p̄D atoms in realistic potentials

The p̄D atoms are studied in various realistic, popular N̄N potentials. The small energy shifts and decay widths of the atoms, which stem from the short-ranged strong interactions between the antiproton and deuteron, are evaluated in a well-established, accurate approach based on the Sturmian functions. The investigation reveals that none of the employed potentials, which reproduce the N̄N scattering data quite well, is able to reproduce the experimental data of the energy shifts of the 2p p̄D atomic states. The energy shifts of the 2p p̄D atomic states are very sensitive to the N̄N strong interactions, hence the investigation of the p̄D atoms is expected to provide a good platform for refining the N̄N interaction, especially at zero energy. © 2007 Elsevier B.V. All rights reserved. PACS: 36.10.Gv; 13.75.Cs; 03.65.Ge


Introduction
The second simplest antiprotonic atom is the antiprotonic deuteron atompD, consisting of an antiproton and a deuteron bound mainly by the Coulomb interaction but distorted by the short range strong interaction. The study of thepD atom is much later and less successful than for other exotic atoms like the protonium and pionium. Experiments were carried out at LEAR just in very recent years to study the properties of thē pD atom [1,2]. Even prior to the experiments some theoretical works [3][4][5] had been carried out to study thepD atomic states in simplifiedpD interactions. Recently, a theoretical work [6] proposed a mechanism explaining the unexpected behavior, of the scattering lengths ofNN andpD system, that the imaginary part of the scattering length does not increase with the size of the nucleus.
In the theoretical sector, one needs to overcome at least two difficulties in the study of thepD atom. First, the interaction between the antiproton and the deuteron core should be derived from realisticNN interactions, for example, the ParisNN potentials [7][8][9], the Dover-RichardNN potentials I (DR1) and II (DR2) [10,11], and the Kohno-WeiseNN potential [12]. Even if a reliablepD interaction is in hands, the accurate evaluation of the energy shifts and decay widths (stemming for the strongpD interactions) and especially of the nuclear force distorted wave function of the atom is still a challenge. It should be pointed out that the methods employed in the works [3][4][5] are not accurate enough for evaluating the wave functions of thepD atoms.
In the present work we study thepD atom problem employing a properly adapted numerical method based on Sturmian functions [13]. The method accounts for both the strong short range nuclear potential (local and non-local) and the long range Coulomb force and provides directly the wave function of thē pD system with complex eigenvalues E = E R − i Γ 2 . The protonium and pionium problems have been successfully investigated [14,15] in the numerical approach. The numerical method is much more powerful, accurate and much easier to use than all other methods applied to the exotic atom problem in history. ThepD interactions in the work are derived from various realisticNN potential, which is state-dependent. The work is organized as follows. ThepD interactions are expressed in Section 2 in terms of theNN interactions. In Section 3 the energy shifts and decay widths of the 1s and 2ppD atomic states are evaluated. Discussions and conclusions are given in Section 3, too.

2.pD interactions in terms ofNN potentials
We start from the Schrödinger equation of the antiprotondeuteron system in coordinate space where λ and ρ are the Jacobi coordinates of the system, defined as M ρ = M/2 and M λ = 2M/3 are the reduced masses. Here we have assigned, for simplicity, the proton and neutron the same mass M. Eq. (1) can be expressed in the form, where the strong interaction is expressed in the isospin basis, where V S and V C stand for the nuclear interaction and Coulomb force, respectively, and take the forms V 0 and V 1 in Eq. (4) are the isospin 0 and 1 nuclear interactions, respectively. Note that we have assigned r 12 as the relative coordinate of the deuteron core. One may express the interactions V C and V S in Eqs. (4) and (5) in terms of the interactions of certainNN states. In the |J MLS basis of thepD states (7) are respectively the Coulomb force and strong interaction between the antiproton and deuteron, and V 0 NN the interaction between the proton and neutron in the deuteron core. W C and W S are derived explicitly as where x = cos θ with θ being the angle between λ and ρ. In Eq. (10) |P ≡ |J MLS and |P ≡ |J ML S are as defined in Eq. (6) while the states |Q and |Q are Here σ and γ are also the Jacobi coordinates of the system, defined as So defined the states |Q and |Q is based on the consideration that theNN interactions can be easily expressed in the |J σ M σ L σ S 13 basis of theNN states. Note that P |Q depends on not only the quantum numbers of the states |P and |Q , but also λ, ρ and the angle θ between λ and ρ resulting from the projection of the orbital angular momenta between different Jacobi coordinates. We listed the integral kernels in Eq. (10), Q,Q P |Q Q|V ( r 13 )|Q Q |P , for the lowestpD states in the approximation that the deuteron core is assumed in the S-state, as follows: Table 1 The energy shifts E and decay widths of the 1s and 2p antiproton-deuteron atomic states in the approximation of undistorted deuteron core. The minus sign of the energy shifts means that the strong interaction is repulsive.
where |P ≡ |J MLS and |P ≡ |J ML S are thepD atomic states. Both thepD andNN states in Eq. (16) are labelled as 2S+1 L J with S, L and J being respectively the total spin, total orbital angular momentum and total angular momentum. The potentials VN N , being functions of r 13 = λ 2 + ρ 2 /4 − ρλx, stand for theNN interactions for variousNN states as indicated in the brackets. The F 1 , F 2 and F 3 in Eq. (16) are functions of only λ and ρ, taking the forms α, β, γ, x) is the hypergeometric function and Artanh(x) the inverses hyperbolic tangent function.

Energy shifts and decay widths ofpD atoms
It is not a simple problem to accurately evaluate the energy shifts and decay widths, especially wave functions of exotic atoms like protonium, pionium and antiproton-deuteron atoms, which are mainly bound by the Coulomb force, but also effected by the short range strong interaction. In this work we study thē pD atoms in the Sturmian function approach which has been successfully applied to our previous works [14,15]. Employed for theNN interactions are various realisticNN potentials, namely, the ParisNN potentials of the 1994 version (Paris84), 1998 version (Paris98) and 2004 version (Paris04), the Dover-RichardNN potentials I (DR1) and II (DR2), and the Kohno-WeiseNN potential (KW). In this preliminary work, we just limit our study to the approximation of undistorted deuteron core. However, one may see that the main conclusions of the work are free of this approximation.
Shown in Table 1 are the energy shifts and decay widths, which stem from the Paris98, DR2 and KWNN interactions, in the approximation of undistorted deuteron core. The theoretical results for other interactions like Paris84, Paris04 and DR1 are quite similar to the ones listed in Table 1. The wave function of the undistorted deuteron core is evaluated in the Bonn OBEPQ potential [16]. It is found that the theoretical results for the 1s pD atomic states are more or less the same by all the employed NN potentials. The predicted energy shifts are roughly as twice large as the experimental data. However, one may expect that the predictions of the potentials in question could be improved to some extent by solving thepD dynamical equation in Eq. (7) without any approximation. A better treatment of the deuteron core will yield lower 1spD atomic states, hence smaller energy shifts. The theoretical results for the decay widths of the 1spD atoms are also larger than the experimental data though not as far from the data as for the energy shifts. The predictions for the decay widths are also expected to be improved by treating the deuteron core more properly.
The theoretical predictions for the energy shifts of the 2p pD atomic states are totally out of line for all theNN potentials employed. The experimental data show that the averaged energy level of the 2ppD atoms is pushed up by the strong interaction, the same as for the 1spD atoms, but the theoretical results uniquely show the averaged energy level shifting down. It is unlikely to improve, by treating the deuteron core more accurately, the theoretical predictions of theNN potentials in question for the 2ppD energy shifts since a more accurate treatment of the deuteron core will lead to deeper 2p pD atomic states.
All theNN potentials employed in the work reproduceNN scattering data reasonably, but badly fail to reproduce the energy shifts of the 2ppD atoms. The investigation of thepD atoms may provide a good platform for refining theNN interaction, especially at zero energy since the energy shifts of the 2ppD atomic states are very sensitive to theNN strong interactions.
The research here is just a preliminary work, where a frozen, S-state deuteron is employed. The work may be improved at two steps, considering that the numerical evaluation is timeconsuming. One may, at the first step, solve thepD dynamical equation in Eq. (7) by expanding thepD wave function in a biwave basis of the Sturmian functions, where a realistic nucleonnucleon potential is employed but the deuteron core is assumed to be at the S-state. Such an evaluation is still manageable at a personnel computer but it may take a week or longer. We may compare the results of the improved work with the results here to figure out how important an unfrozen deuteron core is.
One may also consider, at the second step, to solve thepD dynamical equation in Eq. (7) by expanding thepD wave function in a bi-wave basis of the Sturmian functions without any approximation, where realistic nucleon-nucleon and nucleonantinucleon potentials are employed and the deuteron core is allowed to be at both the Sand D-waves. It is certain that the numerical calculation will take longer time but, anyway, we will do it after we complete the first-step improvement.