Structure and Formation of the Deeply Bound $\bar{p}$ atoms

TL;DR

This work addresses the problem of understanding deeply bound antiprotonic atoms and their formation mechanisms. It combines a phenomenological $\bar{p}$-nucleus optical potential with the Klein-Gordon equation to compute binding energies and widths for both nuclear and atomic states, including electromagnetic corrections. The formation of these states is analyzed with the effective number approach for the $^{12}$C, $^{16}$O, and $^{31}$P targets in $(\bar{p},p)$ reactions, showing that atomic states appear as discrete peaks in the emitted-proton spectra due to the small momentum transfer at forward angles. The results indicate that atomic states, including the deepest $1s$ level, can be observed experimentally, especially using $^{31}$P as a target, providing valuable insight into $\bar{p}$-nucleus interactions and finite-density hadron dynamics.

Abstract

We study theoretically the structure and formation of the deeply bound $\bar{p}$ atoms. We find that the widths of the atomic states are narrower than the level spacing even for deeply bound states so that the well-isolated deeply bound $\bar{p}$ atoms are expected to exist. We also find the $\bar{p}$-nuclear states with huge widths. For the observation of the deep $ {\bar p}$-atomic states, we investigate theoretically the $(\bar{p}, p)$ reactions for $^{12}$C, $^{16}$O, and $^{31}$P target nuclei. We find that the momentum transfer of the $( {\bar p},p)$ reaction is small and the formation of the $ {\bar p}$-atomic states can be observed as the discrete peak structures in the $( {\bar p},p)$ spectrum. We conclude that the $(\bar{p}, p)$ reactions are very much suited for the $\bar{p}$ atom formation and the spectra of the reaction are expected to provide new valuable information on the $ {\bar p}$ atoms and $ {\bar p}$-nucleus interaction.

Figures (6)

• Figure 1: The antiproton-$^{11}$B optical potential as functions of the radial coordinate $r$. The potential is determined in Ref. Batty:1997zp as shown in Eqs. \ref{['eq:Optical']} and \ref{['eq:b0']} in the text. The solid and dashed lines show the real and imaginary parts of the potential, respectively.
• Figure 2: The antiproton-$^{11}$B electromagnetic potential $V_{\rm em}(=V_{\rm FS}+V_{\rm VP})$ described in Eqs. \ref{['eq:Coulomb_FS']} and \ref{['eq:Coulomb_VP']} as the function of the radial coordinate $r$.
• Figure 3: Density distributions of ${\bar{p}}$ in the $s$-states of ${\bar{p}}-^{11}$B bound system. The solid lines represent the densities of 1st and 2nd $s-$states ($1s$ and $2s$) which are considered as the nuclear states. The dashed line shows the density of the 3rd $s-$state ($3s$), which is interpreted as the atomic 1$s$ state. The vertical solid line indicates the root-mean-square radius $\ev*{r^{2}}^{1/2}=2.42$ fm of $^{11}$B.
• Figure 4: Formation cross sections of ${\bar{p}}-^{11}$B bound systems in $^{12}\text{C}(\bar{p},p)$ reactions at $p_{\bar{p}}=1$ GeV/c plotted as functions of the emitted proton kinetic energies at (a) $\theta_{\rm lab}=0^{\circ}$ and (b) $\theta_{\rm lab}=5^{\circ}$. The vertical solid line indicates the threshold energy of the ${\bar{p}}$ bound states formation with the ground state of the daughter nucleus. Main subcomponents of the spectra are indicated in the figure as the combinations of the ${\bar{p}}$ atomic state $(n\ell)_{ {\bar{p}}}$ and the proton-hole state $(n\ell_{j})_{h}$ in the final system. The contributions from ${\bar{p}}$ nuclear states and $1s_{1/2}$ proton-hole state do not show any structures in these figures because of their large widths. The contributions of $d$ and $f$ states of ${\bar{p}}$ atom are not seen in the spectrum (see text).
• Figure 5: Same as Fig. \ref{['fig:12C(pbar,p)11B_pbar_spectra']} except for ${\bar{p}}-^{15}{\rm N}$ bound systems in $^{16}{\rm O}( {\bar{p}},p)$ reactions. The contributions from the $(1p_{3/2})_{h}$ state are not shown in these figures because they appear at 6.3 MeV lower proton energies in the area outside the figure range (see text).