Antiproton-nucleus reactions at intermediate energies

Antiproton-nucleus reactions at intermediate energies 1 2 corresp first-author Corresponding author, larionov@fias.uni-frankfurt.de Corresponding author, larionov@fias.uni-frankfurt.de larionov@fias.uni-frankfurt.de Frankfurt Institute for Advanced Studies, Frankfurt am Main D-60438, Germany Frankfurt Institute for Advanced Studies, Frankfurt am Main D-60438, Germany National Research Centre, Kurchatov Institute, Moscow 123182, Russia National Research Centre, Kurchatov Institute, Moscow 123182, Russia Antiproton-induced reactions on nuclei at the beam energies from hundreds MeV up to several GeV provide an excellent opportunity to study interactions between the antiproton and secondary particles (mesons, baryons and antibaryons) with nucleons. The antiproton projectile is unique in the sense that most of the annihilation particles are relatively slow in the target nucleus frame. Hence, the prehadronic effects do not much influence their interactions with the nucleons of the nuclear residue. Moreover, the particles with momenta less than about 1 GeV/c are sensitive to nuclear mean field potentials. This paper discusses the microscopic transport calculations of the antiproton-nucleus reactions and is focused on three related problems: (i) antiproton potential determination, (ii) possible formation of strongly bound antiproton-nucleus systems, and (iii) strangeness production. Antiproton-nucleus reaction GiBUU model Relativistic mean field Optical potential Compressed nuclear configuration I. MOTIVATION 1 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 <math display="block" id="M1"><mrow><msub><mi>V</mi><mrow><mtext>opt</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn><mi>π</mi><msqrt><mi>s</mi></msqrt></mrow><mrow><msub><mi>E</mi><mover accent="true"><mi>p</mi><mo>¯</mo></mover></msub><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><msub><mi>f</mi><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>ρ</mi><mi/><mn>,</mn></mrow></math> where at threshold <math id="M2"><mrow><msqrt><mi>s</mi></msqrt><mo>≃</mo><mn>2</mn><msub><mi>m</mi><mi>N</mi></msub></mrow></math> , <math id="M3"><mrow><msub><mi>E</mi><mover accent="true"><mi>p</mi><mo>¯</mo></mover></msub><mo>≃</mo><msub><mi>m</mi><mi>N</mi></msub></mrow></math> , <math id="M4"><mrow><msub><mi>f</mi><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>p</mi></mrow></msub><mo>≃</mo><mo stretchy="false">(</mo><mo>−</mo><mn>0.9</mn><mo>+</mo><mi>i</mi><mn>0.9</mn><mo stretchy="false">)</mo></mrow></math> fm [ 1 1 ]. Being extrapolated to the normal nuclear density ρ 0 =0.16 fm -3 , Eq.(1) predicts the repulsive antiproton-nucleus potential, Re V opt 75 MeV. In contrast, the <math id="M5"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> -atomic X-ray and radiochemical data analysis [ 2 2 ] favors the strongly attractive antiproton-nucleus potential, Re V opt -100 MeV in the nuclear center. Thus the <math id="M6"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> optical potential is not a simple superposition of vacuum <math id="M7"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> interactions. The strongly attractive <math id="M8"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> 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 <math id="M9"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> -nuclei becomes possible [ 3 3 , 4 4 ]. Another very interesting aspect is <math id="M10"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> -annihilation in the nuclear interior. This results in a large energy deposition 2 m N in the form of mesons, mostly pions, in a volume of hadronic size 1-2 fm [ 4 4 , 5 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 5 , 6 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 7 ], kaons and hyperons [ 8 8 ], charmonia [ 9 9 , 10 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 <math id="M11"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> interactions [ 6 6 , 7 7 , 11 11 - 13 13 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 14 - 16 16 14-16 ] and within the Lanzhou quantum molecular dynamics (LQMD) model [ 17 17 , 18 18 ]. In the present paper, I will report some results of the GiBUU calculations for <math id="M12"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> -nucleus interactions at p lab 1.5-15 GeV/c. II. GIBUU MODEL 1 The GiBUU model [ 19 19 , 20 20 ] solves a coupled set of kinetic equations for baryons, antibaryons, and mesons. In a RMF mode, this set can be written as [ 21 21 , 22 22 ] <math display="block" id="M13"><mtable columnalign="left"><mtr><mtd><mo stretchy="false">(</mo><msup><mi>p</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mo>[</mo><mrow><msup><mi>p</mi><mrow><mo>*</mo><mi>μ</mi></mrow></msup><msub><mo>∂</mo><mi>μ</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>p</mi><mi>μ</mi><mo>*</mo></msubsup><msubsup><mi>F</mi><mi>j</mi><mrow><mi>α</mi><mi>μ</mi></mrow></msubsup><mo>+</mo><msubsup><mi>m</mi><mi>j</mi><mo>*</mo></msubsup><msup><mo>∂</mo><mi>α</mi></msup><msubsup><mi>m</mi><mi>j</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>p</mi><mrow><mo>*</mo><mi>α</mi></mrow></msup></mrow></mfrac></mrow><mo>]</mo></mrow><msub><mi>f</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msup><mtext>p</mtext><mo>*</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>I</mi><mi>j</mi></msub><mo stretchy="false">[</mo><mo>{</mo><mi>f</mi><mo>}</mo><mo stretchy="false">]</mo><mi/><mn>,</mn></mtd></mtr></mtable></math> where α =1,2,3, μ =0,1,2,3, x =( t , r ); <math id="M14"><mrow><mi>j</mi><mo>=</mo><mi>N</mi><mn>,</mn><mtext> </mtext><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mn>,</mn><mtext> </mtext><mi>Δ</mi><mn>,</mn><mtext> </mtext><mover accent="true"><mi>Δ</mi><mo>¯</mo></mover><mn>,</mn><mtext> </mtext><mi>Y</mi><mn>,</mn><mover accent="true"><mi>Y</mi><mo>¯</mo></mover><mn>,</mn><mtext> </mtext><mi>π</mi><mn>,</mn><mtext> </mtext><mi>K</mi><mn>,</mn><mtext> </mtext><mover accent="true"><mi>K</mi><mo>¯</mo></mover></mrow></math> etc.. f j ( x , p * ) is the distribution function of the particles of sort j normalized such that the total number of particles of this sort is <math display="block" id="M15"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext> </mtext><mfrac><mrow><msub><mi>g</mi><mi>j</mi></msub><msup><mi>d</mi><mn>3</mn></msup><mi>r</mi><msup><mi>d</mi><mn>3</mn></msup><msup><mi>p</mi><mo>*</mo></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><msub><mi>f</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msup><mtext>p</mtext><mo>*</mo></msup><mo stretchy="false">)</mo><mi/><mn>,</mn></mrow></math> with g j 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 <math id="M16"><mrow><msubsup><mi>m</mi><mi>j</mi><mo>*</mo></msubsup><mo>=</mo><msub><mi>m</mi><mi>j</mi></msub><mo>+</mo><msub><mi>S</mi><mi>j</mi></msub></mrow></math> , where S j = g σj is a scalar field; the field tensor <math id="M17"><mrow><msubsup><mi>F</mi><mi>j</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msup><mo>∂</mo><mi>μ</mi></msup><msubsup><mi>V</mi><mi>j</mi><mi>ν</mi></msubsup><mo>−</mo><msup><mo>∂</mo><mi>ν</mi></msup><msubsup><mi>V</mi><mi>j</mi><mi>μ</mi></msubsup></mrow></math> , where <math id="M18"><mrow><msubsup><mi>V</mi><mi>j</mi><mi>μ</mi></msubsup><mo>=</mo><msub><mi>g</mi><mrow><mi>ω</mi><mi>j</mi></mrow></msub><msup><mi>ω</mi><mi>μ</mi></msup><mo>+</mo><msub><mi>g</mi><mrow><mi>ρ</mi><mi>j</mi></mrow></msub><msup><mi>τ</mi><mn>3</mn></msup><msup><mi>ρ</mi><mrow><mn>3</mn><mi>μ</mi></mrow></msup><mo>+</mo><msub><mi>q</mi><mi>j</mi></msub><msup><mi>A</mi><mi>μ</mi></msup></mrow></math> is a vector field, τ 3 =+1 for p and <math id="M19"><mover accent="true"><mi>n</mi><mo>¯</mo></mover></math> , τ 3 =-1 for <math id="M20"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> and n ; and the kinetic four-momentum <math id="M21"><mrow><msup><mi>p</mi><mrow><mo>*</mo><mi>μ</mi></mrow></msup><mo>=</mo><msup><mi>p</mi><mi>μ</mi></msup><mo>−</mo><msubsup><mi>V</mi><mi>j</mi><mi>μ</mi></msubsup></mrow></math> satisfying the effective mass shell condition <math id="M22"><mrow><msup><mi>p</mi><mrow><mo>*</mo><mi>μ</mi></mrow></msup><msubsup><mi>p</mi><mi>μ</mi><mo>*</mo></msubsup><mo>=</mo><msubsup><mi>m</mi><mi>j</mi><mo>*</mo></msubsup><msup><mrow/><mn>2</mn></msup></mrow></math> . In the present calculations, the nucleon-meson coupling constants g σN , g ωN , g ρN and the self-interaction parameters of the σ -field have been adopted from a non-linear Walecka model in the NL3 parameterization [ 23 23 ]. The latter gives the compressibility coefficient K =271.76 MeV and the nucleon effective mass <math id="M23"><mrow><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><mo>=</mo><mn>0.60</mn><mtext> </mtext><msub><mi>m</mi><mi>N</mi></msub></mrow></math> at ρ = ρ 0 . The antinucleon-meson coupling constants have been determined as <math display="block" id="M24"><mrow><msub><mi>g</mi><mrow><mi>ω</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></msub><mo>=</mo><mo>−</mo><mi>ξ</mi><msub><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow></msub><mn>,</mn><mi/><msub><mi>g</mi><mrow><mi>ρ</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></msub><mo>=</mo><mi>ξ</mi><msub><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow></msub><mn>,</mn><mi/><msub><mi>g</mi><mrow><mi>σ</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></msub><mo>=</mo><mi>ξ</mi><msub><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow></msub><mi/><mn>,</mn></mrow></math> 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, <math id="M25"><mrow><msub><mi>U</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></msub><mo>=</mo><mo>−</mo><mn>660</mn><mtext>​</mtext><mtext> MeV</mtext></mrow></math> . The empirical choice of ξ will be discussed in the following section. <math display="block" id="M26"><mrow><msub><mi>U</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></msub><mo>=</mo><msub><mi>S</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></msub><mo>+</mo><msubsup><mi>V</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>V</mi><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mn>0</mn></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>m</mi><mi>N</mi></msub></mrow></mfrac></mrow></math> The GiBUU collision term 3 3 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 4 NN NNπ ; NN ΔΔ ; NN NR ; N ( Δ , N * ) N ( Δ , N * ) → N ( Δ ) YK ; YN → YN ; Ξ N → ΛΛ ; Ξ N → Λ∑ ; ΞN → ΞN . For invariant energies <math id="M27"><mrow><msqrt><mi>s</mi></msqrt><mo>></mo><mn>2.6</mn></mrow></math> GeV the inelastic production B 1 B 2 → B 3 B 4 (+ mesons) is simulated via the PYTHIA model. • Antibaryon-baryon collisions: annihilation <math id="M28"><mrow><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mi>B</mi><mo>→</mo></mrow></math> mesons 5 5 ; EL and CEX scattering <math id="M29"><mrow><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mi>B</mi><mo>→</mo><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mi>B</mi></mrow></math> ; <math id="M30"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mi>N</mi><mo>↔</mo><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mi>Δ</mi></mrow></math> (+ c.c.); <math id="M31"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> ; <math id="M32"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover accent="true"><mi>Δ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mi>N</mi><mo stretchy="false">(</mo><mover accent="true"><mi>Δ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover><mo>∑</mo></mrow></math> (+ c.c.); <math id="M33"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover accent="true"><mi>Δ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo>→</mo><mover accent="true"><mi>Ξ</mi><mo>¯</mo></mover><mi>Ξ</mi></mrow></math> . For invariant energies <math id="M34"><mrow><msqrt><mi>s</mi></msqrt><mo>></mo><mn>2.4</mn></mrow></math> GeV (i.e. p lab > 1.9 GeV/c for <math id="M35"><mrow><mover accent="true"><mi>N</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> ) the inelastic production <math id="M36"><mrow><msub><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mn>1</mn></msub><msub><mi>B</mi><mn>2</mn></msub><mo>→</mo><msub><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mn>3</mn></msub><msub><mi>B</mi><mn>4</mn></msub></mrow></math> (+ mesons) is simulated via the FRITIOF model. • Meson-baryon collisions: MN R (baryon resonance excitations and decays, e.g., πN Δ and <math id="M37"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>↔</mo><msup><mi>Y</mi><mo>*</mo></msup></mrow></math> ); π ( ρ ) Δ R ; πN → πN ; πN → ππN ; πN → ηΔ ; πN → ωN ; πN → ϕN ; πN → ωπN ; πN → ϕπN ; π ( η , ρ , ω ) N → YK ; <math id="M38"><mrow><mi>π</mi><mi>N</mi><mo>→</mo><mi>K</mi><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> ; πN → YKπ ; πΔ → YK ; <math id="M39"><mrow><mi>K</mi><mi>N</mi><mo>→</mo><mi>K</mi><mi>N</mi></mrow></math> (EL, CEX); <math id="M40"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> (EL, CEX); <math id="M41"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>↔</mo><mi>Y</mi><mi>π</mi></mrow></math> ; <math id="M42"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>↔</mo><msup><mi>Y</mi><mo>*</mo></msup><mi>π</mi></mrow></math> ; <math id="M43"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mi>Ξ</mi><mi>K</mi></mrow></math> . At <math id="M44"><mrow><msqrt><mi>s</mi></msqrt><mo>></mo><mn>2.2</mn></mrow></math> GeV the inelastic meson-baryon collisions are simulated via PYTHIA. • Meson-meson collisions: M 1 M 2 M 3 (meson resonance excitations and decays, e.g., ππ ρ and Kπ K * ); <math id="M45"><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>M</mi><mn>2</mn></msub><mo>↔</mo><mi>K</mi><mover accent="true"><mi>K</mi><mo stretchy="true">¯</mo></mover></mrow></math> , <math id="M46"><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>M</mi><mn>2</mn></msub><mo>↔</mo><mi>K</mi><msup><mrow><mover accent="true"><mi>K</mi><mo stretchy="true">¯</mo></mover></mrow><mo>*</mo></msup></mrow></math> (+ c.c.). III. ANTIPROTON ABSORPTION AND ANNIHILATION ON NUCLEI 1 Without a mean field acting on an antiproton, the GiBUU model is expected to reproduce a simple Glauber model result for the <math id="M47"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> -absorption cross section on a nucleus (left, Fig. 1 Fig. 1 ): <math display="block" id="M48"><mrow><msubsup><mi>σ</mi><mrow><mtext>abs</mtext></mrow><mrow><mtext>Glauber</mtext></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>2</mn></msup><mi>b</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mtext> </mtext><mi>e</mi><msup><mtext> </mtext><mrow><mo>−</mo><msub><mrow><mover accent="true"><mi>σ</mi><mo stretchy="true">¯</mo></mover></mrow><mrow><mtext>tot</mtext></mrow></msub><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mrow><mo>+</mo><mi>∞</mi></mrow></munderover></mrow></mstyle><mtext> </mtext><mi>d</mi><mi>z</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>b</mi><mn>,</mn><mi>z</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mo>)</mo></mrow><mi/><mn>,</mn></mrow></math> where <math id="M49"><mrow><msub><mrow><mover accent="true"><mi>σ</mi><mo stretchy="true">¯</mo></mover></mrow><mrow><mtext>tot</mtext></mrow></msub></mrow></math> is the isospin-averaged total <math id="M50"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> cross section. The attractive mean field bends the <math id="M51"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> trajectory to the nucleus (right, Fig. 1 Fig. 1 ). Thus, the absorption cross section should increase. Figure 2 Figure 2 shows the GiBUU calculations of antiproton absorption cross sections on 12 C, 27 Al and 64 Cu in comparison with experimental data [ 26 26 - 29 29 26-29 ] and with the Glauber formula (6). Indeed, GiBUU calculations without mesonic components of the <math id="M52"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 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 <math id="M53"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> mean field, ( ξ > 0) noticeably increases the absorption cross section at p lab < 3 GeV/c. The best fit of the KEK data [ 26 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 <math id="M54"><mrow><mtext> </mtext><mtext>Re</mtext><mtext> </mtext><msub><mi>V</mi><mrow><mtext>opt</mtext></mrow></msub><mo>≡</mo><msub><mi>U</mi><mover accent="true"><mi>p</mi><mo>¯</mo></mover></msub><mo>≃</mo><mo>−</mo><mo stretchy="false">(</mo><mn>150</mn><mo>±</mo><mn>30</mn><mo stretchy="false">)</mo></mrow></math> MeV at ρ = ρ 0 . The corresponding imaginary part is <math display="block" id="M56"><mrow><mtext> </mtext><mtext>Im</mtext><mtext> </mtext><msub><mi>V</mi><mrow><mtext>opt</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo><</mo><msub><mi>v</mi><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>N</mi></mrow></msub><msub><mrow><mover accent="true"><mi>σ</mi><mo stretchy="true">¯</mo></mover></mrow><mrow><mtext>tot</mtext></mrow></msub><mo>></mo><mi>ρ</mi><mi/><mn>.</mn></mrow></math> At ρ = ρ 0 this gives Im V opt -(100-110) MeV independent on the choice of ξ . It is interesting that the BNL [ 27 27 ] and Serpukhov [ 28 28 ] data at p lab =1.6-20 GeV/c favor ξ =1, i.e. Re V opt -660 MeV at ρ = ρ 0 . This discrepancy needs to be clarified, which could be possibly done at FAIR. Figure 3 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 <math id="M57"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> mean field. The best agreement is reached for ξ =0.3, i.e., for Re V opt -(220±70) MeV. IV. SELFCONSISTENCY EFFECTS 1 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 <math display="block" id="M59"><mrow><mo stretchy="false">(</mo><msub><mo>∂</mo><mi>μ</mi></msub><msup><mo>∂</mo><mi>μ</mi></msup><mo>+</mo><msubsup><mi>m</mi><mi>ω</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><msup><mi>ω</mi><mi>ν</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>N</mi><mn>,</mn><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></munder></mstyle><msub><mi>g</mi><mrow><mi>ω</mi><mi>j</mi></mrow></msub><mo>〈</mo><msub><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>γ</mi><mi>ν</mi></msup><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>〉</mo><mn>,</mn></mrow></math> <math display="block" id="M60"><mrow><mo stretchy="false">(</mo><msub><mo>∂</mo><mi>μ</mi></msub><msup><mo>∂</mo><mi>μ</mi></msup><mo>+</mo><msubsup><mi>m</mi><mi>ρ</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><msup><mi>ρ</mi><mrow><mn>3</mn><mtext> </mtext><mi>ν</mi></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>N</mi><mn>,</mn><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></munder></mstyle><msub><mi>g</mi><mrow><mi>ρ</mi><mi>j</mi></mrow></msub><mo>〈</mo><msub><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>γ</mi><mi>ν</mi></msup><msup><mi>τ</mi><mn>3</mn></msup><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>〉</mo><mn>,</mn></mrow></math> <math display="block" id="M61"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msup><mo>∂</mo><mi>μ</mi></msup><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>d</mi><mi>U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>σ</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>N</mi><mn>,</mn><mover accent="true"><mi>N</mi><mo>¯</mo></mover></mrow></munder></mstyle><msub><mi>g</mi><mrow><mi>σ</mi><mi>j</mi></mrow></msub><mo>〈</mo><msub><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>〉</mo><mn>,</mn></mrow></math> with <math id="M62"><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>m</mi><mi>σ</mi><mn>2</mn></msubsup><msup><mi>σ</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi>g</mi><mn>2</mn></msub><msup><mi>σ</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>g</mi><mn>3</mn></msub><msup><mi>σ</mi><mn>4</mn></msup></mrow></math> , 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 Figure 4 shows the density profiles of nucleons and an antiproton at different times when the <math id="M63"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> implanted at t =0 in the center of the 40 Ca 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 2 ρ 0 . Figure 5 Figure 5 displays the time evolution of the central nucleon density. The <math id="M64"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> annihilation is simulated at the time t ann . The t ann =0 corresponds to the usual annihilation of a stopped <math id="M65"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 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 < 0.6 ρ 0 ), where the sound velocity squared <math id="M66"><mrow><msubsup><mi>c</mi><mi>s</mi><mn>2</mn></msubsup><mo>=</mo><mo>∂</mo><mi>P</mi><mo>/</mo><mo>∂</mo><msub><mi>ρ</mi><mrow><mo>|</mo><mi>s</mi><mo>=</mo><mtext> </mtext><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mtext> </mtext></mrow></msub></mrow></math> becomes negative 6 6 . This should result in the breakup of the residual nuclear system into fragments. A possible observable signal of the <math id="M69"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> annihilation in a compressed nuclear configuration is the total invariant mass M inv of emitted mesons <math display="block" id="M70"><mrow><msubsup><mi>M</mi><mrow><mtext>i</mtext><mi>n</mi><mi>v</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><msub><mi>p</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mi/><mn>.</mn></mrow></math> For the annihilation of a stopped antiproton on a proton at rest in a vacuum, M inv =2 m N . In a nuclear medium, the proton and antiproton vector fields largely cancel each other 7 7 . Therefore, it is expected that in nuclear medium the peak will appear at M inv 2 m N * . This simple picture is illustrated by GiBUU calculations in Fig. 6 Fig. 6 . In calculations where t ann =0, we clearly see a sharp medium-modified peak shifted downwards by 200 MeV from 2 m N . 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 16 O the spectrum shift is nearly 500 MeV [ 14 14 ]). V. STRANGENESS PRODUCTION 1 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 31 ], the cold QGP formation has been suggested to explain the unusually large ratio <math id="M71"><mrow><mi>Λ</mi><mo>/</mo><msubsup><mi>K</mi><mi>S</mi><mn>0</mn></msubsup><mo>≃</mo><mn>2.4</mn></mrow></math> measured in the reaction <math id="M72"><mrow><msup><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mrow><mn>181</mn></mrow></msup></mrow></math> Ta at 4 GeV/c [ 32 32 ]. On the other hand, in Refs. [ 8 8 , 11 11 , 16 16 - 18 18 16-18 , 33 33 - 35 35 33-35 ] most features of strangeness production in <math id="M73"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> reactions have been explained by hadronic mechanisms. Figure 7 Figure 7 presents the rapidity spectrum of ( Λ + ∑ 0 ) hyperons, <math id="M74"><mrow><msubsup><mi>K</mi><mi>S</mi><mn>0</mn></msubsup></mrow></math> mesons and <math id="M75"><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover><mo>+</mo><msup><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mn>0</mn></msup><mo stretchy="false">)</mo></mrow></math> antihyperons for the collisions <math id="M76"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><msup><mrow><mo stretchy="false">(</mo><mtext> </mtext><mn>4</mn><mtext> </mtext><mi>G</mi><mi>e</mi><mi>V</mi><mo>/</mo><mi>c</mi><mtext> </mtext><mo stretchy="false">)</mo></mrow><mrow><mn>181</mn></mrow></msup></mrow></math> Ta in comparison with the data [ 36 36 ] and the intranuclear cascade (INC) calculations [ 11 11 ]. The GiBUU model underpredicts hyperon yields at small forward rapidities y 0.5 and overpredicts <math id="M77"><mrow><msubsup><mi>K</mi><mi>S</mi><mn>0</mn></msubsup></mrow></math> 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 16 ] shows that 72% of Y and Y * production rates in GiBUU are due to the antikaon absorption processes <math id="M78"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>B</mi><mo>→</mo><mi>Y</mi><mi>X</mi></mrow></math> , <math id="M79"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>B</mi><mo>→</mo><msup><mi>Y</mi><mo>*</mo></msup></mrow></math> , and <math id="M80"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>B</mi><mo>→</mo><msup><mi>Y</mi><mo>*</mo></msup><mi>π</mi></mrow></math> . The second largest contribution, 23The antibaryon-baryon (including the direct <math id="M81"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> 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 <math id="M82"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> cross sections, in-particular, due to the problematic K - n channel 8 8 . The possible in-medium enhancement of the hyperon production in antikaon-baryon collisions is also not excluded. As shown in Fig. 8 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. 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 <math id="M89"><mrow><mi>s</mi><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></math> 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 Fig. 9 shows the rapidity spectra of the different strange particles in <math id="M90"><mrow><msup><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mrow><mn>197</mn></mrow></msup></mrow></math> 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 <math id="M91"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mi>Y</mi><mi>π</mi></mrow></math> with slow <math id="M92"><mover accent="true"><mi>K</mi><mo>¯</mo></mover></math> . In contrast, the second largest, 18%, contribution to the Ξ production is given by endothermic double strangeness exchange reactions <math id="M93"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mi>Ξ</mi><mi>K</mi></mrow></math> 9 9 . Since the threshold beam momentum of <math id="M94"><mover accent="true"><mi>K</mi><mo>¯</mo></mover></math> for the process <math id="M95"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi><mo>→</mo><mi>Ξ</mi><mi>K</mi></mrow></math> is 1.05 GeV/c, which corresponds to the <math id="M96"><mrow><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mi>N</mi></mrow></math> 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 31 ], the rapidity spectra of all strange particles would be peaked at the same rapidity. VI. SUMMARY 1 This work was focused on the dynamics of a coupled antiproton-nucleus system and the strangeness production in <math id="M100"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> interactions. The calculations were based on the GiBUU transport model. The main results can be summarized as: • The reproduction of experimental data on <math id="M101"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> 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 <math id="M102"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> optical potential, Re V opt -(150-200) MeV at ρ = ρ 0 . • As a response of a nucleus to the presence of an antiproton, the nucleon density can be increased up to ρ (2-3) ρ 0 locally near <math id="M103"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> . Annihilation of the <math id="M104"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 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 <math id="M105"><mrow><msubsup><mi>K</mi><mi>S</mi><mn>0</mn></msubsup></mrow></math> - 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 <math id="M106"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> reactions. The GiBUU code is constantly developing. Thus the actual version may include more channels. 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DOI: 10.1103/PhysRevC.38.2788 http://doi.org/10.1103/PhysRevC.38.2788 Neutral strange particle production and inelastic cross section in p¯+Ta reaction at 4 GeV/c The author is grateful to Gaitanos T, Greiner W, Mishustin I N, Mosel U, Pshenichnov I A, and Satarov L M for the collaboration on the GiBUU studies of <math id="M107"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>A</mi></mrow></math> reactions. 10.13538/j.1001-8042/nst.26.S20506 Supported by HIC for FAIR within the framework of the LOEWE program 2015-01-03 2015-01-19 2015-04-20 2015-04-20 2021-04-11 2015 Nuclear Science and Techniques 2 26