arXiv:0910.4853v2 [nucl-th] 19 Apr 2010
Nuclear suppression at low energy heavy ion collisions
Santosh K Das, Jan-e Alam, Payal Mohanty and Bikash Sinha
Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar , Kolkata - 700064
(Dated: April 20, 2010)
The effects of non-zero baryonic chemical potential on the drag and diffusion coefficients of heavy
quarks propagating through a baryon rich quark gluon plasma have been studied. The nuclear
suppression factor, RAA for non-photonic single electron spectra resulting from the semileptonic
decays of hadrons containing heavy flavours have been evaluated for low energy collisions. The role
of non-zero baryonic chemical potential on RAA has been highlighted.
PACS numbers: 12.38.Mh,25.75.-q,24.85.+p,25.75.Nq
The nuclear collisions at low energy RHIC run [1,
2] and GSI-FAIR  is expected to create a ther-
mal medium with large baryonic chemical potential
(µB) and moderate temperature (T).
flavours namely, charm and bottom quarks may play
a crucial role in understanding the properties of such
medium because they do not constitute the bulk
part of the system and their thermalization time
scale is larger than the light quarks and gluons and
hence can retain the interaction history very effec-
tively. The perturbative QCD (pQCD) calculations
indicate that the heavy quark (Q) thermalization
is larger [4, 5] than the light quarks and
gluons thermalization scale τi.
malized even before up and down quarks [6, 7]. In
the present work we assume that the Quark Gluon
Plasma (QGP) is formed at time τi.
the interaction of the non-equilibrated heavy quarks
with the equilibrated QGP for the time interval
τi< τ < τQ
Fokker-Planck (FP) equation [8, 9] i.e. the heavy
quark can be thought of executing Brownian mo-
tion [4, 6, 10–17] in the heat bath of QGP during
the said interval of time.
Gluons may ther-
i can be treated within the ambit of the
As the relaxation time for heavy quarks of mass
M at a temperature T are larger than the corre-
sponding quantities for light partons by a factor of
M/T(> 1)  i.e. the light quarks and the gluons
get thermalized faster than the heavy quarks, the
propagation of heavy quarks through QGP (mainly
contains light quarks and gluons) therefore, may be
treated as the interactions between equilibrium and
non-equilibrium degrees of freedom. The FP equa-
tion provide an appropriate frame work for such pro-
cesses. In case of low energy collisions the radiative
energy loss of of heavy quarks will be much smaller
than the loss due to elastic processes. Moreover, the
thermal production of charm and bottom quarks can
be ignored for the range of temperature and bary-
onic chemical potential under study. Therefore, the
FP equation is better applicable in the present situ-
The paper is organized as follows. In the next
section a brief description of Fokker Planck equa-
tion and the T and µq dependence of the drag and
diffusion co-efficients are outlined, the non-photonic
electron spectra is discussed in section III, the initial
conditions and the space time evolution have been
discussed in section IV, section V is devoted to the
discussions on nuclear suppression and finally sec-
tion VI contains the summary and conclusions.
II. THE FOKKER PLANCK EQUATION
The Boltzmann transport equation describing a
non-equilibrium statistical system reads:
where p and E denote momentum and energy, ∇x
(∇p) are spatial (momentum space) gradient and
f(x,p,t) is the phase space distribution (in the
present case f stands for heavy quark distribution).
The assumption of uniformity in the plasma and ab-
sence of any external force leads to
The collision term on the right hand side of the above
equation can be approximated as (see [11, 15] for
where we have defined the kernels
for | p |→ 0, Ai → γpi and Bij → Dδij where
γ and D stand for drag and diffusion co-efficients
respectively. The function ω(p,k) is given by
ω(p,k) = g
where f′is the phase space distribution, in the
present case it stands for light quarks and gluons,
v is the relative velocity between the two collision
partners, σ denotes the cross section and g is the
statistical degeneracy. The co-efficients in the first
two terms of the expansion in Eq. 3 are compara-
ble in magnitude because the averaging of ki in-
volves greater cancellation than the averaging of the
quadratic term kikj. The higher power of ki’s are
With these approximations the Boltzmann equa-
tion reduces to a non-linear integro-differential equa-
tion known as Landau kinetic equation:
The nonlinearity is caused due to the appearance of
f′in Aiand Bijthrough w(p,k). It arises from the
simple fact that we are studying a collision process
which involves two particles - it should, therefore,
depend on the states of the two participating parti-
cles in the collision process and hence on the prod-
uct of the two distribution functions. Considerable
simplicity may be achieved by replacing the distri-
bution functions of one of the collision partners by
their equilibrium Fermi-Dirac or Bose-Einstein dis-
tributions (depending on the statistical nature) in
the expressions of Aiand Bij. Then Eq. 6 reduces
to a linear partial differential equation - usually re-
ferred to as the Fokker-Planck equation describing
the interaction of a particle which is out of thermal
equilibrium with the particles in a thermal bath of
light quarks, anti-quarks and gluons. The quantities
Aiand Bij are related to the usual drag and diffu-
sion coefficients and we denote them by γiand Dij
respectively (i.e. these quantities can be obtained
from the expressions for Ai and Bij by replacing
the distribution functions by their thermal counter-
The evolution of the heavy quark distribution (f)
is governed by the FP equation 
where γi and Dij are the drag and diffusion coef-
ficients. The elastic collisions of heavy quarks (Q
stands for charm and bottom) with thermal light
quarks (q), anti-quarks (¯ q) and gluons (g) i.e. Qq →
Qq, Q¯ q → Q¯ q and Qg → Qg have been used to eval-
uate the drag and diffusion coefficients as indicated
in [11, 18]. The thermal distribution of the quarks
and the anti-quarks are responsible for the T and µq
dependence of the drag and diffusion co-efficients.
The gluon distribution introduces the T dependence
to these quantities. chemical potential dependence
of the drag and diffusion co-efficients originate from
the thermal phase space of quarks and anti-quarks.
A.The drag and diffusion co-efficients
At low√sNN the net baryon density at mid-
rapidity is non-zero and its value could be high de-
pending on the value of√sNN. Therefore, we need
to solve the FP equation for non-zero µB. The drag
and diffusion coefficients are functions of both the
thermodynamical variables: µBand T.
The energy dependence of the chemical potential
has been obtained from the parametrization of the
experimental data on hadronic ratios as  (see
µB(sNN) = a(1 +√sNN/b)−1
where a = 0.967± 0.032 GeV and b = 6.138± 0.399
GeV. The parametrization in Eq. 8 gives the values
of µB at the freeze-out. The corresponding values
at the initial condition are obtained from the baryon
number conservation equation. The initial baryonic
chemical potential carried by the quarks µq(= µB/3)
are shown in table 1 for various√sNNunder consid-
In the present work we take αs= 0.3 because the
dependence of the strong coupling on the temper-
ature and baryonic chemical potential is not accu-
rately known yet. The sensitivity of collisional en-
ergy loss on the running αs is studied in Ref. 
in detail. The variation of the drag coefficients of
charm quarks (due to its interactions with quarks
and anti-quarks) on the baryonic chemical potential
for different T are displayed in Fig. 1. The drag co-
efficient for the process : Qg → Qg is ∼ 8.42×10−3
fm−1(1.86 × 10−2fm−1) for T = 140 MeV (190
MeV) (not displayed in Fig. 1). The T and µq de-
pendence of the drag and diffusion co-efficients may
be understood as follows. The drag may be defined
as the thermal average of the square of the invari-
ant transition amplitude weighted by the momentum
transfer for the reactions qQ → qQ, Q¯ q → Q¯ q and
gQ → gQ. As the temperature of the thermal bath
increases the light quarks (q) and the gluons move
faster and gain the ability to transfer larger momen-
tum during their interaction with the heavy quarks
- resulting in the increase of the drag of the heavy
quarks propagating through the partonic medium.
Since the average momentum of the quarks increases
with µq, similar behaviour is expected in the varia-
tion of drag with baryonic chemical potential. This
trend is clearly observed in the results displayed in
Fig. 1 for charm quark. The drag due to the pro-
cess Qq → Qq is larger than the Q¯ q → Q¯ q interac-
tion because for non-zero chemical potential, the Q
propagating through the medium encounters more q
than ¯ q at a given µq. For vanishing chemical poten-
tial the contributions from quarks and anti-quarks
In the same way it may be argued that the dif-
fusion coefficient involves the square of the momen-
tum transfer - which should also increase with T and
µq as observed in Fig. 2. The diffusion co-efficient
for charm quarks due to its interaction with gluons
is given by ∼ 1.42 × 10−3GeV2/fm (4.31 × 10−3
GeV2/fm) for T = 140 MeV (190 MeV). It may
be mentioned here that the drag increases with T
when the system behave like a gas. In case of liq-
uid the drag may decrease with temperature (ex-
cept very few cases) - because a substantial part of
the thermal energy goes in making the attraction
between the interacting particles weaker - allowing
them to move more freely and hence making the drag
force lesser. The drag co-efficient of the partonic
system with non-perturbative effects may decrease
with temperature as shown in Ref.  - because in
this case the system interacts strongly more like a
liquid. The heavy quark momentum diffusion co-
efficient has been computed  at next to leading
order within the ambit of hard thermal loop approxi-
mations. For T ∼ 400 MeV the momentum averaged
pQCD value (for µq= 0) of the diffusion co-efficient
obtained in the present work is comparable to the
value obtained in  in the leading order approxi-
mation for the same set of inputs (e.g. strong cou-
pling constant, number of flavours etc). The drag
and diffusion coefficients for bottom quarks are dis-
played in Figs. 3 and 4 respectively, showing qual-
itatively similar behaviour as charm quarks. The
drag co-efficients for bottom quarks due to the pro-
cess Qg → Qg is given by ∼ 3.15 × 10−3fm−1and
6.93×10−3fm−1at T = 140 MeV and 190 MeV re-
spectively. The corresponding diffusion coefficients
are ∼ 1.79×10−3GeV2/fm and 5.38×10−3GeV2/fm
at T=140 MeV and 190 MeV respectively.
III.THE NON-PHOTONIC ELECTRON
After obtaining the drag and diffusion coefficients
we need the initial heavy quark momentum distri-
butions for solving the FP equation. For low col-
lision energy rigorous QCD based calculations for
heavy flavour production is not available (for higher
√sNN = 200 GeV see Ref.  for rigorous QCD
calculations). In the present work this is obtained
from pQCD calculation [25, 26] for the processes:
gg → Q¯Q and q¯ q → Q¯Q. Here we intend to deal
with the nuclear suppression factor, RAA, which in-
volves the ratio of the momentum distribution func-
tions. Therefore, the final results may not be too
sensitive to the initial distributions because of some
cancellations that may take place in the ratio.
00.1 0.2 0.30.4
FIG. 1: Variation of the drag coefficient of charm quark
due to its interactions with light quarks and anti-quarks
as a function of µq for different temperatures.
FIG. 2: Variation of the diffusion coefficient of charm
quark due to its interactions with light quarks and anti-
quarks as a function of µq for different temperatures.
With the initial condition mentioned above the
FP equation has been solved for the heavy quarks.
We convolute the solution with the fragmenta-
tion functions of the heavy quarks to obtain the
pT distribution of the heavy (B and D) mesons
(dND,B/qTdqT). For heavy quark fragmentation we
use Peterson function  given by:
for charm quark ǫc= 0.05. For bottom quark ǫb=
(Mc/Mb)2ǫcwhere Mc(Mb) is the charm (bottom)
FIG. 3: Same as Fig. 1 for bottom quark.
0 0.1 0.2
FIG. 4: Same as Fig. 2 for bottom quark.
The non-photonic single electron spectra originate
from the decays of heavy flavour mesons - e.g. D →
Xeν at mid-rapidity (y = 0) can be obtained as
F(pT,qT) = ω
where M is the mass of the heavy mesons (D or B),
ω = 96(1 − 8m2+ 8m6− m8− 12m4lnm2)−1M−6
(m = MX/M) and g(Ee) is given by
(M − 2Ee)
related to the rest frame spectrum for the decay D →
Xeν through the following relation 
= ωg(Ee). (13)
We evaluate the electron spectra from the decays
of heavy mesons originating from the fragmentation
of the heavy quarks propagating through the QGP
medium formed in heavy ion collisions. In the same
way the electron spectrum from the p-p collisions
can be obtained from the charm and bottom quark
distribution which goes as initial conditions to the
solution of FP equation.
quantities gives the nuclear suppression, RAAas :
The ratio of these two
called the nuclear suppression factor, will be unity in
the absence of any medium. In the above equation
Ncoll denotes the number of nucleon nucleon colli-
sions in Au+Au interaction. However, the experi-
mental data [31, 32] at RHIC energy (√sNN=200
GeV) shows substantial suppression (RAA< 1) for
pT ≥ 2 GeV indicating substantial interaction of
the plasma particles with charm and bottom quarks
from which electrons are originated through the
process: c(b) (hadronization)−→ D(B)(decay)−→
e+X. The loss of energy of high momentum heavy
quarks propagating through the medium created in
Au+Au collisions causes a depletion of high pT elec-
IV.THE INITIAL CONDITIONS FOR THE
The nuclear suppression for heavy quarks depend
on the parameters like initial temperature (Ti), ther-
malization time (τi), equation of state (EOS) and
the transition temperature (Tc).
We assume that the system reaches equilibration
at a time τi after the collision at temperature Ti
which are related to the produced hadronic (predom-
inantly mesons) multiplicity through the following
Net suppression(µq=0.2 GeV)
FIG. 5: The nuclear suppression factor RAA as a func-
tion of pT due to the interaction of the charm quark
(solid line) and anti-quark (dashed-dot line) for µq = 200
MeV. The net suppressions including the interaction
of quarks, anti-quarks and gluons for µq = 200 MeV
(dashed line) and µq = 0 (with asterisk) are also shown.
where RA is the radius of the system, ζ(3) is the
Riemann zeta function and aeff= π2geff/90 where
geff (= 2 × 8 + 7 × 2 × 2 × 3 × NF/8) is the degen-
eracy of quarks and gluons in QGP, NF=number of
The value of the multiplicities for various√sNN
have been calculated from the Eq. below ;
(1 − x)< Npart>
+ x < Ncoll>
Ncoll is the number of collisions and contribute x
fraction to the multiplicity dnpp/dy measured in pp
collision. The number of participants, Npart con-
tributes a fraction (1−x) to dnpp/dy, which is given
= 2.5 − 0.25ln(s) + 0.023ln2(s)(17)
The values of Npart and Ncoll are estimated for
(0 − 5%) centralities by using Glauber Model. The
value of x depends very weakly on√sNN , in
the present work we have taken x = 0.1 for all the
The time evolution of the temperature and the
baryon density (nB) have been obtained by solving
the following equations:
in (1+1) dimension with boost invariance along the
longitudinal direction . In the above equation
TABLE I: The values center of mass energy , dN/dy,
initial temperature (Ti) and quark chemical potential -
used in the present calculations.
Tµν= (ǫ + P)uµuν− gµνP, is the energy momen-
tum tensor and nµ
where ǫ is the energy density, P is the pressure,
and uµis the hydrodynamic four velocity. The ra-
dial co-ordinate dependence of T and nBhave been
parametrized as in Ref. . The velocity of sound
for the QGP phase is taken as cs = 1/√4. Some
comments on the effects of the radial flow are in
order here. The radial expansion will increase the
size of the system and hence decrease the density of
the medium. Therefore, with radial flow the heavy
quark will traverse a larger path length in a medium
of reduced density. These two oppositely competing
effects may have negligible effects on the nuclear sup-
pression (see also ). Moreover, at lower collision
energies (as in the present case) the amount of ra-
dial flow will not be as substantial as in√sNN= 200
The total amount of energy dissipated by a heavy
quark in the QGP depends on the path length it tra-
verses. Each parton traverses different path length
which depends on the geometry of the system and
on the point where its is produced. The probabil-
ity that a parton is created at a point (r,φ) in the
plasma depends on the number of binary collisions
at that point which can be taken as :
B= nBuµis the baryonic flux,
R2)θ(R − r) (19)
where R is the nuclear radius. A parton created at
(r,φ) in the transverse plane propagate a distance
?R2− r2sin2φ − rcosφ in the medium. In the
present work we use the following equation for the
averaging of the drag coefficient:
where v is the velocity of the propagating partons.
Similar averaging has been performed for the dif-
fusion co-efficient. For a static system the T and
µqdependence of the drag and diffusion co-efficients
of the heavy quarks enter via the thermal distribu-
tions of light quarks, anti-quarks and gluons through
which it is propagating. However, in the present
scenario the variation of the temperature and the
baryon density with time are governed by the equa-
tion of state or the velocity of sound of the thermal-
ized system undergoing hydrodynamic expansion. In
such a scenario the quantities like Γ (Eq. 20) and
hence RAAbecomes sensitive to velocity of sound in
FIG. 6: Nuclear suppression factor, RAA as function of
pT for various√sNN. Inset: the nuclear suppression
factor due to the interaction of D meson in a thermal
medium of pions and nucleons.
V. THE NUCLEAR SUPPRESSION
To demonstrate the effect of non-zero baryonic
chemical potential we evaluate RAA for µq = 200
MeV and µq= 0 for a given Ti= 200 MeV. The re-
sults are displayed in Fig. 5 - representing the com-
bined effects of temperature and baryon density on
the viscous drag and diffusion. The viscous drag on
the heavy quarks due to its interaction with quarks
is larger than that of its interactions with the anti-
quarks (Fig.1). Resulting in larger suppression in
the former case than the later. The net suppression
of the electron spectra from the Au+Au collisions
compared to p+p collisions is effected by quarks,
anti-quarks and gluons.
pressions are displayed for µq = 200 MeV (dashed
line) and µq= 0 (with asterisk). The experimental
detection of the non-zero baryonic effects will shed
light on the net baryon density (and hence baryon
stopping) in the central rapidity region. However,
whether the effects of non-zero baryonic chemical po-
tential is detectable or not will depend on the overall
The results for RAAare shown in Fig. 6 for vari-
ous√sNNwith inputs from table I. We observe that
The results for net sup-
at large pT the suppression is similar for all energies
under consideration. This is because the collisions
at high√sNNare associated with large temperature
but small baryon density at mid-rapidity- which is
compensated by large baryon density and small tem-
perature at low√sNN collisions. Low pT particles
predominantly originate from low temperature and
low density part of the evolution where drag is less
and so is the nuclear suppression.
So far we have discussed the suppression of the
non-photonic electron produced in nuclear collisions
due to the propagation of the heavy quark in the the
partonic medium in the pre-hadronization era. How-
ever, the suppression of the D mesons in the post
hadronization era (when both the temperature and
density are lower than the partonic phase) should in
principle be also taken into account. We have esti-
mated RAAfor D mesons due to ts interaction with
pions  and nucleons  and found that it has a
value value closer to unity, indicating the fact that
the hadronic medium (of pions and nucleons) is un-
able to drag the D mesons. Therefore, the measured
depletion in RAAfor the non-photonic electron will
indicate the presence of partonic medium and the
amount of depletion may the used to characterize
the thermal medium.
It has been shown in  that a large enhance-
ment of the pQCD cross section is required for the
reproduction of experimental data on elliptic flow at
RHIC energies. In our earlier work  we have eval-
uated the RAAfor non-photonic single electron spec-
tra resulting from the semileptonic decays of hadrons
containing heavy flavours and observed that the data
from RHIC collisions at√sNN= 200 GeV are well
reproduced by enhancing the pQCD cross sections
by a factor 2 and with an equation of state P = ǫ/4.
In the same spirit we evaluate RAA with twice en-
hanced pQCD cross section and keeping all other
quantities unaltered (Fig. 7). The results in Fig. 7
show stronger suppression as compared to the re-
sults displayed in Fig. 6, but it is similar in all the
energies under consideration.
VI. SUMMARY AND CONCLUSIONS
We have studied the effects of baryonic chemical
potential and temperature on the drag and diffusion
coefficients of heavy quarks moving in a thermalized
system of quarks and gluons. We have observed that
both the drag and diffusion coefficients increase with
temperature and chemical potential. When we have
enhanced the pQCD cross section for the interac-
tion of the heavy quarks with the thermal system
by a factor of two - the resulting suppressions in
RAAare between 20% − 30% for√sNN = 39 − 7.7
GeV. The radiative energy loss [39–43] (see  for
FIG. 7: Same as Fig. 6 with enhancement of cross section
by a factor of 2.
a review) of heavy quarks is suppressed due to dead
cone effects and has been neglected in the present
work. Moreover, at low collisions energies the colli-
sional loss [21, 45, 46] is dominant over its radiative
counter part (see  for details). It may be men-
tioned here that the theoretical formalism, the FP
equation is applicable better for heavy quarks than
light quarks and gluons (because of their frequent
productions and annihilations). However, the pro-
duction of charm and bottom quarks are smaller at
low energy collisions making the measurements of
non-photonic single electron spectra and hence RAA
for heavy quarks difficult. The detection of the non-
zero baryonic chemical potential effects observed in
the present work through the nuclear suppression
factor will help in determining the net baryon den-
sity (and hence baryon stopping) in the mid-rapidity
region. However, whether such effects is detectable
experimentally or not will depend on the overall ex-
Acknowledgment: We thank Bedangadas Mo-
hanty and Jajati K Nayak for useful discussions.
This work is supported by DAE-BRNS project Sanc-
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