximation for ν e , χ and χ . We havesimilar expression for the process ν e ν e → χν e + χν e . The cross-section for these processes hasbeen evaluated in Eq. (4.2).The core density lies anywhere between × to × gm / cc . At a core temperatureof about
40 MeV , the electron chemical potential is µ e ≈
200 MeV and µ e − µ ν e ≈
50 MeV .In our estimate of the energy loss, we consider the core density to be × GeV with acore temperatures
30 (50) MeV and electron and neutrino chemical potentials 200 (150) MeVand 150 (100) MeV, respectively, and evaluate the energy loss integral numerically.Constraints from supernova cooling are obtained by numerical integration of the emissiv-ity expression (5.2) for the process e + e − → χν e + χν e at T = 30 MeV and electron chemicalpotential µ e = 200 MeV , we obtain ˙ (cid:15) ( e + e − → χν e + χν e ) ρ core = 2 . × C (cid:20) Λ1 GeV (cid:21) − ergs / gm / s (5.3)– 8 –here we have taken the core density ρ core to be about × gm / cc . Requirement of ˙ (cid:15)ρ core < ergs / gm / cc constrains C (cid:20) Λ1 GeV (cid:21) − ≤ . × − . (5.4)Core temperature of
50 MeV and µ e = 250 M eV results in a somewhat tighter constraint C [Λ / − ≤ − . The contribution from the process ν e ν e → χν e + χν e is totallynegligible being roughly 10 orders of magnitude smaller compared to the annihilation process. We summarize the constraints on the parameters of our . keV spin- / dark matter particle χ from cosmological and astrophysical observations. We observe that the ratio of the couplingand the square of the cut-off scale C/ Λ associated with spin- / particle χ of mass . keVcan be constrained as • C (cid:2) Λ1 GeV (cid:3) − (cid:46) − from the consideration of χ as a WDM candidate accounting forthe entire observed relic dark matter density Ω χ h = Ω DM = 0 . , • C (cid:2) Λ1 GeV (cid:3) − (cid:46) . × − from the the rapid cooling of the supernova through theemission of χ and, • C (cid:2) Λ1 GeV (cid:3) − ≈ . × − from the lifetime of χ through its decay χ → ν e γ .These combined constraints on the parameter space of coupling C and the cut-off scale Λ arising from its appropriate lifetime, contribution to relic density and supernova cooling areshown in figure 2. The curves marked f = 0 . and f = 1 correspond to the life time τ χ required for the observed X-ray flux for WDM χ contribution Ω χ h = 0 . × Ω DM and Ω χ h = Ω DM respectively. We find that the constraints from the cooling of supernova 1987Aand the DM relic density Ω χ h = 0 . enclose an allowed band (shaded with yellow lines inthe figure) in the parameter space spanned by C and Λ . The parameter space shaded in greenis forbidden.We thus see that a minimal extension of the SM by adding a spin- / SM singlet withmass . keV can account for the dark matter in the Universe, while at the same time ex-plaining the . keV X-ray line in the galactic X-ray spectrum through its decay χ → νγ . Recently, superconducting detectors are proposed for direct detection of light DM particles ofmass as low as keV through electron recoil from DM-electron scattering in superconductors[47]. It will be worthwhile to study the DM model discussed in this article to compute the DMscattering rates with electrons in a superconducting environment where electrons are highlydegenerate and the scattering is inhibited by the Pauli blocking and to explore the feasibilityof detecting the proposed DM particle. We leave this for the future work.– 9 – -10 -9 -8 -7 S N A f = . Ω D M = . f = Ω D M = . S N A f = . Ω D M = . f = Ω D M = . C Λ in GeV ForbiddenAllowed
Figure 2 . Combined constraints on the coupling C and cut-off scale Λ from contribution to therelic density as WDM, the rapid cooling of supernova SN 1987 A and decay of the spin- / particle.Allowed region (with yellow lines) of the parameter space is bounded by the maximum DM relic densityconstraint Ω DM = 0 . and from supernova cooling of SN 1987 A. The region in green is forbidden. Acknowledgments
We would like to thank the referee for drawing our attention to a recent analysis of the X-rayspectrum lines observed by XMM-Newton [29] and for his constructive suggestions to improvethe manuscript. SD would like to thank IUCAA, Pune for hospitality where part of this workwas completed. AG would like to acknowledge CSIR (ES) Award for partial support. SKacknowledges financial support from the DST project FTP/PS-123/2011.
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