PHYSICAL REVIEW C 73, 034913 (2006)
Heavy-quark probes of the quark-gluon plasma and interpretation of recent data taken at
the BNL Relativistic Heavy Ion Collider
Hendrik van Hees,
1
Vincenzo Greco,
2
and Ralf Rapp
1
1
Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-3366, USA
2
Laboratori Nazionali del Sud INFN, via S. Sofia 62, I-95123 Catania, Italy
(Received 31 August 2005; published 29 March 2006)
Thermalization and collective flow of charm (c) and bottom (b) quarks in ultrarelativistic heavy-ion collisions
are evaluated based on elastic parton rescattering in an expanding quark-gluon plasma (QGP). We show that
resonant interactions in a strongly interacting QGP (sQGP), as well as parton coalescence, can play an essential
role in the interpretation of recent data from the BNL Relativistic Heavy-Ion Collider (RHIC), and thus illuminate
the nature of the sQGP and its hadronization. Our main assumption, motivated by recent findings in lattice quantum
chromodynamics, is the existence of D-andB-meson states in the sQGP, providing resonant cross sections for
heavy quarks. Pertinent drag and diffusion coefficients are implemented into a relativistic Langevin simulation to
compute transverse-momentum spectra and azimuthal asymmetries (v
2
)ofb-andc-quarks in Au-Au collisions
at RHIC. After hadronization into D-andB-mesons using quark coalescence and fragmentation, associated
electron-decay spectra and v
2
are compared to recent RHIC data. Our results suggest a reevaluation of radiative
and elastic quark energy-loss mechanisms in the sQGP.
DOI: 10.1103/PhysRevC.73.034913 PACS number(s): 12.38.Mh, 24.85.+p, 25.75.Nq
I. INTRODUCTION
Recent experimental findings at the BNL Relativistic
Heavy-Ion Collider (RHIC) have given intriguing evidence for
the production of matter at unprecedented (energy-) densities
with surprisingly large collectivity and opacity, as reflected
by (approximately) hydrodynamic behavior at low transverse
momentum (p
T
) and a substantial suppression of particles with
high p
T
. This has led to the notion of a “strongly interacting
Quark-Gluon Plasma” (sQGP), whose microscopic properties,
however, remain under intense debate thus far.
Heavy quarks (HQs) are particularly valuable probes of the
medium created in heavy-ion reactions, as one expects their
production to be restricted to the primordial stages. Recent
calculations of radiative gluon energy-loss of charm (c) quarks
traversing a QGP in central Au-Au collisions at RHIC have
found nuclear suppression factors R
AA
0.3–0.4 [1,2], com-
parable to the observed suppression of light hadrons at high p
T
,
and in line with preliminary (nonphotonic) single-electron (e
±
)
decay spectra [3–5]. The latter also exhibit a surprisingly large
azimuthal asymmetry (v
2
) [6–8] in semicentral Au-Au which
cannot be reconciled with radiative energy loss, especially
if c-quarks are hadronized into D-mesons via fragmentation.
While the underlying transport coefficients [2] exceed their
predicted values from perturbative quantum chromodynamics
(pQCD) by at least a factor of ∼5 [9], energy loss due to
elastic scattering parametrically dominates toward low p
T
(by
a factor 1/
√
α
s
[10]). But elastic pQCD cross sections [11,12]
also have to be upscaled substantially to obtain c-quark v
2
and
R
AA
reminiscent to preliminary e
±
data, as shown in a recent
Langevin simulation for RHIC [10]. In addition, contributions
of bottom (b) quarks [13,14] will reduce the effects in
the electron-R
AA
and -v
2
at high p
T
. Quark coalescence
approaches suggest that an e
±
-v
2
in excess of 10% can only be
obtained if [15] (i) light quarks impart their v
2
on D-mesons
(see also Refs. [16,17]), (ii) the c-quark v
2
is comparable to
that of light quarks.
We are thus confronted with marked discrepancies between
pQCD energy-loss calculations and semileptonic heavy-quark
(HQ) observables at RHIC. The resolution of this issue is
central to the understanding of HQ interactions in the QGP
in particular, and to the interpretation of energy loss in
general. HQ rescattering also has direct impact on other key
observables such as heavy quarkonium production (facilitating
regeneration) and dilepton spectra (where c
¯
c decays compete
with thermal QGP radiation).
In this paper we investigate the HQ energy-loss problem
by introducing resonant HQ interactions into a Langevin
simulation of an expanding QGP. Our calculations imple-
ment a combined coalescence+fragmentation approach for
hadronization, as well as bottom contributions, to allow for a
quantitative evaluation of pertinent e
±
-spectra (v
2
and R
AA
)
which is mandatory for a proper interpretation of recent RHIC
data. Our main assumption of resonant D- and B-like states in
the sQGP has been shown [18] to reduce HQ thermalization
times by a factor of ∼3 compared to pQCD scattering.
Theoretical evidence for resonances in the sQGP derives from
computations of heavy and light meson correlators within
lattice QCD (lQCD) [19,20], as well as applications of lQCD-
based heavy-quark potentials within effective models [21–24].
Except for the mass and width of these states, no further free
parameters enter our description, with degeneracies based on
chiral and HQ symmetry.
II. HEAVY-QUARK INTERACTIONS IN THE QGP
Following Ref. [18] our description of HQ interactions in
the QGP focuses on elastic scattering, mediated by resonance
excitations on light antiquarks (
¯
q) as well as (nonresonant)
0556-2813/2006/73(3)/034913(4)/$23.00 034913-1 ©2006 The American Physical Society
HENDRIK VAN HEES, VINCENZO GRECO, AND RALF RAPP PHYSICAL REVIEW C 73, 034913 (2006)
leading order pQCD processes dominated by t-channel gluon
exchange. The latter correspond to Born diagrams [25] regu-
larized by a gluon-screening mass m
g
= gT with a strong cou-
pling constant, α
s
= g
2
/(4π) = 0.4. The key assumption [18]
is that a QGP at moderate temperatures T
2T
c
sustains strong
correlations in the lowest-lying color-neutral D- and B-meson
channels. Support for the relevance of such interactions stems
from quenched lQCD computations of euclidean mesonic
correlation functions, which, after transformation into the
timelike regime, exhibit resonance structures for both (heavy)
Q-
¯
Q and (light) q-
¯
q states [19,20]. In addition, applications of
lQCD-based Q-
¯
Q potentials have revealed both bound [21,22]
and resonance states [23] with dissolution temperatures of
∼2T
c
, quite compatible with the disappearance of the peak
structures in the lQCD spectral functions. Here, we do not
attempt a microscopic description of these correlations but cast
them into an effective lagrangian with
¯
q-Q- vertices ( =
D, B), at the price of two free parameters: the masses of t he
meson-fields, fixed at m
D(B)
= 2(5) GeV, i.e., 0.5 GeV above
the Q-
¯
q threshold (with quark masses m
c(b)
= 1.5(4.5) GeV,
m
u,d
= 0), and their width, , obtained from the one-loop
self-energy (which, in turn, dresses the -propagator) with the
pertinent coupling constant varied to cover a range suggested
by effective quark models [26,27], = 0.4–0.75 GeV. The
multiplicity of states follows from chiral and HQ symmetries
alone, implying degenerate J
P
= 0
±
and 1
±
states. We empha-
size again that, besides the mass and width of the states, no
other free parameters (or scale factors) are introduced.
The matrix elements for resonant and pQCD scattering
are employed to calculate drag and diffusion coefficients of
HQs in a Fokker-Planck approach [11]. Resonances reduce
the thermalization times for both c- and b-quarks by a factor
of ∼3 compared to pQCD scattering alone [18].
III. LANGEVIN SIMULATION
To evaluate thermalization and collective flow of HQs in
Au-Au collisions we perform relativistic Langevin simulations
[10] embedded into an expanding QGP fireball. In the local
rest frame of the bulk matter, the change in position (x) and
momentum ( p)ofc- and b-quarks during a time step δt is
defined by
δ x =
p
E
δt, δ p =−A(t, p + δ p) pδt + δ
W (t, p + δ p)
(1)
(E: HQ energy), where δ
W represents a random force which
is distributed according to Gaussian noise [34],
P (δ
W ) ∝ exp
−
ˆ
B
jk
δW
j
δW
k
4δt
. (2)
The drag coefficient (inverse relaxation time), A, and the
inverse of the diffusion-coefficient matrix,
B
jk
= B
0
(δ
jk
−
ˆ
p
j
ˆ
p
k
) + B
1
ˆ
p
j
ˆ
p
k
, (3)
are given by the microscopic model of Ref. [18], including p-
and T-dependencies (the latter converts into a time dependence
using the fireball model described below). The longitudinal
diffusion coefficient is set in accordance with Einstein’s
dissipation-fluctuation relation to [10]B
1
= TEA, to ensure
the proper thermal equilibrium limit. The latter also requires
care in the realization of the stochastic process in Eq. (1); we
here use the so-called H
¨
anggi-Klimontovich realization [34]
which approaches a relativistic Maxwell distribution if the
Einstein relation is satisfied (in the Ito realization, e.g., an
extra term has to be introduced in A [35]). Finally, t he HQ
momenta are Lorentz-boosted to the laboratory frame with the
velocity of the bulk matter at the actual position of the quark,
as determined by the fireball flow profile (see below).
The time evolution of Au-Au collisions is modeled by an
isentropically expanding, isotropic QGP fireball with a total
entropy fixed to reproduce measured particle multiplicities at
hadro-chemical freezeout which we assume to coincide with
the phase transition at T
c
= 180 MeV [28]. The temperature
at each instant of time is extracted from an ideal QGP
equation of state with an effective flavor degeneracy of
N
f
= 2.5. Radial and elliptic flow of the bulk matter are
parameterized to closely resemble the time-dependence found
in hydrodynamical calculations [29], assuming a flow profile
rising linearly with the radius. We focus on semicentral
collisions at impact parameter b = 7 fm, with an initial spatial
eccentricity of 0.6 and a formation time of τ
0
= 0.33 fm/c,
translating into an initial temperature of T
0
= 340 MeV. The
evolution is t erminated at the end of the QGP-hadron gas
mixed phase (constructed via standard entropy balance [28])
after about 5 fm/c, at which point the surface flow velocity and
momentum anisotropy have reached v
⊥
= 0.5c and v
2
= 5.5%
(variations in τ
0
by a factor of two affect the c-quark v
2
and
R
AA
by 10–20% (less for the e
±
spectra), while a reduction
in the critical temperature to 170 MeV increases (decreases)
v
c
2
(R
AA
) somewhat less. D-meson rescattering in the hadronic
phase [30] is neglected).
To specify initial HQ p
T
-distributions, P
ini
(p
T
), and es-
pecially the relative magnitude of c- and b-quark spectra
(essential for the evaluation of e
±
spectra), we proceed
as follows: we first use modified PYTHIA c-quark spectra
with δ-function fragmentation to fit D and D
∗
spectra in
d-Au collisions [31]. These spectra are decayed to single-e
±
which saturate pertinent data from p − p and d-Au up to
p
e
T
3.5 GeV [32,33]. The missing yield at higher p
T
is
attributed to contributions from B-mesons, resulting in a cross
section ratio of σ
b
¯
b
/σ
c
¯
c
5 · 10
−3
(which is slightly smaller
than for pQCD predictions [13] and implies a crossing of c-
and b-decay electrons at p
T
5 GeV, as compared to 4 GeV
in pQCD).
Figure 1 summarizes the output of the Langevin sim-
ulations for the HQ nuclear modification factor, R
AA
=
P
fin
(p
T
)/P
ini
(p
T
)(P
fin
: final p
T
-distributions), and elliptic
flow, v
2
=(p
2
x
− p
2
y
)/(p
2
x
+ p
2
y
)
p
T
(evaluated at fixed p
T
).
For c-quarks and pQCD scattering only, our results are in fair
agreement with those of Ref. [10] (recall that the Debye mass in
our calculations is given by m
g
= gT while in Ref. [10] it was
set to m
g
= 1.5T independent of α
s
). However, both R
AA
and
v
2
exhibit substantial sensitivity to the inclusion of resonance
contributions, increasing the effects of pQCD scattering by a
factor of ∼3–5. Also note the development of the plateau in
v
2
(p
T
>3 GeV) characteristic for incomplete thermalization of
HQs in the bulk matter.
034913-2
HEAVY-QUARK PROBES OF THE QUARK-GLUON . . . PHYSICAL REVIEW C 73, 034913 (2006)
FIG. 1. (Color online) Nuclear modification factor (upper panel)
and elliptic flow (lower panel) of charm and bottom quarks in b =
7fmAu-Au(
√
s = 200 GeV) collisions based on elastic rescattering
in the QGP. Red (green) and blue lines are for c-(b-) quarks with
and without resonance rescattering, respectively, where the bands
encompass resonance widths of = 0.4–0.75 GeV.
IV. HADRONIZATION AND SINGLE-ELECTRON
SPECTRA
Semileptonic single-e
±
spectra are a valuable tool to
investigate heavy-meson spectra in ultrarelativistic heavy-ion
collisions, since their decay kinematics largely conserves the
spectral properties of the parent particles [15,36]. To com-
pare our results to measured single-e
±
in Au-Au collisions,
the above HQ spectra have to be hadronized. To this end we
employ the coalescence approach of Ref. [15] based on earlier
constructed light-quark spectra [37]. Quark coalescence has
recently enjoyed considerable success in describing, e.g., the
“partonic scaling” of elliptic flow and the large p/π ratio in
Au-Au at RHIC [37–39], as well as flavor asymmetries
in D-meson production in elementary hadronic collisions
[40]. Whereas at low p
T
most of the HQs coalesce into
D- and B-mesons, this is no longer the case at higher
p
T
where t he phase space density of light quarks rapidly
decreases. Therefore, to conserve HQ number in the B-
and D-meson spectra, the remaining c- and b-quarks are
hadronized using δ-function fragmentation. Finally, single-e
±
p
T
- and v
2
-spectra are computed via B- and D-meson 3-body
decays, and compared to experiment in Fig. 2. We find
that the effects of r esonances are essential in improving the
agreement with data, both in terms of lowering the R
AA
and
increasing v
2
.TheB-meson contribution reflects itself by
limiting R
AA
and v
2
to values above 0.4 and below 10%,
respectively, as well as the reduction of v
2
above p
T
3GeV.
To better illustrate the effects of coalescence we plot in
Fig. 3 calculations where all HQs are fragmented into D- and
FIG. 2. (Color online) R
AA
(upper panel) and v
2
(lower panel)
of semileptonic D-andB-meson decay-electrons in b = 7fm
Au-Au(
√
s = 200 GeV) using HQ coalescence supplemented by
fragmentation (line identification as in Fig 1). The data [4,6–8] are
for minimum bias collisions, the R
AA
from STAR [5] for 10–40%
central collisions.
FIG. 3. (Color online) Same as Fig. 2 but using HQ fragmentation
only.
034913-3
HENDRIK VAN HEES, VINCENZO GRECO, AND RALF RAPP PHYSICAL REVIEW C 73, 034913 (2006)
B-mesons. While R
AA
is significantly reduced, most notably
in the p
T
1–2 GeV region, v
2
also decreases reaching at
most 6%, which is not favored by current data. It is, however,
conceivable that modifications in the fraction of coalescence
to fragmentation contributions, as well as improvements in
our schematic (δ-function) treatment of fragmentation, will
be necessary once more accurate experimental data become
available. Additional corrections may also arise from a more
precise determination of the b/c ratio and nuclear shadowing.
V. CONCLUSION
We have investigated thermalization and collective flow
of c- and b-quarks within a r elativistic Langevin approach
employing elastic scattering in an expanding QGP fireball in
semicentral Au-Au collisions at RHIC. Underlying drag and
diffusion coefficients were evaluated assuming resonant D-
and B-meson correlations in the sQGP, enhancing heavy-quark
rescattering. Corresponding p
T
-spectra and elliptic flow of
c-quarks exhibit a large sensitivity to the resonance effects,
lowering R
AA
down to 0.2 and raising v
2
up to 10%, while the
impact on b-quarks is small. Heavy-light quark coalescence
in subsequent hadronization significantly amplifies the v
2
in
single-electron decay spectra, but also increases their R
AA
,
especially in the p
T
2 GeV region. Bottom contribu-
tions dominate above 3.5 GeV reducing both suppression
and elliptic flow. The combined effects of coalescence and
resonant heavy-quark interactions are essential in generating
a v
e
2
of up to 10%, together with R
e
AA
0.5, supplying a
viable explanation of current electron data at RHIC without
introducing extra scale factors. Our analysis thus suggests
that elastic rescattering of heavy quarks in the sQGP is an
important component for the understanding of heavy-flavor
and single-electron observables in heavy-ion reactions at
collider energies. While induced gluon-radiation is expected
to be the prevalent interaction with the medium at sufficiently
high p
T
[1,2,14], it may not be the dominant effect below
p
T
6 GeV or so. A complete picture should clearly include
both elastic and inelastic rescattering mechanisms.
ACKNOWLEDGMENTS
One of us (H.vH.) thanks the Alexander von Humboldt
foundation for support within a Feodor Lynen fellowship.
This work was supported in part by a U.S. National Science
Foundation CAREER award under grant PHY-0449489.
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![FIG. 2. (Color online) RAA (upper panel) and v2 (lower panel) of semileptonic D- and B-meson decay-electrons in b = 7 fm Au-Au( √ s = 200 GeV) using HQ coalescence supplemented by fragmentation (line identification as in Fig 1). The data [4,6–8] are for minimum bias collisions, the RAA from STAR [5] for 10–40% central collisions.](/figures/fig-2-color-online-raa-upper-panel-and-v2-lower-panel-of-28r5iuju.webp)
