The pD atoms are studied in various realistic, popular NN 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 NN scattering data quite well, is able to reproduce the experimental data of the energy shifts of the 2p pD atomic states. The energy shifts of the 2p pD atomic states are very sensitive to the strong interactions, hence the investigation of the pD atoms is expected to provide a good platform for refining the NN interaction, especially at zero energy

Amand, Faessler

C., Kobdaj

K., Khosonthongkee

P., Suebka

Th., Gutsche

V.E., Lyubovitskij

Y., Yan

Suranaree University of Technology Intellectual Repository

Physics Letters B 659 (2008) 555–558www.elsevier.com/locate/physletbp̄D atoms in realistic potentialsY. Yan a,∗, K. Khosonthongkee a, C. Kobdaj a, P. Suebka a, Th. Gutsche b, Amand Faessler b,V.E. Lyubovitskij ba School of Physics, Suranaree University of Technology, 111 University Avenue, Nakhon Ratchasima 30000, Thailandb Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle 4, D-72076 Tübingen, GermanyReceived 2 November 2007; received in revised form 22 November 2007; accepted 29 November 2007Available online 5 December 2007Editor: J.-P. BlaizotAbstractThe p̄D atoms are studied in various realistic, popular N̄N potentials. The small energy shifts and decay widths of the atoms, which stemfrom the short-ranged strong interactions between the antiproton and deuteron, are evaluated in a well-established, accurate approach based on theSturmian functions. The investigation reveals that none of the employed potentials, which reproduce the N̄N scattering data quite well, is able toreproduce the experimental data of the energy shifts of the 2p p̄D atomic states. The energy shifts of the 2p p̄D atomic states are very sensitiveto 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.GeKeywords: p̄D atom; N̄N interaction; Sturmian function; Accurate numerical approach1. IntroductionThe second simplest antiprotonic atom is the antiprotonicdeuteron atom p̄D, consisting of an antiproton and a deuteronbound mainly by the Coulomb interaction but distorted by theshort range strong interaction. The study of the p̄D atom ismuch later and less successful than for other exotic atoms likethe protonium and pionium. Experiments were carried out atLEAR just in very recent years to study the properties of thep̄D atom [1,2]. Even prior to the experiments some theoreticalworks [3–5] had been carried out to study the p̄D atomic statesin simplified p̄D interactions. Recently, a theoretical work [6]proposed a mechanism explaining the unexpected behavior, ofthe scattering lengths of N̄N and p̄D system, that the imagi-nary part of the scattering length does not increase with the sizeof the nucleus.* Corresponding author.E-mail address: yupeng@sut.ac.th (Y. Yan).0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.11.085In the theoretical sector, one needs to overcome at least twodifficulties in the study of the p̄D atom. First, the interactionbetween the antiproton and the deuteron core should be derivedfrom realistic N̄N interactions, for example, the Paris N̄N po-tentials [7–9], the Dover–Richard N̄N potentials I (DR1) and II(DR2) [10,11], and the Kohno–Weise N̄N potential [12]. Evenif a reliable p̄D interaction is in hands, the accurate evalua-tion of the energy shifts and decay widths (stemming for thestrong p̄D interactions) and especially of the nuclear force dis-torted wave function of the atom is still a challenge. It shouldbe pointed out that the methods employed in the works [3–5]are not accurate enough for evaluating the wave functions ofthe p̄D atoms.In the present work we study the p̄D atom problem employ-ing a properly adapted numerical method based on Sturmianfunctions [13]. The method accounts for both the strong shortrange nuclear potential (local and non-local) and the long rangeCoulomb force and provides directly the wave function of thep̄D system with complex eigenvalues E = ER − i Γ2 . The pro-tonium and pionium problems have been successfully investi-gated [14,15] in the numerical approach. The numerical method556 Y. Yan et al. / Physics Letters B 659 (2008) 555–558is much more powerful, accurate and much easier to use thanall other methods applied to the exotic atom problem in his-tory. The p̄D interactions in the work are derived from variousrealistic N̄N potential, which is state-dependent. The work isorganized as follows. The p̄D interactions are expressed in Sec-tion 2 in terms of the N̄N interactions. In Section 3 the energyshifts and decay widths of the 1s and 2p p̄D atomic states areevaluated. Discussions and conclusions are given in Section 3,too.2. p̄D interactions in terms of N̄N potentialsWe start from the Schrödinger equation of the antiproton–deuteron system in coordinate space(P 2ρ2Mρ+ P2λ2Mλ+ V12(r2 − r1) + V13(r3 − r1)(1)+ V23(r3 − r2))Ψ (λ, ρ) = EΨ (λ, ρ)where λ and ρ are the Jacobi coordinates of the system, definedas(2)λ = r3 − r1 + r22, ρ = r2 − r1,Mρ = M/2 and Mλ = 2M/3 are the reduced masses. Here wehave assigned, for simplicity, the proton and neutron the samemass M . Eq. (1) can be expressed in the form, where the stronginteraction is expressed in the isospin basis,(3)(P 2ρ2Mρ+ P2λ2Mλ+ VS + VC)Ψ (λ, ρ) = EΨ (λ, ρ)where VS and VC stand for the nuclear interaction and Coulombforce, respectively, and take the forms(4)VS = V 0NN(r2 − r1) +14[V 0N̄N(r3 − r1) + V 0N̄N (r3 − r2)]+ 34[V 1N̄N(r3 − r1) + V 1N̄N (r3 − r2)],(5)VC = 12[VC(r3 − r1) + VC(r3 − r2)]V 0 and V 1 in Eq. (4) are the isospin 0 and 1 nuclear inter-actions, respectively. Note that we have assigned r12 as therelative coordinate of the deuteron core.One may express the interactions VC and VS in Eqs. (4) and(5) in terms of the interactions of certain N̄N states. In the|JMLS〉 basis of the p̄D states(6)|JMLS〉 = ∣∣(Lρ ⊗ Lλ)L ⊗ (S12 ⊗ S3)S 〉JMwe derive(H0 + WC(λ,ρ) + V 0NN(ρ) + WS(λ,ρ))Ψ (λ,ρ)(7)= EΨ (λ,ρ)with(8)H0 =P 2ρ2Mρ+ P2λ2MλWC and WS in Eq. (7) are respectively the Coulomb force andstrong interaction between the antiproton and deuteron, andV 0NN the interaction between the proton and neutron in thedeuteron core. WC and WS are derived explicitly as(9)WC(λ,ρ) = 121∫−1dx VC(r13),WS(λ,ρ)(10)= 121∫−1dx∑Q,Q′〈P |Q〉〈Q|VN̄N(r13)|Q′〉〈Q′|P ′〉with(11)VN̄N(r13) =12V 0N̄N(r13) + 32V 1N̄N(r13),(12)r13 ≡ |r1 − r3| =(λ2 + ρ24− λρx)1/2where x = cos θ with θ being the angle between λ and ρ. InEq. (10) |P 〉 ≡ |JMLS〉 and |P ′〉 ≡ |JML′S〉 are as defined inEq. (6) while the states |Q〉 and |Q′〉 are(13)|Q〉 = ∣∣(Lσ ⊗ S13)Jσ ⊗ (Lγ ⊗ S2)Jγ 〉JM,(14)|Q′〉 = ∣∣(L′σ ⊗ S13)Jσ ⊗ (Lγ ⊗ S2)Jγ 〉JM.Here σ and γ are also the Jacobi coordinates of the system,defined as(15)γ = r2 − r1 + r32, σ = r3 − r1.So defined the states |Q〉 and |Q′〉 is based on the consider-ation that the N̄N interactions can be easily expressed in the|Jσ Mσ Lσ S13〉 basis of the N̄N states. Note that 〈P |Q〉 dependson not only the quantum numbers of the states |P 〉 and |Q〉,but also λ, ρ and the angle θ between λ and ρ resulting fromthe projection of the orbital angular momenta between differentJacobi coordinates. We listed the integral kernels in Eq. (10),∑Q,Q′ 〈P |Q〉〈Q|V (r13)|Q′〉〈Q′|P ′〉, for the lowest p̄D statesin the approximation that the deuteron core is assumed in theS-state, as follows:|P 〉 = |P ′〉 = ∣∣2S1/2〉: 34VN̄N(1S0) + 14VN̄N(3S1),|P 〉 = |P ′〉 = ∣∣4S3/2〉: VN̄N (3S1),|P 〉 = |P ′〉 = ∣∣2P1/2〉:F 21 ·[112VN̄N(3P0) + 34VN̄N(1P1) + 16VN̄N(3P1)],|P 〉 = |P ′〉 = ∣∣4P1/2〉: F 21 ·[23VN̄N(3P0) + 13VN̄N(3P1)],|P 〉 = |P ′〉 = ∣∣2P3/2〉:F 21 ·[34VN̄N(1P1) + 124VN̄N(3P1) + 524VN̄N(3P2)],|P 〉 = |P ′〉 = ∣∣4P3/2〉: F 21 ·[5VN̄N(3P1) + 1VN̄N (3P2)],6 6Y. Yan et al. / Physics Letters B 659 (2008) 555–558 557Table 1The 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 ofthe energy shifts means that the strong interaction is repulsive. The units are eV and meV for 1s and 2p states, respectivelyParis98 DR2 KW Data	E Γ 	E Γ 	E Γ 	E Γ2S1/2 −2445 1781 −2673 2380 −2478 24504SD3/2 −2680 2822 −2668 2390 −2503 24692P1/2 −186 584 17 896 99 6574P1/2 265 402 47 846 101 7852P3/2 −128 515 14 897 98 6434PF3/2 282 477 21 887 97 6484PF5/2 244 814 21 877 101 660	Ē1s , Γ̄1s −2602 2475 −2670 2387 −2494 2463 −1050±250 [1] 1100±750 [1]2270±260 [2]	Ē2p , Γ̄2p 124 602 22 883 99 668 −243±26 [2] 489±30 [2]|P 〉 = |P ′〉 = ∣∣4P5/2〉: F 21 · VN̄N (3P2),|P 〉 = |P ′〉 = ∣∣4D3/2〉: F 23 ·[12VN̄N(3D1) + 12VN̄N(3D2)],|P 〉 = |P ′〉 = ∣∣2F3/2〉: F 22 · VN̄N (3F2),|P 〉 = |P ′〉 = ∣∣4F5/2〉: F 22 ·[49VN̄N(3F2) + 59VN̄N(3F3)],|P 〉 = ∣∣4P3/2〉, |P ′〉 = ∣∣4F3/2〉: F1F2 · 1√6VN̄N(3PF2),|P 〉 = ∣∣4P5/2〉, |P ′〉 = ∣∣4F5/2〉: F1F2 · 23VN̄N(3PF2),|P 〉 = ∣∣4S3/2〉, |P ′〉 = ∣∣4D3/2〉:(16)F3 ·[1√2VN̄N(3SD1) + 1√2VN̄N(3SD2)]where |P 〉 ≡ |JMLS〉 and |P ′〉 ≡ |JML′S〉 are the p̄D atomicstates. Both the p̄D and N̄N states in Eq. (16) are labelled as2S+1LJ with S, L and J being respectively the total spin, totalorbital angular momentum and total angular momentum. Thepotentials VN̄N , being functions of r13 =√λ2 + ρ2/4 − ρλx,stand for the N̄N interactions for various N̄N states as indi-cated in the brackets.The F1, F2 and F3 in Eq. (16) are functions of only λ and ρ,taking the forms(17)F1 ={1 − 112 ρ2λ2, ρ < 2λ,4λ3ρ , ρ > 2λ,(18)F2 ={(1 − ρ24λ2)2, ρ < 2λ,0, ρ > 2λ,(19)F3 =⎧⎨⎩ 2F 1(1,− 32 , 32 , ρ24λ2), ρ < 2λ,58 − 3ρ232λ2+ Artanh( 2λρ)[ 3λ4ρ − 3ρ8λ + 3ρ364λ3], ρ > 2λ,where 2F 1(α,β, γ, x) is the hypergeometric function andArtanh(x) the inverses hyperbolic tangent function.3. Energy shifts and decay widths of p̄D atomsIt is not a simple problem to accurately evaluate the energyshifts and decay widths, especially wave functions of exoticatoms like protonium, pionium and antiproton–deuteron atoms,which are mainly bound by the Coulomb force, but also effectedby the short range strong interaction. In this work we study thep̄D atoms in the Sturmian function approach which has beensuccessfully applied to our previous works [14,15]. Employedfor the N̄N interactions are various realistic N̄N potentials,namely, the Paris N̄N potentials of the 1994 version (Paris84),1998 version (Paris98) and 2004 version (Paris04), the Dover–Richard N̄N potentials I (DR1) and II (DR2), and the Kohno–Weise N̄N potential (KW). In this preliminary work, we justlimit our study to the approximation of undistorted deuteroncore. However, one may see that the main conclusions of thework are free of this approximation.Shown in Table 1 are the energy shifts and decay widths,which stem from the Paris98, DR2 and KW N̄N interactions, inthe approximation of undistorted deuteron core. The theoreticalresults for other interactions like Paris84, Paris04 and DR1 arequite similar to the ones listed in Table 1. The wave function ofthe undistorted deuteron core is evaluated in the Bonn OBEPQpotential [16]. It is found that the theoretical results for the 1sp̄D atomic states are more or less the same by all the employedN̄N potentials. The predicted energy shifts are roughly as twicelarge as the experimental data. However, one may expect thatthe predictions of the potentials in question could be improvedto some extent by solving the p̄D dynamical equation in Eq. (7)without any approximation. A better treatment of the deuteroncore will yield lower 1s p̄D atomic states, hence smaller energyshifts. The theoretical results for the decay widths of the 1s p̄Datoms are also larger than the experimental data though not asfar from the data as for the energy shifts. The predictions forthe decay widths are also expected to be improved by treatingthe deuteron core more properly.The theoretical predictions for the energy shifts of the 2pp̄D atomic states are totally out of line for all the N̄N poten-tials employed. The experimental data show that the averagedenergy level of the 2p p̄D atoms is pushed up by the strong558 Y. Yan et al. / Physics Letters B 659 (2008) 555–558interaction, the same as for the 1s p̄D atoms, but the theoret-ical results uniquely show the averaged energy level shiftingdown. It is unlikely to improve, by treating the deuteron coremore accurately, the theoretical predictions of the N̄N poten-tials in question for the 2p p̄D energy shifts since a moreaccurate treatment of the deuteron core will lead to deeper 2pp̄D atomic states.All the N̄N potentials employed in the work reproduce N̄Nscattering data reasonably, but badly fail to reproduce the en-ergy shifts of the 2p p̄D atoms. The investigation of the p̄Datoms may provide a good platform for refining the N̄N inter-action, especially at zero energy since the energy shifts of the2p p̄D atomic states are very sensitive to the N̄N strong inter-actions.The research here is just a preliminary work, where a frozen,S-state deuteron is employed. The work may be improved attwo steps, considering that the numerical evaluation is time-consuming. One may, at the first step, solve the p̄D dynamicalequation in Eq. (7) by expanding the p̄D wave function in a bi-wave basis of the Sturmian functions, where a realistic nucleon–nucleon potential is employed but the deuteron core is assumedto be at the S-state. Such an evaluation is still manageable at apersonnel computer but it may take a week or longer. We maycompare the results of the improved work with the results hereto figure out how important an unfrozen deuteron core is.One may also consider, at the second step, to solve the p̄Ddynamical equation in Eq. (7) by expanding the p̄D wave func-tion in a bi-wave basis of the Sturmian functions without anyapproximation, where realistic nucleon–nucleon and nucleon–antinucleon potentials are employed and the deuteron core isallowed to be at both the S- and D-waves. It is certain that thenumerical calculation will take longer time but, anyway, we willdo it after we complete the first-step improvement.AcknowledgementsThis work is supported in part by the National ResearchCouncil of Thailand through Suranaree University of Technol-ogy and the Commission on Higher Education, Thailand (CHE-RES-RG Theoretical Physics).References[1] M. Augsburger, et al., Phys. Lett. B 461 (1999) 417.[2] D. Gotta, et al., Nucl. Phys. A 660 (1999) 283.[3] S. Wycech, A.M. Green, J.A. Niskanen, Phys. Lett. B 152 (1985) 308.[4] G.P. Latta, P.C. Tandy, Phys. Rev. C 42 (1990) R1207.[5] G.Q. Liu, J.-M. Richard, S. Wycech, Phys. Lett. B 2602 (1991) 15.[6] V.A. Karmanov, K.V. Protasov, A.Yu. Voronin, Eur. Phys. J. 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pD atoms in realistic potentials

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pD atoms in realistic potentials

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