I. MOTIVATION

It is difficult to produce antiproton beams. However, antiproton-nucleus interactions have attracted experimentalists and theorists since about 30 years when the KEK and LEAR data appeared. Since this time significant progress has been done to describe these data on the basis of optical and cascade models. Still, antiproton interactions inside nuclei need to be better understood. One example is the antiproton-nucleus optical potential. According to the low-density theorem, it can be expressed as

where at threshold $\sqrt{s}\simeq 2m_N$, $E_{\overline{p}}\simeq m_N$, $f_{\overline{p}p}\simeq (-0.9+i0.9)$ fm [1]. Being extrapolated to the normal nuclear density ρ₀=0.16 fm^-3, Eq.(1) predicts the repulsive antiproton-nucleus potential, Re V_opt 75 MeV. In contrast, the $\overline{p}$-atomic X-ray and radiochemical data analysis [2] favors the strongly attractive antiproton-nucleus potential, Re V_opt -100 MeV in the nuclear center. Thus the $\overline{p}A$ optical potential is not a simple superposition of vacuum $\overline{p}N$ interactions. The strongly attractive $\overline{p}A$ potential is consistent with the Relativistic Mean Field (RMF) models and has a consequence that a nucleus may collectively respond to the presence of an implanted antiproton. The formation of strongly bound $\overline{p}$-nuclei becomes possible [3, 4].

Another very interesting aspect is $\overline{p}$-annihilation in the nuclear interior. This results in a large energy deposition 2 mN in the form of mesons, mostly pions, in a volume of hadronic size 1-2 fm [4, 5]. After the passage of annihilation hadrons through the nuclear medium a highly excited nuclear residue can be formed and even experience explosive multifragment breakup [5, 6]. The annihilation of an antiproton at p_lab < 5 GeV/c on a nuclear target gives an excellent opportunity to study the interactions of secondary particles (pions [7], kaons and hyperons [8], charmonia [9, 10]) with nucleons. This is because most annihilation hadrons are slow (γ < 2) and have short formation lengths. Thus their interactions are governed by usual hadronic cross sections.

Over the last decades, several microscopic transport models have been developed to describe particle production in $\overline{p}A$ interactions [6, 7, 11-13]. Nowadays there is a renaissance in this field, since the antiproton-nucleus reactions at p_lab 1.5-15 GeV/c will be a part of the PANDA experiment at FAIR. The most recent calculations are done within the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) model [14-16] and within the Lanzhou quantum molecular dynamics (LQMD) model [17, 18]. In the present paper, I will report some results of the GiBUU calculations for $\overline{p}$-nucleus interactions at p_lab 1.5-15 GeV/c.

II. GIBUU MODEL

The GiBUU model [19, 20] solves a coupled set of kinetic equations for baryons, antibaryons, and mesons. In a RMF mode, this set can be written as [21, 22]

where α=1,2,3, μ=0,1,2,3, x=(t,r); $j=N\mathrm{,}\overline{N}\mathrm{,}\Delta \mathrm{,}\overline{\Delta }\mathrm{,}Y\mathrm{,}\overline{Y}\mathrm{,}\pi \mathrm{,}K\mathrm{,}\overline{K}$ etc.. fj(x,p^*) is the distribution function of the particles of sort j normalized such that the total number of particles of this sort is

with gj being the spin degeneracy factor. The Vlasov term (the l.h.s. of Eq.(2)) describes the evolution of the distribution function in smooth mean field potentials. The collision term (the r.h.s. of Eq.(2)) accounts for elastic and inelastic binary collisions and resonance decays. The Vlasov term includes the effective (Dirac) mass $m_j^{*}=m_j+S_j$, where Sj=gσj is a scalar field; the field tensor $F_j^{\mu \nu }={\partial }^{\mu }{V}_j^{\nu }-{\partial }^{\nu }{V}_j^{\mu }$, where $V_j^{\mu }=g_{\omega j}{\omega }^{\mu }+g_{\rho j}{\tau }^3{\rho }^{3\mu }+q_j{A}^{\mu }$ is a vector field, τ³=+1 for p and $\overline{n}$, τ³=-1 for $\overline{p}$ and n; and the kinetic four-momentum $p^{*\mu }=p^{\mu }-V_j^{\mu }$ satisfying the effective mass shell condition $p^{*\mu }{p}_{\mu }^{*}=m_j^{*}{}^2$.

In the present calculations, the nucleon-meson coupling constants gσN, gωN, gρN and the self-interaction parameters of the σ-field have been adopted from a non-linear Walecka model in the NL3 parameterization [23]. The latter gives the compressibility coefficient K=271.76 MeV and the nucleon effective mass $m_N^{*}=0.60m_N$ at ρ=ρ₀. The antinucleon-meson coupling constants have been determined as

where 0 < ξ 1 is a scaling factor. The choice ξ=1 corresponds to the G-parity transformed nuclear potential. In this case, however, the Schrödinger equivalent potential becomes unphysically deep, $U_{\overline{N}}=-660\text{}\text{ MeV}$. The empirical choice of ξ will be discussed in the following section.

The GiBUU collision term³ includes the following channels: (notations: B – nonstrange baryon, R – nonstrange baryon resonance, Y – hyperon with S=-1, M – nonstrange meson):

• Baryon-baryon collisions:

elastic (EL) and charge-exchange (CEX) scattering BB → BB; s-wave pion production/absorption ⁴

NN NNπ; NN ΔΔ; NN NR; N(Δ,N^*)N(Δ,N^*) → N(Δ)YK; YN → YN; ΞN → ΛΛ; ΞN → Λ∑; ΞN → ΞN.

For invariant energies $\sqrt{s}>2.6$ GeV the inelastic production B₁B₂ → B₃B₄ (+ mesons) is simulated via the PYTHIA model.

• Antibaryon-baryon collisions:

annihilation $\overline{B}B\to$ mesons⁵; EL and CEX scattering $\overline{B}B\to \overline{B}B$; $\overline{N}N\leftrightarrow \overline{N}\Delta$ (+ c.c.); $\overline{N}N\to \overline{\Lambda }\Lambda$; $\overline{N}(\overline{\Delta })N(\overline{\Delta })\to \overline{\Lambda }\sum$ (+ c.c.); $\overline{N}(\overline{\Delta })N(\Delta )\to \overline{\Xi }\Xi$.

For invariant energies $\sqrt{s}>2.4$ GeV (i.e. p_lab > 1.9 GeV/c for $\overline{N}N$) the inelastic production ${\overline{B}}_1{B}_2\to {\overline{B}}_3{B}_4$ (+ mesons) is simulated via the FRITIOF model.

• Meson-baryon collisions:

MN R (baryon resonance excitations and decays, e.g., πN Δ and $\overline{K}N\leftrightarrow Y^{*}$); π(ρ)Δ R; πN → πN; πN → ππN; πN → ηΔ; πN → ωN; πN → ϕN; πN → ωπN; πN → ϕπN; π(η,ρ,ω) N → YK; $\pi N\to K\overline{K}N$; πN → YKπ; πΔ → YK; $KN\to KN$ (EL, CEX); $\overline{K}N\to \overline{K}N$ (EL, CEX); $\overline{K}N\leftrightarrow Y\pi$; $\overline{K}N\leftrightarrow Y^{*}\pi$; $\overline{K}N\to \Xi K$.

At $\sqrt{s}>2.2$ GeV the inelastic meson-baryon collisions are simulated via PYTHIA.

• Meson-meson collisions:

M₁M₂ M₃ (meson resonance excitations and decays, e.g., ππ ρ and Kπ K^*); $M_1{M}_2\leftrightarrow K\overline{K}$, $M_1{M}_2\leftrightarrow K{\overline{K}}^{*}$ (+ c.c.).

III. ANTIPROTON ABSORPTION AND ANNIHILATION ON NUCLEI

Without a mean field acting on an antiproton, the GiBUU model is expected to reproduce a simple Glauber model result for the $\overline{p}$-absorption cross section on a nucleus (left, Fig. 1):

Fig. 1.Full-screen View

Left panel – the straight-line propagation of an antiproton in the absence of a mean field. Right panel – an illustration of the curved trajectory of an antiproton due to an attractive mean field.

where ${\overline{\sigma }}_{\text{tot}}$ is the isospin-averaged total $\overline{p}N$ cross section. The attractive mean field bends the $\overline{p}$ trajectory to the nucleus (right, Fig. 1). Thus, the absorption cross section should increase.

Figure 2 shows the GiBUU calculations of antiproton absorption cross sections on ¹²C, ²⁷Al and ⁶⁴Cu in comparison with experimental data [26-29] and with the Glauber formula (6). Indeed, GiBUU calculations without mesonic components of the $\overline{p}$ mean field, i.e., with scaling factor ξ=0, are very close to Eq.(6) at p_lab > 0.3 GeV/c. At a lower p_lab, the Coulomb potential makes the difference between GiBUU (ξ=0) and Glauber results. Including the mesonic components of the $\overline{p}$ mean field, (ξ > 0) noticeably increases the absorption cross section at p_lab < 3 GeV/c. The best fit of the KEK data [26] at p_lab=470-880 MeV/c is reached with ξ=0.21±0.03. This produces the real part of the antiproton-nucleus optical potential $\text{Re}{V}_{\text{opt}}\equiv U_{\overline{p}}\simeq -(150\pm 30)$ MeV at ρ=ρ₀. The corresponding imaginary part is

Fig. 2.Full-screen View

(Color online) Antiproton absorption cross section on the ¹²C, ²⁷Al, and ⁶⁴Cu nuclei vs the beam momentum. The GiBUU results are shown by the lines marked with the value of a scaling factor ξ. Thin solid lines represent the Glauber model calculation, Eq.(6). For the $\overline{p}$+¹²C system, a calculation with ξ=0 without annihilation is shown by the dotted line.

At ρ=ρ₀ this gives Im V_opt -(100-110) MeV independent on the choice of ξ. It is interesting that the BNL [27] and Serpukhov [28] data at p_lab=1.6-20 GeV/c favor ξ=1, i.e. Re V_opt -660 MeV at ρ=ρ₀. This discrepancy needs to be clarified, which could be possibly done at FAIR.

Figure 3 displays the calculated momentum spectra of positive pions and protons for antiproton interactions at p_lab=608 MeV/c with the carbon and uranium targets. GiBUU reproduces a quite complicated shape of the pion spectra which appears due to the underlying πN Δ dynamics. The absolute normalization of the spectra is weakly sensitive to the $\overline{p}$ mean field. The best agreement is reached for ξ=0.3, i.e., for Re V_opt -(220±70) MeV.

Fig. 3.Full-screen View

(Color online) Momentum differential cross sections of π⁺ and p production in $\overline{p}$ annihilation at 680 MeV/c on ¹²C and ²³⁸U. The different lines are denoted by the value of a scaling factor ξ. The data points are from [30].

IV. SELFCONSISTENCY EFFECTS

The strong attraction of an antiproton to the nucleus has to influence on the nucleus itself. This back coupling effect can be taken into account by including the antinucleon contributions in the source terms of the Lagrange equations for ω-, ρ-, and σ-fields

with $U(\sigma )=\frac{1}{2}{m}_{\sigma }^2{\sigma }^2+\frac{1}{3}{g}_2{\sigma }^3+\frac{1}{4}{g}_3{\sigma }^4$, or, in other words, by treating the meson fields selfconsistently. As follows from Eqs. (4) and (8)–(10), nucleons and antinucleons contribute with the opposite sign to the source terms of the vector fields ω and ρ, and with the same sign – to the source term of the scalar field σ. Hence, repulsion is reduced and attraction is enhanced in the presence of an antiproton in the nucleus.

Figure 4 shows the density profiles of nucleons and an antiproton at different times when the $\overline{p}$ implanted at t=0 in the center of the ⁴⁰Ca nucleus. As a consequence of the pure Vlasov dynamics of the coupled antiproton-nucleus system (annihilation is turned off), both the nucleon and the antiproton densities grow quite fast. At t 10 fm/c the compressed state is already formed, and the system starts to oscillate around the new equilibrium density ρ 2ρ₀.

Fig. 4.Full-screen View

(Color online) The density of nucleons (thick lines) and antiproton (thin lines) as a function of coordinate on z-axis drawn through the nuclear center (z=0).

Figure 5 displays the time evolution of the central nucleon density. The $\overline{p}$ annihilation is simulated at the time t_ann. The t_ann=0 corresponds to the usual annihilation of a stopped $\overline{p}$ in the nuclear center. In this case, the nucleon density remains close to the ground state density. However, if the annihilation is simulated in a compressed configuration (t_ann > 0), then the residual nuclear system expands. Eventually the system reaches the low-density spinodal region (ρ < 0.6ρ₀), where the sound velocity squared $c_s^2=\partial P/\partial {\rho }_{|s=const}$ becomes negative⁶. This should result in the breakup of the residual nuclear system into fragments.

Fig. 5.Full-screen View

(Color online) The central nucleon density as a function of time. The annihilation of $\overline{p}$ with the closest nucleon into mesons is simulated at the time moment t_ann as indicated. The calculations without annihilation and for the ground state nucleus (without $\overline{p}$) are also shown.

A possible observable signal of the $\overline{p}$ annihilation in a compressed nuclear configuration is the total invariant mass M_inv of emitted mesons

For the annihilation of a stopped antiproton on a proton at rest in a vacuum, M_inv=2mN. In a nuclear medium, the proton and antiproton vector fields largely cancel each other⁷. Therefore, it is expected that in nuclear medium the peak will appear at M_inv 2mN^*. This simple picture is illustrated by GiBUU calculations in Fig. 6. In calculations where t_ann=0, we clearly see a sharp medium-modified peak shifted downwards by 200 MeV from 2mN. The final state interactions of mesons create a broad maximum at M_inv 1 GeV. For annihilation in compressed configurations (t_ann=10 and 60 fm/c), the total spectrum further shifts by about 100 MeV to the smaller M_inv. This effect becomes stronger with the decreasing mass of the target nucleus (e.g., for ¹⁶O the spectrum shift is nearly 500 MeV [14]).

Fig. 6.Full-screen View

(Color online) Annihilation event spectrum on the total invariant mass (11) of emitted mesons. Calculations are done for three different values of annihilation time t_ann.

V. STRANGENESS PRODUCTION

Originally, the main motivation of experiments on strangeness production in antiproton-nucleus collisions was to find signs of unusual phenomena, in-particular, a multinucleon annihilation and/or a quark-gluon plasma (QGP) formation. In Ref. [31], the cold QGP formation has been suggested to explain the unusually large ratio $\Lambda /K_S^0\simeq 2.4$ measured in the reaction ${\overline{p}}^{181}$Ta at 4 GeV/c [32]. On the other hand, in Refs. [8, 11, 16-18, 33-35] most features of strangeness production in $\overline{p}A$ reactions have been explained by hadronic mechanisms.

Figure 7 presents the rapidity spectrum of (Λ+∑⁰) hyperons, $K_S^0$ mesons and $(\overline{\Lambda }+{\overline{\Sigma }}^0)$ antihyperons for the collisions $\overline{p}{(4GeV/c)}^{181}$Ta in comparison with the data [36] and the intranuclear cascade (INC) calculations [11]. The GiBUU model underpredicts hyperon yields at small forward rapidities y 0.5 and overpredicts $K_S^0$ yields. In the GiBUU calculation without hyperon-nucleon scattering, the (Λ+∑⁰) spectrum is shifted to forward rapidities. However, the problem of underpredicted total (Λ+∑⁰) yield remains. A more detailed analysis [16] shows that 72% of Y and Y^* production rates in GiBUU are due to the antikaon absorption processes $\overline{K}B\to YX$, $\overline{K}B\to Y^{*}$, and $\overline{K}B\to Y^{*}\pi$. The second largest contribution, 23The antibaryon-baryon (including the direct $\overline{p}N$ channel) and baryon-baryon collisions contribute only 3% and 2%, respectively, to the same rate. The underprediction of the hyperon yield in GiBUU could be due to the used partial $\overline{K}N$ cross sections, in-particular, due to the problematic K^-n channel⁸. The possible in-medium enhancement of the hyperon production in antikaon-baryon collisions is also not excluded.

Fig. 7.Full-screen View

(Color online) Rapidity spectra of (Λ+∑⁰), $K_S^0$, and $(\overline{\Lambda }+{\overline{\Sigma }}^0)$ from ${\overline{p}}^{181}$Ta collisions at 4 GeV/c. See text for details.

As shown in Fig. 8, at higher beam momenta the agreement between the calculations and the data on neutral strange particle production becomes visibly better. The exception is again the region of small forward rapidities y 0.5 where both GiBUU and INC calculations underpredict the (Λ+∑⁰) yield.

Fig. 8.Full-screen View

(Color online) Rapidity spectra of Λ, $K_S^0$ and $$ from ${\overline{p}}^{64}$Cu collisions at 8.8 GeV/c. The data and INC calculations are from Ref. [33].

Finally, let us discuss the Ξ (S=-2) hyperon production. The direct production of Ξ in the collision of nonstrange particles would require to produce two $s\overline{s}$ pairs simultaneously. Thus, Ξ production could be even stronger enhanced in a QGP as compared to the enhancement for the S=-1 hyperons. Fig. 9 shows the rapidity spectra of the different strange particles in ${\overline{p}}^{197}$Au collisions at 15 GeV/c. Even at such a high beam momentum, the S=-1 hyperon spectra still have a flat maximum at y 0 due to exothermic strangeness exchange reactions $\overline{K}N\to Y\pi$ with slow $\overline{K}$. In contrast, the second largest, 18%, contribution to the Ξ production is given by endothermic double strangeness exchange reactions$\overline{K}N\to \Xi K$⁹. Since the threshold beam momentum of $\overline{K}$ for the process $\overline{K}N\to \Xi K$ is 1.05 GeV/c, which corresponds to the $\overline{K}N$ c.m. rapidity of 0.55, the rapidity spectra of Ξ’s are shifted forward with respect to the Λ rapidity spectra. However, in the QGP fireball scenario [31], the rapidity spectra of all strange particles would be peaked at the same rapidity.

Fig. 9.Full-screen View

(Color online) The rapidity spectra of (Λ+∑⁰), $K_S^0$, $(\overline{\Lambda }+{\overline{\Sigma }}^0)$, ^-, and ⁰ from ${\overline{p}}^{197}$ Au collisions at 15 GeV/c.

VI. SUMMARY

This work was focused on the dynamics of a coupled antiproton-nucleus system and the strangeness production in $\overline{p}A$ interactions. The calculations were based on the GiBUU transport model. The main results can be summarized as:

• The reproduction of experimental data on $\overline{p}A$ absorption cross sections at p_lab < 1 GeV/c and on π⁺ and p production at p_lab=608 MeV/c requires a strongly attractive $\overline{p}A$ optical potential, Re V_opt -(150-200) MeV at ρ=ρ₀.

• As a response of a nucleus to the presence of an antiproton, the nucleon density can be increased up to ρ (2-3)ρ₀ locally near $\overline{p}$. Annihilation of the $\overline{p}$ in such a compressed configuration can manifest itself in the multifragment breakup of the residual nuclear system and in the substantial (300-500 MeV) shift of an annihilation event spectrum on the total invariant mass of produced mesons M_inv toward low M_inv.

• GiBUU describes the data on inclusive pion and proton production fairly well. Still, the strangeness production remains to be better understood (overestimated $K_S^0$ - and underestimated (Λ+∑⁰) - production).

• Ξ hyperon forward rapidity shift with respect to Λ is suggested as a test of hadronic and QGP mechanisms of strangeness production in $\overline{p}A$ reactions.