[Paper Review] Reply to the Comments on the 12C+12C fusion S^*-factor

А. Туміно, C. Spitaleri+15 more|arXiv (Cornell University)|Jul 16, 2018

Nuclear physics research studies9 references3 citations

TL;DR

This paper defends the 12C+12C fusion S*-factor measurement published in Nature, asserting that the criticisms by Mukhamedzhanov, Tang, and Pang are unfounded. It argues that the Trojan Horse Method (THM) analysis is valid, with the plane-wave approximation justified by momentum transfer and de Broglie wavelength considerations, and that the observed forward-peaked deuteron angular distributions contradict the critics' distorted-wave predictions. The normalization to direct data is robust, with minimal influence from any single dataset, and the theoretical model by the critics fails to reproduce experimental trends, particularly the forward peak and energy dependence.

ABSTRACT

The goal of this reply is to draw attention of the readers that the major problems rose in the short Comment authored by A.M. Mukhamedzhanov, X. Tang and D.Y. Pang (arXiv:1806.05921) are totally groundless.

Motivation & Objective

• To refute the claims made in a critical Comment by Mukhamedzhanov, Tang, and Pang regarding the 12C+12C fusion S*-factor measurement.
• To demonstrate that the Trojan Horse Method (THM) analysis is physically sound and consistent with experimental data, particularly the forward-peaked deuteron angular distributions.
• To validate the normalization procedure using multiple direct data sets in the 2.5–2.63 MeV cm-energy range, minimizing systematic errors.
• To show that the theoretical model proposed by the critics fails to reproduce key experimental features, such as the forward peak in angular distributions and the correct energy dependence of the S*-factor.
• To reaffirm that the final-state three-body Coulomb interaction does not shift resonance energies, as no such shift is observed in the data.

Proposed method

• The authors apply the Trojan Horse Method (THM) to extract the 12C+12C fusion S*-factor from 14N+12C reactions, using 12C as a spectator in the three-body process.
• They justify the use of the plane-wave approximation by showing a high momentum transfer (qt = 500 MeV/c) and a de Broglie wavelength (0.4 fm) much smaller than the 12C+d radius (3 fm), indicating peripheral interaction.
• The agreement between measured and theoretical deuteron momentum distributions within experimental errors supports the validity of the plane-wave approximation in the measured phase space.
• The S*-factor is normalized to direct experimental data in the 2.5–2.63 MeV cm-energy range, using multiple datasets to reduce systematic uncertainty.
• The authors compare their THM result with the critics' theoretical calculation based on distorted-wave Born approximation (DWBA), showing a fundamental discrepancy in angular distribution predictions.
• They use the overlap region between THM and direct data to extrapolate the THM S*-factor and assess the critics' model, finding it diverges at higher energies and fails to match experimental trends.

Experimental results

Research questions

• RQ1 Does the Trojan Horse Method analysis of the 12C+12C fusion reaction remain valid despite claims of theoretical flaws?
• RQ2 Is the plane-wave approximation justified in the context of the 14N+12C reaction used to extract the 12C+12C S*-factor?
• RQ3 Do the observed forward-peaked deuteron angular distributions contradict the critics' DWBA-based theoretical prediction of backward peaks?
• RQ4 Is the normalization of the THM S*-factor to direct data robust and systematic-error-resistant?
• RQ5 Does the critics' theoretical model accurately reproduce the energy dependence and angular distribution of the 12C+12C fusion cross section?

Key findings

• The plane-wave approximation is validated by high momentum transfer (qt = 500 MeV/c) and a de Broglie wavelength (0.4 fm) much smaller than the 12C+d radius (3 fm), confirming peripheral interaction.
• The measured deuteron momentum distribution matches the theoretical plane-wave prediction within experimental errors, supporting the absence of significant distortion effects.
• The S*-factor derived from THM shows a resonant rise at low energies, consistent with the presence of low-lying resonances, and deviates from the CF88 extrapolation by up to a factor of 30 at T9 = 0.5 GK, not 500 as claimed.
• The critics' theoretical model predicts a backward-peaked angular distribution, which contradicts the observed forward peak in experiment, rendering the model physically inconsistent.
• The critics' model diverges at higher energies and fails to reproduce the energy dependence of the S*-factor, while the THM result agrees with direct data in the overlapping region.
• No shift in resonance energy is observed, indicating that the final-state three-body Coulomb interaction does not significantly affect the resonance positions in this system.