JHEP10(2007)015 10 hep102007015.pdf July 19, 2007 May 22, 2007 with masses October 3, 2007 September 17, 2007 establish a lower for singlet leptons Revised: Received: Published: the properties of neutral strength of interaction of MSM. The search of sin- nal in kaon decays would searches similar to CERN to cosmologically interest- ν Accepted: oton beams. We argue that article physics experiments. nd Big Bang Nucleosynthe- ion couplings and to find or MSM? aneously neutrino oscillations, nd from the data on neutrino a ences, f kinematics of more than 10 ν in the decays of different mesons http://jhep.sissa.it/archive/papers/jhep102007015 /j 5 GeV would require a considerable increase of the − Published by Institute of Physics Publishing for SISSA Beyond Standard Model, Neutrino Physics, Weak Decays. An extension of the Standard Model by three singlet fermions collisions. We study in detail decays of neutral leptons and [email protected] [email protected] pp Institute for Nuclear Research of60th the October Russian Anniversary Academy prospect 7a, ofE-mail: Moscow Sci 117312, Russi Institut Physiques, de des Th´eorie Ph´enom`enes deEcole Lausanne, Polytechnique F´ed´erale CH-1015 Lausanne, Switzerland E-mail: SISSA 2007 c ° intensity of proton acceleratorsB-meson or decays. the detailed analysis o Keywords: bound on their mass comingsis. from existing We experimental argue dataallow a that to the strengthen search considerably fordefinitely the a exclude bounds specific them on missing below neutraling the energy ferm parameter kaon sig range threshold. forPS191 masses To experiment above would enter kaon be in mass needed thethe with dedicated the use use of of CNGS, intensivebelow pr NuMI, charm T2K in or a NuTeV large beams portion could allow of to the search parameter space of the glet fermions in the mass interval 2 Dmitry Gorbunov Mikhail Shaposhnikov Abstract: smaller than the electroweakdark scale matter allows and to baryon explain asymmetryleptons simult in of this the model Universe. and theWe We ways establish, discuss they in can be particular, searched aneutral for lower in leptons p and coming an from upperoscillations. cosmological bound We analyse considerations on the the a productionand of in neutral leptons How to find neutral leptons of the JHEP10(2007)015 1 13 15 22 29 32 34 leptons 10 est possible number use different types of scale of the new neutral MSM (neutrino Minimal d scale). An example of ν phenomena that cannot be (such as supersymmetry or ructure as the quark sector, glet scalar field allows to have e for dark matter particle – the rpart. This model is consistent e neutrinos interchangeably) are in the baryon asymmetry of the ch of these particles. Let us review MSM 6 ν – 1 – singlet right-handed fermions (we will be using MSM. ν light A crucial feature of this theory is the relatively small mass guidelines. A possiblefit strategy to is the toof SM attempt new to by particles explain minimal withoutextra the means, adding dimensions) that any or new issuch new physical by a energy principles introducing theory scales the is (like small the the renormalizable Grand extension Unifie of the SM, the 1. Introduction In a search for physics beyond the Standard Model (SM) one can 2. The Lagrangian and parameters of the Contents 1. Introduction 4. Decays of heavy neutral leptons 5. Production of heavy neutral6. leptons Prospects for future experiments 7. Conclusions A. Sterile neutrino decays B. Decays into sterile neutrino Standard Model) [1, 2], where three 3. Laboratory and BBN constraints on the properties of heavy also the names neutral fermions,introduced. or heavy The leptons, leptonic or steril sectori.e. of every the left-handed theory fermion haswith has the the its same data st right-handed on counte neutrinolightest oscillations, singlet provides fermion a (sterile candidat Universe neutrino), [2]. and A further can extensioninflation expla of in this the model Early by Universe a [3]. light sin leptonic states, which opens a possibilityshortly for the a physical direct sear applications of the JHEP10(2007)015 f ron greatly exceeding decays of different 1 N β τ s [1, 4, 5, 7, 6]: one of ntum [16]. However, no the quark sector. This ure formation and to the ticle [8 – 11]. Dark matter ium exactly because their n with allowed values for atisfied. Similarly to the Dark Matter cosmological gical considerations of the ermions, 3 Dirac masses as- can be potentially observed which is characterized by 9 and sterile neutrinos [8, 9]; y pattern (and in particular and 3 CP-violating phases). his particle can be produced ogenesis, associated with the 1 ) eV. The choice of the small MSM contains 6 CP-violating exist. An astrophysical lower imits on the strength of their 5 ir free streaming length at the ded neutrinos, 6 mixing angles he Yukawa coupling constants, hermal equilibrium for temper- neutrino can be searched for in N ν pplications [36]. − nematics of (10 O MSM contains extra degrees of free- MSM contains 18 new parameters in ν ν (10) keV region. The arguments leading The baryon (B) and lepton (L) numbers , may have a life-time O 1 The N – 2 – scale of active neutrino masses can be established in MSM does not have any extra stable particle in compari- ν absolute MSM. The radiative decays of GeV. As for CP-breaking, the MSM. The lepton number is violated by the Majorana neu- ν 3 keV, following from the analysis of the rotational curves o , which is crucial for explanation of dark matter and baryon ν . is broken by the electroweak anomaly. As a result, the sphale 12 12 L 10 − + 10 Though the B − 6 X-ray observationsinteraction [10, with 18], active and neutrinos [19, theonset 20, stringent of 5, l cosmological 21 – structure 27]bound and formation on the [28 – their 31] mass already is 0 the age of the Universesterile and neutrino thus play is a likely roleto to of the have a a keV dark matter mass massproblems par in for of the dark missing matter satellitesmodels neutrino [12 and are – 15]; cuspy related the to profilesproduction keV struct in of scale the dark is matter Cold alsowarm DM due favoured may to by help the transitions toupper cosmolo solve between the limit active problem on of the galacticin mass angular interactions mome of beyond sterile the neutrino exists [17, 3] as t phases in the leptonmakes sector two and of a theSM, Kobayashi-Maskawa this Sakharov phase theory conditions in the does [38] Higgs not for mass have baryogenesis [39],non-equilibrium an making s impossible bubble electroweak the phase expansion. electroweak transitio bary However, the dom - sterile neutrinos - which may be out of thermal equilibr comparison with SM. They are:sociated 3 with Majorana the masses mixing for between singletand left-handed f 6 and CP-violating right-han phases.the These observed parameters one) can of describeparameters masses an only and (3 mixings active ofInspite neutrino active of masses, neutrinos, this 3 freedom, mixingthe the angles, son with the SM, the lightest singlet fermion, 1. Neutrino masses and oscillations. mass scale for singlet fermionsat leads the to the level small 10 values of t the active neutrinos must have a mass smaller than asymmetry of the Universe. 2. Dark matter. JHEP10(2007)015 inflaton lness of the light MSM requires ν ated through CP- MSM couplings can ν rticle, while two other ameters and to have a MeV, and the coupling hance the CP-violating parameter space is quite ron transitions. Roughly nsfers of leptonic number be much smaller than the ng constants. At the same mion masses should be in s plus the total helicity of out in [2]. The effects to tter fact is a key point for ged leptons. For masses of hat they are stable against nflaton- of the Universe. of a lepton number symmetry, struction of sterile and active t due to interaction with the iggs mass parameter can be set os becomes different from zero mechanism for the dark matter Universe’s evolution. However, ecific mass-coupling pattern for nt experiments and methods. ton asymmetry at the sphaleron aryon asymmetry. In addition, e the electroweak scale. GeV intermediate energy scale y cosmological and astrophysical 15 na masses the total lepton number 10 , which are sometimes invoked as the − 10 MSM may be extended by a ν 10 ∼ – 3 – The phenomenological success of the MSM can provide simultaneous solution to the problem experimental facts ν 20 GeV the mechanism does not work as the sterile neutrinos MSM. ∼ ν MSM, ensuring the validity of the third Sakharov condition. ν In [3] it was proposed that the -boson mass and the Planck mass, really requires it. The smal W (10) keV region to provide a candidate for the dark-matter pa In [40] it was proposed that the baryon asymmetry can be gener The kinetics of sterile neutrino oscillations and of the tra In [2] it was shown that the To summarize, none of the O violating sterile neutrino oscillations.of For small the Majora system, definedsterile as neutrinos, the is conserved lepton and numberbecause equal of of to active oscillations zero neutrino the duringand the lepton gets number transferred of active tospeaking, neutrin the the resulting baryon baryon asymmetry number isfreeze-out. equal due to the to lep rapid sphale between active and sterilebe neutrino taken into sectors account hasneutrinos, include been coherence oscillations, worked in creationmedium, sterile and dynamical de neutrino asymmetries sector insterile and active neutrinos neutrinos exceeding its and los char equilibrate. The temperature of baryogenesis is right abov be taken to be scale invariantto at zero. the classical The level mass andof the of the H the lightest inflaton sterile neutrino can toproduction. be it as may serve small as as an few efficient hundreds 5. Fine-tunings in the of neutrino oscillations, dark matter and baryon4. asymmetry Inflation. in order to accommodatecommon source inflation. for the To Higgs reduce and sterile the neutrino number masses of the i par a number of finethe tunings. In particular, one of the singlet fer the baryogenesis in the Yukawa couplings to ordinary fermions are very small. The la between the masses must be mucheffects larger in but the almost sterilethe degenerate neutrino Yukawa coupling oscillations [2, of leading 40] theYukawa to couplings to dark of the en the matter b heavier sterile singletconstrains neutrino fermions, [1]. to must satisf Theseradiative corrections. fine-tunings Moreover, are in [41] “natural”the was in shown singlet that fermions, a a described sp sense above,slightly can t broken be by a the consequence Majoranatime mass not terms all and 18 Yukawa coupli large new to parameters yield are variety of fixed: signatures the to allowed be region tested in with differe arguments for the existence of the large JHEP10(2007)015 or the d of N ± e → leptons and ± MSM we will → ν K MSM). The aim , ν use of the reasons lectroweak scale is N ± µ eir decay modes have mber of experiments for of work in this direction → energies (see e.g. [43] for ities in the xperimental search of these ± wa couplings rather than in mass of the beauty mesons, pectra of electrons or muons e add new general results for in this paper we will consider inglet fermions participate in he thermal leptogenesise [42], replaced by the baryogenesis K ually with some stable SUSY 49] (“nothing” low nglet fermions responsible for m its mass. work of the uppressed roughly by a factor sewhere. , therefore, what would be the es of the SM is almost fixed by at the production and the decay an be associated with the light this mass range and in what kind plitudes) would allow to compute . The existence of the U(1) lepton GeV, with the perturbation power is replaced by an ordinary neutrino. 13 N 10 ∼ (1) GeV mass of these singlet leptons [41]. O MSM. Finding these particles and studying ν MSM predictions for the branching ratios. – 4 – ν changes when N of rare meson decays can constrain the strength of the -mesons have been studied. The second strategy is to + are the Dirac and Majorana masses correspondingly. Since ∓ K e M + ± M of the baryon asymmetry of the Universe theoretically along π - and π → kinematics and (100) GeV or with an axion, can well be a much lighter sterile O D L,S MSM without addition of the inflaton; we will discuss what kin M ν K magnitude , where 2 ) and the M 5 GeV, considering this mass range as the most plausible beca /M ∼ < Putting all the physics beyond the Standard Model below the e Naturally, several distinct strategies can be used for the e Clearly, to find the best way to search for neutral leptons, th sign D N M active neutrino masses may find its explanation in small Yuka neutrino, practically stable on the cosmological scales. T large energy scale.partner The dark of matter the particle, mass associated us working well only at large masses ofthrough Majorana light fermions, can singlet b fermioninflaton oscillations. field rather than The with inflation that with c the mass a discussion of neutrinoless double beta-decay in the frame of this paper isbaryon to asymmetry analyse of the the possibilities Universe to in search the for si spectrum coming from inflaton self-coupling rather than fro not harmless, as it can be confronted with experiment at their properties in detail (inthe particular, CP-violating am differences one can expect if the light inflaton isparticles. included The el first oneall is reactions related the to ordinary their( neutrinos production. do The with s a probability s they are massive, the kinematics of, say, two body decays lines of [2] and confrontsymmetry this prediction provides with an observations argument in favour of In addition, the structure of their couplings to the particl the data on neutrinoexperimental oscillations. signatures of It the is neutralof singlet interesting experiments fermions to they in know could bethe found. variant To answer of this the question, the search of neutraloriginating leptons in in the decays past of [44, 45], where the s coupling of heavy leptons. This strategy has been used in a nu three-body decays Therefore, the study of look for the decays of neutral leptons inside a detector [46 – to be identified andhas branching been ratios already must done in bethree refs. body estimated. [51 – meson 54] A decays. for lot the To general analyze case; the w corresponding quant hadrons). Finally, these two strategiesoccurs can be inside unified, the so same th detector [50]. presented above. We will use the specific constrain ourselves by theM singlet fermion masses below the JHEP10(2007)015 is ble already teresting exist. However, s or find neutral So, the search for ) detector (like the search of decays of e the cosmologically avy neutral leptons. et fermions with the 4 7]. To this end quite . on its energy, pions, nalysis of the equirement that these d some distance away S and T2K experiment To enter into cosmolog- . In short, an intensive (related, e.g. to CNGS, asis of the proton beams synthesis (BBN) [55, 56] r a detailed study of the at further constraints on ar detectors. experimental data of KLOE are already disfavoured on the 450 MeV, where strong bounds π e points. , where current bounds are weak < M K N M < M In addition, the NA48/3 (P326) experi- N 1 -meson thresholds is very hard if not impossi- At the same time, the existing setups of the < M is perfectly allowed from the cosmological and B – 5 – MSM for 3 ν K The search for the missing energy signal, specific for < M 2 8 GeV. . N 1 ∼ < factories, is unlikely to gain the necessary statistics and the search for the missing energy signal, potentially possi < M N τ π D M M < M -meson but still below N D < M K M ble with present or planned proton machines or B-factories. decays of neutral fermionsbeam is of the protons most effectivestrange, hitting charmed opportunity the and fixed beauty mesonsA target, that part creates, decay of and depending produce thesefrom he the leptons collision then point. decay TheNuMI inside dedicated or experiments a NuTeV on at detector, the FNAL, b parameter situate CNGS range at for CERN, or JPARC can touch a very in ment at CERN would allow to find or to exclude completely singl mass below that of the kaon. at beauty, charm and are quite promising for heavypossibility neutrino of searches neutral and fermions calls detection fo at (possible) ne (s.f. [57]). The record intensity of the neutrino beam at CNG neutral fermions, as was donea in CERN number PS191 of experiment already [46,MiniBoone, 4 existing MINOS or or T2K), planned complementedone neutrino by of a facilities near CERN (dedicated PS191) can be used. the experiments mentioned above, can be complemented by the very difficult if not impossible at hadronic machines like LHC experimental points of view. Moreover, it is not excluded th the couplings of singletexisting fermions but can never becollaboration considered derived from and from of this the the point E949 rea of experiment. view MiniBooNE or MINOS experiments wouldinteresting unlikely allow parameter to space prob of the on the parameters comingMiniBooNE from and CERN MINOS PS191fermions can experiment in already possibly the mass improve region the 450 existing MeV limit basis of existing experimentalparticles data do of not spoil [46, the 47](s.f. predictions and [57]). of from the Big the Bang r Nucleo We arrived at the following conclusions. We thank Augusto Ceccucci forWe discussion thank Francois of Vannucci this for point. We discussion thank of Tasuya this Nakada point. for discussion of this point. We thank Gino Isidori and Yury Kudenko for discussion of thes 2 3 4 1 (i) The singlet fermions with the masses smaller than (ii) The mass interval (iv) Going above (iii) For JHEP10(2007)015 , 1 3 12 N − N (2.1) MSM 10 ν . , ∼ < and field can c | . 1 2 1 -lepton. In α N N τ F +h studying the | J choice of [58], but e N c I ¯ e in parameterization N ion without tilde for 5 nomaly [60]. This type of IJ 2 ection 5 we consider the M of this symmetry is small relevant part of the MSM ase of the present intensity is and explanation of dark ∆ -mesons and of is the common mass of two ν of the masses of perties of these particles. In B e sterile neutrino − M ions. In section 3 we analyze d by the X-ray observations [5] these particles coming from the 3 he explanation of the small values of ay. This leads to the singlet fermion existing and future experiments. e N c 2 - and ¯ e der of the Grand Unified scale. The theory MSM ) are the Higgs and lepton doublets, N scenario assumes that the Yukawa coupling D ymmetry of the Universe but does not give metry of the universe. Also, the MiniBooNE s to fix the Majorana masses of sterile neu- ν lar couplings of the charged leptons or quarks M -, MSM paradigm is to determine the Lagrangian K − ν ment that it should explain neutrino oscillations, e,µ,τ ˜ Φ I = e N α – 6 – α ( ¯ L α 0 the Lagrangian (2.1) has a global U(1) lepton L αI F → B-mesons. − 2 10 α I are related to the mass of the lightest sterile neutrino e F N µ , Φ and IJ ∗ j γ 0, µ Φ M ij i∂ → ǫ I ¯ e than the electroweak scale, in the contrast with the see-saw N = IJ i + M ˜ Φ is a matrix of Yukawa coupling constants, smaller MSM L F . The Yukawa coupling constants of the dark matter neutrino = 10 eV energy range (eV see-saw) [59] to accommodate the LSND a M are the right-handed singlet leptons (we will keep the notat − I e ≪ N MSM of, say, CNGSkinematics beam of by more two than 10 orders of magnitude or producing and ically interesting parameter space would require the incre ν The paper is organized as follows. In section 2 we discuss the In the limit ∆ IJ L Of course, this Lagrangian is not new and is usually used for t 5 M Lagrangian and specify the predictionsdata for the on couplings neutrino of the oscillations present and experimental cosmological andsection considerat cosmological 4 limits we on analyze theproduction the of pro heavy decay neutral modes leptons in of decays singlet of fermions. In s parameters from available observations,dark i.e. from matter require and baryon asymmetry of the universe in a unified w Majorana masses constants of the singletand fermions that are the of Majorana thewith order masses this of of choice the singlet simi of fermions are parameters of can the also or explain the baryon as neutrino masses via the see-saw mechanism [58]. The see-saw a candidate for atrinos dark in matter 1 particle. Another suggestion i theory, however, cannot explain dark matter and baryon asym experiment [61] did not confirm the LSND result. The much larger than few eV, as in the eV see-saw of [59]. For our aim it is more convenient to use the Lagrangian of the section 6 we analyze theWe possibilities conclude of in their section detection 7. in 2. The Lagrangian and parameters of the where of ref. [41]: mass eigenstates), respectively, heavy neutral fermions, ∆ ∆ responsible for dark matter and produce the small splitting are strongly bounded by cosmologicaland considerations can [1] be an safely neglected for the present discussion and th be omitted from the Lagrangian. symmetry [41]. In thisnot paper only we in will the assume mass that the sector breaking (which is required for baryogenes JHEP10(2007)015 · κ · our | (2.2) (2.3) (2.4) (2.5) 6 . 2 0 β eutrino F ∼ > -lepton a ǫ ) with the τ 10 m, and 3 | ∼ | N ∼ 3 α situated on the − l 1 the coherence has a particular . , F l c 2 | , 3 . ∼ 2 N N s may be important , the explicit form of +h 3 IJ and φl/L by the unitary transfor- N 2 , c M . As was shown in [41], (2 keV) 3 † 3 , N ¯ ) state with the probability 2 F ∼ > 3 V N e 2 can be expressed through the N · os for different reactions can 3 ifferent mesons or , which go away if the lepton N ≡ 2 ) α , M M 2 − f 2 iδ 2 ! 2 F − N e 1 2 3 2 iφ is the energy of the neutral fermion, . For the case when N iφ | − c e 2 ,m 2 e ¯ E 1 β , and 2 N iδ F . To present the corresponding relations, ii 2 ] 2 oscillations can be safely neglected. Only symmetry breaking, we introduce a small 1 ν iφ N , e 2 F − 2 3 R iφ M L M † e must be almost the same (baryogenesis con- e U N | ≪ | F − − 3 ,m 3 , but smaller and larger degeneracy works well – 7 – 1 = 2 α [2, 40, 41]. The baryon asymmetry generation ˜ à ↔ Φ M F ) m = [ e | 2 0 N 2 3 2 2 iφ i N will appear ( N M √ e and 5 F φ + diag( − is completely adequate, while if terms describing the interactions of ( (2 keV) 2 ≃ · 2 2 3 ǫ 10 ∗ 1 coherence effects are not essential and the description of without tilde) are related to M R ≃ N N , and sin ) will be created, while in a detector of size ( ∼ V 3 U < 3 2 3 , α | ≫ 2 2 ∼ ¯ 3 N = L 10 has the form M coming from the baryon asymmetry of the Universe, N and α ν 1, where + M ǫ ∼ > P f R 2 2 = 1(2) for the case of normal(inverted) hierarchy in active n 2 M U φl/L 1 the interaction of the mass eigenstates − N 1 N ≪ √ κ 2 2 following ref. [62]: can be expressed through the elements of ∆ ≪ φl/L ǫ ν M k /F | φ ≃− 3 . The masses M | )). For F 2 = N α 2 matrix E L 2 = F | (4 × ǫ M from the creation point an admixture of the ( = L/ GeV), where 2 L α f M M/ 100 GeV, then ( The mass eigenstates ( The fact that two heavy fermions are almost degenerate in mas To characterize the measure of the U(1) As a result, for As it was demonstrated in [41], the coupling constants 4 − =∆ mation, where the 2 sector. 10 matter), but also in the Yukawa sector, occurs most effectively if ∆ where the phases φ straint), ∆ for analysis of the experimental constraints. In decays of d also. effects are important, and order simple form, general conclusions remain thechange. same, In but this the work branchingnumber we rati symmetry also is neglect exact. all CP-violating effects where parameter (in the relativistic limit) distance coherent combination ( the process in terms of we parameterize particles of the SM must be included. Numerically, if ∆ which is irrelevant for us. there is a lower bound on elements of the active neutrino mass matrix this case will be considered in what follows. E < JHEP10(2007)015 o < + = αX atm 2 µ 1 13 U f (2.8) (2.6) (2.7) (2.9) m θ ( and to / 2 e ) 3 f < m δ − 3 al hierarchy 2 δ m leptons of the − 1 . The couplings ron and its neu- δ 1 ( i , δ 4 ie | x 35 . hown in [1, 4 – 6], the ± 0 e solar mass difference, 1 re almost degenerate and | the coupling of a neutral ≃ 1 2 . x : 1(2) for the case of normal is of experimental signatures | be derived for the case second and third generations ctions to relations (2.7), (2.8) α 2 | ≃ U 2 as before, we arrive at , x κ 2 1 12 − : is characterized by quantities θ | ≡ | 2 1 1 2 | , α α 2 : 1 V | : M 3, leading to p p ǫ . and for inverted hierarchy 2 2 = 0 | v − | 3 atm x 2 2 1 1+ ≃ α m can be expressed through the elements of the ± ) the active neutrino mixing matrix [63], and for the case of normal hierarchy and to U ≈ κ 12 – 8 – 1 | α | θ < m 12 2 f τ for the case of inverted hierarchy. θ ≃ 2 sol = 12 f 2 θ 1 the ratios of the coupling constant can be quite 2 | : m 1 2 F )R( 2 µ atm α , and all combinations of signs are admitted. For a ∼ f V 13 sin | m < m : 12 ǫ θ 2 θ 3 2 2 e / = 1 MSM there are two neutral leptons with almost identical f m m | ν 2 sol m 1 cos 1)R( α , ≈ m 2 3 3 U 2 δ m m | ). Taking the same value of τ i = f e 12 : , 05 eV and to q ) and for θ . ) 01 eV, so that other active neutrino masses are simply equal t ǫ . 2 3 0 µ ( δm δ 0 f is given by [41]: − O : ≃ 2 sin(2 δ ≃ F 2 generations are almost identical, but the coupling to elect e 1), so that 1 − 05. In other words, the coupling of the singlet fermion to the )diag(1 f . δ τ 1 δ 2 atm 0 23 2 sol ( ≪ i θ 25), depending on the value of unknown CP-violating phase m ǫ sin ( m ∼ 4, which is in agreement with the experimental data. For norm ie and − ∆ R ) ∆ ± to charged or neutral currents of flavour 2 τ µ π/ = = 174 GeV is the vev of the Higgs field and p 04 . In the case of the f q = = . to X = v x + p (0 = αX 2 3 µ = V , V f 2 The relations (2.6), (2.7), (2.8) form a basis for our analys The ratios of the Yukawa couplings The coupling For the case of inverted hierarchy two out of four solutions a 23 ∼ ( . All active neutrino masses are taken to be positive. As was s N / ) atm 2 sol 2 2 e τ , θ with where where trino can be enhanced orare suppressed of considerably. the The corre order of with a mass splitting choose for normal hierarchy f numerical estimate one can take [62] sin of where of heavy neutral leptons.lepton In most of the works the strength of different from those in eqs. (2.7), (2.8). (inverted) hierarchy. m first generation is suppressed,are whereas close the to couplings each to other. the one has [41]: and couplings (if active neutrino mixing matrix [41]. A simple expression can there are possibilities: m one of the activem neutrino masses must be much smaller than th f 0 JHEP10(2007)015 F . In 2 (2.11) (2.10) (2.12) . U g M below that to the third ) they were from BBN, the N /M . I we simply increase . This is always true, (GeV , h makes the search for hysics experiments. = 2 13 ations between Yukawa pling of , ogenesis the constant peak − , II and III with ratios of T = 2 for relatively small fermion sterile neutrinos are coming n, choosing the appropriate ed. This leads to the upper 10 = 1 come to thermal equilibrium antitative predictions we will d ): es peaks roughly at the tem- , κ (2.7), (2.8). × 3 3 . ree “extreme hierarchies”, when uld only happen if the hierarchy . 2 . , κ N CP-violating phases in the active can be derived as well. The max- , 8 , κ ¶ 2 > . 2 2 U ¶ 2 v and 2 M U M 2 GeV 2 F M N GeV µ 061 : 1 : 4 . = µ 0 1 : 16 : 3 52 : 1 : 1 11 2 8 | − ≈ ≈ ≈ − α 10 2 2 2 U τ τ τ – 9 – | 10 f f f × : : : × α κ 2 2 2 X GeV and for µ µ µ κ , otherwise 3 f f f . 2 6 3 ≡ 1 : : : − / < 2 1 2 2 2 e e e > are taken to be as small as possible compared to another 10 f f f 2 U 2 β U × f U MSM, a lower bound on , characterizing the breaking of the U(1) leptonic symmetry 2 , GeV) . ǫ ν α 1 f M/ ∼ < MSM be a very challenging problem, especially for large F ν 10 ( model I : model II : ∼ , which thus mostly determines the overall strength of mixin model III : γ 6= peak 1 GeV. On the basis of inequalities (2.11), (2.12) and limits MSM the constraint (2.12) is required to be valid. We will see T β ν < 6= N α M , γ f In the framework of the It is the smallness of the required strength of coupling whic As it was found in [41](see also [40, 2]), for successful bary Further cosmological constraints on the couplings of heavy The relations (2.7), (2.8) still allow a lot of freedom in rel Let us explain how these numbers were obtained. For the model = 1. This results in ǫ MSM can be probed (either confirmed or ruled out) in particle p imal value of the parameter neutral leptons of the is The overall strength of the coupling is given by above the electroweak scale andbound the baryon asymmetry is eras whereas the relations between different flavours follow from must be small enough, from BBN. The cosmologicalperature production rate [55] of these particl in thermal equilibrium in some region of temperatures aroun since in the the BBN constraints aremasses in fact stronger than those of (2.12) generation of leptons is stronger than to the others. This co ν couplings to different leptonic flavours, since theneutrino Majorana mass matrix are not known.consider three Therefore, sets to of present Yukawa couplings qu correspondingvalue to of th Yukawa constants in a maximal way thecombination value of of the signs coupling in constant to eq. electro (2.8). In case of model III the cou what follows we willcoupling refer constants to which these can be sets read as off benchmark from models eqs. I (2.7), (2.8 one JHEP10(2007)015 3 . ixing (2.13) (2.14) one can see leptons f magnitude 1. A special up to 450 MeV. x | µ ≪ U e ǫ U | ntitative predictions . and in the region 250 MeV hus provide non-trivial s for 2 e 9 71 U eqs. (2.11), (2.12) give at the maximal values of the − ents devoted to the search s of neutrino mixing (see, ≃ MSM predictions in case of ons reveals that for the mass 10 x within their error bars one 1 ν real and positive − ∼ 450 MeV are coming from the < ng Nucleosynthesis constraints in model II. thank F. Vannucci for providing us a 2 ! | 2 esting parameter range defined by atuions. For a given process, they ¶ e,µ MSM, constrainted already by cos- . 2 U ν M < | | 2 x x f [64] are ignored, our plots should be modified ¶ clusion plots for − x x 1 1+ , though not as strong as in [47], was presented up | 2 − e is given by U µ 1 1+ 2 roughly · | µ MSM. µ 6 – 10 – 12 ν ≃ θ /f . τ 2 9 2 2 | | f − 450 MeV, and phenomenologically viable region expands. | 2 GeV [48], whereas the NOMAD [49] and L3 LEP τ µ sin f f < 10 | | 3 ≃ · 2 N of CERN SPS experiment [47] contain the exclusion plots up to m 5 m 2 M < M à 35. This means that the ratio (2.13) can be as large as 4 450 MeV none of the past or existing experiments enter into . /κ < ≃ 0 2 2 2 | | ≈ e τ f f M > < U | | x 12 − in the region MSM region defined by eq. (2.11). The NuTeV upper limit on the m 10 7 ν · is given by − 2 | e 450 MeV (the NuTeV limit in this mass range is some two orders o /f τ f | The best constraints in the small mass region, The analysis of the published works of different collaborati These benchmark models are choosen to show the variety of qua 1, when relations (2.7) and (2.8) become invalid. The most recent published results = 2 GeV: 6 6 ∼ CERN PS191 experiment [46, 47], giving astrophysics, and observations ofshould neutrino be oscill confined between numbers given for benchmark model JHEP10(2007)015 , 6 − (3.1) 10 · are the 17 K . 5 − < M 1 s, to make a bounds on the . = N ,τ 2 M leptons (the details 140 MeV, for higher s lower utrino of Dirac type, < imental and BBN con- = f passing that it would be ed the life-time of sterile 140 MeV in order to have exceeds 0 . N roducts could change the ,µ nset of the BBN and the [55], expressed in terms of s ∓ for the models I-III. nsidered, while we discuss 2 > s µ nels for extra factor 2 in comparison 2 take into account that there l just require (conservatively) ve flavor. Here we took into ing from [46, 47] (this exper- ± he admitted window for the , xing angles, the same number N < M ´ studied in [55] and we will use U 5 -III described in section 2. π s (3.1) are valid in the models ,β M − 2 s → 10 ll below the equilibrium one so that its · + α β This means that we have exactly the 05 140 MeV the constraint on the mixing -meson. , N . π 1 7 ∓ of the heavy lepton must be smaller than e − N ± MeV) is in fact 2 times weaker than that of [55]. τ π = 2 3 = 568 → 2 Iβ − ,τ U N 1 = s τ = α = ,µ 1 µ s α 4, . 1 s for neutral fermions heavier than . 0 = 140 070 and . of sterile neutrino are applicable to our case. < 3 ,e 1 N − s τ Ref. [55] studied in detail only the mass range 10 MeV For the masses in the interval 10 MeV The successful predictions of the BBN are not spoiled provid First, we note that [55] considered the case of one sterile ne For various patterns of neutrino mixing we present the exper To consider higher masses we computed the life-time of heavy The concentration of the dark matter sterile neutrinos is we = 7 1 s to definitely avoid any situation when heavy lepton decay p e . 0 standard BBN pattern ofextremely light interesting element to abundances. repeat the We computation note of in [55] for masses these authors argued that the life-time existence may be safely neglected at this time. lifetime same number of degrees of freedom and that the constraints of a robust BBN constraints in this massthat range; meanwhile we wil limits on the parameters of the considerations coming from BBN allow to establish a number o couplings of neutralcouplings leptons and which masses of decrease the considerably neutral leptons. t neutrinos is short enough.products of their Then decays neutrinos thermalize.the decay This results question before of has their the been general o analysis for the casewhereas of we Models have I two Majorana sterile neutrinos. angle, based on a fit to numerical BBN computations [55], read with α one extra neutrino species, as explained in [55]); the limit where sterile neutrino mixaccount predominantly that with in only ref. oneneutrino acti [55] of Majorana neutrinos type, of hencewith Dirac the the total type Dirac width have contains case been an and co the constraint of conservative exclusion plot. The most important decay chan straints in figure 1.iment Note presented that 90% in confidence extracting the levelare limits exclusion two on plot) degenerate mix we neutrinos also in the two-body semileptonic ones for Dirac type sterile neutrinos. For the same value of the mi The limits (3.1) can be converted into limits on the mixing of computation can be found in section 4) and required that it JHEP10(2007)015 meson π rarchy and specific sses below the ologically allowed region of nel. Hence, the constraints 191 experiment (upper bound). c and Majorana cases, but in the for three benchmark models (I-III from left to right) – 12 – 2 | µ U | and | µ U . are in fact by a factor 2 stronger, since the number of decay 4 || | e 2 | U U | | µ , U 2 | | e U | and | µ U || e Limits on U | , 2 | e One can see that depending on the type of the neutrino mass hie U | of sterile neutrino helicity states are created in both Dira from BBN (lower bound) andBlank from regions direct are searches phenomenologically in allowed. the CERN PS former case only halfon of states contribute to each decay chan Figure 1: events is proportional to branching ratios in theparameter benchmark space models I-III can the be phenomen reduced or enlarged. Moreover, the ma JHEP10(2007)015 . . . + being 2 GeV 1 X < N N → 3 , 2 < M N π pion are certainly al- M decay with ude below the direct gnitude improvement clusion is true for the X of neutral fermions for + sible future experiments ne planned in JPARC, or ase. For heavier leptons imits to the upper limits π I and final states become impor- sponding branching ratios. We would also like to stress → n literature. For convenience e formulae for Dirac neutrinos s are also accounted resulting 9 + r prove or completely rule out endix A. Most of them (but not or neutral fermion masses above ted in Brookhaven and Frascati. K stics of thousand of billions charged ], where the coupling of sterile neutrino to aper contains a factor 4 error. In addition, are open; the modes like MSM with sterile neutrinos lighter than MSM with sterile neutrinos lighter than decay is not correct, see discussion in section 4. ν α ν ν ν 0 ν, πµ, Ke, Kµ,ην,ρν,... − 10 MeV) are unstable, since decay channels to π − e – 13 – µ + → & + e N N → M but still there are models where small regions of the N 8 1 can be quite different from (2.7), (2.8) leading to extra , ν, πe, µ β 0 ∼ ν α ǫ ν α ¯ ν µeν, π → and neutrino lifetime and plotted them in figure 4. → N 2 N U MSM with light inflaton BBN bounds are weaker and masses below ν In the next two sections we discuss the decays and production Above pion mass, the BBN limits are down to two order of magnit Decays of sterile neutrinos have been exhaustedly studied i Neutrino decays branching ratios for benchmark models I-II The improvement required to test the Note that our exclusion plot is different from that of ref. [57 For the 9 8 generation was not considered. Moreover, eq. (3.1) of this p MSM with sterile neutrinos lighter than 450 MeV. The same con lowed [41]. mass are excluded inparameter-space above most the cases pion mass are perfectly allowed. that the branching ratios for hypothetical long-lived neutral particle [65].kaons, With available stati in this experiment,ν one can expect to eithe are strongly suppressed. Hereafterin charge double conjugated rates mode formore Majorana decay modes neutrinos are as relevant, compared to Dirac c third stage of CERN NA48 experiment. a mass range up tothat could 5 GeV, allow to to enter understand into400 the MeV. interesting requirements parameter to space f pos 4. Decays of heavy neutral leptons Heavy neutral leptons we consider ( uncertainties. limits form CERNis PS191 required experiment, to thus either one-two confirm orders or of disprove ma the we present explicit formulae for relevantall) decay can rates be in be app obtainedpresented straightforwardly by in making ref. use [54], of th which we found to be correct. light active leptons, τ on overall mixing 450 MeV. For the three benchmark models we transfered these l special analysis of the availableIn data particular, on E787/E949 kaon Collaboration decays reported collec limit on 450 MeV can be done with either new kaon experiments, such as o the formula (21) of [56] for the probability of are plotted in figures 2,tant, 3. and one can For heavier use neutrino spectator many-hadron quarks to calculate the corre JHEP10(2007)015 , GeV N M (dashed light gray), (long-dashed black), πµ Kµ width is less than 10%. saw constraint (2.12) for , b) II, c) III; different lines 1 s may happen to be too as functions of neutrino mass . tween benchmark numbers. ovided careful study of pro- a given neutrino mass these 0 these tightest limits are used er limits and tightest among r the three benchmark models per limits) and dashed (lower henomenologically viable mod- < r-of-magnitude upper limit on XY N n overall strength of mixing. The ng dashed one(s) is disfavoured. → τ I MSM predictions beyond benchmark N ν plotted in figure 4b: in phenomenolog- 2 (short-dashed black), U , GeV N πe (short-dashed dark gray), M – 14 – ν ′ η (long-dashed gray). ρµ (solid black), is confined by corresponding solid and dashed lines. One 2 πν U 1 c, which guarantees that the results of standard BBN remain . 0 (solid dark gray), 140 MeV. In a given model the range of neutrino mass, where the < ην N & τ (short-dashed gray), N M ρe a) b) c) , GeV Branching ratios of neutrino two-body decays N M 1 GeV. However, it is worth noting that the limit for models with the same hierarchy in mixing as in models: a) I These limits imply limits on overall mixing Note in passing that as we already mentioned the (solid light gray), (solid gray), ∼ < N Figure 2: correspond to different modes: M Ke ρν Below 2 GeV theNeutrino contribution lifetime of is these constrained by modesresults limits for to models (2.11), I, (2.12) total II o and neutrino els III neutrino are presented lifetime in figure islimits) 4a: confined lines. in by p corresponding Theneutrino solid life horizontal (up time, solid lineintact indicates [55] the for orde corresponding solid line(s) is(are) above the correspondi ically viable models mixing can see that theM constraint from BBN is stronger than the see- conservative and can presumably becesses relaxed in primordial to plasma some in extent BBNwe epoch. pr give In upper what and follows, fo lowerlimits limits are on saturated various neutrino respectivelylower rates. by limits For on the neutrino tightest mixing,below. among presented in upp figure 4b. Only models could deviate to some extent from a naive interplay be JHEP10(2007)015 , , Q Q,L (5.1) p enta- (solid − e and + , GeV ross section νe N H,T p M , , H,L 2 Q,T p and heavy quark , b) II, c) III; different dir Q as functions of neutrino dp 10 H upper and lower bounds dσ Q,L reated in beam-beam and vy neutral leptons are the dp ABC · ) → re those which are stable with z ( I ymmetry. N (short-dashed gray). in a given experiment is determined H,Q − µ N D · + , differential cross section of their direct ) -quark production, νµ Q subsequently decaying into heavy leptons. H Q , GeV zp H N M − – 15 – MSM, which allows it to be falsified. Q ν p ( (sum over all invisible modes, solid black), δ · ννν dz 1 0 Z = 2 H,T dir ) describes the details of hadronization. The differential c H dp z ( dσ is a part of hadron momentum carried by heavy quark and a fragm H,L can be estimated by use of the factorization theorem H,Q H dp dir H D , GeV zp a) b) c) is differential cross section of direct N dσ Branching ratios of neutrino three-body decays (sum over two modes, long-dashed gray), M for models with the same hierarchy in mixing as in models: a) I dir Q N dσ νeµ are longitudinal and transverse spatial momenta of hadron M The spectrum of outgoing heavy neutral leptons We assume non-polarized beam(s) and target and hence axial s 10 Q,T tion function respectively; mostly by the spectrum of produced hadrons where p Figure 3: Since relevant hadrons contain one heavy quark production gray), mass lines correspond to different modes: ¯ At the same timeon for neutrino any set rates is of a parameters general the feature presence of of both 5. Production of heavy neutral leptons In high-energy experiments the most powerful sources of hea kinematically allowed weak decaysbeam-target of collisions. mesons (and Obviously,respect the baryons) to relevant c strong hadrons and a electromagnetic decays. JHEP10(2007)015 es for sterile q. (2.12), thick nerator PYTHIA al approximations ect tiny imprints of , GeV N M trino lifetime in models I lisions. The distribution b) , r hadrons, which give indirect t we are interested in hadrons ) hus apart of direct production perturbative QCD, while func- acc 2 H,T p ·L + ick solid black line indicates the upper 2 itude BBN bound on neutrino lifetime, H from eqs. (2.12) and (2.11), respectively. are accounted. b) Lower (solid lines) and . reads M nd to limits from direct searches for sterile 2 2 H,T ( · − 2 b z dir H dir H dp U − e dσ dN · H,L in models I (black), II (dark gray) and III (light a 2 2 Q dp 58 GeV ) . U – 16 – z m · = b − = 0 1+ b z (1 2 H,T dir ∝ H dp ) z dN 3 and ( H,L . D dp = 0 a 1 c, suggested in ref. [55], thin solid lines are limits from e , GeV . a) 140 MeV, thin dashed lines refer to limits from direct search N 0 > M < N N τ M ) comprises non-perturbative information. There are sever is an integrated luminosity of a given experiment and we negl z a) Upper (solid lines) and lower limits (dashed lines) on neu ) in literature, e.g. commonly used in high energy physics ge ( z ( acc 1 c for . H,Q L 0 H,Q D The rate of hadron production depends on the intensity of col < , s D N N τ adopts modified Lund fragmentation function [66] to gray). Thin solid lines and thick dashed lines depictτ limits Thick solid lines indicate lower limits from order-of-magn neutrinos discussed in section 3. entering the integrand in eq.(5.1)tion can be calculated within real bunch structure on outgoingstable hadronic with spectra. respect Note to tha they strong emerge and due electromagnetic to strong decays, and t electromagnetic decays of othe with default parameters dashed lines refer to eq. (2.11), thin dashed lines correspo neutrinos discussed in section 3; charge-conjugated modes upper limits (dashed lines) on overall mixing Figure 4: (black), II (dark gray)limit and from III BBN, (light gray); a horizontal th where of total number of directly produced hadrons JHEP10(2007)015 is a (5.3) (5.4) (5.2) H dN forwardly , aying kaons, 2 H,T H MSM) neutrinos dp -factor of hadrons can be correlated ν γ dN d class of the lightest H,L N agnetic decays travel p dp . This contribution is in enhancement of pure · f heavy neutral leptons. . H p ¶ by charged lepton masses. 11 used to calculate weak de- 2 ame and sum up all contri- N 2 H,T if typical ssed by phase volume factor; ature (see, e.g., refs. [52, 53]). frame there can be additional M oduction. As we explained in and ind ; the corresponding differential H dp π − ted in appendix B. Contribution N dN N M factors governing semileptonic width are H,L p 2 N ) is a differential inclusive branching E ≃ dp dE . · q N . . . + I ) MSM are likely to be heavier than pion. · are speed, lifetime and boost factor of a , ν N M + γ I . . . ± α n H 2 H,T l N N γ + ∓ dir − H ± α dp π l -boson. → N H N – 17 – dN W H,L p and → → H → ( dE L ± − dp H τ H H K K N ( = , p -boson (charged current): the only difference is that in dB H H µ β W ( δ 2 H,T dB · H H dp γ Z γ 1. Hence, in the laboratory frame, the distribution of heavy n · · dN 3 H,L d H H ≫ τ τ dp Z · H H × H X into heavy neutrino. These branching ratios can be straight β . The distribution of the total number of produced hadrons /M = H ind H H E is lighter than kaon, the dominant source of neutrinos is dec 2 N,T I dN = N N dp H γ dN N,L dp Produced hadrons stable with respect to strong and electrom The heavier the quark the lower its production rate; hence, a Also, in models with heavy neutrinos values of hadronic form 11 MSM neutrinos are massive. For heavy neutrinos this results The two-body decays (5.3) haveFor been convenience, already the studied differential in branching liter of ratio is three-body presen decays (5.4)as to the largest neutrino impact production they is give suppre a few per cent at If neutrino smearing with growth of statistics and can be also neglected branching ratio is presented in appendix B. ratio of hadron leptons over spatial momentum is given by distances of about given hadron) and thenIn decay the weakly, producing hadron some rest amount frame o the spatial momentum of heavy lepton is large, where we integrate overbutions unit from sphere all boosted relevant hadrons; to laboratory fr changed in accordance with shift in virtuality of leptonic decay modes which are strongly suppressed in the SM kinematically allowed hadrons saturatessection heavy 3, neutrinos neutrino in pr phenomenologically viable ν contribution are produced mostly via virtual obtained for each hadroncays with in help the of the framework standard of technique the SM. Indeed, in both models (MSM an with the hadron total spin.to Consequently, Lorenz in the boost laboratory contribution to correlations between sum of both contributions, JHEP10(2007)015 ts , are (5.5) (5.6) I and is -meson N α N D α l , and larger Λ g simple ap- Kπl | bu . Semileptonic → → V , dominate over I 5 GeV, neutrinos c I N D . ± α lN l ) N ∗ | ≫ | ( o mostly leptonic and armed hadrons, decays bc is more promising, but M t in general. For suffi- → D V | I tor and can be neglected. . ± on to heavy neutrino pro- . e decay Λ des, e.g. → r neutrinos leptonic modes lN D N sion of differential branching are not suppressed by CKM x. For neutrinos lighter than , , s (5.5) of charmed mesons are I B I by general formulae presented M artial width to heavy neutrinos I ) → 700 MeV, because the N N ve functions in the meson required to , N c α α l l α . ) B ∗ , which in turn can be estimated as -meson decays into heavy neutrinos N... N H K is given by the integration of eq. (5.2) Kl . → H -mesons, B 0 s -meson semileptonic decay modes give M → N D → → → ,f D D N . + H Q f K , D Br ( Br ( ≪ M · · I , D – 18 – B H I Q at N α N 700 MeV, N N I l /M ) α ′ N (5.6) is also presented. Both the rest of kinemati- ( B = . H -meson decays, α -production in hadronic collisions. Differential branch- φl η f X l ∗ s H D N = D → → 12 N → M -leptons (if kinematically allowed), which emerge as resul s s N φ, K s τ D D . N D = K because of both larger CKM-mixing, V I M -mesons. lN B -meson which leptonic decays → s . For order-of-magnitude estimates one can use the followin -boson in case of leptonic decay. D - and B N s W E D being a total number of produced hadrons and H γ N n In models with neutrino heavier than kaon but lighter than ch The total number of produced heavy leptons Note, that additional, but always subdominant, contributi In models where neutrino masses are within the range 2 GeV This is a consequence of strong overlapping between quark wa -production in hadron collisions is suppressed. For heavie 12 c produce virtual of those latter dominate neutrinois production. exhibited The by largest p mixing angles as compared to similar decays of three-body decay modes duction comes from decays of of decays of proximation, over leptonic mode ing ratios of the leptonicprovided by decays general and formulae the in semileptonicratio appendix decay B, to where the vector expres mesons are strongly suppressed by off-diagonalabout entries 2.5 of GeV CKM semileptonic matri modes to charm mesons, e.g. with are unsuppressed by CKM-mixingciently light as neutrinos, well, and are sub-dominan strongly suppressed by eitherThe CKM largest mixing contribution or from phase charmed volume baryons fac comes from th dominate. The baryon contribution is subdominant at any total production dominates over B cally allowed three-body decay modes and four-body decay mo values of hadronic form factors, contribution comparable to are produced mostlysemileptonic in decays, decays which of branchingin beauty ratios appendix are mesons. B. described These As compared are to als negligibly small for heavy neutrino production. JHEP10(2007)015 · on 4 . ) = 0 , GeV + N D . from figure 4b, M ) is a relative 2 → (dashed lines) as 15 H U . c . I 1 → . µN Q ) = 0 s mass region, where the → neutrino mass for three D ) = 0 s imits on branching ratios K B → c and Br ( → b Q Br ( nes: inside these regions all limits , the branching ratios are confined and lower limits on K Br ( , ) obtain for relevant hadrons M c 5 . , Λ . 4 . → N (solid lines) and ) = 0 c , GeV I M 0 = 0 N ¢ D eN . 0 M – 19 – B → π ). Following ref. [67] we set Br ( → c L ) = Br ( M → s K K b D in models: a) I, b) II, c) III. In a phenomenologically viable Br ( ¡ -quark hadronization. For strange meson the reasonable → N → Q s , c M 2 MSM the interesting branching ratios are confined between in = Br . ν the dominant contribution to heavy neutral lepton producti ¢ H + Q = 0 B ¢ → ) = Br ( + − → Q D K b ¡ → → Br c s ¡ , GeV N Br a) b) c) ) and assuming Br ( Branching ratios of decays is a total number of produced heavy quarks M 0 Q D N plotted in figure 4 are fulfilled, in a given model the neutrino → 2 For each heavy quark c U Figure 5: model and heavy neutrino mass within functions of heavy neutrino mass between corresponding thin and thick lines which show upper respectively. where Br ( weight of the channel estimate is Br ( corresponding thin (upper limit) andon thick (lower limit) li For beauty mesons we use [68] comes from leptonic andfor semileptonic relevant decays decays are ofbenchmark plotted mesons. models. in The figures Within l 5-15 as function of JHEP10(2007)015 (black I τN , GeV N → , the branching M s D (gray long-dashed D M I sting models where . µN N 2 GeV. In models with M → . eir heights are correlated . . D N π and it is a challenging task 2 N very clean signature of heavy M atistics is smaller. Contrary, M ics of billions charmed hadrons M . in models: a) I, b) II, c) III. In a MSM: produced charged leptons ne, is disfavoured. Rate doubling − ν thin es which show upper and lower limits 2 N ¶ M 2 l M − (gray solid lines), , GeV H (black long-dashed lines) and 2 N I N I M M 2 M – 20 – eN µN + → 2 H → M s D D µ s = | l p | MSM with neutrino of masses 0.5 GeV ν (black solid lines), I , GeV eN -factories. Note that the set of phenomenologically intere a) b) c) N Branching ratios of decays B → M s D from figure 4b, respectively. 2 The two-body decays can be searched for to probe U lighter neutrinos kaon decaysin are models important with and required heavierfor st neutrinos future statistics has to be larger The positions of theseobviously for peaks different modes in and charged mesons.leptons. lepton These From the features spectra plots is in a andis figure needed 6 th to one probe concludes that statist phenomenologically viable model and heavy neutrino mass wi short-dashed lines) as functions of heavy neutrino mass lines), Figure 6: corresponding thin line is below the corresponding thick li ratios are confined betweenon corresponding thin and thick lin due to heavy neutrino degeneracy is taken into account. are monochromatic with spatial momenta JHEP10(2007)015 in N M ss , GeV (gray long- N collisions is I , M 10 µN in and thick lines total pp (black long-dashed -10 → ) σ 7 I · c 5 B µN − N... s sections at large ener- 10 → on, 10 l neutrino production we → tely, as it requires billions ∼ B tion, but spectra of outgoing b ss be less promising probe of ys of strange, charmed and H → d be four orders of magnitude pp Br ( del and heavy neutrino mass within · ) H , σ ) within relevant ranges of neutrino (gray solid lines), I → total pp eN Q s, c, b σ · , GeV (black solid lines), → = 3 N Br ( I c − M – 21 – Q B ≡ eN 10 from figure 4b, respectively. ∼ → Q,H 2 c B U → pp , ξ , σ Q,H (gray short-dashed lines) as functions of heavy neutrino ma ξ I total pp H X τN σ · ≡ → 7 c / Q 1 ξ , the branching ratios are confined between corresponding th B (black short-dashed lines), B ∼ I M s , GeV a) b) c) τN . → . N Branching ratios of decays N pp → M N σ M M B With a reasonable estimate of strange, charm and beauty cros To illustrate the relative weight of different mesons in tota Semileptonic decays also contribute to heavy lepton produc . π MSM heavy neutrinos. Figure 7: required, while for heavier neutrinos the statistics shoul lines), dashed lines) and models: a) I, b)M II, c) III. In a phenomenologically viable mo one concludes that to produce a few neutrinos lighter than ka gies [67] masses which show upper and lower limits on plot in figure 16 the quantity leptons and mesons are notν monoenergetic, making this proce neutrinos are produced inof kaon kaons decays and can collected be World examined statistics comple is much larger. (where all consideredbeauty above mesons leptonic are and taken semileptonic into deca account, JHEP10(2007)015 4 GeV) (dashed I , GeV . N N πµN from figure 4b, M 2 M → U . K MSM with neutrinos ν ell as decays with more conclude that statistics e, if neutrino masses are to neutrinos in the frame- tios become too small, so rinos can be searched for: ore SM particles. se additional contributions to , the branching ratios are confined (solid lines) and K and lower limits on I M . πeN N → M , GeV K N . π M M in models: a) I, b) II, c) III. In a phenomenologically – 22 – N M 2 GeV) or even eight orders of magnitude (2 GeV . N , GeV N M a) b) c) M Branching ratios of semileptonic decays . In section 5 we presented plots with hadron branching ratios Note in passing that in our considerations baryon decays as w Figure 8: lines) as functions of heavy neutrino mass viable model and heavy neutrino mass within than three particles in aneutrino final production state are have expected been neglected. to be The insignificant. 6. Prospects for future experiments Generally, there are twoneutrino types production hadron of decays processes and where neutrino decays heavy into neut larger. works of the threeexpected benchmark at models. proposed From Super these plots B-factories one give can a chance to expl between corresponding thin and thick lines which show upper lighter than about 1 GeVin and 1-2 probe GeV some range. part of For parameter spac heavier neutrinos typical branching ra respectively; form factors are taken from refs. [68]. (0.5 GeV JHEP10(2007)015 ÷ nos 1 − (black I model and , GeV N πµN M → D am-target experiments etector aimed at searches from figure 4b, respectively; 2 ons. Heavy neutrinos from ncertainties in prediction of iate small distance from the U dump experiment secondary nce and then decay into SM ontribution to production of ivergence. In what follows we h lifetime in the range 10 rget, assuming that produced in (gray dashed lines) as functions of (black solid lines), ts on r interaction inside the target. So, I I KµN πeN → → , GeV D N D M , the branching ratios are confined between corre- D – 23 – M . N M . π , which estimate requires additional study. Heavier neutri M K (gray solid lines) and in models: a) I, b) II, c) III. In a phenomenologically viable I N < M M KeN N → M , GeV D N a) b) c) M Branching ratios of semileptonic decays s neutrino covers a distance in exceed of one kilometer, so a d For heavier neutrinos the most promising experiments are be 5 − Figure 9: dashed lines), heavy neutrino mass within heavy neutrino mass for neutrino decay signaturestarget should to be avoid placed decrease at ofconsider an statistics the appropr due experimental setup to with neutrinobeam-target appropriately beam collision thin d hadrons ta decay freely withoutthis furthe is not akaons classical interact beam-dump in setup. materialneutrinos For before with classical decay, beam- that change their c 10 sponding thin and thick lines which show upper and lower limi even with large number of available hadrons actually small u form factors are taken from refs. [69]. background can make any searches insensitive. with high intensity of adecays beam of and numerous secondary highparticles energy hadrons of with will incident branching travel prot some ratios dista discussed in section 4. Wit JHEP10(2007)015 in N M s , GeV N M (black, dark gray, in and thick lines , ) ,K ′ E ( η, η pp = M · , X ) ) , f the target is straightforward E I ( E ( e suppose that all beam protons del and heavy neutrino mass within Q XlN χ POT incident upon a target protons with N ≡ → · s ) ) ) POT ) is a total multiplicity (average number ) D E E E N ( , GeV E E ( ( ( ( Q N Q pp → M → total pp pp total pA M σ – 24 – pA σ σ σ . Assuming as a reasonable approximation at large · ≈ from figure 4b, respectively; form factors are taken from our results obtained below are valid for beam-dump Q ) 2 ) ξ POT U E K -mesons, which even in beam-dump setup decay before E ( N ( Q B → u,d,s,... X > M total pA pA = σ σ Q N - and M (solid and dashed lines), as functions of heavy neutrino mas D )= E e, µ ( N = N , the branching ratios are confined between corresponding th l D , GeV M Hence, for N a) b) c) Branching ratios of semileptonic decays . M is given by refers to the target material and N E A M The total number of neutrinos produced by . π where Figure 10: light gray lines), of secondary particles ininteract proton-proton once collision). in Here the w and target; results the in account effective of decrease in finite thickness o E models: a) I, b)M II, c) III.which In show a upper and phenomenologically lower viable limits mo on ref. [70]. are produced mostly by energy interaction. experiment as well. JHEP10(2007)015 I µN ∗ , GeV 58 24 10 82 i K B L p → model and h , GeV N D M , GeV 58 24 10 82 i D L p h . ) esented in table 1. To rameters of several experi- E 2K setup) and TeVatron , GeV from figure 4b, respectively; ( 44 24 68 8.5 i 2 pp K L y beams available today or -mesons we make use of the p U (gray dashed lines) as functions h (black solid lines), M B I · 8 7 I ) − − 10 12 [67] ts on eN − − E 10 10 φµN ∗ ( b · · 10 10 χ - and 3 2 K → D 3 s POT → − 4 5 3 D − − − N of 10 D · [67] · , GeV 10 10 10 i ) c · · · N χ 45 1 1 1 E . H,L ( M 0 , the branching ratios are confined between corre- p Q h D – 25 – χ 7 7 7 7 [66] · M / / / / 1 1 1 1 s Q χ . ξ N [68] s,c,b M 7 13 11 15 X = pp Q . (gray solid lines) and M I π 19 )= M in models: a) I, b) II, c) III. In a phenomenologically viable φeN E ( , 10 N 5 1 4.5 → N 100 M s N POT D N , GeV 50 400 120 800 , GeV N a) b) c) Branching ratios of semileptonic decays E M Adopted values of relevant for heavy neutrino production pa T2K [75] NuMi [74] CNGS [73] NuTeV [76] Experiment Table 1: ments. we arrived at Figure 11: Below we present thewill numerical be estimates available for in four the(NuTeV high nearest setup). energ future: The CNGS,estimate NuMi, relevant the JPARC parameters mean (T of longitudinal momenta these beams are pr (black dashed lines), heavy neutrino mass within of heavy neutrino mass sponding thin and thick lines which show upper and lower limi form factors are taken from refs. [71, 70]. JHEP10(2007)015 he = 3 for , GeV c N M model and heavy (black dashed lines), s) as functions of heavy l neutrino production KN gets for average neutrino → , with neutrino longitudi- τ . , , ¶ = 400 GeV [79] and from figure 4b, respectively. 2 2 H,L max H,L ¶ H N p p 2 E pp M M U E ≡ M F 1+ + 9 for . µ i , GeV N , x i · D,L (black solid lines), = 4 c M p ) h c H,L F – 26 – p µ πN h x 1 2 (sum over all active neutrino species, dark gray dashed 2 1 − → I = = τ , the branching ratios are confined between corresponding (1 i τ H νeN ∝ i M K,L → F . p N,L h τ dσ p N dx h M . π = 50 GeV as an factor-of-two estimate. In case of kaons we use t = 800 GeV [77, 78], M E E in models: a) I, b) II, c) III. In a phenomenologically viable (sum over all active neutrino species, light gray solid line N I , GeV M N 7 for a) b) c) Branching ratios of decays . M νµN (dark gray solid lines), → = 7 τ c ρN As we show in section 5, the dominant contribution to the tota = 120 GeV and → E estimate parameterization neutrino mass within neutrino mass with thin and thick lines which show upper and lower limits on in collisions come mostlynal from momentum two-body uniformly hadron distributedmomentum decays. in in hadron laboratory Thus rest frame frame one τ Figure 12: lines), JHEP10(2007)015 (dark -meson I B lN , GeV s N D from figure 4b, M → 2 s U B . . ) H H i N N L M → p H N h ǫ ide the detector · Q · , (black lines), to total neutrino production, H , the branching ratios are confined ) I i Br ( Q,H B N · ξ H and lower limits on · M ) N,L M DlN N... p E Q . is estimated as h ( , the total number of neutrino decays χ l → (solid, long dashed and short dashed lines) Q N → H ) χ H B M X · E X , GeV Q,H H · ( ) · . N l N e, µ, τ l E N π M N ( ∆ τ Br ( N ∆ τ = · M · pp – 27 – l · ) ) M E pp E in models: a) I, b) II, c) III. In a phenomenologically · ( ( ) M H N N · E N N ( M = = POT POT H N N ǫ N -meson decays are taken to be equal to the form factors for s = decays B N (light gray lines), -meson decays we adopted form factors from ref. [72]. )= c N I E B ( lN decays N c H η N N → c , GeV B N a) b) c) Branching ratios of semileptonic decays M being a relative contribution of a given hadron H N ǫ inside the fiducial volume is If neutrino decay length exceeds detector length ∆ decays from ref. [71], for between corresponding thin and thick lines which show upper respectively; form factors for viable model and heavy neutrino mass within as functions of heavy neutrino mass gray lines) and where the number of produced hadrons of a type with Figure 13: Finally we obtain for the total number of neutrino decays ins JHEP10(2007)015 of (dark -meson I onding B lN = 5m the , GeV ∗ s placed at a l N D with account from figure 4b, M T l → 2 s U B of off-axis detector. ble 1 and ∆ o beam spreading due to . 1 (black lines), . articular neutrino decay mode. I , the branching ratios are confined benchmark models T D l l lN and lower limits on (solid, short dashed and long dashed ∗ M ∆ D · . H → N i e, µ, τ H both lower and upper bounds scale with M B N L , GeV M = p N . one has to consider the realistic distributions h . With a detector of width ∆ l i π have to be corrected by a suppression factor M ∼ M in models: a) I, b) II, c) III. In a phenomenologically – 28 – N,T T p decays l N l N h ∆ N M · H H i i N T N L p p h h -meson decays are taken to be equal to the form factors for (light gray lines), s ≡ I B ζ -meson decays we adopted form factors from ref. [72]. c B J/ψlN → c B , GeV . One has to multiply these numbers by the value of the corresp 4 N a) b) c) Branching ratios of semileptonic decays U M from a target this is justified if . Similar suppression factor should be accounted for in case 2 l /ζ For the four available beams with parameters presented in ta Note in passing that in these estimates we neglected neutrin distance mixing as of all experimental and theoretical constraints; branching ratio (see plots in section 4), if interestednonzero in average a transverse p momentum quantitative predictions are given in figure 17 for the three decays from ref. [71], for between corresponding thin and thick lines which show upper Figure 14: lines) as functions of heavy neutrino mass Otherwise the predictions for neutrino signal order 1 Likewise, as the next approximation to respectively; form factors for viable model and heavy neutrino mass within gray lines) and JHEP10(2007)015 (dark I , GeV ρlN N → from figure 4b, M 2 B U ved pattern of neutrino In the present paper we ches for heavy neutrinos, . This studies are beyond es. Finally, in case of real (black lines), duction in decays of different ptons is an essential ingredient I tudies; they can be as large as , the branching ratios are confined e to focusing system. For a given D and lower limits on πlN (solid, long dashed and short dashed MSM, which can be used for their M ν → . N B e, µ, τ M , GeV = N . l π M in models: a) I, b) II, c) III.In a phenomenologically M – 29 – N M (light gray lines), I lN ∗ K → s B , GeV N Branching ratios of semileptonic decays a) b) c) M MSM. In this model these particles are responsible for obser ν lines) as functions of heavy neutrino mass between corresponding thin and thick lines which show upper 7. Conclusions A pair of relatively light and almost degenerate Majorana le respectively; form factors are taken from refs. [71, 70]. of hadron and neutrinoneutrino momenta experiment, additional instead suppression of can emerge theirexperiment du average with valu fixed geometrythese and suppression detector factors not canone-two tuned be orders obtained to of after sear dedicated magnitude,the s depending scope on of this the work. neutrino mass of the viable model and heavy neutrino mass within gray lines) and masses and mixings andanalysed for the baryon properties asymmetry of of the the singlet Universe. fermions of the experimental search. In particular, we discussed their pro Figure 15: JHEP10(2007)015 from 2 U , GeV de range of N M ary leptons is MSM the interesting ν , charm (dark gray lines) ound on the strength of xperiment, together with ormed some 20 years ago action the whole region of from neutrino experiments ed the latter constraint in -meson would require more t were able to enter into the terestingly, with an order of π gical considerations. l leptons should be larger than of neutral fermions, described in d c) III; within how upper and lower limits on , GeV N M – 30 – by cosmology (baryon asymmetry of the Universe) and from above collisions. We studied the decays of singlet fermions in a wi pp , GeV N MSM the strength of the coupling of singlet fermions to ordin a) b) c) Inclusive heavy lepton production by strange (black lines) ν M by neutrino oscillation experiments. In addition, a lower b The only particle physics searches for neutral fermions tha In the theoretical (BBN analysis) andmagnitude experimental improvement work. of the Quite bounds in on the strength of inter interaction comes from Bigsome Bang detail and Nucleosynthesis. showed thatfor We it masses analys is smaller stronger than than the 1 GeV. one coming cosmologically interesting region of massesthis and paper, couplings is theand CERN since PS191 then experiment no improvements [46,the of 47]. the BBN considerations, bounds It indicates were was made. thatthe perf the This pion masses e mass, of neutra though definite exclusion of the region below and beauty (light gray lines) hadrons in models: a) I, b) II an rates are between corresponding thin and thick lines which s bounded both from their masses and other parameters, consistent with cosmolo figure 4b, respectively. mesons and in below Figure 16: JHEP10(2007)015 , GeV , GeV N N M M . Black, dark gray and N volume for a) CNGS, b) b) d) M could be either ruled out mass nt improvement of the results ng thin (upper limits) and thick ectively; in phenomenologically viable – 31 – , GeV , GeV N N c) a) M M Number of sterile neutrino decays within 5m-length fiducial NuMI, c) T2K, d) NuTeV beams as a function of sterile neutrino Figure 17: masses below kaon mass can be scanned and the neutral leptons light gray lines refer tomodels benchmark the models number I, of(lower II limits) decay and lines. III, events resp are confined by correspondi or found in this mass range. It looks likely that the significa JHEP10(2007)015 imits should ! ¶ 2 2 l N ase volume factor M M esons would require . Vannucci for useful NGS, NuMI, T2K or , 1+ gs for the lepton mass responsible for neutrino µ ! ve accelerators would be 2 ettle down the question. 2 2 rse. Clearly, to study the ) , but with higher intensity N H l g machines can be used for cked them and found to be data of KLOE and of E949 straightforwardly by making ional Science Foundation, by 3 (DG), by the grants of the M M M 2 N dentical in Dirac and Majorana nment contract 02.445.11.7370) n beams. − stics would be required, calling + M 2 H ¶ 2 M 2 l N existing ( , , M M − 2 2 1 − ! ¶ 1 2 2 2 η 2 π N N µ !à M M M M 2 à ) · – 32 – l − − 3 N M 1 1 2 N M µ à have not been considered there. Our result for decay − · · 2 M H H f 3 3 ρν N N 2 | M M M ( ), and corresponding limit on neutrino mixing angle should H → 2 2 η π V 2 N | f f − N 2 F 2 F 2 F 1 /M G G G à 2 π 2 2 2 | | | and π π π v u u t M α α α ν 16 32 32 U U U 0 × − | | | π (1 = = / 2. Hence, the neutrino life-time used in ref. [55] to get BBN l → ) = / ¢ ¢ α α − α l N ν 0 ην + π H differs from the estimate used in ref. [55] by an additional ph → → ν → N 0 N π N ¡ Γ ( The search for the neutral fermions that are heavier than K-m In conclusion, it is quite possible that the already existin Two-body decay modes are ¡ Γ → Γ N and by a factor 1 experiments, and that the third stage of NA48 would allow to s of [46, 47] can be achieved by the reanalysis of the dedicated experiments similar toand the higher CERN PS191 energy experiment protonNuTeV beams. beams can We argued allowbelow that to charm, the whereas touch use for thenecessary. of going interesting the above range C charm of much more mixin intensi the search of theoscillations, physics dark beyond matter the andCP-violation Minimal baryon in Standard interactions asymmetry Model, of offor neutral the the leptons intensity Unive (rather more than stati energy) increase of theAcknowledgments proto We thank A. Dolgov,discussions. S. Eidelman, This work Yu. was supported Kudenko,the in P. part Russian Pakhlov by Foundation the and of Swiss F Nat BasicPresident of Research the grants Russian 05-02-1736 Federation NS-7293.2006.2and (Gover MK-2974.2006.2 (DG). A. Sterile neutrino decays Formulae for most decay rates presenteduse here can of be the obtained formulaecorrect. for Formulae Dirac for decay neutrinos rates fromcases. into fixed ref. Decays final [54]; states are we i che be divided by this factor. be multiplied by 2 JHEP10(2007)015 2 l [68]; x ¶ !! 4 2 L 2 ρ − ¢ , 1 1 M 2 ¢ 2 N q − 1 , ¢ − 4 l M 4 l ¢ lements. − 102 GeV x 2 l . 2 l x ¡ w x 4 l M θ 12 , x c 2 × = 0 cb − l log ) N B 2 l V ρ 480 1+ 4 l x g M , αβ M x + 12 à 2sin δ , , 2 ¡ 2 l · s π 12 2 2 ρ ≡ N us x ! f ! w 3 B l V [68], while for meson decay 230 4 θ − 2 2+10 2 M 2 M ρ C N ) 45 ¡ 2 8 l l . − 2 l M x 0 M + 1 + M x sin ub )+ 2 2 − N , x − − V µ B 190 2 1 q + ¶ 6 l αβ ) M , 1 =    ¢ ρ δ 2 x 2 l N 6 l 2 l = ′ à # αβ x η x M − · M 2 δ s M ( ¶ f cs , 4 + 8 · 12 D V L ! , (1 2 l 4 − 2 − − 280.1 · π ´ ¢ 2 ρ 2 x − N C 1 2 l ! 4 l f 1 1 1 8 ′ 4 l 2 x x µ , M 2 2 . M η N C x 4 q )+ − ( , 2 à 2 + | ¢ cd M M !à " · 1 − + 2 2 l ¤ αβ α V · 2 D ¡ = 1 − , C δ β x – 33 – 2 l 1 222.6 ) 3 l U N − 2 2 l 1 + 2 | 2 l | | η x ¢ − 1 − x q f 2 α α M à M w 2 2 α N · 1 à U U θ , M | (1 14 X − + ¡ · − us α N 4 · 3 · | · | · M l N 1+ ud 1 ρ V − K 3 − 2 N ¡ V ³ M 159.8 5 5 5 M M | N N N 3 3 3 1 2 l 4 l C M 2 l £ M ¡ π π π 2 F 2 F ( x x ′ x M M M 2 η 3 µ G G 2 F 2 F 2 F + f + 8sin ud − 2 2 ρ ρ 192 192 192 2 2 π − ρ ρ + 6 × + 4 ( V 2 F 130 G G G max 1 w g g 1 θ M M G à = = = = 2 2 2 2    | | | π π l v u u t π ´ ´ α α α x   8 β − β 16 32 U U U β × l | | | ν H 4sin ν H V + β + β , MeV = log β l l − = = = ¯ ν H α β L 1 α f ¢ ¢ ¢ ν = The values of meson decay constants and relevant CKM matrix e ¡ ν α α − − 6 α α l l ν ν 4 1 ′ → 0 + η α,β ρ X ρ = → is Fermi coupling constant, N 1 ³ → → → → N F C Γ ³ G N Table 2: N N N ¡ Γ ¡ ¡ Three body decay modes read  Γ Γ Γ Γ with and where hereafter and for CKM matrixconstants elements we we used use values most from recent ref. values from refs. [68, 80]. JHEP10(2007)015 n ¶ × 2 ¶¶ ¢ 2 (B.1) (B.2) ω 2 2 l H 2 q V , − M M ¢ − M . 2 2 2 ¶ N q Ω − ¢ 2 N 2 M 1 + ω M µ − + 1 2 N 2 l + 2 2 l + w N M 2 θ l H 2 M l M M 2 + M + M ¶ M + 2 2 l 2 K − 2 − l 2 H ´ ω 2sin M M 2 2 2 V ¡ N esented above multiplied 2 H M ¢ M − N − 2 w l ileptonic decays one has M M 2 M ¶ θ E H 2 ¢ 4 +2 M 2 Ω modes. − q ¶ 2 ¡ − 2 , M 2 2 2 2 q H N − c decays into sterile neutrinos is corresponding entry of CKM µ ω sin q q − N N ) ! ′ 2 M M + N 2 µ 2 1 E E 2 + − V ¶ q 4 4 2 2 − N 2 ( 2 ¢ µ M 2 HH 2 = V M Ω 2 f 1 q 2 ¡ l 4 + δ − V 4 M 4 f · µ µ ′ Ω M − M − · · 2 µ 4 2 + H 2 2 2 ¢ l N H 2 ¢ one obtains − 2 2 | µ l dq l 2 ¢¢ l M α M M M ¢ 2 2 N 2 V M M 2 4 q l U M Z | + M − 2 − + M − − | + × 2 2 F N 2 ¢ − H 2 2 2 , C H K H N 2 – 34 – 2 N N G 2 2 F 2 M V ¶ H 2 M | ¢ ω q | M M M G M M 2 ′ 2 2 l 3 H ¡ 2 w 2 M N N | ¡ − θ − π ¢¡ 2 3 ¶µ − M M E 2 2 q 2 2 4 l 2 HH M N π 2 q 4 64 q HV ³ V Ω q H | − M M · 32 V ¡ − − | π − · ¡ ) M + ¢¡ ) 2 2 8 · N H 2 2 2 π π + 2 2 2 l 2 2 | V H H q 2 4 2 q + 8sin | ( M M α f M M ( M ω M Ω α 2 M 2 − 2 F U w − 4 2 f U θ N N f + G · | ) are dimensionless hadronic form factors [68] can be found i + 2 − ) 2 µ 1+ 2 à · | E · 2 M 2 N 2 q H 2 2 4 2 µ K K ¡ q V ( τ H ¡ H is momentum of leptonic pair, ( 5 M τ dq − τ M M s 2 M + l f ¡ 2 f = 2 2 2 ) 3 3 f 2 = = ¡ ¡ Z 5 × f f M 2 ) N ), f ) ) p 1+4sin 2 − × + +2 × N ¡ q + +2 +2 N N ( + + α 4 1 l α l + α + ′ l p f = V l H N is the meson life-time [68]. N → N 3 = ( → → C + 2 H dE dE dE τ q H H H For three-body decays into vector mesons The Majorana neutrino total decay rate is a sum of all rates pr For differential branching ratios of pseudoscalar meson sem Br ( Br ( Br ( d d d literature. matrix and by a factor of 2, which accounts for charge-conjugated decay B. Decays into sterile neutrino Differential branching ratio ofreads pseudoscalar meson leptoni where where JHEP10(2007)015 ¶ ¢ !! 2 l (B.4) (B.3) 2 ρ M M + 2 2 2 N τ ¶ 2 N ! ¢ − M M , ) and vector 2 M ¶ , 2 2 q N ; form factors ¡ 4 − 2 q 2 τ N 2 f 2 ( V − M q 2 N 2 q M M 2 + 2 l M E − × 2 3 , A − M ) + f 1+ q 2 ´ A V 2 ¶ 2 1+ 2 ¢ 2 à q − H τ , = ¢ 2 · M l N ( µ ¶ 2 2 2 ρ τ l 1 5 2 · N M 2 M E τ N 2 ! 2 τ M − H M M 2 M 2 ! − M , A M M ) 2 M − H M ¡ ) 2 − − , f ) + 2 N N = 2 − 2 N q M 2 − 2 N 2 E l 2 3 N ( 1 M 2 2 2 N q τ 2 0 ¶ M 2 -lepton one has = M τ M M · ¡ +Ω E 2 ¶ 2 A + 2 2 τ l τ τ N − M ¡ 2 ¢ 2 + M − ¢¢ 2 τ )) N q H M 2 , f M 2 M 2 M N , − l N 1 ρ +2 2 N M 1 M 2 ¶ 4 l M A M M M − M q ( τ A M ( M − ¶¶ ³ 1 − · , + , and Ω − − 2 M − 2 ) 1 4 2 V 2 µ − 2 V q 2 τ l 2 N N 1 ¶ 4 V N µ Ω ! A 1+ 1 à M M ( M E ω M 2 · E N 2 M à µ N M 2 2 ¡ ¡ 3 · H + τ H !à − 2 M N 2 2 M !à + − l 3 − ¢ – 35 – τ + 2 2 ω M 1 M M 2 E 2 ) ¢ l + ¢ 2 ) + 2 H M 2 · N 2 M µ | l 2 τ τ N l 2 2 N +Ω M 2 ρ 2 H 2 τ H τ + 2 M )+ M − ud M M M M ¢ f M 2 2 − τ 2 M τ 2 V M M 2 2 2 M 2 2 M l | + | 2 + A N − ω − = ( 2 F M Ω 2 F 2 F − M − 2 2 − τ V H M 2 2 2 N G − M G ρ H 2 − G 2 V τ τ − q | ¡ 2 M M 2 1 − 2 2 ρ | 3 | 3 M 2 2 ρ M M 2 F M M − -lepton into heavy neutrino and meson we obtain 2 τ α A ( ( 2 V π g Ω π 2 ω − N M , f τ G − 2 4 2 U U 2 2 Ω N − | | − 1 − 2 M 2 − − ¡ 2 M ω | | V π · · 2 q 0 µ π 2 2 τ τ M N 1 1 N ¡ 8 τ τ ω q ¡ + A M Ω U 16 U E τ E τ × Ã Ã ¡ 4 | | 2 µ 2 µ V (2 2 · · µ à Ω + V Ω 5 v u u t v u u t 3 f = = δ δ 2 τ τ V f µ 1 M f H 2 · × · × τ τ 2 2 ) ) 2 ω f 1 M f f f − M ) as N N = = 2 + +2 +2 + + α α 2 H l l q ) ) = = ( ( τ α -lepton life-time. For three-body decays of M 1 4 ν ¯ ν V τ N f f N ρN HN = → is → N dE dE N → 2 τ τ τ → ω τ τ dE dE τ ) can be expressed via standard axial form factors For two-body decays of Br ( 2 Br ( d q Br ( d ( Br ( d i d form factor where f which can be found in literature. where JHEP10(2007)015 , . , , ¶ B 639 (1994) ]. ! 2 N Astrophys. (1999) 19. 72 , M τ , Astropart. M − 524 es , 3 How to find a 2 N Phys. Lett. E , + 2 ) N astro-ph/0603660 Phys. Rev. Lett. E ntribute to Majorana neu- The masses of active (2006) 133 τ Astrophys. J. , (2006) 1073 , − M 83 ov, τ and I. Tkachev, 368 M Does the Fornax dwarf spheroidal ( ]. ¶ à 2 l ]. ¶ ]. M τ ]. (2006) 261302 [ 2 τ JETP Lett. + M ]. ]. M , 97 2 N N The nuMSM, dark matter and neutrino masses E M 2 Lightest sterile neutrino abundance within the On the hadronic contribution to sterile neutrino 2 Sterile neutrino hot, warm and cold dark matter l − − – 36 – hep-ph/0505013 1 2 M N ]. µ ]. 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