Journal of Modern Physics, 2013, 4, 945-949
http://dx.doi.org/10.4236/jmp.2013.47127 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Temperature Dependent Diquark and Baryon Masses
A. Chandra1, A. Bhattacharya1, B. Chakrabarti2
1Department of Physics, Jadavpur University, Kolkata, India
2Department of Physics, Jogamaya Devi College, Kolkata, India
Received March 21, 2013; revised April 22, 2013; accepted May 19, 2013
Copyright © 2013 A. Chandra et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Temperature dependence of diquark mass has been investigated in the frame work of the quasi particle diquark model.
The effective mass of the diquark has been suggested to have a temperature dependence which shows a power law be-
havior. The variation of the diquark mass with temperature has been studied. A decrease in effective mass at tempera-
ture T < Tc, where Tc is the critical temperature has been observed. Some features of the phase transition have been dis-
cussed. The phase transition is found to be of second order. Temperature variation of baryon masses has also been stud-
ied. The results are compared and discussed with available works.
Keywords: Diquark; Quasi Particle; Rushbrooke Inequality; Power Law
1. Introduction
Quasi particles are particle like entities which arise in
some system of interacting particles. The existence of
quasi particles is mostly known in condensed matter
physics. The quantum phases like superfluidity and su-
perconductivity are largely described by the properties of
such low lying excitations which behave like quasi parti-
cle simulating many body interactions in the system. An
electron in crystal lattice behaves like a quasi particle. It
is well known that the behavior of electron and other
particles depends upon the environment. The interaction
inside can alter the collective properties which produce a
new particle called quasi particle. The quantum Hall
effect which is an emerging area of research also sug-
gests that the collective behavior of an electron results in
a new particle which may have fractional charges but
seems to be the constituent of electron. Quasi particles
and electrons are same entity in this picture having no
distinction as expected between an elementary particle
and a quasi particle. The concept of quasi particles has
been widely used in describing the system of fermions,
superconductivity, superfluidity. Khodel et al. [1] have
investigated the properties of fermion system and corre-
sponding phase transition of fermion condensate. With
the advent of low temperature experimental technique the
quantum critical point phenomena and quantum fluctua-
tion have become area of interest and intense research.
Abrahams et al. [2] have investigated critical quasi parti-
cle theory and have discussed the scaling behavior asso-
ciated with the quantum critical point (QCP). The effect
of temperature dependent quasi particle mass which is
effective on the surface impedance of a crystal has been
studied by Cassinese et al. [3] in the context of two fluid
model. Nawa et al. [4] have studied the BEC condensa-
tion of composite diquark in the quark matter (color su-
perconductivity) using a quasi chemical theory at a low
density in a region near the de-confinement phase transi-
tion. They have argued that the dynamical quark-pair
fluctuation can be described as bosonic degrees of free-
dom which are diquarks. Schneider et al. [5] have sug-
gested that at very high temperature the QGP may be
considered to be consisted of quasi free quarks and glu-
ons. They have considered a temperature dependent
quasi particle effective mass by parametrizing
 
0
~1
c
mT GTTT



 . Koh [6] has pointed out that


the heavy massive quasi particle becomes superconduct-
ing in heavy fermion system and has suggested that the
phase transition is of second order. The study of hadrons
and their masses at finite temperature is extremely im-
portant for understanding the phase transition from had-
ronic phase to QGP phase. It has been suggested that the
nucleon mass depends substantially on the temperature
variation of quark condensate and relevant interaction
process. A number of works have been done on the
possible variation of nucleon mass at finite temperature.
In the current work we have studied the temperature
dependence of effective mass of diquarks and baryons in
C
opyright © 2013 SciRes. JMP

A. CHANDRA ET AL. 946
quasi particle approach. Recently we have proposed a
quasi particle model for diquark where diquark has been
described as a quasi particle which behaves like low ly-
ing excited state resembling the hypothetical phonon
particle in quasi quantum systems. We have suggested
relation between the temperature and the effective mass
of diquark which follows a power law type of behavior.
The power law is a useful tool for studying the critical
behaviour near the transition point. The quasi particle
critical behaviour has been studied by a number of au-
thors. It would be interesting to study how does the mass
of the diquark behaves which is also described as a quasi
particle in the current work. The baryons are studied in
the frame work of diquark-quark system and the varia-
tion of masses of baryons with temperature have also
been investigated. Critical exponents of different ther-
modynamic co-ordinates have been studied and they are
found to show Rushbrooke inequality indicating a second
order phase transition for deconfinement.
2. Quasi Particle Model of Diquark and
Temperature Dependence of Mass
We have suggested a model [7] in which two quarks are
assumed to be correlated to form a low energy configura-
tion, forming a diquark and behaving like a quasi particle
in an analogy with an electron behaving as a quasi parti-
cle in the crystal lattice [8]. It is well known that a quasi
particle is a low-lying excited state whose motion is
modified by the interactions within the system. An elec-
tron in a crystal is subjected to two types of forces,
namely, the effect of the crystal field (grad V) and an
external force (F) which accelerates the electron [8].
Under the influence of these two forces, an electron in a
crystal behaves like a quasi particle having velocity v
whose effective mass m* reflects the inertia of electrons
which are already in a crystal field such that:
d
d
v
mF
t
 (1)
The bare electrons (with normal mass) are affected by
the lattice force-grad V (where V is the periodic potential)
and the external force F so that:
dd
dd
vV
mF
x
x


m
(2)
Hence the ratio of the normal mass (m) to the effective
mass can be expressed as:
1
1
mV
F
x

m










(3)
An elementary particle in vacuum may be suggested to
be in a situation exactly resembling that of an electron in
a crystal. We have proposed a similar type of picture for
the diquark

0
ud as a quasi particle inside a nucleon.
We assume that the diquark is an independent body
which is under the influence of two types of forces. One
is due to the background meson cloud which is repre-
sented by the potential
 
23 r   s V, where αs
being the strong coupling constant, and this potential
resembles the crystal field on a crystal electron. On the
other hand for the external force we have considered an
average force F = −ar, where a is a suitable constant,
which is of confinement type. It has been assumed that
under the influence of these two types of interactions the
diquark is behaving like a quasi particle, a low lying ex-
cited state and its mass gets modified. The ratio of the
constituent mass and the effective mass of the diquark


D
m

can be expressed by using the same formalism as
in Equation (3) and is obtained as:
3
12
qq
D
mm
mar



mm

(4)
Here qq


represents the normal constituent mass
of the diquark and mD is the effective mass of the diquark,


23 ,0.58  
ss [9] and the strength parameter a
= 0.003 GeV3 [10] for the light and heavy-light diquarks,
V being the average value of the one gluon exchange
type of potential. “r” is the radius parameter of the
diquark. To calculate the effective mass of the diquark
from the above expression we need the radius parameter
r of the diquark. To calculate the effective mass of the
diquark from the above expression we need the radius
parameter “r” of the diquark. The radii parameter of the
scalar diquarks have been used from existing literature
[11-15] and constituent masses of quarks are taken from
Karliner et al. [16]. We have estimated the masses of the
diquarks in the framework of the quasi particle and the
results that obtained are displayed in Table 1. The
diquarks which are described as the elementary excita-
tion simulating many body interactions behaves like sca-
lar boson and may be regarded as separate entity. Two
diquarks should be antisymmetric in color so that a total
color singlet state is obtained. We presume that such a
system of low lying excitations behaving like quasi par-
ticles does not interact among themselves and as in an
ideal gas their energies are additive [17]. Thus the mass
of a baryon in a diquark-quark system can be represented
as:
M
B
Dq
mm


 
(5)
We have parametrized the temperature dependent effec-
tive mass of diquark by a power law such that:
01
DD
c
T
mT mT







(6)
where “ϵ” is critical exponent. The critical exponent de-
scribes the temperature dependence of the system near

A. CHANDRA ET AL. 947
Table 1. Diquark masses (m*(0)).
Diquark
Quark Content Radius (GeV−1) Mass Computed (GeV)
[ud]0 5.38 [11] 0.509
[us]0, [ds]0 6.06 [12] 0.698
[ss]0 3.65 [13] 0.985
[uc]0, [dc]0 5.5 [14] 1.491
[sc]0 5.4 [13] 1.596
[sb]0 4.1 [14] 2.887
[ub]0, [db]0 4.4 [14] 3.079
[cc]0 2.88 [15] 3.287
[cb]0 2.415 [15] 6.161
[bb]0 1.75 [15] 8.556
the critical point. We have assumed to be 32. The
mass variation of diquark with temperature has been dis-
played in Figure 1. We have considered the critical ex-
ponent as 32 from the concept of statistical model of
hadron which is described in details in [18-20]. The
probability density of baryon in the ground state is ex-
pressed as [18]:
 
32
00
rrr  

rr  
2
92
0
315
64 π
rr
r
  (7)
where r0 is the radius parameter of the corresponding
hadron and 0 represents the step function. It is
interesting to note here that the wave function or the
probability density possesses a fractional power 32 in-
dicating a non analytical behavior [19,20]. In the current
work we have assumed that the temperature dependence
of the diquark mass can be represented by a power law
with a critical exponent 32 in an analogy with the
power law bahaviour of the statistical model wave func-
tion as discussed earlier. It would be interesting to inves-
tigate how does the above mentioned parameterization
yield the mass variation and the thermodynamic proper-
ties of the diquarks and the baryons. We define an order
parameter by

12
1c TT 
and arrived at:
12
12
1
c
Ttt
T



 

 (8)
where 12

 and

1c TT t. The specific heat C
runs:
72
Ct t



1
Ttt
(9)
The coefficient of volume expansion is obtained as:




22
(10)
We find that

Figure 1. The temperature dependent diquark mass below
the critical point.
tive mass with temperature has been estimated for the
diquark and the baryons using Equations (5) and (6). The
masses of the various baryons in the context of the quasi
particle model have been estimated using Equation (5)
with temperature dependent diquark mass from Equation
(6) with different “a” such as a = 0.04 GeV3 for the [ss]0
diquark (fitting the experimental mass of c
0

from
PDG2010), a = 0.2 GeV3 for doubly heavy diquarks. The
results are displayed in Figure 2 for light baryons, Fig-
ure 3 for charm sector, Figure 4 for bottom sector and
Figure 5 for triply heavy baryons. The masses are found
to decrease below Tc.
3. Results and Discussions
In the current work we have investigated the temperature
dependent effective mass of diquark incorporating a
power law type of behavior with critical exponent 32
TT
inspired by the power law behavior of wave function
obtained in context of the statistical model [18]. It has
been suggested that diquark is a fundamental constituent
of hadron and quasi particle in nature formation of which
is favoured by the quantum behavior (superfluid) of the
vacuum. The mass of diquark has been found to decrease
at c

. The decrease in mass may be attributed to the
fact that as the temperature of the system falls below
critical value, more and more hypothetical virtual di-
quarks have been created making the system ordered
which changes the interaction of the system. This results
in the suppression of the mass enhancement in contrary
to the normal state. Koh et al. [6] have pointed out that
the quasi particles in heavy fermion system become su-
perconducting with anomalously high mass near transi-
tion temperature and the mass decreases with decreasing
temperature from Tc. They have suggested that the phase
transition is of second order. Similar observation has
been made in the current work. The mass is found to in-
crease anomalously near transition temperature. Results
 
22
which satisfy Rush-
brooke inequality

  suggesting a second
order phase transition for diquark. The variation of effec-

A. CHANDRA ET AL. 948
Figure 2. Baryon masses for light-sector as a function of
temperature below critical temperature (T = Tc).
Figure 3. Baryon masses for charm-sector as a function of
temperature below critical temperature (T = Tc).
Figure 4. Baryon masses for bottom-sector as a function of
temperature below critical temperature (T = Tc).
are displayed in Figures 2-5. The transition from had-
ronic to deconfinement phase is suggested to be of sec-
ond order as the dissolution of diquark indicates non ex-
istence of baryon in the current scheme. It may be men-
tioned that temperature dependent hadron masses have
Figure 5. Triply heavy baryon masses as a function of tem-
perature below critical temperature (T = Tc).
been investigated by a number of authors [21-25] in the
context of hot dense hadronic matter. Results obtained
are divergent in nature and somewhat inconclusive. Some
predicts increase in masses with temperature [21,22]
whereas some observed just the opposite [23-25]. Eletsky
et al. [26] have investigated current correlator in QCD at
finite temperature. They have pointed out that the study
of current correlator could yield a clear signal for phase
transition between QGP at high temperature and hadronic
phase at low temperature. At c they have observed
a decrease in mass with temperature for ρ and a1 meson.
The mass shift is found to vary as T4. Zakout [27] has
studied nucleon bound state at finite temperature in di-
quark-quark scheme using Bethe Sal-peter equation with
an interaction via exchange quark. The modification of
interaction has been approximated by imaginary time
formalism in propagator and then an adiabatic approxi-
mation. The nucleon mass has been suggested to de-
crease with temperature as
TT
 



12
22
01 c MTMb TT . It may be mentioned
that the critical like divergent equations have been sug-
gested and studied widely in many systems in condensed
matter physics [28-30].
4. Conclusion
In the current work the temperature variation of baryon
masses has been studied considering a power law be-
havior with critical exponent 32 in an analogy with
the power law behaviour of the probability density of
hadrons obtained in the context of statistical model [19,
20]. The phase transition has been found to be of second
order. The power law behavior assigned in the current
investigation leads to the results which agree well with
the observations made by other authors with quasi parti-
cle approach in different context of hadrons and super

A. CHANDRA ET AL. 949
physics. It may be mentioned that the power law behav-
iors are the manifestation of the dynamics of complex
system whose striking feature is the showing of universal
laws which are characterized by exponents in scale in-
variant distribution and they are basically independent of
the detailed of the microscopic dynamics [31]. Hadron
itself is a complex system. It appears that it may not be
far from reality to describe such a complex system by
chaos, fractal which shows power law behavior. Further
insight in this direction would be done in our future
works.
5. Acknowledgements
Authors are thankful to University Grants Commission
(UGC), New Delhi, India for financial assistance. Ref No.
F.No.37-217/2009 (SR).
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