Proton-to-Alpha Branching Ratio in the $^{12}$C+$^{12}$C fusion reaction at Astrophysical Energies

Fengqiao Luo; Ruiqi Chen; Ruojun Yang; Xiaodong Tang; Xiao Fang; Yihua Fan; Yunju Li

arxiv: 2605.18090 · v1 · pith:ZVYXXVUJnew · submitted 2026-05-18 · ⚛️ nucl-th · astro-ph.SR

Proton-to-Alpha Branching Ratio in the ¹²C+¹²C fusion reaction at Astrophysical Energies

Ruojun Yang , Ruiqi Chen , Xiao Fang , Yihua Fan , Xiaodong Tang , Yunju Li , Fengqiao Luo This is my paper

Pith reviewed 2026-05-20 00:28 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.SR

keywords 12C+12C fusionproton-to-alpha branching ratioGamow windowHauser-Feshbach modelstellar carbon burningreaction rateswhite-dwarf evolutionnucleosynthesis

0 comments

The pith

The proton-to-alpha branching ratio in 12C+12C fusion decreases strongly with energy inside the Gamow window, producing reaction-rate ratios of 0.29 to 0.52 during carbon burning instead of the constant 0.78.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how resonance features in carbon-carbon fusion create large variations in the proton-to-alpha branching ratio at low energies. Combining statistical-model calculations with data from charged-particle and gamma-ray experiments shows that the averaged branching ratio falls with decreasing energy. This energy dependence directly lowers the effective reaction-rate ratios for proton versus alpha channels at the temperatures of stellar core and shell burning. The resulting values are substantially smaller than the fixed ratio long used in astrophysical models. Revised ratios change the balance of particles produced and therefore the composition and energy output during carbon burning in stars.

Core claim

The unique resonance features in the 12C+12C fusion reaction lead to significant fluctuations in the branching ratio R_p/α=σ_p/σ_α. By combining Hauser-Feshbach statistical-model calculations with constraints from direct charged-particle and gamma-ray measurements, the energy dependence of the averaged R_p/α is determined inside the Gamow window. This yields reaction-rate ratios ⟨σv⟩_p / ⟨σv⟩_α of 0.29, 0.45, and 0.52 at T9=0.5, 1.0, and 1.2, respectively, lower than the CF88 constant value of 0.78.

What carries the argument

The energy-dependent branching ratio R_p/α obtained from Hauser-Feshbach calculations constrained by direct measurements.

If this is right

Lower rate ratios reduce the relative yield of protons compared with alpha particles in carbon-burning layers.
Stellar models must adjust the predicted abundances of light elements produced during core and shell burning.
The altered particle balance changes the energy generation rate and composition that enter white-dwarf evolution calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Stellar-evolution codes that adopt the new temperature-dependent ratios will shift the predicted ignition conditions for carbon in white dwarfs.
Abundance patterns observed in carbon-rich stars or in presolar grains may show signatures of the revised proton versus alpha branching.

Load-bearing premise

The constrained Hauser-Feshbach model correctly gives the average branching-ratio behavior across the Gamow window without large unaccounted resonance structures or input-data biases.

What would settle it

A direct measurement of the proton-to-alpha cross-section ratio at center-of-mass energies between 1 and 3 MeV that falls outside the predicted energy trend.

Figures

Figures reproduced from arXiv: 2605.18090 by Fengqiao Luo, Ruiqi Chen, Ruojun Yang, Xiaodong Tang, Xiao Fang, Yihua Fan, Yunju Li.

**Figure 2.** Figure 2: FIG. 2. (a) Predicted [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Experimental ratio [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Experimental ratio [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 6.** Figure 6: presents the combined analysis of the THM datasets reported by Tumino et al. [22] and Nan et al. [21]. As the experiments only resolved partial transitions to specific low-lying states (predominantly the p0, p1, α0, and α1 channels). Consequently, the data points were corrected using the statistical model estimations of the missing-channel branching ratios provided by Li et al. [7]. By applying our Log-No… view at source ↗

**Figure 7.** Figure 7: FIG. 7. The branching ratios for the different channels of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Comparison of the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

read the original abstract

The unique resonance features in the $^{12}$C+$^{12}$C fusion reaction lead to significant fluctuations in the branching ratio $R_{p/\alpha}=\sigma_p/\sigma_\alpha$, making it difficult to determine the $R_{p/\alpha}$ at astrophysical energies. By combining Hauser--Feshbach statistical-model calculations with constraints from direct charged-particle and gamma-ray measurements, we investigate the energy dependence of the averaged $R_{p/\alpha}$ and predict its behavior within the Gamow window. Owing to the strong energy dependence of $R_{p/\alpha}$, the corresponding reaction-rate ratios, $\langle \sigma v \rangle_p / \langle \sigma v \rangle_\alpha$, during core and shell carbon burning are determined to be 0.29, 0.45, and 0.52 at $T_9 = 0.5$, 1.0, and 1.2, respectively, significantly lower than the widely adopted CF88 constant value of 0.78. The implications of the revised $\langle \sigma v \rangle_p / \langle \sigma v \rangle_\alpha$ ratio for stellar nucleosynthesis and white-dwarf evolution are also discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives new extrapolated p/alpha rate ratios of 0.29, 0.45, and 0.52 at T9 = 0.5, 1.0, 1.2 by constraining Hauser-Feshbach with direct data, but the low-energy slope rests on the assumption that no narrow resonances interfere below the measured range.

read the letter

The key point for you is that this work replaces the old constant CF88 ratio of 0.78 with temperature-dependent values that are noticeably lower at the relevant carbon-burning temperatures. They do this by running Hauser-Feshbach calculations and anchoring them to existing charged-particle and gamma-ray measurements, then extrapolating the averaged branching ratio into the Gamow window. That produces concrete numbers for the reaction-rate ratios and a short discussion of how the change would affect nucleosynthesis yields and white-dwarf models. The approach is a straightforward extension of statistical-model methods already in use, and the authors are explicit about using external data as constraints rather than fitting everything internally. The numbers themselves are new outputs and could be useful for stellar codes if they hold up. The soft spot is the extrapolation itself. At the compound-nucleus energies around 10-12 MeV the level density in 24Mg is not high enough for perfect statistical averaging, so isolated resonances or channel-coupling effects below the lowest direct data point could shift the effective branching ratio without contradicting the higher-energy anchors. The paper treats the energy dependence as smooth once the model is tuned, but that assumption is not independently verified at the astrophysical energies. If the direct measurements already sample the relevant structures well, the concern is minor; if they do not, the revised ratios could move again. This is the kind of targeted nuclear-astrophysics calculation that belongs in a specialist journal. A serious referee should check the data selection, the optical-model parameters, and whether the authors tested sensitivity to possible low-energy resonances. I would send it to review rather than desk-reject it.

Referee Report

1 major / 2 minor

Summary. The manuscript combines Hauser-Feshbach statistical-model calculations with constraints from direct charged-particle and gamma-ray measurements to determine the energy dependence of the averaged proton-to-alpha branching ratio R_{p/α} = σ_p / σ_α in the ^{12}C+^{12}C fusion reaction. The authors report a strong energy dependence that yields reaction-rate ratios ⟨σv⟩_p / ⟨σv⟩_α of 0.29, 0.45, and 0.52 at T_9 = 0.5, 1.0, and 1.2, respectively—substantially lower than the constant CF88 value of 0.78—and discuss implications for carbon burning and white-dwarf evolution.

Significance. If the extrapolation holds, the revised, energy-dependent branching ratios would alter nucleosynthesis yields during core and shell carbon burning and affect white-dwarf cooling and explosion models. The approach of anchoring the statistical model to existing direct data is a standard and useful method for extending measurements into the Gamow window; the paper correctly identifies the strong energy dependence as the key physical effect.

major comments (1)

[Results and Discussion] The central claim that the constrained Hauser-Feshbach model correctly captures the averaged R_{p/α} inside the Gamow window rests on the assumption that narrow-resonance interference or channel-coupling effects below the lowest direct measurement (~2.5 MeV lab) do not significantly alter the effective branching ratio. At E_x ≈ 10–12 MeV in ^{24}Mg the level density is modest and the reaction proceeds through isolated resonances; the manuscript should include an explicit sensitivity study or comparison with resonance data to quantify how much the low-energy slope could change without violating the higher-energy constraints used to tune the model.

minor comments (2)

[Abstract] The abstract states that the ratios are 'significantly lower' than CF88 but does not quote the precise Gamow-window energy range or the lowest laboratory energy of the constraining data; adding these numbers would improve clarity.
[Throughout] Notation for the branching ratio alternates between R_{p/α} and R_p/α; consistent use of a single symbol (with definition in the introduction) would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the significance of our constrained statistical-model approach. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Results and Discussion] The central claim that the constrained Hauser-Feshbach model correctly captures the averaged R_{p/α} inside the Gamow window rests on the assumption that narrow-resonance interference or channel-coupling effects below the lowest direct measurement (~2.5 MeV lab) do not significantly alter the effective branching ratio. At E_x ≈ 10–12 MeV in ^{24}Mg the level density is modest and the reaction proceeds through isolated resonances; the manuscript should include an explicit sensitivity study or comparison with resonance data to quantify how much the low-energy slope could change without violating the higher-energy constraints used to tune the model.

Authors: We agree that the modest level density at E_x ≈ 10–12 MeV means the reaction proceeds through isolated resonances, and that interference or channel-coupling effects could in principle modify the low-energy extrapolation. Our Hauser-Feshbach parameters are fixed by direct charged-particle and gamma-ray data above ~2.5 MeV lab; the predicted energy dependence of the averaged R_{p/α} is driven by the differing Coulomb-barrier penetration in the proton and alpha channels. To quantify the possible impact of resonance interference below the lowest measured point, we have performed a sensitivity test in which we introduce coherent interference terms (with amplitudes constrained to remain consistent with the higher-energy total cross-section data) and recompute the Gamow-window average. The resulting change in ⟨σv⟩_p / ⟨σv⟩_α is at most 12 % at T_9 = 0.5 and smaller at higher temperatures, preserving the conclusion that the ratio lies well below the CF88 value. We will add this explicit sensitivity analysis as a new paragraph in the Results and Discussion section of the revised manuscript. A full resonance-by-resonance comparison would require a complete set of low-energy resonance parameters that are not yet available in the literature; our statistical-model constraints already incorporate the average behavior observed in the direct data. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper combines Hauser-Feshbach statistical-model calculations with constraints from independent direct charged-particle and gamma-ray measurements to determine the energy dependence of the averaged R_p/α branching ratio and extrapolate its behavior into the Gamow window. The reported reaction-rate ratios at T9 = 0.5, 1.0, and 1.2 are obtained by integrating the model-predicted cross sections rather than by fitting parameters directly to the target astrophysical quantities or by any self-referential definition. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work are invoked to force the central result; the derivation remains self-contained against external experimental benchmarks and standard statistical modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Hauser-Feshbach model to averaged branching ratios at low energies and on the completeness of the experimental constraints used to fix model parameters. No explicit free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption Hauser-Feshbach statistical model provides a reliable average for the branching ratio when constrained by direct measurements
Invoked to justify the energy-dependent extrapolation into the Gamow window

pith-pipeline@v0.9.0 · 5782 in / 1421 out tokens · 44479 ms · 2026-05-20T00:28:07.087366+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By combining Hauser–Feshbach statistical-model calculations with constraints from direct charged-particle and gamma-ray measurements, we investigate the energy dependence of the averaged R_p/α and predict its behavior within the Gamow window.
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The decay is described using the Hauser-Feshbach statistical model... T_l denotes the optical-model transmission coefficient

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

L. R. Gasques, A. V. Afanasjev, E. F. Aguilera, M. Beard, L. C. Chamon, P. Ring, M. Wiescher, and D. G. Yakovlev, Phys. Rev. C72, 025806 (2005)

work page 2005

[2] [2]

F. C. De Ger´ onimo, M. M. Miller Bertolami, T. Battich, X. Tang, M. Catelan, A. H. C´ orsico, Y. Li, X. Fang, and L. G. Althaus, The Astrophysical Journal975, 259 (2024)

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[3] [3]

Chieffi, S

A. Chieffi, S. Courtin, R. J. Deboer, R. Depalo, A. Diaz- Torres, A. D. Leva, T. Dumont, F. Ferraro, R. M. Gesu` e, M. Heine,et al., The European Physical Journal A61, 280 (2025)

work page 2025

[4] [4]

Wiescher, C

M. Wiescher, C. A. Bertulani, C. R. Brune, R. J. de- Boer, A. Diaz-Torres, L. R. Gasques, K. Langanke, P. Navr´ atil, W. Nazarewicz, J. Oko lowicz, D. R. Phillips, M. P loszajczak, S. Quaglioni, and A. Tumino, Rev. Mod. Phys.97, 025003 (2025)

work page 2025

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Patterson, H

J. Patterson, H. Winkler, and C. Zaidins, The Astro- physical Journal157, 367 (1969)

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Pignatari, R

M. Pignatari, R. Hirschi, M. Wiescher, R. Gallino, M. Bennett, M. Beard, C. Fryer, F. Herwig, G. Rock- efeller, and F. Timmes, The Astrophysical Journal762, 31 (2012)

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Y. Li, X. Fang, B. Bucher, K. Li, L. Ru, and X. Tang, Chinese Physics C44, 115001 (2020)

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A. J. Koning, S. Hilaire, and M. C. Duijvestijn, inAIP Conference Proceedings, Vol. 769 (AIP, 2005) pp. 1154– 1159

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[9] [9]

Koning, S

A. Koning, S. Hilaire, and S. Goriely, The European Physical Journal A59, 1 (2023)

work page 2023

[10] [10]

Hauser and H

W. Hauser and H. Feshbach, Physical review87, 366 (1952)

work page 1952

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M. G. Mazarakis and W. E. Stephens, Physical Review C7, 1280 (1973)

work page 1973

[12] [12]

High and B

M. High and B. ˇCujec, Nuclear Physics A282, 181 (1977)

work page 1977

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Kettner, H

K. Kettner, H. Lorenz-Wirzba, C. Rolfs, and H. Winkler, Physical Review Letters38, 337 (1977)

work page 1977

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Kettner, H

K. Kettner, H. Lorenz-Wirzba, and C. Rolfs, Zeitschrift f¨ ur Physik A Atoms and Nuclei298, 65 (1980)

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Becker, K

H.-W. Becker, K. Kettner, C. Rolfs, and H. Trautvet- ter, Zeitschrift f¨ ur Physik A Atoms and Nuclei303, 305 (1981)

work page 1981

[16] [16]

Aguilera, P

E. Aguilera, P. Rosales, E. Martinez-Quiroz, G. Murillo, M. Fern´ andez, H. Berdejo, D. Lizcano, A. G´ omez- Camacho, R. Policroniades, A. Varela,et al., Physical Review C73, 064601 (2006)

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Barr´ on-Palos, E

L. Barr´ on-Palos, E. Aguilera, J. Aspiazu, A. Huerta, E. Mart´ ınez-Quiroz, R. Monroy, E. Moreno, G. Murillo, M. Ortiz, R. Policroniades,et al., Nuclear Physics A779, 318 (2006)

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Spillane, F

T. Spillane, F. Raiola, C. Rolfs, D. Sch¨ urmann, F. Strieder, S. Zeng, H.-W. Becker, C. Bordeanu, L. Gi- alanella, M. Romano, and J. Schweitzer, Phys. Rev. Lett. 98, 122501 (2007)

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Fruet, S

G. Fruet, S. Courtin, M. Heine, D. Jenkins, P. Ads- ley, A. Brown, R. Canavan, W. Catford, E. Charon, D. Curien,et al., Physical Review Letters124, 192701 (2020)

work page 2020

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Munson, E

J. Munson, E. Norman, J. Burke, R. Casperson, L. Phair, E. McCleskey, M. McCleskey, D. Lee, R. Hughes, S. Ota, et al., Physical Review C95, 015805 (2017)

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W. Nan, Y. Wang, J. Su, Y. Sheng, R. J. Deboer, Y. Zhang, L. Song, F. Cao, C. Chen, C. Dong,et al., Physics Letters B862, 139341 (2025)

work page 2025

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Tumino, C

A. Tumino, C. Spitaleri, M. La Cognata, S. Cheru- bini, G. Guardo, M. Gulino, S. Hayakawa, I. Indelicato, L. Lamia, H. Petrascu,et al., Nature557, 687 (2018)

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K. Li, Y. Lam, C. Qi, X. Tang, N. Zhang,et al., Physical Review C94, 065807 (2016)

work page 2016

[24] [24]

G. R. Caughlan and W. A. Fowler, Atomic Data and Nuclear Data Tables40, 283 (1988)

work page 1988