Role of Sterile Neutrino Warm Dark Matter in Rhenium and Tritium Beta Decays
H. J. de Vega, O. Moreno, E. Moya de Guerra, M. Ramon Medrano, N. Sanchez
aa r X i v : . [ h e p - ph ] S e p Role of Sterile Neutrino Warm Dark Matter in Rhenium and Tritium Beta Decays
H.J. de Vega
LPTHE Universit´e Pierre et Marie Curie (Paris VI),Laboratoire Associ´e au CNRS UMR 7589, Tour 24, 5eme. ´etage,Boite 126, Place Jussieu, 75252 Paris, cedex 05, France andObservatoire de Paris, LERMA. Laboratoire Associ´e au CNRS UMR 8112.61, Avenue de l’Observatoire, 75014 Paris, France.
O. Moreno and E. Moya de Guerra
Departamento de F´ısica At´omica, Molecular y Nuclear,Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain
M. Ram´on Medrano
Departamento de F´ısica Te´orica I, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain
N. G. S´anchez
Observatoire de Paris, LERMA. Laboratoire Associ´e au CNRS UMR 8112.61, Avenue de l’Observatoire, 75014 Paris, France. (Dated: August 21, 2018)Sterile neutrinos with mass in the range of one to a few keV are important as extensions of theStandard Model of particle physics and are serious dark matter (DM) candidates. This DM massscale (warm DM) is in agreement with both cosmological and galactic observations. We study therole of a keV sterile neutrino through its mixing with a light active neutrino in Rhenium 187 andTritium beta decays. We pinpoint the energy spectrum of the beta particle, 0 . T e . ( Q β − m s ),as the region where a sterile neutrino could be detected and where its mass m s could be measured.This energy region is at least 1 keV away from the region suitable to measure the mass of the lightactive neutrino, located near the endpoint Q β . The emission of a keV sterile neutrino in a betadecay could show up as a small kink in the spectrum of the emitted beta particle. With this inview, we perform a careful calculation of the Rhenium and Tritium beta spectra and estimate thesize of this perturbation by means of the dimensionless ratio R of the sterile neutrino to the activeneutrino contributions. We comment on the possibility of searching for sterile neutrino signaturesin two experiments which are currently running at present, MARE and KATRIN, focused on theRhenium 187 and Tritium beta decays respectively. PACS numbers: 23.40.-s, 14.60.St, 14.60.Pq, 95.35.+dKeywords: keV sterile neutrinos, beta decay, warm dark matter
I. INTRODUCTION
It is well known that dark matter (DM) is not described by the Standard Model (SM) of particle physics. Manyextensions can be envisaged to include DM particles, coupled weakly enough to the SM particles to fulfill all particleexperimental constraints, namely the fact that DM has not been detected so far in any particle physics experiment.On the other hand, cosmological and astrophysical constraints such as the ones coming from the dark matter densityand the galaxy phase space density, or alternatively, the universal galaxy surface density, lead to DM candidates inthe keV mass scale, namely warm DM (WDM), refs. [1–7]. A keV mass scale sterile neutrino is the front runningcandidate for WDM. Other possible WDM candidates in the keV mass scale are gravitinos, light neutralinos andmajorons [1, 8].Considering the first WDM candidate, sterile neutrinos can be naturally embedded in the SM of particle physics.They do not participate in weak interactions, and hence they are singlets of color, weak SU(2) and weak hypercharge.One sterile neutrino per lepton family could be expected, of which the lightest one (i.e. electron family) would havea lifetime of the order of the Hubble time and could be considered a DM candidate.In this work, we consider the role played by a 1-2 keV sterile neutrino in Rhenium 187 and Tritium beta decayexperiments. The left-handed neutrino flavor state ν e (and equivalently for ¯ ν e ) will be a mixing of two mass eigenstates:one light active neutrino mass state ( ν l ) and one keV scale sterile neutrino mass state ( ν s ). Other neutrino massstates will not be taken into account for the time being. The mass m l of the lightest active neutrino state is negligible( m l ≪ eV ) in comparison with the mass m s of the keV sterile mass state. The smallness of the mixing angle ζ makessterile neutrinos difficult to detect.Sterile neutrinos in the beta decay of Rhenium 187 are currently searched for by the Microcalorimeter Arrays fora Rhenium Experiment (MARE) [9]. In this decay the available energy is Q β ( Re) ≃ .
469 keV. The beta decay ofRhenium 187 into Osmium 187 is a first forbidden unique Gamow-Teller process (5 / + → / − ).Up to now, the non observation of keV scale sterile neutrinos in the beta decay of Rhenium 187 gave an upper boundon the mixing angle ζ < .
095 for 1 keV steriles [10], which is compatible with the cosmological constraints on themixing angle, ζ < − , appropriate to produce enough sterile neutrinos to account for the observed DM. However,the amount of the sterile neutrinos that could be produced in the early universe also depends on the productionmechanism, which is model dependent. We refer for that to the original references [3].The Karlsruhe Tritium Neutrino Experiment (KATRIN) is currently studying the Tritium beta decay [13] and,if suitably adapted, it could study the presence of a sterile neutrino as well. In this decay the available energy is Q β ( H ) ≃ . / + → / + ) with Fermiand Gamow-Teller contributions. Clearly, KATRIN has in principle the potential to detect sterile neutrinos withmass up to 18 keV. However, the main difficulty in detecting WDM sterile neutrinos comes from the smallness of themixing angle between the active and sterile neutrino ζ < − . Such range of values for ζ are too small for the presentexperimental sensitivities [13, 14] and would require a source with a large stability to reduce the systematic errors.Detection of massive neutrinos by β -decay has been proposed in Ref. [15]. Other methods proposed to detect sterileneutrinos include measurements of the nuclear recoil [16, 17] and sterile neutrino capture on β -decaying nuclei [18].In 1985 evidence for the emission of a 17 keV mass neutrino in Tritium beta decay was reported by J. J. Simpson[19]. The evidence was hotly debated, new experiments gave clear negative results and by 1993 the general conclusionwas reached that there are no 17 keV neutrinos [20, 21]. The experiments in the nineties using Ni, S and othernuclei yielded an upper bound ζ < .
03 [20]. This bound is not restrictive for DM because the cosmological constraintsbased on the observed average DM density indicate for the currently popular models of DM sterile neutrinos a muchlower bound, ζ < − [6, 7].Sections II and III deal with keV dark matter from the cosmological and galactic point of view. WDM (DM particlemass between 1 keV and 10 keV, and decoupling temperature T d ∼
100 GeV) produces the observed small (galactic)structures, as well as the large scale and cosmological structures, the observed cored density profiles and the rightsurface density value, while GeV WIMPS ( m ∼
100 GeV, and T d ∼ s - and p -waves [11, 12]. The electron kinetic energy range T e suitable for the detection of sterile neutrinos lies between 0 and ( Q β − m s ), where m s is the mass of the keV sterileneutrino. On the contrary, the electron kinetic energy region close to the endpoint energy Q β is the one suitable forthe detection of light active neutrinos. Systematic uncertainties such as Beta Environmental Fine Structure (BEFS)are not considered here [9]. In order to analyze the sterile neutrino effect, we introduce the dimensionless ratio R ofthe sterile neutrino contribution to the active neutrino contribution. It allows us to compare two regions of the samespectrum: the one where the keV neutrino imprints a kink on the spectrum, and the one near the endpoint where theactive light neutrino effect shows up. The Kurie function is also analyzed and expressed in terms of the ratio R .In Section V we study the role of sterile neutrinos in Tritium decay [13], where the emitted electrons are purely s -wave. Analogously to the Rhenium beta decay, the kinetic energy region relevant for the sterile neutrino detectionis the low energy range 0 ≤ T e ≤ ( Q β − m s ), while for the active neutrino it is the one close to the endpoint energy Q β .Finally, in Section VI we present our conclusions. Natural units ~ = c = 1 are used all over this paper. II. DARK MATTER
Although dark matter was noticed seventy-five years ago [22, 23], its nature is not yet known. Dark matter (DM)is needed to explain the observed structures in the Universe, in particular galaxies. DM particles must have beennon-relativistic by the time of structure formation in order to reproduce the observed small structure at ∼ − l fs [24]. This is the distance that the DM particles can freely travel. Structures at scalessmaller than l fs are erased by free-streaming and hence l fs provides a lower bound on the size of DM dominatedstructures. WDM particles with mass in the keV scale give l fs ∼
100 kpc while 100 GeV cold dark matter (CDM)particles produce an extremely small l fs ∼ . l fs ∼
100 kpc is in nice agreement with the astronomicalobservations of galaxies [25] (smaller objects like stars are made up of baryons, not of DM), as well as at cosmologicalscales.The GeV CDM free-streaming length l fs is a million times smaller and would lead to the existence of a host of CDMsmaller scale structures till the size of the solar system. No structure of such type has ever been observed. LighterDM particles in the eV scale (hot dark matter, HDM) have a free-streaming length l fs ∼ Mpc and hence would eraseall existing structures below the Mpc scale in contradiction with all observations. This is why HDM has been ruledout [26].The reason why CDM does not work is simple: CDM particles in the GeV scale are too slow (too cold), whichprevents them to erase the small scale structure, while the eV particles (HDM) are excessively fast, which erases allstructures. In between, WDM keV particles are able to produce the observed structures.Astronomical observations strongly indicate that dark matter halos have cored profiles till scales below 1 kpc. Onthe contrary, CDM simulations (particles heavier than 1 GeV) always give cusped profiles. No cusped profiles havebeen ever observed. Linear profiles computed from the Boltzmann-Vlasov equation turn out to be cored for WDMand cusped for CDM indicating that WDM does reproduce the astronomical observations [5].The surface density in DM-dominated galaxies is defined by µ ≡ ρ r where ρ is the central core density and r is the core radius. µ turns out to be universal, taking the same value up to ±
10% for galaxies of different sizes,morphologies, Hubble types and luminosities [27]. The surface density value predicted by CDM simulations is 1000times larger than the observed value [28], while the surface density for keV WDM computed from the Boltzmann-Vlasov equation is in full agreement with the observed value of 120 (MeV) , indicating again that WDM does reproducethe astronomical observations [5].Constraints of the DM particle mass to the keV range are obtained from combining theoretical analyses withthe observed values of dark matter densities and phase space densities today (density over the cube of the velocitydispersion) of dwarf spheroidal galaxies.Recent radioastronomy observations of velocity widths in galaxies from 21cm HI surveys clearly favours WDMover CDM [29]. WDM simulations contrasted to astronomical observations suggest a WDM particle mass slightlyabove 1 keV. Constraints from large scale structure give this value too [30]. Recent cosmological WDM N-bodysimulations with keV sterile neutrino WDM clearly show the agreement of the predicted small scale structures withthe observations, while CDM simulations do not agree with observations at such scales [31].None of the predictions of CDM simulations at small scales (cusps, substructures, dark disks, ...) have beenobserved. Here are some examples. The CDM satellite problem, namely that CDM simulations predict too manysatellites in the Milky Way and only 1/3 of satellites predicted by CDM simulations around our galaxy are observed.The surface density problem, which consists of the galaxy surface density for CDM simulations being 1000 larger thanobserved [5, 28]. And the voids problem and the size problem, that have to do with the fact that CDM simulationsdo not produce big enough galaxies [32–34]. Further WDM properties are discussed in [35].Notice that all DM observable effects discussed above only arise from the gravitational behaviour of the DM. Galaxyproperties are independent of the non-gravitational couplings of the DM particles, provided that their couplings aresmall enough.DM may decouple at or out of thermal equilibrium. The distribution function freezes out at decoupling. Whetherthey decouple at or out of equilibrium depends on the non-gravitational couplings of the DM particle. Normally,sterile neutrinos are so weakly coupled that they decouple out of thermal equilibrium. The functional form of the DMdistribution function depends on the DM particle couplings and is therefore model dependent.Sterile neutrinos can decay into an active-like neutrino and a monochromatic X-ray photon with an energy half themass of the sterile neutrino. Observing the X-ray photon provides a way to observe sterile neutrinos in DM halos[36, 37].WDM keV sterile neutrinos can be copiously produced in the supernovae cores. Supernovae (SN) stringentlyconstrain the neutrino mixing angle squared to be . − for m >
100 keV, in order to avoid excessive energy lost.However, for smaller masses the SN bound is not so direct. Within the models worked out till now, mixing angles areessentially unconstrained by SN in the keV mass range [38].Sterile neutrinos are produced out of thermal equilibrium and their production can be non-resonant, in the absenceof lepton asymmetries, or resonantly enhanced, if lepton asymmetries are present. keV sterile neutrino WDM inminimal extensions of the Standard Model is consistent with Lyman-alpha constraints within a wide range of themodel parameters. Lyman-alpha observations give a lower bound for the sterile neutrino mass of 4 keV only for sterileneutrinos produced in the case of a non-resonant (Dodelson-Widrow) mechanism [39, 40]. The Lyman-alpha lowerbounds for the WDM particle mass are smaller in the Neutrino Minimal Standard Model, where sterile neutrinos areproduced by the decay of a heavy neutral scalar, and for fermions in thermal equilibrium. Moreover, the number ofobserved Milky-Way satellites indicates lower bounds between 2 and 13 keV for different models of sterile neutrinos.In summary, contrary to CDM, WDM essentially works , reproducing in a natural way the astronomical observationsof structures over all scales, small as well as large and cosmological scales. The sterile neutrino with mass in the keVscale appears as a serious candidate for WDM. Galaxy observations alone cannot determine the DM particle propertiesother than the mass and the decoupling temperature. A direct particle detection is necessary to pinpoint and determinewhich particle candidate describes DM. Beta decay is a promising way to detect DM sterile neutrinos.
III. DARK MATTER AND KEV STERILE NEUTRINOS
As it is known, DM is not described by the Standard Model (SM) of particle physics. However, many extensionsof the SM can be envisaged to include a DM particle with mass in the keV scale and coupled weakly enough to theStandard Model particles so as to fulfill all particle physics experimental constraints, coming mainly from the factthat DM has not been detected so far in any particle physics experiment. Besides sterile neutrinos, possible DMcandidates in the keV mass scale are gravitinos, light neutralinos, majorons, etc. [8].As particle physics motivations for sterile neutrinos one can advance that there are both left- and right- handedquarks (with respect to chirality) while active neutrinos are only left-handed. It is thus natural to have right-handedneutrinos ν R besides the known left-handed active neutrinos. This argument is called ‘quark-lepton similarity’.Sterile neutrinos can be naturally embedded in the SM of particle physics with the symmetry group SU (3) color ⊗ SU (2) weak ⊗ U (1) weak hypercharge . Leptons are singlets under color SU(3) and doublets under weak SU(2) in the SM.Sterile neutrinos ν R do not participate in weak interactions. Hence, they must be singlets of color SU(3), weak SU(2)and weak hypercharge U(1).Let us consider a simple embedding of the sterile neutrino in the Standard Model. More elaborated sterile neutrinomodels have been put forward [43]. The SM Higgs Φ is a SU(2) doublet with a nonzero vacuum expectation valueΦ . This allows a Yukawa-type coupling with the left- and right-handed leptons: L Y uk = y ¯ ν L ν R Φ + h.c. , (1)where y is the Yukawa coupling, and Φ = (cid:18) v (cid:19) , v = 174 GeV . (2)These terms in the Lagrangian induce a mixing (bilinear) term between ν L and ν R allowing for transmutations ν L ⇔ ν R . Mixing and oscillations of particle states are typical of low energy particle physics. Further well knownexamples are: (i) flavor mixing: e- µ neutrino oscillations, which explain solar neutrinos, (ii) K − K , B − B and D − D meson oscillations in connection with CP-violation.As a consequence of the Lagrangian in Eq. (1), the neutrino mass matrix takes the form(¯ ν L ¯ ν R ) (cid:18) m D m D M (cid:19) (cid:18) ν L ν R (cid:19) (3)where M is the mass term of the right-handed neutrino ν R , and m D = y v with M ≫ m D .The masses of the active and sterile neutrinos are given by the seesaw mechanism. The mass eigenvalues in thissimple model take the form: m D /M (active neutrino) and M (sterile neutrino), with eigenvectors ν active ≃ ν L − m D M ν R (active neutrino) and ν sterile ≃ ν R + m D M ν L , M ≫ m D /M (sterile neutrino). Choosing M ∼ m D ∼ . m D /M about 10 − eV, consistent with observations. This corresponds to a mixing angle ζ ∼ m D /M about10 − and would be appropriate to produce enough sterile neutrinos to account for the observed DM. However, noticethat the amount of the sterile neutrinos produced in the early universe also depends on the production mechanism,which is model dependent. The smallness of the mixing angle ζ makes sterile neutrinos difficult to detect.One sterile neutrino per lepton family could be expected, of which the lightest one (i.e. electron family) would havea lifetime of the order of the Hubble time and could be considered a DM candidate. In summary, the empty slot ofright-handed neutrinos in the Standard Model of particle physics could be filled in a fully consistent way by keV-scalesterile neutrinos describing the DM. IV. RHENIUM 187 BETA DECAY AND STERILE NEUTRINO MASS
As a probe to detect possible mixing of keV sterile neutrinos with light active neutrinos, we consider in this sectionthe beta decay of Rhenium 187 (
Re; Z = 75 , A = 187) into Osmium 187 ( Os; Z = 76 , A = 187), Re → Os + e − + ¯ ν e (4)The neutrino flavor eigenstate ν e (and equivalently for ¯ ν e ) can be written as a combination of light active (subscript i ) and heavy sterile mass eigenstates as [6, 15] | ν e i = X i U ei | ν i i + X s U es | ν s i (5)where the quantities U belong to the unitary leptonic mixing matrix. For the purpose of this paper, we approximatethis combination as a mixing of two mass eigenstates given by [6] | ν e i = cos ζ | ν l i + sin ζ | ν s i (6)where ζ is the mixing angle between a light neutrino mass state ν l , and the heavy sterile neutrino mass state ν s .Other neutrino mass states will not be taken into account in this work. An effective mass m l can be used for theformer combination of light mass active neutrinos, but its value ( m l . eV) is negligible in comparison with the sterileneutrino mass in the keV scale. As for the mixing angle ζ , the cosmological constraints based on the observed averageDM density suggest [6, 7] sin ζ ∼ − , ζ ∼ . o . (7)We should keep in mind that these constraints on the value of ζ depend both on the sterile neutrino model and onthe sterile neutrino production mechanism. Eq. (7) corresponds to currently popular models of DM sterile neutrino[6, 7]. Re is a long half-life isotope ( t / ≃ . · years), with ground state spin-parity assignment J π = 5 / + ,that has a single β − -decay branch mode to the ground state 1 / − of Os with an endpoint energy Q β ≃ .
469 keV( Q β = T e + m ν + T ν , where T e and T ν are kinetic energies of the electron and the neutrino respectively).In this transition, the change of total angular momentum is ∆ J = 2 and there is also a change of parity (∆ π = − ).Therefore we are dealing at best with a first forbidden Gamow-Teller process. The lepton system ( e − ¯ ν ) carries anorbital angular momentum L = 1 (first forbidden transition) and a spin S = 1 (unique Gamow-Teller transition),that couple to the total angular momentum J = 2. The two possible angular momentum components of the system,[( l j ) e ( l j ) ¯ ν ] J =2 , are [( p / ) e ( s / ) ¯ ν ] J =2 and [( s / ) e ( p / ) ¯ ν ] J =2 . Therefore, as noted in [11], the total differential decayrate d Γ /dE e is a sum of the two contributions corresponding to the emission of electrons in p -wave and in s -wave d Γ dE e = d Γ p / dE e + d Γ s / dE e (8)Following Eq. (6), we write the theoretical spectral shape of the electron in an ( l j )-wave as a sum of the contributionsfrom light (l) and sterile (s) neutrinos, d Γ l j dE e = d Γ ll j dE e cos ζ + d Γ sl j dE e sin ζ (9)where d Γ χl j dE e = C B Re R Re p e p ν χ E e ( E − E e ) F ( Z, E e ) S l ( p e , p ν χ ) θ ( E − E e − m χ ) , (10)for χ = l, s . Z stands for the atomic number of the daughter nucleus, F ( Z, E e ) is the Fermi function and θ ( E − E e − m χ ) is the step function. R Re is the nuclear radius [44], B Re is the dimensionless squared nuclear reducedmatrix element (r.m.e.) and C is a constant to be defined later on. In the above expression, E e , E and p e = p E e − m e are the total energy, maximum total energy and momentum of the emitted electron respectively, and p ν = p ( E − E e ) − m ν is the momentum of the emitted neutrino.Being Q β the endpoint energy, we have E = m e + Q β , and the kinematical ranges of E e , p e and p ν for zeroneutrino mass are as follows m e ≤ E e ≤ m e + Q β ; 0 ≤ p e ≤ q Q β + 2 m e Q β ; 0 ≤ p ν ≤ Q β . (11)The shape factor S l ( p e , p ν ) appears in forbidden decays. For the case of interest here, a first forbidden decay, l takesthe value l = 0 for the s -wave and l = 1 for the p -wave electrons, with shape factors S ( p e , p ν ) = 13 p ν and S ( p e , p ν ) = 13 p e F ( Z, E e ) F ( Z, E e ) . (12) e -m e [keV]012345 ( d Γ / d E e ) / Γ [ k e V - ] m = 0 keVm = 1 keVm = 1.5 keVm = 2 keV 0 0.5 1 1.5 2 2.5E e -m e [keV](a) s wave (b) p wave FIG. 1: Contributions of s -wave (left) and p -wave (right) electrons to the normalized differential decay rate of the process Reto
Os plotted against the electron kinetic energy E e − m e . Selected values of the sterile neutrino mass are used, m s = 1,1.5, 2 keV (dashed, dashed-dotted and dotted lines, respectively), compared to the light neutrino case m l = 0 (solid line). The relativistic Fermi functions F ( Z, E e ) and F ( Z, E e ) account for the Coulomb interaction between the residualnucleus ( Z = 76 in our case) and the emitted electron in the s and p -waves respectively. They are defined as F k − = (cid:20) Γ(2 k + 1)Γ( k ) Γ(1 + 2 γ k ) (cid:21) (2 p e R ) γ k − k ) | Γ( γ k + iz ) | e πz (13)and depend on the strength of the Coulomb interaction, given by the fine structure constant α ≃ / .
03, through γ k = p k − ( αZ ) and z = α Z E e p e , (14) k = 1, 2 in our case. We note that the Fermi functions in Eq. (13) satisfy F k − ( Z → , E e ) → α Z → k ≥
1. The constant factor C in Eq. (10) is given by C ≡ G F V ud c V π ≃ × − (keV) − , (15)where G F is the Fermi constant, V ud the element of the Cabibbo-Kobayashi-Maskawa matrix ( | V ud |≃ . c V ≃ B Re , can be computed directly from the experimental Re mean-life τ = t / / ln 2 as [see Eq. (10)] B − Re = τ C R Re Z E m e p e p ν E e ( E − E e ) F ( Z, E e ) S ( p e , p ν ) dE e , (16)and it takes the value B Re ≃ . × − for a value of the nuclear radius R Re approximated as R Re ≃ . × (187) / fm ≃ .
86 fm. Microscopic calculations of these quantities are in progress.In Fig. 1 we represent the s -wave (left) decay rates, d Γ ls / /dE e and d Γ ss / /dE e , and the p -wave (right) decay rates d Γ lp / /dE e and d Γ sp / /dE e , normalized to one. We plot the sterile neutrino contribution for s and p -wave outgoingelectrons and for selected values of the sterile neutrino mass, m =1, 1.5 and 2 keV (dashed, dash-dotted and dottedline respectively), compared to the light neutrino case with m = 0 (solid line). e -m e [keV]012345678 d Γ / d E e [ C R e k e V ] m = 0 keVm = 1 keVm = 1.5 keVm = 2 keV 0 0.5 1 1.5 2 2.5E e -m e [keV] 012345678 d Γ / d E e [ C R e k e V ] (a) s wave (b) p wave FIG. 2: Contributions of s -wave (left) and p -wave (right) electrons to the process Re to
Os as in Fig. 1 but for unnormalizeddecay rates, in units of 10 C Re keV (left) and of 10 C Re keV (right). The maximum differential decay rate for s -waveelectrons is of the order of 10 , whereas for p -wave electrons it is 10 . The p -wave dominates by four orders of magnitude forboth light and sterile neutrino emission and so the spectral shape of beta decay is dictated by the r.h.s. panel. In Fig. 2, we represent the unnormalized decay rates, on the left d Γ ls / /dE e and d Γ ss / /dE e , and on the right d Γ lp / /dE e and d Γ sp / /dE e . The same choices of neutrino masses as in Fig. 1 are considered. As seen in the plots,the maximum differential decay rate for s -wave electrons is of the order of 10 whereas for p -wave electrons it is 10 (in units of C Re keV ≡ C B Re R Re keV ). This dominance by four orders of magnitude of the p -wave that wasnoticed both theoretically [11] and experimentally [12] for the light neutrino emission, holds also for sterile neutrinoemission. This is why the spectral shape of beta decay is dictated by the curves shown in the r.h.s. panel of Fig. 2.The effect of the sterile neutrino emission on the electron spectral shape is represented in Fig. 3 by comparing thedifferential decay rate d Γ /dE e for m = 1 keV with (solid line) and without (dashed line) neutrino contribution for ζ = 0.01 o . The two curves start to deviate at the step point T e = Q β − m s = 1 .
469 keV, where the sterile neutrinostarts to contribute, and the difference grows as T e goes to zero. The region where the kink appears is shown in theinset on a magnified scale as a function of E e − m e − Q β + m s in eV. The chosen value of the mixing angle (0.01 o ) isinspired on Fig. 8 in Ref. [37], where a plot is made of upper bounds for the mixing angle as a function of the sterileneutrino mass based on X-ray observations of dwarf spheroidal galaxies. For different values of the mixing angle theeffect scales as the function R that we define below.In order to analyze the possible effect of a sterile neutrino, we introduce the dimensionless function R ≡ d Γ s /dE e d Γ l /dE e tan ζ (17)which is the ratio between the sterile and light neutrino contributions to the total decay rate times the tangent squareof the mixing angle. The function R is largest for p e (or T e ) going to zero. This procedure is useful because we arecomparing two regions of the same spectrum: the region where ( E e − m e ) < ( Q β − m s ) and the emitted neutrinohas enough energy for the sterile neutrino ( m s ∼ keV) to imprint an effect on the spectrum, and the region where( E e − m e ) > ( Q β − m s ) and the sterile neutrino effect does not show up. Clearly, there is a step in the spectrum for E e − m e = Q β − m s which could be observed if the experimental relative error in this energy region is lower than theheight of the step.The ratio R , Eq. (17), is shown in Fig. 4 as a function of the electron momentum p e for a mixing angle ζ = 0 . o ,and for different values of the neutrino masses, m s = 0, 1, 1.5 and 2 keV, corresponding to the solid, dashed, dash-dotted and dotted lines respectively. As can be seen in this figure, the ratio is different from zero in the range E e -m e [keV] d Γ / d E e [ C R e k e V ] -1 -0.5 0E e -m e -Q β +m s [eV]0.7390.740 ζ ~ 0.01 o x 10 -5 FIG. 3: The
Re to
Os beta particle spectrum in units of 10 C Re keV , for a mixing angle ζ = 0.01 o and sterile mass m s = 1 keV. The curves with (solid) and without (dashed) sterile neutrino contribution are indistinguishable in the main plot,but are shown in the inset as a function of E e − m e − Q β + m s in eV, with a separation between them magnified as indicated. e [keV]10 -9 -8 R s ζ = 0.01 o FIG. 4: Ratio R , Eq. (17), of the sterile neutrino to the active neutrino contributions of the process Re to
Os vs. theelectron momentum for a fixed mixing angle ζ = 0 . o and different sterile neutrino masses, m s = 0, 1, 1.5 and 2 keV (solid,dashed, dashed-dotted and dotted lines, respectively). R increases with decreasing m s . R is nonzero in a range 0 < p e < p max and p max decreases as m s increases. e [keV]10 -10 -9 -8 -7 R o o o ζ m s = 1 keV FIG. 5: The ratio R as in Fig. 4 but for a fixed sterile neutrino mass m s = 1 keV and different mixing angles, ζ = 0 . o , . o , . o (solid, dashed, dashed-dotted lines, respectively) of the process Re to
Os. R is almostconstant in the range 0 < p e < p max and increases with ζ . ≤ p e < ( p e ) max , with ( p e ) max = [( Q β − m s )( Q β − m s + 2 m e )] / . For example, for m s = 2 keV, ( p e ) max ≃ m s = 1 keV, ( p e ) max ≃ m s increases, R decreases.Similarly, in Fig. 5 we show the ratio R as a function of the electron momentum p e but for a fixed sterile neutrinomass of m s = 1 keV and different light-sterile mixing angles ζ = 0 . o , . o , . o . Figure 5 shows not only theincrease of R with increasing mixing angle for a fixed value of m s , but also shows the fact that for fixed values of m s and ζ , the ratio R is almost constant in the region 0 < p e < ( p e ) max .From Eq. (8) and Eq. (9), we write d Γ dE e = d Γ l dE e [1 + R ] cos ζ , (18)where d Γ /dE e is the total differential decay rate when neutrino mixing is present ( ζ = 0), and d Γ l /dE e is thedifferential decay rate for the light neutrino (no mixing: ζ = 0). For small mixing angle ζ , the differential decay rate d Γ /dE e [Eq. (18)] normalized to d Γ l /dE e is d Γ /dE e d Γ l /dE e ≃ R , (19)which shows that for small mixing angle the ratio between the differential decay rates with mixing and without mixingis given by 1 + R . This ratio is larger for p e or T e going to zero.We want to emphasize that the energy region suitable for creation and detection of the keV sterile neutrino corre-sponds to low p e or T e . On the contrary, information on active neutrinos should be obtained from the region of T e close to the endpoint energy Q β .From eqs. (8), (10) and (12), we can write for the function R the explicit expression R = p ν s p ν l θ ( Q β − T e − m s ) 1 + p e p ν s F ( Z, E e ) F ( Z, E e )1 + p e p ν l F ( Z, E e ) F ( Z, E e ) tan ζ (20)0In order to analyze the ratio p e F ( Z, E e ) /F ( Z, E e ), it is worth to define F k − ( Z, E e ) as F k − ( Z, E e ) ≡ C k − d k − (cid:18) m e p e (cid:19) k − (cid:18) E e m e α Z (cid:19) γ k − (21)where C k − ≡ π (2 m e R ) γ k − k ) (cid:20) Γ(2 k + 1)Γ( k ) Γ(1 + 2 γ k ) (cid:21) ; d k − ≡ π (cid:18) E e p e α Z (cid:19) − γ k (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) γ k + i α Z E e p e (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e π α Z Eepe (22) γ k is defined by Eq. (14) and d k − ( α Z E e /p e ) → p e → k ). The above definitions yield again F k − → α Z → k ). In this respect, eqs. (21) and (22) differ from references [11] and [41, 42]. Finally,the ratio in the shape factor of Eq.(12) for l = 1 is given by p e F ( Z, E e ) F ( Z, E e ) = C C d d m e (cid:18) E e m e α Z (cid:19) γ − γ ) (23)and for p e → p e F ( Z, E e ) F ( Z, E e ) p e → ∼ . m e , (24)[ d /d → p e → R p e → ∼ p ν s p ν l tan ζ . (25)For p e → p ν s ) max ∼ .
45 keV, 2 .
26 keV for m s ≃ p ν l ) max = 2 .
469 keV for m l = 0 keV.We have shown the relevance of the function R to the analysis of the sterile neutrino effect. One can also studythe effect of a sterile neutrino through the difference between the decay rate with mixing ( ζ = 0) and the referencecase without mixing ( ζ = 0). This difference is very small, since the mixing is in any case small, as it is expressed inthe following ratio, R ∗ = (cid:20) d Γ dE e (cid:21) ζ =0 − (cid:20) d Γ dE e (cid:21) ζ =0 (cid:20) d Γ dE e (cid:21) ζ =0 = (cid:18) − d Γ s /dE e d Γ l /dE e (cid:19) sin ζ = d Γ /dE e d Γ l /dE e − , (26)which can be written as well as a function of RR ∗ = − sin ζ + R cos ζ (27)In Fig. 6 we plot the quantity R ∗ for a fixed sterile mass m s = 1 keV and for different mixing angles.The Kurie function is defined as K ( y ) = s d Γ /dE e p e E e F ( Z, E e ) S ( Z, E e ) , y ≡ E − E e = Q − T e ≥ . (28)Considering the mixing between the light and sterile neutrinos, K ( y ) can be written as K ( y ) = q K l ( y ) cos ζ + K s ( y ) sin ζ (29)where K χ ( y ) = s d Γ χ /dE e p e E e F ( Z, E e ) S ( Z, E e ) , χ = l, s . (30)1 e [keV]-3-2-10 R * [ - ] o o o m s = 1 keV ζ FIG. 6: Same as in Fig. 5 but for R ∗ , Eq. (26 or Eq. (27), in units of 10 − for fixed m s and different mixing angles of theprocess Re to
Os. The small p e region is always the best for the sterile neutrino detection. For ζ = 0 (no sterile-light neutrino mixing), and due to the introduction of F ( Z, E e ) S ( Z, E e ) in the denominator, K ( y ) vs. y is a straight line for m l ≃
0. This follows straightforwardly from Eq. (28) for K ( y ) and from Eq. (10) for d Γ /dE e , setting m l = 0 and p ν = E − E e = y , and it therefore follows that K ( y ) ≃ const × y . K ( y ) can be writtenas well in terms of R Eq. (17), K ( y ) = K l ( y ) √ R cos ζ . (31)Finally, in Fig. 7 we present the Kurie plot K considering several neutrino masses, and K l (solid line). V. TRITIUM BETA DECAY AND STERILE NEUTRINO MASS
Let us now consider the beta decay of Tritium ( H; Z = 1; A = 3) H → He + e − + ¯ ν e (32)as a probe to detect a possible mixing of keV sterile neutrinos with active neutrinos. Tritium beta decay would allowthe detection of sterile neutrinos heavier than in the Rhenium beta decay case, within the 1 to 10 keV range suggestedby cosmological and galactic observations. Tritium, H, is a hydrogen isotope going to the helium isotope He, witha half-life t / ≃ .
33 years, endpoint energy Q β ≃ .
59 keV, and a spin-parity transition 1 / + → / + .For the Tritium decay case there is no change in angular momentum and parity corresponding to an allowedtransition ( L = 0) with Fermi ( S = 0) and Gamow-Teller ( S = 1) components. Therefore, the electron is emitted in s -wave and the differential decay rate is simply d Γ dE e = d Γ s / dE e (33)Similarly to Eq. (9) and Eq. (10) we have d Γ s / dE e = d Γ ls / dE e cos ζ + d Γ ss / dE e sin ζ (34)2 -2.5 -2.0 -1.5 -1.0 -0.5 0E e -E [keV]00.51.01.52.02.5 K [( C R e ) / k e V ] m = 0 keVm = 1 keVm = 1.5 keVm = 2 keV FIG. 7: Kurie plot K of the process Re to
Os for different neutrino masses, m = 0, 1, 1.5 and 2 keV. and d Γ χs / dE e = CB T p e p ν χ E e ( E − E e ) F ( Z, E e ) θ ( E − E e − m χ ) , χ = l, s , (35)as the shape factor S ( p e , p ν ) is 1 for allowed decays. The relativistic Fermi function F ( Z, E e ) was defined in Eq. 13.The squared r.m.e. for the allowed decay of Tritium (T) is B T = B F T + B GT T , where B F and B GT are the Fermi andGamow-Teller decay strengths respectively, given by: B F T = 12 |h He(1 / + ) k A =3 X j =1 τ + j k H(1 / + ) i| (36)and B GT T = g A |h He(1 / + ) k A =3 X j =1 τ + j ~σ j k H(1 / + ) i| (37)where g A = c A /c V ≃ .
26 is the axial-to-vector strength ratio of the charged weak interaction.From the experimental mean-life of Tritium the decay strength can be obteined from B − T = τ C Z E m e p e p ν E e ( E − E e ) F ( Z, E e ) dE e , (38)which yields a value B T ≃ .
61. In Fig. 8 we plot the differential decay rates of the process H to He for neutrinosof different masses. As an illustration of the heavy neutrino contribution to the total decay rate of the Tritium betadecay, we plot the electron spectrum in Fig. 9 with (solid line) and without (dashed line) sterile neutrino contributionwith mass m s = 1 keV and mixing angle 0.01 o . The effect is made visible in the inset thanks to the indicatedmagnification. Finally, in Fig. 10 we plot the ratio R vs. p e for a fixed sterile neutrino mass ( m s = 1 keV) anddifferent mixing angles.3 e -m e [keV]10 d Γ / d E e [ C T k e V ] m = 0 keVm = 1 keVm = 1.5 keVm = 2 keV FIG. 8: Differential decay rate for the process H going to He with the emission of active ( m = 0) or sterile ( m = 1, 1.5, 2keV) neutrinos, in units of C T keV . The vertical axis is in logarithmic scale. e -m e [keV]10 d Γ / d E e [ C T k e V ] x 10 -5 ζ ~ 0.01 o FIG. 9: The H to He beta particle spectrum with (solid line) and without (dashed line) sterile neutrino contribution forsterile mass m s = 1 keV and ζ = 0.01 o . Axis labels of the inset are the same as in the main plot, where a magnification of theseparation between curves is magnified as indicated.
40 60 80 100 120 140 160p e [keV]10 -10 -9 -8 -7 R o o o ζ m s = 1 keV FIG. 10: Ratio R , Eq. (17), vs. the electron momentum for a fixed sterile neutrino mass m s = 1 keV and different mixingangles, for the process H going to He.
VI. CONCLUSIONS
The detection of sterile neutrinos is not only important from the point of view of particle physics for the extensionof the SM, but also from the point of view of cosmology and astrophysics as a serious candidate for dark matter inthe keV mass range. With the relevance of the possible detection of keV scale dark matter candidates in mind, wehave studied Rhenium 187 and Tritium beta decays. The low electron energy domain of the beta spectrum is theregion where a sterile neutrino could be detected and its mass measured, the expected mass being in the keV scale(1 to 10 keV) as constrained from cosmological and galactic observations and theoretical analysis. The electron lowenergy region that is suitable to detect the sterile neutrino, 0 . T e . ( Q β − m s ), is away from the endpoint energyregion suitable for the detection of the active neutrino mass.Two experiments are running at present, MARE and KATRIN, dealing with Rhenium 187 and Tritium beta decaysrespectively. The MARE experiment will provide the entire shape of the electron differential decay rate, giving datafor both the sterile and the active neutrino detection regions. KATRIN so far concentrates on the region near theendpoint of the electron spectrum but it would be extremely interesting to look for data in the region where keVsterile neutrinos could show up [14]. In this paper, we have carried out the study of the role of sterile neutrinos in betadecay spectra, within the expected keV mass range and considering different mixing angles, according to astronomicaland cosmological observations and experiments.For Re the electrons can be emitted in p -wave and in s -wave, the former dominating the decay by a factor 10 over the latter. For Tritium the electrons are emitted in s -wave only. The spectra of the electrons emitted in thesewaves have been carefully computed from the experimental beta decay half-lives using relativistic Fermi functions.Results for different neutrino masses (light and in the 1-2 keV range) have been obtained separately and mixed. Wehave also computed for both decays the ratio R of the light to the heavy component of the mixing. It is differentfrom zero in an electron momentum range 0 ≤ p e < ( p e ) max , where ( p e ) max decreases with increasing sterile neutrinomass m s (equivalently for electron kinetic energy range). In the vicinity of ( p e ) max the ratio R drops off sharply, butin the rest of the range it exhibits an almost constant plateau with a slight increase as p e goes to zero. It increaseswith the mixing angle and decreases with the sterile neutrino mass m s .In order to detect the small deviation in the experimental spectrum due to the sterile neutrino mixing, the relativeexperimental random error (inversely proportional to the square root of the number of measured events, ǫ ∼ N − / β )must be as small as possible. To this end, the number of detected events N β must increase by choosing, for instance,a beta decay with a small Q β value or by increasing the time of data acquisition. For MARE, the typical number of5events is 10 − for 10 years of data acquisition, 8 arrays and 400 gr of natural Rhenium [9]. We found that atits largest value, the ratio R of the sterile neutrino to the active neutrino contributions is about 10 − using a realisticmixing angle. Therefore the sterile neutrino probability R × N β is about 10 − , which is not negligible. It impliesfinding 10 − sterile neutrinos within 10 − events. These numbers increase one order of magnitude forthe MARE option of 10 events for 10 years of data acquisition, 16 arrays and 3.2 kg of natural Rhenium [9]. Asimple estimate requires the Poisson error ǫ to be smaller than the ratio R . Namely, N β > / R ∼ − . Ofcourse, in order to assess a precise prediction of the detection probability one should include a careful analysis of thesystematic errors and instrument parameters, but such study goes beyond the scope of the present paper. The smalleffect expected on the electron spectrum calls for sources with larger stability to reduce the systematic errors, whichpose at present a difficult challenge on the detection capabilities of these experiments. Furthermore, for R = 10 − there would be one sterile neutrino event for one hundred million active neutrino events.The main purpose of this paper has been to guide future experimental searches for sterile neutrinos. From thepoint of view of particle physics, one is talking about an extension of the Standard Model. From the point of view ofcosmology, one is looking for a keV candidate for DM (mass range favoured by cosmological observations). Therefore,we show in this paper the relevant energy range where experimentalists should focus on as well as the order ofmagnitude of the expected signal, both in absolute terms and with respect to the background. Acknowledgments
We are grateful to Peter Biermann, Angelo Nucciotti (MARE) and Christian Weinheimer (KATRIN) for usefuldiscussions. O.M and E.M.G acknowledge the Spanish Ministry of Science and Education for partial financial support(FIS 2008-01301, FIS 2011-23565 and FPA 2010-17142). M.R.M. acknowledges the financial support of the SpanishMinistry of Science and Education (FIS 2008-01323), and the kind hospitality of the Observatoire de Paris. [1] J. R. Bond, A. S. Szalay, ApJ 274 (1983) 443. J R Bond, A S Szalay, M S Turner, Phys. Rev. Lett. 48 (1982) 1636.[2] C. J. Hogan, J. J. Dalcanton, Phys. Rev. D 62 (2000) 063511. J. J. Dalcanton, C. J. Hogan, ApJ 561 (2000) 35.[3] S. Dodelson, L. M. Widrow, Phys. Rev. Lett. 72 (1994) 17. X. Shi, G. M. Fuller, Phys. Rev. Lett. 82 (1999) 2832. K.Abazajian, G. M. Fuller, M. Patel, Phys. Rev. D 64 (2001) 023501; K. Abazajian, G. M. Fuller, Phys. Rev. D 66 (2002)023526; G. M. Fuller et. al. , Phys. Rev. D 68 (2003) 103002; K. Abazajian, Phys. Rev. D 73 (2006) 063506. P. L. Biermann,A. Kusenko, Phys. Rev. Lett. 96 (2006) 091301; T. Asaka, M. Shaposhnikov, A. Kusenko; Phys. Lett. B 638 (2006) 401. A.Kusenko, F. Takahashi, T. T. Yanagida, Phys. Lett. B 693, 144 (2010). K. Petraki, A. Kusenko, Phys. Rev. D77: 065014(2008).[4] H.J. de Vega and N. S´anchez, Mon. Not. R. Astron. Soc. 404 (2010) 885, astro-ph/0901.0922. D. Boyanovsky, H.J. de Vegaand N. S´anchez, Phys. Rev. D 77 (2008) 043518.[5] H.J. de Vega and N. S´anchez Int. J. Mod. Phys. A 26 (2011) 1057. H.J. de Vega, P. Salucci and N. S´anchez, New Astronomy,17, 653 (2012).[6] A. D. Dolgov, Phys. Rept. 370 (2002) 333. A. Kusenko, Phys. Rept. 481 (2009) 1.[7] F. Munyaneza, P. L. Biermann, Astron. and Astrophys. 458 (2006) L9. D. Boyanovsky, C. M. Ho, JHEP (2007) 0707.[8] F. D. Steffen, Eur. Phys. J. C 59 (2009) 557.[9] MARE collaboration, http://mare.dfm.uninsubria.it/frontend/exec.php; A. Nucciotti, Neutrino 2010, arXiv:1012.2290. A.Nucciotti, on behalf of the MARE collaboration,
The MARE experiment and its capabilities to measure the light (active)and heavy (sterile) neutrinos , lecture at the Chalonge Meudon Workshop 2011: Warm Dark Matter in the Galaxies, Theoryand Observations, available at http://chalonge.obspm.fr/[10] M. Galeazzi et al. , Phys. Rev. Lett. 86 (2001) 1978.[11] R. Dvornick´y, K. Muto, F. ˇSimkovic and A. Faessler, Phys. Rev. C 83 (2011) 045502.[12] C. Arnaboldi et al. et al. , Phys. Rev. D 46 (1992) R6. G. Finocchiaro, R. E. Shrock,Phys. Rev. D 46 (1992) R888.[17] G. Finocchiaro and R R Shrock, Phys. Rev D46, R888 (1992). F. Bezrukov, M. Shaposhnikov, Phys. Rev. D 75 (2007)053005. S. Ando, A. Kusenko Phys. Rev. D81, 113006 (2010). [18] W. Liao, Phys. Rev. D 82 (2010) 073001. Y. F. Li, Zhi-Zhong Xing, Phys. Lett. B 695 (2011) 205 and JCAP 1108 (2011)006.[19] J. J. Simpson, Phys. Rev. Lett. 54 (1985) 1891.[20] F. E. Wietfeldt, E. B. Norman, Phys. Rep. 273 (1996) 149.[21] A. Franklin, Rev. Mod. Phys. 67 (1995) 457.[22] F. Zwicky, Helv. Phys. Acta 6 (1933) 124.[23] J. H. Oort, ApJ 91 (1940) 273. See S. van den Bergh, astro-ph/0005314 for a history of the research on dark matter.[24] E. W. Kolb, M. S. Turner, The Early Universe , Addison-Wesley (1990).[25] D. Boyanovsky, H.J. de Vega and N. S´anchez, Phys. Rev. D 78 (2008) 063546.[26] S. Dodelson,
Modern Cosmology , Academic Press, London (2003).[27] J Kormendy, K C Freeman, IAU Symposium, Sydney, 220 (2004) 377, arXiv:astro-ph/0407321. M. Spano et al., MNRAS,383 (2008) 297. G. Gentile et al., Nature, 461 (2009) 627. F. Donato et al., MNRAS 397 (2009) 1169.[28] Y. Hoffman et al. , ApJ 671 (2007) 1108.[29] J. Zavala et al. , ApJ 700 (2009) 1779. E. Papastergis et al. , ApJ 739 (2011) 38, arXiv:1106.0710.[30] R. E. Smith, K. Markovic, Phys. Rev. D84 (2011) 063507. K. Markovic et al. , JCAP 1101 (2011) 022.[31] A. Kamada, N. Yoshida, keV-mass sterile neutrino dark matter and the structure of galactic halos , lecture at the 15th ParisCosmology Colloquium 2011: From CDM to WDM in the Standard Model of the Universe: Theory and Observations,available at http://chalonge.obspm.fr/[32] H.J. de Vega and N. S´anchez,
Highlights and Conclusions of the Chalonge Meudon Workshop 2010, Dark Matter in theUniverse , astro-ph/1007.2411. H.J. de Vega and N. S´anchez,
Highlights and Conclusions of the Chalonge Meudon Workshop2011, Warm Dark Matter in the Galaxies , astro-ph/1109.3187.[33] H.J. de Vega, M.C. Falvella and N. S´anchez,
Highlights and Conclusions of the Chalonge 14th Paris Colloquium: TheStandard Model of the Universe: Theory and Observations , astro-ph/1009.3494.[34] V. Tikhonov, S. Gottloeber, G. Yepes and Y. Hoffman, MNRAS 399 (2009) 1611.[35] K. Petraki, Phys. Rev. D77: 105004 (2008). J Wu, C-M Ho, D. Boyanovsky, Phys. Rev. D80, 103511 (2009). D. Boyanovsky,J Wu, Phys. Rev. D83: 043524 (2011).[36] M. Loewenstein, A. Kusenko, P. L. Biermann, ApJ 700 (2009) 426. M. Loewenstein, A. Kusenko, ApJ 714 (2010) 652.[37] M. Loewenstein, A. Kusenko, ApJ 751 (2012) 82.[38] G. G. Raffelt and Shun Zhou, Phys. Rev. D 83 (2011) 093014.[39] M. Viel et al. Phys. Rev. D 71 (2005) 63534.[40] A. Boyarsky, O. Ruchayskiy, M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. 59 (2009) 191. A. Boyarsky, J. Lesgourgues, O.Ruchayskiy, M. Viel, Phys. Rev. Lett. 102 (2009) 201304.[41] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. (Supp.) 83 (1985) 1.[42] M. Doi and T. Kotani, Prog. Theor. Phys. 87 (1992) 1207.[43] Y. Chikashige, R. N. Mohapatra, R. D. Peccei, Phys. Lett. B 98 (1981) 265. J. Schechter, J.W.F. Valle, Phys. Rev. D 25(1982) 774. R. R. Volkas, Prog. Part. Nucl. Phys. 48 (2002) 161. M. Shaposhnikov, I. Tkachev, Phys. Lett. B 639 (2006)414. F. Bezrukov, H. Hettmansperger, M. Lindner, Phys.Rev. D 81 (2010) 085032. M. Lindner, A. Merle, and V. Niro,JCAP 1101 (2011) 034. A. Merle and V. Niro, JCAP 07 (2011) 023.[44] H. Behrens and W. Buhring,