An allowed window for heavy neutral leptons below the kaon mass
Kyrylo Bondarenko, Alexey Boyarsky, Juraj Klaric, Oleksii Mikulenko, Oleg Ruchayskiy, Vsevolod Syvolap, Inar Timiryasov
PPrepared for submission to JHEP
An allowed window for heavy neutral leptons below the kaon mass
Kyrylo Bondarenko, a,b
Alexey Boyarsky, c Juraj Klaric, b Oleksii Mikulenko, c,e
Oleg Ruchayskiy, d Vsevolod Syvolap, d and Inar Timiryasov ba Theoretical Physics Department, CERN, 1 Esplanade des Particules, Geneva 23, CH-1211, Switzerland b Institute of Physics, Laboratory for Particle Physics and Cosmology,´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland c Intituut-Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands d Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2010, Copenhagen,Denmark e Department of Physics, Taras Shevchenko National University of Kyiv, 64 Volodymyrs’kastr., Kyiv 01601, Ukraine
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The extension of the Standard Model with two gauge-singlet Majo-rana fermions can simultaneously explain two beyond-the-Standard-model phenom-ena: neutrino masses and oscillations, as well as the origin of the matter-antimatterasymmetry in the Universe. The parameters of such a model are constrained bythe neutrino oscillation data, direct accelerator searches, big bang nucleosynthesis,and requirement of successful baryogenesis. We show that the combination of allthese constraints still leaves an allowed region in the parameter space below thekaon mass. This region can be probed by the further searches of NA62, DUNE, orSHiP experiments. a r X i v : . [ h e p - ph ] J a n ontents The observed light neutrino masses and the baryon asymmetry of the Universe (BAU)remain some of the biggest hints pointing at physics beyond the Standard Model(SM). One of the simplest answers to both questions could be the existence of gauge-singlet Majorana fermions — also known as right-handed neutrinos, sterile neutrinos,or heavy neutral leptons (HNLs). HNLs can provide neutrino masses via the seesawmechanism [1–6]. Right-handed neutrinos can also be responsible for the generationof the BAU through the process known as leptogenesis , see, e.g. [7, 8] for reviewsand [9] for a recent update. The neutrino oscillation data requires at least two right-handed neutrinos. It turns out that the same HNLs with masses in MeV–GeV rangecan successfully generate the BAU [9–11]. This model can be viewed as a part of the ν MSM, where the third singlet fermion plays a roleof dark matter candidate [10, 12]. Dark Matter in the ν MSM can be produced resonantly [13–17], which requires large lepton asymmetry (see, e.g. the recent work [17] and references therein).Alternatively, it can be produced during preheating [18, 19]. – 1 –s a result of the seesaw mechanism, the SM neutrinos (flavor eigenstates) mix withthe light ( ν i ) and heavy ( N I ) mass eigenstates: ν Lα = V PMNS αi ν i + θ αI N cI , (1.1)where V PMNS αi is the PMNS matrix and the matrix θ αI characterizes the mixing be-tween the HNLs and flavor states. Somewhat surprisingly, the elements of θ αI arenot bounded from above by the neutrino oscillation data provided that certain can-cellations between these elements ensure the smallness of neutrino masses [20–28].However, these mixings are subject to other constraints. Let us briefly discuss theseconstraints using (1.1) as a starting point.First, not all values of θ αI are compatible with the neutrino oscillation data. Secondly,the mixing θ αI makes HNLs interact with intermediate vector bosons similarly toneutrinos, with couplings suppressed by θ αI . This opens the possibility for directlaboratory searches of HNLs at both intensity and energy frontier experiments [29–66]. Next, via the same interaction, the HNLs are produced in the early Universe.Their subsequent decays can affect Big Bang Nucleosynthesis (BBN). Recently it hasbeen realized in [67] that even a small fraction of mesons produced from HNL decaysleads to the over-production of the primordial helium. This in turn significantlystrengthens the BBN bounds. Let us stress that both experimental bounds andbounds from BBN are direct consequences of eq. (1.1) and thus come on an equalfooting . Finally, as we already mentioned, HNLs can be responsible for the generationof the BAU. In the past years, leptogenesis in this model has attracted significantinterest of theoretical community and several groups have performed studies of theparameter space [9, 16, 68–81].It is common to consider the case of two HNLs having nearly degenerate masses. Thereason for this choice is twofold. First, mass degeneracy allows for sizable mixingsin a technically natural way [20–28]. Secondly, low-scale leptogenesis in the case oftwo HNLs works for | ∆ M | (cid:28) M , see, e.g. the most recent analysis [9]. In this workwe also consider two degenerate HNLs N , and denote the average mass by M N .Since it is complicated to distinguish N and N , it makes sense to sum over themintroducing flavored mixing U α = | θ α | + | θ α | .Experimental constraints on HNLs from accelerators and astrophysics are usuallygiven in the form of pure mixing angles, which means that two out of three U α are considered to be equal to zero. However, this approach contradicts the mainHNL motivation: explanation of the neutrino oscillation data for which some specificmixing patterns are required. Therefore, to correctly combine constraints on HNLsone needs to reanalyze different experimental data within a specific HNL model.In this paper we perform such an analysis with two degenerate HNLs finding mini-– 2 –al experimentally allowed mass M N and allowed regions of the individual mixingangles U α .The paper is organized as follows: in Section 2 and 3 we discuss constraints on themodel from the neutrino oscillation data and accelerator searches. In Section 4 wediscuss constraints from BBN, taking into account the effects of mesons producedfrom the HNL decays, while in Section 5 we present the parameter region where BAUcould be successful. We combine these limits in Section 6 and give a discussion inSection 7. The experimentally observed neutrino oscillations cannot be explained within theStandard Model of particles physics where neutrinos are massless and the flavorlepton number is conserved. One of the possible ways to solve this problem is to addtwo right-handed neutrinos to the model. In addition to the Dirac masses m D = vF ,which couple the left-handed and right-handed neutrinos, the right-handed neutrinos,being gauge-singlet states, can also have Majorana masses M M , unrelated to the SMHiggs field. To find the physical states we need to diagonalise the full mass matrixof the left- and right-handed neutrinos L ⊃ (cid:0) ν L ν cR (cid:1) (cid:18) m D m TD M M (cid:19) (cid:18) ν cL ν R (cid:19) . (2.1)If the Dirac masses are small compared to the Majorana masses, we can block-diagonalise this matrix to find two sets of masses, m ν (cid:39) m D M − M m TD and M M (cid:39) (cid:18) M N M N (cid:19) . (2.2)This is the famous seesaw formula [1–6], in which the smallness of the light neutrinomasses m ν is explained by the parametrically small ratio of the Dirac and Majoranamasses m D /M M (cid:28)
1. Another consequence of the seesaw mechanism is that theheavy states (henceforth heavy neutral leptons – HNLs) are mixtures of the left-handed and right-handed neutrinos, and can interact with the rest of the SM – inparticular with the W and Z bosons. The strength of this interaction is given by the mixing angle : θ (cid:39) m D M − M m ν = θM M θ T . (2.3)The modulus squared of the mixing angle quantifies how suppressed the HNL inter-actions are compared to the interactions of the light neutrinos. It is often useful to– 3 –ntroduce the quantities U αI ≡ | θ αI | , U I ≡ (cid:88) α U αI , (2.4a) U α ≡ (cid:88) I U αI , U ≡ (cid:88) αI U αI , (2.4b)which quantify the overall suppression of the HNL interactions. If the HNLs are de-generate in mass, it is also useful to consider the sum over the HNL flavors (2.4b).It is important to note that the size of the observed neutrino masses m ν does notconstrain the mixing angles θ , nor the Majorana mass M M , but only their combina-tion from Eq. (2.3). This suggests that the seesaw mechanism does not imply a massscale for the heavy neutrinos. Nonetheless, using the seesaw relation (2.2), we canconnect the HNL mixing angles θ to the known neutrino oscillation data through theCasas-Ibarra parametrization [82]: θ = iV PMNS (cid:0) m diag ν (cid:1) / R (cid:16) M diag M (cid:17) − / , (2.5)where V PMNS is the PMNS matrix, m diag ν is the light neutrino mass matrix with m = 0 for normal hierarchy (NH), and m = 0 for inverted hierarchy (IH). Thecomplex matrix R satisfies the relation R T R = × , and depends on the neutrinomass hierarchy R NH = ω sin ω − ξ sin ω ξ cos ω , R IH = cos ω sin ω − ξ sin ω ξ cos ω , (2.6)where ω is a complex-valued angle, and ξ = ± U αI [84], in the degenerate mass limit it gives a lower bound on the summed mix-ing angles, see e.g. [79, 85]. If the HNL mixing angle is large U α (cid:29) m ν /M N , theexpression for U α is given by [14, 73, 74, 85, 86]: U α = | U α | + | U α | ≈ e ω M N (cid:16) m | V PMNS α | + m | V PMNS α | (2.7) − √ m m Im[ V PMNS α ( V PMNS α ) ∗ ] (cid:17) , (2.8)for the NH while for the IH case the r.h.s. of the corresponding equation is obtainedby replacing 2 → →
2. When we normalize the flavored mixing angles U α tothe total mixing angle U , the dependence on the unknown HNL parameters drops In the model with two HNLs which we consider here the lightest active neutrino is massless attree level [83] and therefore we use the term hierarchy rather than ordering. – 4 –H IHrel. param. min value rel. param. min value x e θ , θ θ , ∆ m sol . x µ θ , δ CP θ , δ CP . · − x τ θ , δ CP θ , δ CP Table 1 . Minimal values of x α = U α /U allowed by neutrino oscillation data for bothnormal (NH) and inverted (IH) hierarchies. The column “rel. param.” shows the mostrelevant neutrino oscillation parameters that change the minimal x α values. out, and the ratios U α /U depend only on the PMNS parameters [73, 87]. In whatfollows we will encounter these ratios very often, so we introduce x α ≡ U α /U , x e + x µ + x τ = 1 . (2.9)The Majorana phases entering the PMNS matrix, α and α (in PDG conventions),which also affect the ratios (2.9), cannot be determined in oscillation experiments,but could instead be measured indirectly through neutrinoless double beta decayexperiments in the near future (see, e.g. [88, 89]). In the limit of two HNLs, thereis effectively only one Majorana phase which we denote η . The phase η is equalto ( α − α ) / α / δ CP , angle θ and η , as for the remaining parameters the experimental un-certainty is sufficiently smaller and their variation only slightly change the allowedparameter space. For each point in the x e – x µ plane we find the smallest possible∆ χ ( δ CP , θ ) and take only the points with ∆ χ < x e can reach small values, while for inverted hierarchy all three x α can be small. Aswe will see later, the results for the minimal allowed HNL mass depend on thesesmall numbers, so one needs to determine them with a high accuracy. Therefore, weanalyzed minimal values of all x α , using the two-dimensional ∆ χ projection fromnuFIT data for the two most relevant parameters for each case. The minimal valueswithin 2 σ bounds are given in Table 1, c.f. [85].– 5 – igure 1 . 95% bounds for x α = U α /U for normal hierarchy (left) and inverted hierarchy(right) in the e ω (cid:29) η ∈ [0 , π ), while ∆ χ istaken for the measured values of the PNMS angles θ and δ , that affect the region moststrongly. Gray area corresponds to the forbidden region of the parameter space x e + x µ > x e and x µ , whereas the lower panel shows all three ratios,owing to x e + x µ + x τ = 1. There exist two types of accelerator experiments capable of searching for MeV-GeV mass HNLs. The first type is missing energy experiments searching for decays π/K → e/µ + (invisible) . The probability of these decays depends solely on U e/µ ,directly probing mixing angles independently on the mixing pattern. The boundsobtained in this type of experiments are generally stronger than for other types of– 6 –xperiments, however, they can only constraint HNLs with mass lower than kaonmass. In addition, they are not sensitive to combinations U α U β ( α (cid:54) = β ) and cannotconstrain U τ (because of large tau-lepton mass m τ > m K ). We use explicit boundsfrom: PIENU [91], TRIUMPH [92] ( π → e ), KEK [93], NA62 [64, 65] ( K → e/µ ),E949 [33] ( K → µ ). For NA62 K → µ decay only 30% of the current data has beenprocessed [65].The second type of experiments is displaced vertices search for appearance of SMparticles in the decays of long-lived HNLs. This type of experiments can probecombinations U α U β because production and decay channels can be governed by dif-ferent mixing angles. The relevant experiments are PS-191 [94, 95], CHARM [96],NuTeV [97] as well as DELPHI [98]. The experimental bounds for pure U e and U µ mixings are shown in Fig. 2. - - - - m N , GeV U e NA62 DUNEPS191 CHARM DELPHIPIENU &TRIUMPH KEK - - - - m N , GeV U μ E949PS191 NuTeV DELPHIKEK NA62 DUNE
Figure 2 . Accelerator bounds for U e (left panel) and U µ (right panel) for the HNL massbelow 5 GeV. Also, the expected DUNE sensitivity [99] is shown (dashed line). The displaced vertex experiments typically report bounds only on some mixings. Thereanalysis including bounds was done using GAMBIT in [100] for the general case of 3HNLs. We use these results from m N > . m N ≈ . − .
14 GeVin the U e bound we have included explicitly PS-191 results reanalyzed following theprescriptions given in [85]. The full set of bounds used in this work is shown inFig. 3.To combine accelerator limits with other constraints for a given mixing pattern x α we estimate the actual upper bounds on U . To find it we need to take into accountthat U τ is typically less constrained compared to U e , U µ , however large values of U τ (cid:29) U e,µ (i.e. x τ ≈
1) are not allowed by neutrino oscillation data. Therefore, foreach mixing pattern we compute the maximal mixing angle that does not contradictto any of the U α U β bounds using: U ( x α ) = min (cid:18) U e, acc x e , ( U e U µ ) acc √ x e x µ , U µ, acc x µ , . . . (cid:19) (3.1)– 7 – e2 U μ U τ U e U μ U e U τ U μ U τ - - - - m N , GeV U α U e2 U μ U τ U e U μ U e U τ U μ U τ - - - - m N , GeV U α Figure 3 . Full set of bounds used in this work for normal hierarchy (left panel) and forinverted hierarchy (right panel). For the U α U β (cid:54) = α and U τ bounds we use only GAMBITresults [100] starting from m N = 0 . Accelerator searches provide upper bound on HNL mixing angles (3.1). On the otherhand, a requirement that the presence of HNLs in the primordial plasma would notlead to the over-production of light elements (Deuterium, Helium-4) provides a lower bound on the HNL mixing angles. For HNLs heavier than π ± -mesons the strongestBBN bound of the HNL lifetime is due to n ↔ p meson driven conversion [67]. Pionsand kaons produced in HNL decays at the time when free neutrons are present inthe plasma modify the resulting freeze-out ratio of neutron to proton abundances,leading to a larger values of He abundance as compared to the Standard ModelBBN. If meson production is kinematically allowed, the following constraint can bederived [67]: τ N (cid:46) .
023 s1 + 0 .
07 ln (cid:18) P conv . N → h . Y N ζ − (cid:19) , (4.1)where P conv is the probability for meson to interact before decaying, Br N → h is thebranching fraction of semileptonic HNL decays producing a given meson h , Y N is theinitial HNL abundance, and ζ ≡ ( a SM /a SM+HNLs ) < P conv Br N → h a summation over meson species is assumed. Note the loga-rithmic dependence on these parameters, since for τ N (cid:28) . N → h only, since Y N ζ depends on processes at high temperature, where all lepton species are inequilibrium, and P conv is solely related to mesons. The value of Y N ζ varies in10 − − − , therefore we use the conservative lower bound Y N ζ = 10 − [67]. Interms of ( U e , U µ , U τ ) = U ( x e , x µ , x τ ), the branching ratio can be parametrized in– 8 – , mesons μτ e, no mesons. μτ τ HN L , s Excluded U e2 :U μ :U τ - :1:010 - :1:010 - :1:0 e, mesons μτ e, no mesons μτ τ HN L , s U e2 :U μ :U τ - :0:110 - :0:110 - :0:1 e, mesons μτ e, no mesons μτ τ HN L , s Figure 4 . Top panel : the lifetime bounds for the pure mixing cases.
Left panel : muonmixing with a small contribution of U e . Right panel : tau mixing with a small contributionof U e . the following way:Br N → h = (cid:88) X ∈ states with h n h ( X ) x e Γ( N e → X ) + x µ Γ( N µ → X ) + x τ Γ( N τ → X ) x e Γ( N e ) + x µ Γ( N µ ) + x τ Γ( N τ ) (4.2)where the notation N α corresponds to an HNL with the mixing angles U α = 1 and U β (cid:54) = α = 0, Γ( N α ) is the total decay width, Γ( N α → X ) is the HNL decay width intostate X , and n h ( X ) is the meson h multiplicity for the final state X . For a givenmixing pattern it is straightforward to compute the corresponding P conv Br N → h andsubstitute in (4.1).For pure mixing cases the bound is applicable only for HNL masses exceeding mesonproduction threshold: m N > m π + m e ≈
130 MeV for electron mixing, m N > m π + m µ ≈
240 MeV for muon mixing, m N > m η ≈
550 MeV for tau mixing.However, even small fraction of U e can relax this restriction to m N > m π + m e dueto logarithmic dependence on the total branching ratio, see examples in Fig. 4.– 9 –or the parameter region where the meson constraint does not work we use a con-servative estimate τ N < . Taking this into account the resultingexpression for the lower bound for U is U ( x α ) = 1 τ BBN N ( x α ) · (cid:80) α x α Γ( N α ) (4.3)where τ BBN N is given by the minimal value between the r.h.s. of (4.1) and 0 . The smallness of the light neutrino masses is not the only problem HNLs can solve,they can also explain the observed BAU through leptogenesis [102]. The conditionof reproducing the observed BAU [103, 104], n B n γ = (5 . − . × − , (5.1)imposes further constraints on the properties of the HNLs. When combined withthe bounds from the seesaw mechanism, leptogenesis imposes a strong constraint onmass spectrum of the HNLs, namely it forbids hierarchical HNL masses if the lightestmass is below 10 GeV [105]. This implies that (in the minimal model) any HNLswe can observe in the near future are degenerate in mass, and that leptogenesis isrealized either via a resonant enhancement in HNL decays [106–115], or via HNLoscillations [10, 116]. If we combine these two mechanisms, leptogenesis is possiblefor all HNL masses larger than ∼
100 MeV [9].Nonetheless, leptogenesis can also provide other interesting constraints on the HNLproperties. Phenomenologically, the most important constraint is the limit on themaximal size of the HNL mixing angles U [9, 14, 16, 72–74, 77, 79]. This limit arisesfrom the fact that for large mixing angles the HNL interactions become too fast, andthe lepton number reaches thermal equilibrium before the sphalerons freeze-out at T ∼
130 GeV.
Allowed flavor mixing patterns.
The upper bounds on U can have a strongdependence on the choice of flavor mixing pattern [73, 77, 79]. A tiny mixing with aparticular lepton flavor means that that lepton flavor will equilibrate more slowly inthe early Universe, and can thus prevent complete equilibration of lepton number.The allowed mixing patterns are almost completely determined by the low-energyphases as shown in Fig 1. This means that the leptogenesis bounds can also shiftas the neutrino oscillation data is updated. For example, in the case of inverted The actual estimate on the HNL lifetime depends on the maximally admissible value of∆ Y He /Y He . Here we use ∆ Y He /Y He ≤ .
35% used in [67] and adopted in this work. Ref. [101]refers a twice stronger bound, because it adopts tighter margin for Y He . – 10 –ierarchy, the choice of optimal phases corresponded to δ CP = 0 [73, 79], which isdisfavored by the latest fits of the light neutrino parameters [90]. U e / U U / U log U U e / U U / U log U Figure 5 . Flavor patterns consistent with both the neutrino oscillation data (as in Fig. 1),and leptogenesis for M N ∼
140 MeV. The upper bound on U depends on the ratios U α /U ,as this can prevent a large washout of the lepton asymmetries. The color coding indicatesthe maximal U for which baryogenesis via leptogenesis remains possible. We note herethat the experimental and BBN bounds on the mixing angles are not included in thesefigures, as in this range of HNL masses they completely dominate over the constraintsfrom leptogenesis, as shown in fig. 9. The HNL Mass splitting.
The mass splitting between the HNLs is one of the keyparameters determining the size of the BAU. The condition of successful leptogenesisconstrains the maximal size of the mass splitting (see, e.g. [9, 16, 72–74, 77, 79]),which can have direct consequences for the various lepton number violating signaturesat direct search experiments [28, 37, 117–122], or for the indirect signatures suchas neutrinoless double beta decay [74, 88]. Since the allowed parameter space for M N ≈
140 MeV is already quite constrained by the combination of past experimentsand leptogenesis, the condition of successful leptogenesis to determine the allowedmass splittings as shown in Fig. 6.
Our results are present in Figures 7 (for the normal hierarchy) and 8 (for theinverted hierarchy). For each mass and each flavor we show a set of admissiblemodels (green points). A model is selected as viable (a green point) if it explainsneutrino oscillations, provides correct baryon asymmetry of the Universe and satisfiesaccelerator and BBN constraints, see Section 6.3 for details. The blue curve shows minimal U α ( U ) compatible with BBN in the model with 2 HNLs explaining neutrino– 11 – M N [MeV]10 U Figure 6 . The allowed range of HNL mass splittings consistent with leptogenesis for abenchmark mass M N = 140 MeV. All points are consistent with the experimental con-straints. Interestingly, relatively large (1 MeV) mass splittings are allowed, which couldpotentially be resolved at experiments. Note that all mass splittings are large enough thatthe rates of lepton number violating and conserving decays are approximately equal. masses and oscillations. The blue curve does not take into account whether othermixing angles pass selection conditions or whether BBN curve is below the acceleratorcurve. The red curve shows the upper limit on U α (correspondingly, U ) compatiblewith accelerator searches and neutrino oscillations.Several comments are in order:1. Although most of the parameter space below ≈ −
360 MeV is closed, thereremains an open window of viable models for the normal hierarchy of neutrinomasses (see insets in Fig. 7). The corresponding HNLs have masses0 .
122 GeV ≤ M N ≤ .
143 GeV (6.1)and the mixing angles (without describing the actual shape of the region)10 − ≤ U e ≤ · − · − ≤ U µ ≤ · − − ≤ U τ ≤ · − The shape of the region can be seen in Fig. 7, while the specific (benchmark)points are listed in Table 2. – 12 – N [GeV] U e U µ U τ NH 0.122 3 · − · − · − · − · − · − .
33 8 · − · − · − IH 0 .
36 2 · − · − · − Table 2 . Benchmark models for the boundary masses of the allowed regions - - - - - m N , GeV U SeesawBBNAcc. exp.Baryogenesis NH - - - - - - - - m N , GeV U e - - - - - - - m N , GeV U μ - - - - - - - - m N , GeV U τ - - Figure 7 . The parameter space of the model with two HNLs. Green points are consistentwith all experimental bounds, explain neutrino data for the normal neutrino mass hier-archy (NH) and generate the correct BAU. Independent bounds for each flavor from theaccelerator experiments (red) and BBN (blue) are also shown.
2. Apart from this window, the lower mass of viable HNLs is given by M N > .
33 GeV normal hierarchy M N > .
36 GeV inverted hierarchy (6.2)3. For each individual flavor there are regions where accelerator bounds (red) areabove the BBN limits (blue), yet the point is white. This means that such amass is excluded by the combination of lower and upper boundaries for someother flavor. – 13 – .05 0.10 0.50 1 510 - - - - - m N , GeV U SeesawBBNAcc. exp.Baryogenesis IH - - - - - m N , GeV U e - - - - - m N , GeV U μ - - - - - m N , GeV U τ Figure 8 . The parameter space of the model with two HNLs. Green points are consis-tent with all experimental bounds, explain neutrino data for the inverted neutrino masshierarchy (IH) and generate the correct BAU. Other notations are the same as in Fig. 7. U e / U U / U log U Δχ Figure 9 . Left: the allowed mixing angles in the open window M N ≈ m π for NH, withall of the constraints applied. The point color code shows the values of U . Right: ∆ χ distribution in ( δ, θ ) planes. The region inside the red curve corresponds to the allowedHNL models. The CP-violating angle δ is in degrees. – 14 – .05 0.10 0.50 1 510 - - - - - m N , GeV U SeesawBBN A cc . + N A ( x ) Baryogenesis NH - - - - - m N , GeV U SeesawBBN A cc + N A ( x ) Baryogenesis IH Figure 10 . The allowed region of the parameter space for the model with two HNLs,including the projected increase of the sensitivity of the NA62 experiment. Green pointsrepresent models explaining neutrino oscillations, the BAU and are consistent with allexperimental constraints (taking into account the eight-fold increase of the amount ofdata collected by the NA62). The results are for the normal hierarchy (left) and invertedhierarchy (right). The minimal mass after the pion mass window M N ≈ m π (for NH)becomes M N = 0 . .
39) GeV for NH(IH). - - - - - m N , GeV U SeesawBBN A cc . + DUN E Baryogenesis NH - - - - - m N , GeV U SeesawBBN A cc + DUN E Baryogenesis IH Figure 11 . The allowed region of the parameter space for the model with two HNLs,including the projected increase of the sensitivity due to the DUNE experiment. Greenpoints are consistent with all experimental bounds, explain neutrino data for the normal(NH) or inverted (IH) mass hierarchy, and generate correct BAU. The projections for theDUNE experiment are based on [99]. The region around pion mass will be fully exploredand the minimal mass will be pushed up to M N (cid:39) .
39 GeV for both hierarchies.
To estimate the future sensitivity of the NA62 experiment, we assume that theexperiment will collect 8 times more data than has been published. Assuming thatboth data collection and analysis strategy will not significantly change in the futureand that no HNLs will be detected, the current limit can be scaled down as √
8, takinginto account that the HNL analysis is background dominated [64]. The result of this Indeed, the goal of NA62 is to collect 80 rare kaon decay K + → π + ν ¯ ν events [123]. The existingHNL constraint [64] are based on the dataset where only 9 . – 15 – .05 0.10 0.50 1 510 - - - - - m N , GeV U SeesawBBNAcc. + SHiPBaryogenesis NH - - - - - m N , GeV U SeesawBBNAcc. + SHiPBaryogenesis IH Figure 12 . Parameter space of the models with two HNLs including prospects of the SHiPexperiment. Green points are consistent with all experimental bounds, explain neutrinodata for the normal (NH) or inverted (IH) mass hierarchy, and generate correct BAU. Theaccelerator bounds include projections for the The minimal mass is M N ≈ . .
60) GeVfor NH(IH). procedure is shown as a red line in Figs. 10. We see that for the normal hierarchyfuture NA62 measurement will not explore the HNL mass “window” beyond the pionmass. The remainder of the allowed parameter space is pushed to a lower mass of M N (cid:38) . .
39) GeV for NH(IH).The DUNE near detector will be very sensitive to HNLs [99]. In particular, it willbe able to push the lower bound to M N = 0 .
39 GeV for both hierarchies and coverthe open window at lower masses. When estimating the sensitivity for DUNE wetook U e , U µ bounds as reported in [99] and derived U , U τ bounds consistent withoscillation data.The SHiP experiment [125] at CERN will provide unprecedented sensitivity for heavyneutral leptons in the mass range of interest. Using the sensitivity matrix, providedby the SHiP collaboration [49] we have performed a full scan in the ( M N , U , x e , x µ )space to find the allowed region (determined by the number of events n events < . M N (cid:38) . .
60) GeV for NH(IH). We note thatthis is a conservative estimate and the actual sensitivity will be even higher as ouranalysis only included HNLs coming from D-mesons [49], while the HNLs originatingfrom kaon decays will significantly increase the sensitivity [126].
Our procedure of finding viable HNLs models (green points) is as follows. We considertwo HNLs degenerate in mass that pass all of the following constraints :1.
The mixing angles U α ( x ) are chosen such that neutrino oscillation data issatisfied. This is ensured by the Casas-Ibarra parametrization (2.7). By varying– 16 –he CP phase δ and θ within their 95% confidence region (∆ χ < .
0) and bychanging the unconstrained Majorana phase η ∈ [0 , π ) we determine the regionof parameters ( x e , x µ ) admissible by the neutrino oscillation data.2. All U α ( x ) must be smaller than the corresponding accelerator limitsfor the flavor α . To ensure this we scan over the points in the ( x e , x µ ) planeconsistent with neutrino oscillation data and for each mass M N compute the upperbound U ( x ) from the accelerator experiments U ( x α ) = min (cid:18) U e, max x e , ( U e U µ ) max √ x e x µ , U µ, max x µ , . . . (cid:19) (6.3)The admissible mixing angles U α should be below x α U ( x α ).3. All U α ( x ) must be larger than the corresponding BBN bounds for thegiven flavor. To this end we find U ( x α ) = 1 τ BBN N ( x α ) · (cid:80) α x α Γ( N α ) (6.4)(where the quantities in Eq. (6.4) are defined in Section 4) and compute admissible U α ( x ) = x α U ( x )4. All U α are minimized/maximized independently with respect to x α .5. When we start to approach the seesaw line ( M U α ≈ (cid:80) i m i ) two HNLs mayin principle have different mixing angles, i.e. U α (cid:54) = U α and, correspondingly,different lifetimes. To probe the region near the seesaw bound, i.e. when U M ∼ ( m + m ) or, equivalently, when 2 Im ω → ω ∈ [0 , ln 100] , Re ω ∈ [0 , π )to ensure that the above conditions are satisfied by each of the HNLs.6. Finally we ensure that the observed value of BAU can be reproduced .To this end we numerically solve the quantum kinetic equations of ref. [9]. The idea that new particles need not be heavier than the electroweak scale, but rathercan be light and feebly interacting draws increasing attention of both theoreticaland experimental communities [see e.g. 57, 127, 128]. In particular, the idea thatheavy neutral leptons are responsible for (some of the) beyond-the-Standard-Modelphenomena has been actively explored in recent years, see e.g. [31, 40, 127, 129] andrefs. therein. This idea is motivated in the first place by the type-I seesaw model thatexplains neutrino oscillations. Furthermore, the same HNLs with nearly degeneratemasses in MeV–TeV range can explain the BAU [see e.g. 9] and refs. therein.– 17 –owever, while theoretical developments have been focusing on the models withtwo or more HNLs that are mixing with different flavors, the experimental searcheswere concentrating on a model with a single HNL mixing with a single flavor [32–34, 36, 38, 44, 47, 49, 50, 62, 64]. Such a model is simple for analysis and providesa number of useful benchmarks. Nevertheless, taken at face value it is incompatiblewith the observed neutrino masses and cannot generate BAU.In this paper we address this issue. We recast the existing accelerator and cosmolog-ical bounds to the model with 2 HNLs with degenerate masses. We perform a scanover all parameter sets of the two HNL model, that simultaneously: (a) explain neu-trino oscillations; (b) are consistent with all previous non-detections at accelerators; (c) do not spoil predictions of Big Bang nucleosynthesis; (d) allow for the generationof the baryon asymmetry of the Universe.Our main findings are as follows.1. For the inverted neutrino mass hierarchy, BBN and accelerator bounds overlapand leave no viable models for masses M N (cid:46) .
36 GeV.2. For the normal neutrino mass hierarchy, there is an open window for masses0 . − .
143 GeV and then for M N (cid:38) .
33 GeV.3. Future experiments, DUNE or SHiP, will be able to fully cover the region ofparameter space 0 . − .
143 GeV for all values of the mixing angle.4. The upper mass limit above 300 MeV will be pushed only slightly by DUNE orNA62, but will be moved beyond the kaon threshold by the SHiP experiment.
Acknowledgments
We would like to thank M. Ovchynnikov and M. Shaposhnikov for useful discusisons.This project has received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (GA 694896)and from the Carlsberg Foundation.
A BBN constraints on long-lived HNLs
If HNLs possess semi-leptonic decay channels and have lifetimes τ N (cid:38) .
02 sec, themesons from HNL decays completely dominate n-p conversion rates, driving neutron-to-baryon ratio X n (cid:39) . The resulting abundance of Helium-4 Y p (cid:39) X n is then Y p ≈
1, incompatible with observations that give Y p < . O (10 − ) sec. For such lifetimes the HNL decayproducts may not only affect the neutron abundance, but also destroy already syn-– 18 –hesized light elements (whose production starts at Hubble times around 40 sec) –the case that has not been analyzed in [67].Below we demonstrate that for all values of HNL masses/lifetimes compatible withneutrino oscillations, such HNLs lead to an overproduction of Helium-4 or other lightelements and therefore the region τ N (cid:38)
40 sec and M N > m π is also excluded fromBBN. The details of the analysis will be presented elsewhere [130].Indeed, all the neutrons in the primordial plasma will either decay or bind into lightelements (deuterium, Helium-3, Helium-4, etc). The presence of pions in the plasmaeffectively “prevents” neutrons from decaying because the rate of n + π + → p + π exceeds both the Hubble expansion rate and the decay rate n → p + e − + ¯ ν e . As aresult, decays (and other weak processes) can be ignored until Hubble times ∼ sec,leading to the following equation of neutron balance: X (free) n + X D + X + 2 X + · · · ≈
12 (A.1)The cross-sections of all reactions that change abundances in (A.1) ( n ↔ p con-version by pions, nucleosynthesis, dissociation of light nuclei by pions, etc) are ofthe same order. Therefore, the rates of various reactions are determined solely bythe concentrations. Without going into details (see [130]) there are two qualitativeregimes: if the instantaneous concentration of pions is n π (cid:38) n B – the pions willefficiently destroy the synthesized nuclei and all terms in the l.h.s. of Eq. (A.1) willend up being of the same order X (free) n ∼ X D ∼ X ∼ X ∼ O (1). If, on theother hand, the instantaneous concentration of pions is small, n π < n B , – most ofthe neutrons will bind into the nuclei, leading to X ∼ X (free) n (cid:28)
1. Bothcases are incompatible with experimentally observed abundances X D ∼ X ∼ − and X ∼ . Finally, few words should be said about long-lived HNLs with M N < m π . Theinfluence of such particles on BBN has been analyzed in a number of recent works [seee.g. 101, 131], providing an upper bound on the HNL lifetime that is below the seesawlimit. Near the seesaw boundary the HNLs are long-lived, so they can survive tillthe onset of nuclear reactions and their decay products can dissociate light nuclei.The recent analysis of [132] based on [133] demonstrated that MeV mass HNLs withlifetimes exceeding the seesaw bound are excluded from cosmological observations(BBN plus CMB) and therefore no “open window” exists below the seesaw line butabove the limits of [101, 131]. We remind that the mass fraction Y p = 4 X = 2 X n . – 19 – eferences [1] P. Minkowski, µ → eγ at a Rate of One Out of Muon Decays? , Phys. Lett. B (1977) 421–428.[2] T. Yanagida, Horizontal gauge symmetry and masses of neutrinos , Conf. Proc. C (1979) 95–99.[3] M. Gell-Mann, P. Ramond, and R. Slansky,
Complex Spinors and Unified Theories , Conf. Proc. C (1979) 315–321, [ arXiv:1306.4669 ].[4] R. N. Mohapatra and G. Senjanovic,
Neutrino Mass and Spontaneous ParityNonconservation , Phys. Rev. Lett. (1980) 912.[5] J. Schechter and J. Valle, Neutrino Masses in SU(2) x U(1) Theories , Phys. Rev. D (1980) 2227.[6] J. Schechter and J. Valle, Neutrino Decay and Spontaneous Violation of LeptonNumber , Phys. Rev. D (1982) 774.[7] S. Davidson, E. Nardi, and Y. Nir, Leptogenesis , Phys. Rept. (2008) 105–177,[ arXiv:0802.2962 ].[8] L. Canetti, M. Drewes, and M. Shaposhnikov,
Matter and Antimatter in theUniverse , New J. Phys. (2012) 095012, [ arXiv:1204.4186 ].[9] J. Klari´c, M. Shaposhnikov, and I. Timiryasov, Uniting low-scale leptogeneses ,[ arXiv:2008.13771 ].[10] T. Asaka and M. Shaposhnikov, The ν MSM, dark matter and baryon asymmetry ofthe universe , Phys. Lett. B (2005) 17–26, [ hep-ph/0505013 ].[11] L. Canetti and M. Shaposhnikov,
Baryon Asymmetry of the Universe in theNuMSM , JCAP (2010) 001, [ arXiv:1006.0133 ].[12] T. Asaka, S. Blanchet, and M. Shaposhnikov, The nuMSM, dark matter andneutrino masses , Phys. Lett. B (2005) 151–156, [ hep-ph/0503065 ].[13] X.-D. Shi and G. M. Fuller,
A New dark matter candidate: Nonthermal sterileneutrinos , Phys. Rev. Lett. (1999) 2832–2835, [ astro-ph/9810076 ].[14] M. Shaposhnikov, The nuMSM, leptonic asymmetries, and properties of singletfermions , JHEP (2008) 008, [ arXiv:0804.4542 ].[15] M. Laine and M. Shaposhnikov, Sterile neutrino dark matter as a consequence ofnuMSM-induced lepton asymmetry , JCAP (2008) 031, [ arXiv:0804.4543 ].[16] L. Canetti, M. Drewes, T. Frossard, and M. Shaposhnikov,
Dark Matter,Baryogenesis and Neutrino Oscillations from Right Handed Neutrinos , Phys. Rev.D (2013) 093006, [ arXiv:1208.4607 ].[17] J. Ghiglieri and M. Laine, Sterile neutrino dark matter via coinciding resonances , JCAP (2020) 012, [ arXiv:2004.10766 ]. – 20 –
18] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov,
Late and early timephenomenology of Higgs-dependent cutoff , JCAP (2011) 001,[ arXiv:1106.5019 ].[19] M. Shaposhnikov, A. Shkerin, I. Timiryasov, and S. Zell, Einstein-Cartan Portal toDark Matter , [ arXiv:2008.11686 ].[20] D. Wyler and L. Wolfenstein,
Massless Neutrinos in Left-Right Symmetric Models , Nucl. Phys. B (1983) 205–214.[21] R. Mohapatra and J. Valle,
Neutrino Mass and Baryon Number Nonconservationin Superstring Models , Phys. Rev. D (1986) 1642.[22] G. Branco, W. Grimus, and L. Lavoura, The Seesaw Mechanism in the Presence ofa Conserved Lepton Number , Nucl. Phys. B (1989) 492–508.[23] M. Gonzalez-Garcia and J. Valle,
Fast Decaying Neutrinos and Observable FlavorViolation in a New Class of Majoron Models , Phys. Lett. B (1989) 360–366.[24] M. Shaposhnikov,
A Possible symmetry of the nuMSM , Nucl. Phys. B (2007)49–59, [ hep-ph/0605047 ].[25] J. Kersten and A. Y. Smirnov,
Right-Handed Neutrinos at CERN LHC and theMechanism of Neutrino Mass Generation , Phys. Rev. D (2007) 073005,[ arXiv:0705.3221 ].[26] A. Abada, C. Biggio, F. Bonnet, M. Gavela, and T. Hambye, Low energy effects ofneutrino masses , JHEP (2007) 061, [ arXiv:0707.4058 ].[27] M. Gavela, T. Hambye, D. Hernandez, and P. Hernandez, Minimal Flavour SeesawModels , JHEP (2009) 038, [ arXiv:0906.1461 ].[28] M. Drewes, J. Klari´c, and P. Klose, On Lepton Number Violation in HeavyNeutrino Decays at Colliders , JHEP (2020) 032, [ arXiv:1907.13034 ].[29] A. Atre, T. Han, S. Pascoli, and B. Zhang, The Search for Heavy MajoranaNeutrinos , JHEP (2009) 030, [ arXiv:0901.3589 ].[30] G. Cvetiˇc, C. Kim, and J. Zamora-Sa´a, CP violations in π ± Meson Decay , J. Phys.G (2014) 075004, [ arXiv:1311.7554 ].[31] M. Drewes, The Phenomenology of Right Handed Neutrinos , Int. J. Mod. Phys. E (2013) 1330019, [ arXiv:1303.6912 ].[32] Belle
Collaboration, D. Liventsev et al.,
Search for heavy neutrinos at Belle , Phys.Rev. D (2013), no. 7 071102, [ arXiv:1301.1105 ]. [Erratum: Phys.Rev.D 95,099903 (2017)].[33] E949
Collaboration, A. Artamonov et al.,
Search for heavy neutrinos in K + → µ + ν H decays , Phys. Rev. D (2015), no. 5 052001, [ arXiv:1411.3963 ].[Erratum: Phys.Rev.D 91, 059903 (2015)].[34] LHCb
Collaboration, R. Aaij et al.,
Search for Majorana neutrinos in – 21 – − → π + µ − µ − decays , Phys. Rev. Lett. (2014), no. 13 131802,[ arXiv:1401.5361 ].[35] G. Cvetiˇc, C. Kim, and J. Zamora-Sa´a,
CP violation in lepton number violatingsemihadronic decays of
K, D, D s , B, B c , Phys. Rev. D (2014), no. 9 093012,[ arXiv:1403.2555 ].[36] CMS
Collaboration, V. Khachatryan et al.,
Search for heavy Majorana neutrinosin µ ± µ ± + jets events in proton-proton collisions at √ s = 8 TeV , Phys. Lett. B (2015) 144–166, [ arXiv:1501.05566 ].[37] G. Cvetic, C. Kim, R. Kogerler, and J. Zamora-Saa,
Oscillation of heavy sterileneutrino in decay of B → µeπ , Phys. Rev. D (2015) 013015,[ arXiv:1505.04749 ].[38] ATLAS
Collaboration, G. Aad et al.,
Search for heavy Majorana neutrinos withthe ATLAS detector in pp collisions at √ s = 8 TeV , JHEP (2015) 162,[ arXiv:1506.06020 ].[39] G. Cvetic, C. Dib, C. Kim, and J. Zamora-Saa, Probing the Majorana neutrinosand their CP violation in decays of charged scalar mesons π, K, D, D s , B, B c , Symmetry (2015) 726–773, [ arXiv:1503.01358 ].[40] F. F. Deppisch, P. Bhupal Dev, and A. Pilaftsis, Neutrinos and Collider Physics , New J. Phys. (2015), no. 7 075019, [ arXiv:1502.06541 ].[41] A. Caputo, P. Hernandez, M. Kekic, J. L´opez-Pav´on, and J. Salvado, The seesawpath to leptonic CP violation , Eur. Phys. J. C (2017), no. 4 258,[ arXiv:1611.05000 ].[42] J. Zamora-Saa, Resonant CP violation in rare τ ± decays , JHEP (2017) 110,[ arXiv:1612.07656 ].[43] SHiP
Collaboration, P. Mermod,
Prospects of the SHiP and NA62 experiments atCERN for hidden sector searches , PoS
NuFact2017 (2017) 139,[ arXiv:1712.01768 ].[44] V. V. Gligorov, S. Knapen, M. Papucci, and D. J. Robinson,
Searching forLong-lived Particles: A Compact Detector for Exotics at LHCb , Phys. Rev. D (2018), no. 1 015023, [ arXiv:1708.09395 ].[45] A. Das, P. S. B. Dev, and C. Kim, Constraining Sterile Neutrinos from PrecisionHiggs Data , Phys. Rev. D (2017), no. 11 115013, [ arXiv:1704.00880 ].[46] NA62
Collaboration, E. Cortina Gil et al.,
Search for heavy neutral leptonproduction in K + decays , Phys. Lett. B (2018) 137–145, [ arXiv:1712.00297 ].[47] A. Izmaylov and S. Suvorov,
Search for heavy neutrinos in the ND280 neardetector of the T2K experiment , Phys. Part. Nucl. (2017), no. 6 984–986.[48] S. Antusch, E. Cazzato, and O. Fischer, Sterile neutrino searches via displacedvertices at LHCb , Phys. Lett. B (2017) 114–118, [ arXiv:1706.05990 ]. – 22 – SHiP
Collaboration, C. Ahdida et al.,
Sensitivity of the SHiP experiment to HeavyNeutral Leptons , JHEP (2019) 077, [ arXiv:1811.00930 ].[50] CMS
Collaboration, A. M. Sirunyan et al.,
Search for heavy neutral leptons inevents with three charged leptons in proton-proton collisions at √ s =
13 TeV , Phys.Rev. Lett. (2018), no. 22 221801, [ arXiv:1802.02965 ].[51] D. Curtin et al.,
Long-Lived Particles at the Energy Frontier: The MATHUSLAPhysics Case , Rept. Prog. Phys. (2019), no. 11 116201, [ arXiv:1806.07396 ].[52] G. Cvetiˇc, A. Das, and J. Zamora-Sa´a, Probing heavy neutrino oscillations in rare W boson decays , J. Phys. G (2019) 075002, [ arXiv:1805.00070 ].[53] M. Drewes, J. Hajer, J. Klaric, and G. Lanfranchi, NA62 sensitivity to heavyneutral leptons in the low scale seesaw model , JHEP (2018) 105,[ arXiv:1801.04207 ].[54] S. Tapia and J. Zamora-Sa´a, Exploring CP-Violating heavy neutrino oscillations inrare tau decays at Belle II , Nucl. Phys. B (2020) 114936, [ arXiv:1906.09470 ].[55] I. Boiarska, K. Bondarenko, A. Boyarsky, S. Eijima, M. Ovchynnikov,O. Ruchayskiy, and I. Timiryasov,
Probing baryon asymmetry of the Universe atLHC and SHiP , [ arXiv:1902.04535 ].[56] C. Dib, J. Helo, M. Nayak, N. Neill, A. Soffer, and J. Zamora-Saa,
Searching for asterile neutrino that mixes predominantly with ν τ at B factories , Phys. Rev. D (2020), no. 9 093003, [ arXiv:1908.09719 ].[57] J. Beacham et al.,
Physics Beyond Colliders at CERN: Beyond the Standard ModelWorking Group Report , J. Phys. G (2020), no. 1 010501, [ arXiv:1901.09966 ].[58] G. Cvetiˇc, A. Das, S. Tapia, and J. Zamora-Sa´a, Measuring the heavy neutrinooscillations in rare W boson decays at the Large Hadron Collider , J. Phys. G (2020), no. 1 015001, [ arXiv:1905.03097 ].[59] D. Bryman and R. Shrock, Constraints on Sterile Neutrinos in the MeV to GeVMass Range , Phys. Rev. D (2019) 073011, [ arXiv:1909.11198 ].[60] P. Ballett, T. Boschi, and S. Pascoli,
Heavy Neutral Leptons from low-scale seesawsat the DUNE Near Detector , JHEP (2020) 111, [ arXiv:1905.00284 ].[61] T2K
Collaboration, K. Abe et al.,
Search for heavy neutrinos with the T2K neardetector ND280 , Phys. Rev. D (2019), no. 5 052006, [ arXiv:1902.07598 ].[62]
ATLAS
Collaboration, G. Aad et al.,
Search for heavy neutral leptons in decays of W bosons produced in 13 TeV pp collisions using prompt and displaced signatureswith the ATLAS detector , JHEP (2019) 265, [ arXiv:1905.09787 ].[63] M. Hirsch and Z. S. Wang, Heavy neutral leptons at ANUBIS , Phys. Rev. D (2020), no. 5 055034, [ arXiv:2001.04750 ].[64]
NA62
Collaboration, E. Cortina Gil et al.,
Search for heavy neutral lepton – 23 – roduction in K + decays to positrons , Phys. Lett. B (2020) 135599,[ arXiv:2005.09575 ].[65] E. Goudzovski, “Search for heavy neutral lepton production at the NA62experiment.” Preliminary results presented at ICHEP 2020, 28 July 2020 to 6August 2020, Prague.[66] J.-L. Tastet, E. Goudzovski, I. Timiryasov, and O. Ruchayskiy,
Projected NA62sensitivity to heavy neutral lepton production in K + → π e + N decays ,[ arXiv:2008.11654 ].[67] A. Boyarsky, M. Ovchynnikov, O. Ruchayskiy, and V. Syvolap, Improved BBNconstraints on Heavy Neutral Leptons , [ arXiv:2008.00749 ].[68] L. Canetti, M. Drewes, and M. Shaposhnikov,
Sterile Neutrinos as the Origin ofDark and Baryonic Matter , Phys. Rev. Lett. (2013), no. 6 061801,[ arXiv:1204.3902 ].[69] B. Shuve and I. Yavin,
Baryogenesis through Neutrino Oscillations: A UnifiedPerspective , Phys. Rev. D (2014), no. 7 075014, [ arXiv:1401.2459 ].[70] A. Abada, G. Arcadi, V. Domcke, and M. Lucente, Lepton number violation as akey to low-scale leptogenesis , JCAP (2015) 041, [ arXiv:1507.06215 ].[71] P. Hern´andez, M. Kekic, J. L´opez-Pav´on, J. Racker, and N. Rius, Leptogenesis inGeV scale seesaw models , JHEP (2015) 067, [ arXiv:1508.03676 ].[72] M. Drewes, B. Garbrecht, D. Gueter, and J. Klaric, Leptogenesis from Oscillationsof Heavy Neutrinos with Large Mixing Angles , JHEP (2016) 150,[ arXiv:1606.06690 ].[73] M. Drewes, B. Garbrecht, D. Gueter, and J. Klaric, Testing the low scale seesawand leptogenesis , JHEP (2017) 018, [ arXiv:1609.09069 ].[74] P. Hern´andez, M. Kekic, J. L´opez-Pav´on, J. Racker, and J. Salvado, TestableBaryogenesis in Seesaw Models , JHEP (2016) 157, [ arXiv:1606.06719 ].[75] T. Hambye and D. Teresi, Baryogenesis from L-violating Higgs-doublet decay in thedensity-matrix formalism , Phys. Rev. D (2017), no. 1 015031,[ arXiv:1705.00016 ].[76] A. Abada, G. Arcadi, V. Domcke, and M. Lucente, Neutrino masses, leptogenesisand dark matter from small lepton number violation? , JCAP (2017) 024,[ arXiv:1709.00415 ].[77] S. Antusch, E. Cazzato, M. Drewes, O. Fischer, B. Garbrecht, D. Gueter, andJ. Klaric, Probing Leptogenesis at Future Colliders , JHEP (2018) 124,[ arXiv:1710.03744 ].[78] J. Ghiglieri and M. Laine, GeV-scale hot sterile neutrino oscillations: a numericalsolution , JHEP (2018) 078, [ arXiv:1711.08469 ]. – 24 –
79] S. Eijima, M. Shaposhnikov, and I. Timiryasov,
Parameter space of baryogenesis inthe ν MSM , JHEP (2019) 077, [ arXiv:1808.10833 ].[80] J. Ghiglieri and M. Laine, Precision study of GeV-scale resonant leptogenesis , JHEP (2019) 014, [ arXiv:1811.01971 ].[81] S. Eijima, M. Shaposhnikov, and I. Timiryasov, Freeze-in generation of leptonasymmetries after baryogenesis in the ν MSM , [ arXiv:2011.12637 ].[82] J. Casas and A. Ibarra,
Oscillating neutrinos and µ → e, γ , Nucl. Phys. B (2001) 171–204, [ hep-ph/0103065 ].[83] S. Davidson, G. Isidori, and A. Strumia,
The Smallest neutrino mass , Phys. Lett. B (2007) 100–104, [ hep-ph/0611389 ].[84] M. Drewes,
On the Minimal Mixing of Heavy Neutrinos , [ arXiv:1904.11959 ].[85] O. Ruchayskiy and A. Ivashko,
Experimental bounds on sterile neutrino mixingangles , JHEP (2012) 100, [ arXiv:1112.3319 ].[86] T. Asaka, S. Eijima, and H. Ishida, Kinetic Equations for Baryogenesis via SterileNeutrino Oscillation , JCAP (2012) 021, [ arXiv:1112.5565 ].[87] A. Caputo, P. Hernandez, J. Lopez-Pavon, and J. Salvado, The seesaw portal intestable models of neutrino masses , JHEP (2017) 112, [ arXiv:1704.08721 ].[88] M. Drewes and S. Eijima, Neutrinoless double β decay and low scale leptogenesis , Phys. Lett. B (2016) 72–79, [ arXiv:1606.06221 ].[89] T. Asaka, S. Eijima, and H. Ishida,
On neutrinoless double beta decay in the ν MSM , Phys. Lett. B (2016) 371–375, [ arXiv:1606.06686 ].[90] I. Esteban, M. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou,
The fate ofhints: updated global analysis of three-flavor neutrino oscillations , JHEP (2020)178, [ arXiv:2007.14792 ].[91] PIENU
Collaboration, A. Aguilar-Arevalo et al.,
Improved search for heavyneutrinos in the decay π → eν , Phys. Rev. D (2018), no. 7 072012,[ arXiv:1712.03275 ].[92] D. Britton et al., Improved search for massive neutrinos in pi+ — > e+ neutrinodecay , Phys. Rev. D (1992) 885–887.[93] T. Yamazaki et al., Search for Heavy Neutrinos in Kaon Decay , Conf.Proc.C (7, 1984) 262.[94] G. Bernardi et al.,
Search for Neutrino Decay , Phys. Lett. B (1986) 479–483.[95] G. Bernardi et al.,
FURTHER LIMITS ON HEAVY NEUTRINO COUPLINGS , Phys. Lett. B (1988) 332–334.[96]
CHARM
Collaboration, F. Bergsma et al.,
A Search for Decays of HeavyNeutrinos in the Mass Range 0.5- { GeV } to 2.8- { GeV } , Phys. Lett. B (1986)473–478. – 25 –
NuTeV, E815
Collaboration, A. Vaitaitis et al.,
Search for neutral heavy leptonsin a high-energy neutrino beam , Phys. Rev. Lett. (1999) 4943–4946,[ hep-ex/9908011 ].[98] DELPHI
Collaboration, P. Abreu et al.,
Search for neutral heavy leptons producedin Z decays , Z. Phys. C (1997) 57–71. [Erratum: Z.Phys.C 75, 580 (1997)].[99] P. Coloma, E. Fern´andez-Mart´ınez, M. Gonz´alez-L´opez, J. Hern´andez-Garc´ıa, andZ. Pavlovic, GeV-scale neutrinos: interactions with mesons and DUNE sensitivity ,[ arXiv:2007.03701 ].[100] M. Chrzaszcz, M. Drewes, T. E. Gonzalo, J. Harz, S. Krishnamurthy, andC. Weniger, A frequentist analysis of three right-handed neutrinos with GAMBIT , Eur. Phys. J. C (2020), no. 6 569, [ arXiv:1908.02302 ].[101] N. Sabti, A. Magalich, and A. Filimonova, An Extended Analysis of Heavy NeutralLeptons during Big Bang Nucleosynthesis , JCAP (2020) 056,[ arXiv:2006.07387 ].[102] M. Fukugita and T. Yanagida, Baryogenesis Without Grand Unification , Phys.Lett. B (1986) 45–47.[103]
Planck
Collaboration, N. Aghanim et al.,
Planck 2018 results. VI. Cosmologicalparameters , Astron. Astrophys. (2020) A6, [ arXiv:1807.06209 ].[104]
Particle Data Group
Collaboration, P. Zyla et al.,
Review of Particle Physics , PTEP (2020), no. 8 083C01.[105] S. Davidson and A. Ibarra,
A Lower bound on the right-handed neutrino mass fromleptogenesis , Phys. Lett. B (2002) 25–32, [ hep-ph/0202239 ].[106] J. Liu and G. Segre,
Reexamination of generation of baryon and lepton numberasymmetries by heavy particle decay , Phys. Rev. D (1993) 4609–4612,[ hep-ph/9304241 ].[107] M. Flanz, E. A. Paschos, and U. Sarkar, Baryogenesis from a lepton asymmetricuniverse , Phys. Lett. B (1995) 248–252, [ hep-ph/9411366 ]. [Erratum:Phys.Lett.B 384, 487–487 (1996), Erratum: Phys.Lett.B 382, 447–447 (1996)].[108] M. Flanz, E. A. Paschos, U. Sarkar, and J. Weiss,
Baryogenesis through mixing ofheavy Majorana neutrinos , Phys. Lett. B (1996) 693–699, [ hep-ph/9607310 ].[109] L. Covi, E. Roulet, and F. Vissani,
CP violating decays in leptogenesis scenarios , Phys. Lett. B (1996) 169–174, [ hep-ph/9605319 ].[110] L. Covi and E. Roulet,
Baryogenesis from mixed particle decays , Phys. Lett. B (1997) 113–118, [ hep-ph/9611425 ].[111] A. Pilaftsis,
CP violation and baryogenesis due to heavy Majorana neutrinos , Phys.Rev. D (1997) 5431–5451, [ hep-ph/9707235 ].[112] A. Pilaftsis, Resonant CP violation induced by particle mixing in transitionamplitudes , Nucl. Phys. B (1997) 61–107, [ hep-ph/9702393 ]. – 26 – Heavy Majorana neutrinos and baryogenesis , Int. J. Mod. Phys. A (1999) 1811–1858, [ hep-ph/9812256 ].[114] W. Buchmuller and M. Plumacher, CP asymmetry in Majorana neutrino decays , Phys. Lett. B (1998) 354–362, [ hep-ph/9710460 ].[115] A. Pilaftsis and T. E. Underwood,
Resonant leptogenesis , Nucl. Phys. B (2004)303–345, [ hep-ph/0309342 ].[116] E. K. Akhmedov, V. Rubakov, and A. Smirnov,
Baryogenesis via neutrinooscillations , Phys. Rev. Lett. (1998) 1359–1362, [ hep-ph/9803255 ].[117] G. Anamiati, M. Hirsch, and E. Nardi, Quasi-Dirac neutrinos at the LHC , JHEP (2016) 010, [ arXiv:1607.05641 ].[118] J. Gluza and T. Jeli´nski, Heavy neutrinos and the pp → lljj CMS data , Phys. Lett.
B748 (2015) 125–131, [ arXiv:1504.05568 ].[119] P. S. Bhupal Dev and R. N. Mohapatra,
Unified explanation of the eejj , dibosonand dijet resonances at the LHC , Phys. Rev. Lett. (2015), no. 18 181803,[ arXiv:1508.02277 ].[120] S. Antusch, E. Cazzato, and O. Fischer,
Resolvable heavy neutrino–antineutrinooscillations at colliders , Mod. Phys. Lett.
A34 (2019), no. 07n08 1950061,[ arXiv:1709.03797 ].[121] J.-L. Tastet and I. Timiryasov,
Dirac vs. Majorana HNLs (and their oscillations)at SHiP , JHEP (2020) 005, [ arXiv:1912.05520 ].[122] S. Antusch and J. Rosskopp, Heavy Neutrino-Antineutrino Oscillations inQuantum Field Theory , [ arXiv:2012.05763 ].[123] A. Ceccucci et al.,
Proposal to measure the rare decay K + → π + ν ¯ ν at the CERNSPS , Tech. Rep. CERN-SPSC-2005-013. SPSC-P-326, CERN, Geneva, Apr, 2005.[124] NA62 Collaboration
Collaboration, C. NA62, , Tech. Rep. CERN-SPSC-2020-007. SPSC-SR-266, CERN, Geneva,Mar, 2020.[125]
SHiP
Collaboration, C. Ahdida et al.,
The experimental facility for the Search forHidden Particles at the CERN SPS , JINST (2019), no. 03 P03025,[ arXiv:1810.06880 ].[126] D. Gorbunov, I. Krasnov, Y. Kudenko, and S. Suvorov, Heavy Neutral Leptonsfrom kaon decays in the SHiP experiment , Phys. Lett. B (2020) 135817,[ arXiv:2004.07974 ].[127] S. Alekhin et al.,
A facility to Search for Hidden Particles at the CERN SPS: theSHiP physics case , Rept. Prog. Phys. (2016), no. 12 124201,[ arXiv:1504.04855 ].[128] R. K. Ellis et al., Physics Briefing Book , [ arXiv:1910.11775 ]. – 27 – The Role of sterile neutrinos incosmology and astrophysics , Ann. Rev. Nucl. Part. Sci. (2009) 191–214,[ arXiv:0901.0011 ].[130] A. Boyarsky, M. Ovchynnikov, O. Ruchayskiy, and V. Syvolap, Cosmologicalconstraints on very long-lived HNLs , to appear (2021).[131] O. Ruchayskiy and A. Ivashko, Restrictions on the lifetime of sterile neutrinosfrom primordial nucleosynthesis , JCAP (2012) 014, [ arXiv:1202.2841 ].[132] V. Domcke, M. Drewes, M. Hufnagel, and M. Lucente,
MeV-scale Seesaw andLeptogenesis , [ arXiv:2009.11678 ].[133] M. Hufnagel, K. Schmidt-Hoberg, and S. Wild,
BBN constraints on MeV-scaledark sectors. Part II. Electromagnetic decays , JCAP (2018) 032,[ arXiv:1808.09324 ].].