2301 00187

Prévia do material em texto

Collective modes in the anisotropic collisional hot QCD medium in the presence of
finite chemical potential
Mohammad Yousuf Jamal1, ∗
1Indian Institute of Technology Goa, Ponda, Goa, 403401, India
The collective modes that are the collective excitations in the hot QCD medium produced in
the heavy-ion collision experiments have been studied. These modes being real or imaginary and
stable or unstable affect the evolution of the medium. To go into the more profound analysis we
incorporated several aspects of the medium such as anisotropy, the presence of medium particle
collisions, and finite baryonic chemical potential. The first two facets have been studied several
times from different perspectives but the inclusion of finite chemical potential is a missing part.
Therefore, to have a close analysis, we have combined these effects. The medium particle collision
has been included using BGK- collisional kernel, the anisotropy, and finite chemical potential enter
via quarks, anti-quarks and gluons distribution functions. Our results indicate that the presence
of finite chemical potential enhances the magnitude of unstable modes which may affect the fast
thermalization of the hot QCD medium. Moreover, it is interesting to see the consequences of
finite chemical potential along with the other aspects of the created medium from the viewpoint
of low-energy heavy-ion collision experiments that operate at low temperatures and finite baryon
density.
Keywords: Quark-Gluon-Plasma, Collective excitations, Collective modes, particle distribu-
tions function, BGK collisional kernel, Anisotropic QCD, Gluon self-energy, finite chemical
potential.
I. INTRODUCTION
In the relativistic heavy-ion collision (HIC) experi-
ments we obtain such an extremely high energy density
than any we know of in the present natural universe and
where the QCD predicts a new form of matter consisting
of an extended volume of interacting quarks, antiquarks,
and gluons known as quark-gluon plasma (QGP) [1, 2].
Such a high energy density phase, however, is thought
to have existed a few microseconds after the Big Bang.
In this phase, the effective degrees of freedom are quarks,
anti-quarks, and gluons rather than the hadronic matter.
The historic discovery of the J/Ψ, a bound state of the
charmed quark and its anti-quark in the year 1974 first
time proved the existence of the heavy quarks which indi-
cated that the nucleus–nucleus collisions at high energy
are very different from a simple superposition of nucleon-
nucleon interactions. As it mimics the cosmic Big Bang,
the term mini-bang has been coined to describe these in-
teractions, in which nuclear collisions are thought to pro-
ceed in a series and cause the formation of matter. The
presumed line of events begins with an extremely high-
temperature region inhabited by the two nuclei at the
moment of collision, as a large fraction of their kinetic
energy is converted into heat. At thermal equilibrium
it forms a high-temperature plasma medium consisting
of quarks, anti-quarks, and gluons that instantly begins
to expand and cool, passing down through the critical
temperature at which the QGP condenses into a system
of mesons, baryons, or simply hadrons. On further ex-
pansion, the system reaches its “freeze-out” density, at
∗ mohammad@iitgoa.ac.in
which the hadrons no longer interact with each other and
stream into the detectors.
There are several theoretical investigations based on
various approaches such as semi-classical transport or ki-
netic, hard-thermal loop (HTL), ASD-CFT, holographic
theories have been invoked to study the produced mat-
ter [3, 4]. There have been several signatures observed
that support the formation of the QGP in these experi-
ments such as suppression of heavy quarkonia (a colorless
and flavorless bound state of a heavy quark and its anti-
quark) yields at high transverse momentum, color screen-
ing [5–8], Landau damping [9, 10], energy loss or gain of
heavy quarks [11–15], etc. The observables that we see
at the detector are affected by the presence of the QGP
medium that in turn, is influenced by several plasma as-
pects. One of the important factors among them is the
collective excitations or collective modes produced in the
hot QGP medium. It has been explored in previous stud-
ies that these modes can be real or imaginary, stable or
unstable [16–21]. Each of them plays a significant role
in altering the observed outcomes at the detector.
In many-body systems, the spectrum of collective exci-
tations is a fundamental part as it possesses information
regarding the thermodynamic and transport properties
of that system along with the temporal evolution of a
non-equilibrium system [22]. The investigation begins
with the analogical idea from Quantum Electro Dynam-
ics (QED) that says that the evolving medium produced
in the HIC could also contain instabilities that are sim-
ilar to Weibel or filamentation instabilities [23] present
in electromagnetic plasma (EMP) that may contribute
to accomplishing the fast thermalization [24]. In par-
ticular, at very high-temperature, QCD resembles QED
due to the small coupling constant. The most common
ar
X
iv
:2
30
1.
00
18
7v
1 
 [
nu
cl
-t
h]
 3
1 
D
ec
 2
02
2
mailto:mohammad@iitgoa.ac.in
2
feature of EMP and QGP is collective behavior that sug-
gests the QGP could also have a very rich spectrum of
collective modes they are there in EMP [25, 26]. In the
present context, these modes could refer to plasma parti-
cles that could be real or imaginary and stable or unsta-
ble. The stable modes have infinite lifetimes whereas the
unstable modes decay quickly. The stable modes have a
long-range interaction with the heavy quarks traversing
through the QGP medium whereas the unstable modes
are only found in anisotropic plasma and may help in the
fast thermalization/equilibration of the system, a lesser-
known puzzle that attracts the curiosity of investigating
the unstable modes and hence, the prior is less studied
and discussed in the literature though the latter has been
discussed widely.
In several analyses the question about the formation,
existence, and effects of the collective modes have been
discussed considering various plasma aspects such as
anisotropy, medium particle collisions, etc, [27–34]. Here,
we are interested in a crucial aspect which is to under-
stand the inclusion and effects of finite baryon density in
the study of these modes. Thrusting to very high baryon
density at low temperatures, i.e., squeezing nuclei with-
out heating them takes us into another fascinating region
of the QCD phase diagram. The high baryon density gen-
erally means the surplus of quarks over antiquarks in the
hot system parameterized by the baryon chemical poten-
tial, µ. The zero chemical potential refers to the equal
densities of quarks and antiquarks which is a good ap-
proximation for the matter produced at midrapidity at
very high energy HIC such as RHIC and LHC, and ex-
ceptionally good for the early Universe but not at the
lower energies. It is essential from the viewpoint of the
experimental facilities which perform at moderate tem-
peratures along with finite baryon density such as the
Super Proton Synchrotron (SPS) at CERN, Switzerland,
and also upcoming planned experimental facilities such
as the Facility for Antiproton and Ion Research (FAIR)
in Darmstadt, Germany and the Nuclotron-Based Ion
Collider Facility (NICA) in Dubna, Russia, and at the
Japan Proton Accelerator Research Complex (J-PARC)
in Tōkai, Japan. To have a comprehensive study of the
collective modes we shall also include the momentum
anisotropy and medium particle collisions along with the
finite chemical potential. The momentum anisotropy and
the chemical potential will enter via particle distribution
functions and for medium particle collisions we shall con-
sider the Bhatnagar–Gross–Krook (BGK) collisional ker-
nel in the Boltzman transport equation. The dispersion
relations for the modes are acquired from the poles of
the (gluon)propagator. Based on their solutions it can
be identified if the modes are stable or unstable and real
or imaginary which we shall discuss in detail in the up-
coming sections.
The current manuscript is arranged as follows. In sec-
tion II, the methodology to obtain the dispersion rela-
tions of the modes will be discussed. Section III, con-
tains a detailed discussion of the results. Finally, sec-
tion IV, is dedicated to the summary and conclusions of
the present work along with the potential future prob-
lems. Here, natural units are used throughout the text
with c = kB = ~ = 1. We use a bold typeface to dis-
play three vectors and a regular font to indicate four
vectors. The centre dot depicts the four-vector scalar
product with gµν = diag(1,−1,−1,−1).
II. METHODOLOGY
The formation of collective modes in the QGP medium
can be understood as if one assumes the QGP initially
in a homogeneous and stationary state with no net lo-
cal color charges and no net currents. Next, suppose
this state is slightly perturbed by either a random fluc-
tuation or some external field that induces local charges
or currents in the plasma that in turn generate chromo-
electric and chromo-magnetic fields. These fields interact
back with colored partons that contributed to their for-
mation. If the wavelength of the perturbation surpasses
the typical inter-particle spacing, the plasma undergoes a
collective motion known as collective modes/excitations.
The collective modes can be specified by the nature of
the solutions of their dispersion relations (ω(k), k being
the wave vector) that can be fetched from the poles of the
gluon propagator. The solutions to the dispersion equa-
tions, ω(k) could be complex-valued. For =(ω(k)) = 0
there exist only stable modes. If =(ω(k)) > 0 the am-
plitude of the mode exponentially grows with time as
e=(ω(k))t that represents the instability. If =(ω(k)) < 0
the mode is damped and the mode amplitude exponen-
tially decays with time as e−=(ω(k))t. Likewise, if we have,
−=(ω(k)) ≥ |<(ω(k))| the modes will be over damped.
To obtain the gluon propagator we first derive the gluon
polarization tensor that contains the medium informa-
tion such as medium anisotropy, finite chemical poten-
tial, the effect of medium particle collision, etc that we
shall discuss next.
A. Gluon polarization tensor
The gluon polarization tensor or gluon self-energy
(Πµν), bears the information of the QCD medium as it
represents the interactions term in the effective action
of the QCD. Before we start our derivation, we spec-
ify that our working scale is the soft momentum scale
where the plasma aspects first appear, i.e., k ∼ gT � T ,
( g being the strong coupling constant). At this scale,
one can obtain that the strength of field fluctuations,
Aµ ∼ O(√gT ), and the derivatives, ∂x ∼ O(gT ). Now,
if we examine the field strength tensor,
Fµνa = ∂
µAνa − ∂νAµa − ig [A
µ
b , A
ν
c ] , (1)
we notice that the order of the non-abelian term is O(g2)
which is smaller than the order of the first two terms i.e.,
O(g3/2) in Fµν and hence can be neglected. Now we end
3
up with the abelian part of Fµν . Next, in the abelian
limit, the linearized semi-classical transport equations for
each color channel are given as [16, 28, 31, 35],
vµ∂µδf
i
a(p,X) + gθivµF
µν
a (X)∂
(p)
ν f
i(p) = Cia(p,X),
(2)
where the index ‘i‘ refers to the particle species (quark,
anti-quark, and gluon), ‘a‘, ‘b‘, and‘c‘ are the color index
and θi ∈ {θg, θq, θq̄} and have the values θg = θq = 1
and θq̄ = −1. The four vectors xµ = (t,x) = X and
vµ = (1,v) = V , are respectively, the four space-time
coordinate and the velocity of the plasma particle with
v = p/|p|. The ∂µ, ∂(p)ν are the partial four derivatives
corresponding to space and momentum, respectively. As
mentioned earlier the collisional term Cia(p,X), is the
BGK - kernel [28, 35–37] that describes the effects of
collisions between hard particles in a hot QCD medium
given as,
Cia(p,X) = −ν
[
f ia(p,X)−
N ia(X)
N ieq
f ieq(|p|)
]
, (3)
where,
f ia(p,X) = f
i(p) + δf ia(p,X), (4)
are the distribution functions of quarks, anti-quarks and
gluons, f i(p) is equilibrium part while, δf ia(p,X) per-
turbed part of the distribution function and
N ia(X) =
∫
d3p
(2π)3
f ia(p,X),
N ieq =
∫
d3p
(2π)3
f ieq(|p|) =
∫
d3p
(2π)3
f i(p). (5)
is the particle number and its equilibrium value. The
BGK - kernel [36] depicts the equilibration of the system
due to the collisions in a time proportional to ν−1. Here,
we consider the collision frequency ν to be independent of
momentum and particle species. We preferred to choose
BGK -kernel because it conserves the particle number
instantaneously as,∫
d3p
(2π)3
Cia(p,X) = 0. (6)
At finite baryon or quark chemical potential µ, the
momentum distributions of gluon, quark, and anti-quark
are given as,
fg =
exp[−βEg](
1− exp[−βEg]
) ,
fq =
exp[−β(Eq − µ)](
1 + exp[−β(Eq − µ)]
) ,
fq̄ =
exp[−β(Eq + µ)](
1 + exp[−β(Eq + µ)]
) . (7)
To include the momentum anisotropy, we track the strat-
egy given in Ref. [31]. In this method, the anisotropic dis-
tribution function was obtained from the isotropic distri-
bution function given in Eq. (7) by rescaling (stretching
and squeezing) in one direction of the momentum space
as follows,
f(p) ≡ fξ(p) = f(
√
p2 + ξ(p · n̂)2), (8)
where, n̂ is an unit vector (n̂2 = 1) showing the di-
rection of momentum anisotropy and ξ the strength of
anisotropy. It refers to squeezing when ξ > 0 and stretch-
ing when −1 < ξ < 0 of the distribution function in the
n̂ direction.
Next, the gluon polarization tensor can be obtained
from the current induced due to the change in the parti-
cles distribution functions in the Fourier space as,
Πµνab (K) =
∂Jµind a(K)
∂Abν(K)
, (9)
where the induced current is given by [28, 31, 35, 38],
Jµind,a(X) = g
∫
d3p
(2π)3
V µ{2Ncδfga (p,X) +Nf [δfqa(p,X)
− δf q̄a(p,X)]}. (10)
Rewriting Eq. (2) as,
vµ∂µδf
i
a(p,X) + gθivµF
µν
a (X)∂
(p)
ν f
i(p) =
ν
(
f ieq(|p|)− f i(p)
)
− νδf ia(p,X)
+
νf ieq(|p|)
N ieq
(∫
d3p′
(2π)3
δf ia(p
′, X)
)
. (11)
On solving Eq.(11) for δf ia(p,X) in the Fourier space we
obtained,
4
δf ia(p,K) =
−igθivµFµνa (K)∂
(p)
ν f i(p) + iν(f ieq(|p|)− f i(p)) + iνf ieq(|p|)
(∫
d3p
(2π)3 δf
i
a(p
′,K)
)
/Neq
ω − v · k + iν
, (12)
where, δf i(p,K) and Fµν(K), are the Fourier trans-
forms of δf i(p,X) and Fµν(X), respectively. Where,
K = kµ = (ω,k) Now taking the Fourier transform of the
induced current from Eq. (10) and employing δf ia(p,K),
from Eq. (12) we get,
Jµind,a(K) = g
2
∫
d3p
(2π)3
vµ∂(p)ν f(p)Mνα(K,V )D−1(K,v, ν)Aαa + giν{2NcSg(K, ν) +Nf (Sq(K, ν)− S q̄(K, ν))}
+g(iν)
∫
dΩ
4π
vµD−1(K,v, ν)
∫
d3p′
(2π)3
[
g∂(p
′)
ν f(p
′)Mνα(K,V ′)D−1(K,v′, ν)W−1(K, ν)Aαa
+iν(feq(|p′|)− f(p′))D−1(K,v′, ν)
]
W−1(K, ν) , (13)
where,
D(K,v, ν) = ω + iν − k · v,
Mνα(K,V ) = gνα(ω − k · v)−Kνvα,
f(p) = 2Ncf
g(p) +Nf
[
fq(p) + f q̄(p)
]
,
feq(|p′|) = 2Ncfgeq(|p′|) +Nf
[
fqeq(|p′|) + f q̄eq(|p′|)
]
,
W(K, ν) = 1− iν
∫
dΩ
4π
D−1(K,v, ν),
Si(K, ν) = −
∫
d3p
(2π)3
vµ[f i(p)− f ieq(|p|)]D−1(K,v, ν).
(14)
Implying Eq. (9) in Eq.(13), we get
Πµνab (K) = δabg
2
∫
d3p
(2π)3
vµ∂
(p)
β f(p)M
βν(K,V )
×D−1(K,v, ν) + δabg2(iν)
∫
dΩ
4π
vµ
×D−1(K,v, ν)
∫
d3p′
(2π)3
∂
(p′)
β f(p
′)
Mβν(K,V ′)D−1(K,v′, ν)W−1(K, ν).
(15)
Rewriting Eq. (15), in temporal gauge for anisotropic hot
QCD medium at finite chemical potential,
Πij(K) = m2D
∫
dΩ
4π
vi
vl + ξ(v · n̂)nl
(1 + ξ(v · n̂)2)2
[
δjl(ω − k · v) + vjkl
]
D−1(K,v, ν) + (iν)m2D
∫
dΩ′
4π
(v′)i
×D−1(K,v′, ν)
∫
dΩ
4π
vl + ξ(v · n̂)nl
(1 + ξ(v · n̂)2)2
[
δjl(ω − k · v) + vjkl
]
D−1(K,v, ν)W−1(K, ν), (16)
where the squared Debye mass is given as,
m2D = −
g2
2π2
∫ ∞
0
dp p2
dfiso(p)
dp
, (17)
In the limit µ/T � 1, we have
m2D = 4παsT
2
[
Nc
3
+
Nf
6
+
µ2
T 2
Nf
2π2
]
, (18)
and the strong coupling constant [39], αs = g
2/4π at
finite chemical potential is given as,
αs =
6π −
18π(153−19Nf ) ln
(
2 ln
(√(
T
ΛT
)2
+ µ
2
π2T2
))
(33−2Nf )2 ln
(√(T
ΛT
)2
+ µ
2
π2T2
)
(33− 2Nf ) ln
(√(
T
ΛT
)2
+ µ
2
π2T 2
) .
(19)
5
where ΛT � T is the QCD scale parameter. Here,
Tc = 0.155 GeV
T=2Tc
T=4Tc
0 20 40 60 80 100
0
10
20
30
40
μ (GeV)
m
D
[μ
,T
]
(G
eV
)
FIG. 1. Variation of screening mass with respect to chemical
potential at different temperatures.
Eq. (17) is fully solved numerically and the variation
of screening mass with respect to the chemical poten-
tial at different temperatures is shown in Fig. 1. It has
been found that mD increases with the chemical poten-
tial whereas the effect of different values of temperature
is negligible. Next, we shall discuss the derivation of the
gluon propagator.
B. Gluon Propagator
To obtain the gluon propagator, we initiate with
Maxwell’s equation in the Fourier space,
−ikνF νµ(K) = Jµind(K) + J
µ
ext(K). (20)
Here, Jµext(K) is the external current. The induced cur-
rent, Jµind(K) can be described in terms of self-energy,
Πµν(K) as follows,
Jµind(K) = Π
µν(K)Aν(K). (21)
The Eq.( 20) can also be noted as,
[K2gµν − kµkν + Πµν(K)]Aν(K) = −Jµext(K). (22)
The quantity in the square bracket is the needed gluon
propagator. Now assuming the temporal gauge, the
above equation can be written as,
[∆−1(K)]ijEj = [(k2 − ω2)δij − kikj + Πij(K)]Ej
= iωJ iext(k), (23)
where Ej is the physical electric field and
[∆−1(K)]ij = (k2 − ω2)δij − kikj + Πij(K), (24)
is the inverse of the propagator. The dispersion equations
for collective modes can be obtained from the poles of the
propagator, [∆(K)]ij . Since, it is a tensorial equation
and hence, can not be simply integrated. To solve it we
first need to construct an analytical form of Πij using the
available symmetric tensors such as,
Πij = αP ijT + βP
ij
L + γP
ij
n + δP
ij
kn, (25)
where, P ijT = δ
ij − kikj/k2, P ijL = kikj/k2, P ijn =
ñiñj/ñ2 and P ijkn = k
iñj + kj ñi [31, 33, 40], where,
ñi = (δij − k
ikj
k2 )n̂
j is a vector orthogonal to, ki ,i.e.,
ñ · k = 0. The structure functions α, β, γ and δ, can be
determined by taking the appropriate projections of the
Eq.(25) as follows,
α = (P ijT − P
ij
n )Π
ij , β = P ijL Π
ij ,
γ = (2P ijn − P
ij
T )Π
ij , δ =
1
2k2ñ2
P ijknΠ
ij . (26)
The structure functions mainly depend on k, ω, ξ,
k · n̂ = cos θn, and ν. We confined our analysis in the
small anisotropy limit, ξ < 1, where all the structure
functions can be calculated analytically up to linear or-
der in ξ, given as
α = (P ijT − P
ij
n )Π
ij , β = P ijL Π
ij ,
γ = (2P ijn − P
ij
T )Π
ij , δ =
1
2k2ñ2
P ijknΠ
ij . (27)
The structure functions mainly depend on k, ω, ξ ν
and k · n̂ = cos θn. In the small anisotropy limit, ξ < 1,
all the structure functions can be calculated analytically
up to linear order in ξ, given as
α (K) =
m2D
48k
(
24kz2 − 2kξ
(
9z4 − 13z2 + 4
)
+ 2iνz
(
ξ
(
9z2 − 7
)
− 12
)
− 2ξ cos 2θn
(
k
(
15z4 − 19z2 + 4
)
+iνz
(
13− 15z2
))
+
(
z2 − 1
) (
3ξ
(
kz
(
5z2 − 3
)
+ iν
(
1− 5z2
))
cos 2θn + kz
(
−7ξ + 9ξz2 − 12
)
+iν
(
ξ − 9ξz2 + 12
) )
ln
z + 1
z − 1
)
, (28)
6
β (K) = −m
2
D
k
2(kz − iν)2
ν ln z+1z−1 + 2ik
(
1− 1
2
z ln
z + 1
z − 1
+
1
12
ξ (1 + 3 cos 2θn)
(
2− 6z2 +
(
3z2 − 2
)
z ln
z + 1
z − 1
))
,
(29)
γ (K) = −m
2
D
12k
ξ
(
k
(
z2 − 1
)
− iνz
)(
4− 6z2 + 3
(
z2 − 1
)
z ln
z + 1
z − 1
)
sin2 θn, (30)
δ (K) =
m2D
24k2
ξ
(kz − iν)
2k − iν ln z+1z−1
(
k
(
88z − 96z3
)
+ 8iν
(
6z2 − 1
)
+ ln
z + 1
z − 1
×
(
12k
(
4z4 − 5z2 + 1
)
− 10iνz − 3 iν
(
4z4 − 5z2 + 1
)
ln
z + 1
z − 1
))
cos θn, (31)
where z = ω+iνk , and
ln
z + 1
z − 1
= ln
|z + 1|
|z − 1|
+ i
[
arg
(
z + 1
z − 1
)
+ 2πN
]
. (32)
where, N - corresponds to the number of Riemannian
sheets. Now, we can rewrite Eq.(24) as,
[∆−1(K)]ij = (k2 − ω2 + α)P ijT + (−ω
2 + β)P ijL
+γP ijn + δP
ij
kn. (33)
Next, we know that both a tensor and its inverse lie in
a space spanned by the same basis vectors or projection
operators. Hence, [∆(K)]ij can also be expanded as its
inverse, as,
[∆(K)]ij = aP ijL + bP
ij
T + cP
ij
n + dP
ij
kn. (34)
Now, using the relation [∆−1(K)]ij [∆(K)]jl = δil, we
obtained the following expressions for the coefficients a,
b, c and d,
a = ∆G(k
2 − ω2 + α+ γ), b = ∆A
c = ∆G(β − ω2)−∆A, d = −∆Gδ (35)
where,
∆A = (k
2 − ω2 + α)−1
∆G = [(k
2 − ω2 + α+ γ)(β − ω2)− k2ñ2δ2]−1. (36)
Considering the linear ξ, approximation we ignore δ2, as
it will be of order ξ2, we have
∆G = [(k
2 − ω2 + α+ γ)(β − ω2)]−1. (37)
Rewriting, Eq. (34) as,
[∆(K)]ij = ∆A(P
ij
T − P
ij
n ) + ∆G
[
(k2 − ω2 + α+ γ)P ijL
+(β − ω2)P ijn − δP
ij
kn
]
, (38)
This is a simplified structure of the gluon propagator
that could be easily used to obtain the poles.
C. Modes dispersion relation
As noted earlier, the dispersion relation of the collec-
tive modes can be acquired from the poles of the gluon
propagator. The poles of Eq. (38) are given as,
∆−1A (K) = k
2 − ω2 + α = 0, (39)
∆−1G (K) = (k
2 − ω2 + α+ γ)(β − ω2) = 0. (40)
∆−1G (K), can further splits as,
∆−1G (K) = ∆
−1
G1(k) ∆
−1
G2(k) = 0. (41)
Thus, we have two more dispersion equations,
∆−1G1(K) = k
2 − ω2 + α+ γ = 0, (42)
∆−1G2(K) = β − ω
2 = 0. (43)
Note that we have got three dispersion equations (39),
(42), and (43). We call these A-, G1- and G2-mode
dispersion equations, respectively. Now, we have three
dispersion relations for the collective modes. Based on
their solutions, ωk we can identify if the modes are real
or imaginary, stable or unstable which we shall discuss
in the next section.
III. RESULTS AND DISCUSSIONS
We shall examine the results by solving the disper-
sion equations (39), (42) and (43) numerically. We
normalize the frequency ω and wave vector k by the
plasma frequency in the absence of chemical potential
(ωp = mD(µ = 0)/
√
3). To avoid the bulk of plots that
are already available in the literature [19, 20, 27, 28, 31],
we shall primarily focus on the effects of finite chemical
potential and to have a comparative study we shall also
highlight the anisotropy and medium particles collisions
effects in the context of collective modes. We have fixed
the temperature as T = 2Tc, where Tc = 0.155 GeV the
7
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
10
12
ω/ωp
ℛ[
α]
/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
-8
-6
-4
-2
0
ω/ωp
ℑ[α
]/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
0
5
10
ω/ωp
ℛ[
β]
/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
-14
-12
-10
-8
-6
-4
-2
0
ω/ωp
ℑ[β
]/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
ω/ωp
ℛ[
γ]/
ω p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
ω/ωp
ℑ[γ
]/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
ω/ωp
ℛ[
δ]
/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
-0.4
-0.2
0.0
0.2
0.4
0.6
ω/ωp
ℑ[δ
]/ω
p
k = ωp, ν = 0.3ωp, ξ = 0.3, θn =
π
6
FIG. 2. Variation of real and imaginary parts of structure functions at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where
Tc = 0.155GeV at different µ.
direction, θn = π/6, the strength of anisotropy, ξ = 0.3, and the finite collision frequency, ν = 0.3ωp.
8
μ = 0
μ = 5 GeV
μ = 10GeV
0 2 4 6 8 10
0
2
4
6
8
10
k/ωp
ℛ
[ω
A
]/ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0 2 4 6 8 10
0
2
4
6
8
10
k/ωp
ℛ
[ω
G
1]
/ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0 1 2 3 4 5
0
1
2
3
4
5
k/ωp
ℛ
[ω
G
2]
/ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
FIG. 3. Variation of real stable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ.
We shall start with the discussion of structure func-
tions given in Eq. (28) to Eq. (31) as they are theimportant parts of the gluon selfenergy, gluon propaga-
tor and hence, important for collective modes. We have
plotted all the structure functions, i.e., α, β, γ and δ
with respect to ω normalized by ωp at different values of
chemical potential but at a fixed collisions frequency and
momentum anisotropy as shown in Fig. 2. It can be no-
ticed that with the increase in the chemical potential, the
magnitudes of both the real and imaginary parts of the
structure functions grow. The behavior of all the struc-
ture functions are changing at ω ∼ ωp. At very high ω,
only the real parts of structure functions survive whereas,
their imaginary parts vanish.
The stable modes are long-range modes. As shown in
Fig. 3, the real stable modes are found to increase with
the k at mentioned parameters. The stable G2- mode is
highly, whereas the G1- mode is marginally suppressed
as compared to stable A- mode. The magnitude of these
modes rise with the chemical potential.
The imaginary parts are emerging only because of the
collisional effects. Here, the finite ν generates the imag-
inary modes with Im(ω(k)) < 0, i.e., the modes are
damped and their amplitudes exponentially decay with
time as e−Im(ω(k))t whereas it does not further satisfy
the over-damped condition. If we consider the collision-
less plasma, i.e., ν = 0 these modes disappear. This
fact has already been shown in the studies by different
groups [19, 27, 31]. These results for the collisionless
plasma remain untouched in our case as well. In Fig. 4
the imaginary stable modes are plotted. The magnitude
of imaginary A- and G1- modes are growing with the
chemical potential and their magnitudes are not very dis-
tinguishable. An explanation for that is the difference
between the A- and G1- modes are only the structure
functions, γ whose magnitude as compared to α is very
small and hence does not carry a high impact that could
make it distinguishable. The imaginary G2- mode on the
other hand shows the opposite behavior. This is because
G2- mode depends on β whose magnitude at µ = 10 GeV
is very high.
A crucial aspect in the current study is the unsta-
ble modes that are shown in Fig. 5 for weakly squeezed
plasma,ξ = 0.3 and non-zero collisions frequency, ν =
0.3ωp at a fixed angle, θn = π/6 for different values of
chemical potential. We have encountered only two un-
stable modes, i.e., unstable A- and G1- modes whereas,
the unstable G2- mode is missing. This is because the
unstable modes are the imaginary solutions of ω, in the
dispersion equations (39), (42) and (43) and to obtain
their solutions we consider ω, to be purely imaginary, i.e.,
ω = iΓ. In this substitution, we marked that β > 0 and
consequently Γ2 + β = 0 from Eq. (43) will never be sat-
isfied. A similar case was discussed in Ref. [19, 27, 28, 31]
in the absence of medium particle collision. These modes
are short-lived though the presence of chemical potential
enhances their magnitude which may further support the
fast thermalization of the created hot medium.
To comprehend the dependence of instability on
anisotropy and chemical potential and medium particle
collisions, we obtained kmax and νmax at which the un-
stable modes were completely suppressed at fixed values
of the other parameters. The results are shown in Fig. 6
and Fig. 7. The values of kmax, at which the unstable A-
and G1- modes are completely suppressed are obtained
by substituting ω = 0, in Eq. (39) and (42), respectively.
In Fig. 6 for A- mode (left panel) and G1- mode (right
panel), it is shown that kmax grows with the chemical po-
tential and the existence of anisotropy further enhances
it at fixed θn and ν. The corresponding values of G1-
mode are suppressed as compared to A- mode. This sup-
ports our prior discussion that the unstable modes exist
up to higher values of k at a large chemical potential.
Following the similar procedure discussed above, we
obtained νmax, at the point where the unstable A- and
G1- modes are fully suppressed. In Fig. ??, the behavior
of νmax/ωp, for A- mode (left panel) and G1- mode (right
panel) with respect to µ are shown for different ξ (ξ =
0.2, 0.4 and 0.6) at fixed θn and k. Again it is found that
with the increase in chemical potential, the maximum
values of collision frequency increase, and the results are
further enhanced with the momentum anisotropy.
9
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
-0.20
-0.15
-0.10
-0.05
0.00
k/ωp
ℑ[
ω
A
]/
ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
-0.20
-0.15
-0.10
-0.05
0.00
k/ωp
ℑ[
ω
G
1]
/ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
-0.4
-0.3
-0.2
-0.1
0.0
k/ωp
ℑ[
ω
G
2]
/ω
p
ν = 0.3ωp, ξ = 0.3, θn =
π
6
FIG. 4. Variation of imaginary stable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ.
μ = 0
μ = 5 GeV
μ = 10GeV
0.0 0.5 1.0 1.5 2.0
-0.04
-0.02
0.00
0.02
0.04
k/ωp
Γ A
/ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
μ = 0
μ = 5 GeV
μ = 10 GeV
0.0 0.5 1.0 1.5 2.0
-0.04
-0.02
0.00
0.02
0.04
k/ωp
Γ G
1/
ω
p
ν=0.3ωp, ξ = 0.3, θn =
π
6
FIG. 5. Variation of unstable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ.
ξ=0.2
ξ = 0.4
ξ = 0.6
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
μ (GeV)
k m
ax
/ω
p
ν=0.3ωp, θn =
π
6
ξ=0.2
ξ = 0.4
ξ = 0.6
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
μ (GeV)
k m
ax
/ω
p
ν=0.3ωp, θn =
π
6
FIG. 6. Variation of the maximum values of propagation vector for unstable A- mode (left) and G1- mode (right) at ν = 0.3ωp,
θn = π/6 and T = 2Tc where Tc = 0.155GeV at different ξ.
Moreover, it can be seen that in both the cases, i.e.,
kmax and νmax, respectively in Fig. 6 and Fig. 7, the
results for G1- mode are suppressed as compared to A-
mode. It is mainly because of the behavior of the struc-
ture function, γ that tries to suppress the G1- mode.
This indicates that if we have a high baryon density, the
collision frequency among the medium particles also in-
creases, and hence, one can infer that at a large baryon
10
ξ=0.2
ξ = 0.4
ξ = 0.6
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
μ (GeV)
ν m
ax
/ω
p
k = ωp, θn =
π
6
ξ=0.2
ξ = 0.4
ξ = 0.6
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
μ (GeV)
ν m
ax
/ω
p
k = ωp, θn =
π
6
FIG. 7. Variation of the maximum values of collision frequency for unstable A- mode (left) and G1- mode (right) at k = ωp,
θn = π/6 and T = 2Tc where Tc = 0.155GeV at different ξ.
density, the medium will thermalize faster.
IV. SUMMARY AND CONCLUSIONS
In this manuscript, we attempted to assimilate the fi-
nite chemical potential in the study of collective excita-
tion present in the hot QGP medium and examine its
effect in the presence of medium particle collision and
small but finite momentum anisotropy. We begin with
the derivation of gluon self-energy in terms of structure
functions where we included the finite chemical potential
at the particle distribution level along with the momen-
tum anisotropy straightforwardly either by stretching or
squeezing the distribution function in one of the direc-
tions (denoted as n̂). We solved the Boltzman equation
after incorporating the medium particle collision via the
BGK collisional kernel to obtain the change in the distri-
bution function of each species viz., quarks, anti-quarks
and gluons. Next, using the Yang-Mills equation or clas-
sical Maxwell’s equation, we formulated the gluon propa-
gator. From the poles of this propagator, we fetched the
dispersion equation for the collective modes. As men-
tioned in the introduction section, we classified the modes
as real or imaginary and stable or unstable based on the
solution of the dispersion equations.
In the present formalism, one can analyze the modes
at any angular direction by changing θn, (i.e.,) the an-
gle between k and n̂ at different values of anisotropic
strength in the given range, −1 < ξ < 1. Just to avoid
the bulk of plots that are available in the priorliterature,
we did not display the variation of θn and ξ. Entertain-
ing the anisotropy in the analysis is important otherwise
there will be no unstable modes appearing. As shown
in Ref. [28] that the maximum possible value of collision
frequency could be 0.62 mD, i.e., 0.36 ωp, we fixed the
value of collision frequency, ν = 0.3ωp.
We first investigate the structure functions of the self-
energy and found that they intensely depend on the
chemical potential. We noticed that at θn = 0, δ dis-
appears whereas at θn = π/2, γ vanishes. From the
poles of the propagator, we got three dispersion equations
for collective modes. Based on their solutions, we found
three real and imaginary stable modes. Whereas, only
two unstable modes were found. The imaginary stable
modes disappear in the absence of medium particle col-
lision whereas the unstable modes vanish in the absence
of anisotropy. Also, the two unstable modes overlap at
θn = π/2 as the structure function γ goes to zero. In the
collisionless isotropic medium, we obtain only three real
stable modes. It has been found that all kinds of modes
rely strongly on the chemical potential and mainly en-
hance their magnitude.
We also investigate the suppression of unstable modes
through kmax and νmax via plotting them against µ. It
has been observed that with µ the maximum values of
the wave vector and collision frequency increase which
further enhance with anisotropy.
In the near future, the current project can be extended
through the inclusion of a non-local BGK kernel. More-
over, an essential aspect that is missing in the study of
modes is the study of group velocity as these modes are
nothing but plasma waves. The non-abelian term of the
field strength tensor could also be included to make the
analysis more realistic and closer to the experimental ob-
servations.
V. ACKNOWLEDGEMENTS
I would like to that SERB for providing NPDF (project
ID 2022/F/SKD/014). I would also like to acknowledge
the people of INDIA for their generous support for the
research in fundamental sciences in the country.
11
[1] J. Adams et al. (STAR Collaboration), Nucl. Phys. A
757, 102 (2005); K. Adcox et al. PHENIX Collaboration,
Nucl. Phys. A 757, 184 (2005); B.B. Back et al. PHOBOS
Collaboration, Nucl. Phys. A 757, 28 (2005); I. Arsene et
al. BRAHMS Collaboration, Nucl. Phys. A 757, 1 (2005).
[2] K. Aamodt et al. (The Alice Collaboration), Phys. Rev.
Lett. 105, 252302 (2010); Phys. Rev. Lett. 105, 252301
(2010); Phys. Rev. Lett. 106, 032301 (2011).
[3] S. Ryu, J. F. Paquet, C. Shen, G. S. Denicol, B. Schenke,
S. Jeon, C. Gale, Phys. Rev. Lett. 115, 132301 (2015).
[4] G. Denicol, A. Monnai, B. Schenke, Phys. Rev. Lett. 116,
212301 (2016).
[5] M. C. Chu and T. Matsui, Phys. Rev. D 39, 1892 (1989).
[6] V. K. Agotiya, V. Chandra, M. Y. Jamal and I. Nilima,
Phys. Rev. D 94, no.9, 094006 (2016)
[7] M. Y. Jamal, I. Nilima, V. Chandra and V. K. Agotiya,
Phys. Rev. D 97, no.9, 094033 (2018).
[8] M. Y. Jamal, Springer Proc. Phys. 250, 135-139 (2020).
[9] L.D.Landau and E.M.Lifshitz, Electrodynamics of con-
tinuous media, Butterworth-Heinemann, (1984).
[10] M. Y. Jamal, S. Mitra and V. Chandra, J. Phys. G 47,
no.3, 035107 (2020).
[11] Y. Koike, AIP Conf. Proc. 243, 916 (1992) .
[12] M. Y. Jamal, S. K. Das and M. Ruggieri, Phys. Rev. D
103, no.5, 054030 (2021).
[13] M. Y. Jamal and B. Mohanty, Eur. Phys. J. C 81, no.7,
616 (2021).
[14] M. Y. Jamal and B. Mohanty, Eur. Phys. J. Plus 136,
no.1, 130 (2021).
[15] M. Yousuf Jamal and V. Chandra, Eur. Phys. J. C 79,
no.9, 761 (2019).
[16] S. Mrowczynski, Phys. Lett. B 314, 118 (1993).
[17] S. Mrowczynski, Phys. Rev. C 49, 2191 (1994).
[18] S. Mrowczynski, Phys. Lett. B 393, 26 (1997).
[19] M. Y. Jamal, S. Mitra and V. Chandra, Phys. Rev. D
95, 094022 (2017).
[20] A. Kumar, M. Y. Jamal, V. Chandra and J. R. Bhatt,
Phys. Rev. D 97, no.3, 034007 (2018).
[21] Avdhesh Kumar, Jitesh. R. Bhatt, Predhiman. K. Kaw,
Phys. Letts. B 757, 317-323 (2016).
[22] M. Le. Bellac, Thermal Field Theory, Cambridge Univer-
sity Press, Cambridge, UK (2000).
[23] E. S. Weibel, Phys. Rev. Lett. 2, 83 (1959).
[24] S. Mrowczynski, A. Rebhan and M. Strickland, Phys.
Rev. D 70, 025004 (2004).
[25] Daniel F. Litim, C. Manual, Phys. Rep. 364, 451 (2002).
[26] J.-P.Blaizot and E.Iancu, Phys. Rep. 359, 355 (2002).
[27] M. E. Carrington, K. Deja and S. Mrowczynski, Phys.
Rev. C 90, no.3, 034913 (2014).
[28] Bjoern Schenke, Michael Strickland, Carsten Greiner,
Markus H. Thoma, Phys. Rev. D 73, 125004 (2006).
[29] P. B. Arnold, J. Lenaghan and G. D. Moore, JHEP 0308,
002 (2003).
[30] S. Mrowczynski, Acta Phys. Polon. B 37, 427 (2006).
[31] P. Romatschke and M.Strickland, Phys. Rev. D 68,
036004 (2003).
[32] B. Karmakar, R. Ghosh and A. Mukherjee, Phys. Rev.
D 106, no.11, 116006 (2022).
[33] P. Romatschke and M.Strickland Phys. Rev. D 70,
116006 (2004).
[34] B. Schenke and M. Strickland, Phys. Rev. D 76, 025023
(2007).
[35] Bing-feng Jiang, De-fu Hou and Jia-rong Li, Phys. Rev.
D 94, 074026 (2016).
[36] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev.
94, 511 (1954).
[37] M. Carrington, T. Fugleberg, D. Pickering, and M.
Thoma, Can. J. Phys. 82, 671 (2004).
[38] S. Mrowczynski and M. H. Thoma, Phys. Rev. D 62,
036011 (2000) .
[39] M. Laine and Y. Schöder, JHEP 0503, 067 (2005).
[40] R. Kobes, G. Kunstatter and A. Rebhan, Nucl. Phys. B
355, 1 (1991).
	Collective modes in the anisotropic collisional hot QCD medium in the presence of finite chemical potential
	Abstract
	I Introduction
	II Methodology
	A Gluon polarization tensor
	B Gluon Propagator
	C Modes dispersion relation
	III Results and Discussions
	IV Summary and Conclusions
	V Acknowledgements
	 References