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What the data say

As far as the motivation for studying kaon production is concerned, the following points can summarize the discussion :

• Because kaons are the lightest (and therefore the most abundant) strange particles, the strangeness content of the final state can not be understood without measuring kaons.
• On the other hand, the kaon abundance alone does not permit to draw conclusions about presence or absence of the deconfined phase in the evolution of the system.

For the former reason, and in the light of the latter caveat, we have carried out a study of kaon production in $Pb+Pb$ collisions at SPS [26].

Tables 5.1 and 5.2 give the $m_T$ slope parameters and values of $dN/dy$ for kaons and pions. The measured distributions for charged kaons of both signs in transverse kinetic energy and rapidity, are shown on Fig. 5.1

Table 5.1: Inverse slope parameters T.

PID	$y$ interval	$T$ (MeV)	$\sigma(T)$ stat., syst. (MeV)
$K^+$	2.3-2.6	230	$\pm$ 8 $\pm$ 14
$K^+$	2.4-2.9	254	$\pm$ 4 $\pm$ 7
$K^-$	2.3-2.6	259	$\pm$ 8 $\pm$ 12
$K^-$	2.4-2.9	245	$\pm$ 7 $\pm$ 6

Table 5.2: Particle distributions in rapidity for top 4% centrality. Every spectrometer setting provides an independent measurement. Settings overlapping in $y$ are listed separately. Statistical and systematic errorbars are added in quadrature to form $\sigma(\,dN/\,dy)$ listed.

PID	$y$ interval	$\,dN/\,dy$	$\sigma(\,dN/\,dy)$	PID	$y$ interval	$\,dN/\,dy$	$\sigma(\,dN/\,dy)$
$K^-$	2.7-2.9	21.5	$\pm$ 7.5	$K^+$	2.7-2.9	37.1	$\pm$ 5.4
	2.3-2.5	18.7	$\pm$ 1.9		2.3-2.6	27.2	$\pm$ 2.5
	3.1-3.4	15.4	$\pm$ 4.1		3.1-3.4	29.7	$\pm$ 5.6
	2.6-2.8	14.8	$\pm$ 1.4		2.6-2.8	33.6	$\pm$ 3.1
$\pi^-$	3.3-3.7	176	$\pm$ 14	$\pi^+$	3.3-3.7	160	$\pm$ 15
	2.6-2.9	193	$\pm$ 12		2.6-2.9	153	$\pm$ 10
	3.5-4.0	173	$\pm$ 12		3.5-4.0	145	$\pm$ 10
	2.6-2.9	173	$\pm$ 15		2.6-2.9	164	$\pm$ 13

Figure 5.1: Measured transverse kinetic energy distributions of positive and negative kaons for the 4% and 10% most central of Pb+Pb collisions. Two spectrometer angle settings meet at $m_T-m=0.35$ GeV. The fits follow the form $1/m_{T}\,dN/\,dm_{T} \propto exp(-m_{T}/T)$, where $m_{T} = (m^{2}+p_{T}^{2})^{1/2}$. $y$ ranges of the fits are given in Table 2 and are indicated by the horizontal errorbars in the inserts. RQMD predictions for $\vert y-y_{CM}\vert<0.6$ (i.e., within NA44 acceptance) are shown as histograms.

The $1/m_T$ scaled spectra appear exponential in accordance with the behaviour typical for thermalized ensembles of interacting particles, or for particles in whose production the phase-space constraints played the dominant role [55]. The spectra were fit with an exponential in $(m_T - m)$, and the resulting slopes are shown in the inserts in Fig. 5.1. The inverse slopes of the $K^+$ and $K^-$ spectra are the same, within errors. Our event selection is sufficiently central that the slopes show no dependence on multiplicity.

In Fig. 5.2, it is clear that many fewer kaons are produced than pions, as was observed in $p+p$ collisions. There are approximately twice as many positive as negative kaons produced. This is typical for baryon rich systems, and was also observed in $p+p$ collisions. PreliminaryNA49 measurements of $K^+$ and $K^-$ $\,dN/\,dy$ in $Pb+Pb$ [56] are consistent with those reported here.

Figure 5.2: Comparison of measured charged kaon and pion yields with RQMD predictions. The vertical error bars indicate statistical and systematic errors, added in quadrature; the horizontal ones - $y$ boundaries of the acceptance used for $p_T$ integration in each spectrometer setting. Open symbols represent spectrometer settings whose $y$ position is shown mirror-reflected around midrapidity (2.92); their solid analogs - the actual settings. RQMD: solid line - standard mode, dashed line - no rescattering.

Both Fig. 5.1 and 5.2 compare the data with predictions of the transport theoretical approach RQMD [57]. While RQMD tends to overpredict both the $K^+$ and $K^-$ yields, for $K^-$ the discrepancy appears to be larger. Running RQMD in the mode which does not allow the hadrons to rescatter (shown by the dashed line on the figure) decreases the number of kaons produced. This result illustrates the importance of the secondary scattering to the total kaon yields. Measurements of proton production at midrapidity[48] and of the $p-\bar{p}$ rapidity distribution[58] indicate that RQMD somewhat overpredicts the degree of baryon stopping. Because $\pi N$ inelastic collisions can produce kaons, an increase in stopping translates naturally into kaon enhancement at midrapidity. The data show that the hadron chemistry via secondary scattering, as implemented in RQMD, successfully reproduces the general trends in the hadron distribution. However, the hadron chemistry in the model is not quantitatively correct.

Figure 5.3: $K/\pi$ ratios in symmetric systems at midrapidity $\vert y-y_{CM}\vert\le \vert y_{proj}-y_{targ}\vert/8$. The solid line shows full solid angle $K/\pi$ in $p+p$ collisions from the interpolation [59]. The data points from other experiments result from an interpolation in $y$ to the midrapidity interval. The E866 data points [60] are also interpolated in the number of participants, for comparison with the SPS data.

Strangeness enhancement compared to the interpolated [59] $pp$ collision data, shown as the line, is seen in Fig. 5.3. The solid point, corresponding to ISR data at midrapidity, indicates the extent of the enhancement due to the midrapidity cut on the particles. The figure shows that $K^+/\pi ^+$ is enhanced in high multiplicity heavy ion collisions, but $K^-/\pi^-$ is consistent with $p+p$ values. Higher multiplicity, or more central collisions, yields larger enhancement, independent of $\sqrt{s}$.

Secondary hadronic interactions of the type $\pi + N \leftrightarrow Y + \overline{K}$ are important for the strangeness production [57,61], and their rate is proportional to the product of the participant's effective concentrations.

Figure: Comparison of measurements with RQMD predictions: $K^+/\pi ^+$ ratio in the specified rapidity interval around mid-rapidity, as a function of the product of pion and proton $dN/dy$, obtained in the same rapidity interval, in symmetric collisions. $\diamond$ - E866 AuAu, $\bullet$ - NA44 SS, $\circ$ - NA44 PbPb. RQMD: solid line - standard mode, dashed line - no rescattering.

Fig. 5.4 shows the dependence of the $K^+/\pi ^+$ ratio on the product of rapidity densities of the two ingredients of the associated strangeness production, $N$ (represented by $p$) and $\pi^+$ in the AGS [62] and SPS [63] data, and RQMD calculations. This `` $p\times\pi$'' product serves as an observable measure of the strangeness-enhancing rescattering. The rate of change in the $K^+/\pi ^+$ ratio with this rescattering observable is initially very high. However, $K^+/\pi ^+$ nearly saturates after this initial rise. The figure shows why the enhancement is large as soon as the multiplicity becomes appreciable. The values of `` $p\times\pi$'' reached at the SPS and AGS are comparable, explaining the similarity of the kaon enhancement despite the different energies. RQMD reproduces the trend of the data very well, and the dotted lines (illustrating no rescattering) along with the shape of the rise with `` $p\times\pi$'' underscore the role of hadronic rescattering in kaon yields. The quantitative agreement of RQMD with the data is not as good, but the final results are undoubtedly quite sensitive to the magnitude of the cross sections used in the model.

When comparing $AA$ data to $pp$, even if only particle ratios are considered, it is important to bear in mind that there are effects which distinguish $AA$ from $pp$ and which do not qualify as QGP signals. The time interval between two $NN$ interaction in the target rest frame is of the order of $fm/c$, but due to the Lorentz time dilation, the intermediate object created in an $NN$ interaction has no time to hadronize and is involved in the next collision and all further ones. RQMD[57] and VENUS[64] take this into account. Comparison to $pA$, rather than $pp$, is more credible, but if the intermediate partonic objects can involve constituents from more than one projectile nucleon, a similar argument still holds. And according to the Lorentz invariance, the intermediate partonic objects can involve more than one projectile nucleon since, as we have seen, they can involve more than one target nucleon, whereas such kind of discussion should not depend on the choice of reference frame. We therefore conclude that, qualitatively speaking, a comparison with a lighter system can not be done in a completely model-independent way, even though, quantitatively speaking, there are different degrees of credibility among the existing methods. In making the claim about the enhancement of (multi)strange (anti)baryons, WA85 compared $SW$ with $pW$[65], NA49 - $PbPb$ with $pp$[66], WA97 - $PbPb$ with $pPb$ system[67], but notably, the latter experiment devoted a special paper to the RQMD and VENUS comparisons[69].

Some strangeness production in RQMD goes through the excitation of the nucleon resonances - these are not considered secondaries, they are propagated and can be re-excited and de-excited[68]. Some of their decay channels contain strange mesons and baryons. For this reason, there is a difference between $K^+/\pi ^+$ ratio in $pp$ and $K^+/\pi ^+$ ratio in RQMD without rescattering (as seen from comparison between Fig.5.3 and Fig.5.4). This difference looks larger for SPS than for AGS.

There are two processes in the RQMD mode without rescattering that affect the $K/\pi$ ratio differently [68]:

1. slowing down of the original nucleon as it passes through the medium. This works to reduce $K/\pi$ ratio, compared with $pp$ collisions at the original $\sqrt{s}$.
2. excitation of resonance nucleon states some of which decay into $\Lambda + K$ - this enhances $K/\pi$.

Because at the AGS energy the slowing down is significant, these two processes tend to balance each other. At SPS, slowing down is not so significant, and the resonances win.

WA97 Collaboration measured yields of $K^0_S$, $\Lambda$, $\Xi$, and $\Omega$ (both particles and antiparticles) at midrapidity for $p+Pb$ and $Pb+Pb$ collisions [67]. It was found that the enhancement factor with respect to $p+Pb$ is larger for $\bar{s}\bar{s}\bar{q}$ and $ssq$ than for $\bar{s}\bar{q}\bar{q}$ and $sqq$ baryons. However, the measured enhancement for antibaryons is smaller than for baryons.

RQMD predictions for strange and antistrange baryon yields in $Pb+Pb$ are available [57] to compare with. The microscopic cascade method of RQMD does not involve the notion of the deconfined quark-gluon soup, even though the partonic degrees of freedom are involved via color strings and ropes. Elastic and inelastic rescattering is simulated. The publication [57], based on RQMD 2.1, contains predictions for all the hyperons measured in [67], except for $\Omega$ and $\bar{\Omega}$, as $\,dN/\,dy$ histograms and total number yield per central event. With reasonable accuracy, one can draw meaningful conclusions from comparing WA97's $\Delta N/\Delta y$ within $\Delta y=1$ around midrapidity in the most central sample with RQMD's $\,dN/\,dy$. It turns out that RQMD [57] overpredicts $K^0_S$, overpredicts $\Lambda$, does a good job on $\bar{\Lambda}$, and considerably overpredicts $\Xi^-$ and $\bar{\Xi}^+$ yields reported for $Pb+Pb$ by the WA97 [67]. The same work includes predictions for RQMD runs with ropes, but without rescattering, and with no ropes and no rescattering. Whereas ropes are the main effect responsible for the birth of strange (anti)hyperons, rescattering depletes their abundance by redistributing (anti)strange quarks into mesons. The latter is a generic hadrochemistry feature not unique to RQMD, as has been discussed earlier.

The WA97 Collaboration made a dedicated comparison of their data with VENUS and RQMD 2.3[69], and concluded that VENUS[64] (based on Gribov-Regge theory with rescattering simulation via pre-hadron clusters) overpredicts yields of $\bar{\Lambda}$, $\Xi^-$, $\bar{\Xi}^+$, $\Omega^-$ and $\bar{\Omega}^+$ in $pPb$ and $PbPb$, whereas RQMD 2.3 does a good job for $K^0_S$, $\Lambda$, $\bar{\Lambda}$, $\Xi^-$ and $\bar{\Xi}^+$, but underpredicts $\Omega^-$ and $\bar{\Omega}^+$.