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

TL;DR

The authors present a non-perturbative real-space Green's function formalism to describe sub-barrier proton-nucleus resonances, addressing how resonance energies, widths, and cross-sections emerge from tunneling through a Coulomb barrier with a short-range strong interaction modeled as a delta-shell impurity. By solving the Dyson equation exactly and calibrating the barrier via a WKB mapping to a square barrier, they obtain analytic expressions for the barrier propagator and the full propagator, with resonances determined by $1-\\gamma \\mathcal{D}(x_1,x_1; \\mathcal{E}_{\\text{res}})=0$ and lifetimes from $\\tau=\\hbar/\\Gamma$. Applying the method to $p+^7$Li, $p+^{14}$N, and $p+^{23}$Na, they find a dichotomy: lighter systems host threshold resonances near the continuum ($E_{\\text{res}}\\approx0.489$ MeV for Li and $1.067$ MeV for N) with strong cross-section enhancements, while $^{23}$Na exhibits a saturated resonance at $E_{\\text{res}}\\approx2.11$ MeV set by barrier geometry, in close agreement with experiment. The study defines a domain of validity up to $Z\\le 18$, beyond which the proton tends toward bound-state behavior, indicating the need for alternative treatments for heavier systems. These results advance a rigorous, analytic framework for interpreting resonances in stellar reaction rates and proton-capture processes.

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.

Figures (4)

• Figure 1: Diagrammatic summation of quantum paths for the 1D potential barrier. (a) The potential profile $V(x)$. (b) Paths contributing to the transmission Green's function (Eq. (\ref{['eq:G_tr_1']})). (c) Paths contributing to the reflection Green's function (Eq. (\ref{['eq:G_ref_1']})). (d) Infinite internal reflections constituting the barrier propagator $\mathcal{D}(x,x'|E)$ (Eq. (\ref{['eq:Green_D']})), used in the impurity Dyson equation.
• Figure 2: The dependence of the proton resonance energy $E_{\text{res}}$ on the surface impurity strength $\gamma$ for $^7$Li, $^{14}$N, and $^{23}$Na.
• Figure 3: The relative resonant cross-section $\sigma(E)$ for $p+^7$Li (blue), $p+^{14}$N (red), and $p+^{23}$Na (green). (a) In the strong-coupling regime, resonances occur at the saturation "plateau" energies. This regime correctly reproduces the physical $^{23}$Na resonance at $\approx 2.1$ MeV, though with a relatively broad, low-amplitude signature. (b) In the weak-coupling regime, the Li and N resonances shift to their physical energies ($0.49$ MeV and $1.07$ MeV) and exhibit a massive enhancement in amplitude, characteristic of sharp threshold states. The vertical dotted lines indicate the classical Coulomb barrier heights.
• Figure 4: The transition from quantum scattering to classical stability. For light nuclei ($Z \le 18$), the $p+\text{Nucleus}$ system supports broad scattering resonances with lifetimes on the order of $10^{-21}$ s (blue circles). A distinct stability limit is reached at Argon ($Z = 18$). Beyond this vertical cutoff, the Gamow suppression becomes severe, causing the lifetime to diverge to macroscopic scales. This indicates that for $Z > 18$, the proton is effectively trapped, forming a bound state rather than a transient scattering resonance.