Spectra and radial flow at RHIC with Tsallis statistics in a Blast-Wave description
We have implemented the Tsallis statistics in a Blast-Wave model and applied it to mid-rapidity transverse-momentum spectra of identified particles measured at RHIC. This new Tsallis Blast-Wave function fits the RHIC data very well for 3 GeV/. We observed that the collective flow velocity starts from zero in p+p and peripheral Au+Au collisions growing to 0.470 0.009() in central Au+Au collisions. The parameter, which characterizes the degree of non-equilibrium in a system, changes from in p+p to in central Au+Au collisions, indicating an evolution from a highly non-equilibrated system in p+p collisions toward an almost thermalized system in central Au+Au collisions. The temperature and collective velocity are well described by a quadratic dependence on . Two sets of parameters in our Tsallis Blast-Wave model are required to describe the meson and baryon groups separately in p+p collisions while one set of parameters appears to fit all spectra in central Au+Au collisions.
Identified particle spectra in transverse momenta provide pillars in the discoveries in relativistic heavy ion collisions Adams et al. (2005a); Adcox et al. (2005); Gyulassy and McLerran (2005); Muller (2004). In Au+Au collisions at RHIC, identified particle yields per unity of rapidity integrated over the transverse momentum range have provided information on chemical potential and temperature at the chemical freeze-out in a statistical analysis Adams et al. (2005a, 2004a). The transverse momentum ( ) distributions of particles with different masses can be described in a Boltzmann-Gibbs Blast-Wave (BGBW) model with a compact set of parameters of temperature (), flow velocity ( ) and flow profile () Adams et al. (2005a); Schnedermann et al. (1993). With the BGBW model applied to the data in a limited range, a large radial flow 0.6 is obtained in central Au+Au collisions, while much smaller flow is obtained in peripheral Au+Au collisioins Adams et al. (2004a); Retiere and Lisa (2004). The description of heavy ion collisions using these freeze-out conditions have been used as evidence of collective motion in relativistic heavy ion collisions Adams et al. (2005a); Adcox et al. (2005); Voloshin et al. (2008); Adams et al. (2004b); Adler et al. (2004) (which has further allowed extraction of drag and diffusion coefficient of heavy quarks in the expanding bulk Teaney (2006); Rapp and van Hees (2008); Abelev et al. (2008a); Zhao and Rapp (2008)), provided necessary interplay with elliptic flow effect for the observed mass ordering in Voloshin et al. (2008), and as a necessary condition for explaining the ridge phenomenon in many models Voloshin (2006); Armesto et al. (2005); Dumitru et al. (2008); Gavin et al. (2008); Shuryak (2007).
However, such BGBW descriptions have limitations. Modeling nuclear collisions in ideal hydrodynamics is limited to low ( 1 GeV/) because it is generally believed that the equilibrium description fails at high , where particle production may be dominated by non-equilibrium or hard processes and exhibits a characteristic power-law tail Wilk and Wlodarczyk (2000). The blast-wave model has a strong assumption on local thermal equilibrium so that a Boltzmann distribution can be applied Schnedermann et al. (1993). This results in an arbitrary choice of range of the spectra where the function is able to fit the data and requires low and high cuts Adams et al. (2004a); Retiere and Lisa (2004). The BGBW model also lacks non-extensive quantities to describe the evolution from p+p to central A+A collisions. For example, one would expect that the energy deposited at mid-rapidity for particle production fluctuates significantly from event to event in p+p collisions while the current BGBW treats all p+p events the same with a heat bath at a fixed temperature. The resulting finite flow velocity 0.2 in p+p collisions from the same fitting procedure obscures the interpretation that collective flow is large and unique in A+A collisions Adams et al. (2004a). In Au+Au collisions, the fluctuations at initial impact due to Color-Glass Condensate (CGC) formation or individual nucleon-nucleon collision may not be completely washed out by subsequent interactions at either the QGP phase or hadronic phase Drescher et al. (2002); Mishra et al. (2008); Broniowski et al. (2007). All these effects leave footprints in the spectra at low and intermediate ( a few GeV/). For example, the volution of and spectra from an power-law (Levy) function in p+p and peripheral Au+Au collisions to an exponential (Boltzmann) function in central Au+Au collisions can be clearly observed Adler et al. (2002); Adams et al. (2005b, c); Abelev et al. (2007a), where is the transverse mass of particle with mass at a given .
With its development and success of Tsallis statistics in dealing with complex systems in condensed matter, many authors have utilized Tsallis statistics to understand the particle production in high-energy and nuclear physics De et al. (2007); Wilk and Wlodarczyk (2008); Alberico et al. (2000); Osada and Wilk (2008); Biro and Muller (2004). Although the implications and understanding of the consequences of such an application are still under investigation, the function is relatively easy to understand. The usual Boltzmann distribution in an exponential form is re-written as an power-law function:
where the left-hand side is the invariant differential particle yield and is a parameter characterizing the degree of non-equilibrium. The distribution can be derived from the usual procedure in statistical mechanics, starting from a non-equilibrium -entropy Tsallis (1988). If particle production is not distributed at a fixed temperature but rather as a system where varies with total energy (E) as , this will result in a negative binomial distribution (NBD) for particle number and temperature fluctuations. Their relations to are the skewness in NBD and the fluctuation of temperature Beck (2002). Regardless of the physical interpretation, Eq. 1 does provide a necessary power-law behavior at high and an exponential behavior at low . This is exactly what has been observed in p+p and peripheral Au+Au collisions at RHIC for and , whose measurable range can reach low and high Adler et al. (2002); Adams et al. (2005b, c); Abelev et al. (2007a). When , Eq. 1 becomes the familiar Boltzmann distribution again.
In addition to the features of the spectral evolution necessary to describe the spectra observed at RHIC, the physical interpretation of the results can provide quantitative insight into the non-equilibrium processes in relativistic heavy ion collisions and whether the system has thermalized in central Au+Au collisions. We expect that individual nucleon-nucleon collisions inside a nucleus-nucleus collision are in non-equilibrium at the initial impact and/or have a large energy fluctuation or a large value, which produce a large power-law tail in the spectra. Alternatively, in a CGC scenario, the system as a whole produces a strong color field with large fluctuations Dumitru et al. (2008); Gavin et al. (2008); Sorensen (2008). This can be treated as many hot spots in a nucleus-nucleus collision at the initial impact. In a viscous hydrodynamic evolution, the hot spots are smoothed (dissipated) into producing collective flow, creating more particles and increasing temperature Kolb and Heinz (2003). It has been argued in the Tsallis statistics that the increase of temperature and flow velocity during the evolution is connected to the decrease of ) by (shear and bulk ) viscosity in linear or quadratic proportion Wilk and Wlodarczyk (2000, 2008), such as . This can provide quantitative insight into the bulk viscosity, which is predicted to peak at the phase transition and is much larger than the shear viscosity Kharzeev and Tuchin (2008); Karsch et al. (2008).
In this paper, we present the procedure of implementing Tsallis statistics in the Blast-Wave model (TBW) and use it to fit the identified particle spectra at mid-rapidity at RHIC. The physics implications are discussed and preparation for future work is also presented. Good TBW fits can also provide a practical experimental tool to extract particle yields () by extrapolating to unmeasured kinematic ranges since most of the experimental measurements only cover a limited range for any given particle.
Ii Implement Tsallis Statistics into Blast-Wave model
To take into account collective flow in both longitudinal and transverse directions in relativistic heavy ion collisions, a simple Tsallis distribution needs to be embedded in the framework of hydrodynamic expansion Wilk and Wlodarczyk (2008). We follow the recipe of the Blast-Wave model provided by Schnedermann et al. Schnedermann et al. (1993). The formula has been adopted by many authors to implement a fit to the data in relativistic heavy ion collisions Retiere and Lisa (2004); Adams et al. (2004a); Adler et al. (2004); Abelev et al. (2008b). It is relatively trivial to change sources of particle emission from a Boltzmann distribution to a Tsallis distribution in the Blast-Wave model.
where is the flow profile growing as -th power from zero at the center of the collisions to at the hard-spherical edge () along the transverse radial direction (), and is the average flow velocity. We have used in this study. However, the integrations after the replacement are hypergeometric functions and the integrals over rapidity () and azimuthal angle () cannot be decoupled into two Bessel functions as in BGBW Schnedermann et al. (1993). The computing program typically used to provide conventional Blast-Wave fits to RHIC data Abelev et al. (2008b) has been modified to numerically calculate the integration in the above spectral function and fit the reslting distributions to the data. There are a few assumptions which do not change when a Tsallis distribution replaces a Boltzmann distribution:
Bjorken longitudinal expansion is assumed so that the measured particle yield does not depend on rapidity due to the integration over rapidity of the source Schnedermann et al. (1993). This is approximately true at mid-rapidity at RHIC or LHC Arsene et al. (2005). On the other hand, this assumption can be lifted in future analyses with a more complicated integration provided an emitting source along rapidity is known.
Isotropic emission in azimuth is assumed for each local source. However, the distribution of the source can have an azimuthal dependence in reality Ackermann et al. (2001). In the future, this can be implemented as has been done in the conventional BGBW model to fit the azimuthal dependence of HBT radius and the identified particle elliptic flow Retiere and Lisa (2004).
The emission source everywhere has the same density and degree of non-equilibrium () at the time of kinetic freeze-out. This may not be true since the high- particles (jet) tend to have surface emission Loizides (2007); Zhang et al. (2007). This kind of corona effect has been implemented in many other models and can be adopted in this framework as well.
Resonance decay contributions to the stable particle yields have been treated as part of the source emission. The detailed decay kinematics and its effect on the spectra have been studied in Ref. Schnedermann et al. (1993); Abelev et al. (2008b). Incorporation of resonance effects on spectra can be made in future improvements.
The goal of this paper is to provide a first implementation of the TBW model to exercise a trial case with the RHIC data. Future work can change the above assumptions to resemble more realistic conditions by comparing the assumptions to data and hydrodynamic calculations.
Iii Results of fit to RHIC data
The STAR Collaboration has published a series of particle spectra at mid-rapidity. The most complete set is for p+p and Au+Au collisions at = 200 GeV. The identified particle spectra include , , , , , , , , , , and Adams et al. (2005a, b); Abelev et al. (2007a, b, 2006, c); Adams et al. (2005d, c, 2007, 2006). The invariant differential yields were measured as . In the new TBW model, three parameters are common for all particles: temperature , non-equilibrium parameter , and maximum flow velocity where and average flow velocity is bounded to the range Abelev et al. (2008b) to aid in fit convergence and avoid non-physical results. An additional parameter provides the overall normalization of for each species. We choose the Minuit in Root 5.20/00 (2008) to perform a least- fit used in ref. Adams et al. (2005a). Figure 1 shows the spectrum data together with our fit results in two selected centrality bins (0-10% and 60-80%) in Au+Au collisions. The fit parameters and /DoF are tabulated in Tab. 1. As stated earlier in our model’s third assumption, surface emission could become important at high ; we limit our fits to GeV/ to avoid this region, which still extends the fit range well beyond previous BGBW fits. The curves from our model generally describe the data very well, especially in central Au+Au collisions. For peripheral Au+Au collisions, the meson spectra are well described by the model while the baryons are in general over-predicted at higher . On the other hand, the /DoF show good fits in all cases. The main results are:
(), a measure of the degree of non-equilibrium, decreases by a factor of 5 from 0.086 to 0.018. This means the power in the power-law increases from about 12 to 56, attaining an almost Boltzman distribution.
, the average temperature of the local source, shows a small increase from 114 MeV to 122 MeV. This trend is in contrast to the conventional BGBW result, where a decrease of temperature was observed Abelev et al. (2008b).
, the average flow velocity, increases from 0 in peripheral to 0.47 in central Au+Au collisions. That the minimum /DoF is found at the lower bound of 0 for peripheral collisions indicates that either the model is incomplete (some approximations may not be sufficiently true), or that no flow has developed in peripheral collisions within the context of the model. This also coincides with a large q-1, indicating a very non-equilibrated system if the description as a unified system applies.
Figure 2 shows the temperature and flow velocity versus () for Au+Au collisions. Each shaded region represents a one- contour from the error matrix obtained from the TBW fit for a given centrality. The dependence is clearly non-linear and has a negative correlation. There is a jump of flow velocity from zero in p+p and 60-80% Au+Au centrality to 0.28 at 40-60% Au+Au centrality, coinciding with the transition behavior in several other observables Sorensen (2008). We fit the distributions with a quadratics and obtain and , as shown in the figure.
Since the TBW model can be used to describe systems at non-equilibrium, it is natural to extend the fit to p+p collisions. However, a very poor /DoF was obtained if we include all of the mesons and baryons in a common fit. Instead, two separate groups of mesons and baryons show good fits. Figure 3 shows the results of the fits together with the data points. In both cases, the flow velocity was set at the lower limit of as was also independently verified if was set to be a free parameter. In Fig. 3.a, the proton and anti-proton spectra are presented together with the predicted curves from the TBW fit to the meson group. The need to separate mesons and baryons can be seen from spectral shape of the proton data, which matches more closely that of the kaons than the (steeper at low ) despite being closer in mass to the . The TBW model restricts spectral shapes to vary monotonically with mass for fixed (, , ), so there is no allowance for this artifact. Weak-decay effects from are investigated based on the results from the baryon-only fit and decay kinematics and do not explain this large steepening. On the contrary, a significantly smaller flattens the spectra at low and softens them at high , which is necessary and sufficient to achieve a good fit for the baryons (, , ) and anti-baryons. This implies that not only mass plays an important factor in the particle yield at a given , the particle species also significantly affects the outcome in p+p collisions. This characteristic baryon versus meson grouping in p+p data has been seen previously in scaling analyses of the same data Abelev et al. (2007b), and our results confirm this observation that baryon number plays an important role in hadron production in p+p collisions.
Iv Discussions and Outlooks
Modifying the Blast-Wave model to utilize Tsallis statistics instead of the conventional Boltzmann-Gibbs statistics has allowed high quality fits (see /DoF in Table 1) over a broader transverse momentum range and has altered the conclusions which can be drawn from the fits. The extended range is enabled by the Tsallis statistics’ capacity to evolve from an exponential source into a power law through increasing values, though the physical interpretation of this statistical model in the context of high energy nuclear collisions remains to be fully understood.
For central Au+Au collisions at = 200 GeV, the TBW fits find a value of approaching unity, which implies results very similar to those from BGBW with a large radial flow velocity Adams et al. (2004a). The progression of the TBW fit parameters with centrality is generally smooth but is strikingly different from BGBW fits, culminating in a notably non-equilibrium description of + and peripheral Au+Au collisions where a preference is found for zero collective flow. Qualitatively, increasing centrality produces a strong increase in flow velocity, a mild increase in temperature, and a dramatic decrease in (). This is consistent with a picture of increased thermalization with centrality but disfavors complete thermal equilibrium in all systems, a requirement for applicability of the BGBW model. The BGBW model appears to translate the non-equilibrium features, indicated by our model, into non-zero collective flow velocity and higher temperatures in + and peripheral Au+Au collisions Retiere and Lisa (2004); Adams et al. (2004a)
Hydrodynamics with space-time evolution from an initial condition Kolb and Heinz (2003) is so far the most realistic simulation of what happens in relativistic heavy ion collisions. However, even with implementation of initial conditions and an interface to hadronic cascade models at late stage of the evolution, it is hard to obtain an intuitive picture, in contrast to that offered by an analytical parametrization in the Blast-Wave model. It seems unlikely that hydrodynamics is applicable at all for + and very peripheral A+A collisions at RHIC. Being able to provide a systematic comparison between p+p and central A+A collisions in one model framework is still valuable and in some cases may be necessary.
We have additionally observed that + spectra continue to be well described by our TBW curves from 3 to 10 GeV/, well beyond the range used for the fits. This implies that the origin of the power law behavior may be the same from low to high , whether their underlying connection is fragmentation, parton evolution, or parton interaction cross-section. However, the high behavior in central Au+Au collisions is notably different: the experimental spectra have a reduced power-law tail relative to the binary scaled + spectra, but still significantly above the extension of the fit from GeV/. In the near future, we plan to implement a corona-like radius-dependent () which could accommodate larger power-law tails in central Au +Au collisions and allow the fit range to be extended to higher . This could provide additional information on jet quenching as characterized by surface emission.
It has been argued Wilk and Wlodarczyk (2000, 2008) that the dependence of and on () is related to bulk viscosity. If this viscosity is very large at phase transition, a systematic study of TBW model fits from AGS to SPS and RHIC energies may help locate the critical point. This, however, requires measurements of many identified particles to at least intermediate (3 GeV/) at all energies in a systematic fashion. These data can be provided by a RHIC beam energy scan. At LHC, one expects even larger () value in + collisions than that at RHIC due to increased relative contributions of hard and semi-hard processes; the conventional BGBW is not expected to be very meaningful for + and peripheral Pb+Pb collisions. If () is not larger for LHC + collisions and a non-zero flow velocities is observed, the question of thermalization or even QGP in such collisions might be raised. The results of TBW fits at LHC will certainly be informative.
In summary, we have implemented the Tsallis statistics in a Blast-Wave model and applied it to sets of identified particle spectra versus transverse momenta at mid-rapidity at RHIC. This new TBW function fits the RHIC data quite well for 3 GeV/. We observe that the collective flow velocity starts from zero in + and peripheral Au+Au collisions and rises to 0.4700.009 in central Au+Au collisions. The parameter (), which characterizes the degree of non-equilibrium in a system, changes from to systematically from + to central Au+Au collisions, indicating an evolution from a highly non-equilibrated system in + collisions toward an almost thermalized system in central Au+Au. TBW fits using all species from multiple centralities demonstrate a quadratic dependence of the temperature and collective flow velocity on (), diverging into two different fits for mesons and baryons necessary to best describe + spectra.
The authors would like to thank Drs. Aihong Tang, Bedanga Mohanty, James Dunlop, Paul Sorensen, Hank Crawford and Mike Lisa for valuable discussions. We thank the STAR Collaboration and the RCF at BNL for their support. This work was supported in part by the Offices of NP and HEP within the U.S. DOE Office of Science under the contracts of DE-FG02-88ER40412 and DE-AC02-98CH10886; Authors Yichun Xu and Zebo Tang are supported in part by National Natural Science Foundation of China under Grant No. 10610286 (10610285), 10475071, 10575101 and 10805046 and Knowledge Innovation Project of Chinese Academy of Sciences under Grant No. KJCX2-YW-A14. Lijuan Ruan thanks the Battelle Memorial Institute and Stony Brook University for the support in the form of the Gertrude and Maurice Goldhaber Distinguished Fellowship. Zhangbu Xu is supported in part by the PECASE Award.
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