Int J Theor PhysDOI 10.1007/s10773-014-2417-x

Thermodynamics and Geometry of Strange QuarkMatter

H. Gholizade ·A. Altaibayeva ·R. Myrzakulov

Received: 1 June 2014 / Accepted: 10 November 2014© Springer Science+Business Media New York 2014

Abstract We study thermodynamic of strange quark matter (SQM) using the analyticexpressions of free and internal energies. We investigate two regimes of the high densityand low density separately. As a vital program, in the case of a massless gluon and masslessquarks at finite temperature, we also present a geometry of thermodynamics for the gluonand Bosons using a Legendre invariance metric ,it is so called as geometrothermodynamic(GTD) to better understanding of the phase transition. The GTD metric and its second orderscalar invariant have been obtained and we clarify the phase transition by study the singular-ities of the scalar curvature of this Riemannian metric. This method is ensemble dependenceand to complete the phase transition, meanwhile we also investigate enthalpy and entropyand internal energy representations. Our work exposes new pictures of the nature of phasetransitions in SQM.

Keywords Strange quark matter · Phase transition · Geometrical methods · Riemanniangeometry

1 Introduction

Quark matter idea introduced by Budmer [1] and Witten [2], and the thermodynamics ofquark matter investigated by Jaff and Farhi [3] and several authors [4]. Only strange quark

H. Gholizade (�)Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finlande-mail: hosein.gholizade@gmail.com

A. Altaibayeva · R. MyrzakulovEurasian International Center for Theoretical Physics and Department of General & TheoreticalPhysics, Eurasian National University, Astana 010008, Kazakhstane-mail: aziza.ltaibayeva@gmail.com

R. Myrzakulove-mail: rmyrzakulov@gmail.com

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matter (SQM) - up, down and strange quarks- can be stable and non strange quark matter-up and down quarks- exists at an exited state respected to nuclear matter [3]. The quarkmatter can be formed in Heavy Ions Colliders (HIC), neutron stars and early universe [11].

Quark matter can be in different phases due to values of density and temperature of thesystem. At high temperature the quark matter is in ”Quark Gluon Plasma” (QGP) phase,without spontaneous symmetry breaking [37]. This phase is expected to exist in the earlyuniverse and also in HIC (small and short lived) [37]. If we denote by T absolute tempera-ture and μ chemical potential, at T � μ the spontaneous phase transition is happened. Weexpected that this region exists in the neutron star’s core. At high density and low temper-ature region, we have quark liquid (degenerate Fermi gas). In this region system can be incolor superconductive phase and ferromagnetic phase. [37]

From another point of view, according to equivalence principle of general relativityany matter field gravitates and so SQM can be used as the source of gravity [38]. It isone direction on how SQM gravitates. But in this work we are interested to describe thegeometry of SQM in a simple case of massless gluon using the same geometrical meth-ods which are used to construct general relativity. Based on the Riemannian geometry weare able to construct metric for thermodynamic of SQM. We present a general formalismin the next sections. We use a Legendre’s invariant metric to describe the geometry of glu-ons. By study singularities of scalar curvatures we investigate the possibility of interactionin gluons.

Our plan in this paper is as the following: In Section 2, we study thermodynamic of SQM.In Section 3, Fermi gas at finite temperature in relativistic regime will be studied. In Sec-tion 4, We study massive Fermions in low temperature regime μ >> T . In Section 5, basicproperties of massless Fermions in regime T

m→ ∞ is calculated. In Section 6, Gluons at

finite T are investigated in details. In Section 7, is devoted to study a geometric approach tothe phase transitions in a Legendre’s invariant formalism. We study the Riemannian geom-etry of Gluons, Bosons and Fermions. By a careful analysis of singularities, we find thecritical points of these systems. We conclude in the final section.

2 SQM Thermodynamics

In this section we present a short review of the thermodynamics of quark matter. Thethermodynamic potential density of massless quarks in second the order is written as thefollowing [26]:

� = − 1

4π2

Nf∑

i=1

μ4i

[Nc

3− Ng

αc

π

]. (1)

The ideal massive S quark thermodynamic potential is obtained as follows:

�s = − Nc

3π2

⎡

⎣1

4μs

(μ2

s − m2s

) 12(

μ2s − 5

2m2

s

)+ 3

8m4

s ln

⎡

⎣μs + (μ2s − m2

s

) 12

ms

⎤

⎦

⎤

⎦ . (2)

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The first order perturbative term αc in thermodynamic potential has the meaningof the exchange interaction. For massive quarks exchange interaction contribution inthermodynamic potential is written as the below:

�ex = Ngαc

π3

⎧⎪⎨

⎪⎩3

4

⎡

⎣μs

(μ2

s − m2s

) 12 − m2

s ln

⎡

⎣μs + (μ2s − m2

s

) 12

ms

⎤

⎦

⎤

⎦2

− 1

2

(μ2

s − m2s

)2

⎫⎪⎬

⎪⎭.

(3)

At zero temperature and free SQM we obtain following results for thermodynamicspotentials density:

�u = − μ2u

4π2,�d = − μ2

d

4π2,

�s = − μ2s

4π2

[(1 − λ2

)1/2(

1 − 5

2λ2)

+ 3

2λ4 ln

[1 + (1 − λ2

)1/2

λ

]]. (4)

λ = ms

μsand m is mass of strange quarks. After some algebraic manipulations, we obtain the

total internal energy s:

U(Nb, V ) = BV +5m2V

√4π4 − 9m2V 2

Nb2

16π4−

Nb2(−8π2 +

√4π4 − 9m2V 2

Nb2

)

18V

−27m4V 3 ln

⎡

⎣− 2π2

−2π2+√

4π4− 9m2V 2

Nb2

⎤

⎦

32Nb2π6

. (5)

Finite temperature calculations for massless quarks are simple, but for massive quarks wecan use the low temperature results. At finite temperature we must add the gluons effectsin thermodynamics, because their thermodynamic potentials at finite temperature are notzero. All results of this section are motivated due to the leakage of analytical expressions inliterature.

3 Massive Fermions at Low Temperature

The particle number density of the relativistic Fermi system is:

n = g

2π2

∫ ∞

0

[1

1 + e(ε−μ)/T− 1

1 + e(ε+μ)/T

]p2dp. (6)

The second term in brackets is anti-Fermions number density. At the low temperature limit(μ � T ) its contribution to the Fermions number density is negligible, so we can write

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following relations for thermodynamics quantities [34]:

N = gV

2π2

∫ ∞

0

(x2 + 2m2

) 12 (x + m)

exp[

x−μT

]+ 1dx (7)

U = gV

2π2

∫ ∞

0

x(x2 + 2m2

) 12 (x + m)

exp[

x−μT

]+ 1dx (8)

p = gV

6π2

∫ ∞

0

(x2 + 2m2

) 32

exp[

x−μT

]+ 1dx. (9)

Expanding the integrals at the low temperature limit we have [34]:

μ = μ0

⎧⎪⎪⎨

⎪⎪⎩1 − π2

6

(T

μ0

)2(1 + 2ξ 2

) [(1 + ξ 2

) 12 − 1

]

ξ 2(1 + ξ 2

) 12

− π4

360

(T

μ0

)4(5ξ 2 + 4

) (4ξ 2 + 9

) [(1 + ξ 2

) 12 − 1

]3

ξ 6(1 + ξ 2

) 32

⎫⎪⎪⎪⎬

⎪⎪⎪⎭,

ξ = 2π

m

(3n

4πg

) 13

, μ0 = m

[(1 + ξ 2

) 12 −1

]. (10)

Using the similar low temperature expansion method we can write following relations forother thermodynamic potentials [34]:

U = 3Nm

⎡

⎣ε(ξ) + 2π2

12

(T

m

)2 (1 + ξ 2) 1

2

ξ 2+ 6

(T

m

)4 (1 + ξ 2) 1

2

ξ 6

{7π4

720

(2ξ 2 − 1

)− π4

144

(2ξ 2 + 1

)2

1 + ξ 2

}](11)

and the heat capacity at low temperature is: [34]:

Cv = ∂U

∂T|v = 12N

T

m

⎡

⎣π2(1 + ξ 2

) 12

ξ 2+ 6

(T

m

)2 (1 + ξ 2) 1

2

ξ 6

{7π4

720

(2ξ 2 − 1

)− π4

144

(2ξ 2 + 1

)2

1 + ξ 2

}]. (12)

For massive Fermions up to order two for isothermal compressibility, we have:

κ =54gmV 2

(1 + 62/3π4/3

(NgV

)2/3)3/2

√

1 + 62/3π4/3(

NgV

)2/3

m2

A (13)

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Where

A = 108 61/3π8/3N2(

N

gV

)1/3√

1 + 62/3π4/3

(N

gV

)2/3

+18 62/3gπ4/3V N

(N

gV

)2/3√

1 + 62/3π4/3

(N

gV

)2/3

+9gπ2T 2V N

√√√√1 +

62/3π4/3(

NgV

)2/3

m2

+61/3gπ2/3T 2V N

√

1 + 62/3π4/3(

NgV

)2/3

m2

(NgV

)2/3

+6 62/3gπ10/3T 2V N

(N

gV

)2/3

√√√√1 +

62/3π4/3(

NgV

)2/3

m2. (14)

It is obvious that in zero temperature limit T = 0, we have

κT =0 =31/3m

√

1 + 62/3π4/3(

NgV

)2/3

m2

22/3gπ4/3(

NgV

)5/3(15)

4 Massless Fermions at Finite Temperature

By following the method was proposed in [32, 33], the internal energy of massless Fermionsis:

U = 3pV = gV

8π2

[μ4 + 2π2μ2T 2 + 7π4

15T 4

]. (16)

The derivative of internal energy with respect to the temperature at constant volume iswritten as the following:

Cv(T , V ,μ) =gV(

28π4T 3

15 + 4π2T μ2 + 4π2T 2μμ′ + 4μ3μ′)

8π2,

μ′ = ∂μ

∂T= 31/3g5π8/3T 5V 5

√729g4N2V 4 + 3g6π2T 6V 6

(27g2NV 2 +√729g4N2V 4 + 3g6π2T 6V 6

)2/3

+ 9g7π10/3T 7V 7

√729g4N2V 4 + 3g6π2T 6V 6

(81g2NV 2 + 3

√729g4N2V 4 + 3g6π2T 6V 6

)4/3

− 2gπ4/3T V(

81g2NV 2 + 3√

729g4N2V 4 + 3g6π2T 6V 6)1/3

. (17)

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The results for massless Fermions at finite temperature are valid for all temperature ranges,but in the next section , we shall calculate massive Fermions at low temperature. In thatregime, the equations only valid at the low temperature limit and in this region we shallignore the anti-Fermions contributions in the equation of state.

5 Massless Gluons at Finite Temperature

At finite temperature we must consider the gluons contribution in thermodynamics parame-ters. In this section we ignore the gluons condensation effects and higher order corrections.The thermodynamic potentials of massless Bosons at finite temperature are [26]:

�(T ,V ) = − 8

45π2T 4V. (18)

Heat capacity at finite temperature can written as:

Cgluonv (T ,V ) = 32π2

15T 3V. (19)

Collecting all results we can write the equation of state of free SQM at the low temperaturelimit. Because the thermodynamic potentials are extensive thermodynamic quantities, wecan add them in different phases. Therefore the total grand thermodynamic potential ofSQM that composed of massive strange quarks and massless up, down quarks and gluons atthe low temperature limit is:

�SQM =3∑

i

�i + �gluon + BV,

CSQMv =

3∑

i

Civ + Cgluon

v . (20)

For SQM we have g = 3 × 2 = 6. Again, like zero temperature case, we can calcu-late the interaction contributions in the equation of state, but the effects of all perturbationcorrections can collected into an effective bag pressure [14].

Isothermal compressibility for massless gluons by definition is:

κ = 0. (21)

The thermodynamic relation between heat capacities is written as the following:

Cp = Cv + T Vα2

κ.

About isobaric expansion coefficient we mention here that since the pressure depends onlyon temperature, at constant pressure, temperature must remain constant, so the heat capac-ity at finite pressure (Cp) is meaningless. For free massless gluons constant pressure andconstant temperature are equivalent.

The Isothermal expansion coefficient for massless gluons are also zero, but in the limit

of T V α2

κit is important for us to discuss about Cp. The constant heat capacity for gluons

can be evaluated if we insert the interactions in the equation of state with medium dependentcoupling constant. So we ignore discussion about Cp, and only investigate other responsefunctions.

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All the thermodynamics potentials are extensive parameters and we can add them invarious phases. But isothermal expansion coefficients and compressibility can not be addedin different phases (intensive parameters). All differentiations are with respect to pressure

or temperature of the system. The volume of the system is: ∂HSQM

∂PSQM, where enthalpy and

pressure of SQM are the sum of enthalpy and pressure of all phases (massive and masslessFermions and massless gluons). So we can not add expansion coefficients in separate phasesto each other.

Unlike isothermal expansion coefficients and compressibility, the heat capacity atconstant volume is an extensive parameter and we can simply calculate it for SQM.

6 Legendre Invariant Riemannian Geometry of Massless Gluons and Bosons atFinite Temperature

The relation between thermodynamic and gravitation is discussed in detail in several works,especially in the relation of the recent observational data in cosmology [43]. There are twotypes of problems can be solved using differential geometry: one is how to describe matterfields in the presence of gravitational field using statistical mechanics. In this case youneed to find the Green’s functions of matter fields in curved space-time. Second is how todescribe phase transitions in particle physics using geometry. In this sense, we need to finda unique metric associated with a set of thermodynamic variables and study the dynamics ofthis metric using Riemannian geometry. This type of problem has been investigated in thispaper for quark matter. We use a Legendre invariant metric of the thermodynamic space toaddress the phase transitions. Let us start with some preliminary definitions:

We describe a thermodynamic system with n degree of the freedoms in an equilibriumstate by a set of n extensive variables Ea, with a = 1, ..., n. It means we have an equilibriummanifold and in any point of it , we can label a coordinate system. The fundamental equationwhich is needed is an energy function � = �(Ea), here � has the meaning of the thermo-dynamic potential [44]. We have different possibilities to identify this energy function forexample the entropy S or the internal energy U of the system in equilibrium. The differen-tial geometric structure is not a unique and we can write several kinds of these forms. Oneof the first and simple and historically former is a Hessian metric as the following:

gH = ∂2�

∂Ea∂EbdEadEb , (22)

With this metric the manifold becomes Riemannian. It was presented in different ensembleforms previously [45] . The Gauss second order form gH looks like as a scalar under diffeo-morphism of coordinates Ea → Ea = Ea(Ea). It is possible to change the potential usingthe unique Legendre transformation [50] :

{�,Ea, Ia

} −→{�, Ea, I a

}, (23)

� = � − δklEkI l , Ei = −I i , Ej = Ej , I i = Ei , I j = I j , (24)

where Ia denotes a set of n intensive variables. It is known the Hessian metrics are notinvariant under Legendre transformations. The most general Legendre’s invariant metric canbe written as the following:

G = (d� − IadEa)2 +

(ξabE

aIb)(

χcddEcdId)

, (25)

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Here ξab and χab are diagonal constant tensors, and denotes an arbitrary Legendreinvariant function. We choice the metric to be in this form :

GII =(d� − δabI

adEb)2 +

(δabE

aIb)(

ηcddEcdId)

, (26)

By computing the pullback we have:

gII =(

Ea ∂�

∂Ea

)(ηc

b

∂2�

∂Ec∂EddEbdEd

)(27)

It is the metric for the equilibrium manifold, here ηcb = diag(−1, 1, ..., 1). This formalism is

called as the geometrothermodynamics (GTD) [51]. We need the fundamental equation � =�(Ea). We believe the metric given by (27) is a useful tool to study the phase transitions inthermodynamical systems and the black holes and using this approach most of the criticalpoints and even those ones that cannot be obtained by Davis’s approach [52] can be obtainedas well [53].

Metric for Bosons in S representation with Ea = {U, V }, γ = 38 51/3

(3

2π

)2/3is written:

gIIS = 3

16γ 3/2√

UV 3

(V 2dU2 − U2dV 2

)(28)

Here curvature scalar and Riemannian tensor read:

RIIS = 0, RIJKL

S = 0 (29)

So the geometry is flat and it indicates on no interaction. In fact the metric gIIS is

conformally flat if we rewrite it in the following coordinates:

u = log U, v = log V. (30)

So, metric transforms to form:

gIIS = 3

16γ 3/2e(v+3u)/2

(du2 − dv2

). (31)

This metric is trvially conformal to another two dimensional flat metric η = du2 − dv2.Vanishing of the curvature is also a result of this conformal flat duality.

Similarly for U representation in which Ea = {S, V }:

gIIU = 4γ 2

9

(S

V

)2/3(

S2dV 2

V 2− dS2

). (32)

By following the same technique as S representation, we can show that this metric is alsoconformally flat under these coordinate transformations:

s = log S, v = log V. (33)

So, metric transforms to form:

gIIS = 4γ 2

9e(8s−2v)/3

(−ds2 + dv2

). (34)

Here curvature scalar

RIIU = 0. (35)

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So there exists no singularity in curvature. In fact, it is useful to check whether the metric isflat or not. So we compute Riemannian tensor RIJKL, to find:

RIJKLU = 0. (36)

Consequently the geometry of Boson fields defined a flat manifold. So, according tothe GTD interpretation of interaction as geometrical curvature, flatness of manifold indi-cates to the no interaction system. So in the case of the massless Bosons the system has nointeraction as GTD metric also predicts (Fig. 1).

Just to show that GTD analysis depends to different ensembles (or not) we study GTDmetric for H(S, p), so we have

gIIH−SS = 0. (37)

So metric degenerates and we cannot apply GTD metric gII here.It is adequate to deal with the case of Hessian metric given by (22). We have the following

expressions for gH :

gHS = 3

16γ 3/4

⎡

⎣− U 3/4

V 7/41

4√UV 3/4

14√

UV 3/4− U 3/4

V 7/4

⎤

⎦ , gHU = 4γ

9

⎡

⎣S4/3

V 7/3 − 3√S

V 4/3

− 3√S

V 4/31

3√V S2/3

⎤

⎦ . (38)

So the curvature R is written as the following:

RHS = −4

3

γ 3/4U5/4(−15 V 2 + 7 U2

)

4√

V(U2 − V 2

)2 , (39)

RHU = undefined. (40)

Fig. 1 Ricci scalar for (5). It shows that for massless quarks the geometry is non trivial. We have singularitiesin R. So, quark matter enters the phase transition smoothly. It is one of the important evidences of GTD forquark matter

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Fig. 2 Ricci scalar for RHS . It shows that for massless quarks the geometry is non trivial. We have

singularities in R

Thus, curvature singularities exist only for RHS and gH

U is singular. Singularities of RHS

locate at U = V . Figure 2 shows a normalized behavior of RHS .

Just to keep the motivation we plot the R for (5) in Fig. 2. This case defines GTD structureof massless quark. The system thermodynamically has phase transition. Geometry is nonflat. Full expression of metric is so complicated and we shall not write them here. UsingGR-TENSOR package for metric, we can obtain Ricci scalar R. We plot it in Fig. 1. TheRicci scalar has singularities near some critical points (N∗, V ∗). Since GTD metric is notflat , we have interaction in the system. Further information has come from singularities inwhich R diverges. GTD tool here predicts phase transition in the system as well as existenceof interaction.

7 Conclusion

In this work we studied briefly basic thermodynamic properties of quark matters. In case ofmassless and massive we presented general expressions of energy functions and responsefunctions. Our results are valid in both zero temperature and finite temperature limits. TheGluons and Bosons thermodynamics have been studied analytically. Further for the firsttime in literatures, we study geometry of some quark matter cases using a Legendre invari-ant formalism. For massless Gluons and Bosons and quarks in zero temperature, we studygeometric metric. For Gluons and Bosons the geometry is conformal flat and we have notany type of interaction. But for the case of zero temperature, massless quark geometry hasa more complicated non flat form. We computed Ricci scalar as the first scalar invariant ofthe associated metric. It showed that there exists interaction and also finite numbers of phys-ical singularities in which R diverges. This divergence predicts that system enters phasetransitions.

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