aa r X i v : . [ nu c l - t h ] N ov Antibaryon–nucleus bound states
J. Hrt´ankov´a , , J. Mareˇs Nuclear Physics Institute, 25068 ˇReˇz, Czech Republic Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering,Bˇrehov´a 7, 115 19 Prague 1, Czech RepublicE-mail: [email protected]
Abstract.
We calculated antibaryon ( ¯ B = ¯ p , ¯Λ, ¯Σ, ¯Ξ) bound states in selected nucleiwithin the relativistic mean-field (RMF) model. The G-parity motivated ¯ B –meson couplingconstants were scaled to yield corresponding potentials consistent with available experimentaldata. Large polarization of the nuclear core caused by ¯ B was confirmed. The ¯ p annihilationin the nuclear medium was incorporated by including a phenomenological imaginary part ofthe optical potential. The calculations using a complex ¯ p –nucleus potential were performedfully self-consistently. The ¯ p widths significantly decrease when the phase space reduction isconsidered for ¯ p annihilation products, but they still remain sizeable for potentials consistentwith ¯ p –atom data.
1. Introduction
The study of antibaryon–nucleus interactions has attracted increasing interest in recent yearsat the prospect of future experiments at the FAIR facility [1]. In particular, much attention hasbeen devoted to the antiproton–nucleus interaction and the possibility of formation of ¯ p –nucleusbound states [2, 3, 4]. Exploring the ¯ p –nucleus interaction could provide valuable informationabout the behavior of the antiproton in the nuclear medium as well as nuclear dynamics. One ofthe motivations for our study of ¯ p –nucleus bound states is the conjecture that the considerablesuppression of the phase space for the ¯ p annihilation products in the nuclear medium could leadto relatively long living ¯ p inside the nucleus [2].In this contribution, we report on our recent fully self-consistent calculations of antibaryon–nucleus bound states within the relativistic mean-field model [5]. The behavior of an antibaryonin the nuclear medium and the dynamical effects caused by the presence of the antibaryon in thenucleus were studied for several selected nuclei. Special attention was devoted to the ¯ p –nucleusinteraction including ¯ p absorption in the nucleus.In Section 2, a brief description of the underlying model is given. Few representative resultsof our calculations are presented and discussed in Section 3.
2. Model
In the present work, antibaryon–nucleus bound states are studied within the framework of theRMF approach applied to a system of nucleons and one antibaryon ( ¯ B = ¯ p , ¯Λ, ¯Σ, ¯Ξ). Theinteraction among (anti)baryons is mediated by the exchange of the scalar ( σ ) and vector ( ω µ , ~ρ µ ) meson fields, and the massless photon field A µ . The standard Lagrangian density L N fornucleonic sector is extended by the Lagrangian density L ¯ B describing the antibaryon interactionith the nuclear medium (see ref. [6] for details). The variational principle yields the equationsof motion for the hadron fields involved. The Dirac equations for nucleons and antibaryon read:[ − i~α ~ ∇ + β ( m j + S j ) + V j ] ψ αj = ǫ αj ψ αj , j = N, ¯ B , (1)where S j = g σj σ, V j = g ωj ω + g ρj ρ τ + e j τ A (2)are the scalar and vector potentials. Here, α denotes single particle states, m j standsfor (anti)baryon masses and g σj , g ωj , g ρj , and e j are (anti)baryon coupling constants tocorresponding fields. The presence of ¯ B induces additional source terms in the Klein–Gordonequations for the meson fields:( −△ + m σ + g σ + g σ ) σ = − g σN ρ SN − g σ ¯ B ρ S ¯ B ( −△ + m ω + dω ) ω = g ωN ρ V N + g ω ¯ B ρ V ¯ B ( −△ + m ρ ) ρ = g ρN ρ IN + g ρ ¯ B ρ I ¯ B −△ A = e N ρ QN + e ¯ B ρ Q ¯ B , (3)where ρ S j , ρ V j , ρ I j and ρ Q j are the scalar, vector, isovector and charge densities, respectively, and m σ , m ω , m ρ are the masses of considered mesons. The system of coupled Dirac (1) and Klein–Gordon (3) equations represents a self-consistent problem which is to be solved by iterativeprocedure.The values of the nucleon–meson coupling constants and meson masses were adopted from thenonlinear RMF model TM1(2) [7] for heavy (light) nuclei. We used also the density–dependentmodel TW99 [8, 9] in which the couplings are a function of the baryon density. The hyperon–meson coupling constants for the ω and ρ fields are obtained using SU(6) symmetry relations.The coupling constants for the σ field are constrained by available experimental data — Λhypernuclei [10], Σ atoms [11], and Ξ production in ( K + , K − ) reaction [12].In the RMF approach, the nuclear ground state is well described by the attractive scalarpotential S ≃ −
350 MeV and by the repulsive vector potential V ≃
300 MeV. The resultingcentral potential acting on a nucleon in a nucleus is then approximately S + V ≃ −
50 MeVdeep. The ¯ B –nucleus interaction is constructed from the B –nucleus interaction with the helpof the G-parity transformation: the vector potential generated by the ω meson exchange thuschanges its sign and becomes attractive. As a consequence, the total potential acting on a ¯ B will be strongly attractive. In particular, the ¯ p –nuclear potential would be S − V ≃ −
650 MeVdeep. However, the G-parity transformation should be regarded as a mere starting point todetermine the ¯ B –meson coupling constants. Various many-body effects, as well as the presenceof strong annihilation channels could cause significant deviations from the G-parity values inthe nuclear medium. Indeed, the available experimental data from ¯ p atoms [13] and ¯ p scatteringoff nuclei [14] suggest that the depth of the real part of the ¯ p –nucleus potential is in the range − (100 − ξ ∈ h , i for the antibaryon–meson coupling constants which are in the following relation to the baryon–meson couplings: g σ ¯ B = ξ g σN , g ω ¯ B = − ξ g ωN , g ρ ¯ B = ξ g ρN . (4)The annihilation of an antibaryon inside the nuclear medium is an inseparable part of anyrealistic description of the ¯ B –nucleus interaction. In our calculations, only the ¯ p absorption inthe nucleus has been considered so far. Since the RMF model does not address directly the .8 1 1.2 1.4 1.6 1.8s (GeV)0.20.40.60.81.0 f s (-) ππωπρ ηωρω ω π π π π π πη πρ πω Figure 1.
The phase space suppression factors f s as a function of the c.m. energy √ s .absorption of the ¯ p in the nucleus we adopted the imaginary part of the optical potential in a‘ tρ ’ form from optical model phenomenology [13]:2 µ Im V opt ( r ) = − π (cid:18) µm N A − A (cid:19) Im b ρ ( r ) , (5)where µ is the ¯ p –nucleus reduced mass. While the density ρ was treated as a dynamical quantitydetermined within the RMF model, the parameter b was constrained by fits to ¯ p -atomic data[13]. The global fits to the ¯ p -atomic data give a single value for the imaginary part of b ,Im b = 1 . p annihilation in the nuclear medium is usually expressed as √ s = m ¯ p + m N − B ¯ p − B N , where B ¯ p and B N is the ¯ p and nucleon binding energy, respectively.Therefore, the phase space available for the annihilation products is considerably suppressed forthe deeply bound antiproton.The phase space suppression factors ( f s ) for two body decay are given by [15] f s = M s s [ s − ( m + m ) ][ s − ( m − m ) ][ M − ( m + m )][ M − ( m − m ) ] Θ( √ s − m − m ) , (6)where m , m are the masses of the annihilation products and M = m ¯ p + m N .For channels containing more than 2 particles in the final state the suppression factors f s were evaluated with the help of Monte Carlo simulation tool PLUTO [16]. In Figure 1,we present the phase space suppression factors as a function of the center-of-mass energy √ s for considered annihilation channels. As the energy √ s decreases many channels becomesignificantly suppressed or even closed which could lead to much longer lifetime of ¯ p in a nucleus.
3. Results
We applied the formalism introduced in the previous section to self-consistent calculations of¯ p , ¯Λ, ¯Σ, ¯Ξ bound states in selected nuclei across the periodic table. First, we did not considerabsorption of an antibaryon in a nucleus. Our calculations within the TM model confirmedsubstantial polarization of the nuclear core caused by the antibaryon embedded in the nucleus [2]. S + V ( M e V ) O p16 O Λ O Σ O Ξ O p16 O Λ O Σ O Ξ (a) (b) Figure 2.
The B –nucleus (a) and ¯ B –nucleus(b) potentials in O, calculated dynamically for ξ = 0 . -40-35-30-25-20-15-10-50 E ( M e V ) Ξ + Ξ − Li C O Ca Zr Pb Figure 3.
Single particle energies of Ξ − and¯Ξ + for ξ = 0 . B and the totalbinding energies increase considerably, as well. The nucleon densities in ¯ p nuclei reach 2 − p nuclei within the density–dependent model TW99,as well. We obtained very similar results as for the TM model and thus confirmed only smallmodel dependence of our calculations.Figure 2 shows the total potential acting on an extra baryon (a) and on an extra antibaryon(b) in 1 s state in O, calculated dynamically in the TM2 model. The scaling parameter is chosento be ξ = 0 . p potential comparable with the available experimentaldata. We assume the same scaling parameter also for antihyperons, since there is no reliableexperimental information on the in-medium antihyperon potentials. The potentials acting onantibaryons are fairly deep in the central region of the nucleus in contrast to the baryon potentials(notice that the potential for Σ is even repulsive while the potential for ¯Σ is strongly attractive).Such strongly attractive potentials yield deeply bound states of antibaryons in atomic nuclei.Figure 3 presents a comparison between the Ξ − and ¯Ξ + s single particle energies in variousnuclei across the periodic table, calculated dynamically in the TM model. The ¯Ξ + couplingconstants are scaled by factor ξ = 0 .
2. The binding energy of Ξ − is increasing with the numberof nucleons in the nucleus. The ¯Ξ + binding energy follows the opposite trend and in Pb it iseven less bound than Ξ − . This can be explained by enhanced Coulomb repulsion felt by ¯Ξ + inheavier nuclei.We performed calculations of ¯ p nuclei using a complex potential describing the ¯ p annihilationin the nuclear medium. The results of static as well as dynamical calculations with the realpotential, complex potential, and complex potential with the suppression factors f s for ¯ p boundin O are presented in Table 1. The scaling of the ¯ p –meson coupling constants is chosen tobe ξ = 0 .
2. The static calculations, which do not account for the core polarization effects, giveapproximately the same values of the ¯ p single particle energy for all three cases. The singleparticle energies calculated dynamically are larger, which indicates that the polarization of the able 1. The 1 s single particle energies E ¯ p and widths Γ ¯ p (in MeV) in O ¯ p , calculateddynamically (Dyn) and statically (Stat) with the real, complex and complex with f s potentials(TM2 model), consistent with ¯ p –atom data.Real Complex Complex + f s Dyn Stat Dyn Stat Dyn Stat E ¯ p ¯ p - - 552.3 293.3 232.5 165.0core nucleus is significant (even if the ¯ p absorption is taken into account). When the effect ofthe phase space suppression is considered the ¯ p annihilation width is substantially suppressed(compare 552 . . p width is stillconsiderable for relevant ¯ p potentials consistent with the ¯ p data.The annihilation of the ¯ p with a nucleon takes place in a nucleus. Therefore, the momentumdependent term in Mandelstam variable s = ( E N + E ¯ p ) − ( ~p N + ~p ¯ p ) is non-negligible in contrastto two body frame [17]. Our self-consistent evaluation of √ s by considering the momenta ofannihilating partners leads to an additional downward energy shift. As a consequence, the ¯ p width in O ¯ p is reduced by additional ≈
50 MeV. We conclude that even after taking intoaccount the phase space suppression corresponding to self-consistent treatment of √ s includingthe ¯ p and N momenta, the ¯ p annihilation widths in nuclei remain sizeable. The correspondinglifetime of the ¯ p in the nuclear medium is ≃ Acknowledgements
This work was supported by GACR Grant No. P203/12/2126. We thank Pavel Tlust´y for hisassistance during Monte Carlo simulations using PLUTO. J. Hrt´ankov´a acknowledges financialsupport from CTU-SGS Grant No. SGS13/216/OHK4/3T/14.
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