Revisiting Cornell potential model of the Quark–Gluon plasma

https://doi.org/10.1016/j.physa.2020.124921Get rights and content

Highlights

  • Semi-classical equation of state of Quark–Gluon Plasma is derived.

  • Cornell potential is the interaction between the partons.

  • Mayer’s theory of Plasma is modified.

  • Include the quantum effects of bound states of quarks and gluons near the transition temperature.

  • The results compared with the lattice QCD results, we get a good fit.

Abstract

The equation of state (EoS) of Quark–Gluon plasma (QGP) is revisited with Mayer’s theory of plasma. We derived a semi-classical EoS of QGP including the contribution of quantum effects near the transition temperature Tc. The cluster expansion (CE) method allows us to take into account the interaction between the partons, i.e, Cornell potential. The results obtained within our approach agrees with the lattice QCD results.

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 Tc 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 Tc. 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, T150MeV 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 TTc, 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 Tc.

The characteristic feature of quantum cluster expansion method [29] is to introduce cluster function Uˆl. In the presence of interactions, we expect that the functions Uˆl would be quite appropriate for carrying out high temperature expansions. The partition function for the N particle system is, QN(V,T)=1N!λ3N{ml}̀p[U1...U2][U2...U2]..d3Nrwhere the restricted summation goes over all the sets {ml} with the condition l=1Nlml=N;ml=0,1,2...The pressure and particle density in a canonical ensemble system in terms of the cluster integral bl at zero chemical potential system is, PT=1λ3l=1Nbl(T) n=1λ3l=1Nlbl(T)where λ=2πmT is the thermal wave length.

Consider up to second term in Eqs. (3), (4) and combine the equations, we get, PT=n1+nB22,where n1=b1λ3 and nB2=2b2λ3.

The cluster integral bl 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 l-body cluster operator Ul, bl(V,T)=1l!λ3(l1)VUl(r1,r2,r3,.,rl)d3lrLet us represent cluster integral as the sum of two terms, bl=bl(0)+bl(i),l>1Where bl(0) are the cluster integral for the ideal system and bl(i) appear due to the particle interaction b1=1 b2=λ3Vie(βEi(2))ie(βEi(0))where Ei(2), Ei(0) 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 b2b2(0)=8(12)BeβEB+8(12)πl(2l+1)0eβħ2k2mηl(k)kdk for bound states b2b2(0)=8(12)BeβEBThe value of b2(0) is 12(52) for Fermions and +12(52) for Bosons.

The cluster integral b2 related to irreducible cluster integral β1 [29] via, β1=2b2where the restricted summation is with the condition, k=2l(k1)mk=l1;mk=0,1,2...

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 PT=ini+DiniDniin natural units. For gluons plasma let the number density ni be represented by ng and for quarks–antiquarks plasma nq, nq̄ respectively. D is the Sum over all irreducible cluster integrals β(NiNj) is defined as D=N2NiNjβ(NiNj)niNinjNjwhere N=Ni+Nj. The D can be simplified as follows after using Newton’s binomial formula, D=14π20l2dl(κ2Vl)22+(κ2Vl)33+.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 PT4 versus TTc for three flavor, two-flavor, and pure gauge QGP along with lattice results. For each system, the σ, αs and δ are adjusted, so that we get a good fit with the lattice results. Our EoS fit with the lQCD results for α = 0.12, 0.1 and 0.09 and σ = 0.34 GeV2, 0.75 GeV2 and 0.8 GeV2 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 P and energy density ϵ 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, qq and gg 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.

References (42)

  • AdamsJ.

    Nuclear Phys. A

    (2005)
  • ShuryakE.

    Nuclear Phys. A

    (2005)
  • BannurV.M.

    Phys. Lett. B

    (2007)
  • BannurV.M.

    Phys. Lett. B

    (1995)
  • KarschF. et al.

    Phys. Lett. B

    (2000)
  • ShuryakEdward

    Nuclear Phys. A

    (2005)
  • AamodtK.

    Phys. Rev. Lett.

    (2010)
  • BannurV.M.

    Eur. Phys. J. C

    (2007)
  • SimjiP. et al.

    Internat. J. Modern Phys. A

    (2013)
  • BannurV.M.

    J. Phys. G: Nucl. Part. Phys.

    (2006)
  • GelmanBoris A. et al.

    Phys. Rev. C

    (2006)
  • UdayanandanK.M. et al.

    Phys. Rev. C

    (2007)
  • PrasanthJ.P. et al.

    Physica A

    (2018)
  • AgotiyaVineet et al.

    Phys. Rev. C

    (2009)
  • MayerJ.E. et al.

    Statistical Mechanics

    (1977)
  • UshcatsM.V.

    Phys. Rev. Lett.

    (2012)
  • UshcatsM.V.

    Phys. Rev. E

    (2013)
  • UshacatsM.V.

    Phys. Rev. E

    (2015)
  • Syam KumarA.M. et al.

    Physica A

    (2015)
  • KarshF.

    Nuclear Phys. A

    (2002)
  • LaermannE. et al.

    Annu. Rev. Nucl. Part. Sci.

    (2003)
  • Cited by (0)

    View full text