Interaction of antiprotons with nuclei\recdate
May 23, 2016 \abstWe report on self-consistent calculations of quasi-bound states in selected nuclei performed within the relativistic mean-field (RMF) model employing coupling constants tuned to reproduce -atom data. We confirmed considerable polarization of the nuclear core induced by the antiproton. The annihilation was treated dynamically, taking into account the reduced phase space in the nuclear medium as well as the compressed nuclear density. The energy available for the annihilation products was evaluated self-consistently, considering additional energy shift due to particle momenta in the -nucleus system. Next, we constructed the -nucleus optical potential using scattering amplitudes related to the latest version of the Paris potential. We explored energy dependence of the potential and the implications for -nucleus quasi-bound states. \kwordantiproton-nucleus interaction, antiproton annihilation, RMF model, Paris potential
The -nucleus interaction has attracted renewed interest at the prospect of future experiments with beams at FAIR . The possibility of formation of –nuclear bound states has been extensively studied within the RMF approach in recent years [2, 3, 4]. The considerations about their existence are supported by a strongly attractive potential that feels in the nuclear medium. However, the –nucleus interaction is dominated by annihilation which significantly influences propagation of the antiproton in nuclear matter. The data from experiments with atoms  and scattering off nuclei at low energies  could be well fitted by a strongly attractive and strongly absorptive -nuclear optical potential, imaginary part of which outweighs the real part. Still, the phase space for the annihilation products is significantly suppressed for the antiproton deeply bound in the nuclear medium, which could lead to relatively long living inside the nucleus .
In this contribution, we present our fully self-consistent calculations of –nuclear bound states. First, we discuss the -nucleus interaction derived within the RMF model  using G-parity transformed proton-meson coupling constants, properly scaled to fit -atom data . We demonstrate for selected nuclei dynamical effects in the nuclear core caused by the antiproton and the role of proper treatment of the annihilation in the nuclear medium. Next, we analyze energy and density dependence of the in-medium scattering amplitudes constructed from the Paris potential  and apply them in calculations of -nuclear quasi-bound states.
In Section 2, we briefly introduce the underlying model. Few selected representative results of our calculations are discussed in Section 3 and conclusions are summarized in Section 4.
The interaction with nuclei is studied within the RMF approach. The (anti)nucleons interact among themselves by the exchange of the scalar () and vector (, ) meson fields, and the massless photon field . The standard Lagrangian density for nucleonic sector is extended by the Lagrangian density describing the antiproton interaction with the nuclear medium:
where denotes the mass of the antiproton; , , are the masses of the considered meson fields; , , and are the couplings to corresponding fields.
The equations of motion for the hadron fields are derived within the variational principle employing the mean-field and no-sea approximations. The Dirac equations for nucleons and antiproton read:
are the scalar and vector potentials, and denotes single particle states. The Klein–Gordon equations for the boson fields acquire additional source terms due to the presence of :
where and are the scalar, vector, isovector, and charge densities, respectively. The values of the nucleon–meson coupling constants and meson masses are adopted from a particular RMF parametrization. In this work, we present results for the nonlinear RMF model TM1(2)  for heavy (light) nuclei, the nonlinear NL-SH model  and the density dependent model TW99 . In the case of the density dependent model, the couplings are a function of baryon density
2.1 -nucleus interaction
First, the –nucleus interaction is constructed within the RMF model from the –nucleus interaction using the G-parity transformation: the vector potential generated by the meson changes its sign and becomes attractive. As a consequence, the total potential would be excessively attractive. The G-parity transformation is surely a valid concept for long- and medium-range potential, however, at short distances the interaction is dominated by strong annihilation. To take into account possible deviations from G parity due to the absorption as well as various many-body effects in the nuclear medium , we introduce a scaling factor for the –meson coupling constants:
The annihilation in the nuclear medium is described by the imaginary part of the optical potential in a ‘’ form adopted from optical model phenomenology :
where is the –nucleus reduced mass. The density is evaluated dynamically within the RMF model, while the parameter Im fm as well as are determined by fitting the atom data . It is to be noted that the effective scattering length Im describes the absorption at threshold. In the nuclear medium, the energy available for annihilation is reduced due to the binding of the antiproton and nucleon. Consequently, the phase space accessible to annihilation products is suppressed. The absorptive potential then acquires the form
where is the phase space suppression factor and is the branching ration for a given channel (see ref.  for details).
Next, the S-wave scattering amplitudes derived from the latest version of the Paris potential  are used to develop a optical potential. The amplitudes are modified using the multiple scattering approach of Wass et al.  in order to account for Pauli correlations in the medium. The in-medium isospin 1 and 0 amplitudes are of the form
Here, denotes the free-space amplitude as a function of ; is the nuclear core density distribution and is defined as follows
where is Spherical Bessel function, is Fermi momentum, and is the momentum. The S-wave optical potential is expressed as
where () is the proton (neutron) density distribution and the factor transforms the in-medium amplitudes to the -nucleus frame.
The energy available for the annihilation in the nuclear medium is given by Mandelstam variable
where and , with being the average binding energy per nucleon and the binding energy. In the two-body c.m. frame and Eq. (12) reduces to
However, when the annihilation of the antiproton with a nucleon takes place in a nucleus, the momentum dependent term in Eq. (12) is no longer negligible  and provides additional downward energy shift. Taking into account averaging over the angles , Eq. (12) can be rewritten as
where is the average kinetic energy per nucleon and represents the kinetic energy. The kinetic energies were calculated as the expectation values of the kinetic energy operator , where is the (anti)nucleon reduced mass.
For comparison, we applied in our calculations also another form of which was originally used in the studies of -nuclear potentials [13, 14]. The momentum dependence in was transformed into the density dependence. The nucleon kinetic energy was approximated within the Fermi gas model by , where MeV, and the kaon kinetic energy was expressed within the local density approximation by , where and is the Coulomb potential, which led to the expression
We applied the RMF formalism introduced above in calculations of quasi-bound states in various nuclei. First, we did not consider the absorption and studied dynamical effects in a nucleus due to the presence of . We confirmed a large polarization of the nuclear core caused by the antiproton. The energies calculated dynamically are substantially larger than those calculated statically (i.e., without source terms in the r.h.s of Klein-Gordon equations (4)). Moreover, the polarization of the nuclear core varies with the applied RMF model due to different values of nuclear compressibility .
The absorption in a nucleus was described by the imaginary part of phenomenological optical potential (Eq. (8)) and treated self-consistently. In Fig. 1, we present binding energies (left panel) and widths (right panel) in selected nuclei calculated in the TM models for different forms of . The energies do not deviate much from each other. However, the widths, which are sizable in all nuclei considered, exhibit much larger dependence on the applied form of . The largest widths are predicted for M in the two body frame (see Eq. (13)). The widths are significantly reduced after including the momentum dependent terms in (Eq. (14)).
In order to study the effects of the medium, the (anti)nucleon kinetic energies were calculated with constant as well as reduced (anti)nucleon masses. As a result, the kinetic energies calculated with reduced masses ( Jr) are larger and consequently the widths are smaller than those calculated using constant masses ( Jc). The widths calculated using K (Eq. (15)) and Jr are comparable.
Next, we calculated the -nuclear quasi-bound states using the optical potential derived from the S-wave scattering amplitudes of the Paris potential (Eq. (11)). In Fig. 2, the energy dependence for the free-space and in-medium amplitudes for fm is shown. The peaks of the in-medium amplitudes (solid line) are lower in comparison with the free amplitudes and shifted towards threshold. In the case of the in-medium amplitude evaluated for the momentum (dashed line), both the real and imaginary part are substantially reduced and become smooth in the whole energy region.
Fig. 3 shows the and binding energies and widths of the antiproton in O calculated dynamically with the S-wave Paris potential for Jr, compared with the phenomenological RMF approach. The S-wave Paris potential yields similar spectrum of bound states as the phenomenological potential, however the energies and widths are larger than those calculated within the RMF model, particularly in the state. It is to be noted that the Paris potential contains a sizable -wave interaction which should be taken into account. Calculations involving the P-wave term in the optical potential are currently in progress and will be published elsewhere.
In this work, -nucleus quasi-bound states in selected nuclei were studied. The -nucleus interaction was constructed using two different approaches: a) the RMF model with G-parity motivated coupling constants, properly scaled to fit -atom data, and a phenomenological absorptive part; b) the model based on in-medium scattering amplitudes derived from the latest version of the Paris potential. We explored dynamical effects caused by the presence of the strongly interacting in selected nuclei across the periodic table and confirmed sizable changes in the nuclear structure. The dependence of the energies and widths on the applied form of was discussed. We evaluated self-consistently additional downward energy shift due to the and momenta, which leads to significant suppression of the widths in the nuclear medium. However, the widths remain still sizable. We calculated the spectrum of bound states in O using the S-wave Paris potential for the first time. The calculated binding energies and widths are larger than those obtained by the RMF approach.
We wish to thank E. Friedman, A. Gal and S. Wycech for valuable discussions, and B. Loiseau for providing us with the amplitudes. This work was supported by the GACR Grant No. P203/15/04301S.
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