Collective Neutrino Oscillations and Heavy-element Nucleosynthesis in Supernovae: Exploring Potential Effects of Many-body Neutrino Correlations

Here, we shall describe how the charged-current neutrino and antineutrino capture rates are calculated in this simplified neutrino problem. These rates depend on the neutrino number fluxes across the energy spectrum and the corresponding cross sections. For example, for ν_e capture on neutrons, one has

$$\begin{eqnarray}&&{\lambda }_{{\nu }_{e}n}({\boldsymbol{r}},t)={\int }_{{\boldsymbol{p}}}{\sigma }_{{\nu }_{e}n}({\boldsymbol{p}}){{dn}}_{{\nu }_{e}}({\boldsymbol{r}},t,{\boldsymbol{p}}),\end{eqnarray} \tag{ 1 }$$

where ${\sigma }_{{\nu }_{e}n}$ is the capture cross section and ${{dn}}_{{\nu }_{e}}({\boldsymbol{r}},t,{\boldsymbol{p}})$ is the differential number density of electron neutrinos with momentum p , at location r and time t. The limits of integration on the neutrino momentum directions in Equation (1) are dependent on the geometry of the neutrino source. In the rest of the discussion, we shall assume that there is no explicit time dependence in the problem, so that the geometry and neutrino emission characteristics are fixed with respect to time. ¹⁰ Hence, the label t shall be omitted henceforth. The neutrino capture rates do still have an implicit time dependence, encoded in the motion r(t) of a fluid element as it travels outward from the PNS.

Here, it is useful to define the quantity ${{dn}}_{\nu ,\alpha }({\boldsymbol{r}},{\boldsymbol{p}})$ as the differential number density of the subset of neutrinos at location r with momentum p , which were emitted specifically with an initial flavor α. In other words, ${{dn}}_{\nu ,\alpha }({\boldsymbol{r}},{\boldsymbol{p}})$ would represent the differential neutrino number density in flavor α, in the absence of any flavor evolution. One can then write ${{dn}}_{{\nu }_{e}}$ in terms of ${{dn}}_{\nu ,\alpha }$ for each flavor and the corresponding flavor conversion probabilities P_{α e} as follows:

$$\begin{eqnarray}&&{{dn}}_{{\nu }_{e}}({\boldsymbol{r}},{\boldsymbol{p}})=\displaystyle \sum _{\alpha }{{dn}}_{\nu ,\alpha }({\boldsymbol{r}},{\boldsymbol{p}}){P}_{\alpha e}({\boldsymbol{r}},{\boldsymbol{p}}).\end{eqnarray} \tag{ 2 }$$

In what follows, we shall assume a spherically symmetric neutrino source with a sharp decoupling surface at r = R_ν and isotropic emission in all outward directions from each point on the surface (i.e., the neutrino bulb geometry from Duan et al. 2006). Under this assumption, the differential number densities ${{dn}}_{\nu ,\alpha }$ lose their dependence on the direction of the neutrino momenta and can instead be written simply in terms of the neutrino energy as

$$\begin{eqnarray}&&{{dn}}_{\nu ,\alpha }(r,E)=\displaystyle \frac{1}{2{\pi }^{2}{\left({\hslash }c\right)}^{3}}\displaystyle \frac{d\cos \vartheta \,d\phi }{4\pi }{f}_{\nu ,\alpha }(E){E}^{2}{dE},\end{eqnarray} \tag{ 3 }$$

where f_ν,α (E) is the initial occupation number as a function of energy, and ϑ and ϕ are the polar and azimuthal angles, respectively, relative to the radial direction at distance r from the center of the neutrino source. With these symmetry assumptions, d ϕ integrates to 2π, and the limits of integration for $\cos \vartheta$ depend on r (e.g., see Figure 1). Henceforth, $\hslash$ and $c$ are omitted from all the equations (natural units are used).

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Flavor oscillations in dense neutrino streams may be substantially impacted by coherent ν–ν forward scattering. This interaction causes the flavor evolution histories of neutrinos with different energies/momenta to become coupled to one another. Moreover, these interactions may also lead to quantum entanglement among neutrinos. In essence, an interacting neutrino system undergoing flavor evolution is a quantum many-body system described by a Hamiltonian with one- and two-body interactions, the latter of which are long range in momentum space. The "one-body" terms describe vacuum oscillations of each neutrino as well as neutrino coherent forward scattering with ordinary matter (electrons and nucleons; Wolfenstein  1978; Mikheyev & Smirnov 1985), whereas the two-body term represents ν–ν interactions (Pantaleone 1992a, 1992b). In regimes where ν–ν interactions dominate, the neutrino-matter forward scattering term can often be "rotated away" with a suitable change-of-basis transformation, at the expense of an in-medium modification to the mixing angle and mass-squared splitting. For simplicity, we neglect this term in our calculation. For a system of N neutrinos with discrete momenta (and, therefore, discrete energies) quantized as plane waves in a box of volume V, this Hamiltonian in a two-flavor approximation can be written as

$$\begin{eqnarray}&&H=\displaystyle \sum _{{\boldsymbol{p}}}{\omega }_{{\boldsymbol{p}}}\,\vec{B}\cdot {\vec{J}}_{{\boldsymbol{p}}}+\displaystyle \sum _{{\boldsymbol{p}},{\boldsymbol{q}}}{\mu }_{{\boldsymbol{pq}}}\,{\vec{J}}_{{\boldsymbol{p}}}\cdot {\vec{J}}_{{\boldsymbol{q}}},\end{eqnarray} \tag{ 4 }$$

where ω _p ≡ δ m²/(2∣ p ∣) are the vacuum oscillation frequencies of the neutrinos (with δ m² being the mass-squared splitting). ${\vec{J}}_{{\boldsymbol{p}}}$ is a vector of operators forming an SU(2) algebra and representing the neutrino "isospin" in the flavor or mass basis (for details, see Balantekin & Pehlivan 2007; Pehlivan et al. 2011). Additionally, ${\mu }_{{\boldsymbol{pq}}}\equiv (\sqrt{2}{G}_{F}/V)\,(1-\cos {\theta }_{{\boldsymbol{pq}}})$ is the interaction strength between the neutrinos, ¹¹ which in general depends on the intersection angle θ _pq between their momenta, and $\vec{B}\,=\,(0,0,-1)$ in the mass basis and $(\sin 2\theta ,0,-\cos 2\theta )$ in the flavor basis, where θ is the neutrino mixing angle. Note that we use bold symbols to denote vectors in position/momentum space and overhead arrows to denote vectors in neutrino isospin space.

We assume the neutrinos to be produced in an initially uncorrelated state at the PNS surface, and therefore their N-body wave function has a direct product form:

$$\begin{eqnarray}&&| {\rm{\Psi }}(r={R}_{\nu })\rangle =\mathop{\displaystyle \bigotimes }\limits_{{\boldsymbol{p}}}| {\psi }_{{\boldsymbol{p}}}\rangle .\end{eqnarray} \tag{ 5 }$$

Since the neutrinos are produced by weak interactions, one typically assumes that each ∣ψ _p 〉 is initially in a flavor state (i.e., ∣ν_e 〉 or ∣ν_x 〉 in the two-flavor approximation, where ∣ν_x 〉 is a superposition of ∣ν_μ 〉 and ∣ν_τ 〉). The wave function ∣Ψ〉 evolves in time according to the Schrödinger equation: ¹²

$$\begin{eqnarray}&&i\displaystyle \frac{d}{{dt}}| {\rm{\Psi }}\rangle =H| {\rm{\Psi }}\rangle ,\end{eqnarray} \tag{ 6 }$$

where H is the many-body neutrino Hamiltonian from Equation (4). In general, the ν–ν interaction term of the Hamiltonian can drive the evolution of the wave function away from a separable product state, resulting in development of nontrivial quantum entanglement between the constituent neutrinos. In this situation, a complete description of the many-body evolution as a function of time requires a Hilbert space that is 2^N -dimensional, where N is the number of neutrinos.

The flavor evolution of individual neutrinos in the ensemble can be quantified in terms of the expectation values of certain one-body operators within each individual neutrino subspace—for instance, the components of the isospin ${\vec{J}}_{{\boldsymbol{q}}}$ in the flavor or mass basis. One can define the neutrino "polarization vectors" ${\vec{P}}_{{\boldsymbol{q}}}=2\langle {\vec{J}}_{{\boldsymbol{q}}}\rangle$, which contain information about the flavor composition of the neutrino, with P_z denoting the net lepton number in the flavor or mass basis and the P_x and P_y components representing the coherence between different flavor or mass eigenstates. The relation between the polarization vector components in the flavor (f) and mass (m) bases is given by

$$\begin{eqnarray}&&{P}_{z}^{(f)}=\cos (2\theta )\,{P}_{z}^{(m)}+\sin (2\theta )\,{P}_{x}^{(m)},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{P}_{x}^{(f)}=\cos (2\theta )\,{P}_{x}^{(m)}-\sin (2\theta )\,{P}_{z}^{(m)},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{P}_{y}^{(f)}={P}_{y}^{(m)},\end{eqnarray} \tag{ 9 }$$

where θ is the neutrino mixing angle. The electron flavor survival probability, which is required for calculating the (anti-)neutrino capture rates, can then be obtained using ${P}_{\alpha e}=(1+\mathrm{sign}(\omega )\,{P}_{z}^{(f)})/2$. Here, sign(ω) = + 1 for neutrinos and −1 for antineutrinos.

The exponential scaling of the complexity of this many-body system as a function of particle number makes this problem numerically intractable even for a relatively small number (∼a few tens) of neutrinos in the most general case. To circumvent this difficulty, a frequently adopted approach involves neglecting any nontrivial quantum correlations among the neutrinos, thereby enforcing that the neutrino wave function (initially in a product state) forever remains in a product state. ¹³ Effectively, this simplification amounts to replacing operator products in the Hamiltonian with one-body terms proportional to an operator times an expectation value. One can then reduce the Hamiltonian to an effective one-body operator form (Balantekin 2018b):

$$\begin{eqnarray}&&H=\displaystyle \sum _{{\boldsymbol{p}}}{\omega }_{{\boldsymbol{p}}}\,\vec{B}\cdot {\vec{J}}_{{\boldsymbol{p}}}+\displaystyle \sum _{{\boldsymbol{p}},{\boldsymbol{q}}}{\mu }_{{\boldsymbol{pq}}}\,\left[{\vec{J}}_{{\boldsymbol{p}}}\cdot \left\langle {\vec{J}}_{{\boldsymbol{q}}}\rangle +\langle {\vec{J}}_{{\boldsymbol{p}}}\rangle \cdot {\vec{J}}_{{\boldsymbol{q}}}-\langle {\vec{J}}_{{\boldsymbol{p}}}\rangle \cdot \langle {\vec{J}}_{{\boldsymbol{q}}}\right\rangle \right].\end{eqnarray} \tag{ 10 }$$

Ehrenfest's theorem then dictates that the expectation values of the isospin operators $\langle {\vec{J}}_{{\boldsymbol{q}}}\rangle \equiv \langle {\rm{\Psi }}|{\vec{J}}_{{\boldsymbol{q}}}|{\rm{\Psi }}\rangle$ evolve according to

$$\begin{eqnarray}&&\displaystyle \frac{\text{d}\langle {\vec{J}}_{{\boldsymbol{q}}}\rangle }{\text{d}t}=-i\langle [{\vec{J}}_{{\boldsymbol{q}}},H]\rangle .\end{eqnarray} \tag{ 11 }$$

Using the SU(2) commutation relations for the isospin operators, together with the mean-field condition from Equation (10), one obtains the mean-field evolution equations for the neutrino polarization vectors, ${\vec{P}}_{{\boldsymbol{q}}}$:

$$\begin{eqnarray}&&\displaystyle \frac{d{\vec{P}}_{{\boldsymbol{q}}}}{{dt}}={\omega }_{{\boldsymbol{q}}}\,\vec{B}\times {\vec{P}}_{{\boldsymbol{q}}}+\displaystyle \sum _{{\boldsymbol{p}}}{\mu }_{{\boldsymbol{pq}}}\,{\vec{P}}_{{\boldsymbol{p}}}\times {\vec{P}}_{{\boldsymbol{q}}}.\end{eqnarray} \tag{ 12 }$$

Certain practical differences between the many-body and mean-field treatments can be quantified by comparison of their predictions for the evolution of polarization vectors. In the many-body treatment, one can use Equation (6), together with the Hamiltonian from Equation (4), to evolve the many-body wave function ∣Ψ〉 and then compute the polarization vectors ${\vec{P}}_{{\boldsymbol{q}}}=2\langle {\vec{J}}_{{\boldsymbol{q}}}\rangle$ at each time step. These results can then be compared to the polarization vectors of the corresponding mean-field solution, obtained by integrating Equation (12).

Neutrinos play a vital role in supernova neutrino-driven wind nucleosynthesis processes through capture reactions on free nucleons:

$$\begin{eqnarray}&&{\nu }_{e}+n\rightleftharpoons p+{e}^{-},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\bar{\nu }}_{e}+p\rightleftharpoons n+{e}^{+}.\end{eqnarray} \tag{ 18 }$$

These capture reactions shape the neutron-to-proton ratio (n_n /n_p ) and the electron fraction Y_e = 1/(1 + n_n /n_p ) in the initial phase of the outflow (at a high temperature and density), which determine what type of nucleosynthesis is possible in the ejecta and influence the availability of free nucleons to capture as element synthesis proceeds. We start each nucleosynthesis calculation from T ∼ 10 GK, when both electron and positron capture rates are very small compared to neutrino capture rates and the ejecta are in a weak equilibrium stage. At this stage, the electron fraction Y_e can be approximated as

$$\begin{eqnarray}&&{Y}_{e}\approx \displaystyle \frac{1}{1+{\lambda }_{{\bar{\nu }}_{e}}/{\lambda }_{{\nu }_{e}}}.\end{eqnarray} \tag{ 19 }$$

In this work, we have investigated the nucleosynthesis outcomes for different choices of initial neutrino energies (or, equivalently, oscillation frequencies ω_q ), roughly based on supernova simulations (e.g., Keil et al. 2003; Mirizzi et al. 2016; Horiuchi et al. 2018). Broadly, these choices are classified as "symmetric" (sym) or "asymmetric" (asym), based on whether the initial ${\bar{\nu }}_{e}$ average energies are equal to or greater than the initial ν_e average energies. For the calculations with N = 4 neutrino modes, we assign one mode each to the ν_e , ${\bar{\nu }}_{e}$, ν_x , and ${\bar{\nu }}_{x}$ species, respectively (N_ν,α = 1 for each α). For the N = 8 case, we assign two modes to each of the species (N_ν,α = 2). Table 2 summarizes the parameters for each of these initial neutrino distribution choices: the energies E_ν,α and the corresponding luminosities L_ν,α , such that Equation (14) is satisfied, given the values of R_ν and μ(R_ν ) from Table 1. The resulting initial electron fractions to be used for the heavy-element nucleosynthesis simulations are also included in the final column.

Table 2. Initial Conditions for Neutrino Energy Distributions and Electron Fraction

Label	Eν,e	${E}_{\bar{\nu },e}$	Eν,x	${E}_{\bar{\nu },x}$	Lν,e	${L}_{\bar{\nu },e}$	Lν,x	Ye
	(MeV)	(MeV)	(MeV)	(MeV)	(erg s−1)	(erg s−1)	(erg s−1)	${\left(1+{\lambda }_{{\bar{\nu }}_{e}}/{\lambda }_{{\nu }_{e}}\right)}^{-1}$
sym	10	10	20	20	9.091 × 1051	9.091 × 1051	1.818 × 1052	0.634
asym2	10	12.5	20	20	9.091 × 1051	1.136 × 1052	1.818 × 1052	0.504
asym2.1	10	13	20	20	9.091 × 1051	1.182 × 1052	1.818 × 1052	0.482
asym3	10	14.28	20	20	9.091 × 1051	1.298 × 1052	1.818 × 1052	0.427
asym4	10	16	20	20	9.091 × 1051	1.455 × 1052	1.818 × 1052	0.366
sym-4nu	10, 11.11	10, 11.11	16.67, 20	16.67, 20	9.591 × 1051	9.591 × 1051	1.667 × 1052	0.634
asym2.1-4nu	10, 11.11	12.8, 14.3	16.67, 20	16.67, 20	9.591 × 1051	1.232 × 1052	1.667 × 1052	0.482

Note. We initiate the oscillations at r_i ≃ 100 km, where μ_i = 100 ω₀ (50 ω₀ for the $4\nu +4\bar{\nu }$ cases: sym-4nu and asym2.1-4nu).

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To test the potential effects of neutrino interactions on the nucleosynthesis results, for each set of calculations, we consider four different neutrino treatments: no neutrinos involved for T < 10 GK (nn), no neutrino oscillations (nosc), mean-field neutrino oscillations (mf), described in Section 2.3, and many-body neutrino oscillations (mb), described in Section 2.2. The default neutrino number used for our calculations is 2 ν + 2 $\bar{\nu };$ we also tested a symmetric  and an asymmetric scenario (sym-4nu and asym2.1-4nu, respectively, as shown in Table 2) with an increased neutrino number: 4 ν + 4 $\bar{\nu }$. Here, we adopted the normal mass ordering for the neutrino flavor evolution as default (i.e., ω _p > 0 for neutrinos and <0 for antineutrinos); the effect of the inverted mass hierarchy scenario is also examined, as discussed in Section 5.1.2.

Figure 2 shows an example comparison between the mb and mf calculations for the flavor evolution of a 2 ν + 2 $\bar{\nu }$ neutrino system in the sym configuration. In particular, we show the evolution (i.e., with distance r) of neutrino polarization vector component P_z (ω) in the mass basis, evolution of the average ν_e and ${\bar{\nu }}_{e}$ energies, as well as of the ν_e and ${\bar{\nu }}_{e}$ capture rates. The vacuum oscillation frequencies of the neutrinos, calculated using their average energy through ω_j = δ m²/2E_j , are ω₁, ω₂, ω₃, and ω₄ for neutrinos initially in the ${\bar{\nu }}_{e}$, ${\bar{\nu }}_{x}$, ν_x , and ν_e states, respectively. We can see that the onset of high-amplitude oscillations occurs earlier for the mb case, and the two cases have distinct evolution with distance from the PNS, which we anticipate will influence the ultimate nucleosynthesis yields.

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In this work, we adopt the nuclear reaction network code Portable Routines for Integrated nucleoSynthesis Modeling (PRISM) (Mumpower et al. 2018; Sprouse et al. 2020). The four neutrino treatments summarized in Section 4.1 (nn, nosc, mf, mb) are implemented into the nucleosynthesis calculations in the form of external ν_e + n and ${\bar{\nu }}_{e}+p$ capture rates for each of the neutrino energy configurations from Table 2. At T ∼ 10 GK, the nuclear composition is determined by nuclear statistical equilibrium (NSE). Here, we obtain the initial composition at ∼10 GK through an NSE calculation with Y_e obtained from Equation (19) and the density taken from the chosen matter trajectory. We dynamically evolve the composition and the electron fraction from T ∼ 10 GK onwards. This naturally incorporates a possible "alpha effect"—an increase in the number of alpha particles and, subsequently, seeds and corresponding decrease in free nucleons available for capture—due to neutrino interactions on free nucleons during alpha particle formation (Fuller & Meyer 1995; Meyer et al. 1998). For the network calculation, we use REACLIB reaction rates (Cyburt et al. 2010) along with NUBASE β-decay properties (Kondev et al. 2021). The available REACLIB data are supplemented with neutron-capture rates, β-decay rates, and fission properties in the actinide region based on the FRDM/FRLDM model (Mumpower et al. 2018), though we find the nucleosynthesis calculations in this work largely do not extend into this region of the nuclear chart.

To test the nucleosynthetic outcomes of both the symmetric and asymmetric neutrino calculations, here we adopt two types of parameterized supernova neutrino-driven wind trajectories: trajectories parameterized as per the "standard model" in Wanajo et al. (2011, hereafter Wanajo2011) with a range of entropies, and a high-entropy trajectory from Duan et al. (2011, hereafter Duan2011). The details of these trajectories are summarized in Table 3. We also tested the shocked trajectories from Arcones et al. (2007); the resulting abundance patterns lie between those for the Wanajo2011 trajectory with entropy per nucleon in units of the Boltzmann constant s/k = 50, and those for the Duan2011 trajectory, so for brevity these results do not appear in this work.

The Wanajo2011 trajectories adopted the wind solution for the "standard model" in Wanajo et al. (2011) with M_ns = 1.4 M_⊙ and L_ν = 1 × 10⁵² erg s⁻¹, followed by a subsonic expansion after wind termination at 300 km using a standard parameterization with the time evolution of temperature T and density ρ as follows: ρ(t) ∝ t⁻² and T(t) ∝ t^−2/3. We assume an adiabatic expansion throughout the evolution, with constant values for the entropy per baryon set to s/k = 50, 100, and 150. Higher (lower) values of the initial entropy result in lower (higher) mass fractions of seed nuclei produced, with more (fewer) free nucleons remaining for capture. The Duan2011 trajectory is more extreme, with faster expansion and a higher entropy of s/k ∼ 200. The density initially falls exponentially, ρ(t) ∝ e^−t/3τ , with τ ∼ 12 ms for 3 GK ≤ T ≤ 6 GK and τ ∼ 18 ms for 1.5 GK ≤ T ≤ 3 GK, before switching to a power law ρ(t) ∝ t⁻² when the temperature is roughly 1 GK. This ∼t⁻² drop-off, rather than the much faster ρ ∝ t⁻³ that one expects in the event of free expansion, can be attributed to the fact that the neutrino-driven wind is expected to encounter the outer layers of the supernova ejecta/envelope as it emerges from the inner "hot-bubble" region, slowing down its expansion. The entropy of this trajectory is somewhat outside of those expected from modern supernova simulations. However, this combination of fast expansion and high entropy maximizes the ratio of free nucleons to seeds, which is ideal for emphasizing the nucleosynthetic impact of the differences in neutrino treatments.

Figure 3 illustrates the evolution of the neutrino capture rates as a function of the temperature along the matter trajectory for the Duan2011 and Wanajo2011 s/k = 50 cases, respectively. The neutrino fluxes are taken from the sym case with mb and mf treatments. In all cases, the onset of oscillations results in a steep rise to the capture rates, which occurs within the temperature window relevant for nucleosynthesis. The degree and duration of the rise are expected to shape the formation of seed nuclei and the subsequent nucleosynthesis. Note that, in each of our calculations, we assume steady-state neutrino emission from the source, resulting in an overestimation of about 20% in the integrated ${\bar{\nu }}_{e}$ capture rates for the Wanajo2011 trajectories, and about 2% for the Duan2011 trajectory, when compared with corresponding calculations that used exponentially decaying neutrino luminosities with a time constant of 3 s. In each case, this approximation does not qualitatively affect any of the results and conclusions.

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Symmetric neutrino energies and luminosities between the ν_e and ${\bar{\nu }}_{e}$ species will produce charged-current neutrino capture rates that are faster than antineutrino rates, thanks to the neutron–proton mass difference. As a result, the resulting nucleosynthetic conditions in the supernova neutrino-driven wind will tend to be proton-rich. If a sufficient excess of free protons is available following seed formation, along with a sizeable ${\bar{\nu }}_{e}$ flux to convert a fraction of the free protons into neutrons for (n,p) and (n, γ) reactions, then a ν p-process will result.

5.1.1. s/k ∼ 50: ν p-process Nucleosynthesis

In the proton-rich neutrino-driven winds, the ν p-process starts with seed nuclei around ⁵⁶Ni (or possibly ≥64), synthesized in nuclear quasi-static equilibrium (QSE) at high temperatures (T ≥ 3 GK). As the temperature drops below ∼3 GK, QSE freezes out and, in the absence of neutrinos, the nuclear reaction flow paths to higher A via proton capture will be stalled by the long β⁺-decay lifetimes of bottleneck (mainly even–even N = Z) nuclei. However, in the presence of neutrinos, antineutrino capture on free protons will produce a small increase in free neutrons. The bottlenecks can then be bypassed by neutron capture, allowing synthesis of increasingly heavy nuclei beyond iron (Fröhlich et al. 2006a; Pruet et al. 2006; Thielemann et al. 2010). The ν p-process yields depend on the hydrodynamic state of neutrino-driven winds, nuclear reaction rates, and the neutrino physics of the astrophysical site (Wanajo et al. 2011; Arcones et al. 2012; Fujibayashi et al. 2015; Bliss et al. 2018a; Nishimura et al. 2019; Rauscher et al. 2019; Jin et al. 2020; Vincenzo et al. 2021; Sasaki et al. 2022, 2023). The symmetric neutrino parameters we adopt from Table 2 result in initially proton-rich conditions with an equilibrium electron fraction Y_e,eq = 0.634.

We follow the nucleosynthesis for T < 10 GK in four distinct neutrino treatments, as elaborated in Section 4. To begin, we first test the effects of neutrino interactions for the Wanajo2011 model with s/k = 50, which is a typical ν p-process trajectory. The abundance pattern results are shown in Figure 4. We can see that the final abundance pattern in the no oscillations case shows typical ν p-process features, with element synthesis up to A ∼ 100, while the case with no neutrino interactions results in only iron-peak nuclei. Both cases are consistent with the results of Wanajo et al. (2011). For the abundance yields in the cases with neutrino oscillations included, the enhanced neutrino capture rates shown in Figure 3 create more neutrons throughout the ν p-process. Thus, the ν p-process can proceed to heavier nuclei (Figure 4). As the material cools below temperatures required for proton capture, some of the remaining free protons continue to be converted to neutrons, producing a small amount of excess neutrons whose captures shift some material slightly neutron-rich. The maximum extent of the reaction flow for the sym-mb case is illustrated in the bottom panel of Figure 4. Nuclei up to the Z = 50 shell closure are populated, with a handful of species on the neutron-rich side of stability. The final abundances for the sym-mf and sym-mb cases remain those of a robust ν p-process.

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We also tested the effect of neutrino interactions adopting the inverted mass ordering on the nucleosynthesis yields. In the two-flavor framework, which our calculations are based on, changing the mass ordering amounts to changing ω_p → − ω_p , where ω_p = Δm²/(2p) for neutrinos and −Δm²/(2p) for antineutrinos. This difference leads to some differences in the oscillation behaviors. In the normal mass ordering, the extent of conversion is greater in the many-body case than in the mean-field case. In the inverted ordering, the opposite is true: the mean-field case exhibits a greater degree of flavor swapping than the many-body case. Additionally, the inverted-ordering scenario results in a greater degree of flavor conversion as a result of bipolar oscillations (e.g., Samuel 1996; Pastor et al. 2002; Duan et al. 2006; Hannestad et al. 2006), resulting in higher neutrino average energies, higher capture rates, and, consequently, a more robust synthesis of heavier elements. This behavior is illustrated in the top panel of Figure 4. The results obtained using the normal mass ordering, which are the main focus of this work, could thus be considered conservative in that sense.

5.1.2.  s/k ≳ 100: Emergence of a ν i-process

The initial entropy of the astrophysical trajectory plays an important role in the resulting heavy-element nucleosynthesis. Of most relevance here, entropy augments the ratio of free nucleons to seed nuclei at T ∼ 3 GK when QSE ends. This effect can be understood as follows: a higher entropy implies a lower density at a given temperature. Therefore, the reaction rates involved in the assembly of seed nuclei, which are strongly density dependent, are lower at higher entropies, resulting in reduced seed formation—and thereby a higher free nucleon-to-seed ratio.

We examined variations in the initial entropy of the Wanajo2011 trajectories from s/k = 50 to 100 and 150, with the same set of four distinct neutrino treatments. The results for s/k = 100 and 150 are shown in Figure 5. When the entropy is increased to s/k = 100, all cases that include neutrino interactions generate heavier elements compared to the s/k = 50 example discussed above. Furthermore, the higher free proton-to-seed ratio provides greater leverage for the influence of neutrino oscillations. The higher neutrino capture rates of the sym-mf and sym-mb cases produce dramatically different nucleosynthetic outcomes from those of the sym-nosc case, with the sym-mb case producing the heaviest nuclear species. At s/k = 150, this influence is even more significant. Interestingly, in these examples, there are even more free protons available at low temperatures. Consequently, more neutrons are available at late times, such that neutron capture becomes not only a means to bypass certain slow β⁺ half-lives but actually robust enough to drive the reaction flow across the valley of stability and over to the neutron-rich side for Z ≳ 50. The nucleosynthesis path in the right panels of Figure 5 shows that, for sym-mb cases, the reaction flow shifts to the neutron-rich side of the valley of stability at high A and synthesis proceeds to heavy nuclei beyond the Z = 50 and N = 82 shell closures. In the sym-mb case, the final abundances reach A ∼ 160 for s/k = 100 and A ∼ 190 for s/k = 150.

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To understand the mechanism by which this transition to neutron-capture nucleosynthesis occurs, we show the temperature evolution of the abundances of neutrons (top panel, log scale) and protons (bottom panel, linear scale) in Figure 6. Without neutrino interactions on free nucleons, iron group nuclei are synthesized and additional proton capture is blocked by the slow β bottlenecks, shown in the red lines of each of the panels of Figure 6. When neutrino interactions are included (without oscillations), shown by the cyan lines of Figure 6, the proton abundance continues to decrease as the temperature drops due to the conversion of protons to neutrons by neutrinos and the resulting increase in proton captures to higher A. Both of these effects are enhanced by neutrino oscillations, with the sym-mb case (blue lines) showing a larger effect than sym-mf (orange lines). Comparing the s/k = 150 examples to s/k = 50, we see the influence of the oscillations at an earlier stage of the nucleosynthesis, since we take the onset of oscillations to occur at a fixed radius, which corresponds to a higher temperature for the s/k = 150 outflow. The neutrons produced at T > 6 GK end up bound into alpha particles, with enhanced production of alpha particles for the sym-mb and sym-mf cases. Still, the higher entropy ensures that fewer seed nuclei are assembled from the alpha particles, so the free proton-to-seed ratio remains high. The production of neutrons and their subsequent capture is therefore enhanced. This effect allows late time, low (T < 1.5 GK) temperature capture of neutrons, and reaction flow to heavier, neutron-rich nuclei. As with the s/k = 50 case, the effect is stronger for the inverted mass ordering calculations than for normal ordering.

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The neutron-rich portions of the nucleosynthetic pathways illustrated in Figure 5 are similar to those that characterize another nucleosynthesis process: the i-process. The i-process is an intermediate neutron-capture process that results from neutron-rich conditions in which the neutron number density is greater than that of the slow neutron-capture (s) process but significantly less than required for an r-process, e.g., N_n ∼ 10¹³–10¹⁵ cm⁻³ (Cowan & Rose 1977). It has been suggested that neutron number densities within this range can be attained in, for example, the helium flash of asymptotic giant branch (AGB) and post-AGB stars (Cowan & Rose 1977; Malaney 1986; Jorissen & Arnould 1989; Cristallo et al. 2016) and rapidly accreting white dwarfs (Denissenkov et al. 2017, 2019), and there is tentative evidence that points to i-process contributions to some unusual stellar abundance patterns (Dardelet et al. 2014; Hampel et al. 2019). The process that produces this i-process-like nucleosynthetic pathway here is clearly distinct from an i-process that occurs in mildly neutron-rich conditions. Instead, the heavy elements here are created via neutron capture where the neutrons are a product of neutrino interactions on excess protons in a proton-rich wind. The suggestion that a robust ν p-process could shift to neutron-rich species at late times was first pointed out in Wanajo et al. (2011), Arcones et al. (2012), and also Nishimura et al. (2019), though the details were not explored. A similar process (with a smaller maximum mass number ∼160) was also observed in a hypernova neutrino-driven wind with higher neutrino luminosities L_ν ∼ 10⁵³ erg s⁻¹ (Fujibayashi et al. 2015). We find neutrino oscillations amplify the shift from proton-rich to neutron-rich nucleosynthesis and can result in a full intermediate neutron-capture process. We dub this novel nucleosynthesis process a "ν i-process."

5.1.3. s/k ∼ 200: ν i-process Nucleosynthesis

Section 5.1.2 shows that the influence of different neutrino treatments on proton-rich nucleosynthesis grows with the entropy of the matter trajectory. Here, we adopt a supernova neutrino-driven wind trajectory with faster evolution and higher entropy (s/k = 200) from Duan2011 to test the potential extent of the ν i-process in a more extreme case.

The abundance pattern results for the Duan2011 trajectory with the four distinct neutrino treatments are shown in Figure 7. We can see that, compared to the Wanajo2011 s/k = 150 case, there is even more robust production of heavy species and a clearing out of light species via neutron capture, which proceeds to high A (∼205 for the mb case). Also, the differences in the nucleosynthetic outcomes among the neutrino treatments are enhanced, with the sym-mb showing the most robust production of heavy species. This behavior is further illustrated in Figure 8, which shows the nucleosynthetic pathway at the time when the crossover to the neutron-rich side of stability occurs (at temperature T ∼ 2.5 GK). The sym-mb case shows the greatest extent in A, reaching almost up to the stability gap.

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Given the dramatic difference between the sym-mf and sym-mb results in this scenario, we explore the possibility that the difference manifested in this extreme case is due to the uncertainties inherent in our simplified many-body neutrino treatment. We therefore also examined a many-body calculation with increased neutrino numbers (4nu: 4 ν + 4 $\bar{\nu }$). A comparison of the nucleosynthesis results with the default 2nu (2 ν + 2 $\bar{\nu }$) calculation are shown in Figure 7. We can see that the abundance patterns with different neutrino treatments for the default 2nu calculations and the 4nu case are very similar. This trend suggests that the differences we see in the sym-mf and sym-mb results are due to the distinct oscillation patterns in the two cases in our setup, a difference that is amplified by the high entropy/fast evolution of the Duan2011 matter trajectory.

We explore the details of the sym-mb case in Figure 9, which shows the evolution of the nucleosynthetic pathway as a function of time. At early times/high temperatures (T > 3.5 GK), the path is on the proton-rich side of stability, as for a typical ν p-process. By T ∼ 3 GK, the production of neutrons by neutrino interaction and their subsequent capture via (n,γ), (n,p), and (n,α) have shifted the pathway to the valley of stability. At lower temperatures, T < 2 GK, charged particle capture rates slow and neutron capture takes over in earnest.

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The dominant nuclear reaction flows at T = 2 GK are shown in Figure 10. Similar to a "traditional" i-process, depicted in, for example, Choplin et al. (2020, 2021, 2022), the predominant reaction is neutron capture. Here, the charge-changing interactions include both (p,n) and β decay, with the latter increasing in importance with increasing A.

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At its most neutron-rich extent, the subsequent pathway runs parallel to and roughly ∼8 neutrons away from the valley of stability for Z > 40, with kinks at the neutron closed shells N = 82 and N = 126. Unlike the Wanajo2011 s/k > 100 cases, here most of the pathway is shifted neutron-rich, even for the lighter species. The resulting abundance pattern is largely featureless, with small excesses at A ∼ 135 and A ∼ 200 associated with the neutron closed shells and with the original ν p pattern erased.

Figure 11 shows the abundances of neutrons, protons, and alpha particles for the sym-nn, sym-nosc, sym-mf, and sym-mb calculations with the Duan2011 trajectory. Comparing to Figure 6, we note that the influence of the neutrino oscillations sets in at a later stage in the nucleosynthesis, owing to the fast expansion of the Duan2011 trajectory. Still, neutrino oscillations have greater leverage to impact the nucleosynthetic outcome here, as fewer alphas are assembled into seeds, leading to a larger free nucleon-to-seed ratio and more robust free nucleon capture. Neutron capture is in fact so robust that the transition of the nucleosynthetic pathway from proton-rich to neutron-rich occurs at a higher temperature, when photon-induced reactions are still prominent. The rise in neutron abundance after its initial dip is the establishment of an (n, γ)-(γ, n) equilibrium on the neutron-rich side of stability. As a result, the nucleosynthetic pathway depicted in the third panel of Figure 9 shows some similarities to the shape of an r-process path—for example, "kinks" in the path at the N = 82 and N = 126 closed shells—though these nuclear features are encountered much closer to stability than in an r-process. Notably, we find that the nuclear flow does not proceed past the N = 126 shell closure for any of the ν i-processes considered in this work, including those that adopt the inverted ordering for the oscillation calculations. Indeed, for the Duan2011 trajectory with higher entropy, the nucleosynthesis yields with symmetric neutrino oscillations calculations adopting inverted mass ordering are similar to the yields with the normal ordering sym-mb case.

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5.1.4.  νp- to νi-process Trends

As discussed above, we find that the influence of neutrino oscillations grows with the increase of initial entropy, and the ultimate extent and yields of the element synthesis vary from a typical ν p-process to an unusual mixture of light proton-rich nuclei and heavy neutron-rich nuclei to a full and robust ν i-process. To examine this trend, Figure 12 shows the comparison of the final abundance yields from the Wanajo2011 trajectories with varying entropy as well as the Duan2011 trajectory, adopting the symmetric many-body (sym-mb) neutrino calculations that result in the greatest influence of the neutrino interactions. Also shown (as points) are the abundances of the proton-rich p nuclei synthesized in each calculation, to highlight the fraction of the material that ultimately shifts neutron-rich. Note that the abundance pattern of the Wanajo2011 s/k = 50 case follows a typical ν p-process where p nuclei are dominantly produced, while the abundance patterns resulting from larger initial entropy values shift from a ν p-process at lower mass to a neutron-rich pattern for heavier nuclei (A ≳ 115 for s/k = 100, A ≳ 100 for s/k = 150, A ≳ 70 for Duan2011).

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To further explore the features of these novel ν i-process patterns, we also compare the resulting abundance patterns of the simulations with the Duan2011 matter trajectory and sym-mb neutrino calculations to the solar system separated s-process and r-process patterns (Sneden et al. 2008) in Figure 13. Given that its nucleosynthetic pathway is between those of the s-process and r-process, the ν i-process abundances are distinct from those of both the solar s-process and r-process, showing shifted neutron closed shell features and a distinctly higher lanthanide production than the s-process.

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In summary, from the nucleosynthesis tests of the supernova neutrino-driven wind trajectories with various neutrino treatments, using a neutrino energy distribution symmetric in ν_e and ${\bar{\nu }}_{e}$ (to ensure proton-rich conditions), we see the emergence of a novel type of heavy-element nucleosynthesis: a ν i-process. From this first exploratory study, we conclude that a ν i-process can occur in high-entropy (s/k ≳ 100), neutrino-rich environments with initially proton-rich conditions. This process results in abundance yields that can be a mixture of a ν p-process-like pattern at lower mass and an i-process-like pattern for heavier elements, or an entirely i-process-like pattern for all elements. Since the transition to neutron-capture nucleosynthesis results from the continued production of neutrons by neutrino reactions, this process is highly sensitive to neutrino oscillation physics.

The asymmetric neutrino calculations adopted in Table 2 result in capture rates for antineutrinos that are similar to or higher than for neutrinos, producing equilibrium electron fractions that are closer to or less than 0.5. As already noted, the supernova neutrino-driven wind could be a mildly neutron-rich environment that allows for a limited or weak r-process, with the nucleosynthesis extent sensitive to the neutrino physics adopted. So, here we investigate a possible r-process in the neutrino-driven wind and the effect of the neutrino physics treatments described above on the r-process yields. We examine the nucleosynthetic impact of neutrinos with asymmetric energy distributions and luminosities with four different choices of neutrino energy distributions, i.e., asym2, asym2.1, asym3, and asym4, corresponding to equilibrium Y_e values ranging from 0.504 to 0.366, as listed in Table 2.

Three of the four adopted sets of asymmetric neutrino energies result in initially neutron-rich conditions. During alpha particle formation, all of the protons are consumed to produce alpha particles. This reaction leaves few free protons to convert to neutrons via antineutrino capture, while any excess free neutrons are still subject to neutrino captures. The protons produced as a result combine promptly with free neutrons to continue forming alphas. Thus, neutrino interactions act to increase the number of alpha particles (and, subsequently, seed nuclei) and reduce the number of free neutrons available for capture on the seed nuclei, which is the so-called alpha effect (Fuller & Meyer 1995; McLaughlin et al. 1996; Meyer et al. 1998). With the influence of collective neutrino oscillations included, the effective energies of ν_e and ${\bar{\nu }}_{e}$ are raised. The corresponding increases to the neutrino capture rates cause the alpha effect to become more significant and the production of heavy nuclei is hindered.

We first test the nucleosynthesis calculations with the asymmetric neutrino treatment asym4, which results in the most neutron-rich conditions of the four cases and therefore is most likely to produce r-process species. The corresponding abundance patterns and nucleosynthesis paths are shown in Figure 14 for the s/k = 50, 100, and 150 Wanajo2011 matter trajectories. All simulations result in at most a weak r-process, and only the s/k = 150 case robustly produces the second (A ∼ 130) r-process peak. The influence of neutrinos and their oscillations only act to hinder the r-process through an enhanced alpha effect. The asym4-mb case shows the largest influence, though overall the impact of neutrinos on the ultimate nucleosynthetic outcome is much smaller compared to the simulations discussed in Section 5.1. The free nucleon-to-seed ratio is low and the neutrons are efficiently consumed, cutting off the potential influence of neutrinos. This situation is in stark contrast to the proton-rich examples above in which protons are available to convert to neutrons throughout the element synthesis.

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In order to fully explore the potential impact of the different neutrino treatments (nn, nosc, mf, mb) defined in Section 4.1, we also calculate the nucleosynthesis resulting from the Duan2011 matter trajectory. Its fast expansion and high entropy ensure that a robust main r-process—one that reaches A ∼ 195 peak nuclei—is possible for at least some of the neutrino energy distributions explored, and the impact of the neutrino treatments is maximized for all asymmetric configurations explored: asym4, asym3, asym2.1, and asym2.

For the most neutron-rich asym4 case, as seen in the right panel of Figure 15, the nucleosynthesis calculation for the Duan2011 matter trajectory with neutrino interactions turned off for T < 10 GK results in a robust main r-process pattern that reaches the actinide region. Neutrino interactions produce an alpha effect of varying strength, as shown in Figure 16, with the asym4-mb case resulting in the lowest free neutron abundance and highest alpha abundance. As a result, we observe decreased production of heavier nuclei especially around the third-peak region and beyond, with the smallest A ∼ 195 peak produced in the asym4-mb case.

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The asym3 case starts with a slightly higher Y_e value at equilibrium, producing a lower initial neutron abundance and a faster consumption of neutrons than in the asym4 case, as shown in Figure 16. As a result, the abundance pattern for the asym3-nn case just barely fills out the third-peak region and underproduces the actinides. Again, from the left panel of Figure 15, we can see that neutrinos and neutrino oscillations hinder the r-process production of heavier nuclei. The effect of the difference in neutrino treatments on the r-process yields follows the same pattern as the asym4 case, but with more significant deviations shown.

Finally, we examine the effects of the different neutrino treatments on the nucleosynthesis calculations for the Duan2011 matter trajectory with initial Y_e values around 0.5, with the results shown in Figure 17. For the neutrino parameters that result in an electron fraction slightly bigger than 0.5, the nucleosynthesis result is either iron-peak nuclei (for asym2-nn) or a mild ν p-process when neutrino interactions are included. Similar to the symmetric neutrino cases described in Section 5.1.1, the extent of the ν p-process is sensitive to the treatment of neutrino oscillations, with the asym2-mb case producing the highest ν p-process yields compared to the other neutrino treatments. The asym2.1 neutrino distribution, on the other hand, corresponds to an equilibrium electron fraction slightly smaller than 0.5 and results in a weak r-process up to the second r-process peak. Here, the influence of neutrinos is modest, similar to the Wanajo2011 asym4 results shown in Figure 14.

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For the asymmetric neutrino conditions studied here, we find the neutrino physics to be particularly crucial for the nucleosynthesis outcomes that sit at a threshold, either for an r-process that barely reaches the third-peak region, or for initial conditions around Y_e = 0.5. We accordingly expect similar behavior for an r-process that proceeds just beyond the second peak. In these cases, the changes to the neutrino interaction rates that result from the treatment of neutrino oscillation behavior can produce marked changes in the abundance yields. However, in general, the leverage that the neutrinos have on nucleosynthetic outcomes in neutron-rich conditions remains smaller than in very proton-rich conditions. This difference is due to neutrino interactions changing the abundance of the subdominant free nucleon species, e.g., neutrons in proton-rich and protons in neutron-rich nucleosynthesis, as element synthesis proceeds. However, as the temperature drops, only neutrons can be captured (and protons cannot). Thus, the influence of neutrino interactions in proton-rich conditions is extended throughout the element synthesis, resulting in a larger impact on a ν p-process compared to the r-process for the collective oscillation effects considered here.