Antiproton-nucleus quasi-bound states within the 2009 version of the Paris $\bar{N}N$ potential

We studied the ${\bar p}$ interactions with the nuclear medium within the 2009 version of the Paris ${\bar N}N$ potential model. We constructed the $\bar{p}$--nucleus optical potential using the Paris $S$- and $P$-wave ${\bar p}N$ scattering amplitudes and treated their strong energy and density dependence self-consistently. We considered a phenomenological $P$-wave term as well. We calculated $\bar{p}$ binding energies and widths of the $\bar{p}$ bound in various nuclei. The $P$-wave potential has very small effect on the calculated ${\bar p}$ binding energies, however, it reduces the corresponding widths noticeably. Moreover, the $S$-wave potential based on the Paris amplitudes supplemented by a phenomenological $P$-wave term yields the ${\bar p}$ binding energies and widths in very good agreement with those obtained within the RMF model consistent with ${\bar p}$-atom data.


Introduction
The antiproton-nucleus interaction below threshold have been so far studied within phenomenological RMF approaches [1,2]. The G-parity motivatedp coupling constants were used to construct thep-nucleus potential. The absorption ofp was accounted for in terms of a purely phenomenological optical potential. Thep optical potential was confronted withp atom data. It was found that thep coupling constant have to be properly scaled in order to be consistent with the data. Consequently, thep potential was applied in the calculations ofp quasi-bound states in various nuclei [2].
However, it is desirable to study thep interactions with the nuclear medium within other theoretical approaches, such as microscopic models ofNN interaction based on mesonexchange models [3][4][5] or chiralNN interaction models [6,7]. Comparison between thesē NN interaction models could bring valuable information about in-mediump interactions in the direct confrontation with the data fromp atoms andp scattering off nuclei, as well as predictions forp-nuclear quasi-bound states.
Recently, the 2009 version of the ParisNN potential [3] was confronted by Friedman et al. with thep-atom data and antinucleon interactions with nuclei up to 400 MeV/c, including elastic scattering and annihilation cross sections [8]. The analysis revealed the necessity to include the P-wave interaction in order to describe thep atom data. The Paris S -wave potential supplemented by a phenomenological P-wave term was found to fit the data on low-density, near-thresholdp-nucleus interaction. This fact stimulated us to apply it in the present calculations ofp-nuclear quasi-bound states and explore the effect of the P-wave interaction onp binding energies and widths ofp-nuclear states.
In Section 2, we briefly introduce the model applied in our calculations. Section 3 presents few representative results together with the discussion of the main findings of our study.

Methodology
The binding energies Bp and widths Γp ofp quasi-bound states in a nucleus are obtained by solving self-consistently the Dirac equation with the optical potential where mp is the mass of the antiproton and p = −Bp − iΓp/2 (Bp > 0). The S -wavepnucleus optical potential V opt enters the Dirac equation as the time component of a 4-vector and is constructed in a 'tρ' form as follows: Here, Ep = mp − Bp, F 0 and F 1 are isospin 0 and 1 in-medium amplitudes, and ρ p (r) [ρ n (r)] is the proton (neutron) density distribution calculated within the RMF NL-SH model [9]. The in-medium amplitudes F 0 and F 1 entering Eq. (2) account for Pauli correlations in the nuclear medium. They are constructed from the free-spacepN amplitudes using the multiple scattering approach of Wass et al. [10] (WRW) Here, f S pn ( f S pp ) denotes the free-space c.m.pn (pp) S -wave scattering amplitude derived from the ParisNN potential as a function of Mandelstam variable √ s, m N represents the mass of the nucleon and ρ(r) = ρ p (r) + ρ n (r). The factor √ s/m N transforms the amplitudes from the two-body frame to thep-nucleus frame. The Pauli correlation factor ξ k is defined as follows where j 1 (k F r) is the spherical Bessel function, k F is the Fermi momentum and k = ( p + mp) 2 − m 2 p is the antiproton momentum. The integral in Eq.(4) can be solved analytically. The resulting expression is of the form where q = −ik/k F . The analysis ofp atom data [8] revealed that it is necessary to supplement the Paris Swave potential by the P-wave interaction to make the realp potential attractive in the relevant low-density region of a nucleus. To incorporate the P-wave interaction in our model we supplement the r.h.s. of the S -wave optical potential in Eq. (2) [2EpV S opt = q(r)] by a gradient term [8]:  The factor 2l + 1 = 3 in the P-wave part is introduced to match the normalization of the Paris NN scattering amplitudes and Here, f P pp ( √ s) and f P pn ( √ s) represent the Paris P-wavepp andpn free-space c.m. scattering amplitudes, respectively. We do not consider any medium modifications of the P-wave amplitudes since we assume that the P-wave potential should contribute mainly near the surface of the nucleus due to its gradient form.
The analysis of Ref. [8] also revealed that the optical potential constructed from the Paris S -and P-wave amplitudes fails to reproduce thep atom data and that it is mainly due to the P-wave amplitude -its real and imaginary parts had to be scaled by different factors to get reasonable fit. On the contrary, the optical potential based on the Paris S -wave potential supplemented by a purely phenomenological P-wave term with f P pN = 2.9 + i1.8 fm 3 fits the data well. In our calculations, we adopt both P-wave amplitudes, Paris as well as phenomenological, in order to study their effect on the binding energies and widths ofp-nuclear states.
The Paris amplitudes used in our calculations are shown in Fig. 1. There arepp (top) and pn (bottom) medium modified S -wave amplitudes (3) at saturation density ρ 0 = 0.17 fm −3 and free-space P-wave scattering amplitudes plotted as a function of the energy shift δ √ s = E − E th with E th = mp + m N . The S -wave amplitudes vary considerably with energy below threshold. The real in-mediumpp amplitude is attractive in the entire energy region below threshold. The real part of the in-mediumpn amplitude is attractive for δ √ s ≤ −70 MeV with a small repulsive dip near threshold. The imaginary parts of the S -wave amplitudes are comparable or even larger than the corresponding real parts. The energy dependence of the free-space P-wave amplitudes is less pronounced than in the S -wave case. Moreover, the P-wave amplitudes are considerably smaller than the in-medium S -wave amplitudes in the region relevant top-nuclear states calculations.
Strong energy dependence of thepN amplitudes presented in Fig. 1 requires a proper self-consistent scheme for evaluating thep optical potential. The energy argument √ s of the amplitudes is expressed in thep-nucleus frame where the contributions from antiproton and nucleon kinetic energies are not negligible [11] Here, B Nav = 8.5 MeV and T Nav are the average binding and kinetic energy per nucleon, respectively, and Tp represents thep kinetic energy. The kinetic energies are evaluated as corresponding expectation values of the kinetic energy operatorT = − 2 2m . Since the Bp appears as an argument in the √ s, which in turn serves as an argument for V opt , √ s has to be determined self-consistently. Namely, its value obtained by solving Eq. (8) should agree with the value of √ s which serves as input in Eq. (3) and thus Eq. (1), as well.

Results
We performed self-consistent calculations ofp-nuclear quasi-bound states in selected nuclei within the model presented in the previous section. We explored the energy and density dependence of the S -wavep-nucleus potential as well as the role of thepN P-wave interaction, and compared the predictions forp binding energies and widths with the phenomenological RMF approach [2]. ThepN amplitudes are strongly energy and density dependent, as was shown in Fig. 1. Consequently, the depth and shape of thep-nucleus potential depend greatly on the energies and densities pertinent to the processes under consideration. It is demonstrated in Fig. 2 where we present thep potential in 40 Ca calculated for different energies and densities: i) using the Paris free-space S -wave amplitudes at threshold (denoted by 'th free'), ii) using in-medium Paris S -wave amplitudes at threshold (denoted by 'th medium'), iii) using in-medium Paris S -wave amplitudes at energies relevant top atoms (constructed following Ref. [8]), and iv) using in-medium Paris S -wave amplitudes at energies relevant top nuclei [ √ s of Eq. (8)]. Thep potential constructed using the free-space amplitudes has a repulsive real part and fairly absorptive imaginary part. When the medium modifications of the amplitudes are taken into account, thep potential becomes attractive and more absorptive. At the energies relevant tō p atoms, thep potential is more attractive and weakly absorptive. Finally, at the energies relevant top nuclei, thep potential is strongly attractive, however, also strongly absorptive. The figure clearly shows that proper self-consistent evaluation of the energy √ s is essential. Next, we performed static and dynamical calculations ofp binding energies and widths using the ParisNN potential. In the static calculations, the core nucleus is not affected by the presence of extraB. In the dynamical calculations, the polarization of the nuclear core due toB, i.e., changes in the nucleon binding energies and densities, is taken into account. The response of the nuclear core to the extra antiproton is not instant -it could possibly  . 1sp binding energies (left panel) and widths (right panel) in various nuclei, calculated statically (triangles) and dynamically (circles) using S -wave Paris potential (red) and including phenomenological P-wave potential (black). Thep binding energies and widths calculated dynamically using the Paris S + P-wave potential (blue circles) are shown for comparison. last longer than the lifetime ofp inside a nucleus [12,13]. As a result, the antiproton could annihilate before the nuclear core is fully polarized. Our static and dynamical calculations of p binding energies and widths may be thus considered as two limiting scenarios.
In Fig. 3, we present 1sp binding energies (left) and widths (right) as a function of mass number A, calculated statically (triangles) and dynamically (circles) with the Paris S -wave and Paris S -wave + phen. P-wave potentials. We present thep binding energies and widths calculated dynamically using the Paris S + P-wave potential for comparison as well.
In dynamical and static calculations alike, the P-wave interaction does not affect much thep binding energies -they are comparable with the binding energies evaluated using only the S -wave potential. On the other hand, thep widths are reduced significantly when the phenomenological P-wave term is included in thep optical potential. The effect is even more pronounced for the Paris P-wave interaction.
Thep widths calculated dynamically are noticeably larger than the widths calculated statically. It is caused by the increase of the central nuclear density, which exceeds the decrease of thepN amplitudes due to the larger energy shift with respect to threshold (δ √ s ∼ −255 MeV in the dynamical case vs. δ √ s ∼ −200 MeV in the static case). On the other hand, thep binding energies increase only moderately and get closer to each other when the dynamical effects are taken into account. Thep widths exhibit much large dispersion then thep binding energies for the different potentials.
We explored thep excited states in selected nuclei as well and compared the results with those obtained within the RMF approach [2]. Fig. 4 showsp spectra in 40 Ca calculated using the Paris S -wave + phen. P-wave potential and phenomenological RMF approach. The Paris S -wave + phen. P-wave potential yields the 1p and 1d binding energies slightly larger and thus the s-p and s-d level spacing smaller than the RMF approach. It is an effect of a broader p potential well generated by the Paris S -wave + phen. P-wave potential. Nevertheless, both approaches yield comparablep widths as well as energies and the overall agreement is surprisingly good.
It is to be noted that there is no spin-orbit splitting of the p and d levels presented in Fig. 4 since the V opt is a central potential constructed from angular momentum-averaged scattering amplitudes. In the RMF approach, thep binding energies in 1p and 1d spin doublets are nearly degenerate, the difference inp energies (as well asp widths) is up to ∼ 1 MeV. This is in agreement with spin symmetry in antinucleon spectra within the RMF approach [14,15]. In the left panel of Fig. 4 we show the spin-averaged 1p and 1dp binding energies and widths for better comparison with the results obtained with the central Paris potential.
In conclusion, we performed self-consistent calculations ofp-nuclear quasi-bound states using a microscopic potential, namely the ParisNN potential, for the first time. We explored  the effect of the P-wave interaction onp binding energies and widths. We found that the P-wave interaction almost does not affect the binding energies ofp-nuclear states. This is in sharp contrast to the case ofp atoms where it was found necessary to include the P-wave interaction in order to increase attraction of thep optical potential [8]. Moreover, we found good agreement between the results obtained using the phenomenological RMF potential and the Paris S -wave + phenomenological P-wave potential which are the two potentials consistent with antiprotonic atom data andp scattering off nuclei at low energies.