 Journal of Modern Physics, 2013, 4, 13-20 http://dx.doi.org/10.4236/jmp.2013.44A003 Published Online April 2013 (http://www.scirp.org/journal/jmp) A Study on Quark-Gluon Plasma Equation of State Using Generalized Uncertainty Principle Nabil Mohamed El Naggar, Lotfy Ibrahim Abou-Salem, Ibrahim Abdelbasit Elmashad, Ahmed Farag Ali Physics Department, Faculty of Science, Benha University, Benha, Egypt Email: loutfy.Abousalem@fsc.bu.edu.eg, ibrahim.elmashad@fsc.bu.edu.eg, ahmed.ali@fsc.bu.edu.eg Received February 6, 2013; revised March 8, 2013; accepted March 20, 2013 Copyright © 2013 Nabil Mohamed El Naggar 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 The effects of Generalized Uncertainty Principle, which has been predicted by various theories of quantum gravity re- placing the Heisenberg’s uncertainty principle near the Planck scale, on the thermodynamics of ideal Quark-Gluon Plasma (QGP) consisting of two and three flavors are included. There is a clear effect on thermodynamical quantities like the pressure and the energy density which means that a different effect from quantum gravity may be used in en- hancement the theoretical results for Quark-Gluon Plasma state of matter. This effect looks like the technique used in lattice QCD simulation. We determine the value of the bag parameter from fitting lattice QCD data and a physical in- terpretation to the negative bag pressure is introduced. Keywords: Generalized Uncertainty Principle 1. Introduction Since various theories of quantum gravity predict essen- tial modifications in the Heisenberg’s uncertainty princi- ple near the Planck scale, We utilize the proposed gener- alized uncertainty principle (GUP), which proved com- patible with string theory, doubly special relativity and black hole physics. Recently, a new model of GUP was proposed [1-3]. It predicts a maximum observable momentum and a minimal measurable length. Accordingly, (via the Jacobi identity) results in. ,,0 iji j pp xx 3, iji j ppp 22 , ij ij ij ij xp pp ip p (1) where 00pp Mc 2 and p c stand for Planck energy. and is Planck mass and length, respectively. 0 sets on the upper and lower bounds to . Apparently, Equation (1) imply the existence of a minimum measurable length and a maximum measurable momentum min0 , x (2) max 0 c p (3) where min x pp and max . Accordingly, for a particle having a distant origin and an energy scale com- parable to the Planck’s one, the momentum would be a subject of a modification [1-3]. 22 000 12 , ii ppp p (4) where 0ii x and 0 p , satisfy the canonical commu- tation relations 00ij ij pi p and simultaneously fulfil Equation (1). Here, 0i can be interpreted as the momentum at low energies (having the standard repre- sentation in position space, i.e. 00ii pix i p) and as that at high energies. The proposed GUP is assuming that the space is dis- crete, and that all measurable lengths are quantized in units of a fundamental minimum and measurable length. The latter can be as short as the Planck length [1,2]. In order to support the idea of this procedure, we can men- tion that similar quantization of the length (spatial di- mensions) has been studied in context of loop quantum gravity [4]. Furthermore, it has been suggested recently [5] that the GUP implications can be measured directly in quantum optics lab which seems to confirm the theoreti- cal predictions [6-8]. Since the GUP apparently modifies the fundamental commutator bracket between position and momentum operators, then it is natural to expect that this would re- sult in considerable modifications in the Hamiltonian. C opyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. 14 Furthermore, it would affect a host of quantum phenom- ena, as well. It is important to make a quantitative study of these effects. In a series of earlier papers, the effects of GUP was investigated on atomic and condensed matter systems [2,6-8], on the weak equivalence principle (WEP), and on the Liouville theorem (LT) in statistical mechanics [9]. For instance, it has been found that GUP can potentially explain the small observed violations of the WEP in neutron interferometry experiments [10-12]. Also, it can predict the existence of a modified invariant phase space which is relevant to the Liouville theorem.It seems that this approach accordingly modifies almost all mechanical Hamiltonians. Therefore, it can be imple- mented on studying the thermodynamics. In this paper, we present a study for the impact of the GUP on Quark Gluon Plasma (QGP). We calculate the corrections to various thermodynamic quantities, like energy density, pressure, equation of state and entropy. This paper is organized as follows. In Section 2, we review QGP, briefly. In Section 3, we investigate the thermodynamics of QGP and estimate the impact of GUP approach. We give our conclusions in Section 4. 2. Thermodynamics of Quark-Gluon Plasma In this section, we briefly review the thermodynamics of QGP consists of fermions. At finite temperature and chemical potential T , the grand-canonical partition function z for non-interacting massive fermions with internal degrees of freedom is given as [13] 0,1 exp Fl k lE z k T (5) 1exp Fk z Ek T k l (6) where is the momentum of the particle and is the occupation number for each quantum state with energy 22 k mEk . Here the infinite product is taken for all possible momentum states. Then the grand potential of a non-interacting massive fermion gas in a vanishing chemical potential is The value of this integral reads [13] 3 3 0 ,,0 dln 1e 2π Ek T TV k gT V . (7) 22242 4 22 ,,0 7πln , 89042 56ππ TV V MT MM gT C T with 232 0.346C , and , the Euler constant, defined as [13] 12 1 lim ln10.5772156649 nn nkn n n kn For simplicity we consider a non-interacting massless fermion gas in a vanishing chemical potential. Then the grand potential reads 32 4 3 0 d7π ln 1exp. 890 2π Ek k TgT VT P Therefore, pressure and energy density, , of hadronic state can be deduced 22 44 7π7π ,3 890 890 HHH q PgTg T . 3. Thermodynamics of Quark-Gluon Plasma with Effect of GUP For a particle of mass having a distant origin and an energy scale comparable to the Planck’s one, the mo- mentum would be a subject of a tiny modification and so the dispersion relation would too. According to GUP- approach, the dispersion relation in the co moving frame reads 222 24 12 .Ekkck Mc 1c (8) For simplicity we use natural units in which and consider a massless pion gas 12 12Ek kk . (9) For large volume, the sum over all states of single par- ticle can be rewritten in terms of an integral [14] 2 3 324 00 d d. 2π 2π1 k VVkk kk (10) Therefore, the partition function reads 12 22 2 4 00 12 3 12 2 31 0 ln 1 exp1 2 d, 2π 11 1exp1 2 2π31 1exp 12 F Ek kk TT Vg Vg zk kkk kk kk VgkkVgk TT kk T kk k T 24 3 23 ln 1exp ln d 2π ln 1exp12 2π31 (11) 12 2 0 13 d. 12 kk k Copyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. 15 It is obvious that the first term in Equation (11) vanishes. Thus 12 12 3 d 2π31 exp1 21 F k Tk Vg k zk k kk T 3 23 0 11 12 ln. (12) Let 12 12 k k T so that 12 113 dd 12 k k Tk and the integral becomes 3 3 d e1 x x k 2 0 ln 2π31 FVg k z . (13) Apparently, as we are interested in the terms contain- ing the first order of , so can be approximated as follows 12 12 k xk 1. kk TT (14) Then 222 kxTk xTxT 224 2.xTk k (15) when ignoring higher orders of , 1TxT . (16) kx Then the partition function becomes 3 3 1d e1 x xT x xT 33 2 0 1 ln 3 2π1 FVg zx T . (17) It is apparent that the integral contains Maclaurin se- ries, which are 22 2 3 3 3 33 116 1 xT TxT xT xT (18) when ignoring terms containing with order , then 2 34 4 dd . e1 xx x x T 3 2 00 ln 6 6πe1 FVgx x zT (19) The partition function is related to the grand canonical potential, lnF zT, so we have 34 34 2 00 335 4 22 00 45 22 56 dd 6 6πe1 e1 dd , 6ππ e1 e1 50 60 6ππ xx xx gxxxx TT T V Tgx TxgTxx gg TI TI 0n I (20) where are Bose and Fermi integrals. Substituting by the value of these integrals in (20) we have 2 45 2 π745 5 90 82π g gT T V (21) 2 45 2 π7 90 8 TgT V (22) where 22 45 5 2π . But the pressure is directly re- lated to the grand canonical potential, PV, so we obtain 2 45 2 7π 890 Hq q Pg TgT . (23) The energy density is related with the pressure by the relation 3P T and the entropy density is given by the derivative of pressure with respect to temperature 2 45 2 7π 33 890 Hq q . TgT (24) Taking into consideration the relevant degrees of freedom, it is obvious that this set of equations, Equa- tions (23) and (24), is valid in hadronic state. Similarly, the grand-canonical partition function z for non-interacting massive bosons with g internal de- grees of freedom is given as 0 ln exp Bl k Ek zl T (25) ln1 exp Bk Ek zT . (26) Then the grand canonical potential reads 2 45 2 2 45 1 π24 5 90 π π 90 g gT T V gTgT (27) 2 1245 π where . The pressure is directly re- lated to the grand canonical potential, PV . Then, in the hadronic phase 2 45 1 π 90 H PgTgT 3 . (28) In a massless ideal hadronic gas, the energy density is related with the pressure by the equation of state, H P , and the entropy density is given by the de- rivative of pressure with respect to temperature T 2 45 1 π 33. 90 H TgT (29) Copyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. 16 Now we will derive the QGP equation of state consists of free massless quarks and gluons. The total grand ca- nonical partition function of the hadronic matter can be obtained by combining the grand partition functions coming from the contribution of fermions (quarks), bos- ons (gluons) and vacuum. It reads [15] ln ln QGPF ln ln B v ZZZ ln (30) where ,ln B Z and lnv are the grand canonical partition functions of quarks, gluons and vacuum respec- tively. Since the value of vacuum partition function equals ln vVB ZT , so Equation (30) becomes ln ln QGP F ZZ ln BVB Z T . (31) From Equation (30) and equation ln ZT we have gluonsQGP VV quarks vacuum V V . (32) Substituting from Equations (21) and (28) into Equation (32) we have 45 12 5 12 . q q 22 45 2 4 π7π 908 90 7π 890 ggq gqg TgTg T V ggTg gTB gTB Thus, the QGP pressure PV reads 2 π 90 QGP QGPg Pg Tg 45 12q gTB (33) where 7 8 g qQGP gg are the QGP degenercy. Then the energy density reads 2 π 33 90 QGP QGPg 45 12q Tg gTB . (34) 4. Results and Conclusions 4.1. Comparison to Lattice QCD Simulations with Massless Quarks Generally, the value of pressure and the energy density depends on the temperature, the degenercy and the bag pressure. In this work, another factor added to them which originating from quantum gravity effect. The sec- ond term in Equation (23) which includes 2 gives a positive contribution to the pressure. As discussed in [12], the exact bound state on can be obtained by com- paring with observations and experiments [16]. The gamma rays burst would allow us to set an upper value for the GUP-charactering parameter . 1 0.005 GeV We will take the value of hich equals to the half value that corresponding to the upper bound for w 0 [6]. This term give a good results with lattice QCD due to the rapid increase contribution to the pressure. The bag parameter, B, has more than one method to be deter- mined. One of them based on fitting the pressure or the energy density with lattice QCD. The problem appears when one start from fitting the pressure function with Equation (33), one obtains a good qualitative agreement with lattice QCD results for admitting positive values of the bag constant. The positive bag constant needed in Equation (33) to fit the pressure leads, however, to an incorrect behavior of energy density. The same situation occurs when we start to calculate the value of, B, from fitting the energy density with Equation (34), this gives us a negative value of bag constant. Thus the value of, B, depending on whether we start from fitting the pressure or the energy density [17]. To overcome this problem, a modification of bag model was introduced to solve this problem [18]. In this technique, the fundamental ther- modynamical relation between the pressure and the en- ergy density is used which reads [14] d d P TT PT T . (35) Since Equation (35) is a 1st order partial differential equation, so if the energy density function known, then the general solution of Equation (35) gives us an arbi- trary integration constant which surely depends on tem- perature. Then the new pressure function can be fitted from lattice QCD results. In Refs. [17,18], this term had a linear dependence on temperature and the bag model carries the name (A-Bag model). This method was dis- cussed at first time in Ref. [19]. For the energy density in the form of Equation (34), i.e. GUP is included, the gen- eral solution of Equation (35) reads 2 45 12 π3 90 4 QGP QGPgq PgTggTBAT . (36) It is clear that beside the linear term, there is also an- other term, which can be easily obtained from comparing 5 12 4 gq ggT Equations (33) and (36), equals . This term due to quantum gravity effect and gives a negative contribution to the QGP pressure. To get a good fitting, we will consider the suppression factor of the Stefan Boltzmann constant 2 π 390 SB QGP g . The quasi-particle approach [20] is used. In this approach, the system of interacting gluons is treated as a gas of non-interacting quasiparticles with gluon quantum num- bers, but with thermal mass (i.e. m(T)). The modified SB constant equals to [17] Copyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. Copyright © 2013 SciRes. JMP 17 SB aIn Figure 3, the fitting of QGP pressure and energy density from Lattice QCD results [21] Using Equations (34) and (36). We take a TT 37, QGP g 0.150 GeV 4 4c T 0m (37) where has a direct connection with temperature and the thermal mass. The expression (37) was used for all c to keep the high temperature behavior of both pressure and energy density in agreement with lattice QCD. In Figure 1, the fitting of QGP pressure and en- ergy density from Lattice QCD results [21] Using Equa- tions (34) and (36). We take [13], c T and . The fitting of energy density gives us the value of bag constant as shown in Figure 1. 2, f n 8, B 24, g g 0.005 GeV q g 16,g 1 4. 3,47.5, 36, fQGPq ng g 16 g gπ8g [13], c T,0.150GeV and 1 0.005GeV 4 5.15 c BT B 3 8.42 . The fitting of energy density gives us the value of bag constant as shown in Figure 3. Substituting the value of , we obtain the value of A which equals c T. Then Substituting the value of and , we get the behavior of pressure showed in Figure 4 which is in a good qualitative behavior com- paring with lattice QCD. The value of suppressed factor in both cases (i.e. B 2 f n and ) eqals 3 f n Since we take the chiral limit q, then the dominant excitations in the hadronic phase is the mass- less pions, while that in the QGP is the massless quark and gluon. At extremely high temperature, the typical momenta of quarks and gluons are high and the running coupling 0.78a . 4.2. Discussion on the Negative Bag Pressure Since the bag pressure (vacuum pressure) is related mainly to the confinement phenomenon of hadrons, we need to discuss the picture of confinement. It is believed that all strongly interacting particles are made of quarks and gluons which are color charged. However, all ob- servable physical states formed from them are color neu- tral. This means that the true vacuum abhors color [15]. Any vacuum in which colored particles can exist as indi- vidual entities and so move freely is called “perturbative vacuum”. The difference between the perturbative vac- uum and the true vacuum, in which we live, is the amount of energy density in the regions of space [15]. In becomes so weak due to asymptotic freedom. Thus we can assume a free pion gas (free Quark-Gluon) in the low (high) temperature limit as a first approximation [13]. From the phase equilibrium condition cQGPc , the critical point can be obtained by equating Equations (28) and (36) at c. Substituting the value of , we obtain the value of A which equals PTPT B 3 76 c T 6. T. Then Substituting the value of and , we get the behavior of pressure showed in Figure 2 which is in a quit good qualitative behavior comparing with lattice QCD. B T4 T GeV 0 0 0 ε/T 4 T GeV Figure 1. The solid curve gives the fitting of energy density normalized to from lattice QCD results for two flavors . 4 T 2 f n  N. M. EL NAGGAR ET AL. 18 T GeV PT 4 0 0 Figure 2. The solid curve gives the pressure normalized to comparing with lattice QCD results for two flavors 4 T 2 f n. T 4 T GeV 0 0 ε/T 4 T GeV 4 T 3 f n Figure 3. The solid curve gives the fitting of energy density normalized to from lattice QCD results for three flavors . the true vacuum, color-charged quarks and gluons are confined but under extreme conditions of temperature and density, a transition to a deconfinement state of mat- ter is possible. This picture of hadronic interactions is consistent and justifies the perturbative approach to QCD interactions. This allows us to describe hadrons as “bags”. So, we need to melt the confining structure to able to move color charges within a region of space. For a first-order phase transition, the two phases have a differ- ence in energy density, the latent heat per unit volume, B, equivalent to QGPHG [15]. According to the cal- culated frame work, which was proposed by Bololiubov, independent quarks confined by a static Lorentz-scalar otential with infinite walls was considered. Thus, it was p Copyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. 19 PT 4 T GeV 0 0 0 Figure 4. The solid curve gives the pressure normalized to comparing with lattice QCD results for two flavors 4 T 3 f n. understood that the confining potential does not originate from quark-quark interaction, but it arises from the re- pulsion of colored quarks by the structured QCD vacuum state [22]. The positive value of bag pressure is coming from the difference in energy density between the QGP and the Hadronic states of matter. The negative value of bag pressure, in our opinion, may be understood by con- sidering the simplest imagination of QGP formation, in which the bag surface is compressed towards the bag center and so it always reduce the diameter of the bag. Thus the negative sign is not a numerical value to the bag pressure, but it has a physical interpretation. Although we tried to fined a physical origin to the negative bag pressure, it still needs more study. 4.3. Conclusion In this paper, the effects of Generalized Uncertainty Principle, which has been predicted by various theories of quantum gravity replacing the Heisenberg’s uncer- tainty principle near the Planck scale, on the thermody- namics of ideal Quark-Gluon Plasma (QGP) consisting of three massless quark flavors is included. There is a clear effect on the thermodynamical quantities like the pressure and the energy density which means that a dif- ferent effects from quantum gravity may be used in en- hancement the theoretical results for Quark-Gluon Plasma state of matter. This effect looks like the techniqe used in lattice QCD. We determine the value of the bag parameter from fitting lattice QCD data and a physical interpretation to the negative bag pressure is introduced. 5. Acknowledgements The research of NME, LIA, IE and AFA is supported by Benha University. IE would like to thank Prof. Abdel Nasser Tawfik for his fruitful discussion. REFERENCES [1] A. F. Ali, S. Das and E. C. Vagenas, “Discreteness of Space from the Generalized Uncertainty Principle,” Physics Letters B, Vol. 678, No. 5, 2009, p. 497-499. doi:10.1016/j.physletb.2009.06.061 [2] A. F. Ali, S. Das and E. C. Vagenas, “The Generalized Uncertainty Principle and Quantum Gravity Phenome- nology,” 2010. arXiv:1001.2642[hep-th] [3] S. Das, E. C. Vagenas and A. F. Ali, “Discreteness of Space from GUP II: Relativistic Wave Equations,” Phys- ics Letters B, Vol. 690, No. 4, 2010, p. 407-412. arXiv:1005.3368[hep-th] [4] T. Thiemann, “A Length Operator for Canonical Quan- tum Gravity,” Journal of Mathematical Physics, Vol. 39, No. 6, 1998, pp. 3372-3392. doi:10.1063/1.532445 [5] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. Kim and C. Brukner, “Probing Planck-Scale Physics with Quan- tum Optics,” Nature Physics, Vol. 8, 2012, pp. 393-397. doi:10.1038/nphys2262 [6] S. Das and E. C. Vagenas, “Universality of Quantum Gravity Corrections,” Physical Review Letters, Vol. 101, No. 22, 2008, Article ID: 221301. doi:10.1103/PhysRevLett.101.221301 Copyright © 2013 SciRes. JMP  N. M. EL NAGGAR ET AL. 20 [7] S. Das and E. C. Vagenas, “Phenomenological Implica- tions of the Generalized Uncertainty Principle,” Canadian Journal of Physics, Vol. 87, No. 3, 2009, pp. 233-240. doi:10.1139/P08-105 [8] A. F. Ali, S. Das and E. C. Vagenas, “A Proposal for Testing Quantum Gravity in the Lab,” Physical Re view D, Vol. 84, 2011, Article ID: 044013. arXiv:1107.3164[hep-th] [9] A. F. Ali, “Minimal Length in Quantum Gravity, Equiva- lence Principle and Holographic Entropy Bound,” Clas- sical and Quantum Gravity, Vol. 28, 2011, Article ID: 065013. arXiv:1101.4181[hep-th] [10] R. Collela, A. W. Overhauser and S. A. Werner, “Obser- vation of Gravitationally Induced Quantum Interference,” Physical Review Letters, Vol. 34, No. 23, 1975, pp. 1472- 1474. doi:10.1103/PhysRevLett.34.1472 [11] K. C. Littrell, B. E. Allman and S. A. Werner, “Two- Wavelength-Difference Measurement of Gravitationally Induced Quantum Interference Phases,” Physical Review A, Vol. 56, No. 3, 1997, pp. 1767-1780. doi:10.1103/PhysRevA.56.1767 [12] A. Camacho and A. Camacho-Galvan, “Test of Some Fundamental Principles in Physics via Quantum Interfer- ence with Neutrons and Photons,” Reports on Progress in Physics, Vol. 70, 2007, pp. 1-56. arXiv:0810.1325[gr-qc] [13] K. Yagi, T. Hatsuda and Y. Miake, “Quark-Gluon Plasma from Big Bang to Little Bang,” Cambridge University Press, Cambridge, 2005. [14] W.Greiner, L. Neise and H. Stocker, “Thermodynamics and Statistical Mechanics,” 1997. [15] J. Letessier and J. Rafelski, “Hadrons and Quark-Gluon Plasma,” Cambridge University Press, Cambridge, 2004. [16] A. S. Kapoyannis, “The Gibbs Equilibrium Conditions Applied to the QGP—Hadron Transition Curve,” The European Physical Journal C, Vol. 51, No. 4, 2007, p. 1013. doi:10.1140/epjc/s10052-007-0374-8 [17] V. V. Begun, M. I. Gorenstein and O. A. Mogilevsky, “Modified Bag Models for the Quark Gluon Plasma Equation of State,” International Journal of Modern Physics E, Vol. 20, 2011, pp. 1805-1815. arXiv:1004.0953v3[hep-ph] [18] V. V. Begun, M. I. Gorenstein and O. A. Mogilevsky, “Equation of State for the Quark Gluon Plasma with the Negative Bag Constant,” Ukrainian Journal of Physics, Vol. 55, No. 9, 2010, p. 1049. arXiv:1001.3139[hep-ph] [19] M. I. Gorenstein and O. A. Mogilevsky, “On a Non-Per- turbative Pressure Effect in Lattice QCD,” Zeitschrift für Physik C Particles and Fields, Vol. 38, No. 1, 1988, pp. 161-163. doi:10.1007/BF01574531 [20] M. I. Gorenstein and S. N. Yang, “Gluon Plasma with a Medium-Dependent Dispersion Relation,” Physical Re- view D, Vol. 52, No. 9, 1995, pp. 5206-5212. doi:10.1103/PhysRevD.52.5206 [21] F. Karsch, E. Laermann and A. Peikert, “The Pressure in 2, 2 + 1 and 3 Flavour QCD,” Physics Letters B, Vol. 478, No. 4, 2000, pp. 447-455. doi:10.1016/S0370-2693(00)00292-6 [22] P. N. Bogolioubov, “Sur un Modéle à Quarks Quasi- Indépendants,” Annales de l’Institut Henri Poincaré, Vol. 8, No. 2, 1967, pp. 163-189. Copyright © 2013 SciRes. JMP
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