Green's Function Formalism for Impurity-Induced Resonances in Sub-barrier Proton-Nucleus Scattering

Abstract

Motivated by recent experimental refinements of stellar reaction rates, we establish a non-perturbative Green's function formalism based on the exact solution of the Dyson equation for sub-barrier proton-nucleus resonant scattering. By utilizing bare Green's functions to map the quantum tunneling problem onto a scattering formalism, we demonstrate that the summation of infinite quantum paths recovers the exact tunneling coefficients, enabling an analytical solution of the Dyson equation where the strong nuclear force is modeled as a surface delta-shell impurity embedded within the Coulomb field. Applying this framework to the astrophysically relevant $p+{}^7\text{Li}$, $p+{}^{14}\text{N}$, and $p+{}^{23}\text{Na}$ systems, we achieve precise agreement with experimental resonance energies while revealing a fundamental physical distinction in resonance formation. The heavier ${}^{23}\text{Na}$ system is identified as a saturated state, residing on a geometric plateau where the resonance energy becomes insensitive to the interaction strength; our calculated value of $2.11$~MeV aligns remarkably well with the experimental level of $2.08$~MeV. In contrast, the lighter ${}^7\text{Li}$ and ${}^{14}\text{N}$ systems emerge as threshold states in a weak-coupling window, where the resonance energy is highly sensitive to the potential parameters and is sustained near the continuum edge. In this regime, our model yields energies of $0.489$~MeV and $1.067$~MeV, closely reproducing the experimental benchmarks of $0.441$~MeV and $1.058$~MeV, respectively. We demonstrate that these threshold states are characterized by a significant enhancement of the resonant cross-section, driven by the inverse relationship between the tunneling width and the spectral density peak.

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