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Coulomb glory in low-energy antiproton scattering by a heavy nucleus: screening effect of vacuum polarization

Backward scattering of antiprotons by bare uranium is studied theoretically for antiproton energies within the interval 100 eV -- 1 keV. A marked maximum of the differential cross section in the backward direction (Coulomb glory) at some energies of the incident particle is revealed. The effect is due to the screening properties of the vacuum polarization potential and can be regarded as a manifestation of the vacuum polarization in non-relativistic collisions of heavy particles. Experimental observation can become feasible with new facilities for antiproton and ion research at GSI.


The project FAIR (Facility for Antiproton and Ion Research) at GSI in Darmstadt will
give an opportunity to get high-intensity antiproton beams at energies between 30 MeV and 300 keV at a magnetic storage ring and at energies between 300 keV and 20 keV at an electrostatic storage ring. It will be possible to decelerate the antiprotons to ultra-low eV energies by means of heavy ion trap facilities. This will make accessible a large variety of new atomic collision experiments, such as investigations of the antiproton scattering. These investigations open new opportunities in observation of the Coulomb glory effect, which was predicted in Refs. [1,2]. The phenomenon results in a prominent maximum of the differential cross section (DCS) in the backward direction at some energy of the incident particle, provided the interaction with the target is represented by a screened Coulomb attraction potential (the pure Rutherford cross section has a smooth minimum at 180 • irrespectively of the energy).
In the previous paper [3] we investigated the backward scattering of antiprotons by highly charged and neutral uranium (Z = 92). It was shown that the Coulomb glory effect takes place due to screening of the nuclear Coulomb attraction by the electrons. In collisions of antiprotons with bare uranium, the Coulomb glory is also present because of the effect of vacuum polarization (VP), which was accounted for in Ref. [3] within the Uehling approximation. Observation of the Coulomb glory in collisions of antiprotons with bare uranium nuclei can be of particular interest since the screening property of the VP potential in nonrelativistic collisions of heavy particles can be manifested. In the present paper we study the influence of the exact one-loop VP potential on the backward scattering of antiprotons with bare uranium. The calculations have been performed in the framework of semiclassical and quantum theory. Atomic units ( = e = m e = 1) are used in the paper.
The non-relativistic scattering theory can be applied if the kinetic energy of the antiproton is as low as a few hundreds of electron volts. We use the partial wave expansion of the scattering amplitude A(θ), where k is the momentum of the antiproton, ν = −Zmp/k is the Coulomb parameter (mp is the antiproton mass) and P l (cos θ) are the Legendre polynomials. Note that the total scattering amplitude is a sum of the pure Coulomb amplitude and the contribution due to short-range (non-Coulomb) terms in the scattering potential. The phase shifts δ s l are the differences between the total phase shifts δ l and the Coulomb phase shifts δ c l δ s l = δ l − δ c l .
DCS is related to the scattering amplitude as follows: The phase shifts δ s l can be calculated by means of the variable phase method [4,5,6] without solving the radial Schrödinger equation. Within this approach, the variable phase δ s l (k, r) is a solution of a first-order differential equation. In our case, this differential equation can be written as Here F l (k, r) and G l (k, r) are the regular and irregular Coulomb wave functions, respectively [7], and v(r) is the short-range part of the scattering potential V (r): The initial condition for solving (4) is δ s l (k, 0) = 0. The phase shift is the limit of δ s l (k, r) as r → ∞: For the energies of the order of a few atomic units the Coulomb parameter is large (|ν| ≫ 1), and the motion of the antiproton can be described in the framework of the quasiclassical approximation [8]. Then the phase shifts δ s l can be presented as a difference of the two integrals which correspond to the phases of the quasiclassical wave functions in the total scattering potential and in the Coulomb potential: where R 0 and R C are the classical turning points for the two motions, respectively. In our calculations we applied both the variable phase method and the quasiclassical approximation and found that the results are very close to each other for the energy range under consideration (100 eV -1 keV).
The total potential V (r) experienced by the antiproton due to electromagnetic interaction is represented by a sum of the potential of a finite nucleus and the VP potential: The potential of a finite nucleus is given by Here ρ n is the nuclear charge density, normalized to the nuclear charge number Z. We employ the Fermi-like nuclear charge distribution where the parameter a is equal to 2.3/(4 ln 3) fm, the parameters r 0 and N 0 are derived from the root-mean-square nuclear charge radius and the normalization condition [9].
The VP potential is conveniently represented as a sum of the Uehling and the Wichmann-Kroll (WK) potential: The Uehling potential is given by the lowest-order term in the expansion of the one-electronloop vacuum polarization in powers of the Coulomb electron-nucleus interaction. According to the Furry theorem, this term contains one Coulomb interaction in the vacuum loop and is ultraviolet divergent. It becomes finite after charge renormalization. The renormalized expression for the Uehling potential is given by (see, e.g., [9,10]) where c is the speed of light in vacuum.
The Wichmann-Kroll potential V WK (r) accounts for the higher-order terms in the expansion of the vacuum loop in powers of the Coulomb electron-nucleus interaction [11]. Although the WK potential is finite, the regularization is still required in the lowest-order non-zero term due to a spurious gauge-dependent piece of the light-by-light scattering contribution. The spurious term disappears if the WK potential is calculated by summing up the partial-wave differences between the full one-loop contribution and the unrenormalized Uehling term [12,13,14,15]. The calculation formula for the WK potential can be written as [10,16]: Here κ is the relativistic angular momentum quantum number, G ik κ (iω, y, z) and F ik κ (iω, y, z) are the radial Dirac components of the partial-wave contributions to the bound and free electron Green's functions, respectively. We have calculated DCS of elastic scattering of antiprotons by bare uranium nuclei in the energy range from 100 eV to 1 keV. To make the results at different energies comparable, the differential cross section has been scaled according to