Particle production from soft processes

Abstract

Relativistic heavy-ion collisions provide a unique environment to study matter under extreme conditions of high temperature andenergy density. In the soft pT region (< 2 GeV/c) particle production is governed by the bulk properties (e.g. hydrodynamic pressure, freeze-out temperature) of the collisions.Total particle yields as well as their distributions in phase space are strongly dependent on these properties. These bulk properties will be discussed and we will compare our heavy-ion results to measurements in the p+p reference system, where multiparticle processes such as recombination and rescattering are generally thought to be negligible. However, we will show evidence for pi+ pi- rescattering interactions even in p+p collisions.

Soft interaction; Elliptic flow; Mass modification

SOFT INTERACTIONS

Particle production from soft processes

Patricia Fachini

Brookhaven National Laboratory, Upton, NY 11973 USA

ABSTRACT

Relativistic heavy-ion collisions provide a unique environment to study matter under extreme conditions of high temperature andenergy density. In the soft p_T region (< 2 GeV/c) particle production is governed by the bulk properties (e.g. hydrodynamic pressure, freeze-out temperature) of the collisions.Total particle yields as well as their distributions in phase space are strongly dependent on these properties. These bulk properties will be discussed and we will compare our heavy-ion results to measurements in the p+p reference system, where multiparticle processes such as recombination and rescattering are generally thought to be negligible. However, we will show evidence for p⁺p^- rescattering interactions even in p+p collisions.

Keywords: Soft interaction; Elliptic flow; Mass modification

I. ELLIPTIC FLOW

In non-central collisions the initial spatial anisotropy is transformed into an anisotropy in momentum-space if sufficient interactions occur among the constituents within the system. Once the system has expanded enough to quench the spatial anisotropy, further development of momentum anisotropy ceases. This self-quenching process happens quickly, so elliptic flow is primarily sensitive to the early stages of the collisions [1].

A. Hydrodynamics

The elliptic flow v₂ as a function of p_T for , L, f, X, and W is depicted in Fig. 1 [2, 3]. The f, X, and W have low hadronic cross-sections, therefore the large v₂ observed suggest that the elliptic flow is built up in the partonic stage. The expected range of v₂ from hydrodynamic calculations is also shown in Fig. 1.

A more detailed comparison can be seen in Fig. 2, where the mass dependent hydrodynamic results [1] are compared to the v₂measurements of p, K, p, and L [4]. Hydrodynamics describes well the mass dependence observed in the data that is characteristic of a common flow velocity. Since an ideal hydrodynamic fluid is a thermalized system with a zero mean free path that yields the maximum possible v₂, the good agreement between the measured v₂ and the hydro results [1] suggests thermalization in heavy-ion collisions at RHIC.

B. Constituent Quark Scaling

While hydrodynamic calculations keep increasing as a function of p_T, the measured v₂ saturates at p_T > 2GeV/c [2]. The saturation value for mesons is about 2/3 of that for baryons. This separation pattern holds for p, K, L, and X, and seems to hold for f and W[3, 6]. This result and the baryon-meson splitting of the high p _T suppression pattern [7] suggest the relevance of the constituent quark degrees of freedom in the intermediate p_T region [8]. v₂ scaled by the number of valence quarks n as a function of p_T/n is depicted in Fig. 3. The lower panel of Fig. 3 displays the ratio between the measurements and a polynomial fit to all the data. At low p_T/n the observed deviations from the fit follow a mass ordering which is expected from hydrodynamics. At higher p_T, all v₂/n measurements are reasonably close to unity showing the constituent quark scaling.

II. FREEZE-OUT PROPERTIES

The measured particle spectra and yields [9] and event-by-event áp_Tñ fluctuations [10] indicate a nearly chemically and kinetically equilibrated systemat the final freeze-out stage.

A. Chemical and Kinetic Freeze-out Parameters

STAR has now measured hadron distributions at = 200 and 62 GeV [11, 12]. Chemical freeze-out properties were extracted from stable particle ratios within the thermal model [28]. Kinetic freeze-out properties from particle p_T distributions were extracted within the blast wave model [14]. The extracted chemical freeze-out temperature T_ch, the strangeness suppression factor g_s, kinetic freeze-out temperature T_kin and the average radial flow velocity áb_Tñ at = 62 GeV are found to be qualitatively the same as those obtained at = 200 GeV and resonance decays are found to have no significant effect on the extract kinetic freeze-out parameters [12].

T_ch is independent of centrality. T_kin obtained from p, K, and p decreases as a function of centrality,while the corresponding áb_Tñ increases.This is evidence that the system expands between chemical and kinetic freeze-outs, which brings the system to a lower temperature. g_s increases from p+p to peripheral and central Au+Au collisions. In central Au+Au collisions, g_s is ~ 1 suggesting that strangeness is saturated. The T_ch of f, X, and W is higher than that of p, K,and p, while the áb_Tñ is lower.Noting that the f, X, and W have small hadronic cross-sections, they may chemically and kinetically freeze-out atthe same time.

B. Resonances

In order to study different hadronic interactions effects on different resonances in relativistic heavy-ion collisions, one should compare the potential differences in the ratio of the resonance to its corresponding ground state. Figure 4 depicts the K^*/K^-, f/K^-, and r⁰/p^- yield ratios as a functions of dN_rmch/dh for Au+Au collisions at = 200 GeV measured by STAR [34]. All yield ratios have been normalized to the corresponding yield ratio measured in minimum bias p+p collisions at the same c.m. system energy and indicated by the solid line in Fig. 4. The K^*/K^- yield ratio for central Au+Au collisions is significantly lower than the minimum bias p+p measurement at the same .

In addition, the statistical model prediction of K^*0/K^- is 2s higher than the measurement [34]. The lower K^*/K^- yield ratio measured may be due to the rescattering of the K^*0 decay products. Due to the relatively long lifetime of the f meson and the negligible KK cross-section, the rescattering of the f decay products and the phi regeneration should be negligible. The statistical model prediction for the f/K^- yield ratio indeed agrees with the measurement[34].

III. EVIDENCE FOR p⁺p^- SCATTERING IN p+p COLLISIONS

The r⁰ was measured via its hadronic decay channel in minimum bias p+p and Au+Au collisions at RHIC and a mass shift of -40 and -70 MeV/c² of the position of the r⁰ was observed, respectively. While no explanations were explicitly attributed to the mass shift in p+p collisions, the possible explanations for the apparent modification of the r⁰ meson properties in Au+Au collisions were attributed to dynamical interactions with the surrounding matter, interference between various p⁺p^- scattering channels, phase space distortions due to the rescattering of pions forming r⁰ and Bose-Einstein correlations between r⁰ decay daughters and pions in the surrounding matter [16].

The r⁰ meson measured in the dilepton channel probes all stages of the system formed in relativistic heavy-ion collisions because the dileptons have negligible final state interactions with the hadronic environment. Heavy-ion experiments at CERN show an enhanced dilepton production cross section in the invariant mass range of 200-600 MeV/c², showing that the r⁰ is broadened rather than shifted [17, 18].

The modification of the r⁰ properties in heavy-ion collisions has been expected [19], contrary to the modifications of the r⁰ properties in p+p collisions.Similar modifications of that measured in p+p collisions at RHIC has been observed before at CERN-LEBC-EHS and CERN-LEP. Until now, the mass shift measured in p+p at CERN-LEBC-EHS has been attributed to phase space.

A. Discussion r mass average from the Particle DataGroup (PDG)

We will first discuss the r mass average from the PDG[26]. The r⁰ mass average 775.8 ± 0.5 MeV/c² from e⁺e^- was obtained from either e⁺e^- ® p⁺p^- or e⁺e^-® p ⁺p^-p⁰. This meansthat the r⁰ mass average was obtained from exclusive leptonic reactions. Similarly, the r^± mass average 775.5 ± 0.5 MeV/c² was also was obtained from exclusive leptonic reactions.

The r averages reported by the PDG from reactions other than leptonic interactions are systematic lower than the value obtained from leptonic exclusive interactions by ~ 10 MeV/c²[26]. The r production in these hadronic reactions are inclusive and exclusive. In the case of inclusive productions, the phase space was take into account when the r mass was measured (e.g. [20]).

These observations lead us to conclude that the r mass depends on specific interactions, e.g. whether the r is produced in inclusive or exclusive reactions. Since a leptonic reaction and exclusive measurement of the r⁰ lead to a negligible modification of any kind of the r⁰ mass, the average 775.8 ± 0.5 MeV/c² [26] from e⁺e^- should correspond to the r⁰ mass in the vacuum.

B. r⁰ mass shifts at RHIC, CERN-LEBC-EHS, and CERN-LEP

The r⁰ was measured in the hadronic decay channel r⁰ ® p⁺p^- at RHIC, CERN-LEBC-EHS, and CERN-LEP in inclusive production. At RHIC, the STAR collaboration measured the r⁰ at = 200 GeV at midrapidity (|y| < 0.5) and observed mass shifts of the position of the r⁰ peak of about -40 MeV/c² and -70 MeV/c² in minimumbias p+p and peripheral Au+Au collisions, respectively [16]. The r⁰ mass obtained from the BW×PS fit to the invariant mass distributions after the subtraction of the like-sign background is depicted in Fig. 5, where it is clear that the phase space does not account for the measured mass shifts of the position of the r⁰ peak.

At CERN-LEBC-EHS, NA27 measured the r⁰ in minimum bias p+p at = 27.5 GeV for x_F > 0, where x_F is the ratio between the longitudinal momentum and the maximum momentum of the meson, and reported a mass of 762.6 ± 2.6MeV/c² [20]. The r⁰ signal can be seen even before background subtraction. The invariant p⁺p^- mass distribution was fit to the BW(NA27)×PS(NA27) plus a background function [20]. In this analysis, the phase space function (PS(NA27)) used is the same as the combinatorial background (BG). The invariant p⁺p^- mass distribution after subtraction of the mixed-event reference distribution is shown in Fig. 6. The vertical dashed line represent the average of the r⁰ mass 775.8 ± 0.5MeV/c² measured in e⁺e^- [26]. The vertical solid lineis the r⁰ mass 762.6 ± 2.6 MeV/c² reported by NA27 [20]. As shown in Fig. 6, the position of the r⁰ peak is shifted by ~ 30 MeV/c² compared to the r⁰ mass in the vacuum 775.8 ± 0.5 MeV/c² [26].

As mentioned, NA27 obtained the r⁰ mass by fitting theinvariant p⁺p^- mass distribution to the BW×PS function, and they reported a mass of 762.6 ± 2.6 MeV/c²,which is ~ 10 MeV/c² lower than the r⁰ mass in the vacuum. Ideally, the PS factor should have accounted for the shift on the r⁰ peak, and the mass obtained from the fit should have agreed with the r⁰ mass in the vacuum. However, just like in the STAR measurement, this was not the case, since thephase space did not account for the mass shift on the position of the r⁰ peak.

At CERN-LEP, OPAL, ALEPH and DELPHI measured the r⁰ ininclusive e⁺e^- reactions at = 90 GeV [21-24]. Even though OPAL never reported the value of the r⁰ mass, OPAL reported a shift on the position of the r⁰ peak by ~ 70 MeV/c² at low x_p, where x_p is the ratio between the meson and the beam energies,and no shift at high x_p (x_p ~ 1)[21, 22]. OPAL also reported a shift in the position of the r^± peak from -10 to -30 MeV/c², which was consistentwith the r⁰ measurement [23]. ALEPH reported the same shift on the position of r⁰ peak observed by OPAL [24]. Both OPAL and ALEPH used the like-sign method to subtract the background. DELPHI fit the raw invariant p⁺p^- mass distribution to the (BW×PS + BG) function for x_p> 0.05 and reported a r⁰ mass of 757 ± 2 MeV/c² [25], which is five standard deviations below the r⁰ mass in the vacuum (775.8 ± 0.5 MeV/c²). As one can see, similarly to NA27, DELPHI assumed that the phase space was described by the background function. Bose-Einstein correlations were used to describe the shift on the position of r⁰ peak. However; high (even unphysical) chaocity parameters (l ~ 2.5) were needed [21, 22, 24].

C. Phase space in p+p collisions

In p+p collisions, most models assume that particles are born at hadronization according to phase space without any final state interaction. In multiparticle production processes, single-inclusive (e.g. p+p), invariant particle spectra are typically exponential in p_T [35]. The exponential behavior does not require final state interactions and it can be due to phase space population at hadronization. The slope parameter in p+p collisions are independent of the particle species [36]. At RHIC, the r⁰ spectra in minimum bias and high multiplicity p+p are exponential in p_T up to 1.1 GeV/c² with slope parameters of ~ 180 MeV [37].These results are also independent of the multiplicity.

However, the r⁰ mass measured in p+p at RHIC is multiplicity dependent (the mass shift in high multiplicity is higher than in minimum bias p+p collisions) [16], which is opposite to the slope parameter that is multiplicity independent. Note that the systematic errors are correlate among themselves and between the two measurements. We will demonstrate that the mass should be independent of p_T as well as multiplicity at hadronization without final state interactions. According to quantum mechanics, a resonance at rest is described by a probability amplitude that is proportional to a non-relativistic Breit-Wigner distribution. When the energy conservation law is imposed in the partition of string fragmentation into multiple hadrons, a phase space factor, similar to the Boltzmann factor in thermal distribution, has to be added to the non-relativistic Breit-Wigner distribution

where M is the invariant mass (M), exp() is the phase space, m_T equals and T equals 160 MeV [37]. However; as discussed previously, the phase space does not describe the mass shift of the r meson measured at RHIC, CERN, and LEP.

Since the phase space of a non interacting multiparticle state cannot explain the distortion of the r⁰ line shape, we can conclude that the phase space in p+p collisions also accounts for hadrons scattering and forming resonances. In the case of the r⁰, p⁺p^-® r⁰® p⁺p^-. This can be pictured in the string fragmentation particle production scenario, where the string breaks several times, two pions are formed, they scatter, and form a r⁰.Such interactions are significant and modify the r⁰ spectral shape in e⁺e^- and p+p from a relativistic p-wave Breit-Wigner function.

Using the p+p data measured by STAR [16] and the rescattering formalism of [39], we can test the ideas of a mass shift having a p⁺p^- scattering component. The di-pion production is given by Eq. 21 of [39]. For the r⁰ which is p-wave = 1 Eq. 21 becomes

The d₁ is the p-wave phase shift (r⁰) and in our case becomes the Breit-Wigner times phase space equation 1. |D|²becomes the direct production of r⁰ for the p_T range 600 to 800 MeV/c. |A|² is the phase space overlap of di-pions in the = 1 partial waves. The pions emerge from a close encounter in a defined quantum state with a random phase.The emerging pions can re-interact or re-scatter through the p-wave quantum state or a phase shift. The phase space overlap comes from sampling the p spectrum from p+p collisions,where the sum of the p⁺p^- has the correct p_T range. The di-pion mass spectrum for this p_T range falls with increasing di-pion invariant mass. The variable a is related to the real part of the di-pion rescattering diagram and measures the range of interaction [39].

In order the fit the p+p data we need to add the rest of the cocktail of resonances used by STAR [16]. Such a fit is shown in Fig. 7 (solid line), where the direct r⁰ term time the phase space (dotted line) and the rescattered term (dash-dotted line) plus the sum (dashed line) are also depicted.

STAR measured the r⁰ at midrapidity and NA27 measured at the forward region. In addition, there is the energy difference of = 200 and 27.5 GeV, respectively. All these facts are consistent our model, the difference between the r⁰ mass shift measure by STAR and NA27 is due to pion multiplicity. Similar arguments is valid for the r measurements at CERN-LEP at = 90 GeV indicating that there is also significant scattering interactions in e⁺e^- reactions.

IV. CONCLUSIONS

There are various evidences for equilibration of the matter formed at Au+Au collisions at = 200 GeV (e.g. v₂ measurements, relative particle yields described by thermal models, g_S = 1, etc). All these together with the constituent quark scaling are strong signatures of sQGP. I also discussed that the natural mass of the r meson should be measured in exclusive reactions only. I showed that a shift on the position of the r⁰ peak has been measured and that the phase space does not account for the r(770)⁰ mass shift measured at RHIC, CERN-LEBC-EHS, and CERN-LEP. In addition, we discussed the phase space in p+p collisions and concluded that there are significant scattering interactions in p+p reactions.

Acknowledgement

I would like to thank F. Laue, R. Longacre, T. Ullrich, and Zhangbu Xu for the lively discussions.

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Received on 30 October, 2006; revised version received on 21 March, 2007

Publication Dates

• Publication in this collection
  10 Aug 2007
• Date of issue
  June 2007

History

• Received
  30 Oct 2006
• Reviewed
  21 Mar 2007