Antiproton-nucleus reactions at intermediate energies
Antiproton-induced reactions on nuclei at the beam energies from hundreds MeV up to several GeV provide an excellent opportunity to study interactions of the antiproton and secondary particles (mesons, baryons and antibaryons) with nucleons. Antiproton projectile is unique in the sense that most of 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 the nuclear mean field potentials. This paper discusses the microscopic transport calculations of the antiproton-nucleus reactions and is focused on three related problems: (i) the antiproton potential determination, (ii) possible formation of strongly bound antiproton-nucleus systems, and (iii) strangeness production.
It is difficult to produce antiproton beams. However, antiproton-nucleus interactions attract 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 remain 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 , , fm Batty:1997zp (). Being extrapolated to the normal nuclear density fm, Eq.(1) predicts the repulsive antiproton-nucleus potential, MeV. In contrast, the -atomic X-ray and radiochemical data analysis Friedman:2005ad () favors the strongly attractive antiproton-nucleus potential, MeV in the nuclear center. Thus the optical potential is not a simple superposition of vacuum interactions. The strongly attractive potential is consistent with Relativistic Mean Field (RMF) models and has a consequence that a nucleus may collectively respond on the presence of an implanted antiproton. The formation of strongly bound -nuclei becomes possible Buervenich:2002ns (); Mishustin:2004xa ().
Another very interesting aspect is -annihilation in the nuclear interior. This results in a large energy deposition in the form of mesons, mostly pions, in a volume of hadronic size fm Rafelski:1979nt (); Mishustin:2004xa (). After the passage of annihilation hadrons through the nuclear medium a highly excited nuclear residue can be formed and even experience explosive multifragment breakup Rafelski:1979nt (); Cugnon:1986tx (). The annihilation of an antiproton at GeV/c on a nuclear target gives an excellent opportunity to study the interactions of secondary particles (pions Ilinov:1982jk (), kaons and hyperons Ko:1987hf (), charmonia Brodsky:1988xz (); Farrar:1988me ()) with nucleons. This is because most of annihilation hadrons are slow () and have short formation lengths. Thus their interactions are governed by usual hadronic cross sections.
Over last decades several microscopic transport models have been developed to describe particle production in interactions Ilinov:1982jk (); Cugnon:1986tx (); Cugnon:1990xw (); Strottman:1985jp (); Gibbs:1990sf (). Nowadays there is a renaissance in this field, since the antiproton-nucleus reactions at 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 Larionov:2008wy (); Larionov:2009tc (); Larionov:2011fs () and within the Lanzhou quantum molecular dynamics (LQMD) model Feng:2014cqa (); Feng_IWND2014 (). In the present paper I will report some results of GiBUU calculations for -nucleus interactions at GeV/c.
Ii GiBUU model
The GiBUU model Buss:2011mx (); GiBUU () solves a coupled set of kinetic equations for baryons, antibaryons, and mesons. In a RMF mode, this set can be written as (c.f. Refs. Ko:1987gp (); Blaettel:1993uz ())
where , , ; etc.. is the distribution function of the particles of sort normalized such that the total number of particles of this sort is
with 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 , where is a scalar field; the field tensor , where is a vector field, for and , for and ; and the kinetic four-momentum satisfying the effective mass shell condition .
In the present calculations, the nucleon-meson coupling constants and the self-interaction parameters of the -field have been adopted from a non-linear Walecka model in the NL3 parameterization Lalazissis:1996rd (). The latter gives the compressibility coefficient MeV and the nucleon effective mass at . The antinucleon-meson coupling constants have been determined as
where is a scaling factor. The choice corresponds to the -parity transformed nuclear potential. In this case, however, the Schrödinger equivalent potential
becomes unphysically deep, MeV. The empirical choice of will be discussed in the following section.
The GiBUU collision term includes the following channels111The GiBUU code is constantly developing. Thus the actual version may include more channels. This description approximately corresponds to the release 1.4.0. (notations: – nonstrange baryon, – nonstrange baryon resonance, – hyperon with , – nonstrange meson):
elastic (EL) and charge-exchange (CEX) scattering ; s-wave pion production/absorption222Implemented in a non-RMF mode only. ; ; ; ; ; ; ; .
For invariant energies GeV the inelastic production (+ mesons) is simulated via the PYTHIA model.
annihilation mesons333Described with a help of the statistical annihilation model Golubeva:1992tr (); PshenichnovPhD ().; EL and CEX scattering ; (+ c.c.); ; (+ c.c.); .
For invariant energies GeV (i.e. GeV/c for ) the inelastic production (+ mesons) is simulated via the FRITIOF model.
(baryon resonance excitations and decays, e.g. and ); ; ; ; ; ; ; ; ; ; ; ; ; (EL, CEX); (EL, CEX); ; ; .
At GeV the inelastic meson-baryon collisions are simulated via PYTHIA.
(meson resonance excitations and decays, e.g. and ); , (+ c.c.).
Iii Antiproton absorption and annihilation on nuclei
Without mean field acting on an antiproton the GiBUU model is expected to reproduce a simple Glauber model result for the -absorption cross section on a nucleus (left Fig. 1):
where is the isospin-averaged total cross section.
The attractive mean field bends the trajectory to the nucleus (right Fig. 1). Thus the absorption cross section should increase.
Fig. 2 shows the GiBUU calculations of antiproton absorption cross sections on C, Al and Cu in comparison with experimental data Nakamura:1984xw (); Abrams:1972ab (); Denisov:1973zv (); Carroll:1978hc () and with the Glauber formula (6).
Indeed, GiBUU calculations without mesonic components of the mean field, i.e. with scaling factor , are very close to Eq.(6) at GeV/c. At lower , the Coulomb potential makes the difference between GiBUU () and Glauber results. Including the mesonic components of mean field () noticeably increases the absorption cross section at GeV/c. The best fit of the KEK data Nakamura:1984xw () at MeV/c is reached with . This produces the real part of the antiproton-nucleus optical potential MeV at . The corresponding imaginary part is
At this gives MeV independent on the choice of . It is interesting that the BNL Abrams:1972ab () and Serpukhov Denisov:1973zv () data at GeV/c favor , i.e. MeV at . This discrepancy needs to be clarified which could be possibly done at FAIR.
Fig. 3 displays the calculated momentum spectra of positive pions and protons for antiproton interactions at MeV/c with the carbon and uranium targets. GiBUU very well reproduces a quite complicated shape of the pion spectra which appears due to the underlying dynamics. The absolute normalization of the spectra is weakly sensitive to the mean field. The best agreement is reached for , i.e. for MeV.
Iv Selfconsistency effects
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 -, -, and -fields:
with , 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 and , and with the same sign – to the source term of the scalar field . Hence, repulsion is reduced and attraction is enhanced in the presence of an antiproton in the nucleus.
Fig. 4 shows the density profiles of nucleons and of an antiproton at the different time moments for the case of the implanted at in the center of the Ca nucleus. As the consequence of a pure Vlasov dynamics of the coupled antiproton-nucleus system (annihilation is turned off), both the nucleon and the antiproton densities grow quite fast. At fm/c the compressed state is already formed, and the system starts to oscillate around the new equilibrium density .
Fig. 5 displays the time evolution of the central nucleon density. The annihilation is simulated at the time moment . The choice corresponds to the usual annihilation of a stopped 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 (), then the residual nuclear system expands. Eventually the system reaches the low-density spinodal region (), where the sound velocity squared becomes negative444Here, is the pressure and is the entropy per nucleon.. This should result in the breakup of the residual nuclear system into fragments.
A possible observable signal of the annihilation in a compressed nuclear configuration is the total invariant mass of emitted mesons
For the annihilation of a stopped antiproton on a proton at rest in vacuum, . In nuclear medium, the proton and antiproton vector fields largely cancel each other 555The cancellation is exact for the antiproton vector fields obtained by the -parity transformation from the respective proton vector fields, i.e. when .. Therefore, it is expected that in nuclear medium the peak will appear at . This simple picture is illustrated by GiBUU calculations in Fig. 6. In calculations with we clearly see a sharp medium-modified peak shifted downwards by MeV from . The final state interactions of mesons make a broad maximum at GeV. For annihilation in compressed configurations ( and 60 fm/c), the total spectrum further shifts by about MeV to smaller . This effect becomes stronger with decreasing mass of the target nucleus (e.g., for O the spectrum shift is nearly MeV Larionov:2008wy ()).
V Strangeness production
Originally, the main motivation of experiments on strangeness production in antiproton-nucleus collisions was to find the signs of unusual phenomena, in-particular, of a multinucleon annihilation and/or of a quark-gluon plasma (QGP) formation. In Ref. Rafelski:1988wn (), the cold QGP formation has been suggested to explain the unusually large ratio measured in the reaction Ta at 4 GeV/c Miyano:1984dc (). On the other hand, in Refs. Ko:1987hf (); Cugnon:1990xw (); Ahmad:1997fv (); Larionov:2011fs (); Gaitanos:2011fy (); Feng:2014cqa (); Feng_IWND2014 (); Gaitanos:2014bla () most features of strangeness production in reactions have been explained by hadronic mechanisms.
Fig. 7 presents the rapidity spectrum of hyperons, mesons and antihyperons for collisions Ta in comparison with the data Miyano:1988mq () and the intranuclear cascade (INC) calculations Cugnon:1990xw (). The GiBUU model underpredicts hyperon yields at small forward rapidities and overpredicts yields. In the GiBUU calculation without hyperon-nucleon scattering, the spectrum is shifted to forward rapidities. However, the problem of underpredicted total yield remains. A more detailed analysis Larionov:2011fs () shows that 72% of and production rate in GiBUU is due to antikaon absorption processes , , and . The second largest contribution, of the rate, is caused by the nonstrange meson - baryon collisions. The antibaryon-baryon (including the direct 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 cross sections, in-particular, due to the problematic channel666The channel has been improved in recent GiBUU releases, however, after the present calculations were already done.. The possible in-medium enhancement of the hyperon production in antikaon-baryon collisions is also not excluded.
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. Exception is again the region of small forward rapidities where both GiBUU and INC calculations underpredict the yield.
Finally, let us discuss the () hyperon production. The direct production of in the collision of nonstrange particles would require to produce two pairs simultaneously. Thus, production could be even stronger enhanced in a QGP as compared to the enhancement for the hyperons. Fig. 9 shows the rapidity spectra of the different strange particles in Au collisions at 15 GeV/c. Even at such a high beam momentum, the hyperon spectra still have a flat maximum at due to exothermic strangeness exchange reactions with slow . In contrast, the second largest, , contribution to the production is given by endothermic double strangeness exchange reactions 777The main, , contribution to the total yield of ’s at 15 GeV/c is given by decays. The direct channel contributes only.. Since the threshold beam momentum of for the process is GeV/c, which corresponds to the c.m. rapidity of 0.55, the rapidity spectra of ’s are shifted forward with respect to the rapidity spectra. However, in the QGP fireball scenario Rafelski:1988wn (), the rapidity spectra of all strange particles would be peaked at the same rapidity.
This work was focused on the dynamics of a coupled antiproton-nucleus system and on the strangeness production in interactions. The calculations were based on the GiBUU transport model. The main results can be summarized as:
The reproduction of experimental data on absorption cross sections at GeV/c and on and production at MeV/c requires to use a strongly attractive optical potential, MeV at .
As the response of a nucleus to the presence of an antiproton, the nucleon density can be increased up to locally near . Annihilation of the in such a compressed configuration can manifest itself in the multifragment breakup of the residual nuclear system and in the substantial ( MeV) shift of annihilation event spectrum on the total invariant mass of produced mesons toward low .
GiBUU describes the data on inclusive pion and proton production fairly well. Still, the strangeness production remains to be better understood (overestimated - and underestimated - production).
hyperon forward rapidity shift with respect to is suggested as a test of hadronic and QGP mechanisms of strangeness production in reactions.
Acknowledgements.The author is grateful to T. Gaitanos, W. Greiner, I. N. Mishustin, U. Mosel, I. A. Pshenichnov, and L. M. Satarov for the collaboration on the GiBUU studies of reactions. This work was supported by HIC for FAIR within the framework of the LOEWE program.
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