---
abstract: |
    We have used the event generator LUCIAE to analyse NA35 data of the $\bar p$ and $\bar{\Lambda}$ yields, the ratio $\bar{\Lambda}$/$\bar p$, and the transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ in pp and central sulphur-nucleus collisions at 200A GeV. The NA35 data could be reproduced reasonably if one assumes that the s quark pair suppression factor and concerns in nucleus-nucleus collisions are larger than the nucleon-nucleon collision. It seems to indicate that NA35 data might imply the reduction of strangeness suppression in ultrarelativistic nucleus-nucleus collisions comparing to the nucleon-nucleon collision at the same energy. However, the ratio $\bar{\Lambda}$/$\bar p$ approaching to unity in AA collisions comparing to the pp collision does not necessarily mean a flavor symmetry since hadronic rescattering plays a role as well.

    PACS number: 25.75.+r
---

**[Strange Antibaryon Production Data and Reduction of Strangeness Suppression in Sulphur-Nucleus Collisions at 200A GeV]{}\
Sa Ben-Hao$^{1,2,3,5}$ and Tai An$^4$\
**

ttttt = tt = 1. CCAST (World Lab.), P. O. Box 8730 Beijing, China.\
2. China Institute of Atomic Energy, P. O. Box 275 (18),\
Beijing, 102413 China.\
3. Department of Physics, University of Hiroshima,\
Higashi-Hiroshima, 739 Japan.\
4. Institute of High Energy Physics, Academia Sinica,\
P. O. Box 918, Beijing, 100039 China.\
5. Institute of Theoretical Physics, Academia Sinica,\
Beijing China.\

0.8cm =0.3cm =0.3cm Strangeness production is expected to be a powerful probe for the mechanism of nucleus-nucleus collisions. Strangeness enhancement, the increased strangeness particle production in nucleus-nucleus collisions comparing to the nucleon-nucleon collision, is predicted to be a sensitive signature of the QGP formation in the ultrarelativistic nucleus-nucleus collisions \[1\].

The first experimental results of the enhanced production of strange particles in nucleus-nucleus collisions at 200A GeV incident energy was reported six years ago \[2\]. Later on this enhancement was confirmed by more and more experiments \[3-7\].

Strange antibaryon production, in particular, might bring messages of the equilibrium and flavour symmetry of quarks in ultrarelativistic nucleus-nucleus collisions. It has been estimated \[8\] that if there is QGP formed in the ultrarelativistic nucleus-nucleus collisions, strange (antistrange) quark pair might be copiously reproduced, resulting in an approximate flavour symmetry among u, d and s quarks \[9\]. Thus the ratio of $\bar{\Lambda}$/$\bar p$ should approach unity, since $\bar{\Lambda}$ is composed of $\bar u\bar d\bar s$ and $\bar p$ is $\bar u\bar d\bar d$.

Recently published NA35 data of antibaryon production in sulphur-nucleus collisions at 200A GeV \[7\] did really observe that the ratios $\bar{\Lambda}$ /$\bar p$ are all approaching, even over, unity and far exceeding the corresponding value of $\sim$0.25 in the nucleon-nucleon collision at the same energy. Although the NA35 data of the $\bar{\Lambda}$/$\bar p$ has been fairly reproduced by RQMD 1.08 \[10,7\] with fusion of overlpping strings into a color rope and with hadronic rescattering, the data of transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ have not been explained yet, the physics behind the data have not been exposed especially.

Based on the idea that the strangeness enhancement in ultrarelativistic nucleus-nucleus collision compared to the nucleon-nucleon collision at the same energy could be investigated via the reduction of s quark suppression \[11-14\], in this letter the event generator LUCIAE \[15\] is used to analyse the NA35 data and to explore the physics behind the data.

LUCIAE was updated based on FRITIOF 7.02 \[16\] by taking into account the collective string interaction \[17-18\] during the emission of gluon bremsstrahlung and the rescattering of produced particles \[19\]. One knows that in FRITIOF 7.02 \[16\], two colliding hadrons are excited due to the longitudinal momentum transfer and/or a Rutherford parton scattering. The highly excited states will emit bremsstrahlung gluons according to the soft radiation model. The deexcited states are then treated as Lund strings allowing to decay into the final hadronic state due to the Lund fragmentation scheme. However, in the ultrarelativistic nucleus-nucleus collisions there are generally many excited strings formed close by each other. These strings would behave like vortex lines in a color superconducting QCD vacuum and will interact with each other (as the repulsive interaction suffered by the “ordinary” vortex lines in a type II superconductor). This kind of collective interaction among strings, not included in FRITIOF 7.02, is depicted by the firecracker model \[15,18\] dealing with the large p$_t$ gluon (firecracker gluon) production from the collective interaction among strings. The firecracker gluon will work as a gluon-kink excitation on a string, which tends to shorten the longitudinal size of the string and then brings about the increasing of the string tension (thus it should be regarded as an effective string tension). In firecracker model it is assumed that the groups of neighboring strings might form interacting quantum states, the large common energy density (corresponding to the collective interaction among strings) might then affect the emission of gluonic bremsstrahlung \[17-18\].

A rescattering model has been developed to describe the reinteraction of produced particles, from FRITIOF event generator, with each other and with the participant and the spectator nucleons \[19\]. In the model, the produced particles and the participant (wounded) nucleons, from FRITIOF event generator, are distributed randomly in the geometrical overlapping region between the projectile and the target nuclei under a given impact parameter. The target (projectile) spectator nucleons are distributed randomly outside the overlapping region and inside the target (projectile) sphere. A rescattering cascade process has evolved since then, cf. Ref. \[15,19\] for the detail. The considered inelastic reactions, concerning strangeness, are here cataloged into:

tttttttttttttttttttttttttttttttttttttttttttttttttttt= $\pi\pi \rightleftharpoons k\bar{k}$;\
$\pi N \rightleftharpoons kY$, $\pi\bar{N} \rightleftharpoons  \bar{k}\bar{Y}$;\
$\pi Y  \rightleftharpoons k\Xi$, $\pi\bar{Y}  \rightleftharpoons  \bar{k}\bar{\Xi}$;\
$\bar{k}N  \rightleftharpoons  \pi Y$ , $k\bar{N}  \rightleftharpoons  \pi\bar{Y}$;\
$\bar{k}Y  \rightleftharpoons  \pi\Xi$, $k\bar{Y}  \rightleftharpoons  \pi\bar{\Xi}$;\
$\bar{k}N  \rightleftharpoons  k\Xi$, $k\bar{N}  \rightleftharpoons  \bar{k}\bar{\Xi}$;\
$\pi\Xi \rightleftharpoons k\Omega^- $, $\pi\bar{\Xi} \rightleftharpoons  \bar{k}\overline{\Omega^-}$;\
$k\bar{\Xi} \rightleftharpoons \pi\overline{\Omega^-}$, $\bar{k}\Xi \rightleftharpoons \pi\Omega^-$;\
$\bar{N}N$ annihilation;\
$\bar{Y}N$ annihilation;\

where $Y$ refers to the $\Lambda$ or $\Sigma$ and $\Xi$ refers to the $\Xi^-$ or $\Xi^0$. There are 299 inelastic reactions involved altogether. As the reactions introduced above do not make up the full inelastic cross section, the remainder is again treated as elastic scattering \[19\]. The cross section of $\pi\pi \rightarrow k\bar{k}$ is taken to be 2.0 mb as usual \[10\]. The isospin averaged parametrization of Ref. \[9\] is adopted for the cross sections of the reactions $\pi N \rightarrow kY$ and for the other strange quark production reactions. Of course, the difference in threshold energy among reactions is taken into account. Following Ref. \[9\], the cross section of strange quark exchange reaction, $\bar{k}N\rightarrow\pi Y
$ for instance, is assumed to be equal to ten times the value of the cross section of the strangeness production reaction. As for the cross section of the reverse reaction, the detailed balance assumption \[20\] is required.

The cross sections of the inelastic reactions given by the isospin averaged parameterization formulas of \[9\] for the $\pi N \rightarrow kY$ decrease exponentially with the CMS energy of the two colliding particles. But we know that the total inelastic cross section of the $\pi N$ is approximately energy -independent, which means that more inelastic channels would occur like $\pi^-$ + p $\rightarrow K^{*0}$ + $\Lambda \rightarrow$ K + $\pi + \Lambda$ and $\pi^-$ + p $\rightarrow K^{*0}$ + $\Sigma^0 \rightarrow$ K + $\pi + \Sigma^0$ etc., when the CMS energy of the two colliding particles is increasing. In some sense the reactions we listed above should be looked upon as ’representative channels’ of certain types of reactions. Therefore instead of applying energy-dependent inelastic cross section, we could alternatively give a constant ’effective cross section’ to the reaction $\pi N \rightarrow kY$ \[15,19\].

In JETSET routine, which runs together with LUCIAE event generator, there are model parameters parj(2) (or ’s’) and parj(3) which are responsible for the s quark suppression and related to the effective string tension. ’s’ refers to the suppression of s quark pair production in the color field compared to u or d pair production. parj(3) is the extra suppression of strange diquark production compared to the normal suppression of strange quark pair. Besides ’s’ and parj(3) there is parj(1), which stands for the suppression of diquark- antidiquark pair production in the color field compared to the quark-antiquark pair production and is related to the effective string tension as well. Originally, in the LUND fragmentation scheme ’s’ was assumed to be a ’constant’ and this assumption was confirmed by the e$^+$e$^-$ physics from the low energies to Z$^0$ energy. However, there are experimental facts that this parameter is energy dependent when the fragmentation scheme is applied to the Deep Inelastic Scattering (DIS) and hh collisions. In DIS experiments, ’s’ is seen to be rising up with the increase of energy from 0.15 at $\sqrt{S}$=5 GeV to 0.35 at $\sqrt{S}$=20 GeV \[21\]. An energy dependent ’s’ for hh collisions has been known for many years. It varies from 0.2 at the ISR energy to about 0.4 at $\sqrt{S}$= 1.8 TeV \[21\]. In addition, this parameter is also observed strongly phase space dependence in DIS and hh collisions. The corresponding value of ’s’ runs from 0.15 to 0.55 with a mean value close to 0.3 \[22\]. Thus the idea of the reduction of s quark suppression in nucleus-nucleus collisions comparing to the nucleon-nucleon collision is possible to be executed via changing the ’s’ and concerning parameters in JETSET routine.

From LUND string model point of view s quark suppression factor could be related to the effective string tension. It is reasonable to expected that the more violent collision, the stronger collective interaction among strings, the larger effective string tension and then the larger s quark suppression factor. Generally speaking, a nucleus-nucleus collision is more violent than the nucleon-nucleon collision at the same interaction energy. That might be the reason of the reduction of s quark suppression in nucleus- nucleus collisions comparing to the nucleon-nucleon collision at the same energy.

The purpose of this letter is to explore the physics behind the NA35 data and not to fit the data as good as possible. We fix a set of somewhat larger parameters (relative to the defaults: ’s’=0.3, parj(3)=0.4 and parj(1)=0.1) of ’s’=0.4, parj(3)=0.5333 and parj(1)=0.1333 (referred to as parameter set 1, later on) in the calculations of nucleus-nucleus collisions and a set of somewhat smaller parameters of ’s’=0.2, parj(3)=0.2666 and parj(1)=0.06666 ( referred to as parameter set 2) in the calculation of the nucleon-nucleon collision . These results are given in table 1 and figure 1 and obtained from the average over 10$^5$ generated events for pp, 2000 events for $\bar{\Lambda}$ and $\bar{p}$ yield in AA, and 3000 for m$_t$ distribution in AA. The y and p$_t$ acceptances are set to be the same as the experiment \[7\], correspondingly. Of course, the two sets of parameters above imply that one makes an assumption there that those three parameters are linearly proportional to the effective string tension in their responsibilities to the final hadronic production.

Tab. 1 gives the rapidity densities of $\bar{p}$, $\bar{\Lambda}$, and $h^-$ and the ratio $\bar{\Lambda}$/$\bar{p}$ in p + p and S + S, S + Ag, and S + Au central collisions at 200A GeV. The rapidity acceptance is 3 $\leq$ y $\leq$ 4. The results of LUCIAE are comparable with the corresponding NA35 data. The LUCIAE results of $\bar{\Lambda}$ /$\bar{p}$ seem going down monotonously from S + S to S + Au, which might attribute to the fact that one did not consider the possible difference of the reduction of strangeness suppression among them. The results of LUCIAE and RQMD are comparable with each other, since both of them do have contained the collective interaction among strings and the hadronic rescattering \[23\]. The hadronic rescattering, in LUCIAE and RQMD, is expected to play a similar role in the final state distributions though details of the hadronic rescattering are not the same in these two generators. In RQMD the effect of highly dense strings in ultrarelativistic nucleus-nucleus collisions is considered via the fusion of strings into a color rope, which brings the enhanced production of strange antibaryon. In LUCIAE the highly dense strings are considered by the firecracker model \[15,18\], where the effect of highly dense strings is depicted analytically only in the emission of gluon bremsstrahlung. It leaves the possibility to enlarge the s quark pair suppression factor and concerns in JETSET routine to bring the enhanced production of strange antibaryon.

Fig. 1 gives $\bar{p}$ and $\bar{\Lambda}$ transverse mass distributions in central S + S (upper frame), S + Ag (middle frame), and S + Au (lower frame) reactions at 200A GeV, respectively. In this figure, the open squares and open circles are the NA35 data of $\bar{p}$ and $\bar{\Lambda}$, respectively and the corresponding results of LUCIAE are given by full squares and full circles. The solid lines are the exponential fits of the NA35 data, cf. Ref. 7 for details. One sees from this figure that the agreement between the NA35 data and the results of LUCIAE is fair for $\bar{p}$ and reasonably good for $\bar{\Lambda}$.

Table 2 gives the average yield (in full phase space) of $\Lambda$, $\bar{\Lambda}$, K$^+$ and K$^0_s$, in p+p and central S + S and S + Ag collisions at 200A GeV. In this table the data were taken from \[5\] besides p+p, which was taken from \[24\]. One sees from this table that although we are here aiming at analysing strange antibaryon production data, the strange particle production data are also reproduced reasonably good at the same time.

In order to distinguish the role of the s quark pair suppression factor and concerns from the hadronic rescattering and the role of ’s’ and parj(3) from parj(1) one calculates the results of table 3 and figure 2. In table 3 one compares the results of the rapidity densities of $\bar{p}$, $\bar{\Lambda}$, and $h^-$ and the ratio $\bar{\Lambda}$/$\bar{p}$ in S + S central collisions at 200A GeV calculated by using different parameters and with or without rescattering: ’LUCIAE 1’: parameter set 2 and without rescattering; ’LUCIAE 2’: parameter set 2 and with rescattering; ’LUCIAE 3’: parameter set 1 but the value of parj(1) is changed to 0.06666 and with rescattering; and ’LUCIAE 4’: parameter set 1 and with rescattering. One sees from this table that the ’LUCIAE 1’ result of $\bar{\Lambda}$/$\bar{p}$ is close to the corresponding result in the pp collision, as it should be. For results of ’LUCIAE 2’, $\bar{\Lambda}$ yield is close to the corresponding result of ’LUCIAE 1’ but $\bar{p}$ yield is lower than ’LUCIAE 1’, which brings about the larger ratio $\bar{\Lambda}$/$\bar{p}$ in ’LUCIAE 2’ than in ’LUCIAE 1’. Here one knows that the hadronic rescattering seems to play nearly null role for ${\bar\Lambda}$ yield, since ${\bar\Lambda}$ production in rescattering is mainly via $\pi\bar{N} \rightleftharpoons\bar{k}\bar{Y}$ and $k\bar{N} \rightleftharpoons \pi\bar{Y}$, that is nearly canceled by the corresponding inverse reactions and $\bar{Y}N$ annihilations especially. On the contrary, the rescattering tends to reduce $\bar{p}$ multiplicity through the $\bar{p}p$ annihilation. However, relying on hadronic rescattering only is not possible to have the ratio $\bar{\Lambda}$/$\bar{p}$ approaching to unity. Furthermore, comparing the results of ’LUCIAE 3’ with ’LUCIAE 4’ one knows that although by increasing parj(1) the yield of $\bar{p}$ and $\bar{\Lambda}$ are increased the ratio $\bar{\Lambda}$/$\bar{p}$ is hardly affected.

Figure 2 gives the transverse mass distributions of ${\bar\Lambda}$ (upper frame) and $\bar{p}$ (lower frame) in S + S reaction at 200A GeV, respectively. The full triangles, the open squares, and the full circles are calculated individually for the case 1: parameter set 2 and with rescattering; the case 2: parameter set 1 and without rescattering; and the case 3: parameter set 1 and with rescattering. The rapidity acceptance are all set to be 1 $\leq$ y $\leq$ 3. One knows from this figure again that the hadronic rescattering seems to play nearly null role for ${\bar\Lambda}$ production, but hadronic rescattering tends to reduce $\bar{p}$ multiplicity, at lower m$_t$ region especially, through the $\bar{p}p$ annihilation.

In summary, we have roughly reproduced the NA35 data of $\bar p$, $\bar{\Lambda}$ yields , the corresponding ratio $\bar{\Lambda}$/$\bar p$, and the transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ in pp and central S + S, S + Ag, and S + Au collisions at 200A GeV, using event generator LUCIAE and via increasing s quark pair suppression factor and concerns in AA collisions. It seems to be true that the NA35 data \[5,7\] imply the reduction of strangeness suppression in ultrarelativistic nucleus-nucleus collisions comparing to the pp collision at the same energy. Although the ratio $\bar{\Lambda}$/$\bar p$ approaching to unity is the result of reduction of s quark suppression, it does not necessarily mean a flavor symmetry, since reproducing the NA35 data does not require the ’s’ value should be equal to one and the hadronic rescattering also plays a role in enlarging the ratio $\bar{\Lambda}$/$\bar p$. However, it is absolutely needed to have a further study for the microscopic mechanism of the reduction of strangeness suppression before making a conclusion using strangeness enhancement as a signal of QGP.

Acknowledgment

We are grateful to J. Eschke and D. R$\ddot{o}$hrich for providing the NA35 data. Thanks go to O. Miyamura, K. Kumagai and T. Sasaki for discussions and helps. SBH thanks Department of Physics, University of Hiroshima for hospitality and JSPS for financial support staying in Japan to finish most of the calculations. This work is supported by the national Natural Science Foundation of China as a cooperation program between NSFC of China and JSPS of Japan.

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Figure Captions

> Fig. 1 Transverse mass distributions of $\bar p$ (3 $\leq$ y $\leq$ 4) and $\bar\Lambda$ produced in central S+S (upper frame, 1 $\leq$ y $\leq$ 3), S+Ag (middle frame, 1 $\leq$ y $\leq$ 3), and S+Au (lower frame, 3 $\leq$ y $\leq$ 5) collisions at 200A GeV. The open squares and open circles are the NA35 data of $\bar p$ and $\bar\Lambda$, respectively and the full squares and full circles are the corresponding results of LUCIAE.
>
> Fig. 2 Transverse mass distributions (1 $\leq$ y $\leq$ 3) of $\bar\Lambda$ (upper frame) and $\bar p$ (lower frame) in S+S reaction at 200A GeV. The full triangles, the open squares, and the full circles are calculated individually for the case 1: parameter set 2 (’s’=0.2, parj(3)=0.2666 and parj(1)=0.06666) and with rescattering; the case 2: parameter set 1 (’s’=0.4, parj(3)=0.5333, parj(1)=0.1333) and without rescattering; and the case 3: parameter set 1 and with rescattering.

[cccccc]{}\
\
reaction & &$\bar p$ & $\bar\Lambda$ & $\bar\Lambda$/$\bar p$ &h$^-$\
p + p& data & 0.02$\pm$0.02 & 0.005$\pm$0.002 & 0.25$\pm$0.1 & 0.74$\pm$0.04\
&LUCIAE & 0.017 & 0.0035 & 0.21 & 0.60\
&RQMD & 0.015 &0.005 & 0.3 & $-$\
S + S& data & 0.4$\pm$0.1 & 0.76$\pm$0.16 & 1.9$^{+0.7}_{-0.6}$ &25$\pm$1\
central &LUCIAE & 0.65 & 0.66 & 1.03 & 23.7\
&RQMD & 0.7 & 0.75 & 1.1 & $-$\
S + Ag& data & 0.6$\pm$0.2 & 0.75$\pm$0.19 & 1.3$^{+0.7}_{-0.5}$ & 40$\pm$2\
central&LUCIAE & 1.08 & 0.99 & 0.91 & 39.3\
&RQMD & 1.0 & 0.9 & 0.9 & $-$\
S + Au& data & 0.7$\pm$0.2 & 0.75$\pm$0.1 & 1.1$^{+0.4}_{-0.3}$ & 47$\pm$5\
central&LUCIAE & 1.09 & 0.91 & 0.84 & 42.7\
&RQMD & 1.4 & 1.2 & 0.9 & $-$\

[cccccc]{}\
\
reaction & & $\Lambda$ & $\bar\Lambda$ & K$^+$ & K$^0_s$\
p + p& data & 0.096$\pm$0.015 & 0.013$\pm$0.005 & $-$ & 0.17$\pm$0.01\
& LUCIAE & 0.10 & 0.011 & 0.22 & 0.16\
S + S& data & 9.4$\pm$1.0 & 2.2$\pm$0.4 & $-$ & 10.5$\pm$1.7\
central & LUCIAE & 8.0 & 1.9 & 11.9 & 9.83\
S + Ag& data & 15.2$\pm$1.2 & 2.6$\pm$0.3 & $-$ & 15.5$\pm$1.5\
central & LUCIAE & 13.9 & 3.1 & 21.3 & 17.6\

[cccccc]{}\
\
reaction & &$\bar p$ & $\bar\Lambda$ & $\bar\Lambda$/$\bar p$ &h$^-$\
p + p& data & 0.02$\pm$0.02 & 0.005$\pm$0.002 & 0.25$\pm$0.1 & 0.74$\pm$0.04\
&LUCIAE & 0.017 & 0.0035 & 0.21 & 0.60\
&RQMD & 0.015 &0.005 & 0.3 & $-$\
S + S& data & 0.4$\pm$0.1 & 0.76$\pm$0.16 & 1.9$^{+0.7}_{-0.6}$ &25$\pm$1\
central &LUCIAE 1$^a$ & 0.62 & 0.17 & 0.28 & 25.2\
&LUCIAE 2$^b$ & 0.42 & 0.17 & 0.40 & 24.8\
&LUCIAE 3$^c$ & 0.36 & 0.38 & 1.05 & 24.7\
&LUCIAE 4$^d$ & 0.65 & 0.66 & 1.03 & 23.7\
\
\
\
\
