Second Order Critical Point in QCD Phase
Diagram
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
by
JUL 0 7 2009
Wathid Assawasunthonnet
LIBRARIES _
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2009
@ Wathid Assawasunthonnet, MMIX. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
ARCHIVES
Author ........
Department of Physics
May 22, 2009
Certified by.
Krishna Rajagopal
Professor
Thesis Supervisor
Accepted by .....
David E. Pritchard
Senior Thesis Coordinator, Department of Physics
Second Order Critical Point in QCD Phase Diagram
by
Wathid Assawasunthonnet
Submitted to the Department of Physics
on May 22, 2009, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
In this thesis I explore the theoretical model based on Asakawa and Nonaka's idea[l].
I start by arguing that the critical point of the QCD phase diagram is second order
and belongs to the three dimensional Ising model universality class. Then the sigular
part of the equation of state is derived. The singular part and non-singular part
equation of state are glued together to find the general form of the equation of state.
This equation of state includes the critical point. With this equation of state, we
construct the isentropic trajectories. The pathology of these trajectories is discussed.
Moreover the validation of the signature of the critical point suggested by Asakawa
and Nonaka is also discussed.
Thesis Supervisor: Krishna Rajagopal
Title: Professor
Acknowledgments
I would like to thank Professor Rajagopal for his time and effort in guiding me throguh
out the process of doing this thesis. I also would like to thank him for filling in my
theoretical knowledge, necessary for this thesis.
Contents
1
15
Introduction
1.1
Motivations for Studying the QCD Phase Diagram
1.2
Critical Phenomena ...........................
16
17
Ising M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3.1
Construction of the Model . . . . . . . . . . . . . . . . . . .
18
1.3.2
Representation of the Ising Model as Functional Integration
19
21
2 Singular Part of the Entropy Density
... ..
21
Equation of State .......................
... ..
24
Mapping onto QCD Phase Diagram . . . . . . . . . . . . .
. . . . .
29
2.1
Qualitative Reasoning
2.2
2.3
....................
33
3 The Entropy Density and the Baryon Number Density
3.1
3.2
3.3
16
Critical Exponent and Universality Class . . . . . . . . . . .
1.2.1
1.3
. . . . . . . . .
Non-Singular Part of the Entropy Density . . . . . . . . .
. . . . . 33
3.1.1
Hadronic Equation of State
. . . . . . . . . . . . .
. . . . .
33
3.1.2
Quark-Gluon Plasma Equation of State . . . . . . .
. . . . .
36
3.1.3
Ocrit
. . . . . . . . . . . . . . . . . . . . . . . . . .
. .. ..
38
... ..
39
The Full Entropy Density
..................
3.2.1
Formal Analysis ....................
.. ...
39
3.2.2
Parameters Choice ..................
.. ...
42
.. ...
43
Baryon Number Density ...................
4 Isentropic Trajectory and Conclusion
4.1
The Trajectory of the Hadronic Phase and the Quark-Gluon Plasma
Phase
4.2
4.3
45
. . . ...
...
.........
. . ..
..
. . . . ..
. . ..
45
Pathology of the Full Trajectory ....................
48
4.2.1
D = 0.15
48
4.2.2
D=2 .................
4.2.3
Leftward Kick and the Rightward Turn at the Crossover Region 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.....
......
Conclusion . . . . . . . . . . . . . . . . . . . . . .. . . . .
..
. . . ..
49
54
List of Figures
2-1
Potential for continuous 0(2) spontaneous symmetry breaking. The
oscillatoins along the trough in the potential correspond to the massless
ir fields.
2-2
...
...................
..
......... ......
22
QCD phase digram of m,,d = 0. To the left side of the tricritical point
the phase transition is second order. To the right of the critical point
the phase transition is first order. The superconductor phase is not
important in the analysis of the critical point.[2] . ...........
2-3
QCD phase digram of m,,d f 0. The phase transition chages from the
first order to smooth crossover as the system goes through E.[2] . . .
2-4
Plot of M as a function of h at r = -0.1.
order phase transition at h = 0
2-5
The plot illustrates a first
......................
.....................
26
27
Plot of M as a function of h at r = 0.1. The plot illustrates a smooth
crossover of the magnetization at h = 0 . ................
2-7
25
Plot of M as a function of h at r = 0. The plot illustrates a second
order phase transition at h = 0
2-6
23
28
Plot of M as a function of h at r = 0.5. The plot illustrates smooth
crossover at h = 0. The plot is smoother and more inclined compare
to the plot at r = 0.1 ...........................
2-8
29
Three dimensions plot of the magnetization on the h, r plane. The plot
contains all the expected features of the phase diagram including the
critical point. ...............................
30
2-9
Sketch of the r - h axes mapped onto the T - PB plane. The r-axis is
tangential to the first order line at the CEP. The h-axis is perpendicular
to the r-axis.[1]
3-1
..
...
. . . . . . . . ... . . .
. . . .....
..
.
Plot of the hadronic pressure in GeV 4 versus T in GeV generated by
integrating equation. (3.2) numerically. We chose p = 0 ........
3-2
31
Plot of the
3
34
versus T in GeV generated by integrating equation. (3.3)
numerically. We chose p = 0 .......................
35
3-3
Plot of the QGP pressure in GeV 4 versus T in GeV. We chose p = 0
36
3-4
Plot of
37
3-5
The plot of pressure for the quark gluon plasma and hadronic matters
versus T in GeV. We chose p = 0.4 . ............
at p = 0.75 GeV. The crossing illustrates the first order phase thransition. 38
3-6
The plot of Tcit as a function of PB, which is the first order phase
transition line. The result for any temperature higher than 0.6 GeV is
not reliable. . . . . . . . . . .
3-7
. . . . . . . . . . . ... ....
A three dimensional plot of -
as a function of
pick D to be 0.15 ..
. . . . ..
..
...
. ..
PB
. .
. ..
. .
39
and T(GeV). We
..
. . . . .. ..
3-8
A plot of
as a function of T at pB=0. 4 GeV. D is chosen to be 0.15
3-9
A plot of -.
as a function of T at pB=0.4 GeV. D is chosen to be 2.
40
41
The entropy density decays to zero at T=0 and converges to a constant
as T grows larger ..............................
3-10 A plot of
3
42
as a function of T at pB=0.4 GeV. D is chosen to be 2.
This plot describes the first order phase transition.
4-1
. ..........
43
Contour Plot of the hadronic equation of state. The trajectores head
toward the point where T = 0 and PB = mproton=0.938 GeV. ......
4-2
Contour Plot of the QGP equation of state. The contours follow a
straight line of constant
4-3
46
.
.
.
...................
.
47
Contour Plot of the full equation of state using Asakawa and Nonaka's
choice of paremeter, D=0.15. These trajectories are not physical.
. .
48
4-4
Asakawa and Nonaka's result for D=0.15. The triagle represents the
critical point at (0.1547 GeV,0.3678 GeV). This contour plot does not
contain a first order phase transition line.[1]
4-5
49
. .............
Asakawa and Nonaka's model using D=2. The critical point is at
50
(0.1547 GeV,0.3678 GeV). .......................
4-6 This is the plot of !
when PB = 0.3. The area of interest are the local
51
minimum and maximum near the critical temperature. . ........
4-7 This is the plot of the entropy density versus the temperature for ,B
=
0 .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4-8 This is the plot of the baryon number density versus the temperature
for [t B
4-9
=
0.3 ...
. . . . . . . ...
This is a plot of nQ()
. . . . . . . . . . . . . . ... .
zero .....................
53
at /pB = 0.3. It starts from zero then increases
54
to a maxima at T=0.04. .........................
4-10 This is the plot of 2
.
when PB = 0.3. The plot increase steadily from
...............
55
12
List of Tables
3.1
List of Hadronic species used in the calculating the pressure, their
degenracy factors, and their baryon quantum numbers[3]. .......
.
36
14
Chapter 1
Introduction
Quantum chromodynamics (QCD) is a theory that describes a wide range of phenomena such as the mass spectrum of hadrons and deep inelastic collisions. Quantum
chromodynamics also possesses well defined thermodynamic properties. This knowledge of QCD thermodynamics is crucial in understanding phenomena such as compact
stars, heavy-ion collisions, and the early universe. The structure of the QCD phase
diagram is a particularly interesting topic in particle physics. The phase diagram is a
map of the phase of QCD system as a function of temperature and baryon chemical
potential. This paper focuses on the behavior of thermodynamic quantities such as
the entropy density and the baryon numberdensity. In particular,the construction
of isentropic trajectories in the model QCD phase diagram will be analysed. These
trajectories approximate how relativistic heavy ion collision cool. The specific model
that is discussed and explored is the model proposed by Chiho Nonaka and Masayuki
Asakawa[1]. The second chapter discusses about how to calculate the singular part
of the entropy density and explore its behavior near the critical point. The third
chapter describes the construction of the complete expression of the entropy density
in the model of[l]. It contains the expression of the entropy density of the baryonic
matter and of the quark-gluon plasma. Moreover it also states a general assumption
of how the two phases connect. For example it discusses how the entropy density
changes when the system undergoes a first order phase transition as compareed to
the low baryon chemical potential smooth crossover.
1.1
Motivations for Studying the QCD Phase Diagram
Full analytical treatment of QCD is very difficult. Fortunately, in the limit of large
values of the parameters temperature and/or baryon chemical potential, when the
short-distance dynamics dominates, the system can be studied analytically due to
the asymptotic freedom. Furthermore, when the chemical potential equals zero the
system can be solved by using the method of lattice QCD. However, lattice simulation
breaks down in the nonzero chemical potential region. Many theoretical arguments,
for example chiral symmetry argument, suggest the existence of a critical point at
some temperature and nonzero baryon chemical potential. The phase transition from
baryonic matter to quark-gluon plasma is second order at the critical point.
At
lower baryon chemical potential it is a smooth crossover. At higher baryon chemical
potential it is a first order phase transition. However, there is no known analytical
method to locate the critical point and to map the phase diagram in this region of
phase diagram. Investigating the phase diagram in the region near the critical point
is particular interesting and useful since it gives thermodynamic quantities, which
relate to signatures that can be experimentally observed.
1.2
Critical Phenomena
There are three types of phase transitions, first order, second order, and the smooth
crossover. The first order phase transition is a phase transition at which the first
derivative of the free energy with respect to a thermodynamic variable has a discontinuity at T = T, where T, is the critical temperature. The example of this kind
of phase transition is water-ice transition. The water radically change to ice at the
freezing point. The second order phase transition is continuous at the first derivative
but exhibits a discontinuity in the second derivative of the free energy. The smooth
crossover has no discontinuities in the first order or the seconder derivatives. An
example of system that exhibits this class of the phase transition is the ferromag-
netic phase transition, where the magnetization, which is the first derivative of the
free energy with respect to the magnetic field strength increases continuously from
zero as the temperature is lowered below T, which is the Curie temperature. However the magnetic susceptibility which is the second derivative is discontinuous at T'.
Therefore the phase transition at the critical point is second order.
1.2.1
Critical Exponent and Universality Class
The critical phenomena refers to properties of thermodynamic quantities in the region
surrounding the critical point of a second order phase transition. The phase transition
is usually accompanied by a change in symmetry of the system. In the ferromagnetic
system, the two phases have different symmetries. Above the critical temperature
there is no magnetization so the system is rotationally invariant. Below the critical
temperature the magnetization implies that there is a preferred direction. Therefore
the rotational symmetry is broken. Since the symmetry is not present in one of the
phases, extra parameter is needed in order to explain the system. The extra parameter
is called an order parameter. The order parameter is usually only present in one of
the phases and vanishes at the critical point. This implies that the susceptibility
of the order parameter diverges as the system approaches the critical point. From
an earlier discussion about the ferromagnetic system, the order parameter is the
magnetization. Any thermodynamic quantity can be decomposed into singular and
non-singular part. It turns out that the thermodynamic properties at the critical point
can be characterized by parameters called critical exponents. The critical exponents
describe the nature of singularity of thermodynamic quantities at the critical point.
In order to illustrate this, first introduce the quantity r =- Tc.
Let us examine the
behavior of the magnetization, the magnetic susceptibility X and the heat capacity C
near the critical point. When temperature is varied while other parameters are fixed,
the magnetic susceptibility and the heat capacity depend on t as follow:
Moc
|r 3
x oc IrI
C oc Itl-
(1.1)
The exponent /, y and a are called the critical exponents. It is experimentally discovered that there are many different systems that share the same critical exponents.
Sets of systems that share the same critical exponents are described as being in the
same universality classes. The property of universality classes is a prediction of the
renormalization group theory of phase transitions. It states that the thermodynamic
properties near the critical point depends only small number of features such as dimensionality and symmetry. The detail discussion can be found in many standard
statistical mechanics books[4].
1.3
1.3.1
Ising Model
Construction of the Model
This section is intended to be a crude introduction to the Ising Model. It also motivates the connection of the Ising model to field theory and how to calculate the
critical exponent for a simple case. The Ising model is an important model in statistical physics. The initial usage for the model is for explaining the behavior of magnetic
systems. It can explain important properties of ferromagnets including their phase
transition. Moreover, a two-dimensional Ising model yields to an exact treatment
in statistical mechanics.
In the Isig model, the system considered is a system of
n-dimensional lattice (n = 1, 2, 3); each dimension consists of N fixed points called
lattice sites. At each lattice point there is an associated spin value Si(i = 1, ..., N)
which is a number that is either -1 or 1. If Si=l, it is said to have spin up, and if
Si=-1, it is a spin down. The system is specified if the set of Si is given. In this
configuration, the energy of the system is
N
E(S) = - E ijSiSj - hi E Si
ij
i=1
(1.2)
where eij is the interaction between spins and hi is the external magnetic field at site
i. The model can be applied to lattice in higher dimension. The canonical partition
function is Z[Hi] = Es, e-OE(Si). Since the spin can be up or down the summation
will be over 2 ". Here are some notations that will come up often during the analysis:
Kij = Pcij, Hi = phi where p = ' and k is the Boltzmann constant.
1.3.2
Representation of the Ising Model as Functional Integration
All thermodynamic quantities can be derived from this partition function. In the case
of uniform field and translational invariant system, when Hi = H, here are examples
of the thermodynamic quantities:
1
M(H) =
9Z
X=
1
L (Si) =-Z
-Z
OH
1
Hi H=0 N E((jsi)
i
HO
- (S3 )(S))
(1.3)
The terms in the sum can be expressed as correlation functionsg(ri - rj) = ((Si (Si))(Sj - (Sj))). From eq. (1.3), we can write the correlation functions as follows
g(r - r) = Z- 1
Z-1
(1.4)
Since we have translational invariant system we can rewrite the magnetic susceptibility in eq. (1.3) as
X = Eg(R)
(1.5)
R
when R are the vectors of the lattice site relative to a chosen origin. From eq. (1.4),
correlation functions are defined as derivatives of the partition function. Therefore
the partition function can also be thought of as a generating function in functional
integrals. This connection allows the calculation of the critical exponents by using
path integration and other tools from quantum field theory. The rigorous derivation
of this can be found in[5]. The critical exponent in the Ising model can be calculated
using renormalization group theory. On a two dimensional square lattice, the solution
can be calculated exactly. Onsager obtained the expression for the magnetization as
a function of temperature[6]
M = (1- [sinh(log(1 + V/2)-T
)
(1.6)
The expansion of this result near the critical temperature gives us the critical exponents. The critical exponents from the Onsager solution defines the d = 2, n = 1
universality class. The calculation of the critical exponents for the Ising model of
different dimension is theoretically difficult and not necessary for the purpose of this
paper. In the next chapter, some arguments will be presented to convince that the
critical point of the QCD phase diagram belongs to the three dimensional Ising model
universality class. We can compute the Ising model for d = 3, n = 1. The critical
exponents are /3 = 0.325, -y = 1.2405, and a = 0.110[7]
Chapter 2
Singular Part of the Entropy
Density
The goal of this chapter and chapter three is to implement the ideas used by Asakawa
and Nonaka to construct an expression of the entropy density(S(TQcD, p)), the baryon
number density density (nB(TQCD, p)), and to illustrate the isentropic trajectory of
the -B contours. Moreover the aim is to find out if the pathology of the isentropic
trajectory they found is an intrinsic feature of the critical point or due to the modeldependent aspect of the construction of the above thermodynamic quantities.
2.1
Qualitative Reasoning
Based on many models and theoretical argument, at the QCD critical point the phase
transition is second order. In the previous chapter, we learn that the second order
phase transition can be categorized into universality classes based on the critical
exponents. Moreover, the critical exponents only depend on the dimensionality and
symmetry of the system. In QCD, the system dimensionality is translated into degree
of freedom hence the numbers of massless particle present at specific point in the phase
diagram which is dictated by the symmetry of the system. The main symmetry driven
the phase transition in QCD is the QCD chiral symmetry. Phase transition is due to
the mechanism of spontaneous breaking. There is a preferred degree of freedom set by
vacuum expectation value that breaks the symmetry of the Lagrangian. Goldstone's
theorem states that for every spontaneous broken symmetry, the theory must contain
a massless particle. This fact can be illustrated by considering simple linear sigma
model for classical field. The Langrangian involves N real scalar field i (x)
L=
2
(a,i)
2 a22i2
2
2
_ A[(i)2]2
(2.1)
4
a is a mass of the field. This Lagrangian is invariant under the symmetry
(2.2)
-*
R is a rotation matrixt in N dimension. The lowest-energy classical configuration is a
Figure 2-1: Potential for continuous 0(2) spontaneous symmetry breaking. The
oscillatoins along the trough in the potential correspond to the massless 7r fields.
constant field 0' that minimize the potential in eq. (2.1). The potential of an N = 2
case is shown in (figure 2-1). The potential is minimized if the field satisfies
O2
(i)2
=
A
(2.3)
This condition tells the length of the field. Its direction is arbitary. We choose it to
point in the N th direction: 0s = (0,0, ..., 0, v), where v =
. v is called the vacuum
expectation value. We can rewrite our field by separating out the preferred direction
as follows
= (rk, v + U(x)), k = 1,..., N - 1
0'(x)
(2.4)
Putting the new form of the field from eq. (2.4) back into eq. (2.1), the Lagrangian is
seperated into Lagrangian for 7r and a fields.
L
=
2
(Ork)2+
2
()2
2
(2a2)(u)2Va3v/X(1r2(
k2
A
4
(k)2U2-
4_
2
A(k
2l2
4
(2.5)
From eq. (2.5), it is obvious that imposing the vacuum expectation value breaks the
O(N) symmetry leaving O(N -1) symmetry. The fields are also seperated into N -1
massless 7r fields and one massive a field. At high temperature and high baryon
mrn
4 , =0; m. =ool
QGP
1l MeV .
...
Hadronic
N
ear M atter
..--
Figure 2-2: QCD phase digram of m ,d = 0. To the left side of the tricritical point
the phase transition is second order. To the right of the critical point the phase
transition is first order. The superconductor phase is not important in the analysis
of the critical point.[2]
chemical potential, the QCD system is described by quarks and gluons as degrees
of freedom since it is asymptotically free. In the quark-gluon plasma phase, all the
symmmetries of QCD Lagrangian present. At high density, we have seen that the
quarks form a condensate. If the density is increased higher, the chiral symmetry
is broken and new condensate is formed. In order to argue that the QCD critical
point belongs to the three dimensional Ising model universality class, we start by
examining the phase diagram at low baryon chemical potential. When the mass of the
up quark and the down quark are zero (figure 2-2), chiral symmetry is spontaneouly
broken from SU(2)L x SU(2)R to the subgroup SU(2)L+R by the presence of the
chiral condensate which is also the order parameter. This process produces three
massless Goldstone bosons. At low temperature, the order parameter wins over the
entropy and the symmetry stays broken. Above the critical temperature, Tc the order
parameter vanishes and the symmetry is restored. The phase transition is sencond
order belongs to the universality of 0(4) in three dimensions[2]. At the critical point
there are four massless particles and it is a tricritical point. However in nature, the
up quark and the down quark are not massless. The second order phase transition is
replaced by a smooth crossover. Therefore in figure 2-3, there is no sharp line between
the quark-gluon plasma phase and the hadronic phase. Many theoretical arguments
suggest that the phase transition at high baryon chemical potential region is first
order. This implies that there must be a critical end point (Figure2-3) somewhere
between the first order phase transition and the smooth crossover. At the critical
point there is one massless particle and the phase transition is second order in the
three dimensions Ising universality class.
2.2
Equation of State
Within a small critical region near the critical point, the behavior of thermodynamic
quantities is governed by the critical phenomena based on the universality class assumption. The order parameter in the three dimensions Ising model is the magenetization that depends on the reduced temperature, r =T-Tc and the magnetic field,
h. The critical point is at the origin (r, h) = (0, 0) by construction. The equation of
state in the parametric representation is given by[1]
M
=
M oR 3 0
QGP
mT
,,,,
O;
= oo
E
Hadronic
Superconductor
O. The phase transition chages from the
Figure 2-3: QCD phase digram of m,,d
first order to smooth crossover as the system goes through E.[2]
h =
hoR' h(0)
h(0) = (0 - 0.7620103 + 0.0080405)
r = R(1-0
where R > 0, -1.154
2)
(2.6)
< 0 < 1.154. Mo and ho are normalization constants and
the critical exponentsn 3 and 6 are 0.326 and 4.80, respectively. The normalization
constants are obtained by setting M(r = -1, h = +0) = 1 and M(r = 0, h = 1) = 1.
First we start with the Gibbs free energy density G(h, r),
G(h, r) = F(M,r) - Mh
(2.7)
where F(M, r) is the Helmholtz free energy density. Before calculating the entropy
density, we define some other parametric functions to help with the calculation. From
universality assumption, the free energy density is set to be
F(M.r) = hoMor2-ag(O)
(2.8)
M(h)
1.0.
0.5
-1.0
0.5
-0.5
1.0
-1.0
Figure 2-4: Plot of M as a function of h at r = -0.1.
phase transition at h = 0
The plot illustrates a first order
where a critical exponent a is 0.11. From the expression of the free energy and the
relation h = (f)
, the differential equation for g(0) can be written as
h(0)(1 - 02 + 2002) = 2(2 - a)0g(O) + (1 - 02 )g'(0).
(2.9)
Plugging in the expression for h(O), we obtain
g(O) = g(1) + ci(1 - 02) + c 2 (1 - 02)2 + C3 (1- o2)3
(2.10)
where g(1) is determined by using eq. (2.6) and eq. (2.9),
g(1)
2-a
h(1)
(2.11)
and the coefficient cl,c 2 , and c3 are 0.321,-1.203, and -0.0012 respectively. The singular part of the entropy density near the critical end point can be obtained from
differentiating the Gibbs free energy with respect to the temperature TQCD by using
the chain rule.
Sc
(G
(TQCD
)A
26
M(h)
1.0-
0.5
1.0
0.5
-0.5
-1.0
-1.0
Figure 2-5: Plot of M as a function of h at r = 0. The plot illustrates a second order
phase transition at h = 0
(G\
Oh
(G
r aTQCD
-
r
ar h TQCD
(2.12)r
(2.12)
This temperature is the temperature of the QCD phase diagram. It is not to be
confused with the temperature of the Ising model T. The entropy density can be
expressed as sc(h, r) since we can evaluate (I)
()
in the representation space.
Then we can write the expression in terms of h, t by using the parametric equation(eq. (2.6)). The first partial derivative turns out to be very simple. It is given by
differentiating the Gibbs free energy (eq. (2.7)).
-(
=
since
)
= (
F (M, r)
Oh
(
-M
OM
-h-M
Oh)r
(2.13)
)r h. This can be derived easily by using chain rule and
manipulation of thermodynamic variables.
dF = dE- TdS- SdT
= Tds + hdM - TdS - SdT
aF)dM + aF)
dT/
a2M
dT
(2.14)
M(h)
1.0 h
-1.0
-0.5
0.5
1.0
-1.0
Figure 2-6: Plot of M as a function of h at r = 0.1. The plot illustrates a smooth
crossover of the magnetization at h = 0
From the above equations, we can deduce that (
(F(M,
R)
OF
h
) = h and that
((
M)r
=M
Oh
Oh
h
h
(2.15)
(2.15)
The second partial derivative is more complicated since it cannot be written in term
of a simple thermodynamic function. It is easier to compute in the parametric representation (R, 0). Use the expression of the Gibbs free energy (eq. (2.7)), we got
(G)
-Br
=
(OF(M, r)
y
r
h
M )
r h
hoMoRl
2060h(O) + (1 - 02()
-06h'(O)h(o) - 36h(O)g'(0) + 306h 2( 0)].
x [(2 - a)h'()g(O)
[(2 - a)h()g()
(2.16)
These expressions of the two parts of the entropy density are computed in terms of
R, 0. To calculate this two parts of eq. (2.12), we used Mathematica to numerically
invert it to be functions of h and r. The singular part of the equation of state contains
the desire characteristic including the critical end point. A three dimensions plot of
the magnetization is shown in Figure 2-8. The plot shows desired properties of the
order parameter such as the first order phase transition region, the cross over region
and the critical point which agree with the qualitative argument given in section 2.1.
M(h)
-1.0
0.5
-0.5
1.0
-0.5
Figure 2-7: Plot of M as a function of h at r = 0.5. The plot illustrates smooth
crossover at h = 0. The plot is smoother and more inclined compare to the plot at
r = 0.1
It is instructive to look at slices of these plot at different r. Figure 2-4, figure
2-5, figure 2-6, and figure 2-7 show the plot of M(h, r) versus h at different value
of r, which determine what kind of phase transition the plot illustrates: first order,
second order, smooth crossover. (2)
also contains all the features. However M
dominates the shape of the entropy density plot for a reason that will be clear in the
next section.
2.3
Mapping onto QCD Phase Diagram
Now we want to map the three dimensional Ising model result onto the T -
sB (from
this point, we will refer to the QCD temperature as simply T) plane in the QCD phase
diagram. Recall the expression for the entropy density Sc = --(9r)
(Dr)
d'
The mapping between the two planes is essential for determining - and r. Figure29 shows the sketch of how the r - h axes(three dimensional Ising model) are mapped
onto the T - /B(QCD). According to [1], the r-axis is tangential to the first order
phase transition line at the critical end point(CEP). We set the h-axis perpendicular
to the r-axis for simplicity. In figure 2-9, at r < 0 the order of the phase transition is
first and at r > 0 it is analytic crossover. For simplicity, let us consider the case that
-1- 1.0
1.0
Figure 2-8: Three dimensions plot of the magnetization on the h, r plane. The plot
contains all the expected features of the phase diagram including the critical point.
the r-axis is perpendicular to the T-axis. Since the the angle is 90 degrees there are
simple relations for mapping h +-+T and r
+PlB
The most natural thing to do is to
assume that in a small critical region the relations between the scale of temperature
and baryon chemical potential an the scale of r and h is linear. The relations that
we will use are Ar = 1 +-+ AlB,,rit and Ah = 1 +-+ATit. Moreover, there is a
freedom to choose the CEP, we choose it to be (T, sB)=(154.7 MeV, 367.8 MeV) as
in reference[l]. In this simple assumption, the mapping between the Ising model r - h
plane and the QCD T - PB plane is done by substituting
PB -
h =
0.3678
-0.2
T - 0.1547
0.1
(2.17)
into the eq. (2.12) where we use GeV as the unit. The entropy density becomes
1
T - 0.1547 PB - 0. 3 6 7 8
0.1
'
-0.2
ATcrit
T rTit
T TA JBcri
critical region
C
Figure 2-9: Sketch of the r - h axes mapped onto the T - IB plane. The r-axis
is tangential to the first order line at the CEP. The h-axis is perpendicular to the
r-axis. [1]
where the (_)
ar vanishes since the r-axis is perpendicular to the T-axis. This
mapping procedure can be modified to use in a more general case such as when the
r-axis makes an angle Oit with the /B-axis. The mapping relations have to include
cos(Oit) and sin(Oti) terms. In order to connect the singular part of the entropy
density to the non-singular part by using Asakawa and Nonaka's idea, it is convenient
to define the dimensionless singular entropy density
Sc(T, PB) = A(ATerit, ABc-rit)sc(T , B)
where A(ATeit, A/sABr)
(2.19)
+ AlrtL2ATi and D is a dimensionless constant.
= Dx
From eq. (2.12), eq. (2.18), and eq. (2.19), we can write a general expression of the
dimensionless entropy density as
Sr
SC(T,
= (DV
D
) Z
2jt
+
2
M ( (T - 0.1547) cos(Ocrit) (AB - 0.3678) cos(Ocit)
M
0.1
-0.2
( aG
((T - 0.1547) cos(Oc,.t) (lB - 0.3678) cos(Ocit)
-r h
0.1
'
-0.2
sin(,crit)
aAPBcr
cos(Ocrit)
ATrit
(2.20'
where the angle Oit is usually small. This is why the singular part of the entropy density is dominated by M. A detailed discussion of Ocrit will be in chapter three. This
expression is very important because it offers a set of parameters (ATcrit, ABc,,it, D)
that we can adjust in order to explore the properties of Asakawa and Nonaka's model.
Chapter 3
The Entropy Density and the
Baryon Number Density
This chapter focuses on derivations of the entropy density and the baryon number
density in a model of the entire QCD phase diagram.
3.1
Non-Singular Part of the Entropy Density
For the two phases, hadronic phase and the quark-gluon plasma phase, there must be
two different equations of state. The equations of state must be that: one explains
the behavior very well at T and low
AB;
the other explains the behavior at the other
end of the phase diagram. More importantly, the equations of state must allow a
phase transition between the two phases. Let us start with the hadronic equation of
state.
3.1.1
Hadronic Equation of State
Let us consider gas of hadronic matters in volume V assuming local thermodynamic
equilibrium and neglecting interactions between the hadrons such as the isospin,
Coulomb, and surface effect. Therefore the hadronic gas can be explained by using the description we use to explain the quantum ideal gas. However since hadrons
P(GeV 4)
65
4
3
2
10.2
0.4
0.6
0.8
S T(GeV)
1.0
Figure 3-1: Plot of the hadronic pressure in GeV 4 versus T in GeV generated by
integrating equation. (3.2) numerically. We chose p = 0
have finite sizes so we cannot use the volume V in the calculation. The most intuitive
way to implement the finite-size effects is to use the prescription first used by Van
der Waals called the excluded volume approximation. The volume reduced by the
volume occupied by each particle,
V' = V -
viNi
(3.1)
Here vi and Ni are the excluded volume and the number of hadron of type i.
It
should be metioned that the excluded volume approximation is presented here for
completeness of the theory although we will not implement it in our calculations. Let
us start by calculating a pressure of the hadronic gas PH. The pressure for hadronic
gas can be derived by using relativistic Bose-Einstein distribution and Fermi-Dirac
distribution for the mesons and for the baryons respectively[3].
P(T,PB) =
ro
±1
d
62]m
mi
e
T
(3.2)
-l
where mi is the mass of the i - th hadron and gi is the degeneracy factor. The lower
SH
Temperature 3
5040
.
30
20
10
T(Gev)
1.0
0.8
0.6
0.4
0.2
0.0
Figure 3-2: Plot of the H versus T in GeV generated by integrating equation. (3.3)
numerically. We chose p = 0
sign in the right hand side of eq. (3.2) corresponds to mesons(bosons) and the upper
one corresponds to baryons(fermions). Another important point is that the chemical
potential pB is equal to zero when eq. (3.2) is applied to mesons and equal to -PB
when applied to the antibaryons.
Equation (3.2) can be integrated numerically by Mathematica. We obtained the
hadronic pressure as a function of T and tB. The overall pressure is the sum of
the pressure from different hadronic species, mesons, baryons and its corresponding
antibaryons. The list of hadronic species, their masses, and their degeneracy factors
are givin in table 3.1.1. Figure 3-1 shows a slice of the pressure plot at A = 0. The
entropy density is
From the above relation and equation (3.2), the entropy
(PH)
&T
AB
density is
-E
Si(T, AiB) =gi
6r
mi
d
(2
(e
.
-
mi)-2
e
+ 1)2
A
-
AB
T2
(3.3)
SH = E Si
The plot of the entropy density devided by temperature cube versus T at
is illustrated by figure 3-2. The low temperature behavior of -
B
= 0
comes out as one
hadron
7r
rI
p
w
p'
N
N(1440)
A(1232)
A(1520)
E(1530)
mi(GeV)
0.140
0.543
0.776
0.782
0.958
0.939
1.440
1.232
1.520
1.533
gi
3
1
9
3
1
4
4
16
4
8
Bi
0
0
0
0
1
1
1
1
1
Table 3.1: List of Hadronic species used in the calculating the pressure, their degenracy factors, and their baryon quantum numbers[3].
might expect that at low temperature the entropy decreases rapidly and vanishes at
the absolute zero.
3.1.2
Quark-Gluon Plasma Equation of State
QGP pressure (GeV3 )
0.030
0.025
0.020
0.015
0.010
0.005
0.20
0.25
0.30
T(GeV)
Figure 3-3: Plot of the QGP pressure in GeV 4 versus T in GeV. We chose p = 0
The entropy density for the quark-gluon plasma phase can be obtained from the
SQ
3
T
20
15 -
10
5
0.0
0.5
Figure 3-4: Plot of
2.0
1.5
1.0
T(GeV)
versus T in GeV. We chose p = 0.4
equation of state of the QGP phase in the Bag model.
PQ(T,LB)
PQ(T,
)
(32 + 21Ny)r 2 T4+
180
T
+ Nfy
N
f (B
)
2
T2
3
2
)4 -B
(3.4)
where the number of flavor Nf is 2 and the bag constant B is (0.44GeV) 4 . The
enstropy density of the QGP phase is obtained from differentiating eq. (3.4) by T.
We got
SQ (T, PB) = 2 9 %r
3
+2
2
T
(3.5)
Figure3-3 shows the plot of the pressure of the QGP phase versus temperture. It indicates that the pressure starts from the negative pressure at T = 0 and then increases
dramatically as T increases. The negative pressure is non-physical but the crossing
from negative to positive pressure indicates that there is at least one surface intersection between the hadronic pressure and the QGP pressure. Therefore formulating the
non-singular equation of state this way allows a first order first transition between
hadronic matters and the quark-gluon plasma. Figure 3-4 illustrates T3 growth of the
entropy density.
9 converges to a constant. Dividing by T 3 is instructive because
it makes the trajectory near the critical point more transparent. This quark-gluon
plasma entropy density behaves physically sensible at high temperature. At high temperature, QCD system is asymptotically free so its entropy density depends purely
on the quarks and gluons degrees of freedom, which is a constant number.
3.1.3
Ocit
Pressure(GeV 4)
0.04
0.03
0.02
0.01
0.0-
0.10
0.15
0.20
0.25
T(GeV)
0.30
Figure 3-5: The plot of pressure for the quark gluon plasma and hadronic matters at
p = 0.75 GeV. The crossing illustrates the first order phase thransition.
From chapter two, we know that the r-axis (Ising model) is tangential to the first
order phase transition line at the the CEP in QCD phase diagram. This section main
goal is to illustrate how to obtain the first order phase transition line of the QCD
phase diagram. Then O-it can be obtained by evaluating the tangent angle of the
first order phase transition line at the CEP. Pressure plots from the previous chapter
implies that the QCD system allows the first order phase transition between the QGP
phase and the hadronic phase. We can demonstrate this graphically by plotting slices
at the same
AB
of PQ and PH on the same graph. We expected to see a crossing
Tc(GeV)
0.170
0.165
0.160
0.155
0.0
0.4
0.2
Schemical
0.8
0.6
potential(GeV)
Figure 3-6: The plot of Tit as a function of A1B, which is the first order phase
transition line. The result for any temperature higher than 0.6 GeV is not reliable.
of the two pressure functions. An example of such plot is illustrated in figure 3-5.
It is reasonable to say that the line where the first order phase transition occur is
the line when SQ = SH. This can be solve numerically for different values of
B,
we
obtain Teit as a function of AS. Figure3-6 shows the first order phase transition line.
Since we have an expression Trt(PB), tan(Ocit) can be obtained by simply taking a
derivative of Trit(AB)
tan(cit) = dT i t(B)
The value for 0mit for
3.2
3.2.1
B,,ci
(3.6)
= 0.3768 GeV is very small, 0.0226 radians.
The Full Entropy Density
Formal Analysis
The entropy density that describes the whole QCD phase diagram can be constructed
by connecting the entropy density of the quark-gluon plasma from the Bag model to
the hadronic entropy density. As mentioned earlier, the shape of the QGP pressure relative to the hadronic pressure implies the existence of the first order phase
2
2
.4
1
Chemical potential(GeV)
,.2
0.0
Temperature(Ge
0.4
Figure 3-7: A three dimensional plot of
D to be 0.15
as a function of As and T(GeV). We pick
transition. However a complete entropy density has to contain the CEP. Asakawa
and Nonaka's model connects these two entropy densities by using the dimensionless
singular part entropy density Sc(eq. (2.20)). It can be defined as
S(T,
1AB)
=
1
(1 - tanh[Sc(T, 1 aB)) SH(T, AB) +
1
(1 + tanh[Sc(T,PB)I) SQ(T, 1 B).
(3.7)
First let us qualitatively discuss about this entropy density. The entropy density is
composed of two parts: the part that is dominated by 1 - tanh[S] and the part that
is dominated by 1 + tanh[Sc]. At high temperature and/or large chemical potential
-*
1, the term with SQ dominates the expression and the term
with SH vanishes.
On the other hand at low temperature and/or small chemical
region, tanh[Sc]
potential region, tanh[S] -+ -1, the term with SH dominates and the term with SQ
vanishes. The entropy density at the critical point is the average of the two entropy
densities evaluate at (Tcrit, B~,,)
From eq. (2.20), the expression for Ocit(eq. (3.6),
and eq. (3.7), we can calculate the full QCD entropy density with the CEP. The critical
point is chosen to locate at (Trit, /Bcr)=(0.1547 GeV, 0.3678 GeV). To get a better
S
T,
20
15 -
10
5-
0.0
0.1
Figure 3-8: A plot of
0.2
0.3
0.4
0.5
0.6
T(GeV)
as a function of T at PB=0. 4 GeV. D is chosen to be 0.15
understanding of Asakawa and Nonaka's result, we chose the same set of parameters,
(ATit, A/LB r, D) to be (0.1 GeV, 0.2 GeV, 0.15). To explore the pathology of the
entropy density we plot -.
A three dimensional plot and its two dimensional slices
are shown in figure 3-7 and figure 3-8 respectively. These plots show that the entropy
density with this choice of parameters is non-physical at low temperature. The value
at zero is infinity. From plots of the hadronic and the quark-gluon plasma entropy
density(fig.3-2,fig.3-4), we can conclude that the pathology at low temperature of the
entropy density is dominated by the QGP value rather than the hadronic value. On
the other hand the value at the high temperature region is the sum of both(mixing
of phases). This is incorrect since at fixed
[B
the entropy should be explained by
hadronic equation at low temperature and by the QGP equation at high temperature.
The sense of low and high temperature is relative to the critical region. Thus the
modeled used by Asakawa and Nonaka with current choice of parameter is incorrect
at explaining the physics of QCD. The section will demonstrate that by choosing
different value for the parameters, we can get a sensible result.
Parameters Choice
3.2.2
S
3
T
20
15
10
5
_'/
'I'' I '
0.0
'
'
0.1
'I '
' I'
'
'
0.2
'I ' I '
' II '
0.3
I
0.4
I
I
I
0.5
I
I
I
T(GeV)
0.6
Figure 3-9: A plot of -. as a function of T at PB=0. 4 GeV. D is chosen to be 2. The
entropy density decays to zero at T=0 and converges to a constant as T grows larger.
Recall that the current set of parameters (ATrit, /ABcr,, D)=(0.1 GeV, 0.2 GeV,
0.15) does not give a physical result. The source of the phase mixing at high temperature at the divergence at low temperature can be best understood by looking at
the
(1 - tanh[Sc(T,pB)]) SH(T, ,IB)
(1 + tanh[Sc(T,PLB)) SQ(T, pB)
terms. For the construction in eq. (3.7) to be sensible, the entropy density has to
switch from being hadronic matter dominated at the low temperature side of the
critical region to being quark-gluon plasma dominated at the high temperature side
of the critical region. Mathematically, we want to choose S, so that tanh[S.] --+ -1
and tanh[Sc] -
1 when T equal to 0.0547 GeV and 0.2547 GeV respectively. The
evaluation using current parameter choice gives tanh[Sc(0.5,0.1)]= 0.35622.
This
explains why the pathology is non-physical and why the value at high temperature
is the mixing betweeen both phases. However if we change D to 2, tanh[Sc(0.5, 0.1)]
becomes 0.999903. This value gives quark-gluon plasma dominated behavior. To
convince ourselves that the entropy density behaves properly, we plot
T)
(figure
3-9). It worths pointing out that at the temperature approximately equal to the
critical temperature (0.1547 GeV), the entropy density overshoot the QGP value
then decreases to the QGP constant value. There are two possible explainations for
this pathology. The first explaination is that it is an intrinsic feature of the critical
point that can be tested experimentally. The second explaination is that it is a feature
arise through the construction. This similar pathology also appears in plots of the
baryon number density (nB) and in the plot of
B
that will be discussed in chapter
four.
3.3
Baryon Number Density
n
0.6
0.4
0.2
0.05
0.10
0.15
0.20
T(GeV)
-0.2
-0.4
-0.6
Figure 3-10: A plot of E as a function of T at AB=0. 4 GeV. D is chosen to be 2.
This plot describes the first order phase transition.
One can construct baryon number density starting from the entropy density or
the pressure. Baryon number density is given by
OP
nB
1PB
f=
9S(T', PB) dT' + U (0, LB)
(3.8)
where nB(0, LB) is a initial condition which we set to be zero for simplicity. This
equation can be dervied directly from the thermodynamic relation
dP = SdT + ndpB
and that S =
(3.9)
F. However since the construction of the QGP and the hadronic
equation of states is done by gluing the entropy densities so the second line of the
equation is what we will use. Moreover in the first order phase transition region,
in order to take into account the discontinuity in the entropy density and baryon
number density on the first order phase transition boundary, it is necessary to add
the following term to the above expression when T > Tit
Tcrit (S(Tc + E, B) - S(Tc - E,PB))
(3.10)
where Estands for a small temperature deviation from the critical temperature. Note
that this quantity is the same as tan(O,it), we derived in section 3.1.3. This term
takes into account the discontinuity in the integration. The integration in eq. (3.8)
can be solved numerically by Mathematica. Figure3-10 describes the first order phase
transition, the discontinuity is taken into account when T > Tit, Tit=0.1547 GeV.
Note that there is narrow spike of the baryon number density at T=0. This effect
comes from the construction that it is QGP dominated at zero temperature. For T =
.O1GeV, uB = O.1GeV, the contribution from the QGP and hadrons are 1.258 x 10- 7
and 8.105 x 10- 41 respectively.
Chapter 4
Isentropic Trajectory and
Conclusion
This chapter is dedicated to discussing and explaining isentropic trajectories (trajectories with constant 1s). Note that also the choice of the parameters used in
calculating S, is (AiT,
AIC,,,, D)=(0.1 GeV, 0.2 GeV, 2) for the rest of the paper
unless stated otherwise. The critical point is also fixed to be at (T~,it,iBt)=(0.1547
GeV, 0.3678 GeV). In order to make sense of the isentropic trajectory of the full
equation of state, it is crucial to check whether at the extreme regions the trajectory
behaves properly. At low temperature and/or low baryon chemical potential the trajectory should follow the trajectory of the pure hadronic equation of states. At high
temperature and/or large baryon chemical potential, the trajectory should follow the
trajectory of the pure QGP equation of state.
4.1
The Trajectory of the Hadronic Phase and the
Quark-Gluon Plasma Phase
It is instructive to start with the discussion of the trajectories of the hadronic and
the QGP equation of state separately since it is easy to check the validity of the
full QCD trajectories. From the hadronic equation of state (eq. (3.2)), the number
0.30
T(GeV)
0.25-
.20
0.15
.050
0.0
0.1
0.2
0.3
0.4
Baryon
0.5
0.6
emiCa PoteniaI(GeV)
Figure 4-1: Contour Plot of the hadronic equation of state. The trajectores head
toward the point where T = 0 and JIB = mproton=0.938 GeV.
density can be derived by differentiating the hadronic pressure with respect to the
baryon chemical potential. Only the contribution from the baryons and anti-baryons
survive since all the pressure of mesons does not depend on the chemical potential.
The derivative can be taken inside the integration.
IE)
n(T,
_
S
P(T,
[LB)
(T,
2T
ideC
gi(2
67r2T mi
de
1)
2 eT
- 2
m2A2de+IB
e r
±1)2
(eT-
(4.1)
T(GeV)
Figure 4-2: Contour Plot of the QGP equation of state. The contours follow a straight
line of constant
"_B
T
The second term is due to the anti-baryons
(pB --
-AB).
This integration can be
obtained numerically and the contour plot of the hadronic equation is illustrated
in figure 4-1. The shape of the contour at high temperature regime is unphysical
since it should be QGP dominated. The trajectories in the low temperature region is
important. They should head to the point T=O,
sB = mroto, which they all seem to
follow. Now let us look at the the QGP equation of state. The number density can
be obtained easily since the equation of state for the pressure is a simple polynomial
in PBn(T, pB)
=
2PBT 2
(4.2)
From this equation together with the QGP entropy density (eq. (3.5)), it is obvious
that the trajectories can be analytically determined that it should be straight lines
of constant B (see figure 4-2).
4.2
4.2.1
Pathology of the Full Trajectory
D=0.15
0.30
.... :~.. ..-.
i'-" ~ .
0.25
T
(GeV)
0.20
0.15
0.10
0.05
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Bayon Chemial
Potential (GeV)
Figure 4-3: Contour Plot of the full equation of state using Asakawa and Nonaka's
choice of paremeter, D=0.15. These trajectories are not physical.
After we have a good undertanding of the hadronic and the QGP trajectories,
we can discuss the pathology of the insentropic trajectories of the model proposed
by Asakawa and Nonaka. As mentioned earlier that the choice of parameters plays
an important role in this model. This is confirmed by the trajectories of this model
when D is chosen to be 0.15 (figure 4-3). This pathology is very unphysical but
understood. Comparing to figure 4-1 and figure 4-2, it is obvious that the trajectories
57 200
'
'
0.34 (152,490)
5
160
120
0.29 (155,361
0.4 (11922
0.4 (144,48
80
100
200
300
400
500
600
9B [MeV]
Figure 4-4: Asakawa and Nonaka's result for D=0.15. The triagle represents the
critical point at (0.1547 GeV,0.3678 GeV). This contour plot does not contain a first
order phase transition line.[1]
at low temperature and small chemical potential follow the QGP trajectories while the
trajectories in the high temperature region are dominated by the hadronic one. This
contour plot agrees with what we found in chapter three. However the trajectories
obtained by Asakawa and Nonaka in their 2008 paper (figure4-4) illustrates a very
different pathology. It is important to note that the pathology they discovered does
not contain the first order phase transition. Our trajectories contains a first order
phase transition line, represented by the sharp horizontal line, at LB greater than 0.4
GeV.
4.2.2
D=2
We demonstrate in chapter three by using plots of the entropy density and the number
density that the choice of the parameter D is important. It can be chosen such that
the model makes physical sense at the two extreme regions. However its behavior in
0.30
T
(GeV) .. 25
0.20
0.05
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Baryon Chemical Potential(GeV)
Figure 4-5: Asakawa and Nonaka's model using D=2. The critical point is at (0.1547
GeV,0.3678 GeV).
other regions is not obvious. The contour plot is illustrated in figure 4-5. In order
to see whether the results make sense, let us look at this trajectory in detail starting
from top to bottom. describe At very high T, it is a line of constant T'l so, it is
going downward and leftward. As it approaches the crossover regime, the hadronic
contribution starts to come in. that is what turns the trajectories first vertical, and
then in the case of the left most trajectory even rightwards. Then the trajectory hits
the transition, and gets kicked dramatically to the left then, in the hadronic phase,
the trajectory turns to the right and heads toward T=0, LB = mproton. The behaviors
at the two extreme limits agree with the discussion in section 4.1. The trajectories
follow the QGP trajectories at high temperature and larger pB limit. The trajectories
should be diagonal lines of constant ", which they are. Below the transition, in the
hadronic phase, the trajectories should converge to the point of (T = 0, lB
=
mproton),
and they all do. In this limit, we ignore the pathology at very low T, which arises
from quark dominated behavior at extremely low T. The other interesting feature is
the leftward kick at the crossover region. The leftward kick in the first order line in
correct since the hadronic equation of state is turned on abruptly so a sharp left ward
kick is expected. However in the crossover region, this behavior is not yet understood.
It is interesting since it might indicates an intrinsic property of the critical point. In
the subsequent section, this leftward kick will be discussed.
4.2.3
Leftward Kick and the Rightward Turn at the Crossover
Region
nB
S
0.03
0.02
0.01
I''
0.05
0.10
0.15
0.20
0.25
T(GeV)
0.30
Figure 4-6: This is the plot of 2a when PB = 0.3. The area of interest are the local
minimum and maximum near the critical temperature.
To understand this behavior, we look at a cross section " (T) plot within the
crossover region. From figure 4-5, the contour number one from the left comes down
leftward and downward. Then when it passes by the transition region it got turned
to the right, which makes sense since it wants to follow the hadronic trajectory which
heads toward the (T = 0, AB
=
mpoto) point. However it was repelled to go leftward
S
T3
40
30
20
10
0.05
0.10
0.15
0.20
0.25
0.30
T(GeV)
Figure 4-7: This is the plot of the entropy density versus the temperature for
UB
= 0.3.
from the critical point before turn back to follow the hadronic line. The wedge-like
shape implies that there is a local minima in the sf plot, which is correct according
to figure 4-6.
In those plots we ignore the wide valley and the sharp peak near
T=0. These features are due to the quark dominated feature of the number density
at zero temperature as discussed in chapter three. The maxima corresponds to the
rightward turn. It is important to note that this wedge shape behavior near the
critical temperature is what led Asakawa and Nonaka to claim that the critical end
point acts as an attractor of the insentropic trajectories[1]. The signature proposed
by them is also based on this pathology. Now we will argue that these features of
the trajectories are not intrinsic to the critical point but due to the construction of
our equation of state in this specific way. This can be done by using graphical and
simple physical arguments. First we will explain the origin of the rightward turn.
The entropy density and the number density plots are shown in figure 4-7 and 48. We can see that the entropy density decays faster than the number density. At
the temperature approximately equal to 0.13 GeV, the entropy density had almost
dropped to being zero while the number density has only dropped halfway. The
nB
0.20
0.15
0.10
0.05
T(GeV)
0.05
0.10
0.15
0.20
0.25
0.30
Figure 4-8: This is the plot of the baryon number density versus the temperature for
IB = 0.3.
value of the number density catches up with the value of the entropy density again
as T decreases further. Let us go over the shape of this curve again before we go to
discuss the minimum. As temperature increases the rates that the entropy density
and the number density increases start from the same then the entropy undergoes a
sharp rising. Then the number density drops while the entropy stays at the constant
QGP value. The ratio rises first and then fall therefore there is a local maximum.
Now let us talk about the cause for the local minimum in the cross section. Starting
from the ratio !
of the constructed equation of state and a the ratio in the QGP
and the hadronic case, we require that the equation of state should be continuously
connect in the crossover region. To the right side of the minimum, the quark piece
is responsible for most of the value of 2.
4-6, figure 4-9, and figure 4-10, ,
This is obvious from looking at figure
was the around 0.014. The contribution from the
hadronic piece plays almost no role here. This minimum occurs from the attempt to
match the decreasing function with the QGP value on the right. We can conclude
this the minimum was just merely a requirement for the model to be continuous and
nQ
SQ
0.05
0.04
0.03
0.02
0.01
0.05
Figure 4-9: This is a plot of
0.10
0.15
0.20
0.25
0.30
T(GeV)
(T) at iB = 0.3. It starts from zero then increases to
a maxima at T=0.04.
follows the QGP equation of state at high temperature and it does not implies any
intrinsic property of the critical point.
4.3
Conclusion
In this thesis, we discussed the behavior of the thermodynamic quantities near the
critical point in the QCD phase diagram. The analysis of the entropy density in
chapter two suggests that the model proposed by Asakawa and Nonaka is parameterdependent. The results obtained from their choice of parameters does not make sense
physically. The QGP equation of state dominates at low temperature limit and the
mixed state at the high temperature limit. However by choosing a sensible parameters, this model can be made sensible at the extreme regions. The ultimate goal of
this project is to find an intrinsic property of the critical point that can be used as the
signature in a collider experiment. The isentropic trajectory, nB/S derived from this
model with an appropriate choice of parameters, behaves properly at both high temperature and at low temperature. There are some other unique features such as the
sudden leftward kick near the critical temperature of the trajectories in the crossover
region followed by a rightward turn to match up with the hadronic trajectories. These
nH
SH
0.004
0.003
0.002
0.001
T(GeV)
0.05
0.10
0.15
0.20
0.25
0.30
Figure 4-10: This is the plot of E when pB = 0.3. The plot increase steadily from
zero.
features are proved graphically to be the model-dependent features. Ultimately, we
can conclude that the claim by Asakawa and Nonaka regarding the signature that can
be tested experimentally is incorrect. This observation is a motivation for physicists
to come up with a new theoretical model that can truly demonstrate the intrinsic
features of the critical end point of the QCD phase diagram.
56
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