Neutrinos secretly converting to lighter particles to please both KATRIN and the cosmos
NNeutrinos secretly converting to lighter particles to please bothKATRIN and the cosmos a Yasaman Farzan and b Steen Hannestad a School of physics, Institute for Research in Fundamental Sciences (IPM),P.O. Box 19395-5531, Tehran, Iran . b Department of Physics and Astronomy, Aarhus University,8000 Aarhus C, Debmark . Abstract
Within the framework of the Standard Model of particle physics and standard cos-mology, observations of the Cosmic Microwave Background (CMB) and Baryon AcousticOscillations (BAO) set stringent bounds on the sum of the masses of neutrinos. If thesebounds are satisfied, the upcoming KATRIN experiment which is designed to probe neu-trino mass down to ∼ . T ∼ keV in the early Universe,leading to a much less pronounced suppression of density fluctuations compared to thestandard model. e-mail address: [email protected] e-mail address: [email protected] a r X i v : . [ h e p - ph ] O c t Introduction
In recent years various solar, atmospheric, long baseline and reactor neutrino experiments haveshown that the flavor of neutrino beams traveling over relatively large macroscopic distancescan change. Neutrino oscillation within the three neutrino scheme is given by two mass squaresplittings (∆ m and ∆ m ), three mixing angles ( θ , θ and θ ) and a CP-violating phase( δ D ). All these parameters, except for δ D and sign(∆ m ), have already been measured with aremarkable precision (see e.g. [1] for a recent overview). However, the overall scale of neutrinomass or in other words, the mass of lightest neutrino is not yet known.Information on the overall scale of neutrino mass can be obtained by measuring the distor-tion of the endpoint of the electron spectrum emitted in beta decay. The strongest bound sofar was obtained by the Mainz experiment by studying the endpoint of the electron spectrumin Tritium decay ( H → He + ¯ ν e + e ). The Mainz upper bound on neutrino mass is 2 . m ν e downto 0.2 eV at 90 % C.L. with a detection limit of 0.35 eV (5 σ ) [4] .On the other hand, nonzero neutrino mass can dramatically affect cosmological structureformation by suppressing the growth of fluctuation on scales below the free-streaming scale.A sum of neutrino masses saturating the bound from the Mainz experiment would have beeneasily visible in current Cosmic Microwave Background (CMB) and Large Scale Structure data.In fact, data from the Planck satellite mission measurements of the CMB [6] provide an upperlimit on the neutrino mass of (cid:80) i m ν i < .
71 eV, already close to the projected sensitivityof KATRIN. When auxiliary data from measurements of baryon acoustic oscillations (BAO)is also used the bound is strengthened to (cid:80) i m ν i < .
23 eV [6]. Even a sum of neutrinomasses as small as (cid:80) i m ν i ∼ .
06 eV, the minimum allowed in the normal hierarchy, leads toa suppression in power of several percent, enough to be seen by future high precision surveyssuch as EUCLID [7–10].Thus, the cosmological bound on the sum of masses naively implies that KATRIN willnot be able to discern the effect of neutrino masses. A measurement of a non-zero neutrinomass by KATRIN will therefore have profound implications for cosmology and particle physics,making it imperative to reconsider the standard assumptions on cosmic evolution and neutrinoproperties that have been made to derive the cosmological bound on neutrino masses. Inthis paper we propose two possible scenarios which make the relatively large neutrino massesmeasurable at KATRIN compatible with cosmological bounds by introducing new particlescoupled to neutrinos. The scenarios are based on the following mechanism: After the Big BangNucleosynthesis (BBN) era and before recombination epoch (eV < T < T ∼ e V in the early universe. On the other hand, remaining neutrinos and thenew particles should freely stream at the recombination era ( T ∼ . T >
MeV), the new particles will contribute to extra relativistic KATRIN is scheduled to start taking data in 2016 [5]. (cid:28) T (cid:28) T >
T <
10 eV butstill achieve efficient conversion in the interval between these two epochs. In section 3, we willpresent low energy models within which such resonances can occur and discuss the bounds fromvarious cosmological and astrophysical observations as well as terrestrial experiments on theparameters of the model. In section 4, we discuss how many new degrees of freedom are requiredto make cosmological bounds on sum of the masses of neutrinos compatible with relatively largeneutrino mass measurable at KATRIN. Our findings are summarized in section 5.
There are (at least) two possibilities to convert active neutrinos to lighter species at temper-atures T (cid:46) m e : (1) new coannihilation modes of active neutrino pairs and (2) scattering ofneutrinos off dark matter. In this section, we first briefly discuss the conversion mechanism foreach case and then discuss the general effects of back reaction for both cases. In the end, wequantify the effective number of massive neutrinos after conversion.Let us first discuss the case of neutrino pair coannihilation. If the mass of intermediatestate responsible for coannihilation ( m X ) is much larger than the temperature, the coanni-hilation rate will be proportional to T /m X which should be compared to Hubble expansionrate T /M ∗ P l . Coannihilation would be therefore more efficient at higher temperatures whenneutrinos were still in thermal equilibrium and conversion to lighter new states would enhancethe number of extra relativistic degrees of freedom on which there are strong bounds [12]. Onthe other hand, at T (cid:29) m X , the coannihilation rate will be proportional to T . Comparing tothe Hubble expansion rate then implies that the coannihilation becomes more efficient at lowertemperatures so there would be no danger of producing extra relativistic degrees of freedom be-fore neutrino decoupling. However, through the same interactions, neutrinos and new particlesproduced by coannihilation will scatter off each other with rate again given by T . Comparingto the Hubble expansion rate T /M ∗ P l , we find that scattering becomes more important at lowertemperatures. As a result for m X < m X > m X < < m X < X particles can be resonantly produced and subsequently decay intonew light states for T ∼ m X converting a substantial fraction of active neutrinos to lighterstates. In the appendix, using narrow width approximation, we calculate the conversion rate( R ) of a neutrino with a given momentum coannihilating with any other neutrino in a medium.Using the formulas in the appendix it is straightforward to show that the coupling of resonantstates to active neutrinos has to be larger than 5 × − ( m X / keV) / to fulfill the requirementfor efficient conversion; i.e., R · H − >
1. Because of resonance enhancement, coupling sosmall will be enough to efficiently convert neutrinos at T ∼ m X . However, at T (cid:29) m X or at2 (cid:28) m X , this new coupling will be irrelevant because (i) it cannot give rise to a significantdeviation of N eff from 3 and (ii) it cannot hinder the free streaming at recombination era. Inthe next section, we will present two models within which the resonant conversion scenario canbe naturally embedded.Let us now discuss scattering of active neutrinos off background Dark Matter (DM) particles i.e., ν + DM → f + DM where f is the final particle which is even lighter than neutrinos.For DM mass larger than MeV, during epoch of our interest, DM particles are non-relativistic.We generally expect ( e.g., within thermal freeze-out scenario) that the number density of DMparticles has been fixed by T ∼ ρ ∼ keV cm − , the number density of DM particles at T <
MeV is givenby n DM = ρ m DM T T . (1)In general, we expect the scattering cross section ( σ scatt ) to be proportional to E ν / ( m DM − m X ) .The scattering rate will then be given by n DM σ scatt ∝ T / [ m DM ( m DM − m X ) ]. Comparingto H ∼ T /M ∗ P l , we find that the scattering would be more efficient at higher temperatureswhen neutrinos have not decoupled so the scattering would contribute to extra relativisticdegrees of freedom. However, if the splitting between m DM and m X is small (eV < T ∼ ( m DM − m X ) / (2 m DM ) < MeV), there can be resonant production of X which like the caseof coannihilation can satisfy the bounds. However, such fine tuned splitting between X andDM is theoretically difficult to explain, especially that since they couple together to neutrinos,one should be boson and the other should be a fermion. Taking m φ − m DM ∼ m DM /
20 and E res ∼
100 keV, we find m DM ∼ few MeV. Larger m DM and/or smaller E res require higherdegree of fine-tuning between m φ and m DM . Moreover for couplings large enough for efficientconversion, DM pair annihilation at T ∼ MeV can produce and thermalize f particles beforeneutrino decoupling era. The produced f ¯ f will contribute to effective relativistic degrees offreedom on which there are strong bounds.Because of issues enumerated above, we shall not try to build a model to embed the resonantscattering off DM scenario. It is however instructive to discuss the back reaction for thisscenario: i.e., f + DM → ν + DM. During the period m ν (cid:28) T < m DM , the masses of ν and f as well as the recoil energy of DM can be neglected: E ν (cid:39) | (cid:126)p ν | (cid:39) E f (cid:39) | (cid:126)p f | . Moreover, for s -wave interactions, the spin of f and ν have to be the same, too. As a result, the cross sectionof scattering and back scattering will be equal σ ( ν + DM → f + DM) = σ ( f + DM → ν + DM).As a result, the mean free path of ν and f will be equal. Moreover, these interactions donot change the number density of DM. If the number of scatterings that a neutrino undergoesis k , its contribution to ν and f population will be respectively equal to (1 + ( − k ) / − ( − k ) /
2. Suppose during a certain period of time (macroscopically large time scale butmuch smaller than H − ), the average number of interactions that a neutrino undergoes is λ .The distribution of number of scattering k will be given by Poisson distribution so the averageprobability of neutrinos not to be converted will be12 < ¯ p = ∞ (cid:88) k =0 λ k e − k k ! 1 + ( − k e − λ ≤ . For λ (cid:29)
1, ¯ p will quickly converge to 1 /
2. This is the limit that f reaches thermodynamicalequilibrium with ν and as a result, the entropy of neutrinos will be shared with f . Since3hey are both fermions, their share of entropy will be equal so ¯ p = 1 / ν + ( − ) ν → f + ( − ) f ) is more com-plicated. When the temperature just approaches to the resonance ( T → m X ), the density offinal states is still low so back reaction is negligible. Eventually when a significant fraction ofneutrinos convert into f particles, their density will become large enough to make the backreaction efficient. Let us take R to be the rate of scattering of ν off any of neutrinos in theensemble. Three regimes can be distinguished: 1) If (cid:82) R dt (cid:28)
1, the back reaction can beneglected. 2) If (cid:82)
R dt ∼
1, the back reaction is important but thermodynamical equilibriumhas not been reached yet; 3) If (cid:82)
R dt (cid:29)
1, the reaction and back reaction rates will becomeequal. Obviously, in neither of these cases, it is possible to completely remove neutrinos. Wewill focus on the third possibility in this paper. Notice that the energies of initial and finalstates in ν + ( − ) ν → f + ( − ) f are the same. Since each neutrino in the medium undergoes reac-tion, we expect the energy distribution of ( − ) f particles to be similar to those of ( − ) ν . T -reversalsymmetry implies that σ ( ν + ( − ) ν → f + ( − ) f ) = σ ( f + ( − ) f → ν + ( − ) ν ). The equality of reaction andback reaction rates therefore implies that the number density of ν and f should be equal.In all of the above cases, it is possible to further suppress the final density of neutrinosby converting the produced f (or ¯ f ) to other new states that do not interact with neutrinos.Intuitively, this can be understood the following way: If f particles are eliminated before theyfind enough time to reproduce active neutrinos, conversion of neutrinos will be more efficient.Elimination of new states can proceed via a number of processes; e.g., f particles can oscillateto new particles or they can decay into new particles. However, the above argument aboutback reaction applies here, too. If the process is fast enough to remove significant fraction of f particles, the back reaction will be efficient in reproducing them. If all these processes cometo equilibrium, the final density of neutrinos will be reduced by a factor of ρ massive , final ρ massive , initial = 33 + N (2)where N is the number of degrees of freedom that come to equilibrium with neutrinos below T ∼ N .In summary, we discussed the possibility of converting active neutrinos to lighter new par-ticles through resonant neutrino (antineutrino) pair coannihilation or resonant scattering ofneutrinos off the dark matter particles. In case of coannihilation, this requires the intermediatestate to have a mass in the range of 100 eV-100 keV. For neutrino scattering off DM, the inter-mediate state has to be quasi-degenerate with DM with a splitting of 100 eV-100 keV. In eithercase, the back scattering will be non-negligible. In the limit that reaction and back reactioncome to equilibrium, the number density of ordinary active neutrino at recombination will besuppressed by a factor shown in Eq. (2). There will be however no significant contribution toextra relativistic degrees of freedom. That is N eff will remain equal to the value predicted inthe SM ( i.e., N eff = 3 . N eff below resonance4emperature will be given by N massive = 31 + N/ . (3)The rest ( N s = N eff − N massive ) will be in the form of lighter new particles. As we tentatively discussed in the previous section and shall quantify more systematically in thenext section, more than one new particle may be needed to make m ν ∼ . − . ν + ( − ) ν → X ∗ → f i + ( − ) f i For simplicity, we drop the index i . From model building point of view, increasing the numberof final species is straightforward.In this subsection, we first introduce a model for neutrino pair coannihilation via a new gaugeinteraction. We then introduce a Majoron model. In the end, we discuss various observationalbounds. New U (1) (cid:48) gauge interaction : The active neutrinos ( ν L ) as well as the new sterile neutrinos( ν s ) may have an interaction term of the following form with the new gauge boson: g (cid:48) ( e (cid:48) a ¯ ν L γ µ ν L + e (cid:48) s ¯ ν s γ µ ν s ) Z (cid:48) µ . (4)The interaction leads to an s -channel annihilation ν ( k )¯ ν ( k ) → Z (cid:48)∗ → ν s ( p )¯ ν s ( p ) with am-plitude square given by | M | = 8 g (cid:48) ( e (cid:48) a e (cid:48) s ) ( k · p k · p + k · p p · k )( s − m Z (cid:48) ) + Γ Z (cid:48) m Z (cid:48) in which s is the Mandelstam variable and Γ Z (cid:48) is the decay width of Z (cid:48) . The cross section inthe center of mass frame will be given by dσd cos θ a = g (cid:48) ( e (cid:48) a e (cid:48) s p ) (1 + cos θ a )8 πv rel ((4 p − m Z (cid:48) ) + Γ Z (cid:48) m Z (cid:48) ) (5)where θ a is the angle between (cid:126)p and (cid:126)k . Using the above formula and the formula for conversionrate ( R ) derived in the appendix, we find that the condition RH − > g (cid:48) e (cid:48) a > × − ( m Z (cid:48) keV ) / . (6)Since ordinary active neutrinos form a doublet along with the left-handed charged fermions,we in general expect the corresponding charged lepton to be charged under U (1) (cid:48) , too. Inparticular, if ν e couples to Z (cid:48) , we expect the electron to couple to Z (cid:48) , too. There are strongupper bounds ( ∼ − ) on the coupling of the electron to Z (cid:48) of mass (keV) from stellar couplingconsideration [13] which are two orders of magnitude stronger than the values of g (cid:48) e (cid:48) a required5or successful active-sterile conversion (see Eq. (6)). There are two ways to avoid this strongconstraint: (1) Remember that in the SM, photon, being a special linear combination of W µ and B µ , couples to charged leptons but not to the neutrinos. One can in principle invoke asimilar mechanism by mixing the U (1) (cid:48) gauge boson and the neutral component of the SU (2)gauge bosons through the vacuum expectation value of a scalar doublet charged under U (1) (cid:48) to prevent the coupling of Z (cid:48) to charged leptons while e (cid:48) a (cid:54) = 0. We will not however elaboratefurther on this possibility, here. (2) We can assume that the first generation of fermions areneutral under U (1) (cid:48) and do not couple to Z (cid:48) . As a result, the bound from stellar cooling willbe automatically avoided because stars contain only first generation fermions. This possibilityhas been entertained in various anomaly free L µ − L τ model as well as the model presentedin [14]. During T ∼ keV, the time required for oscillation of ν e into ν µ,τ is much shorter thanthe Hubble time ( i.e., ∆ m /T (cid:29) H ). As a result, while ν µ and ν τ convert to ν s , the electronneutrino in the medium will also oscillate into ν µ and ν τ and they will all come to equilibrium.In other words, because of the fast oscillation, the absence of coupling of ν e to Z (cid:48) will notchange the picture. A simple way to avoid anomalies is to take U (1) (cid:48) = L µ − L τ for the SMfermions and to assign opposite U (1) (cid:48) charges to the pairs of ν si .The mass of Z (cid:48) can come either from St¨uckelberg mechanism or from a new scalar ( φ ) singletunder SU (3) × SU (2) × U (1) but charged under U (1) (cid:48) with (cid:104) φ (cid:105) ∼ m Z (cid:48) / ( g (cid:48) e (cid:48) φ ) ∼
10 TeV( e (cid:48) a /e (cid:48) φ ). Majoron interaction : Let us now consider another scenario which converts active neutrinosto lighter sterile neutrinos through resonant production of an intermediate scalar J of keVmass. The effective couplings can be written as( g a ν Ta cν a + g s ν Ts cν s ) J , where c is a 2 × ± O ( m J ), we can have resonant production of J and its subsequent decay. At the center of mass frame, | M | = g a g s (2 k · k )(2 p · p )( s − m J ) + m J Γ J (7)where Γ J is the total decay rate of the intermediate scalar. The cross section is therefore givenby σ ( ν a ( k ) + ν a ( k ) → J ∗ → ν s ( p ) + ν s ( p )) ∼ g a g s π p ( s − m J ) + Γ J m J Similarly to Eq. (6) for efficient conversion of active neutrinos to sterile ones, g a should satisfythe following bound g a > × − ( m J / keV) / . Let us now consider the high energy completion of the model. The effective g a couplingmay come from mixing with a heavy SU (2) triplet ∆: g ∆ L T c(cid:15) ∆ L + g s J ν Ts cν s where (cid:15) , like c , is a 2 × ±
1, butunlike c , acts on the electroweak SU (2) indices. The masses of scalars are given by m J J + m Tr[∆ † ∆] + λ ∆ J H † ∆ (cid:15)H ∗ J. (8)6f course m ∆ should be larger than electroweak scale; otherwise, the components of ∆ wouldhave been discovered by now at colliders. The mixing is given by α (cid:39) λ ∆ J v m − m J and g a (cid:39) g ∆ α .Taking g a ∼ × − , g ∆ > . m ∆ ∼ α m (cid:28) keV so the λ ∆ H term does not considerably change the mass eigenvalues. Taking m J in Eq. (8) to be oforder of (keV) , we will naturally obtain J mass equal to keV without any fine tuning despitethe large hierarchy between m ∆ and m J . The production rate of J via λ ∆ J coupling at hightemperatures is given by λ J T / (4 π ) ∼ T ( g a /g )( m /v ) / (4 π ) which for g a ∼ × − willbe much smaller than H | T = m ∆ . Thus, no extra contribution to relativistic degrees of freedomat big bang nucleosynthesis era is predicted. More observational bounds:
Several observational bounds have been already discussed above.Let us now review other potential bounds. The required values of new coupling of activeneutrinos within this scenario are so small that they can easily avoid all existing bounds.Bounds from supernova cooling consideration are of order of 10 − [15] which are four ordersof magnitude weaker than the required value for coupling (see Eq. (6)). The bounds fromterrestrial experiment (rare meson decay) are even weaker [16]As shown in [11], for the case of massless Majoron, very strong bounds can be obtainedfrom the free streaming of neutrinos at recombination era T ∼ . T ∼ m X freely stream at recombination. Since we have taken the couplings of ν s to be larger than that of active neutrinos, it is enough to check if ν s stream freely duringrecombination ( T ∼ σ scattering ∼ g s T ν πm J and ∆ t ∼ . /T ) years therefore n ν σ scattering ∆ t | T ∼ . ∼ g s (cid:18) keV m J (cid:19) . Thus, for g s (cid:46) . m J / keV), the sterile neutrinos freely stream. For gauge interactions, the g s /m J ratio has to be just replaced by g (cid:48) e (cid:48) s /m Z (cid:48) . The reason why strong bounds found in [11]do not apply here is that while in [11] the Majoron is taken to be massless or very light, in ourcase m J (cid:29) . The minuteness of the coupling between neutrinos and the new scalar has two important impli-cations: First, it means that the sterile neutrinos are not thermalized prior to the decouplingof active neutrinos. Second, the light scalars are never thermalised.Once the active and sterile neutrinos equilibrate at T ∼ m X ∼ keV the total energy densityin neutrinos, sterile neutrinos and scalars is fixed at the standard model value N eff = 3 . N eff from rest mass effects (unlike forexample the neutrinoless universe scenario [17–20]).Furthermore, below the resonance temperature the interaction remains unimportant becausethe coupling is so small and the mass of the scalar is high. This means neutrinos and sterileneutrinos remain weakly interacting and that both neutrinos and sterile neutrinos free streamlike ordinary neutrinos. 7igure 1: 2D marginalized 68% and 95% likelihood contours for the parameters N massive and m ν . The left panel shows CMB data only and the right panel includes BAO data.From the point of view of CMB and structure formation the scenario is therefore the fol-lowing: The total relativistic energy density in neutrinos and sterile neutrinos is given by N eff = N s + N massive , where N massive gives the energy density remaining in the massive standardmodel neutrinos and N s gives the energy density in the massless sterile neutrino component.We have performed a likelihood analysis of current data using CosmoMC [21]. Our benchmarkCMB data set consists of the Planck 2015 high multipole temperature data and low multipolepolarization data (PlanckTT+lowP), implemented according to the prescription of Ref. [22].We have also performed the analysis with Baryonic Acoustic Oscillation (BAO) data,including6dFGS [23], SDSS-MGS [24], BOSS-LOWZ BAO [25] and CMASS-DR11 [26]. The neutrinosector is described by the parameters N massive and the physical neutrino mass m ν .The other cosmological parameters used in the analysis correspond to those in the standardPlanck 2015 analysis of neutrino mass: The baryon density, Ω b h , the cold dark matter density,Ω c h , the angular scale of the first CMB peak, θ , the optical depth to reionization, τ , theamplitude of scalar fluctuations, A s , and the scalar spectral index, n s .The results are shown in Fig. 1. With the inclusion of CMB data only three massiveneutrinos of degenerate mass 0.2 eV are never disfavored at more than 95% C.L. (fitting wellwith the formal Planck 2015 bound from CMB data of (cid:80) m ν < .
72 eV at 95% C.L.), and if N massive is suppressed to 1.2 the scenario is compatible with observations at the 68% C.L.Once BAO data is included massive neutrinos are disfavored at much higher significance,again fitting well with the Planck 2015 bound of (cid:80) m ν < .
21 eV for the standard model case.For the case of N massive = 3 .
046 (the standard model case) a single neutrino mass of 0.2 eV isdisfavored at close to 5 σ . However, provided that N massive is shifted to down approximately 1the model is only disfavored at 95% C.L. (as could be expected because it corresponds to asingle mass state of m ν ∼ . N massive ∼ . m ν = 0 . σ [4] can be made compatible with the 95 %C.L. limits from CMB (CMB+BAO) provided that N massive is lowered down to 1.5 (0.6) which8ccording to Eq. (3) can be achieved if for each active flavors, there are 1 (4) light or masslesssterile neutrinos that are produced in the resonance ( i.e., N = 3 (12)). To relax cosmological bounds on the neutrino mass we have introduced scenarios within whichneutrinos are converted to lighter particles in the era after neutrino decoupling from SM particlesand before recombination. Since the conversion takes place after neutrino decoupling, N eff remains equal to 3.046 as in the SM. The energy distribution of the final particles is similar tothat of neutrinos. The conversion of neutrinos to the new states and the inverse process canequilibrate the new species so that the contribution of active neutrinos to N eff will be suppressedby a factor of 3 / (3+ N ) where N is the number of final light stable states that neutrinos convertinto.We have found that if only CMB measurements are used, there is no significant need fordilution of the massive neutrino states in order to remain within the 95% C.L. limits. Thisconfirms the results of [6]. To make m ν = 0 . N (cid:38) .
5. Adding Baryon Acoustic Oscillation considerationsmore dilution will be required: We need N (cid:38) m ν = 0 . (cid:28) T (cid:28) MeV can satisfy all these threerequirements. We have introduced two classes of possible conversion scenarios: 1) scattering ofactive neutrinos off dark matter. To make the resonant conversion successful, there should bea new particle quasi-degenerate with dark matter with splitting of O ( keV ) and spin differenceof 1/2. 2) Resonant annihilation of neutrino and/or antineutrino pair by production of anintermediate state of mass keV which immediately decays to new lighter states. For successfulconversion, the coupling of neutrinos to the intermediate state should be larger than 5 × − .We have introduced two specific models to realize the second scenario: In the first model the intermediate state is a gauge boson of mass keV which couples to the second and thirdgenerations of leptons but not to the first generation to avoid the stringent bounds from starcooling. The gauge symmetry ( U (1) (cid:48) ) in question can be for example L µ − L τ . To maintainanomaly cancelation the new particles can be scalars or vector-like fermions (or equivalentlypairs of Weyl fermions with opposite U (1) (cid:48) charges). In the second model , the intermediatestate is a scalar of keV mass with a Majoron type coupling to neutrinos. We have shown thatthe model can be naturally UV-completed by introducing a heavy SU (2) triplet scalar mixedwith a light singlet.Finally, we again wish to stress that a detection of a non-zero mass for the active neutrinosby KATRIN will be extraordinarily interesting because new physics must be invoked to makethe measurement compatible with cosmology. 9 Appendix
In the following, we calculate the rate of interaction of a neutrino of four-momentum P µ =( p , , , p ) with any other neutrino in medium at temperature T : R = 12 p (cid:90) (cid:90) (cid:90) d p (2 π ) p
11 + exp p /T d k (2 π ) k d k (2 π ) k (2 π ) δ ( p + p − k − k ) | M | where | M | close to resonance is given by Breit-Wigner function as | M | = A f ( θ a )( q − m X ) + m X Γ X , where A is almost constant and q = p + p is the four-momentum of the intermediate boson. θ a is the angle between momenta of initial ν a and final particle in the center of mass frame.From Eq. (5), we read that in the case of gauge interactions ( X = Z (cid:48) ) A = g (cid:48) ( e (cid:48) a e (cid:48) s ) m Z (cid:48) and f ( θ a ) = 1 + cos θ a . From Eq. (7), we read that in case of Majorana interactions A = g a g s (2 k · k )(2 p · p ) | resonance = g a g s m J and f ( θ a ) = 1 . Using narrow width approximation we find | M | (cid:39) Bf ( θ a ) δ ( q − m X ) where B ≡ A πm X Γ X . Remember that Γ X is the total decay width. If new particles dominate the decay of X ( i.t., if e (cid:48) s (cid:29) e (cid:48) a for gauge interactions or if g s (cid:29) g a for Yukawa interactions), B will be independent ofthe couplings to new states and will be given by the coupling to active neutrinos. That is forgauge interactions, Γ X (cid:39) N ( g (cid:48) e (cid:48) s ) m Z (cid:48) π so B = 8 π ( g (cid:48) e (cid:48) a m Z (cid:48) ) N and for Yukawa interaction Γ X (cid:39) N g s m J π so B = 16 π g a m J N .
In both cases, N is the number of new states coupled to X . We can simplify the calculationby the following convolution: R = (2 π ) B (2 π ) p (cid:90) d p p
11 + exp p /T (cid:90) d qδ ( p + p − q ) S ( q )where S ( q ) ≡ (cid:90) d k k (cid:90) d k k δ ( k + k − q ) f ( θ a ) δ ( q − m X ) = π bδ ( q − m X ) , b = 1(4 /
3) for Majorana (gauge) interaction. To calculate S ( q ), we have used its Lorentzinvariance and have performed calculation in the rest frame of intermediate X . Rememberingthat q = p p (1 − cos θ ) (in which θ is the angle between initial momenta), we can write R = B · b π p (cid:90) p dp d cos θp
11 + exp p /T δ ( p p (1 − cos θ ) − m X ) = B · b · T π p (cid:16) log(1 + exp m X / p T ) − m X / p T (cid:17) . For p T (cid:28) m X , we can write R → B · b · T / (2 π p ) exp − m X / p T . For p T (cid:29) m X , we can write R → log(2) B · b · T / (2 π p ). To check if the active-sterile conversion is effective, R shouldbe compared to H = T /M ∗ P l . The majority of neutrinos have energy of order of temperature p ∼ T so for high temperatures T (cid:29) m X , the conversion of active neutrinos (for majority ofneutrinos in the medium) to sterile neutrinos are not effective. In other words, for T (cid:29) m X and p ∼ T , we expect R/H (cid:28)
1. (Notice that even at high temperatures if p is sufficientlysmall, their conversion rate to sterile neutrinos will be relatively high but such low energyneutrinos comprise only small fraction of neutrinos.) Moreover, at low temperatures for which p T (cid:28) m X , R/H is also small so conversion will be negligible. Conversion can be efficient( R (cid:38) H ) only at T ∼ m X provided that R | T ∼ m X ∼ H | T ∼ m X . Taking m X ∼ g a or g (cid:48) e (cid:48) a (cid:38) × − . (9)Remember that we have assumed that the coupling of X to the new lighter particles are stronger. Acknowledgments
The authors would like to thank Mainz Institute for Theoretical Physics and organizers of“Crossroads of Neutrino Physics” extended workshop where this project started for kind andgenerous hospitality. YF would like to acknowledge partial support from the European UnionFP7 ITN INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442).
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