Decaying neutrinos and implications from the supernova relic neutrino observation

We propose that supernova relic neutrino (SRN) observation can be used to set constraints on the neutrino decay models. Because of the long distance scale from cosmological supernovae to the Earth, SRN have possibility to provide much stronger limit than the present one obtained from solar neutrino observation. Since the currently available data are only the upper limit on the flux integrated over $E_{\bar\nu_e}>19.3$ MeV, the decay models on which we can set constraints is quite restricted; they must satisfy specific conditions such that the daughter neutrinos are active species, the neutrino mass spectrum is quasi-degenerate, and the neutrino mass hierarchy is normal. Our numerical calculation clearly indicates that the neutrino decay model with $(\tau_2/m,\tau_3/m)\alt (10^{10},10^{10})$ [s/eV], where $\tau_i$ represents the lifetime of mass eigenstates $\bar\nu_i$, appears to give the SRN flux that is larger than the current upper limit. However, since the theoretical SRN prediction contains many uncertainties concerning a supernova rate in the universe or simulation of supernova explosions, we cannot conclude that there exists the excluded parameter region of the neutrino lifetime. In the near future, further reduced upper limit is actually expected, and it will provide more severe constraints on the neutrino decay models.


I. INTRODUCTION
A number of ground-based experiments, which observed atmospheric [1], solar [2,3,4], and reactor neutrinos [5], have revealed the nonzero neutrino masses and flavor mixings, i.e., properties beyond the standard model of the particle physics. Fortunately, our current knowledge of the neutrino mass differences and mixing angles further enables us to consider far more exotic properties of the neutrino, such as a nonzero magnetic moment (see Refs. [6,7] and references therein) and neutrino decay. In this Letter we show that neutrino decay of a particular type may be ruled out or severely constrained by the current and the future supernova relic neutrino observation at the Super-Kamiokande (SK) detector [8].
We consider two-body neutrino decays such as ν i → ν j + X, where ν i are neutrino mass eigenstates and X denotes a very light or massless particle, e.g., a Majoron. The strongest limits on this decay modes are obtained from the solar neutrino observation by SK [9] (see also Ref. [10]). It was argued that the limit is obtained primarily by the nondistortion of the solar neutrino spectrum, and the potentially competing distortions caused by oscillations as well as the appearance of active daughter neutrinos were also taken into account. However, owing to the restricted distance scale to the Sun, this limit is very weak, τ /m > ∼ 10 −4 s/eV, and therefore, the possibility of the other astrophysical neutrino decay via the same modes cannot be eliminated. In fact, detectability of the decay of neutrinos from the high-energy astrophysical sources was discussed [11], and it has been concluded that it should be visible by future km-scale detectors such as IceCube, since the neutrino decay strongly alter the flavor ratios from the standard one, φ νe : φ νµ : φ ντ = 1 : 1 : 1, expected from oscillations alone.
Our strategy is basically the same as that of the previous studies [9,11], i.e., the enhancement of the electron neutrino events due to the decay is investigated. As a source of neutrinos, we consider supernovae. Two observational results concerning supernova neutrinos exist; one is the well-known neutrino burst from SN 1987A [12,13], and the other is the recent upper limit on the flux of supernova relic neutrinos (SRN), which is the accumulation of neutrinos from all the past supernovae, by SK [8]. 1 Original discussions concerning the SN 1987A signal have been already given, including the effect of the neutrino decay as well as the pure flavor mixing, in the literatures [14,15] (see also Ref. [16]). In this Letter, we use the latter one (SRN) for obtaining implications for the neutrino decay. The advantage of this approach compared to the former ones is that the neutrinos must transit over very long distance scale from the cosmological supernovae to the Earth, and much longer lifetimes would be probed in principle. The current SRN upper limit is only a factor three larger than theoretical predictions by Ando et al. [17,18] (hereafter AST), which adopted neutrino oscillations using experimentally inferred parameters. Therefore, if some neutrino decay model predicts the SRN flux which is three times larger than the AST prediction, then it is excluded.
This Letter is organized as follows. In Section II, we present formulation for the SRN calculation and adopted models. In particular the detailed discussion concerning the supernova model and the neutrino decay model are given in Section II A and II B, respectively. How the oscillation and decay changes the neutrino spectrum and flux is qualitatively illustrated in Section III and the numerically calculated results are given in Section IV. In Section V, we discuss the various possibilities of the neutrino decay, uncertainties of the adopted models, and future prospects.

II. FORMULATION AND MODELS
The SRNν e flux is calculated by where is a supernova rate per comoving volume at redshift z, and z max is the redshift when the gravitational collapses began (we assume it to be 5).
is the number of emitted neutrinos per unit energy range by one supernova at redshift z. As the supernova rate, we use the most reasonable model to date, which is based on the rest-frame UV observation of star formation history in the universe by the Hubble Space Telescope [19], and the model was also used in AST as "SN1." In this model, the supernova rate exponentially increases with z, peaks around z ∼ 1.5, and exponentially decreases in further high-z region.

A. Supernova model
As original neutrino spectra N 0 ν (E, z), which is not the same as N ν owing to the neutrino oscillation and decay, we adopt the result of a numerical simulation by the Lawrence Livermore group [20], which is the only group that is successful for calculating neutrino luminosities during the entire burst (∼ 10 sec). The average energies are different between flavors, such as E νe ≃ 11 MeV, Eν e ≃ 16 MeV, E νx ≃ 22 MeV, where ν x represents non-electron neutrinos and antineutrinos. This hierarchy of the average energies is explained as follows. Since ν x interact with matter only through the neutral-current reactions in supernova, they are weakly coupled with matter compared to ν e andν e . Thus the neutrino sphere of ν x is deeper in the core than that of ν e andν e , which leads to higher temperatures for ν x . The difference between ν e andν e comes from the fact that the core is neutron-rich and ν e couple with matter more strongly, through ν e n ↔ e − p reaction.
Our calculations presented in Section IV are strongly sensitive to the adopted supernova model, or in particular, the average energy difference betweenν e and ν x . Recently, the Livermore calculation is criticized since it lacks the relevant neutrino processes such as neutrino bremsstrahlung and neutrino-nucleon scattering with nucleon recoils, which are considered to make the mean energy difference between flavors less prominent; it has been actually confirmed by the recent simulations (e.g., Ref. [21]). However, we cannot adopt recent models, even though they include all the relevant neutrino microphysics. This is because the SRN calculation definitely requires the time-integrated neutrino spectrum during the entire burst, whereas all the recent supernova simulations terminate at ∼ 0.5 sec after the core bounce. Since the Livermore group alone is successful to simulate the supernova explosion and calculate the neutrino luminosity during the entire burst, we use their result as a reference model. Again, we should note that our discussions from this point on heavily relies on the adopted supernova model.

B. Neutrino decay model
In this Letter, we consider only the so-called "invisible" decays, i.e., decays into possibly detectable neutrinos plus truly invisible particles, e.g., light scalar or pseudoscalar bosons.
The best limit on the lifetime with this mode is obtained from solar neutrino observations and is τ /m > ∼ 10 −4 s/eV [9], which is too small to set relevant constraints on discussions below. We do not consider other modes such as radiative two-body decay since they are experimentally constrained to have very long life times (e.g., Ref. [22]).
We note that our approach is powerful only when the decay model satisfies specific conditions such that (i) the daughter neutrinos are active species, (ii) the neutrino mass spec- For a while, we assume that the conditions (i)-(iii) are satisfied; all the other possibilities are addressed in detail in Section V A. As discussed in Section V C, we believe that future observational development would enable far more general and model-independent discussions, which are not restricted by the above conditions.

III. NEUTRINO OSCILLATION AND DECAY
Effects of neutrino oscillation and decay are included in Nν e (E, z). We address the problem of flavor mixing by the pure supernova matter effect, first for antineutrinos and second for neutrinos, and then we discuss the neutrino decay.
The state ofν e produced at deep in the core is coincident with the lightest mass eigenstatē ν 1 , owing to large matter potential. This state propagates to the supernova surface without being influenced by level crossings between different mass eigenstates (it is said that there are no resonance points). Thus,ν e at production becomesν 1 at the stellar surface (Nν 1 = N 0 νe ) and the number ofν e there is given by where θ ⊙ is the mixing angle inferred from the solar neutrino observations (cos 2 θ ⊙ ≃ 0.7) [2,3,4,5], N 0 ν represents the neutrino spectrum at production. In the above expression (2), we used the fact, sin 2 2θ atm = 1, and assumed θ 13 = 0. These are justified by the atmospheric [1] and reactor neutrino experiments [23].
The situations changes dramatically for neutrino sector. As the case for antineutrinos, ν e are produced as mass eigenstates owing to large matter potential, however, the difference is that the produced ν e coincide with the heaviest state ν 3 . Since in vacuum ν e most strongly couples to the lightest state ν 1 , there must be two level crossings (or resonance points) between different mass eigenstates during the propagation through supernova envelope; each is labeled by H-and L-resonance, corresponding to whether the density of the resonance point is higher or lower (see, e.g., Ref. [24] for details). It is well-known that at L-resonance the mass eigenstate does not flip (adiabatic resonance) for LMA solution to the solar neutrino problem. However, the adiabaticity of the H-resonance becomes larger than unity when the parameter θ 13 is sufficiently large. Instead, we parameterize the flip probability at the H-resonance by P H , i.e., if the resonance is adiabatic (nonadiabatic), P H = 0(1). Thus, at the stellar surface, the neutrino spectra of mass eigenstates is given by If we include the neutrino decay, the expectedν e flux from each supernova changes drastically. Before giving a detailed discussion, we first place some simplifying assumptions.
Instead of the lifetime, we define "decay redshift" z d i of the mass eigenstate ν i (ν i ); if the source redshift z is larger than the decay redshift z d i , all the neutrinos ν i (ν i ) decay, on the other hand if z < z d i , ν i (ν i ) completely survive. We consider the decaying mode ν 3 (ν 3 ) →ν 1 and ν 2 (ν 2 ) →ν 1 , and z d 2 and z d 3 are taken to be two free parameters. The other case that one of them is stable can be realized if we take z d > z max . With these assumptions and parameterization, the neutrino spectrum which is emitted by the source at redshift z can be obtained. First, we consider the decay modeν i →ν j + X, in which the neutrino helicity conserves. Theν e spectrum is given by where Θ is the step function. On the other hand, if the relevant decay mode is ν i (ν i ) → ν j (ν j ) + X (helicity flips), the expected spectrum becomes Comparing Eqs. (6) and (7) with Eq. (2), the SRN flux with the neutrino decay is expected to be very different from the case of the pure neutrino oscillation. In particular, for the modeν i →ν j + X [Eq. (6)], the SRN spectrum is expected to be hard since it contains a fair amount of ν x . In the case of ν i →ν j + X mode, the ν e spectrum is also included, and then the corresponding upper limit is not as strong as that forν i →ν j + X mode. In the next section, we only consider the model, whose upper limit is the most severe among all the models considered, i.e., the decay modeν i →ν j + X.
In general, the decay redshifts depend on the neutrino energy since the lifetime at the laboratory frame τ lab relates to that at the neutrino rest frame τ via a simple relation τ lab = Eτ /m. However, we believe that it does not make sense in discussing this point strictly, because the estimation of the SRN flux contains many other uncertainties as shown in Section V B. In order to obtain the lifetime of the mass eigenstatesν i from the decay redshifts, the typical neutrino energy E = 10 MeV is assumed with the following formulation: where the Hubble constant H 0 is taken to be 70 km s −1 Mpc −1 , and Λ-dominated cosmology is assumed (Ω m = 0.3, Ω Λ = 0.7). Since the exact value of the neutrino mass m is not known, the relevant quantities is the neutrino lifetime divided by its mass, τ /m. Figure 1 shows the SRN flux for various parameter sets of decay redshifts (z d 2 , z d 3 ) as a function of neutrino energy, which is calculated using Eqs. (1) and (6). The solid curve in This is because the neutrinos from supernovae at redshift larger than z d 3 = 1.0 are affected by theν 3 →ν 1 decay and it results in the increase ofν e . Since the neutrino energies are redshifted by a factor of (1 + z) −1 owing to an expansion of the universe, the decay effect can be seen at low energy alone. When the value of z d 3 is reduced to 10 −2 , the neutrinos even from the nearby sources are influenced by theν 3 →ν 1 decay, resulting in the deviation over the entire energy range as shown by the long-dashed curve in Fig. 1. If we add thē ν 2 →ν 1 decay, it further enhances the SRN flux.

IV. RESULTS
In Table I, we summarize the SRN flux integrated over the energy range of Eν e > 19.3 MeV, for the each decay model. In the second column we placed the lifetime-to-mass ratio which corresponds to each decay redshift, which is obtained using Eq. (8). The corre-    Figure 2 shows a contour plot for a ratio of the predicted flux to the observational upper limit, which is projected against the lifetime-to-mass ratios (τ 2 /m, τ 3 /m). The area below the solid curve labeled as 1.0 is considered to be an excluded parameter region. We can confirm that our approach is very powerful to obtain the constraint on the neutrino lifetimes, because the best observed lower limit thus far ( > ∼ 10 −4 s/eV) is much smaller than that shown in Fig.   2 ( > ∼ 10 10 s/eV), although there are still many uncertainties in this method as discussed in the next section.

A. Other decaying modes
Our discussions until this point were concerned with one specific decaying mode,ν 3 → ν 1 ,ν 2 →ν 1 , where the daughter neutrinos carry nearly the full energy of their parent. We consider the other possible decaying models; firstν 3 →ν 2 ,ν 2 →ν 1 is discussed. Sinceν 2 state containsν e state by less fraction thanν 1 , this decay mode gives smaller SRN flux, resulting in weaker upper limit in general. However, when the lifetime of one mode is much longer (shorter) than the other, i.e., z d 2 ≪ z d 3 or z d 2 ≫ z d 3 , the discussions for our reference decay modesν 3 →ν 1 ,ν 2 →ν 1 are basically applicable to this case.
When the daughter neutrino energy is considerably degraded, which is actually the case when the neutrino masses are strongly hierarchical, the observational upper limit for the each model is not as strong as the previous limit shown in Table I. This is because the energetically degraded daughter neutrinos soften the SRN spectrum.
Finally we consider the case that the daughter neutrinos are sterile species which does not interact with matter. If the lifetime of the mode is sufficiently short, then the obtained spectrum from each supernova can be expressed by which is smaller than the normal oscillation expression (2). Thus, also in this case, the observational upper limit will be looser. In consequence, all the other possible decaying models give weaker upper limit compared to our reference model.

B. Supernova and supernova rate model uncertainties
Again we restrict our discussion to our standard decay mode, and discuss whether the parameter region (τ 2 /m, τ 3 /m) < ∼ (10 10 , 10 10 ) [s/eV] is really ruled out by the current observational data, as shown in Fig. 2.
The theoretical calculation of the SRN flux contains many uncertainties such as the supernova rate as a function of redshift z as well as the original neutrino spectrum emitted by each supernova. As for the supernova rate, since we have inferred it from the rest-frame estimate. Thus for example, if we use the supernova rate model which is larger by a factor of 2 (even if this is actually the case, it is not surprising at all), then almost all the region of lifetime-to-mass ratio (τ 2 /m, τ 3 /m) will be excluded as shown in Fig. 2.
The uncertainty concerning the original neutrino spectrum also gives very large model dependence of the SRN flux calculation. Although we adopted the result of the Livermore simulation, as we have noted above it lacks the relevant neutrino processes, which reduce the average energy of ν x to the value close to that ofν e . If this is the case, the SRN spectrum as a result of the neutrino decay becomes softer compared to that obtained with the Livermore original spectra, which leads to weaker upper limit. In actual, we have calculated the SRN flux for various values of (z d 2 , z d 3 ) assuming the original ν x spectrum is the same as that ofν e , for which we have used the Livermore data. As a result of the calculation for this extreme case, the obtained prediction-to-limit ratio is at most ∼ 0.35, even when both of decay redshifts are sufficiently small. In consequence, considering the uncertainties which is included in the supernova rate or the neutrino spectrum models, we cannot conclude that the parameter region (τ 2 /m, τ 3 /m) < ∼ (10 10 , 10 10 ) [s/eV] is already excluded, although Fig.   2 indicates it.

C. Future prospects
Now, we consider the future possibility to set severer constraint on the neutrino decay models. The largest background against the SRN detection at SK is so-called invisible muon decay products. This event is illustrated as follows. The atmospheric neutrinos produce muons by interaction with the nucleons (both free and bound) in the fiducial volume. If these muons are produced with energies belowČherenkov radiation threshold (kinetic energy less than 53 MeV), then they will not be detected ("invisible muons"), but their decay-produced electrons will be. Since the muon decay signal will mimic theν e p → e + n processes in SK, it is difficult to distinguish SRN from these events. Recent SK limits are obtained by the analysis including this invisible muon background.
In the near future, however, it should be plausible to distinguish the invisible muon signals from the SRN signals; two different methods are currently proposed for the SK detector. One is to use the gamma rays emitted from nuclei which interacted with atmospheric neutrinos [25]. If gamma ray events, whose energies are about 5-10 MeV, can be detected before invisible muon events by muon lifetime, we can subtract them from the candidates of SRN signals. In that case, the upper limit would be much lower (by factor ∼ 3) when the current data of 1,496 days are reanalyzed [25]. Another proposal is to detect signals of neutrons, which are produced through theν e p → e + n reaction, in the SK detector. Actual candidate for tagging neutrons is gadolinium solved into pure water [26]. Additional neutron signals can be used as delayed coincidence to considerably reduce background events. In addition, future projects such as the Hyper-Kamiokande detector is expected to greatly improve our knowledge of the SRN spectral shape as well as its flux. If a number of data were actually acquired, the most general and model-independent discussions concerning the neutrino decay would be accessible. Finally, we again stress that a great advantage to use the SRN is that the cosmologically long lifetime can be probed in principle.

VI. CONCLUSIONS
We obtained the current lower limit to the neutrino lifetime-to-mass ratio using the SRN observation at SK. Since the available data are only the upper limit on the flux integrated over Eν e > 19. 3 MeV, the decay model on which we can set constraints is quite restricted, i.e., the decay models that gives the SRN flux which is comparable to or larger than the corresponding upper limit.
Therefore, our reference decay model in this Letter is the two-body decayν i →ν j + X, with the daughter neutrinos which are active species and carry nearly full energy of their parent. The neutrino mass hierarchy is assumed to be normal (m 1 < m 3 ). The SRN calculation is also very sensitive to the adopted supernova model, i.e., neutrino spectrum from each supernova explosion; we adopted the result of the numerical simulations of the Lawrence Livermore group. Although their calculation is recently criticized since it lacks several relevant neutrino processes, we adopt it as a reference model. This is because it is only the successful model of the supernova explosion and we definitely need fully timeintegrated neutrino spectra.
Our calculations with these models shows that the neutrino decay model with (τ 2 /m, τ 3 /m) < ∼ (10 10 , 10 10 ) [s/eV] apparently gives the SRN flux that is larger than the current upper limit (see Fig. 2). Since this value 10 10 s/eV is much larger than that of the limit obtained by the solar neutrino observation > ∼ 10 −4 s/eV, our approach is shown to be very powerful to obtain the implications for the neutrino decay. At present, however, owing to the large uncertainties such as supernova models, this lower limit 10 10 s/eV cannot be accepted without any doubt. Future experiments with the updated detectors and reanalysis with the refined method is expected to give us greatly improved information on the SRN flux and spectrum. In that case, the most general model-independent discussions concerning the neutrino decay would be possible.