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About the Authors

Zhangbu Xu is physicist at Brookhaven National Laboratory in the STAR group. Yifei Zhang is a recent Ph.D. recipient and is now a postdoc at University of Science and Technology of China.

Brownian Motion in the Color Field: Charmed Hadron Production at Low Transverse Momentum in Au+Au Collisions at RHIC

By Zhangbu Xu and Yifei Zhang

Measurements of charm production at low transverse momentum (pT), in particular radial and elliptic flow, probe the QCD medium at thermal scales and are thus sensitive to bulk medium properties like density and the drag constant or viscosity. Model treatments for low-pT charm production, such as energy loss by collisional dissociation and in-medium transport using a diffusion formalism (in analog to Brownian motion) and resonance cross sections [1], can be used to infer transport properties such as interaction cross sections and the medium density.

In classic Brownian motion [2], the diffusion from the initial input position is related to the coefficient of viscosity (η). Meanwhile, the velocity is connected to the friction coefficient (β). There are two obvious conclusions from this analog:

  1. At long time and small viscosity, the heavy object will be in equilibrium with the system, and E = kT regardless of the viscosity or friction where T is the temperature of the system.
  2. At short time and large viscosity, the heavy object is not in equilibrium with the system; however, dynamic variables (η, β) can be inferred from the spectrum.

This means that in either case very important information can be extracted from heavy-flavor measurements at low pT. However, we cannot observe complete equilibrium and obtain dynamic variables at the same time. In these considerations, we assume the initial spectrum of the probe (heavy-flavor) is known. We need to check if the charm quarks are exclusively produced at the initial impact and can be referenced by p+p or p+A collisions. If charm production is exclusively from the initial hard processes, charm yields should scale with number of initial nucleon-nucleon collisions (Nbin). This is a crucial test.

STAR Collaboration reported measurements of charmed hadron production from hadronic (D → Kπ) and semileptonic (μ and e) decays in 200 GeV Au+Au collisions at RHIC[3]. The muon and hadronic D0 yields are used to constrain the total cross-section since these measurements are limited at very low momentum and are only sensitive to total cross-section. The non-photonic electron (NPE) yield with pT>0.9 GeV/c is sensitive to the spectrum shape once the total cross-section is stringently constrained by other means. Fig.1 shows how the muons are identified at very low pT by a combination of dE/dx with the Time Projection Chamber (TPC) and mass with the Time-of-Flight (TOF) (Fig.1b). Muons from weak decay of hadrons can be statistically subtracted from the distribution of the Distance of Closest Approach (DCA) of tracks in TPC to the collision vertex (Fig.1c). Low-pT muon is an almost-uniform sample of the charm momentum spectrum through charm semileptonic decay[4].Therefore, its yield is proportional to charm total cross section and is insensitive to the details of charm spectrum[4].


Figure 1: mass square distribution from TOF after dE/dx selection of muon candidate. The muon and pion peaks are clearly separated. c) panel shows the DCA distributions from inclusive muons(data points), muons from charm semileptonic decay (dotted line) and muons from pion and kaon decays (histogram).

The first check we can do is to test if charm spectrum in Au+Au collisions is the same as that in d+Au or p+p collisions. The nuclear modification factors (RAuAu/dAu) for μ and NPE are shown in Fig. 2. The RAuAu/dAu are the ratios of the pT spectra in Au+Au over that in d+Au collisions appropriately scaled with the number of binary collisions. It is obvious that the ratio is not a constant at unity. It strongly depends on pT. To study whether charmed hadrons have similar radial flow to light hadrons, we have included curves for the expected nuclear modification factor from a blast-wave model, using the freeze-out parameters for light hadrons [5] (BW3 in Fig. 3) and multi-strange hadrons [6] (BW2). The data and best blast-wave fit (BW1) show large deviations from both these curves for pT > 1 GeV/c. This suggests that the charmed hadron freeze-out and flow are different from light hadrons. These findings, together with the observation of large charm elliptic flow [7], are consistent with the recent prediction from hydrodynamics [8]: elliptic flow is built up at partonic stage, and radial flow dominantly comes from hadronic scattering at later stage where charm may have already decoupled from the system. It means that charm is not completely thermalized with the system throughout the evolution. The good news is that it can be used to probe the dynamic properties of the system. This also implies that the bulk viscosity increases at transition and/or hadron phase, preventing charm from further equilibrating with the rest. Often, people claim that it makes no sense to compare charm spectrum to a simplified thermal+flow distribution (e.g. Blast-Wave) because charm has energy loss and cannot be in equilibrium with the system. The point of comparison is to check how far it is away from equilibrium with the rest of the system (light hadrons, multistrange hadrons). If charm is thermalized and flows with the rest of the system, regardless of how it gets there (radiative energy loss, collisional energy, resonant excitation), the spectrum has to be just like what inferred from the other hadrons.

Figure 2: Nuclear Modification Factor of lepTons from heavy-flavor semileptonic decay for central Au+Au collisions. The results are compared to freeze-out spectra inferred from light hadrons (BW3), and strangeness hadrons (BW2). The best fit prefers high-temperature and low flow velocity (BW1). Model calculations with heavy-flavor collisional energy loss, resonant excitation in medium are also presented for comparison [1].

The missing piece of charm as an ideal probe is to verify that charm is produced only at initial impact. A combined fit to the D0, NPE and μ spectra was used to obtain the mid-rapidity charm cross section per nucleon-nucleon collision (dsNNc¯c /dy), shown in Fig. 3. Yields are an average from the blast-wave and the power-law functions, which are consistent within ±10%. Fig. 3 shows dsNNc¯c /dy as a function of Nbin for minbias d+Au, minbias and central Au+Au collisions. The charm production cross section at mid-rapidity scales with number of binary interactions from d+Au [9] to central Au+Au collisions.

The quality of this scaling can be quantified by fit to all three data points, which is 290 μb with ±15% uncorrelated uncertainty. This indicates that charm quarks are produced in the early stage of relativistic heavy-ion collisions.


Figure 3: Mid-rapidity charm cross-section per underlying nucleon-nucleon collision vs. centrality in A+A collisions. A binary scaling is observed.

These results provide quantitative spectra and cross-sections of charm production from d+Au to central Au+Au collisions. The combination of three techniques provides the most complete kinematic coverage to date for charm production measurements at RHIC. They also allow the study of the charmed hadron spectral shape in order to explore the possibility of charm radial flow in Au+Au collisions. The blast-wave fits and the direct comparisons of the spectra suggest that charmed hadrons interact with and decouple from the system differently from the light hadrons. The total cross sections are found to scale with the number of binary collisions. This confirms the expected scaling of hard production processes with binary interactions among incoming nucleons so that charm quarks can be used as a calibrated probe of the early stage dynamics of the system.

References

[1] A. Adil and I. Vitev, Phys. Lett. B 649, 139 (2007);
R. Rapp and H. van Hees, J. Phys. G 32, S351 (2006);
G. D. Moore and D. Teaney, Phys. Rev. C 71, 064904 (2005)/

[2]E. Nelson (Princeton University), “Dynamical Theories of Brownian Motion”
http://www.math.princeton.edu/~nelson/books/bmotion.pdf

[3]STAR Collaboration, arXiv:0805.0364v2

[4] H.D. Liu et al., Phys. Lett. B 639, 441 (2006)

[5] J. Adams et al., Phys. Rev. Lett. 92, 112301 (2004).

[6] J. Adams et al., Phys. Rev. Lett. 92, 182301 (2004).

[7] A. Adare et al., Phys. Rev. Lett. 97, 252002 (2006);
A. Adare et al., Phys. Rev. Lett. 98, 172301 (2007).

[8] T. Hirano et al., e-Print: arXiv:0710.5795 [nucl-th].

[9] J. Adams et al., Phys. Rev. Lett. 94, 062301 (2005).