Light antiproton-nucleus systems at low energies with the ab initio NCSM/RGM method

Abstract

The availability of low-energy antiproton beams at the CERN Antiproton Decelerator has renewed interest in using antimatter as a probe of nuclear structure and in forming exotic antiprotonic few-body systems. In this work, we extend the ab initio No-Core Shell Model combined with the Resonating Group Method (NCSM/RGM), which was successfully applied to light-nucleus structure and reactions, to antiproton-nucleus dynamics at low energies. The NCSM/RGM formalism is adapted to antiproton projectiles by removing the requirement of antisymmetrization under exchange of target and projectile constituents, while retaining a fully microscopic description of the nuclear target and the relative motion. We focus on the lightest systems, ${\bar p}+d$, ${\bar p}+{}^3 \mathrm{H}$, and ${\bar p}+{}^3\mathrm{He}$, for which benchmarking against exact solutions of the Schrödinger equation enables stringent validation and helps disentangle methodological uncertainties -- e.g., those associated with the choice of configurations included in the NCSM/RGM expansion -- so that the dominant residual uncertainty can be attributed to the $N\bar{N}$ interaction. We compute phase shifts, scattering lengths, cross sections, antiprotonic-atom level shifts and widths, nuclear quasi-bound energies, and annihilation densities. We find that the hard short-range components of the meson-exchange-based $N\bar{N}$ interaction lead to slow convergence of the NCSM/RGM kernels expanded in a harmonic-oscillator basis, requiring exceptionally large model spaces and posing significant numerical challenges. We discuss practical strategies to mitigate these limitations and assess the impact of missing closed-channel configurations, which is a significant source of uncertainties in very light systems.

Figures (16)

• Figure 1: Relative radial HO wave function ($A=2$, $n=40$, $\ell=0$, $\hbar \omega=20$ MeV) before (blue lines) and after (red) applying a Woods–Saxon regulator (green dashed line) with $r_{\text{reg}}=8$ fm.
• Figure 2: Effect of the finite-model-space artifacts and their removal on the real part of the $s$-wave $\bar{p}-d$ phase shift in the ${}^2S^-_{1/2}$ channel. The calculations use the deuteron ground-state and $a_c=20$ fm. For the regulated calculation, we use $r_{\text{reg}}=10$ fm and $r_{\text{reg,c}}=5$ fm for the strong and short-range Coulomb contributions to the kernels, respectively. The blue dashed curve illustrates the oscillation of the phase shift around the physical value due to the presence of the numerical artifacts. The black curve demonstrates suppressing the artifacts by increasing the size of the model space, while the yellow diamonds demonstrate doing so using a regulator.
• Figure 3: Schematic illustration of the naive regularization introduced to mitigate artifacts from the finite HO expansion used in the NCSM/RGM kernels. Filled circles denote the finite-model-space artifacts induced by the hard-core nature of the $N\bar{N}$ interaction, whereas crossed circles represent their removal to obtain the desired smooth asymptotic behavior outside the antinucleon-nucleus interaction region. This cartoon motivates the numerical results shown in \ref{['fig:noise']}.
• Figure 4: Diagonal ($\nu=\nu'$) strong-interaction potential kernel $V^{s}_{\nu \nu}(r',r)$ for antinucleon–deuteron system in the ${}^2S_{1/2}^-$ channel, extracted from the NCSM/RGM calculation with $N_{\text{max}}=20$ and deuteron's ground state. The kernel is shown for $r,r'\le 2$ fm, beyond which it is negligible (consistent with zero on the scale of the plot).
• Figure 5: Real (green) and imaginary (red) parts of the $s$-wave $\bar{p}-d$ phase shift in the ${}^2S^-_{1/2}$ channel, obtained with the regulator parameters $r_{\text{reg}}=12$ fm and $r_{\text{reg,c}}=5$ fm. Top, $N_{\text{max}}$ dependence when only the deuteron ground state is retained. Bottom, dependence on the number of included target states in the deuteron channel at fixed $N_{\text{max}}=70$. Here, $d~ {(\rm g.s.)}$ denotes the phase shift including only the deuteron ground state; $d+d^*$ includes the ground state plus the first pseudo-state in the same (${}^3{S}_1-{}^3{D}_1$) channel, and so on. At this $N_{\text{max}}$, contributions from the other two-body channels are negligible.