# The Quark-Gluon Plasma Equation of State and the Generalized Uncertainty Principle

Authors: L. I. Abou-Salem; N. M. El Naggar; I. A. Elmashad
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

The quark-gluon plasma (QGP) equation of state within a minimal length scenario or Generalized Uncertainty Principle (GUP) is studied. The Generalized Uncertainty Principle is implemented on deriving the thermodynamics of ideal QGP at a vanishing chemical potential. We find a significant effect for the GUP term. The main features of QCD lattice results were quantitatively achieved in case ofnf=0, nf=2, and nf=2+1 flavors for the energy density, the pressure, and the interaction measure. The exciting point is the large value of bag pressure especially in case of nf=2+1 flavor which reflects the strong correlation between quarks in this bag which is already expected. One can notice that the asymptotic behavior which is characterized by Stephan-Boltzmann limit would be satisfied.


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## Body

## 1. Introduction

Essential modifications in Heisenberg’s uncertainty principle are predicted near Planck scale which is called Generalized Uncertainty Principle (GUP). One of the most exciting predictions of some approaches related to quantum gravity [1], perturbative string theory [2], and black holes [3] is the minimal length concept existence. For a recent review, see [4]. These approaches seem to modify almost all mechanical Hamiltonians. Thus, quantum mechanics can be studied in the presence of a minimal length [5–8]. Also, in quantum optics, the GUP implications can be measured directly which confirm the theoretical predictions [9–11]. In particular, exact solutions of various relativistic [12–15] and nonrelativistic problems [16–22] have been obtained in the presence of a minimal length Δx0=ħ(β)1/2. The Generalized Uncertainty Principle [23, 24] was implemented on deriving the thermodynamics of ideal quark-gluon plasma of massless quark flavor [25, 26].In other minimal length formalism [5, 6], the Heisenberg algebra associated with the momentum p^ and the position coordinates x^i is given by(1)x^i,p^j=iħδij1+βp2,where β>0 is the minimal length parameter. The Generalized Uncertainty Principle (GUP) corresponding to it reads(2)ΔxiΔpj≥δij1+βΔp2+βp2which yields a minimal observable length Δx0=ħ(β)1/2. Hence, the momentum would be subject of a modification and becomes(3)pi=poi1+βp02,where xi=x0i and p0i satisfy the canonical commutation relations x0i,p0j=iħδij. Also, p0i can be interpreted as the momentum at low energies and pi at high energies. Since the GUP modifies the Hamiltonian, it is important to study these effects quantitatively. These effects were investigated on condensed matter, atomic systems [10–12, 27], the Liouville theorem (LT) in statistical mechanics [28], and the weak equivalence principle (WKP). Recently, another approach based on super gravity was implemented to QCD and to QGP especially [29–31].In this paper, the effect of the GUP on QGP equation of state of massless quark flavors at a vanishing chemical potentialμ is studied. We calculate the corrections to various thermodynamic quantities, like the energy density and the pressure. Then, these corrections with bag model are used to describe the quark-gluon plasma equation of state and compare it with QCD lattice results. This paper is organized as follows. In Section 2, we derive thermodynamics of QGP consisting of a noninteracting massless bosons and fermions with impact of GUP approach. The results, discussions, and conclusions are given in Sections 3 and 4.


## 2. Thermodynamics of Quark-Gluon Plasma with GUP Effect

In this section, we derive the thermodynamics of QGP in case of bosons and fermions taking into account the GUP impact. Then, the thermodynamical equations such as pressure and energy density of quark-gluon plasma are obtained.


### 2.1. In Case of Bosons

At finite temperatureT and chemical potential μ, the grand canonical partition function zB for noninteracting massive bosons with g internal degrees of freedom is given by [32](4)zB=∏k∑l=0∞exp⁡-lEk-μTg(5)=∏k1-exp⁡Ek-μT-g,where l is the occupation number for each quantum state with energy E(k)=k2+m2 with mass m and k is the momentum of the particle. Here the infinite product is taken for all possible momentum states.For simplicity we consider chiral limit (i.e., vanishing mass) and a vanishing chemical potential, which experiments ensure at high energy. Then the dominant excitation in the hadronic phase is a massless pion, while that in the quark-gluon plasma is massless quarks and gluons. For a particle of massM having a distant origin and an energy comparable to the Planck scale, the momentum would be a subject of a tiny modification so that the dispersion relation would too. According to GUP approach, the dispersion relation reads(6)E2k=k2c21+βk22+M2c4,where M and c are the mass of the particle and the speed of light as introduced by Lorentz and implemented in special relativity, respectively. For simplicity we use natural units in which ħ=c=1. Hence, we have(7)Ek=k1+βk2.For large volume, the sum over all states of single particle can be rewritten in terms of an integral [33](8)∑k⟶V2π3∫0∞d3k⟶V2π2∫0∞k2dk1+βk24.Thus, the partition function, (4), becomes(9)ln⁡zB=-Vg2π2∫0∞k2ln1-exp⁡-Ek/T1+βk24dk=-Vg2π2∫0∞k2ln⁡1-exp⁡-k/T1+βk21+βk24dk=Vg2π2k6β1+βk23ln⁡1-exp⁡-kT1+βk20∞-Vg2π2∫0∞16β1+βk23ln1-exp⁡-k1+βk2T+k1+3βk2T1exp⁡k1+βk2/T-1dk,where “Integration by Parts” was used for solving the above equation. It is obvious that the first term in (9) vanishes. Thus,(10)ln⁡zB=-Vg2π2∫0∞16β1+βk23ln⁡1-exp⁡-k1+βk2T+k1+3βk2T1exp⁡k1+βk2/T-1dk.For solving (10) let x=(k/T)(1+βk2) so that dx=(1+3βk2)dk/T and(11)ln⁡zB=-Vg2π2∫0∞ln⁡1-e-xTdx6β1+βk231+3βk2-Vg2π2∫0∞kdx6β1+βk23ex-1.The momentum k as a function of x variable can be approximated to the first order of β as follows:(12)k=xT-βk3=xT-βxT-βk33=xT-βx3T31-βk3xT3≃xT-βx3T3.Employing the value of k into integrant terms of (11) and approximating them to the first order of β are as follows.For the first term,(13)T6β1+βk231+3βk2=T1-6βk2+9β2k46β≃T6β-x2T3+7β2x4T5.For the second term,(14)k6β1+βk23=k-3βk36β≃xT6β-2x3T33+32βx5T5.Substituting the values of terms in (13) and (14) into (11) and solving them analytically, we have(15)ln⁡zB=π290VgT3-16π4315gβT5.In case of β→0, the above equation is reduced to the partition function for bosons without GUP effect [32].


### 2.2. In Case of Fermions

At finite temperatureT and chemical potential μ, the grand canonical partition function zf for noninteracting massive bosons with g internal degrees of freedom is given by [32] (16)zf=∏k∑l=0,1∞exp⁡-lEk-μTg(17)=∏k1+exp⁡-Ek-μTg,where l is the occupation number for each quantum state with energy E(k)=k2+m2 with mass m and k is the momentum of the particle. Here the infinite product is taken for all possible momentum states.For simplicity we consider chiral limit (i.e., vanishing mass) and a vanishing chemical potential, which experiments ensure at high energy. Then the dominant excitation in the hadronic phase is a massless pion, while that in the quark-gluon plasma is massless quarks and gluons. For a particle of massM having a distant origin and an energy comparable to the Planck scale, the momentum would be a subject of a tiny modification. According to GUP approach, the dispersion relation reads(18)E2k=k2c21+βk22+M2c4,where M and c are the mass of the particle and the speed of light as introduced by Lorentz and implemented in special relativity, respectively. For simplicity we use natural units in which ħ=c=1. Hence, we have(19)Ek=k1+βk2.For large volume, the sum over all states of single particle can be rewritten in terms of an integral [33](20)∑k⟶V2π3∫0∞d3k⟶V2π2∫0∞k2dk1+βk24.Thus, the partition function, (16), becomes (21)ln⁡zf=Vg2π2∫0∞k2ln⁡1+exp⁡-Ek/T1+βk24dk=Vg2π2∫0∞k2ln⁡1+exp⁡-k1+βk2/T1+βk24dk=-Vg2π2kln⁡1+exp⁡-k1+βk2/T6β1+βk230∞+Vg2π2∫0∞ln⁡1+exp⁡-k1+βk2/Tdk6β1+βk23-Vg2π2∫0∞k1+3βk2dk6βT1+βk231+exp⁡k1+βk2/T,where “Integration by Parts” was used for solving the above equation. It is obvious that the first term in (21) vanishes. Thus,(22)ln⁡zf=Vg2π2∫0∞ln⁡1+exp⁡-k1+βk2/Tdk6β1+βk23-∫0∞k1+3βk2dk6βT1+βk231+exp⁡k1+βk2/T.For solving (22) let x=(k/T)(1+βk2) so that dx=(1+3βk2)dk/T and(23)ln⁡zf=Vg2π2∫0∞ln⁡1+e-xTdx6β1+βk231+3βk2-∫0∞k1+3βk2dx6β1+3βk2ex+1.Employing the value of k into integrant terms of (23) and approximating them to the first order of β are as follows.For the first term,(24)T6β1+βk231+3βk2=1-3βk21-3βk26β≃T6β-x2T3+7β2x4T5.For the second term,(25)k6β1+3βk2=k1-3βk2≃xT6β-23x3T3+32βx5T5.Substituting the values of terms in (24) and (25) into (23) and solving them analytically, we have(26)ln⁡zf=78π290VgT3-31π4630VgβT5.In case of β→0, the above equation is reduced to the partition function for fermions without GUP effect [32].Now, the QGP equation of state of free massless quarks and gluons can be derived from the above equations. The total grand canonical partition function of QGP state can be given by adding the grand partition functions coming from the contribution of bosons (gluons), fermions (quarks), and vacuum [39] as follows:(27)ln⁡zQGP=ln⁡zF+ln⁡zB+ln⁡zV,where ln⁡zB, ln⁡zF, and ln⁡zV are the grand canonical functions of gluons, quarks, and vacuum, respectively.In our case, the vacuum pressure can be represented with the bag constantB in case of Bag model [40].Since the value of vacuum partition function equalsln⁡zV=-VB/T and ln⁡z=-Ω/T, using these values (27) reads(28)ΩVQGP=ΩVquarks+ΩVgluons+ΩVvacuum,where Ω is the grand canonical potential. Substituting (15) and (26) into (28), we have(29)ΩVQGP=-gg+78gqπ290T4+gg16π4315+gq31π4630βT6+B.Hence, the QGP equation of state reads(30)PQGP=-ΩVQGP=gg+78gqπ290T4-gg16π4315+gq31π4630βT6-B=σSB3T4-gg16π4315+gq31π4630βT6-B,where σSB is SB constant and is given by gg+(7/8)gqπ2/30. Hence, the energy density of QGP state of matter is given by(31)εQGP=σSBT4-3gg16π4315+gq31π4630βT6+B.


## 2.1. In Case of Bosons

At finite temperatureT and chemical potential μ, the grand canonical partition function zB for noninteracting massive bosons with g internal degrees of freedom is given by [32](4)zB=∏k∑l=0∞exp⁡-lEk-μTg(5)=∏k1-exp⁡Ek-μT-g,where l is the occupation number for each quantum state with energy E(k)=k2+m2 with mass m and k is the momentum of the particle. Here the infinite product is taken for all possible momentum states.For simplicity we consider chiral limit (i.e., vanishing mass) and a vanishing chemical potential, which experiments ensure at high energy. Then the dominant excitation in the hadronic phase is a massless pion, while that in the quark-gluon plasma is massless quarks and gluons. For a particle of massM having a distant origin and an energy comparable to the Planck scale, the momentum would be a subject of a tiny modification so that the dispersion relation would too. According to GUP approach, the dispersion relation reads(6)E2k=k2c21+βk22+M2c4,where M and c are the mass of the particle and the speed of light as introduced by Lorentz and implemented in special relativity, respectively. For simplicity we use natural units in which ħ=c=1. Hence, we have(7)Ek=k1+βk2.For large volume, the sum over all states of single particle can be rewritten in terms of an integral [33](8)∑k⟶V2π3∫0∞d3k⟶V2π2∫0∞k2dk1+βk24.Thus, the partition function, (4), becomes(9)ln⁡zB=-Vg2π2∫0∞k2ln1-exp⁡-Ek/T1+βk24dk=-Vg2π2∫0∞k2ln⁡1-exp⁡-k/T1+βk21+βk24dk=Vg2π2k6β1+βk23ln⁡1-exp⁡-kT1+βk20∞-Vg2π2∫0∞16β1+βk23ln1-exp⁡-k1+βk2T+k1+3βk2T1exp⁡k1+βk2/T-1dk,where “Integration by Parts” was used for solving the above equation. It is obvious that the first term in (9) vanishes. Thus,(10)ln⁡zB=-Vg2π2∫0∞16β1+βk23ln⁡1-exp⁡-k1+βk2T+k1+3βk2T1exp⁡k1+βk2/T-1dk.For solving (10) let x=(k/T)(1+βk2) so that dx=(1+3βk2)dk/T and(11)ln⁡zB=-Vg2π2∫0∞ln⁡1-e-xTdx6β1+βk231+3βk2-Vg2π2∫0∞kdx6β1+βk23ex-1.The momentum k as a function of x variable can be approximated to the first order of β as follows:(12)k=xT-βk3=xT-βxT-βk33=xT-βx3T31-βk3xT3≃xT-βx3T3.Employing the value of k into integrant terms of (11) and approximating them to the first order of β are as follows.For the first term,(13)T6β1+βk231+3βk2=T1-6βk2+9β2k46β≃T6β-x2T3+7β2x4T5.For the second term,(14)k6β1+βk23=k-3βk36β≃xT6β-2x3T33+32βx5T5.Substituting the values of terms in (13) and (14) into (11) and solving them analytically, we have(15)ln⁡zB=π290VgT3-16π4315gβT5.In case of β→0, the above equation is reduced to the partition function for bosons without GUP effect [32].


## 2.2. In Case of Fermions

At finite temperatureT and chemical potential μ, the grand canonical partition function zf for noninteracting massive bosons with g internal degrees of freedom is given by [32] (16)zf=∏k∑l=0,1∞exp⁡-lEk-μTg(17)=∏k1+exp⁡-Ek-μTg,where l is the occupation number for each quantum state with energy E(k)=k2+m2 with mass m and k is the momentum of the particle. Here the infinite product is taken for all possible momentum states.For simplicity we consider chiral limit (i.e., vanishing mass) and a vanishing chemical potential, which experiments ensure at high energy. Then the dominant excitation in the hadronic phase is a massless pion, while that in the quark-gluon plasma is massless quarks and gluons. For a particle of massM having a distant origin and an energy comparable to the Planck scale, the momentum would be a subject of a tiny modification. According to GUP approach, the dispersion relation reads(18)E2k=k2c21+βk22+M2c4,where M and c are the mass of the particle and the speed of light as introduced by Lorentz and implemented in special relativity, respectively. For simplicity we use natural units in which ħ=c=1. Hence, we have(19)Ek=k1+βk2.For large volume, the sum over all states of single particle can be rewritten in terms of an integral [33](20)∑k⟶V2π3∫0∞d3k⟶V2π2∫0∞k2dk1+βk24.Thus, the partition function, (16), becomes (21)ln⁡zf=Vg2π2∫0∞k2ln⁡1+exp⁡-Ek/T1+βk24dk=Vg2π2∫0∞k2ln⁡1+exp⁡-k1+βk2/T1+βk24dk=-Vg2π2kln⁡1+exp⁡-k1+βk2/T6β1+βk230∞+Vg2π2∫0∞ln⁡1+exp⁡-k1+βk2/Tdk6β1+βk23-Vg2π2∫0∞k1+3βk2dk6βT1+βk231+exp⁡k1+βk2/T,where “Integration by Parts” was used for solving the above equation. It is obvious that the first term in (21) vanishes. Thus,(22)ln⁡zf=Vg2π2∫0∞ln⁡1+exp⁡-k1+βk2/Tdk6β1+βk23-∫0∞k1+3βk2dk6βT1+βk231+exp⁡k1+βk2/T.For solving (22) let x=(k/T)(1+βk2) so that dx=(1+3βk2)dk/T and(23)ln⁡zf=Vg2π2∫0∞ln⁡1+e-xTdx6β1+βk231+3βk2-∫0∞k1+3βk2dx6β1+3βk2ex+1.Employing the value of k into integrant terms of (23) and approximating them to the first order of β are as follows.For the first term,(24)T6β1+βk231+3βk2=1-3βk21-3βk26β≃T6β-x2T3+7β2x4T5.For the second term,(25)k6β1+3βk2=k1-3βk2≃xT6β-23x3T3+32βx5T5.Substituting the values of terms in (24) and (25) into (23) and solving them analytically, we have(26)ln⁡zf=78π290VgT3-31π4630VgβT5.In case of β→0, the above equation is reduced to the partition function for fermions without GUP effect [32].Now, the QGP equation of state of free massless quarks and gluons can be derived from the above equations. The total grand canonical partition function of QGP state can be given by adding the grand partition functions coming from the contribution of bosons (gluons), fermions (quarks), and vacuum [39] as follows:(27)ln⁡zQGP=ln⁡zF+ln⁡zB+ln⁡zV,where ln⁡zB, ln⁡zF, and ln⁡zV are the grand canonical functions of gluons, quarks, and vacuum, respectively.In our case, the vacuum pressure can be represented with the bag constantB in case of Bag model [40].Since the value of vacuum partition function equalsln⁡zV=-VB/T and ln⁡z=-Ω/T, using these values (27) reads(28)ΩVQGP=ΩVquarks+ΩVgluons+ΩVvacuum,where Ω is the grand canonical potential. Substituting (15) and (26) into (28), we have(29)ΩVQGP=-gg+78gqπ290T4+gg16π4315+gq31π4630βT6+B.Hence, the QGP equation of state reads(30)PQGP=-ΩVQGP=gg+78gqπ290T4-gg16π4315+gq31π4630βT6-B=σSB3T4-gg16π4315+gq31π4630βT6-B,where σSB is SB constant and is given by gg+(7/8)gqπ2/30. Hence, the energy density of QGP state of matter is given by(31)εQGP=σSBT4-3gg16π4315+gq31π4630βT6+B.


## 3. Results and Discussions

The main features of QCD lattice results show a clearNf-dependance for both energy density and pressure (i.e., they become larger as the number of degrees of freedom increases). As it is clear from the Monte Carlo lattice results [34, 35], the pressure P(T) rapidly increases at T≃Tc which may be agreed with our predictions after adding the GUP effect on QGP equation of state. The bag pressure B can be determined by comparing the obtained equation with that of QCD lattice results.The main problem appears when one starts to adjust the behavior of the energy density, through varying the value of the bag pressure; with the QCD lattice results, one can not obtain a qualitative agreement in case of the pressure using the same bag parameter value [41].For overcoming this problem, the bag model was modified [42]. In this technique, the thermodynamical relation between the energy density and the pressure [33] was used(32)TdPdT-PT=ɛT.Solving the above first-order differential equation, we obtain the QGP equation of state as follows:(33)P=σSB3T4-gg16525+gq311050π4βT6-B+AT,where A is a constant coming from the partial differential equation solution and can be determined from comparing the calculated QGP equation of state with the QCD lattice results.To adjust the high temperature behavior forP(T) and ɛ(T), we will consider the suppression factor of the Stefan Boltzmann constant used in quasi particle approach [43]. In this approach, the system of interacting gluons may be treated as a noninteracting quasi particles gas with gluon quantum numbers, but with thermal mass (i.e., m(T)). The modified SB constant σ reads [41](34)σ=κaσSB,where κ(a) is a suppression factor. For a→0, it follows κ→1. Also, the function κ(a) decreases monotonously and goes to zero at a→∞. Thus, the final form of the bag model equation of state (33) with incorporating the GUP modification is given by(35)P=σ3T4-gg16525+gq311050π4βT6-B+AT.With σ and B being free model parameters(36)ɛ=σT4-3gg16315+gq31630π4βT6+B.In case ofβ→0, the above equations (35) and (36) are reduced to the pressure and the energy density equations obtained in the modified bag model [42] (i.e., without incorporating the GUP modification).The theoretical calculation of the pressure and the energy density of the QGP using (35) and (36) compared to the lattice results in case of nf=0 are given in Figure 1. In this case we used the following parameter values: gQGP=16, gq=0, gg=16, Tc=0.200 GeV, and β=0.0001 GeV−1 as taken in [11, 32, 43].Figure 1
The symbols show the MC LR for the pressure and the energy density in theSU(3) gluodynamics [34, 35]; the line corresponds to the equation of state (35) for the pressure in the left panel and (36) for the energy density in the right panel with σ=4.20719, A=-2.0019Tc3, and B=-1.465358Tc4. The arrows in the above figures correspond to P/ɛ=1/3 (i.e., the SB limit).The theoretical calculation of the pressure and the energy density of the QGP using (35) and (36) compared to the lattice results in case of nf=2 are given in Figure 2. In this case we used the following parameter values: gQGP=37, gq=24, gg=16, Tc=0.152 GeV, and β=0.0001 GeV−1 as taken in [11, 32, 43].Figure 2
The symbols show the MC LR for the pressure and the energy density in case ofnf=2 QCD equation of state [36]; the line corresponds to the equation of state (35) for the pressure in the left panel and (36) for the energy density in the right panel with σ=10.403, A=-8.72895Tc3, and B=-4.9Tc4. The arrows in the above figures correspond to P/ɛ=1/3 (i.e., the SB limit).The theoretical calculation of the pressure and the energy density of the QGP using (35) and (36) compared to the lattice results in case of nf=2+1 are given in Figure 3. In this case we used the following parameter values: gQGP=47.5, gq=36, gg=16, Tc=0.152 GeV, and β=0.0001 GeV−1 as taken in [11, 32, 43].Figure 3
The symbols show the MC LR for the pressure and the energy density in case ofnf=2+1 QCD equation of state [37, 38]; the line corresponds to the equation of state (35) for the pressure in the left panel and (36) for the energy density in the right panel with σ=12.22017, A=-13.8577Tc3, and B=-10.4353Tc4. The arrows in the above figures correspond to P/ɛ=1/3 (i.e., the SB limit).Also, we can calculate the interaction measure(ɛ-3P)/T4 for both nf=0 and nf=2+1 and compare them with lattice results for the interaction measure in the SU(3) gluodynamics [34, 35] and in case of nf=2+1 QCD equation of state [37, 38], respectively. These results can be shown in Figure 4. It can be shown that the inflection point of (ɛ-3P)/T4 is at T=0.1846 GeV for nf=0 and T=0.202 GeV for nf=2+1 which are different from that obtained in [42].Figure 4
The symbols show the lattice results for the interaction measure in theSU(3) gluodynamics (left panel) [34, 35] and in case of nf=2+1 QCD equation of state (right panel) [37, 38].


## 4. Conclusion

In the present work, the effect of the GUP on QGP of massless quark flavors at a vanishing chemical potential,μ, is studied. The equation of state and the energy density are derived for the QGP state consisting of noninteracting massless bosons and fermions with impact of GUP approach. Also, the total grand canonical partition function of QGP state is given. One can conclude that a significant effect for the GUP term exists in case of studying the thermal properties of the QGP state. The main features of QCD lattice results were quantitatively achieved in case of nf=0, nf=2, and nf=2+1 for the equation of state, the energy density, and the interaction measure. The interesting point in our results is the large value of bag pressure especially in case of nf=2+1 flavor. It nearly equals 4.46 times the value obtained in [42] without taking into account the negative sign of it, which reflects the strong correlation between quarks inside the bag which is already expected. The negative sign may be regarded as the tendency of the bag to reduce its volume. One can conclude that the modification of the QGP bag model equation of state using the GUP effect can reproduce the QCD lattice results which stands for the real data of QGP. Finally, with this study, one may be encouraged to implement GUP effect in other thermodynamic properties of QGP.



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