Production of unstable heavy neutrinos in proto-neutron stars

We discuss the production of a class of heavy sterile neutrinos $\nu_h$ in proto-neutron stars. The neutrinos, of mass around $50$ MeV, have a negligible mixing with the active species but relatively large dimension-5 electromagnetic couplings. In particular, a magnetic dipole moment $\mu\approx 10^{-6}$ GeV$^{-1}$ implies that they are thermally produced through $e^+ e^-\to \bar \nu_h \nu_h$ in the early phase of the core collapse, whereas a heavy--light transition moment $\mu_{\rm tr}\approx 10^{-8}$ GeV$^{-1}$ allows their decay $\nu_h\to \nu_i \gamma$ with a lifetime around $10^{-3}$ s. This type of electromagnetic couplings has been recently proposed to explain the excess of electron-like events in baseline experiments. We show that the production and decay of these heavy neutrinos would transport energy from the central regions of the star to distances $d\approx 400$ km, providing a very efficient mechanism to enhance the supernova shock front and heat the material behind it.


Introduction
Neutrinos define a sector of the Standard Model that still presents some important unknowns. The current scheme of mass differences and mixings seems able to explain most of the existing data [1], but the absolute value of their masses, their Dirac or Majorana nature [2] or the presence of additional sterile modes [3,4] are yet to be determined. In particular, the production of sterile neutrinos ν s , through collisions with standard matter or through flavor oscillations has important implications both in particle physics and astrophysics [5][6][7][8]. The mixing with an active neutrino ν may provide sterile modes with small couplings to the W and Z gauge bosons that translate into dimension-6 operators of type In addition, the low-energy effective Lagrangian may also include dimension-5 operators from loops involving heavy particles. Although these operators are usually overlooked, they could mediate the dominant reactions of sterile neutrinos in a star under favorable thermodynamical conditions. Here we will study this possibility in the context of supernova explosions. When a supernova goes off a proto-neutron star can be formed having a typical initial radius of (20-60) km and a mass of (1-1.5) M . It is believed that most of the gravitational binding energy (E grav ≈ 3 × 10 53 erg) is released in a ∼20 second neutrino burst [9]. The neutrino spectrum from supernova SN1987A detected at SuperK and IMB indicated a weak decoupling from baryonic matter, confirming that neutrino transparency sets in as their temperature falls below a few MeV [10] in the dense core. At earlier phases of the collapse, however, computational simulations [11,12] reveal internal peak temperatures exceeding 20 MeV in the central high density regions of the star. At such temperatures and densities the evolution of these astrophysical objects becomes sensitive to the fundamental properties of neutrinos and to the presence of hypothetical weakly-coupled particles. In this context, a lot of effort has been devoted to the cooling through neutrino emission in the nuclear medium [13], to the matter opacity [14] and revival of the stalled shock that arises in the standard paradigm of supernova core collapse [15], or to the synthesis of heavy nuclei taking place in the hot bubble behind the shock [16,17].
In this work we will focus on the astrophysical consequences of the production of a heavy sterile neutrino ν h whose dominant interactions are not the weak ones in Eq. (1) but of electromagnetic kind. This type of particles have been proposed as a possible explanation [18] for the excess of electron-like events in baseline experiments [19]. Let us briefly show how the required couplings could be generated. Consider an SU (2) Let us also suppose that at TeV energies the gauge symmetry is In order to avoid collider bounds [20], the breaking of the leftright symmetry must be such that the charged lepton (E, E c ) gets a mass m E ≥ 300 GeV while ν h remains light. Loop diagrams of heavy gauge bosons and fermions (see Fig. 1) will then generate the operator where A ν is the electromagnetic field and μ is a magnetic dipole moment of order [21] μ ≈ e In addition, the possible mixing of the sterile and the active neutrinos will be parametrized by an angle θ , so that the mass eigenstates read N = cos θ N + sin θ ν and ν = − sin θ N + cos θ ν. This mixing will generate electromagnetic transitions through the same type of diagrams (we drop the prime to indicate mass eigenstates): with μ tr ≈ sin θμ being the transition dipole moment. This operator may imply that the dominant decay mode of the heavy neutrino is ν h → ν γ . Notice also that the presence of additional heavy singlets (ν h with m h ≈ m E ) mixed both with ν and ν h will give additional contributions to μ tr . Therefore, at this point we will treat μ and μ tr as independent parameters.
This type of sterile neutrinos could change substantially the evolution of a proto-neutron star. We will show that sterile pairs can be produced abundantly during the ∼20 second neutrino burst, escape the star core more easily than standard neutrinos, and finally decay within a few hundred km from the core. The very energetic photons from the decay could deposit energy, helping revive the stalled accretion shock formed during the collapse and change the thermal environment in the vicinity of the star. Our scenario could be considered a different realization of the eosphoric neutrino hypothesis proposed in [22].

Decay rate, production and scattering cross sections
Let us first describe the dominant decay and production channels for the heavy neutrino ν h in vacuum. Later we will discuss how the hot and dense medium in a proto-neutron star (including populations of neutrons, protons, electrons and muons) affects these processes.
To be definite in our calculation we will take as reference val- For these values of the mass and the transition moment, the heavy neutrino will decay into νγ with a lifetime We will also assume that the mass mixing, i.e., the active component in ν h , is smaller than sin 2 θ < 10 −3 and only along the muon and/or the tau flavors. In that case, the radiative decay will dominate over the weak processes ν h → νe + e − , νν iνi , which appear with a branching ratio This type of sterile neutrino avoids cosmological bounds since it decays before primordial nucleosynthesis. At colliders it is hardly detectable: even if it were produced in 1% of kaon or muon decays, ν h is too long lived to decay inside the detectors and too light to change significantly the kinematics of the decay [23]. Actually, more elaborate setups with two sterile modes have been proposed to explain the excess of electron-like events at MiniBooNE in terms of the photon that results from its decay [18].
The dominant production channels of ν h will also be electromagnetic. In particular, electron-positron annihilation into ν h pairs, e + e − →ν h ν h , will be mediated by a photon through the magnetic dipole moment coupling in Eq. (4). The differential cross section is given by where α is the fine structure constant, m e is the electron mass, and s and t are the usual Mandelstam variables ( √ s is the center-ofmass energy). In Fig. 2 (left panel) we plot the total cross section for this process. Muon pair annihilation will give an analogous but subleading contribution, since muons are less abundant than electrons in the star core.
The active to sterile transition mediated by a photon can be catalyzed by the presence of charged particles X = p, e: ν X → ν h X (see right panel in Fig. 2). This contribution, however, can be neglected here due to the smaller value of the transition coupling that we have assumed, μ tr ≈ 10 −2 μ. The weak channels which dominate the production of active neutrinos [24] give also a subleading contribution due to the small mixing sin 2 θ < 10 −3 of our sterile, whereas other processes like plasmon decay [25] are irrelevant for heavy neutrino masses around 50 MeV (i.e., much larger than the electron mass).
In addition to its production and decay, the collisions of ν h with charged matter will be essential in order to understand its propagation in the dense medium and estimate how efficiently these  neutrinos escape the proto-neutron star. We need to distinguish between elastic scatterings and absorption reactions of type The differential cross section for the first process reads dσ dt = αμ 2 This is a long distance (photon-mediated) process with a divergent total cross section; if we restrict to collisions substantially changing the direction of the incident ν h (e.g., a scattering angle θ > 30 • in the center-of-mass frame) we obtain the cross section depicted in Fig. 3 (left panel). For the inelastic process that transforms the heavy neutrino into an active one we obtain dσ dt = αμ 2 and its total cross section is also depicted in Fig. 3 (right panel). These cross sections are, in both cases, much smaller than the ones for active neutrinos off nucleons mediated by weak bosons. For example, a neutrino in a gas at T 20 MeV has an average energy E ν π T ≈ 60 MeV, and its lowest order elastic cross section with a neutron is σ (

Production in a proto-neutron star
Let us now calculate the production rate of heavy neutrinos at the astrophysical site. Nucleon and lepton densities in the medium are constrained by electric charge neutrality and baryon and lepton number conservation. Typical baryonic densities in the core of a proto-neutron star are well above nuclear saturation density n B (2-3)n 0 with n 0 = 0.17 fm −3 [9], whereas the lepton and electron fraction evolve dynamically from Y L ≈ 0.31, Y e ≈ 0.27 at t = 0.1 s to Y L ∼ 0.18, Y e ∼ 0.17 at t = 10 s [26]. The extreme conditions in the core are such that quantum effects will be important. The population of baryons and leptons is described by the Fermi-Dirac distribution where μ i (i = n, p, e ± ) denotes the chemical potential of the considered species. Both T and μ i evolve within the star, in particular, the chemical potentials take care of the conservation of charges and quantum numbers in a self consistent way [27]. For a given value of the temperature and the electron chemical potential (μ e + = −μ e − ), the total energy emissivity (energy produced per unit volume and unit time) of heavy neutrino pairs through the dominant process e + e − →ν h ν h , Q E (e + e − →ν h ν h ) = dE dtdV is given by [28,29] where the factor f ( the Pauli blocking factor in the generic reaction 12 → 34 and p i = (E i , p i ) are the 4-momenta. We will consider that the reaction is not affected by the quenching of outgoing sterile states, i.e., (1 − f 3 ) 1 (1 − f 4 ), since heavy neutrinos do not achieve chemical equilibrium and their number density inside the star is always small. The squared matrix element for the interaction defined in Eq. (4) is given by with e the electron charge, s = (p 1 + p 2 ) 2 and t = (p 1 − p 3 ) 2 , see To perform the phase space integral in Eq. (15) we note that there are only four non-trivial independent variables: the initial energies E 1 and E 2 , the angle θ 2 , as defined in Fig. 4, and E 3 .
Any other kinematical variables can be derived from these four or can be trivially integrated. It is convenient to define the 4-vector k = p 1 + p 2 (notice that k 2 = s and | k| 2 = (E 1 + E 2 ) 2 − s) and the angle θ 3 of k with p 3 : After integrating the 4-dimensional Dirac delta that enforces energy and momentum conservation, we obtain We also define The Mandelstam variables in terms of these four quantities read where α (0 ≤ α ≤ π ) is the angle between k and p 1 ,

Transport of energy out of the star core
The possible impact of the heavy neutrino ν h on the evolution of the proto-neutron star will depend on its ability to take a significant amount of energy out of the core. If the sterile neutrinos are abundant inside the star core but unable to reach the surface before decaying into a photon plus an active neutrino, then they become just a state mediating interactions of electrons with neutrinos and photons. We will show that this is not the case and that they could play an interesting role in supernova explosions.
Let us take a temperature T 0 = 25 MeV and an electron chemical potential μ e0 = 100 MeV, which are typical values at the inner central region of a proto-neutron star (see [11,12]). Although these quantities are time and density dependent, the chosen values can be used to estimate the possibilities of our scenario.
Varying the mass m h , the magnetic dipole moment μ, the electron chemical potential μ e and the temperature T and performing a fit of Q E in Eq. (18) For the reference values of all the parameters, the expression above yields Q E ≈ 2 × 10 35 erg/s cm 3 , with an average ν h energy of E h ≈ 103 MeV. This is a very large production rate, ∼10 2 -10 3 times larger than the one obtained in [22] using heavier sterile neutrinos mixed with the active ones. Our neutrinos, however, will not leave the star core unscattered.
We can also compare this production rate with the one of standard neutrinos in early cooling of proto-neutron stars. For example, in the central core the direct URCA process n → peν e provides Q DURCA E ≈ 2.4 × 10 41 R erg/s cm 3 at T = 25 MeV [28,31], being R a factor of order unity [30]. This is five decades over the ν h production rate that we have found. The direct URCA process requires a high proton fraction, Y p 11%, only accessible to large mass objects, however the less demanding modified URCA cooling also gives a much faster rate than for steriles, Q MURCA E ≈ 1.5 × 10 40 R erg/s cm 3 . These active neutrinos will be, to a large extent, trapped (before transparency sets in) inside the star core, whereas ours have weaker interactions with the protons and electrons in the medium.
It is then apparent that we need to consider propagation effects of ν h . Although a precise calculation would require a complete multidimensional simulation that is beyond the scope of this work, we will discuss the qualitative picture and show that the model has enough parameter space to realize it. The first important effect is the diffusion from the center to the surface of the star core, with a radius r ≈ 20 km. Active muon and tau neutrinos scatter there mainly off neutrons with a mean free path λ S ν ∼ 1 m [11,32]. Analogously, ν h will scatter elastically with protons with a cross section ∼40 times smaller, which suggests an interaction length λ S h inside the star longer by the same factor. This implies a larger diffusion coefficient D ≈ λ S h c/3 and a more efficient transport from the core to the outer parts of the star. The typical diffusion time for that process will be τ D ≈ r 2 /(2D) ≈ 5 × 10 −2 s. The second crucial effect in the propagation of the heavy neutrinos is their absorption: the star will capture a fraction of them through the inelastic collisions ν h p → ν p. For μ tr = 10 −8 GeV −1 the absorption length is approximately 2Dτ D 400 km. We estimate that, even if absorptions and decays reduced the number of neutrinos leaving the star from our estimate to 1% of the ones produced (some of them closer to the surface), ν h may still carry a total of 10 51 -10 52 erg and deposit this energy outside the star during the 20 second neutrino burst. Notice that a reduction in the magnetic moment μ would also reduce the production rate of heavy neutrinos, but it would increase the mean free path between elastic scatterings and then the fraction of neutrinos that reach the surface. In addition to the transition moment μ tr , the mass m h is another parameter that could impact the production rate or the decay length of the heavy neutrino.
An interesting variation of the model that depends only on two parameters (m h and μ tr ) would be obtained by suppressing the magnetic moment μ and slightly increasing the transition one, e.g. μ tr ≈ 5 × 10 −8 GeV −1 . In that case the dominant production channel would be where X is any charged particle and the active ν may be any linear combination of ν μ and ν τ . In the star core the heavy neutrinos would be partially absorbed through the inverse reaction (we estimate an absorption length λ A h ≈ 4 km), and the ones reaching the surface would decay with cτ h ≈ 30 km. Since the production would not be so abundant as in the case discussed in the previous section, this possibility provides a safer scenario still able to transport energy to the region near the star surface.
Although the optimal value of these parameters would require a full Monte Carlo simulation, the scenario seems flexible enough to introduce acceptable changes in the dynamics of supernova explosions, with the decay into photon plus active neutrino playing an important role in the enhancement of the supernova shock.

Conclusions
The collapse of a very massive star defines an astrophysical object with extreme conditions where neutrinos determine the thermodynamics. These proto-neutron stars are suitable laboratories to probe the properties of any weakly coupled particles of mass 100 MeV. Here we have proposed a sterile neutrino ν h much heavier than the standard ones (m h ≈ 50 MeV) and with sizable electromagnetic couplings: a magnetic dipole moment μ and dipole sterile-active transition μ tr that mediates its decay ν h → νγ with a lifetime τ h ≈ 10 −3 s. This simple 3-parameter model seems to have interesting implications in supernova explosions. The heavy neutrino is produced in the core at a high rate through e + e − →ν h ν h , it may escape the star more efficiently than active neutrinos and decays depositing a large amount of energy in the outer layers of the star.
We believe that the type of heavy sterile neutrino proposed here could be an essential ingredient to help the progression of the internal shock, which is responsible for the observed supernova events. Full computational simulations could shed more light into the complex energy transport that results from competing processes of scattering, interaction and decay.