Revisiting Cornell potential model of the Quark–Gluon plasma
Introduction
The experiments of creating the quark–gluon plasma (QGP) in Relativistic Heavy-Ion Collider (RHIC) at the BNL [1] and CERN Large Hadron Collider [2], conceived many properties of this state of matter. Several theoretical models were developed to explore the thermodynamics of QGP. In strongly interacting quark–gluon plasma model (sQGP) [3] the effects of the colored and colorless bound states hadron resonances are responsible. The quasi-particle QGP (qQGP) model [4], [5], [6] it is assumed that QGP is consist of quasi-particles with a temperature depended mass. The strongly coupled quark–gluon plasma model (SCQGP) [7], [8] consider QGP is strongly coupled near the transition temperature and used the equation of state (EoS) of strongly coupled quantum electrodynamic plasma (QED) with proper modifications for QCD.
The Cornell potential model for QGP based on Mayer’s theory of plasma [9], [10] is also proposed in the same sense as in classical electrodynamical plasma. The CE method in plasma was taken into account and a detailed derivation of the equation of state and hence the transport coefficients of the QGP was calculated in [11]. In this investigation, the Cornell potential between the partons in the deconfined phase plays a crucial role to reproduce the EoS of the quark–gluon plasma. The screening of color charges of quarks and gluons may modify the Cornell potential at finite temperature and the dissociation of heavy quarkonium states occurred in the hot QCD medium [12]. Mayer’s cluster expansion method [13] is studied and the low-density limitation is considered in [14], [15] and finally, an equation of state has been derived based on the rigorous statistical approach in the high-density region for lattice gas in [16]. Quark–gluon plasma phase transition was studied using the cluster expansion method in [17], where a satisfactory result was obtained for the phase transition and region of condensation of QGP with the modified Cornell potential. In this paper, we obtain the complete EoS for QGP using Cornell potential model and compare with the lattice results [18], [19], [20], [21], [22], [23].
We formulate an equation of state of QGP using Mayer’s CE method, in particular above the critical temperature . Mayer’s theory of plasma was described in [24], where one sums a certain set of infinite diagrams. A generalized equation of state of QGP was computed in [9], [10] using Mayer’s theory of plasma. Their classical cluster expansion method is taken based on the assumption that for higher temperature, i.e, quantum effect can be neglected. Another argument for taking classical treatment is that higher the temperature the thermal wavelength approximately in the range of nuclear forces. But the results deviate from lQCD results when , where Quark–Gluon Plasma is a strongly coupled system. We derive a semi-classical EoS including the quantum aspects in the low temperature region.
In [25] the authors try to develop a description of Quark–Gluon Plasma by considering it as consisting of binary bound states, both hadronlike (colorless) and exotic (colored) bound pairs. Mayer’s cluster expansion method has been employed in the quarkonia dissociation studies [26]. The QCD thermodynamics and transport theory was studied in [27], with Virial Expansion. In [28], investigated the viral expansion of the EoS in terms of quasi-particle number densities. Here we use the quantum cluster expansion to take into account the contribution of binary bound states to obtain a better fit with lattice data at temperatures near .
The characteristic feature of quantum cluster expansion method [29] is to introduce cluster function . In the presence of interactions, we expect that the functions would be quite appropriate for carrying out high temperature expansions. The partition function for the N particle system is, where the restricted summation goes over all the sets with the condition The pressure and particle density in a canonical ensemble system in terms of the cluster integral at zero chemical potential system is, where is the thermal wave length.
Consider up to second term in Eqs. (3), (4) and combine the equations, we get, where and .
The cluster integral in the cluster expansion method includes the interaction potential between the partons, the Cornell potential. The cluster integrals are defined in terms of the trace of the -body cluster operator , Let us represent cluster integral as the sum of two terms, Where are the cluster integral for the ideal system and appear due to the particle interaction where , are appear due to energy states of the two particle system, with and without interaction, respectively.
By separating the different angular momenta and the possible discrete states from the continuum states, Eq. (6) can be transformed into for bound states The value of is for Fermions and for Bosons.
The cluster integral related to irreducible cluster integral [29] via, where the restricted summation is with the condition,
Section snippets
EoS of QGP using Mayer’s theory of CE
According to Mayer’s theory of classical plasma [24], the equation of state for any plasma is given by in natural units. For gluons plasma let the number density be represented by and for quarks–antiquarks plasma , respectively. is the Sum over all irreducible cluster integrals is defined as where . The D can be simplified as follows after using Newton’s binomial formula, The Inverse
Results and discussion
The modified EoS is fits the lattice data [18], [19], [20], [21], [22], [23] on the gluon plasma, and 2-flavor, 3-flavor in the case of a zero chemical potential. In Fig. 1 we plotted the versus for three flavor, two-flavor, and pure gauge QGP along with lattice results. For each system, the , and are adjusted, so that we get a good fit with the lattice results. Our EoS fit with the lQCD results for 0.12, 0.1 and 0.09 and 0.34 , 0.75 and 0.8 for gluon
Conclusions
We revisited the equation of state of QGP using Cornell potential and comparing theoretical results with the lattice data. To calculate the pressure and energy density we have made an equation of state in a semi-empirical way using Mayer’s theory of plasma. We include the binary bound states of quarks and gluon by deriving a complete EoS for quark–gluon phases. We found that the bound state of partons, i.e, and occurs in the temperatures of the order of a few times the critical
CRediT authorship contribution statement
Prasanth J.P.: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. Sebastian Koothottil: Formal analysis, Resources. Vishnu M. Bannur: Resources, Supervision, Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We would like to thank UGC, F.7-180/2007(BSR) New Delhi, for providing financial support for doing research and the organizers of DAE symposium on Nuclear Physics-2017 held at Patiala, India, which inspired us to present this result.
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