Dynamical simulation of bound antiproton-nuclear systems and observable signals of cold nuclear compression
aa r X i v : . [ nu c l - t h ] J un Dynamical simulation of bound antiproton-nuclear systems andobservable signals of cold nuclear compression
A.B. Larionov , , I.N. Mishustin , , L.M. Satarov , , and W. Greiner Frankfurt Institute for Advanced Studies,J.W. Goethe-Universit¨at, D-60438 Frankfurt am Main, Germany Russian Research Center Kurchatov Institute, 123182 Moscow, Russia (Dated: December 12, 2018)
Abstract
On the basis of the kinetic equation with selfconsistent relativistic mean fields acting on baryonsand antibaryons, we study dynamical response of the nucleus to an antiproton implanted in itsinterior. By solving numerically the time-dependent Vlasov equation, we show that the compressedstate is formed on a rather short time scale of about 4 ÷
10 fm/c. This justifies the assumption, thatthe antiproton annihilation may happen in the compressed nuclear environment. The evolutionof the nucleus after antiproton annihilation is described by the same kinetic equation includingcollision terms. We show, that nucleon kinetic energy spectra and the total invariant mass distri-butions of produced mesons are quite sensitive observables to the antiproton annihilation in thecompressed nucleus.
PACS numbers: 25.43.+t; 21.30.Fe; 24.10.Jv; 24.10.Lx . INTRODUCTION As has been shown recently in Refs. [1, 2], an antiproton implanted in a heavy nucleusserves as an attractor for surrounding nucleons that can lead to a sizable increase of thecentral nucleon density. This effect is caused by the strong attractive scalar and vectorpotentials acting on the antiproton, as follows from the G -parity transformation of nuclearpotentials [3]. Correspondingly, the antiproton also creates an attractive potential acting onnucleons. This leads to the concentration of nucleons around the antiproton and, as result,to a considerable increase of the nucleon density.Within the relativistic mean field (RMF) model, the G -parity transformed nuclear opticalpotential is about −
700 MeV at the normal nuclear matter density ρ = 0 .
148 fm − , whilea phenomenological value of an antiproton optical potential is limited within the range of − (100 ÷ σ -, ω - and ρ -meson fields should be reduced withrespect to the values given by the G -parity transformation. The RMF calculations withreduced coupling constants [2] still show quite strong compressional effects for light andmedium nuclei.An important question, which arises here is whether the compression process is fastenough to develop before the ¯ p -annihilation. The total ¯ pp -annihilation cross section invacuum can be parameterized at low relative velocities v rel as σ ¯ pp ann = C + Dv rel , (1)where C = 38 mb and D = 35 mb · c [9]. Using these numbers we can estimate the life timeof an antiproton inside the nuclear matter at normal density: τ ann ≃ ρ σ ¯ pp ann v rel ≃ . (2)This is, of course, a very short time in nuclear scale. However, as argued in Ref. [2], thistime can become much longer, up to 20 fm/c, for deeply bound antiprotons due to the phasespace suppression factors. Therefore, the compression effects can, in-principle, show up in¯ p -nuclear interactions.In the present work, we apply a dynamical transport model in order to study the formationand decay of the compressed ¯ p -nuclear system. Our calculations are based on the Giessen2oltzmann-Uehling-Uhlenbeck (GiBUU) model [10], which has been recently supplementedby the relativistic mean fields [11]. Apart from collision terms, the GiBUU model solves thecoupled (through the mean fields) Vlasov equations for nucleon and antiproton phase spacedistribution functions. As well known [12], the Vlasov equation provides a semiclassical limitof the time-dependent Hartree-Fock calculations. Thus, the compressional effects found inRefs. [1, 2] should also be reproduced as a static solution of the coupled Vlasov equations.It will be demonstrated that the compression process is characterized by the time scalewhich is comparable with the ¯ p life time in nuclear environment. Thus, ¯ p has, indeed, achance to annihilate inside the compressed nucleus. We will show, that the ¯ p -annihilationin a compressed nucleus should lead to the collective expansion of the residual nuclearsystem. The appearance of the high-energy tails in the kinetic energy spectra of the emittednucleons is predicted. The distributions in the total invariant mass of produced mesonsreveal a noticeable shift toward lower invariant masses, when the annihilation takes placeinside the compressed nucleus.The annihilation of slow antiprotons inside heavy nuclei was, first, proposed by Rafelski[13] as a unique opportunity to study nuclear matter in unusual conditions. Later, Cahayet al. [14] studied the ¯ p annihilation inside nuclei within an intranuclear cascade model. InRef. [14], antiproton annihilation events into pions at the center of Ca and
Ag nucleiwere simulated. The mean field effects were, however, completely neglected in [14].In Sect. II, we describe the theoretical model applied in calculations. Sect. III containsthe results of the time evolution study for the compression and explosion dynamics. In Sect.IV, we propose several observable signals sensitive to the ¯ p -annihilation in the compressednucleus. The summary and outlook are given in Sect. V. II. THE MODEL
In calculations, we apply the GiBUU model developed in Giessen University. For thedetailed description and related references, we refer the reader to the web page [10], wherethe new version of the model is presented. Below, we mostly describe the new featuresimplemented in the present work. 3 . Relativistic mean fields
Below we consider a system composed of an antinucleon interacting with baryons. Thissystem is described by the RMF Lagrangian of the following form [2, 15]: L = X j = B, ¯ N ¯ ψ j [ γ µ ( i∂ µ − g ωj ω µ ) − m j − g σj σ ] ψ j + 12 ∂ µ σ∂ µ σ − U ( σ ) − F µν F µν + 12 m ω ω µ ω µ , (3)where ψ j are the baryon ( j = B ≡ N, N ⋆ , ∆ , Y ) and antinucleon ( j = ¯ N ) fields, respec-tively; σ is the isoscalar-scalar meson field ( I G = 0 + , J π = 0 + ); ω µ is the isoscalar-vectormeson field ( I G = 0 − , J π = 1 − ); and F µν ≡ ∂ µ ω ν − ∂ ν ω µ . Here N ⋆ and ∆ denotes, re-spectively, the isospin 1/2 and 3/2 nonstrange baryonic resonances, and Y stands for the S = − ψ j carry also one or more vector indices, whichare dropped in Eq.(3) and below for brevity. When appropriate, the covariant summationis assumed over these indices. For simplicity, the isovector and electromagnetic terms aredisregarded in (3). The selfinteractions of the σ -field are included in (3) via the term U ( σ )in order to avoid an unrealistically high compressibility coefficient of the nuclear matter [16]: U ( σ ) = 12 m σ σ + 13 g σ + 14 g σ . (4)Some comments are in order to gain more insight into Eq. (3). Following Ref. [2],the antinucleon field ψ ¯ N in the Lagrangian density (3) is represented in terms of wavefunctions of physical antinucleons . These wave functions can be obtained by the G -paritytransformation acting on the wave functions of the Dirac sea nucleons (see Ref. [3] fordetails), which appear in the relativistic description of the nucleon [17]. By applying thesame transformation, the nonlinear RMF Lagrangian of Refs. [15, 16] (neglecting termsresponsible for the baryon-antibaryon annihilation) can be expressed as (3) with the followingrelations between coupling constants: g ω ¯ N = − g ωN , g σ ¯ N = g σN . (5)The relations (5) are satisfied if the physical system would be exactly symmetric with respectto the G -parity transformation. However, this is not necessary to be true in a many-bodysystem [2, 3]. The reason is that the concept of the G -parity symmetry is strictly applicable4n the level of the elementary processes only. However, the RMF Lagrangian (3) is dealingwith the effective interactions, which are usually tuned to describe the bulk properties ofthe nuclear medium and/or the properties of some selected nuclei. Due to the many-bodyeffects, such as the Pauli blocking or mixed scalar-vector terms in the scattering amplitudes,these effective interactions may not obey the exact G -parity symmetry anymore. To takeinto account possible deviations from the G -parity symmetry, we introduce an overall scalingof the antinucleon-meson coupling constants with respect to the values given by (5) (see Ref.[2]): g ω ¯ N = − ξg ωN , g σ ¯ N = ξg σN , (6)where 0 < ξ ≤ ξ = 1, motivated by the G -parity, and ξ = 0 .
3, which is ina better agreement with the empirical ¯ p A optical potential. For other baryonic fields weput in the present work, for simplicity, the same coupling constants as for the nucleon : g ωN ∗ = g ω ∆ = g ωY = g ωN , g σN ∗ = g σ ∆ = g σY = g σN .All calculations have been performed emloying the NL3 parameterization [15] of theRMF model. This parameterization provides quite reasonable nuclear matter properties:the binding energy 16.299 MeV/nucleon, the compressibility coefficient K = 271 .
76 MeVand the nucleon effective mass m ∗ N = 0 . m N at ρ . Moreover, the NL3 parameterizationreproduces the ground state properties of spherical and deformed nuclei very well [15].The Dirac equations of motion for baryons have the following form:( γ µ ( i∂ µ − g ωj ω µ ) − m ⋆j ) ψ j = 0 , (7)where m ⋆j = m j + g σj σ (8)is the effective (Dirac) mass.Within the mean field approximation the σ - and ω -fields are treated classically. Theysatisfy the (nonlinear) Klein-Gordon-like equations with the source terms due to couplingto baryons and an antinucleon: ∂ ν ∂ ν σ + ∂U ( σ ) ∂σ = − X j = B, ¯ N g σj ρ Sj , (9)( ∂ ν ∂ ν + m ω ) ω µ = X j = B, ¯ N g ωj j µbj , (10)5here ρ Sj = < ¯ ψ j ψ j > is the partial scalar density and j µbj = < ¯ ψ j γ µ ψ j > is the partial baryoncurrent. Equation (10) has to be supplemented by the four-transversality condition ∂ µ ω µ = 0 . (11) B. Covariant kinetic equations
Instead of solving the Dirac equations (7), we will describe the baryons and antinucleondynamics by the coupled set of the semiclassical kinetic equations [11, 18, 19, 20, 21]:1 p ⋆ " p ⋆µ ∂∂x µ + g ωj p ⋆µ F kµ + m ⋆j ∂m ⋆j ∂x k ! ∂∂p ⋆k f j ( x, p ⋆ ) = I j [ f B , f M ] , (12)where k = 1 , , µ = 0 , , , x ≡ ( t, r ); and f j ( x, p ⋆ ) is the distribution function (DF)in a six-dimensional phase space ( r , p ⋆ ) with p ⋆ being the spatial components of the kineticfour-momentum p ⋆µ = p µ − g ωj ω µ . (13)The baryons and antinucleon are assumed to be on the respective effective mass shells: p ∗ = q ( p ⋆ ) + ( m ⋆j ) . (14)The l.h.s. of Eq. (12) describes the propagation of the j -th type particles in the classical σ -and ω -fields. The r.h.s. of Eq. (12) is a collision integral, which represents the (in)elastic two-body collisions with corresponding vacuum cross sections as well as the resonance decays.The complete description of the collision integral structure, in-particular, the differentialelementary cross sections included into the GiBUU model can be found in [10, 11] and inrefs. therein. The in-medium modification of the baryon-baryon and baryon-meson crosssections is neglected in the present work.We will apply the full kinetic equations, including collision terms, only to describe the post-annihilation evolution of a system. By this reason, the antiproton DF is excludedfrom the collision integral. Instead, we enforce ¯ p to annihilate into mesons at some pre-selected time (see Sect. IIE). Thus, in the present work the collision integral includesthe nucleon, ∆(1232) and higher baryon resonances up to the mass of 2 GeV, which canbe excited in the meson-baryon and baryon-baryon collisions. A possible hyperon forma-tion in the processes πN → Y K and ¯ KN → πY is included too. The “valence mesons”6 ≡ π, η, ρ, σ, ω, η ′ , φ, η c , J/ψ, K, ¯ K, K ∗ , ¯ K ∗ are explicitly taken into account. Theyare assumed to propagate freely between collisions, i.e. we neglect the mean field potentialsacting on these mesons.The scalar density and the baryon current of the j -th type baryons are expressed in termsof DF as follows: ρ Sj ( x ) = g j (2 π ) Z d p ⋆ p ⋆ m ⋆j f j ( x, p ⋆ ) , (15) j µbj ( x ) = g j (2 π ) Z d p ⋆ p ⋆ p ⋆µ f j ( x, p ⋆ ) , (16)where g j is the spin-isospin degeneracy factor ( g N = g ¯ N = 4 , g ∆ = 16 etc.).One can show [21, 22] that the kinetic equations (12) with the σ - and ω -fields evolvingaccording to Eqs. (9),(10) lead to the continuity equations X j = B ∂ µ j µbj = 0 , ∂ µ j µb ¯ N = 0 (17)and the energy-momentum conservation ∂ ν T µν = 0 , (18)where the energy-momentum tensor is written as T µν = X j = B, ¯ N,M g j (2 π ) Z d p ⋆ p ⋆ p µ p ⋆ν f j ( x, p ⋆ ) + ∂ µ σ∂ ν σ − ∂ µ ω λ ∂ ν ω λ − g µν (cid:18) ∂ λ σ∂ λ σ − U ( σ ) − ∂ λ ω κ ∂ λ ω κ + 12 m ω ω (cid:19) . (19)Here we have also included possible contributions of the “valence” mesons M , which can beproduced at the annihilation. It is assumed that p ⋆ = p for the valence mesons. C. Numerical realization
In order to solve Eq. (12) numerically, DF is represented by the set of point-like testparticles: f j ( x, p ⋆ ) = (2 π ) g j n nN j X i =1 δ ( r − r i ( t )) δ ( p ⋆ − p ⋆i ( t )) , (20)where N j is the number of physical particles of the type j and n is the number of testparticles per physical particle (the same for all types j ). The test particle positions r i and7inetic momenta p ⋆i are evolving in time according to the following equations:˙ r i = p ⋆i p ⋆ i , (21)˙ p ⋆ki = g ωj p ⋆iµ p ⋆ i F kµ + m ⋆j p ⋆ i ∂m ⋆j ∂x k (22)with k = 1 , , µ = 0 , , ,
3. It is easy to check that DF (20) with r i and p ⋆i satisfyingEqs. (21),(22) gives a formal solution of the Vlasov equation in the case when the collisionintegral in (12) is equal to zero. Equations (21),(22) are equivalent to the Hamiltonianequations of motion for the test particle positions r i and canonical momenta p i :˙ r i = ∂p i ∂ p i , (23)˙ p i = − ∂p i ∂ r i , (24)where p i = g ωj ω + q ( p ⋆i ) + ( m ⋆j ) is the single-particle energy (see [21, 23]). However, itis more convenient to propagate in time the test particle kinetic momenta rather than thecanonical ones, since then Eq. (9) for the σ -field decouples from Eq. (10) for the ω -field.When the collision integral in Eq. (12) is taken into account, the test particles arepropagated between the two-body collisions using Eqs. (21),(22). All calculations have beenperformed in the parallel ensemble mode. In this mode, the two-body collisions are permittedbetween the test particles belonging to the same parallel ensemble only, while the mean fieldis averaged over n parallel ensembles of the test particles propagated simultaneously (seeEq. (20)). Therefore, a single parallel ensemble can be considered as a physical event.In actual calculations, we have neglected the time derivatives of the meson fields in Eqs.(9),(10). However, the spatial derivatives were treated without any simplifying assumptions.The reason for such a strategy is that we are dealing with nuclear systems which have largedensity gradients, but evolving slowly, as compared with the spatial and temporal scalesinvolved in the mesonic equations of motion. Indeed, including the temporal gradients wouldlead to the frequent oscillations of the mesonic fields with a period of less than 2 π/m , where m is the meson mass. This gives the period of 2.5 fm/c (1.5 fm/c) for the σ - ( ω -) field.By taking into account the finite wave lenghts of these oscillations would further reducethe periods. The treatment of such oscillations would strongly complicate the numericalcalculations, in-particular, due to the classical meson field radiation. On the other hand,the characteristic periods of the oscillations are significantly smaller than the characteristic8ompression times (4 ÷
10 fm/c, see Sect. III A below). Therefore, one can approximatelyaverage-out the mesonic fields with respect to these oscillations, that is actually assumed inour model. The σ - and ω -fields are, therefore, calculated from the equations −△ σ + ∂U ( σ ) ∂σ = − X j = B, ¯ N g σj ρ Sj , (25)( −△ + m ω ) ω µ = X j = B, ¯ N g ωj j µbj . (26)Within the same approximation, the energy-momentum tensor has the following form: T µν = X j = B, ¯ N,M g j (2 π ) Z d p ⋆ p ⋆ p µ p ⋆ν f j ( x, p ⋆ ) + ( ∂ µ σ∂ ν σ − ∂ µ ω λ ∂ ν ω λ )(1 − δ ν ) − g µν (cid:18) −
12 ( ∇ σ ) − U ( σ ) + 12 ∇ ω λ ∇ ω λ + 12 m ω ω (cid:19) . (27)The factor (1 − δ ν ) in Eq. (27) reflects the fact, that due to the omission of the timederivatives of the meson fields in the Lagrangian density, only the first term in the r.h.s.contributes to the three-momentum density T α ( α = 1 , , x = ∆ y = ∆ z .For the systems ¯ p O, ¯ p Ca and ¯ p Pb considered below, the grid covered a cubic volumewith the side of 10, 20 and 30 fm, respectively, centered at the center-of-mass (c.m.) of a ¯ p Asystem. By numerical reasons, the δ -functions in coordinate space, introduced in Eq. (20),have been replaced by the Gaussians of the width L : δ ( r − r i ( t )) ⇒ π ) / L exp ( − ( r − r i ( t )) L ) . (28)The width of the Gaussian and the grid step sizes are pure numerical parameters whichshould resolve the coordinate space nonuniformities of the system. In our case the charac-teristic space scale is given by the radius of the smallest considered nucleus O, i.e. ≃ n oftest particles per physical particle (see Eq. (20)) should be correlated to the width of theGaussian as n ∝ L − . This puts a restriction on too small width due to CPU time increase.As an optimum choice, we fixed in the present work ∆ x = ∆ y = ∆ z = L = 0 . n = 1500 in the most of calculations.The equations of motion (21),(22) have been solved by applying the second-order in timepredictor-corrector method [11] with the time step of 0.1 fm/c. This value is small enoughto resolve the time scale of a few fm/c for the compression processes (see Figs. 3 and 5below). We have checked, that taking smaller time step does not influence the results. Thefull numerical scheme conserves the total energy with the accuracy of about 5% of the initialtotal binding energy of the ¯ p A system.
D. Initialization
The nucleons were distributed in coordinate space according to the Woods-Saxon densityprofile. The momenta of nucleons were sampled according to the local Fermi distribution.The initial antiproton DF was chosen as a Gaussian wave packet in coordinate and mo-mentum space [25, 26] located at the center of a nucleus ( x = y = z = 0): f ¯ N ( t = 0 , r , p ⋆ ) = (2 π ) g ¯ N π exp {− r / (2 σ r ) − σ r p ⋆ } , (29)where σ r is the width in coordinate space. Equation (29) implies, that the antiproton is atrest. The width of the initial antiproton distribution in momentum space is (2 σ r ) − , whichfollows from the uncertainty relation. If not mentioned explicitly, the calculation is donewith the choice σ r = 1 fm. This value agrees with results of the static RMF calculations ofRef. [2]. As for nucleons, the antiproton DF (29) is projected onto test particles accordingto Eq. (20) with the δ -functions in coordinate space replaced by Gaussians. To avoid mis-understanding, we note that one should distinguish the width σ r of the physical antiprotonspatial distribution in Eq. (29) and the width of the test particle Gaussian.10 . Propagation and annihilation After the initialization, the system of nucleons and antiproton was propagated in timeaccording to Eqs. (21),(22). The meson fields have been calculated by Eqs. (25),(26) withthe source terms given by the scalar densities (15) and the baryon currents (16). In sucha way, the evolution of the system toward compressed state has been followed. In thiscalculation, the collision term in the r.h.s. of the kinetic equation (12) has been set to zero,i.e. we considered a pure mean-field Vlasov dynamics. This was done to see most clearly therole of the mean fields. An introduction of the ¯
N N and
N N elastic collisions would mainlylead to a dissipation of the collective energy into heat. As pointed out in Ref. [2], this effectis rather small and, therefore, can not change significantly the compression dynamics.The reason is, that the elastic collisions are not frequent on the time scale of compression(see Figs. 3, 4 and 5). Indeed, the mean time τ coll between nucleon-nucleon collisions can beestimated as τ coll = 1 / ( ρ N σ NN v F ), where σ NN ≃
40 mb is the elastic nucleon-nucleon crosssection (c.f. Refs. [27, 28]) and v F ≃ . c is the Fermi velocity. This gives τ coll = 3 ÷ ρ N = 2 ÷ ρ . The Pauli blocking effect will further increase τ coll . Asimilar estimate can also be done for ¯ N N elastic collisions.At certain time moment t ann , which is an external parameter to our model, we simulatedthe annihilation of an antiproton. This implies that annihilation occurs instantaneously, asa single quantum mechanical transition, in distinction to description of this process via thecollision term in a kinetic equation. In the last case, the antiproton distribution functionwould gradually disappear on the way to the compressed state. The purpose of the presentwork is to look at the strongest possible effect of the nuclear compression on observables.Therefore, we let the compressed system to be formed, and simulate the sudden annihilationafterwards. The ambiguity in the in-medium annihilation cross sections is taken into accountby varying the parameter t ann .In the actual calculations, the annihilation was simulated as follows: For each antipro-ton test particle, the closest in coordinate space nucleon test particle was chosen to be theannihilation partner. At large enough values of the total in-medium c.m. energy √ s of theannihilating ¯ pN pair (see below), the annihilation event of the test particle pair into mesonswas simulated using the quark model [29, 30], which has been already implemented in theGiBUU model [10] earlier. A quark and an antiquark with the same flavour are assumed11o annihilate and transfer their total four-momentum to the remaining (anti)quarks. Theremaining four (anti)quarks form two orthogonal q ¯ q jets with equal energies in the c.m.frame. The jets were hadronized via the Lund string fragmentation model [31] in the JET-SET version included into the PYTHIA 6.225 program package. The applied annihilationmodel corresponds to the R2 type diagram in classification of Ref. [9], i.e. to the quarkrearrangement with one q ¯ q annihilation vertex. In this sense, the model has some similaritywith the two-meson doorway models of Refs. [32, 33]. To illustrate how the model workswe have performed simulations of the p ¯ p annihilation.Fig. 1 shows the pion multiplicity distribution for the p ¯ p annihilation at rest in vacuumcompared to the data compilation from Refs. [9, 34]. The calculated distribution is some-what shifted to smaller pion multiplicities with respect to the data: The calculated averagepion multiplicity < n π > ≃ . ≃ .
0. We wouldlike to remark, that non-vanishing contribution of the n π = 2 channel in calculations iscompletely due to the final states with other particles: ππη (78%), ππK ¯ K (14%), ππηη (6%) and ππ + photons (2% before η decay). In calculations, we took into account η decaysinto 2 γ or into final states with pions and disregarded photons afterwards. However, it isnot clear to us how photons were counted in the data (see also Ref. [32]).Fig. 2 shows the calculated charged pion momentum distributions in the c.m. frame ofthe annihilating p ¯ p pair at rest in vacuum. From the partial contributions of the channelswith various pion multiplicities we observe, as expected, that the hard(soft) part of the totalmomentum distribution is populated mainly by the low(high) pion multiplicity events. Theexperimental data are described reasonably well, except for the momenta 0 . ≤ k ≤ . pN pair can be substan-tially below the vacuum threshold value of 2 m N . This makes the direct application of theJETSET model for the ¯ p -annihilation in nuclei physically and numerically problematic. To12vercome this difficulty, we introduced the corrected invariant energy as follows [11, 35]: √ s corr = √ s ⋆ − m ⋆N − m N ) , (30)where s ⋆ = ( p ⋆ ¯ p + p ⋆N ) . The quantity √ s corr is a vacuum analog of the total in-mediuminvariant c.m. energy √ s with s = ( p ¯ p + p N ) . Provided that √ s > m π , we have used √ s corr in the JETSET simulation in order to produce the mesonic final states. This lowerlimit of √ s is due to the fact that the JETSET model does not generate enough direct 2 π and 3 π annihilation final states.In order to take into account the in-medium effects, in-particular, to ensure the correctin-medium threshold condition √ s > m + m + ... + m n mes , where m , m , ..., m n mes arethe vacuum masses of the produced mesons, the annihilation event was accepted with theprobability P = Φ n mes ( √ s ; m , m , ..., m n mes )Φ n mes ( √ s corr ; m , m , ..., m n mes ) , (31)where Φ n mes ( √ s ; m , m , ..., m n mes ) = Z d k (2 π ) ω Z d k (2 π ) ω · · · Z d k n mes (2 π ) ω n mes × δ (4) ( p ¯ p + p N − k − k − ... − k n mes ) (32)is the invariant phase space volume, k i = ( ω i , k i ) are the four-momenta of the producedmesons satisfying the vacuum mass shell conditions m i = k i , i = 1 , , ..., n mes . Finally, thethree-momenta of the produced mesons in the c.m. frame of the annihilating ¯ pN pair weremultiplied by the common factor adjusted to get the correct in-medium total c.m. energy √ s .In-fact, the way we simulate the in-medium effects Eq.(31) implies using the vacuummatrix elements of the annihilation channels, which are given by the JETSET model, whiletaking into account the in-medium effects in the phase space factors only. Similar procedureshave been applied earlier in Refs. [2, 11, 35].At 2 m π < √ s ≤ m π , the final 2 π or 3 π channel was chosen by Monte-Carlo accordingto the probability ratio P π P π = R Φ ( √ s ; m π , m π )Φ (2 m N ; m π , m π , m π )Φ (2 m N ; m π , m π )Φ ( √ s ; m π , m π , m π ) , (33)where R = 0 .
152 is the ratio of the 2 π and 3 π final state probabilities for the p ¯ p annihilationat rest (see Table VI in Ref. [2]). For the zero total charge Q of the annihilating ¯ pN pair, the13harge states of the outgoing pions were also determined from the data compilation of Ref.[2]. Since for Q = ± π final states were determinedby assuming that the π Q π π and π Q π + π − final channels have equal probabilities. Themomenta of the outgoing pions were distributed microcanonically according to the availabletwo- or three-body phase space.After the annihilation is simulated, the residual nucleons and produced mesons werepropagated in time according to the full kinetic equations (12), including both the baryonicmean fields and collision integrals. This takes into account the entropy production causedby the two-body collisions at the expansion stage. Moreover, important processes of themeson rescattering and absorption, e.g. πN → ∆ → πN or πN → ∆, ∆ N → N N areincluded in the collision integral. These processes influence the observed particle spectra.
III. TIME EVOLUTION OF BOUND ¯ p -NUCLEAR SYSTEMSA. Initial compression stage As demonstrated in Refs. [1, 2] by static RMF calculations, a deeply-bound antiproton-nucleus system can be significantly compressed as compared with a normal nucleus. Nowwe want to study the real dynamics of such a system starting from the unperturbed nuclearground state at t = 0.Fig. 3 (top panels) shows the nucleon and antiproton density profiles calculated at dif-ferent times along the axis z drawn through the center of the ¯ p Ca system. Fig. 3 (bottompanels) also shows the nucleon and antiproton potentials U j ≡ g ωj ω + g σj σ , j = N, ¯ p , alongthe same axis. Left and right panels present results for ξ = 0 . ξ = 1, respectively. Wesee that the initial configuration is unstable and the system starts to shrink. Both nucleonand antiproton central densities grow quite fast, reaching their maxima within several fm/c.In the course of the compression process, the nucleon potential becomes deeper in the caseof the reduced antiproton coupling constants ( ξ = 0 .
3) and does not, practically, change inthe G -parity motivated case ( ξ = 1). The antiproton potential deepens quite strongly withtime for the both sets of the antiproton coupling constants.In the case of ξ = 0 .
3, the first maximum of the central nucleon density ( ρ N = 0 .
30 fm − )is reached at t = 10 fm/c. At later time the system rebounds and oscillates approaching14radually a static configuration with the nucleon density ρ N ≃ .
26 fm − at the center.Since the annihilation is switched off in this calculation, the compressed configuration mayexist, in principle, infinitely long time. However, due to numerical reasons, the stability isdestroyed by a gradual test particle escape from a box in the coordinate space, where themean field is computed. Nevertheless, the numerical accuracy is good enough to trace thestable system up to at least t = 100 fm/c.In the case of ξ = 1, the compression process is much faster than in the case of ξ = 0 . t = 4 fm/c we observe the first maximum of the central nucleon density with ρ N = 0 .
48 fm − . A smaller value ρ N ≃ .
34 fm − is reached asymptotically after someoscillations.In Fig. 4 we compare time evolution of the nucleon density distribution along the centralaxis z for the light (¯ p O) and heavy (¯ p Pb) systems. For ¯ p O, the bell-like shape of thedensity distribution is reached quite fast. However, in the case of ¯ p Pb, we observe a quicklygrowing peak in the center, while peripheral nucleons still do not react on the compression.This leads to the delayed shape equilibration via a complicated compression-decompressioncycle.Fig. 5 shows the time evolution of the central nucleon density for the three systems:¯ p O, ¯ p Ca and ¯ p Pb. The case without annihilation is shown by the dotted lines. Fora comparison, we also present the central density time evolution in the respective groundstate nuclei without an antiproton inside (dashed lines). We see, that at long enough timesof the order of several tens fm/c, the static compressed configuration is indeed reached.The small oscillations of the central density in the compressed system visible at t > p Pb system:the dip in the central nucleon density at t ≃
50 fm/c. This is mostly a consequence of15he delayed shape equilibration mentioned above in discussing Fig. 4. The especially strongdensity drop for ξ = 0 . ÷
20 fm/c, see upper panels in Figs. 3 and 4).Since the compression time is of primary importance, we have also studied the sensitivityof our results to the width σ r of the initial antiproton DF (29). We have found, thatfor a larger (smaller) width the compression time becomes somewhat longer (shorter). In-particular, for the ¯ p O system at ξ = 1 the time needed to reach the first density maximumis 5 fm/c (2.5 fm/c) for σ r = 2 fm ( σ r = 0 . ξ = 0 . σ r = 2 fm ( σ r = 0 . σ r .We have to admit also, that there is some numerical uncertainty in our calculations dueto the choice of the width L of the test particle Gaussian and the grid step size. E.g. in thecalculation with ∆ x = ∆ y = ∆ z = L = 0 .
33 fm for the lightest system ¯ p O the maximumand saturation densities are 20 % higher than in calculation with ∆ x = ∆ y = ∆ z = L = 0 . ξ in ourcalculations. B. Post-annihilation dynamics of residual nuclei
Next, we study the dynamics of a residual nucleus after sudden annihilation of an an-tiproton. The annihilation was simulated as described in Sect. IIE. For each considered ¯ p Asystem and the scaling factor ξ , three different annihilation times t ann have been chosen:They correspond to (i) the early ( t ann = 0) annihilation from a non-compressed ground-state nucleus, (ii) the annihilation at the time moment when the first maximum of thecentral density is reached, and (iii) the late annihilation from an asymptotic compressedconfiguration.One can see from Fig. 5, that the annihilation from a non-compressed ground state nucleus(thin solid lines) does not lead to significant expansion of the residual nuclear system. Thecentral nucleon density stays always below but close to ρ in this case. In the case ofannihilation from the compressed configurations (thick solid and dash-dotted lines), we16bserve that, for the light systems ¯ p O and ¯ p Ca, the central nucleon density decreasessharply after annihilation and reaches values well below ρ . This is a clear indication of thecollective expansion of a system from the initially compressed state. On the other hand,for the heavy ¯ p Pb system, the expansion is not very pronounced at any choice of theannihilation time.It is interesting, that if the annihilation is switched on at the first density maximum(thick solid lines), then after an abrupt falling down the central density stays for sometime ∼ −
20 fm/c close to ρ before decreasing further. This is explained by an inertialcompression: After annihilation, the periphery of a residual nucleus still continues to moveto the center during some time until rebound. In the case of the ¯ p Ca system, this isdemonstrated in Figs. 6 and 7, where we show the baryon density and the radial collectivevelocity, v rad = r · v coll /r, (34)at several times as a function of the radius r . The collective velocity has been determinedas v α coll = T α /T , α = 1 , , . (35)At r > t = 12 fm/c for ξ = 0 . t = 6 fm/c for ξ = 1 reveals the fast outward motion at the center, while the peripheralnucleons still continue to move to the center. This explains plateaus in the central densityevolution near t ≃
20 fm/c in Fig. 5. At t ≃
30 fm/c the whole system starts to expand.This is reflected in the monotonically increasing radial collective velocity with radius. Theespecially strong rise of v rad at large r is due to emission of fast particles. At later times t ≃ ÷
60 fm/c the expansion is replaced by the inward motion of the matter in the centralzone. However, the fast particles are still continuing to escape from the dense region. Weexpect that in reality the system will break-up into fragments before the inward motion willstart (see discussion Sect. IV A). 17
V. OBSERVABLE SIGNALSA. Multifragmentation of residual nuclei
It is presently well established (see [40] and refs. therein) that nuclear matter at lowdensities ( ρ < . ρ ) becomes unstable with respect to small density perturbations, socalled spinodal instability. However, in order these density perturbations to develop intonuclear fragments, the system must stay long enough time ∼
30 fm/c in the spinodal region.One can see from Fig. 5, that the light systems ¯ p O and ¯ p Ca spend a long time in thisregion. Therefore, the residual nuclear systems can undergo a multifragment break-up, if theannihilation happens in the compressed configurations. In other words, the multifragmentbreak-up of nuclei after the ¯ p -annihilation may serve as a signal of the compression prior theannihilation.In Table I we collect the estimated parameters of fragmenting sources for the ¯ p O and¯ p Ca systems. The case of ¯ p -annihilation from the state of maximum central density isconsidered here (see thick solid lines in Fig. 5). The sources have been determined byselecting nucleons in the space region where the baryon density is larger than ρ min = 0 . ρ .They are characterized by the neutron ( N ) and proton ( Z ) numbers, the collective kineticenergy per nucleon ( E collkin ) and the residual excitation energy per nucleon ( E ⋆ res ). Theseparameters are defined as N = Z ρ>ρ min d rρ n ( r ) , Z = Z ρ>ρ min d rρ p ( r ) , (36) E collkin = 1 A Z ρ>ρ min d r (cid:16) T − q T µ T µ (cid:17) , (37) E ⋆ res = 1 A Z ρ>ρ min d r T − E g . s . ( N, Z ) − E collkin , (38)where ρ n and ρ p are the neutron and proton densities, respectively; A = N + Z ; and E g . s . ( N, Z ) is the ground state energy per nucleon of a nucleus with neutron number N and proton number Z computed within our model. The collective kinetic energy (37) iscalculated neglecting pressure effects. Due to the initially isospin-symmetric nuclei and theneglect of the isovector mesons in the Lagrangian density (3), we obtained in all cases N ≃ Z in the source. It turned out also, that the Coulomb interaction optionally included in someof the calculations does not change this result. Thus, only the charge numbers Z are given18n Table I. The time moment for determination of the source parameters has been chosen at35 ÷
40 fm/c, when the central nucleon density is about 1 / ÷ / ρ , i.e. inside the spinodalregion. The calculated residual excitation energy is typically 6 ÷
10 MeV/nucleon thatcorresponds to the temperatures 4 ÷ . ÷ . p O. This is well above the Coulomb energy of the source, which is only about 0.4MeV/nucleon for N = Z = 5. Unfortunately, such a source is too small to experience thereal multifragment break-up, rather a Fermi break-up into small clusters [41]. In the case oflarger sources, produced in ¯ p Ca annihilation the collective kinetic energy is considerablysmaller, 0 . ÷ . ≃ . N = Z = 16. Thus, we expect some signs of collective expansion tobe visible in kinetic energy spectra of produced fragments. B. Knock-out nucleon spectra
Let us now consider other observable effects. Fig. 8 shows the c.m. kinetic energy spectraof the nucleons emitted from the ¯ p O, ¯ p Ca and ¯ p Pb systems after the ¯ p -annihilation. Inorder to separate emitted nucleons from the bound nucleons of a residual nucleus, we useda simple criterion: only those nucleons, both protons and neutrons, were included in thespectra which are separated by at least 3 fm from the other test particles of a given parallelensemble at t = 100 fm/c. One can see, that nucleons with the kinetic energy E kin ≫ E F ,where E F ≃
35 MeV is the Fermi energy of the nuclear matter at ρ , are abundantlyemitted. Such nucleons are knocked-out from the nucleus by the mesons produced after theannihilation [14].In Table II we list the slope parameters T N of the nucleon kinetic energy spectra obtainedby the Maxwell-Boltzmann fit dN nuc dE kin ∝ q E kin exp( − E kin /T N ) (39)in the region of E kin = 200 ÷
500 MeV. We would like to mention, that the authors of Ref.[14] report the slope temperature of about 60 MeV for the kinetic energy spectrum of theemitted protons in the case of ¯ p Ca system, which is not so far from our results for the19nnihilation in the ground-state nucleus at t ann = 0.We want to emphasize that the kinetic energy spectra of nucleons emitted after the ¯ p -annihilation from the compressed ¯ p O, and ¯ p Ca systems are significantly harder thanthe spectra of nucleons from the annihilation at t ann = 0. This can be explained by twoeffects. First, the collective expansion of the outer shell will increase the slope temperature,typically, by several MeV (see Table I and Figs. 6, 7). Second, just after the annihilation thenucleon potential at the center of a nucleus grows suddenly by ∼ ÷
300 MeV. This createsan additional push for the fast nucleons emitted from the nucleus. Although the hardeningeffect is most pronounced for the lightest system ¯ p O, it is also quite visible for ¯ p Ca. Forthe heaviest system ¯ p Pb, we observe almost identical high energy tails of the nucleonspectra for the different annihilation times. The reason is that the collective expansion ispractically absent in this system (see Fig. 5). Also, the yield of fast nucleons is reduced bytheir subsequent rescatterings in the residual nucleus.
C. Mesonic observables
The meson production from the ¯ p -annihilation inside nucleus is influenced both by themean field via the potentials of the annihilating pair and by the final state interactions(FSI), i.e. the two-body collisions and resonance decays. It is instructive to disentangle thecontributions of the mean field effects from the rescattering and absorption effects. To thisaim, we have performed additional calculations by subsequently switching off the FSI andthe mean field. Corresponding results are shown in Figs. 9,10 and 11.Figure 9 shows pion multiplicity distributions for the ¯ p O system. In the case of reducedantiproton coupling constants ( ξ = 0 . n π . For the case of ξ = 1, we observe a strong pionmultiplicity reduction due to smaller √ s for the ¯ pN annihilation, while the FSI effects aremuch weaker. One can also see a quite significant compressional effect for ξ = 1, which isonly very weak for ξ = 0 . p Osystem. FSI strongly modifies these spectra, mostly due to the πN → ∆ → πN processes,which effectively decelerate pions. As we have already observed earlier in Fig. 8, emitted20ucleons gain energy, correspondingly.The effect of the baryonic mean field on the pion momentum spectrum is relatively mod-erate: we observe some depletion of the high-momentum tail, which is more pronounced inthe case of ξ = 1. The compressional effect is visible in the reduction of the pion yield in themomentum range 0 . ≤ k ≤ . n π = 5 and n π = 6 contribute substantially to this momentum range,while the probability of such events is substantially suppressed in a compressed nucleus dueto the reduced annihilation phase space [2].The pion momentum spectrum has clearly the two components: the slow pions, whichhave undergone rescatterings via the ∆-resonance excitation ( k less than about 300 MeV/c),and the high energy pions, which were emitted from the system almost without secondaryinteractions. The similar result has been obtained in earlier intranuclear cascade calculations[14]. Following Ref. [14], we have also fitted the low energy part of the pion spectrum( E − m π = 100 ÷
150 MeV, E = q k + m π ) by the Maxwell-Boltzmann distribution dN π ± dk = k exp( − E/T π ) . (40)This fit has produced the following slope temperatures of the charged pion momentumspectrum for the ¯ p O system: T π ≃
45 MeV and 44 MeV for ξ = 0 . T π ≃
43 MeV and 36MeV for ξ = 1) in the case of early annihilation ( t ann = 0) and annihilation at the time ofmaximum compression, respectively. The extracted T π is smaller than the slope temperatureof 53 MeV of the low energy pions for the ¯ p Ca system reported in Ref. [14]. However, inthe calculation without mean field, we obtain T π ≃
51 MeV, which is in a good agreementwith the result of Ref. [14]. Therefore, the difference between our results and those of Ref.[14] is caused by the mean field acting on the annihilating pair in medium. Moreover, we seethe softening of the pion spectrum in the case of larger antinucleon couplings due to smaller √ s of the annihilating pair.Figure 11 presents the distributions of the annihilation events in the total invariant massof produced mesons from the ¯ p O system. The invariant mass is defined as M inv = (cid:16) (P ) − P (cid:17) / , (41)where P µ mes = P i p µi is the sum of four-momenta of the mesons produced in a given annihila-tion event. The calculations were done for the case of t ann = 0.21n the absence of FSI, M inv should be equal to the invariant energy √ s of the annihilating¯ p N pair. Indeed, without FSI and without mean field, as expected, we get a quite sharppeak at 2 m N only slightly smeared out due to the Fermi motion of nucleons and momentumspread of the initial antiproton DF (29). The baryonic mean field leads to the shift of a peakposition toward smaller M inv and to some broadening of the distribution. The broadening isdue to the spatial spread of the initial antiproton DF (29), which results in different mean-field potentials acting on different annihilating ¯ p N test particle pairs. Additionally, the FSIleads to a very strong broadening of the invariant mass spectrum due to the decelerationand absorption of the annihilation mesons. This is clearly seen in Fig. 11 for the case whenthe RMF was switched off (dotted line). Nevertheless, the full calculation (solid line) showsquite strong softening of the M inv distribution due to the mean-field effects. Obviously, thiseffect is stronger for the case of ξ = 1 as compared with ξ = 0 . p -nucleussystems. Due to the strong reduction of the nucleon effective mass with the scalar density,the meson invariant mass spectra become softer when the annihilation happens in the com-pressed configurations, as compared with the annihilation in the normal state at t ann = 0.The effect is, again, more pronounced for the light systems ¯ p O and ¯ p Ca. In the ¯ p Osystem the shift is almost 500 MeV even in the case of ξ = 0 . V. SUMMARY AND OUTLOOK
We have performed dynamical modeling of possible compression effects in nuclei dueto the presence of an antiproton. The semiclassical transport GiBUU model [10] incorpo-rating the relativistic mean fields for the baryons and antibaryons has been employed incalculations. The model reproduces reasonably well the earlier static calculations of bound¯ p -nuclear systems [1, 2].In this work, we did not consider the stopping process of an incident antiproton in a targetnucleus. This is a rather complicated problem due to the unknown in-medium cross sectionsof the ¯ p -scattering and annihilation. This problem will be addressed in a forthcoming paper.22nstead, we have assumed, that the antiproton has penetrated to the center of the nucleus,stopped there due to an inelastic collision, and then get captured to the lowest energy state.Such events should be very rare, with a probability of the order of 10 − for the centralcollisions [2]. As proposed in Ref. [2], formation of bound antiproton-nucleus states can betriggered by the emission of fast nucleons, pions or kaons. We have shown, that during thetime interval of 4 ÷
10 fm/c after creation of the initial state the central density of the targetnucleus grows up to the values of 2 ÷ ρ depending on somewhat uncertain values of theantiproton coupling constants. We expect that the life time of strongly bound antiprotonscan be long enough to observe this cold compression effect.Detailed kinetic simulations of the post-annihilation evolution of residual nuclei havebeen carried out at different assumptions on the annihilation time. It is shown, that the¯ p -annihilation in compressed light systems, like ¯ p O and ¯ p Ca, leads to the pronouncedcollective expansion of the residual nucleus, which may result in the multifragment break-up.Another clear signature of the nuclear compression is the hardening of the kinetic energyspectra of emitted nucleons. On the other hand, the invariant mass distribution of producedmesons is shifted to smaller invariant masses due to the in-medium reduction of the nucleoneffective mass at high scalar density. Similar phenomena are expected also for the case of¯Λ-nucleus bound states which can be produced via the ¯ pp → ¯ΛΛ reaction on nuclei.Another interesting possibility is that the compressed zone of the nucleus might undergoa deconfinement phase transition. Then one can expect formation of a quark-antiquarkdroplet with a non-zero baryon number and relatively low temperature [2].Our main assumption in the present study was that the annihilation takes place in thecentral region of a nucleus. The experimental selection of the central annihilation eventsis a difficult problem. No clear trigger condition for such events has been invented so far.One suggestion is that the central annihilation events, in average, will be characterised byisotropic emission of secondary particles and high fragment multiplicity [13, 14]. However,further theoretical and experimental efforts are needed to develop a good trigger conditionfor the central annihilation. Despite of these difficulties, we propose to study the abovepredictions in antiproton-nucleus reactions at the future Facility for Antiproton and IonResearch (FAIR) at GSI (Darmstadt). 23 cknowledgments The support by the Frankfurt Center for Scientific Computing is gratefully aknowledged.The authors are grateful to A.S. Botvina, Th.J. B¨urvenich, I.A. Pshenichnov, J. Ritmanand H. St¨ocker for stimulating discussions. This work has been partially supported by theDFG Grant 436 RUS 113/711/0-2 (Germany) and the Grant NS-3004.2008.2 (Russia).24
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TABLE II: Slope temperatures T N (MeV) for the nucleon kinetic energy spectra (see Fig. 8 andEq. (39)). Only the values of T N for t ann = 0 (first number) and for the annihilation at the timeof the maximum compression (second number) are given. Statistical error is ± ξ ¯ p O ¯ p Ca ¯ p Pb0.3 66, 81 67, 71 64, 591 52, 95 46, 79 45, 53 P ( n π ) n π pp- calculationdata fitdata FIG. 1: (color online) Pion multiplicity distribution for p ¯ p annihilation at rest in vacuum. Datapoints are from Ref. [9]. The dashed line represents the data fit [34] with the Gaussian P ( n π ) =exp {− ( n π − < n π > ) / σ n π } / q πσ n π where < n π > = 5 .
01 and σ n π = 1 . d N π ± / d k ( c / G e V ) k (GeV/c)pp- data (arb.units)total2 π π π π π π FIG. 2: (color online) Charged pion momentum distribution for p ¯ p annihilation at rest in vac-uum. The total calculated distribution is shown by the thick solid line. The calculated partialcontributions from events with various pion numbers are also depicted (see key for notations). Thecalculations are normalized to the number of charged pions per annihilation event. Data from Ref.[9] are in arbitrary units and are rescaled to agree with calculations at k = 0 . .00.10.20.30.40.5 ρ N , p - ( f m - ) p- Ca ξ =0.3 0 fm/c, N10 fm/c, N100 fm/c, N0 fm/c, p-10 fm/c, p-100 fm/c, p--0.4-0.3-0.2-0.10.0-10 -5 0 5 10 U N , p - ( G e V ) z (fm) 0.00.20.40.60.8 p- Ca ξ =1 0 fm/c, N4 fm/c, N100 fm/c, N0 fm/c, p-4 fm/c, p-100 fm/c, p--1.5-1.2-0.9-0.6-0.30.0-10 -5 0 5 10 z (fm) FIG. 3: (color online) Nucleon and antiproton densities (top panels) and potentials (bottom panels)vs coordinate z on the axis passing through the center of the ¯ p Ca system at selected timesindicated in the figure. The calculations with the scaling factor ξ = 0 . ξ = 1) are shown in theleft (right) panels. Please, notice different scales on vertical axis. .00.10.20.30.40.50.6 ρ N ( f m - ) p- O ξ =0.3 0 fm/c4 fm/c8 fm/c30 fm/c50 fm/c70 fm/c100 fm/c0.00.10.20.30.40.5-10 -5 0 5 10 ρ N ( f m - ) z (fm) p- Pb ξ =0.3 0 fm/c5 fm/c10 fm/c30 fm/c50 fm/c70 fm/c100 fm/c 0.00.20.40.60.8 p- O ξ =1 0 fm/c4 fm/c10 fm/c30 fm/c50 fm/c70 fm/c100 fm/c0.00.20.40.60.8-10 -5 0 5 10 z (fm) p- Pb ξ =1 0 fm/c4 fm/c10 fm/c30 fm/c48 fm/c70 fm/c100 fm/c FIG. 4: (color online) Nucleon densities along coordinate z at various time moments for ¯ p O and¯ p Pb at ξ = 0 . ξ = 1. .00.20.40.6 ρ N (f m - ) p- O ξ =0.3 w/o ann.ground statet ann =0 fm/c8 fm/c40 fm/c0.00.20.40.6 ρ N (f m - ) p- Ca ξ =0.3 w/o ann.ground statet ann =0 fm/c10 fm/c60 fm/c0.00.20.40.6 0 20 40 60 80 100 ρ N (f m - ) t (fm/c)p- Pb ξ =0.3 w/o ann.ground statet ann =0 fm/c10 fm/c20 fm/c p- O ξ =1 w/o ann.ground statet ann =0 fm/c4 fm/c20 fm/c p- Ca ξ =1 w/o ann.ground statet ann =0 fm/c4 fm/c20 fm/c 0 20 40 60 80 100 t (fm/c)p- Pb ξ =1 w/o ann.ground statet ann =0 fm/c4 fm/c20 fm/c FIG. 5: (color online) Time dependence of the nucleon density at the center of the ¯ p O, ¯ p Caand ¯ p Pb systems for two values of the scaling factor ξ = 0 . ξ = 1 (rightpanels) of the antiproton coupling constants. The dotted line shows the calculation without ¯ p annihilation. The thin solid, thick solid and dash-dotted lines show the results with annihilationsimulated at various times t ann indicated in the figure. The dashed lines show the central nucleondensity evolution for the corresponding ground state nucleus without an antiproton. -3 -2 -1 ρ B ( f m - ) p- Ca ξ =0.3t ann =10 fm/c 0 fm/c3 fm/c10 fm/c12 fm/c16 fm/c20 fm/c30 fm/c40 fm/c60 fm/c-0.10.00.10.2 0 2 4 6 8 v r ad ( c ) r (fm) FIG. 6: (color online) The baryon density (top panel) and the radial collective velocity (bottompanel) as functions of the radial distance for the ¯ p Ca system computed with ξ = 0 .
3. Theannihilation time t ann was set to 10 fm/c. -3 -2 -1 ρ B ( f m - ) p- Ca ξ =1t ann =4 fm/c 0 fm/c2 fm/c4 fm/c6 fm/c10 fm/c20 fm/c30 fm/c40 fm/c60 fm/c-0.10.00.10.2 0 2 4 6 8 v r ad ( c ) r (fm) FIG. 7: (color online) Same as in Fig. 6, but for ξ = 1 and t ann = 4 fm/c. -3 -2 -1 d N nu c / d E k i n ( G e V - ) p- O ξ =0.3t ann =0 fm/c8 fm/c40 fm/c10 -3 -2 -1 d N nu c / d E k i n ( G e V - ) p- Ca ξ =0.3t ann =0 fm/c10 fm/c60 fm/c10 -3 -2 -1 d N nu c / d E k i n ( G e V - ) E kin (GeV) p- Pb ξ =0.3t ann =0 fm/c10 fm/c20 fm/c p- O ξ =1t ann =0 fm/c4 fm/c20 fm/c p- Ca ξ =1t ann =0 fm/c4 fm/c20 fm/c 0.0 0.5 1.0 E kin (GeV) p- Pb ξ =1t ann =0 fm/c4 fm/c20 fm/c FIG. 8: (color online) Kinetic energy spectra of emitted nucleons in the c.m. frame for various ¯ p Asystems and values of the parameter ξ . Different histograms correspond to different values of theannihilation time t ann indicated in the key. .00.20.40.6 P ( n π ) p- O ξ =0.3t ann =0 fm/c, w/o RMF, w/o FSI0 fm/c, w/o FSI0 fm/c8 fm/c40 fm/c0.00.20.40.6 0 1 2 3 4 5 6 7 8 9 10 P ( n π ) n π p- O ξ =1t ann =0 fm/c, w/o RMF, w/o FSI0 fm/c, w/o FSI0 fm/c4 fm/c20 fm/c FIG. 9: (color online) Pion multiplicity distributions for the ¯ p O system. The line with full circlesshows the calculation without mean field and without FSI after the annihilation. The line withfull boxes shows the result with mean field, but without FSI. Other lines show the full calculationat various choices of the annihilation time (shown in the key). The top (bottom) panel presentsresults for ξ = 0 . ξ = 1). d N π ± / d k ( c / G e V ) p- O ξ =0.3t ann =0 fm/c, w/o RMF, w/o FSI0 fm/c, w/o FSI0 fm/c8 fm/c40 fm/c 0 2 4 6 8 100.0 0.2 0.4 0.6 0.8 1.0 d N π ± / d k ( c / G e V ) k (GeV/c) p- O ξ =1t ann =0 fm/c, w/o RMF, w/o FSI0 fm/c, w/o FSI0 fm/c4 fm/c20 fm/c FIG. 10: (color online) Same as in Fig. 9, but for the charged pion momentum distributions in thec.m. frame of the ¯ p O system. -2 -1 d N e v / d M i n v ( G e V - ) p- O ξ =0.3 w/o RMF, w/o FSIw/o RMFw/o FSIfull10 -2 -1 d N e v / d M i n v ( G e V - ) M inv (GeV)p- O ξ =1 w/o RMF, w/o FSIw/o RMFw/o FSIfull FIG. 11: (color online) Distribution of the annihilation events on the total invariant mass of emittedmesons for the ¯ p O system. The calculation without mean field and without FSI is shown by theline with full circles. The results without mean field but with FSI are shown by the dotted line.The line with full boxes shows the result with mean field, but without FSI. The full calculation ispresented by the solid line. Upper (lower) panel corresponds to ξ = 0 . ξ = 1). Only calculationsat t ann = 0 are shown. d N e v / d M i n v ( G e V - ) p- O ξ =0.3 t ann =0 fm/c8 fm/c40 fm/c 0 1 2 d N e v / d M i n v ( G e V - ) p- Ca ξ =0.3t ann =0 fm/c10 fm/c60 fm/c 0 1 2 30.0 1.0 2.0 d N e v / d M i n v ( G e V - ) M inv (GeV) p- Pb ξ =0.3t ann =0 fm/c10 fm/c20 fm/c 0 1 2 p- O ξ =1t ann =0 fm/c4 fm/c20 fm/c 0 2 4 p- Ca ξ =1t ann =0 fm/c4 fm/c20 fm/c 0 2 4 60.0 1.0 2.0 M inv (GeV) p- Pb ξ =1t ann =0 fm/c4 fm/c20 fm/c FIG. 12: (color online) Annihilation event distributions on the total invariant mass of emittedmesons for various ¯ p A systems and values of the parameter ξ . Different histograms correspond todifferent values of the annihilation time t annann