Neutrino mixing and nucleosynthesis in core-collapse supernovae

A B Balantekin^1,2 and H Yüksel¹

¹ Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706, USA
² Physics Department, Tohoku University, Sendai 980, Japan

Email: baha@physics.wisc.edu and yuksel@physics.wisc.edu

Received 5 November 2004
Published 11 February 2005

Contents

• 1. Introduction
• 2. A simple description of core-collapse supernovae
  • 2.1. Neutrino-driven wind
  • 2.2. Neutrino fluxes and luminosities
  • 2.3. Reaction rates
  • 2.4. Electron fraction
  • 2.5. Alpha effect
• 3. Neutrino mixing
• 4. Results and discussion
• 5. Conclusions
• Acknowledgments
• References

1. Introduction

Massive stars, losing energy to radiation and photons, evolve until an iron core is formed (for a recent review see [1]). This core, which has a very low entropy per baryon, is supported by the electron degeneracy pressure. As a consequence, such a core is dynamically unstable and collapses until the matter is mostly neutronized and supernuclear densities are reached. Only when the density significantly exceeds the nuclear density, does the pressure become sufficiently repulsive to stop the collapse. In the current paradigm, the innermost shell of matter reaches to these densities first, rebounds and sends a pressure wave through the rest of the core. Such waves, produced by the subsequent shells and travelling faster than the infalling matter, collect near the sonic point. As that point reaches nuclear density, a shock wave breaks out. The subsequent evolution of this bounce shock is not yet well understood and is subject to much study [2, 3]. Current models fail to explode.

Even though it has not yet been demonstrated that explosion is an outcome of the core-collapse, it is well established that the newly formed hot proto-neutron star cools by neutrino emission. (It was those neutrinos that were observed in Supernova 1987A.) Almost all (99%) of the gravitational binding energy of the neutron star

is radiated away in neutrinos of all flavours. Altogether a star with mass ~8 M will emit ~10⁵⁹ neutrinos. Thus, after the explosion ejects the material from the outer layers, a `neutrino-driven' wind may blow the medium above the neutron star, heating it to an entropy per baryon of several hundreds in units of Boltzmann's constant. For such high entropies, nuclear statistical equilibrium is not established for nuclei heavier than alpha particles.

The rapid neutron-capture process (r-process) is responsible for the formation of a number of nuclei heavier than iron. (For a recent review see [4].) The astrophysical site of the r-process nucleosynthesis is not yet identified. For r-process nucleosynthesis to successfully take place, a large number of neutrons are required to interact in a relatively short time, indicating that r-process sites are associated with explosive phenomena. Indeed, the seminal Burbidge et al paper suggested the neutron-rich ejecta outside the core in a type II supernova as a possible site of the r-process [5]. More recent work pointed to the neutrino-driven wind in the supernovae as a possible site [6]-[8]. Meteoric data and observations of metal-poor stars indicate that r-process nuclei may be coming from diverse sources [9]. Binary neutron star systems were also proposed as a site of the r-process (see e.g. [10]). In outflow models, r-process nucleosynthesis results from the freeze-out from nuclear statistical equilibrium. The outcome of the freeze-out process is determined by the neutron-to-seed ratio. This ratio in the post-core-bounce supernova environment is controlled by the intense neutrino flux radiating from the neutron star.

Neutrino interactions play a crucial role in core-collapse supernovae (for a brief summary, see [11]). Neutrino heating is one of the possible mechanisms for reheating the stalled shock [12]. The neutrino fluxes control the proton-to-neutron ratio in the high-entropy hot bubble. As we describe in the next section, there is a hierarchy of energies for different neutrino flavours. Hence swapping active neutrinos via neutrino oscillations changes the n/p ratio and may alter the r-process nucleosynthesis conditions [13]. Neutrino oscillations in a core-collapse supernova differ from the matter-enhanced neutrino oscillations in the Sun as in the former there are additional effects coming from both neutrino-neutrino scattering [14, 15] and antineutrino flavour transformations [16].

We present a simple description of the core-collapse supernovae in the next section. In this section after summarizing properties of the neutrino-driven wind, we discuss neutrino fluxes, luminosities, reaction rates and the equilibrium electron fraction. A brief description of neutrino mixing and neutrino interactions, relevant to core-collapse supernovae, is given in section 3. In section 4 we calculate values of the electron fraction under several evolution scenarios that may impact r-process nucleosynthesis.

2. A simple description of core-collapse supernovae

2.1. Neutrino-driven wind

A careful treatment of the neutrino-driven wind in post-core bounce supernova environment was given in [17]. Here we present a heuristic description following [18]. One can assume that at sufficiently large radius, above the heating regime, there is hydrostatic equilibrium [19]:

where P is the hydrostatic pressure, G is Newton's constant, M_NS is the mass of the hot proto-neutron star and ρ is the matter density. Using the thermodynamic relation for the entropy at constant chemical potential, μ,

and on integrating equation (2), we can write the entropy per baryon, S, as

where m_B is the average mass of one baryon, which we take to be the nucleon mass. In the region above the neutron star, the material is radiation dominated and the entropy per baryon can be written in the relativistic limit as

where the statistical weight factor is given by

Assuming a constant entropy per baryon, equations (4) and (6) give the baryon density, ρ₃, in units of 10³ g cm ^- 3 as

where S₁₀₀ is the entropy per baryon in units of 100 times Boltzmann's constant, r₇ is the distance from the centre in units of 10⁷ cm, and we assumed that M_NS  =  1.4M. Defining T₉ to be the temperature in units of 10⁹ K, equation (4) takes the form

In figure 1, we present matter density and temperature profiles based on a heuristic description given in this section. Several values of S₁₀₀ can be used to describe stages in the evolution of supernovae. Smaller entropies per baryon, S₁₀₀    0.5, provide a better description of the shock reheating epoch, while larger values, S₁₀₀ 1, describe late times in supernova evolution, namely the neutrino-driven wind epoch. Higher entropy corresponds to less-ordered configurations with smaller baryon densities. In figure 1, the statistical weight factor, g_s, is taken to be 11/2 in the calculation of matter density, since temperature and entropy per baryon are T₉ 4 and S₁₀₀    1.5, respectively. Under these conditions both photons and electron-positron pairs are present in the plasma. When temperature drops, T₉    4, only photons are present in the medium and statistical weight factor has to be taken as g_s ≈ 2.

2.2. Neutrino fluxes and luminosities

We adopt the prescription in [14] for the neutrino fluxes. If we take the density of particles to be ρ, the number of particles that go through an expanding surface of radius r per unit time is

where N is the total number of particles and V  =  4πr³. Assuming a Fermi-Dirac distribution function for the neutrino number densities, ignoring neutrino mass in comparison to its energy and taking the particle velocity to be c, equation (9) gives for the flux of neutrinos emitted from the neutrinosphere as

where μ_ν is the neutrino chemical potential. Choosing the z-direction as the vector that connects the point where the flux is to be calculated (with radial position r) to the centre of the neutron star, one sees that the azimuthal symmetry still holds, but the polar angle is bounded by the finite size of the neutrinosphere. Thus

where

with R_ν being the radius of the neutrinosphere. Within this approximation one then obtains the differential neutrino flux as

Using similar reasoning one can write an expression for the neutrino luminosity. Replacing N in equation (9) by the total energy and the matter density ρ by the energy density one can write down

Again ignoring the neutrino mass in comparison to its energy and using equations (11) and (12) to do the angular integration we obtain

where η  =  μ_ν/kT_ν and F₃(η) is the relativistic Fermi integral

In our calculations, we take η  =  0 and use the value F₃(0)  =  7π⁴/120. Often it is convenient to express the neutrinosphere radius in terms of the neutrino luminosity. Rewriting R_ν in terms of L_ν and inserting it into equation (13), we get

2.3. Reaction rates

The dominant reactions that control the n/p ratio are the capture reactions on free nucleons

and

We take the cross sections for the forward reactions to be [16]

and

where Δ_np ≈ 1.293 MeV is the mass difference between neutron and proton. For simplicity we ignore weak magnetism and recoil corrections which may be important [20]. These corrections cancel for the former cross section, but add for the latter one. The rates of these reactions can be written as [21]

To calculate the neutrino capture rates of nuclei, one needs to include all possible transitions from the parent to the daughter nucleus, including not only the allowed transitions, but also transitions to the isobaric analogue states, Gamow-Teller resonance states, transitions into the continuum as well as forbidden transitions. Aspects of such calculations are discussed in [22, 23] (see also [24]). Careful input of such reaction rates in supernova simulations is especially crucial to assess the possibility of core-collapse supernovae as a site of r-process nucleosynthesis  [11, 25].

Because of their charged-current interactions, electron neutrinos may play a role in reheating the stalled shock as well as regulating the neutron-to-proton ratio. In contrast, since the energies of the muon and tau neutrinos and antineutrinos produced are too low to produce charged leptons, these neutrinos interact only with the neutral-current interactions. Recently, significant attention was directed towards understanding the ν_μ and ν_τ spectra formation. Neutrinos remain in local thermal equilibrium as long as they can participate in reactions that allow exchange of energy and neutrino pair creation or annihilation. It turns out that the neutrino bremsstrahlung process

is more effective than the annihilation process, , at equilibrating neutrino number density [26]. (However, the neutrino-neutrino annihilation process is one of the primary sources of muon and tau neutrinos [28]). The impact of the neutrino bremsstrahlung process on equilibrating the energy spectra seems to be comparable to that of

The most effective process to exchange energy is [27]

which dominates the neutrino spectra formation. Finally, recoil corrections to the νN interactions are very important in the formation of the ν_μ and ν_τ spectra as they permit energy exchange [29].

2.4. Electron fraction

The electron fraction, Y_e, is the net number of electrons (number of electrons minus the number of positrons) per baryon:

where , and n_B are number densities of electrons, positrons and baryons, respectively. Introducing N_j, the number of species of the kind j per unit volume, and A_j, the atomic weight of the jth species, one can write down expressions for the mass fraction, X_j as

and the number abundance relative to baryons, Y_j, as

The electron fraction defined in equation (26) can then be rewritten as

where Z_i is the charge of the species of kind i, and the mass fractions of protons, X_p, alpha particles, X_α and heavier nuclei (`metals'), X_h, are explicitly indicated.

The rate of change of the number of protons can be expressed as

where and are the rates of the forward and backward reactions in equation (18) and and are the rates of the forward and backward reactions in equation (19). Since the quantity does not change with neutrino interactions, one can rewrite equation (30) in terms of mass fractions

In the hot bubble, the rates are usually expressed in terms of the radial velocity field, v(r), above the neutron star, i.e. dY/dt  =  v(r)[dY/dr]. A careful study of the influence of nuclear composition on Y_e in the post-core bounce supernova environment is given in [30] to which we refer the reader for further details.

If no heavy nuclei are present, we can write

Because of the very large binding energy, the rate of alpha particle interactions with neutrinos is nearly zero and we can write dY_e/dt  =  dX_p/dt. Using the constraint X_p  +  X_n  +  X_α  =  1 and equation (32), equation (31) can be rewritten as

where we introduced the total proton loss rate and the total neutron loss rate . It has been shown that when the rates of these processes are rapid as compared to the outflow rate, a `weak chemical equilibrium' is established [13]. The weak freeze-out radius is defined to be where the neutron-to-proton conversion rate is less than the outflow rate of the material. If the plasma reaches a weak equilibrium stage then Y_e is no longer changing, i.e. dY_e/dt  =  0. From equation (33) one can write the equilibrium value of the electron fraction as

Different flavours of neutrinos decouple at different radii. Since ν_μ and ν_τ (and their antiparticles) interact with ordinary matter only through the neutral current interactions, they decouple deeper in the core and have a large average energy. Electron neutrinos and antineutrinos have additional charged-current interactions with neutrons and protons, respectively. Since in the supernova environment there are more neutrons, electron antineutrinos decouple after ν_μ's and ν_τ's, but before electron neutrinos. Consequently, one has a hierarchy of average neutrino energies:

where ν_x stands for any combination of ν_μ's and ν_τ's. However, a more complete description of the microphysics suggests that this hierarchy of average energies may not be very pronounced  [28, 29, 31]. This microphysics is dominated by the inelastic neutrino-nucleon interactions discussed in section 2.3.

In figure 2, we present initial differential neutrino fluxes. In figure 2(a) the typical post-bounce neutrino energies are taken as the representative values of and . (Average neutrino temperature for each flavour can be calculated using the relation, T_ν  = E_ν/3.151.) Here neutrino and antineutrino luminosities are taken to be equal for all flavours. We express the neutrino luminosities in more convenient units of 10⁵¹ ergs s ^- 1. Equal luminosities of L⁵¹_ν  =  1 is typically a good approximation for the neutrino-driven wind epoch. However, at earlier epochs, ν_e and luminosities can be as large as L⁵¹_ν  =  10. Except, through the neutronization burst, for few milliseconds ν_e luminosity can reach values of L⁵¹_ν  =  100, an order of magnitude larger than during the same period. In figure 2(b), we examine a less-pronounced hierarchy of neutrino energies. Here we adopt the representative values of , and . Here the luminosities of ν_x and are taken to be half of the value of the equal luminosities of ν_e and .

Figure 2. Initial differential neutrino fluxes in arbitrary units. Solid, dashed, dotted, and thick lines correspond to the distributions of νe, , νx and . (a) Fluxes with , and . Neutrino and antineutrino luminosities are taken to be equal for all flavours. Distributions of νx, have much longer tails up to 100 MeV not shown in the figure. (b) Fluxes with , and . νe and luminosities are taken to be equal. Luminosities of other flavours are taken to be half of the νe and luminosities.

In our calculations in section 4, we adapt these initial distributions of ν_e, , ν_x and at neutrinosphere, R_ν  =  10 km, with the indicated luminosities and follow the evolution of differential neutrino fluxes (number of neutrinos per unit energy per unit volume).

2.5. Alpha effect

At high temperatures, alpha particles are absent and the second term in equation (34) can be dropped. In the region just below where the alpha particles are formed, ~1 s after the bounce, the temperature is less than ~1 MeV. Here both the electron and positron capture rates are very small and Y_e can be approximated as

As the alpha particle mass fraction increases (when T₉ drops below 8) free nucleons get bound in alphas and, because of the large binding energy of the alpha particle, cease to interact with neutrinos. This phenomenon is called the `alpha effect' [22]. Using equation (36) one can rewrite equation (34) as

Hence if the initial electron fraction is small (Y_e⁽⁰⁾  <  1/2), the alpha effect increases the value of Y_e. Since higher Y_e implies fewer free neutrons, the alpha effect negatively impacts r-process nucleosynthesis [25].

Neutrino oscillations, since they can swap energies of different flavours, may affect the energy-dependent rates in equations (36) and (37), changing the electron fraction. Indeed, transformations between active flavours heat up ν_e's and increase , driving the electron fraction to rather large values. Consequently, a very large mixing between active neutrino flavours would have prohibited the r-process nucleosynthesis in a core-collapse supernova [13, 14].

Actually electron neutrinos radiated from the proto-neutron stars are just too energetic to prevent the alpha effect in most cases. One possibility to reduce is to convert active electron neutrinos into sterile ones which do not contribute to this rate. This possibility is explored in  [18, 32, 33].

3. Neutrino mixing

Neutrino interactions in matter is a rich subject (for a brief review see [34]). While neutrinos are produced through weak interactions in flavour eigenstates, they propagate in mass eigenstates. Mixing angles correspond to rotations describing unitary connection between two bases:

In our discussion, we follow the notation in [35] and denote the neutrino mixing matrix by U_αi, where α denotes the flavour index and i denotes the mass index:

In equation (39) C₁₃, etc is the shorthand notation for cos θ₁₃, etc. The notation S₁₃* was used to indicate (sin θ₁₃)eⁱ, where is a CP-violating phase. We will ignore this phase in our discussion.

We have compelling evidence supporting non-zero neutrino masses and mixings. Two-flavour solar neutrino solution corresponding to θ₁₂ ~ π/6 and δm₁₂² ~ 8 × 10 ^- 5 eV² was identified using the recent Sudbury Neutrino Observatory (SNO) results (cf [36, 37]). Measurement of antineutrinos from nuclear power reactors in Japan by the KamLAND experiment confirmed this solution and improved the limits on the solar mass-squared difference, δm₁₂², significantly [38]. While it is known that the solar neutrino mixing angle is large, but not maximal, the atmospheric mixing angle is large and could be maximal, θ₂₃ ~ π/4, as shown by the Super-Kamiokande experiment [39]. The latter angle is consistent with the KEK-to-Kamioka oscillation experiment, K2K [40]. The corresponding atmospheric mass-squared difference is δm₂₃² ~ 3 × 10 ^- 3 eV².

The size of the last mixing angle, θ₁₃ is currently best limited by the combined CHOOZ [41] and Palo Verde [42] reactor experiments and SK atmospheric data. This is due to the null results from the reactor disappearance over the δm₂₃² distance scale. The upper bound on this angle from KamLAND and the solar neutrino data gets stronger (especially in the region with small atmospheric mass-squared difference where the CHOOZ reactor bound is relatively weak), and even dominates, as these data get refined [43]. Both measurements of the width of the Z boson and oscillation interpretation of the neutrino data, with the notable exception of the LSND signal [44], favour three generations of light active neutrino species. If LSND is confirmed, the most likely explanation could be the existence of a fourth neutrino (a relatively heavier sterile neutrino), since agreement between KamLAND and the combined solar experiments already disfavours the interpretation of the LSND anomaly with CP violation.

In the region above the supernova, core density is still high but steeply decreases. Matter-enhanced oscillations mediated by the solar mass-squared difference are impossible in the region close to the core, which is suitable for the r-process. However, at the late neutrino-driven epoch, baryon density could be low enough to allow resonances through δm₁₃², which is comparable to the atmospheric mass-squared difference. At such late times, neutrino flux is expected to be low. We examine prospects of r-process at such environments in the next section.

We describe neutrino mixing within the density matrix formalism [14], [45]-[47]. We assume that the electron neutrino mixes with a linear combination of μ and τ neutrinos. The neutrino oscillations are mediated by the matter mixing angle for transformation between the first and third mass eigenstates, θ₁₃, and the corresponding atmospheric mass-squared difference, δm₁₃². The mixing between μ and τ neutrino flavours does not have any significant effect on our results as long as total luminosities and corresponding average energies are equal and their mixing is maximal. In this limit, mixing between the second and third mass eigenstates can be rotated away and effectively electron neutrinos oscillate into some linear combination of μ and τ neutrinos [48].

In the rest of the paper, θ and δm² refer to θ₁₃ and δm₁₃². This assumption simplifies the discussion and allows us to write the two-flavour density matrices as

and

where we introduced the polarization vectors for neutrinos and antineutrinos, P_p and . The diagonal elements in these expressions are initially given by the expression in equation (17). Non-diagonal elements are initially zero but may become non-zero during the neutrino evolution.

Equations governing the evolution of neutrinos and antineutrinos can be cast into the following forms [49]

and

where

Integrating equations (42) and (43) exactly in the supernova environment is, at the moment, an unsolved problem. Indeed, the exact solutions of these coupled, nonlinear differential equations are expected to be very complicated. Instead, we adopt the approximation proposed in [14] and also adopted in [49]. In this approximation, one uses flux-averaged values to obtain

and

In these equations J is the polarization integrated over all momentum modes and is the geometrical factor introduced earlier (cf equations (11) and (12)).

4. Results and discussion

In our calculations, we concentrate on the late neutrino-driven wind epoch which is expected to have a larger entropy. We adopt S₁₀₀  =  1.5 and the g_S  =  2, since in this epoch the temperature is too low at later times to have electron-positron pairs to be adequately represented.

In figure 3, we present differential neutrino fluxes at several stages of the evolution in order to provide a better understanding of this mechanism. The mixing parameters are chosen as θ ~ π/10 and δm² ~ 3 × 10 ^- 3 eV². Solid, dashed, dotted, and thick lines correspond to the distributions of ν_e, , ν_x and . In each panel, 1/r² dependence is removed. For this figure, the initial distributions are taken as the values given in figure 2(a). Columns are for L⁵¹  =  0.001, 0.1 and 50 from the left to the right corresponding to very weak, moderate and very strong neutrino self-interaction contributions to the evolution. Rows show neutrino flux at r  =  75, 100 and 150 km, exhibiting the evolution of neutrino distributions along the neutrino path. These should be compared to the initial (at ~10 km) neutrino distributions given in figure 2(a). Electron fractions corresponding to these luminosities are given in the left column of figure 4.

Figure 3. Evolution of differential neutrino fluxes in arbitrary units. Solid, dashed, dotted, and thick lines correspond to the distributions of νe, νμ, and . In each panel, 1/r2 dependence is removed. Columns are for L51  =  0.001, 0.1 and 50 from the left to the right corresponding to very weak, moderate and very strong neutrino self-interaction contributions to the evolution. Rows are calculated at r  =  75, 100 and 150 km showing the evolution of distributions along a neutrino path (see the text for details).

Figure 4. Initial neutrino fluxes and luminosities are taken to be those in figure 2(a) in the left column and those in figure 2(b) in the right column. S100 is taken to be 1.5. In panels (a) and (b), solid, dashed and dotted, lines correspond to the equal luminosities of L51  =  0.001, 0.1 and 50 for all flavours, indicating very weak, moderate and very-strong neutrino self-interaction contributions to the evolution. In panels (d) and (e), νe and luminosities are taken to be L51  =  0.002, 0.2 and 200 (solid, dashed, and dotted lines, respectively). In (d) and (e), the luminosities of other flavours are taken to be L51  =  0.001, 0.1 and 100 (solid, dashed, and dotted lines, respectively). (a, d) Equilibrium electron fraction as a function of the distance from the core with mixing parameters θ13 ~ π/10 and δm132 ~ 3 × 10 - 3 eV2. (b, e) Equilibrium Ye as a function of the distance from the core with mixing parameters θ13 ~ π/20 and δm132 ~ 3 × 10 - 3 eV2. (c) Same as (b) but the impact of alpha particle formation is included according to the equation (34). Ye is shown when Xα  =  0, 0.3 and 0.5 (thin, medium and thick lines) for L51  =  0.001 and 50 (solid and dotted sets of lines). (f) Same as (e) but the impact of alpha particle formation is included as in (c). Ye is shown when Xα  =  0, 0.3 and 0.5 (thin, medium and thick lines) for and 200 (solid and dotted sets of lines).

First column of figure 3, L⁵¹  =  0.001: Initially differential neutrino fluxes for ν_e and ν_x are the same at ~20 MeV as seen in figure 2(a). At energies lower than 20 MeV, differential flux of ν_e is higher than that of ν_x, whereas at energies higher than 20 MeV the numbers reverse. The transformation starts at the low energy tail of the distributions. As we gradually move to lower densities, resonance moves up to higher energies. Until we reach the resonance at 20 MeV, the luminosity of ν_e's decreases. At r  =  100 km electron neutrinos below 20 MeV are mostly swapped with ν_x's, while above 20 MeV there is yet no significant transformation. This point corresponds to the minimum of λ in equation (22) for neutrinos (after the 1/r²-dependence is taken out). Since antineutrinos are not transformed, λ is a constant for them. As a result, r ~ 100 km corresponds to a dip in Y_e (solid line in figure 4(a)). Since above 20 MeV the initial differential flux of ν_e's is smaller than that of ν_x's, transformations after r ~ 100 km would increase the number of high energy ν_e's. As neutrinos travel to regions further away (r  =  150 km, where baryon density is much lower), swapping of the high energy tail of the ν_e and ν_x distributions is completed, and Y_e approaches to the asymptotic fixed value.

Second column of figure 3, L⁵¹  =  0.1: This column describes the same evolution of neutrinos except that neutrino self-interaction effects play a more pronounced role. The chosen value of the luminosity could be representative of such late times in neutrino-driven wind epoch. Because of the small contributions of the self-interactions, the resonance region is relatively wider. No dip is observed in Y_e (dashed line in figure 4(a)) at r  =  100 km because the transformation of low and high energy ends of the distributions happen almost simultaneously. There is also a small transformation between 's and 's.

Third column of figure 3, L⁵¹  =  50: This is an extreme case to illustrate the limit at which neutrino self-interactions dominate. Exchanges of both neutrinos and antineutrinos occur and Y_e reaches its equilibrium value rapidly. After the transformation, electron neutrinos and antineutrinos assume similar luminosities and distributions since they are swapped with ν_x's and 's. Note that this equilibrium value is again different from 0.5 because of the threshold effects (due to the neutron-proton mass difference) in the reaction cross sections (cf equations (20) and (21)).

In the left column of figure 4, initial neutrino fluxes and luminosities are taken to be those in figure 2(a) and in the right column initial neutrino fluxes and luminosities are taken to be those in figure 2(b). In figure 4(a), we present the equilibrium electron fraction, Y_e, as a function of the distance from the core for the three different cases of neutrino flux, L⁵¹  =  0.001, 0.1 and 50, each corresponding to one of the columns in figure 3. In figure 4(a), we use the same neutrino parameters as in figure 3. One could argue that the relatively large mixing angle of π/10 is already disfavoured by CHOOZ. To explore the implications of a smaller mixing angle in figure 4(b), we show electron fractions calculated with the more realistic mixing parameters θ ~ π/20 and δm² ~ 3 × 10 ^- 3 eV². The dip in the Y_e plot is sharper since the resonance region will be much narrower with a smaller mixing angle.

In figures 4(a) and (b), to calculate the equilibrium electron fraction, we ignored possible effects of the alpha particles and used equation (36) to evaluate Y_e. In figure 4(c), we explore possible effects of alpha particles by using equation (34) to calculate Y_e for three different values of X_α. The `alpha effect' is manifest: as X_α gets larger, it pulls the value of the electron fraction closer to 0.5.

In the right column of figure 4, initial neutrino fluxes and luminosities are taken as the generic `almost equal' energies and different luminosities case shown in figure 2(b). It is immediately obvious that the situation is markedly different in this case. Even though the initial ν_e and spectra are almost the same (cf figure 2(b)), Y_e is, initially greater than 0.5 because of the effect of the neutron-proton mass difference in equations (20) and (21). When the effects of the neutrino self-interaction terms are minimal (the low-luminosity case indicated by the solid lines in (d) and (e)) after ν_e's and ν_x's swap, ν_e luminosity decreases significantly causing a big drop in the value of Y_e. When the effects of the self-interaction terms are dominant (the high-luminosity cases indicated by the dotted lines in (d) and (e)), all flavours swap and the electron fraction eventually reaches the asymptotic values. The impact of alpha formation, when the hierarchy of average neutrino energies is less pronounced, is presented in figure 4(f).

5. Conclusions

In this paper we investigated conditions for the r-process nucleosynthesis at late neutrino-driven wind epoch in a core-collapse supernova. We considered the region where the matter-enhanced neutrino transformation is driven by δm₁₃², which we took to be comparable to the mass difference observed in the atmospheric neutrino oscillations.

We found that, when initial luminosities are taken to be equal with a pronounced hierarchy of neutrino energies, the asymptotic value (at large distances from the core) of the electron fraction always exceeds 0.5, hindering r-process nucleosynthesis. In this case, we found that neutrino self-interactions decrease the electron fraction. In general, these self-interaction terms, unlike the MSW effect, tend to transform both neutrinos and antineutrinos. Hence, one expects that when the self-interaction terms are dominant (e.g. when the neutrino luminosities are very large) the electron fraction would reach the value of 0.5. However, we showed that, because of the threshold effects in the neutrino interactions, electron fraction exceeds 0.5 even when the background neutrinos are in abundance. Clearly, these conditions are not favourable for r-process nucleosynthesis. We also found that, when background effects are small, with the parameters we adopted for baryon density and neutrino mixing, there exists a region around 100 km in which the electron fraction is smaller than 0.5. Even in this region, the alpha effect pulls the value of the electron fraction closer to 0.5. Although the conditions are favourable, the r-process in this region could not contribute reasonable quantities of elements since the baryon density is rather low.

In contrast, when the initial luminosities of the ν_e's and 's are taken to be twice the other flavours with a much less-pronounced neutrino hierarchy, we found that the asymptotic electron fraction is less than 0.5 when neutrino self-interactions are negligible. However, once again the baryon density is low and the r-process in this asymptotic region is likely to produce insignificant quantities. When neutrino self-interactions are dominant, the asymptotic electron fraction shows the same behaviour as the first generic case (with a pronounced hierarchy of neutrino energies and equal luminosities) we considered.

It is not surprising that the two cases are markedly different for low neutrino luminosities (late times). If all the initial average neutrino energies and luminosities were the same, neutrino oscillations would clearly have no impact. Our analysis highlights the significance of having a precise knowledge of the muon and tau neutrino spectra and luminosities in assessing core-collapse supernovae as a site of r-process nucleosynthesis. In both of the two generic, but markedly different, cases we studied and found that neutrino self-interactions are crucial in setting the value of the electron-fraction to be rather large. We should emphasize that our treatment of the neutrino self-interactions in equations (45) and (46) is an approximation using flux-averaged values. Hence an exact treatment of the neutrino self-interactions remains the key issue in the proper description of the neutrino transport in core-collapse supernovae.

Acknowledgments

We thank G Fuller and G McLaughlin for useful conversations. ABB also thanks the Nuclear Theory Group at Tohoku University for their hospitality during the latter part of this work. This work was supported in part by the US National Science Foundation grant No PHY-0244384 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation.

References

[1]

Woosley S E, Heger A and Weaver T A 2002 Rev. Mod. Phys. 74 1015 
CrossRef

[2]

Buras R, Rampp M, Janka H T and Kifonidis K 2003 Phys. Rev. Lett. 90 241101 (Preprint astro-ph/0303171)
CrossRefPreprintPubMed

[3]

Mezzacappa A 2004 Preprint astro-ph/0410085
Preprint

[4]

Qian Y Z 2003 Prog. Part. Nucl. Phys. 50 153 (Preprint astro-ph/0301422)
CrossRefPreprint

[5]

Burbidge E M, Burbidge G R, Fowler W A and Hoyle F 1957 Rev. Mod. Phys. 29 29 
CrossRef

[6]

Woosley S E, Wilson J R, Mathews G J, Hoffman R D and Meyer B S 1994 Astrophys. J. 433 229 
CrossRef

[7]

Takahashi K, Witti J and Janka H-T 1994 Astron. Astrophys. 286 857 

[8]

Wanajo S, Kajino T, Mathews G J and Otsuki K 2001 Astrophys. J. 554 578 (Preprint astro-ph/0102261)
IOPsciencePreprint

[9]

Qian Y Z, Vogel P and Wasserburg G J 1998 Astrophys. J. 494 285 (Preprint astro-ph/9706120)
IOPsciencePreprint

[10]

Rosswog S, Liebendoerfer M, Thielemann F K, Davies M B, Benz W and Piran T 1999 Astron. Astrophys. 341 499 (Preprint astro-ph/9811367)
Preprint

[11]

Balantekin A B and Fuller G M 2003 J. Phys. G: Nucl. Part. Phys. 29 2513 (Preprint astro-ph/0309519)
IOPsciencePreprint

[12]

Bethe H A and Wilson J R 1985 Astrophys. J. 295 14 
CrossRef

[13]

Qian Y Z, Fuller G M, Mathews G J, Mayle R, Wilson J R and Woosley S E 1993 Phys. Rev. Lett. 71 1965 
CrossRefPubMed

[14]

Qian Y Z and Fuller G M 1995 Phys. Rev. D 51 1479 (Preprint astro-ph/9406073)
CrossRefPreprint

[15]

Pantaleone J 1995 Phys. Lett. B 342 250 (Preprint astro-ph/9405008)
CrossRefPreprint

[16]

Qian Y Z and Fuller G M 1995 Phys. Rev. D 52 656 (Preprint astro-ph/9502080)
CrossRefPreprint

[17]

Qian Y Z and Woosley S E 1996 Astrophys. J. 471 331 (Preprint astro-ph/9611094)
IOPsciencePreprint

[18]

McLaughlin G C, Fetter J M, Balantekin A B and Fuller G M 1999 Phys. Rev. C 59 2873 (Preprint astro-ph/9902106)
CrossRefPreprint

[19]

Bethe H A 1993 Astrophys. J. 412 192 
CrossRef

[20]

Horowitz C J and Li G 1999 Phys. Rev. Lett. 82 5198 (Preprint astro-ph/9904171)
CrossRefPreprint
Horowitz C J 2002 Phys. Rev. D 65 043001 (Preprint astro-ph/0109209)
CrossRefPreprint
Horowitz C J and Perez-Garcia M A 2003 Phys. Rev. C 68 025803 (Preprint astro-ph/0305138)
CrossRefPreprint

[21]

Fuller G M, Fowler W A and Newman M J 1982 Astrophys. J. 252 715 
CrossRef

[22]

Fuller G M and Meyer B S 1995 Astrophys. J. 453 792 
CrossRef

[23]

McLaughlin G C and Fuller G M 1995 Astrophys. J. 455 202 
CrossRef

[24]

Volpe C 2004 Preprint hep-ph/0409249
Preprint

[25]

Meyer B S, McLaughlin G C and Fuller G M 1998 Phys. Rev. C 58 3696 (Preprint astro-ph/9809242)
CrossRefPreprint

[26]

Suzuki H 1993 Proc. Int. Symp. on Neutrino Astrophysics ed Y Suzuki and K Nakamura (Tokyo: Universal Academy Press) 

[27]

Hannestad S and Raffelt G 1998 Astrophys. J. 507 339 (Preprint astro-ph/9711132)
IOPsciencePreprint

[28]

Buras R, Janka H T, Keil M T, Raffelt G G and Rampp M 2003 Astrophys. J. 587 320 (Preprint astro-ph/0205006)
IOPsciencePreprint

[29]

Raffelt G G 2001 Astrophys. J. 561 890 (Preprint astro-ph/0105250)
IOPsciencePreprint

[30]

McLaughlin G C, Fuller G M and Wilson J R 1996 Astrophys. J. 472 440 (Preprint astro-ph/9701114)
IOPsciencePreprint

[31]

Keil M T, Raffelt G G and Janka H T 2003 Astrophys. J. 590 971 (Preprint astro-ph/0208035)
IOPsciencePreprint

[32]

Caldwell D O, Fuller G M and Qian Y Z 2000 Phys. Rev. D 61 123005 (Preprint astro-ph/9910175)
CrossRefPreprint

[33]

Fetter J, McLaughlin G C, Balantekin A B and Fuller G M 2003 Astropart. Phys. 18 433 (Preprint hep-ph/0205029)
CrossRefPreprint

[34]

Balantekin A B 1999 Phys. Rep. 315 123 (Preprint hep-ph/9808281)
CrossRefPreprint

[35]

Balantekin A B and Yuksel H 2003 J. Phys. G: Nucl. Part. Phys. 29 665 (Preprint hep-ph/0301072)
IOPsciencePreprint

[36]

Ahmed S N SNO Collaborationet al 2004 Phys. Rev. Lett. 92 181301 (Preprint nucl-ex/0309004)
CrossRefPreprintPubMed

[37]

Balantekin A B and Yuksel H 2003 Phys. Rev. D 68 113002 (Preprint hep-ph/0309079)
CrossRefPreprint

[38]

Araki T KamLAND Collaborationet al 2004 Preprint hep-ex/0406035
Preprint

[39]

Ashie Y Super-Kamiokande Collaborationet al 2004 Phys. Rev. Lett. 93 101801 (Preprint hep-ex/0404034)
CrossRefPreprintPubMed

[40]

Ahn M H K2K Collaborationet al 2003 Phys. Rev. Lett. 90 041801 (Preprint hep-ex/0212007)
CrossRefPreprintPubMed

[41]

Apollonio M et al 2003 Eur. Phys. J. C 27 331 (Preprint hep-ex/0301017)
CrossRefPreprint

[42]

Boehm F et al 2001 Phys. Rev. D 64 112001 (Preprint hep-ex/0107009)
CrossRefPreprint

[43]

Balantekin A B, Barger V, Marfatia D, Pakvasa S and Yuksel H 2004 Preprint hep-ph/0405019
Preprint

[44]

Aguilar A LSND Collaborationet al 2001 Phys. Rev. D 64 112007 (Preprint hep-ex/0104049)
CrossRefPreprint

[45]

Sigl G and Raffelt G 1993 Nucl. Phys. B 406 423 
CrossRef

[46]

Raffelt G, Sigl G and Stodolsky L 1993 Phys. Rev. Lett. 70 2363 (Preprint hep-ph/9209276)
CrossRefPreprintPubMed

[47]

Loreti F N and Balantekin A B 1994 Phys. Rev. D 50 4762 (Preprint nucl-th/9406003)
CrossRefPreprint

[48]

Balantekin A B and Fuller G M 1999 Phys. Lett. B 471 195 (Preprint hep-ph/9908465)
CrossRefPreprint

[49]

Pastor S and Raffelt G 2002 Phys. Rev. Lett. 89 191101 (Preprint astro-ph/0207281)
CrossRefPreprintPubMed
Pastor S, Raffelt G G and Semikoz D V 2002 Phys. Rev. D 65 053011 (Preprint hep-ph/0109035)
CrossRefPreprint