Temperature Dependent Diquark and Baryon Masses

doi:10.4236/jmp.2013.47127

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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

Email: [email protected], arpita1chandra@gmail.com, ballari_cha[email protected]

Received March 21, 2013; revised April 22, 2013; accepted May 19, 2013

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



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

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:

 (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:





m

(2)

Hence the ratio of the normal mass (m) to the effective

mass can be expressed as:





















(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



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







can be expressed by using the same formalism as

in Equation (3) and is obtained as:

mar









(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:





 

(5)

We have parametrized the temperature dependent effec-

tive mass of diquark by a power law such that:

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]:





rrr











315

64 π



 (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



TT



and arrived at:

Ttt







 



 (8)

where 12



 and



TT t. The specific heat C

runs:

Ct t







Ttt

(9)

The coefficient of volume expansion is obtained as:









(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



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

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





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

 







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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