Effects of Hoyle state de-excitation on $\nu p$-process nucleosynthesis and Galactic chemical evolution

The partcle-induced hadronic de-excitation of the Hoyle state in $^{12}$C induced by inelastic scattering in a hot and dense plasma can enhance the triple-alpha reaction rate. This prevents the production of heavy nuclei within the neutrino-driven winds of core-collapse supernovae and raises a question as to the contribution of proton-rich neutrino-driven winds as the origin of $p$-nuclei in the solar system abundances. Here we study $\nu p$-process nucleosynthesis in proton-rich neutrino-driven winds relevant to the production of $^{92,94}\mathrm{Mo}$ and $^{96,98}\mathrm{Ru}$ by considering such particle-induced de-excitation. We show that the enhancement of the triple-alpha reaction rate induced by neutron inelastic scattering hardly affects the $\nu p$-process, while the proton scattering contributes to the nucleosynthesis in proton-rich neutrino-driven winds at low temperature. The associated enhanced triple-alpha reaction rate decreases the production of $^{92,94}\mathrm{Mo}$ and $^{96,98}\mathrm{Ru}$ in a wind model of ordinary core-collapse supernovae. On the other hand, the abundances of these $p$-nuclei increase in an energetic hypernova wind model. Hence, we calculate the galactic chemical evolution of $^{92,94}\mathrm{Mo}$ and $^{96,98}\mathrm{Ru}$ by taking account of both contributions from core-collapse supernovae and hypernovae. We show that the hypernova $\nu p$-process can enhance the calculated solar isotopic fractions of $^{92,94}\mathrm{Mo}$ and $^{96,98}\mathrm{Ru}$ and make a significant impact on the GCE of $p$-nuclei regardless of the particle-induced Hoyle state de-excitation.


INTRODUCTION
The excited 0 + state of 12 C at 7.65 MeV (the so-called "Hoyle state") can resonantly enhance the reaction rate of the triple-alpha (3α) process essential for nucleosynthesis inside stars (Burbidge et al. 1957).In a hot and dense plasma, inelastic scatterings of background particles can induce the hadronic de-excitation of the Hoyle state to the ground state or to the excited 2 + state at 4.44 MeV.This enhances the 3α reaction rate (Beard et al. 2017).In this context, the enhancement was calculated based on the statistical Hauser-Feshbach model (Beard et al. 2017), and the contribution of neutrons was recently measured in a neutron inelastic scattering experiment (Bishop et al. 2022).
The 3α reaction is crucial for the nucleosynthesis of heavy elements inside explosive astrophysical sites such as core-collapse supernovae (CCSNe) and neutron star mergers.In particular, the νpprocess nucleosynthesis (Fröhlich et al. 2006), that can occur within the proton-rich neutrino-driven winds of core-collapse supernovae is sensitive to the uncertainty of the 3α reaction rate (Wanajo et al. 2011;Nishimura et al. 2019).The νp-process is induced by the absorption of electron antineutrinos on free protons, p(ν e , e + )n (Fröhlich et al. 2006;Pruet et al. 2006;Wanajo 2006).The free neutrons produced via the neutrino absorption allow for the production of heavier elements beyond the waiting point nucleus 64 Ge and other bottleneck nuclei through (n, p) reactions instead of slower β + decays.
The νp-process is affected also by various nuclear reactions rates.Among them the 59 Cu(p, α) 56 Ni and 7 Be(α, γ) 11 C were recently measured in nuclear experiments (Randhawa et al. 2021;Psaltis et al. 2022).
The νp-process can produce large numbers of p-nuclei that cannot be synthesized through either the slow (s-) or rapid (r-) neutron capture processes.Most p-nuclei can be produced in the γprocess (Woosley & Howard 1978;Hayakawa et al. 2004;Hayakawa et al. 2008) induced by successive photodisintegration reactions on heavier isotopes.However, calculations of the galactic chemical evolution (GCE) of abundant p-isotopes such as 92,94 Mo and 96,98 Ru in models that only include the γ-process in the outer layers of both thermonuclear supernovae (SNe Ia) and ordinary core-collapse supernovae (SNe II) drastically underestimate the solar isotopic fractions (Travaglio et al. 2018).
Moreover, the molybdenum isotopic anomalies in meteorites indicate that the p-isotopes 92,94 Mo and the r-isotope 100 Mo are synthesized in the same star but by different processes (Dauphas et al. 2002;Budde et al. 2016;Poole et al. 2017).A νp-process in core-collapse supernovae where the r-process also occurs could meet such a requirement and be a candidate site for the production of these abundant p-isotopes 92,94 Mo and 96,98 Ru.In particular, a strong νp-process is possible in the proton-rich neutrino-driven winds of very energetic hypernovae (HNe) (Fujibayashi et al. 2015).Such a HN νp-process significantly increases the elemental abundances of Mo and Ru at low metallicity [Fe/H] < −2 (Sasaki et al. 2022).
Recently, Jin et al. (2020) reported that the enhanced 3α reaction induced by the hadronic deexcitation of the Hoyle state should increase the seed nuclei for the production of heavy elements and suppress the νp-process.This raises a question as to the impact of the νp-process on the solar abundances of 92,94 Mo and 96,98 Ru.This seems to be true for a neutrino-driven wind model in SNe II with small entropy per baryon and a large expansion timescale.However, the contribution to the νp-process of such particle-induced Hoyle state de-excitation in HNe with a massive proto-neutron star (PNS) and large neutrino luminosities is still uncertain (Fujibayashi et al. 2015).Neutrinodriven winds in such HNe have larger entropy and a shorter expansion timescale than the SN II wind model.The νp-process in SNe II hardly affects the GCE due to the relatively small production yield of p-nuclei while the HN νp-process can dominantly contribute (Sasaki et al. 2022).Observational quantities such as the solar isotopic fractions and elemental abundances can be affected by the Hoyle state effect in the HN νp-process.
For the present work, we calculate the enahnced 3α reaction in the νp-process by using both SN II and HN wind models.Then, we carry out the GCE calculation with the calculated nuclear yields to demonstrate how the particle-induced Hoyle state de-excitation in the HN νp-process affects the GCE of Mo and Ru.

Models of neutrino-driven winds in supernova and hypernovae
We need hydrodynamic quantities and neutrino fluxes to calculate the νp-process nucleosynthesis within neutrino-driven winds.We calculate the temperature and baryon density profiles of the neutrino-driven wind based upon a model for general-relativistic steady-state, spherically symmetric trajectories (Otsuki et al. 2000).The radius of the PNS is taken to be R PNS = 15 km.The baryon density near the PNS radius is taken to be ρ 0 = 10 11 g/cm 3 , and the temperature at the PNS radius is determined by the condition q = 0, where q is the net heating rate from neutrino interactions (Otsuki et al. 2000).We assume that the neutrino luminosity L ν is independent of neutrino species, and that these neutrinos obey Fermi-Dirac distributions on the surface of the PNS, with the neutrino mean energies fixed to E νe = 13.1 MeV, E νe = 15.7 MeV, and E ν X = 16.3MeV.With these neutrino parameters, we have calculated the rates of neutrino-induced reactions and the electron fraction inside the neutrino-driven winds.The calculated electron fraction is used to determine mass fractions of initial neutrons and protons for the subsequent nuclear network calculation.
To study the effect of the particle-induced Hoyle state de-excitation, we prepare wind trajectories of both ordinary SN II and energetic HN models with different values of the PNS mass M PNS and L ν as shown in Table 1.These parameters for the SN II model are typical values for the late stages of the explosion (e.g.see Burrows et al. (2020); Nagakura et al. (2021)).We assume that a collapse of a rapidly rotating massive star is associated with the energetic explosion mechanism of HNe.For the HN wind model, we consider a proton-rich neutrino-driven wind blown off from the massive PNS toward the polar region before the black hole formation as in Fujibayashi et al. (2015) by employing a large PNS mass (M PNS = 3M ) and a large neutrino luminosity (L ν = 10 53 erg/s) as seen in neutrino radiation hydrodynamic simulations (Sekiguchi et al. 2012;Fujibayashi et al. 2021).

Enhanced 3α reaction rate due to Hoyle state de-excitation
We execute the nuclear network calculation following the numerical setup of Sasaki et al. (2017Sasaki et al. ( , 2022)).We use the LIBNUCNET reaction network engine (Meyer & Adams 2007) with nuclear reaction rates from the JINA Reaclib database (Cyburt et al. 2010).We have included reaction rates for neutrino absorption and e ± capture.However, we ignore the contribution from neutrino oscillations (Ko et al. 2020(Ko et al. , 2022)).The enhanced 3α reaction rate owing to the induced Hoyle state de-excitation is given by Jin et al. (2020), f n (T 9 ) = 75.1e−T 9 + 88.7, (2) where λ 3α is the 3α reaction rate without the enhancement (Fynbo et al. 2005), ρ 6 is the baryon density in units of 10 6 g cm −3 , and T 9 is the temperature in units of 10 9 K.The quantities Y n and Y p are the number abundance fractions of free neutrons and protons, respectively.The second and third terms on the right-hand side of Eq. ( 1) are the contributions from neutron and proton inelastic scattering, and the values of f n and f p are determined by fitting to the statistical Hauser-Feshbach calculation of Beard et al. (2017).This enhanced 3α reaction is utilized for the network calculations in the present SN and HN nucleosynthesis models.

Model for Galactic chemical evolution
We adopt the GCE model of Timmes et al. (1995) which reproduces reasonably well the chemical evolution of the light elements from hydrogen to zinc as well as the model of Ko et al. (2020).The adopted model (Timmes et al. 1995) has already been successfully applied to the GCE of the intermediate-to-heavy mass nuclei including the r-process contributions from magneto-hydrodynamic jet SNe, collapsars and binary neutron star mergers as well as the neutrino-driven wind in core collapse SNe (Yamazaki et al. 2021;Yamazaki et al. 2022) and the νp-process contributions from Type II SNe and Hypernovae (Sasaki et al. 2022).The latter study (Sasaki et al. 2022) includes not only the νp-process, but the γ-(p-), s-, and r-processes in addition to the νp-process.In the present calculations, we follow the same numerical setup except for the input data of the HN νp-process as discussed in the previous section.
We demonstrate the impact of the νp-process on the GCE calculation with the enhanced 3α reaction rate in Eq. ( 1).We focus on the GCE of 92,94 Mo and 96,98 Ru whose total solar isotopic fractions are as high as 24.1% and 7.4%, respectively.A GCE calculation that only includes the γ-process underestimates such large solar abundances and the νp-process potentially resolves this underestimation problem.The HN νp-process can be a main contributor to the GCE of 92,94 Mo and 96,98 Ru (Sasaki et al. 2022) so that the effect of the particle-induced Hoyle state de-excitation on GCE could be well demonstrated by considering the HN νp-process.
We consider the γ-process in both SNe Ia and II and the νp-process in both SNe II and HNe as the astrophysical sources of p-nuclei.We use the HN wind model in Table 1 as the fiducial proton-rich neutrino-driven winds in HNe with Eq. (1).In the HN νp-process, the yield of the nucleus i is estimated by X i Ṁ τ N S where X i and Ṁ are a mass fraction of i and the mass ejection rate inside the HN wind, and τ N S = 1 s is a typical lifetime of the massive PNS (Fujibayashi et al. 2015).The progenitor mass for the HN model is set to 100 M as in Sasaki et al. (2022).

Hydrodynamic and neutrino properties
Table 1 shows the hydrodynamic quantities characterizing properties of the wind models such as the expansion timescale τ dyn , the entropy per baryon S, the initial electron fraction for the network calculation Y (0) e , and the mass ejection rate Ṁ .The values of τ dyn and S are calculated at a high temperature before the production of heavy elements as in (Sasaki et al. 2022).The entropy for the HN wind model is higher than that of the SN II wind model due to the massive PNS mass.The small expansion timescale in the HN model originates from the large neutrino luminosity.Such properties of neutrino-driven winds are consistent with the results of Otsuki et al. (2000); Fujibayashi et al. (2015).

Y
(0) e corresponds to the electron fraction at the beginning of the nuclear network calculation (T 9 = 10).
The value of Y (0) e is larger than 0.5 for the proton-rich neutrino-driven winds, and almost the same in both wind models because we use the same neutrino energies for the neutrino distributions.Y (0) e becomes larger when the difference between the mean neutrino energies, E νe − E νe is small.Ṁ is determined from a supersonic wind solution, and the value increases with the neutrino luminosity.In an ordinary SN explosion, the entropy of the wind increases with the decrease of neutrino luminosity in the later explosion phase (t > 1 s).Although a high entropy is favorable for the production of heavy elements, their total yields in the later wind trajectories are not so large due to the small Ṁ (Sasaki et al. 2022).The massive PNS and the large neutrino luminosity in the HN model simultaneously enable both a high entropy and a large mass ejection rate.and decreasing free neutrons.The large entropy per baryon of the HN wind model results in a small amount of seed nuclei such as 56 Ni and a large production of heavy elements through the νp-process.show the results with and without the Hoyle state effect, respectively, and the difference between them is prominent for nuclei with A > 60.This is because the enhanced 3α reaction suppresses the production of heavy elements through the νp-process.The enhanced 3α reaction rate decreases the production of heavy elements in the higher mass region (A > 140) by 10-60 %.Also, the suppression is less significant than that of the SN II wind model due to the high entropy of the HN wind model.The value of ρ 6 in Eq. ( 1) becomes small at a nearly fixed temperature of the α-particle recombination T 9 ∼ 4 in the high entropy wind.Thus, the enhancement is small for the high entropy case (Jin et al. 2020).The enhanced 3α reaction rate decreases the ratio of free neutrons to the seed nuclei ∆ n from the p(ν e , e + )n reaction (Nishimura The enhanced 3α reaction rate suppresses the production of heavy elements around the νp-process endpoint in both HN and SN II wind models.However, the HN wind model produces sufficient heavy elements beyond A > 100, while the SN II model does not.Therefore, the role of the particle-induced Hoyle state de-excitation in 92,94 Mo and 96,98 Ru depends upon the wind models.The mass fractions of 92,94 Mo and 96,98 Ru in Fig. 3(b) are too small to affect the GCE even without the suppression of the νp-process.Hence, we will focus on neutrino-driven winds in HNe producing a large number of heavy elements, as shown in Fig. 4(b), and we will demonstrate the effects of Hoyle state de-excitation on the solar abundances and the GCE of 92,94 Mo and 96,98 Ru in the next sections.Here, the abundances of p-nuclei are enhanced by up to about 30%.Although the suppression of 92,94 Mo and 96,98 Ru is due to the enhanced 3α reaction rate as reported by Jin et al. (2020), this occurs only in typical neutrino-driven winds of SNe II.On the other hand, such suppression is not necessarily found in the HN neutrino-driven wind based on the energetic supernova explosion model with a massive PNS and large neutrino luminosity as given in (Fujibayashi et al. 2015).

νp-process nucleosynthesis
The Hoyle ON points (circles) on Fig. 5  νp-process is less prominent than for Mo.This is because the solar isotopic fractions of 96,98 Ru are small (7.4%) compared with those of 92,94 Mo (24.1%).Also, the r-process is the main contributor (Bisterzo et al. 2014) to the Ru elemental abundance.To increase the calculated Ru elemental abundance at low metallicity, an increased contribution from the r-process (Yamazaki et al. 2022) may be needed rather than the νp-process.This is because the overestimation of 98 Ru in Fig. 5(a) would not be improved if the yield of the νp-process was increased.
Finally, we note several theoretical uncertainties in our calculation.First, we estimated the yield of the HN νp-process with only one set of neutrino-driven wind trajectories.However, the trajectories should also involve different explosion timescales and different progenitors leading to different massive PNSs.Also, nucleosynthesis calculations employing matter profiles obtained in neutrino radiation hydrodynamic simulations (e.g.Sekiguchi et al. (2012);Fujibayashi et al. (2021)) may provide more reliable yields of the p-nuclei.Another point is that, for simplicity we have ignored the contribution of neutrino oscillations (Ko et al. 2020(Ko et al. , 2022) ) to the νp-process.In particular, fast flavor conversions (Fujimoto & Nagakura 2023) which have been actively studied in recent years can enhance the yields of p-nuclei (Xiong et al. 2020).Additionally, we have ignored the enhancement of the 3α reaction induced by α-particle inelastic scattering in the reaction network.Such reactions may have a non-negligible impact after the α-rich freeze-out at low temperature as in Figs.1(b) and 2(b).
Also, an R-matrix description (Azuma et al. 2010) incorporating experimental results could allow for a more sophisticated evaluation of the particle-induced Hoyle state de-excitation near the energy threshold.This could provide a more accurate 3α reaction rate for nucleosynthesis calculations.We have employed theoretical estimates of nuclear masses for unstable nuclei.Nuclear masses obtained in recent mass measurements (e.g.Zhou et al. (2023)) could make the νp-process calculations more realistic and revise the nuclear yields beyond the waiting-point nucleus 64 Ge.

CONCLUSION
We have analyzed the νp-process in core-collapse supernovae by taking into account the enhanced 3α reaction rates induced by inelastic scatterings of free neutrons and protons on the Hoyle state.
We find a negligible impact from the neutron-induced inelastic scattering but a large effect from the proton-induced scattering.For the SN neutrino-driven wind, the particle-induced Hoyle state de-excitation suppresses the production of 92,94 Mo and 96,98 Ru abundances as reported in previous work.On the other hand, for the HN wind with a massive PNS ∼ 3M , 92,94 Mo and 96,98 Ru are enhanced by the Hoyle state effect although the nuclear yields for A > 140 are reduced.
The calculated abundance yields of the HN wind model were then applied to the GCE calculation of p-nuclei.We found that the HN νp-process significantly contributes to the GCE of 92,94 Mo and 96,98 Ru, regardless of the particle-induced Hoyle state de-excitation.We have demonstrated a possible contribution from the HN νp-process to the origin of p-nuclei in the solar system.Further studies, including an analysis of uncertainties neglected in our calculation, could help substantiate these conclusions. ACKNOWLEDGEMENT

Figure 1
Figure1(a) shows the 3α reaction rates used for the νp-process calculations in the SN II wind model.The solid and dashed lines are respectively the rates with and without the particle-induced Hoyle state de-excitation of neutron and proton scatterings.To see the contribution of the neutroninduced enhancement, we obtained the dotted line by setting f p = 0 in Eq. (1).The 3α reaction rate is enhanced by more than a factor of 100 at higher temperatures (T 9 > 9) due to the large baryon density ρ 6 and the large abundance of free nucleons (Fig.1(b)).The dashed and dotted lines are almost identical at T 9 < 6 and the contribution from the neutron scattering becomes negligible.This is because the mass fraction of free neutrons significantly decreases with the production of seed nuclei around 56 Ni through the α-capture reactions as shown in Fig.1(b).Then, the enhancement of the 3α reaction is mainly caused by proton scattering in the temperature range 2 < T 9 < 6 due to the freeze out of the protons as in Fig.1(b).Finally, as the baryon density decreases, the enhancement is negligible in the low-temperature region, T 9 < 2.

Figure 2 Figure 1 .
Figure2shows the enhanced 3α reaction rates and the evolution of mass fractions in the HN wind model.The results are similar to the case of the SN II wind model.The proton scattering only contributes to the enhancement in Eq. (1) for T 9 < 4.This is due to the freeze out of free protons

Figure 3
Figure 3(a) shows the effect of the particle-induced Hoyle state de-excitation on the nuclear mass fractions of final abundances for various nuclei in the SN II wind model.The solid and dashed lines Figure 3(b)  shows the mass fractions of92,94 Mo and96,98 Ru in the SN II wind model with and without the enhancement of Eq. (1).The results with the enhanced 3α reaction rate (solid line) are smaller than those without it (dashed line) by about a factor of 100.Such significant suppression of the νp-process in the SN II wind model having S ∼ 50 k B nuc −1 , τ dyn ∼ 10 ms is consistent with the results ofJin et al. (2020).

Figure 2 .Figure 3 .
Figure 2. The results of the HN wind model as in Fig. 1.

Figure 4 .
Figure 4.The calculated mass fractions in the HN wind model as in Fig. 3.

Figure 5 .
Figure 5. (a) Comparison between the calculated and observed solar abundances of p-nuclei at [Fe/H] = 0.The square points are observational data (Lodders 2003).The upward-pointing triangles show the calculated result without the HN νp-process.The circles and downward-pointing triangles show respectively the results of the HN νp-process with and without the effects of particle-induced Hoyle state de-excitation.(b) The ratio of Hoyle OFF/Hoyle ON in the top panel.

3. 3 .Figure 6 .Figure 7 .
Figure.5(a)  shows the calculated solar abundances of p-nuclei in the A = 84-102 range.These are compared with observational data(Lodders 2003).All calculated results include the contributions Figure6shows the calculated elemental abundances of Mo normalized to the solar system values at [Fe/H] = 0 and the observational data taken from the SAGA database(Suda et al. 2008).The thick solid line in Fig.6(a) shows the result of Hoyle ON and the thick dashed line shows the result without the contribution from the HN νp-process.These two lines indicate the significant enhancement of the elemental abundances.Moreover, the HN νp-process improves the agreement between the GCE calculation with the observational data.Such impact of the HN νp-process was also found in(Sasaki et al. 2022).

Figure 6
Figure 6(b) shows the impact of the particle-induced Hoyle state de-excitation on the elemental abundance in the shaded region of the top panel.The Hoyle state effect slightly increases the total elemental abundance of Mo with the HN νp-process.The thin solid, dash-dotted, and dotted lines on Fig. 6(a) are partial contributions of the (γ + νp)-processes, the s-process, and the r-process, respectively.At low metallicity [Fe/H] < −2, the γ-process is negligible and the HN νp-process dominantly contributes to the total elemental abundance irrespective of the Hoyle state effect.The results for the Ru elemental abundances are shown in Fig 7. The particle-induced Hoyle state de-excitation is almost unchanged.The Ru elemental abundance and the contribution from the HN