aa r X i v : . [ nu c l - t h ] S e p Thermodynamics of Quasi-Particles
F. G. Gardim a and F. M. Steffens a,ba Instituto de F´ısica Te´orica - Universidade Estadual Paulista,Rua Pamplona 145, 01405-900, S˜ao Paulo, SP, Brazil. b NFC - CCH - Universidade Presbiteriana Mackenzie,Rua da Consola¸c˜ao 930, 01302-907, S˜ao Paulo, SP, Brazil.
Abstract
We present in this work a generalization of the solution of Gorenstein and Yang for aconsistent thermodynamics for systems with a temperature dependent Hamiltonian. Weshow that there is a large class of solutions, work out three particular ones, and discusstheir physical relevance. We apply the particular solutions for an ideal gas of quasi-gluons,and compare the calculation to lattice and perturbative QCD results. Introduction
Lattice QCD suggests [1, 2, 3] that at sufficiently high temperature T and/or quark chemicalpotential µ , the strongly interacting matter exhibits a transition from a hadronic phase to a newstate, where matter is described in terms of fundamental gluon and quark degrees of freedom[4], the Quark-Gluon Plasma (QGP). These lattice simulations suggest that the transitionoccurs for temperatures around T c ∼
190 MeV [3]. Heavy-ion collisions at RHIC, and in thefuture at the LHC, provide us essential experimental data to the quest of the QGP at hightemperatures and small chemical potential. This experimental search of the QGP needs reliabletheoretical estimates of various quantities, such as pressure, entropy, deconfinement temperatureand equations of state (EoS).Perturbative QCD at finite T and µ is one of the theoretical tools to compute the variousQGP quantities. However, strict perturbation theory, which have been pushed up to g s ln(1 /g s )[5], is reasonable only for extremely high temperatures: at temperatures near T c it is, in principle,not applicable, and further treatments appear to be necessary. Moreover, the perturbative seriesseems to be weakly convergent [5, 6]. Specifically, it is expected that when T ≫ T c , the plasmabehaves like an ideal gas of quarks and gluons, but the perturbative series goes very slowlytowards this expectation. A way to attack this problem is through the reorganization of theperturbative series. For instance, there are attempts using resummation based on the Hard-Thermal-Loop (HTL) effective action [7], in the form of so-called HTL perturbation theory[8, 9], and also attempts based on the 2-loop Φ-derivable approximation [10, 11]. The latterapproach, which assumes a massive quasi-particle formalism [12, 13], leads to results whichagree remarkably well with lattice data.The quasi-particle model of the Quark-Gluon plasma (qQGP) is a phenomenological modelwhich assumes non-interacting massive quasi-particles which, with the aid of few parameters,like thermal masses, is able to fit lattice QCD data over a wide range of temperatures [14, 15, 16]:not only at extremely high temperatures as in strict perturbation theory, or when T > T c as in the HTL perturbation theory, but also near T c . Quasi-particle models intend to describedeconfined matter from T c up to T → ∞ . In the usual quasi-particle model for the quark-gluon2lasma [12, 14], statistical mechanics for an ideal massive gas is used, where the mass of eachquasi-particle is dependent on the temperature (and on the chemical potential) of the gas.However, when this idea is naively applied there appear inconsistencies in the thermodynamicsof the system [17]. The in depth study of a general solution for the thermodynamics of a systemcomposed of particles with thermal masses is the main objective of the present work.In Sec. 2 we will formulate the requirements for thermodynamics self-consistency through theuse of statistical mechanics in the canonical ensemble, and explicitly compute three solutions.In Sec. 3 we will use one of these solutions to solve the problem for an ideal gas of quasi-gluons,and obtain the generalized thermodynamics relations as a function of the temperature. In Sec.4 we discuss which solutions have physical meaning, and in Sec. 5 we analyze and summarizeour results. The partition function of a classical system in a canonical ensemble whose Hamiltonian dependson an extra parameter T, which will be identified with the temperature of the system, hasthe same formal structure of the partition function of a system that has a T independentHamiltonian. Specifically: Q N ≡ Z dpdqN !(2 π ) N e − βH , (1)where β ≡ T − , N is the number of particles, H is the system Hamiltonian which depends onthe momentum p , the coordinates q of each particle, and the temperature of the system. Theintegral is computed over all momenta and coordinates of the particle (we use the notation R dpdp ≡ R d N pd N q ).The thermodynamics of the system is obtained from the partition function A ( V, T ) = − T ln Q N , (2)where A is a thermodynamic function to be determined later. For instance, in the Standard Sta-3istical Mechanics ( SM ), one works with a temperature independent Hamiltonian and A ( V, T )is the free energy.In the SM , all the other thermodynamics functions can be found from A ( V, T ) using thethermodynamics relations: P = − ∂A∂V ! T ,S = − ∂A∂T ! V ,U = A + T S, (3)where P is the pressure, S is the entropy, and U is the internal energy.Returning to the case of a temperature dependent Hamiltonian, it can be deduced fromEqs. (1) and (2) the following identity1 N !(2 π ) N Z dpdqe β [ A ( V,T ) − H ( p,q,T )] = 1 . Differentiating it with respect to β on both sides, and averaging the resulting expression, oneobtains A ( V, T ) − T ∂A ( V, T ) ∂T − h H ( p, q, T ) i + T * ∂H ( p, q, T ) ∂T + = 0 . (4)The symbol h g i means the average of the function g in the SM , and it is defined as h g i = (2 π ) N N ! Z dqdpg ( q, p ) ρ, where ρ is the ensemble distribution function, given by ρ = e − βH Q N . In the SM the last term of the Eq. (4) vanishes, and the internal energy is assumed tobe U = h H i . Then, comparing equations (3) and (4), one concludes that A has to be thefree energy, as stated before. As the last term of Eq. (4) does not vanish in general, the usual4efinition can not be used here. Thus, some questions appear for a temperature dependentHamiltonian: which thermodynamic function is A ? Has U the same definition as in the SM ?We will show that there are innumerable possible answers to these questions, with theconstraint that one always has to recover the standard Statistical Mechanics in the limit ofa T independent Hamiltonian. For any of the possible solutions, it is necessary to redefinethe connection between statistical mechanics and thermodynamics, in such a way that thethermodynamics of the system is also consistently built.As Eq. (4) is the one where the problem is explicit, one should start from it. First, one canredefine the free energy as a function of the SM free energy A : A ′ ( V, T, f ( T )) = A ( V, T, f ( T )) + αB ( V, T ) , (5)where B ≡ B ( V, T ) is an extra term which will be chosen to let the thermodynamics formulationconsistent, α is an arbitrary real constant, and f ( T ) is the additional temperature contributionto the thermodynamics functions, which appears because of the T dependent Hamiltonian. Inthe limit of a T independent Hamiltonian, B ( T ) has to be zero, so one recovers all expressions ofthe SM . The redefined free energy A ′ has to satisfy all the thermodynamics relations, meaningthat when replacing Eq. (5) into Eq (3), one finds: P ′ = − ∂A ′ ∂V ! T , (6) S ′ = − ∂A ′ ∂T ! V , (7)and A ′ = U − T S + αB. (8)The quantities without prime are those calculated in the SM , i.e. they are calculated regarding f ( T ) as a constant. In order to let Eq. (8) to have the same form of the third thermodynamicrelation given by Eq. (3), it is necessary to redefine U and S :5 ′ ( V, T, f ( T )) = S ( V, T, f ( T )) − γ B ( V, T ) T ,U ′ ( V, T, f ( T )) = U ( V, T, f ( T )) + ηB ( V, T ) , (9)where γ and η are arbitrary constants, but constrained by α = γ + η . Using the second relationof Eq. (3) and Eqs. (5) and (9), one obtains γ BT = α ∂B∂T + dfdT ∂A∂f ! V,T , and manipulating the partial differential equation, one finds two possible solutions, ∂∂T ( BT − γα ) = − T − γα α dfdT ∂A∂f ! V,T α = 0 ,B = Tγ dfdT ∂A∂f ! V,T α = 0 . (10)Using the definitions of U ′ , S ′ and A ′ in Eq. (4), one obtains γB = T α ∂B∂T + T * ∂H∂T + , or, ∂∂T ( BT − γα ) = − T − γα α * ∂H∂T + α = 0 ,B = Tγ * ∂H∂T + α = 0 . (11)As the entropy in the SM can be written as S = β h H i + ln Q N = −h ln ρ i , so, from Eq. (9), one is able to find the connection between the ensemble distribution functionand the redefined entropy: 6 ′ ( V, T, f ( T )) = −h ln ρ i − γ B ( V, T ) T . (12)With the redefined thermodynamics functions, one then constructs a general model to treatthe case of a T dependent Hamiltonian, where all the thermodynamics relations are satisfied.Hamiltonian functions which depends on the temperature of the system appears in problemswith mean field approximation, as in the theory of nuclear matter [18], or in phenomenologicalmodels of Quark Gluon Plasma [12, 14, 17]. One is able to recover the thermodynamics consis-tency of systems with a T -dependent Hamiltonian if proper care of the extra term B is taken.The natural question is: what is the meaning of B ? As the N ! factor in Eq. (1) can not bejustified classically, the meaning of B is to be interpreted with the aid of quantum mechanics.For a quantum system, the canonical ensemble with a Hamiltonian operator ˆ H has thefollowing definition: Q N ≡ T re − β ˆ H , ˆ ρ = e − β ˆ H Q N , h ˆ g i = T r (ˆ g ˆ ρ ) , (13)where ˆ g is a given hermitian operator and ˆ ρ is the ensemble density operator. Based in theredefined classical internal energy Eq. (9), one is able to write the redefined quantum internalenergy, with the help of Eq. (13), as U ′ = h ˆ H i + ηB = T r [ ˆ ρ ( ˆ H + η ˆ B )] , with ˆ B = B , where is the unitary matrix. Comparing this equation for U ′ with Eq. (13),one is lead to assume the term inside the brackets as the total Hamiltonianˆ H T = ˆ H + ˆ E , (14)where ˆ E = η ˆ B . As B does not depend on the momenta and coordinates, the density operatorfor a general Hamiltonian ˆ H T has exactly the same form as that given by Eq. (13). Also, as7 ρ does not change for the redefined Hamiltonian, the entropy does not change as well. Thequantum statistical mechanics relations are then written as S ′ = −h ln ˆ ρ i − γ BTA ′ = − T ln Q TN + γBU ′ = h ˆ H T i . (15)Here, Q TN is the partition function written in terms of ˆ H T . Note that for γ = 0, the ther-modynamics relations have the same form as those of the SM . The interpretation for B wasfirst given by Gorenstein and Yang in Ref. [17], where they make the observation that in thestandard case of a T independent Hamiltonian, the zero point energy is a constant and it is usu-ally subtracted out because experiments measure only energy differences. In the quasi-particlemodel, the dispersion relation is T dependent, and so is the zero point energy of the system.Thus, it can not be discarded from the energy spectrum. In this sense, ηB is the system energyin the absence of quasi-particle excitations, i.e. the system lowest state energy. Note from Eqs.(10), (11) and (15), that when one of the three constants, α , γ or η are zero, the correspondingthermodynamics relations are independent of the two remaining constants, i.e., they cancel eachother.In this section, it was determined all possible mathematical solutions for the formulationof a consistent thermodynamics for systems with a T dependent Hamiltonian. The constants η and γ can assume any value, consequently α as well, but some of the values of these constantsare of practical use in physics. In the next subsections, three of these particular situations willbe developed. The first solution to be dealt with is the one with γ = 0 and α = η , which implies that entropyis not changed, while the free energy and the internal energy are: A ′ = − T ln T r (cid:16) e − β ˆ H T (cid:17) = − T ln T r (cid:16) e − β ˆ H (cid:17) + B ′ = 1 Q TN T r (cid:16) ˆ H T e − β ˆ H T (cid:17) = 1 Q N T r (cid:16) ˆ He − β ˆ H (cid:17) + B. (16)From Eqs. (10) and (11), one determines the B term: B = B − Z dT * ∂ ˆ H∂T + , (17)where B is an integration constant. Using this equation for B , the redefined thermodynamicsfunctions are S ′ = −h ln ˆ ρ i ,A ′ = − T ln Q N − Z dT * ∂ ˆ H∂T + + B ,U ′ = h ˆ H i − Z dT * ∂ ˆ H∂T + + B . (18)This solution was first used by Gorenstein and Yang [17] to study a gluon plasma witha T dependent Hamiltonian, where the gluon dispersion relation was assumed to have a T dependent gluon mass. In that case, the f ( T ) function is given by the square of the quasi-gluonmass. This solution was used in several works, where the quark-gluon plasma is treated by aquasi-particle model [12, 14, 15, 16], and it can be physically supported: if one views a hadronthrough a bag model, then one deals with a “bag pressure” and a “bag energy”. When heat isadded to the system, there will be some reminiscence from the bag on the energy and on thepressure: B plays this role - generally B is referred to as bag energy or bag pressure. One other possible solution is η = 0 and α = γ . Then, according to Eq. (15), the internal energyis unchanged, and the entropy and the free energy are given by: A ′ = − T ln T r (cid:16) e − β ˆ H T (cid:17) + B = − T ln T r (cid:16) e − β ˆ H (cid:17) + BS ′ = 1 Q TN T r (cid:16) e − β ˆ H T ln ˆ ρ T (cid:17) − βB = 1 Q N T r (cid:16) e − β ˆ H ln ˆ ρ (cid:17) − βB. (19)9qs. (10) and (11) can be rewritten for η = 0 as: ∂∂T (cid:18) BT (cid:19) = − T dfdT ∂A∂f ! V,T = − T * ∂ ˆ H∂T + . (20)Computing B from Eq. (20), the redefined thermodynamics functions are S ′ = −h ln ˆ ρ i + Z dT T * ∂ ˆ H∂T + − B T ,A ′ = − T ln Q N + Z dT T * ∂ ˆ H∂T + − B T ! ,U ′ = h ˆ H i . (21)This second solution was used in Ref. [19]. It describes the quark-gluon plasma by a quasi-particle model satisfying all thermodynamics relations as well. However, here the physics moti-vation is that the whole interaction energy goes to the quasi-particle mass, i.e. there is no extraterm in the usual internal energy expression. A third solution consists in α = 0 and η = − γ . This is simpler than the others, since with α = 0, Eqs. (10) and (11) for B are not partial differential equations. In this solution, the freeenergy is unchanged, while the entropy and the internal energy are: S ′ = 1 Q TN T r (cid:16) e − β ˆ H T ln ˆ ρ T (cid:17) + βB = 1 Q N T r (cid:16) e − β ˆ H ln ˆ ρ (cid:17) + βB,U ′ = 1 Q TN T r (cid:16) ˆ H T e − β ˆ H T (cid:17) = 1 Q N T r (cid:16) ˆ He − β ˆ H (cid:17) + B. (22)The redefined thermodynamics functions are then S ′ = −h ln ˆ ρ i − * ∂ ˆ H∂T + ,A ′ = − T ln Q N ,U ′ = * ∂ ( β ˆ H ) ∂β + . (23)10ote that the internal energy in Eq. (23) is the only one of the three solutions in which theconnection between the partition function and U ′ has the same form as in the SM case [20]: U ′ = − ∂∂β ln Q N . In the quasi-particle model context of the quark-gluon plasma, the third solution can beinterpreted as the following: the quasi-particle changes the zero point energy, such that itbecomes a function of T (just as the quasi-particle mass). This effect occurs because the wholeinteraction energy can not be accommodated in the quasi-particle mass, and part of it is usedby the vacuum to modify its zero point energy, represented here by B ( T ). In this solution,the entropy definition is also modified, which may be seen as a problem. However, as it is wellknown, the entropy of a system is determined up to a constant that is usually subtracted fromthe system entropy. This procedure can not be done here, because for the case of a T dependentdispersion relation, the additive term to the entropy will not be a constant, but also a functionof the temperature. In the previous section, we have constructed a self-consistent thermodynamics for a systemwith a T dependent Hamiltonian. In this section we will apply this theory for one specific case:a plasma composed only by gluons, with a vanishing chemical potential. The quasi-particlemodel implies that we will treat the plasma as a non-interacting gas of massive, temperaturedependent, gluons. This problem has already been solved for two particulars solutions, solution1 [17] and 2 [19] of the previous section, of the general solution given there. To exemplify ourmethod we choose the easiest solution, solution 3, which gives the simplest extra term B . Wewill see that we can obtain an algebraic solution for the thermodynamics function, differentlyfrom solutions 1 and 2. 11 .1 The α = 0 Solution
It is interesting to work in a grand canonical ensemble because the thermodynamics functionsare easier to compute there, but we have developed the theory in a canonical ensemble. Never-theless, for the µ = 0 case it is not necessary to redefine the whole theory in the grand canonicalensemble, because all that is required for this change of ensemble is to use the grand canonicalpartition function, Z , instead of Q N . This connection is justified because both ensembles areequivalent in the thermodynamical limit, and for µ = 0 the additional thermodynamics relationfor N , N = − ∂A/∂µ , disappears. The grand partition function is given by: Z ( z, V, T ) ≡ ∞ X N =0 z N Q N ( V, T ) , (24)where we introduced the fugacity z , z = e βµ . (25)The distribution function ρ is ˆ ρ = e − β ˆ H Z , and the mean value is defined as h A i = P ∞ N =0 Az N Q N ( V, T ) P ∞ N =0 z N Q N ( V, T ) . Using the results given in the Sec. 2.3, one has the redefined thermodynamics functions: S ′ = −h ln ˆ ρ i − * ∂ ˆ H∂T + ,P ′ V = T ln Z,U ′ = * ∂ ( β ˆ H ) ∂β + = − ∂∂β ln Z, (26)where the thermodynamic relation A = − P V was used. From quantum statistical mechanics,the partition function for a free gas is 12 ( V, T ) = Y k − e − βω k , or ln Z = − X k ln(1 − e − βω k ) = − V (2 π ) Z πdkk ln(1 − e − βω ) . where the dispersion relation is ω = k + m ( T ). Therefore P ′ and U ′ are given by Eqs. (26),and take the form P ′ ( T ) = − νT π Z dkk ln(1 − e − βω ) = ν π Z ∞ dkf ( k ) k q k + m ( T ) (27) e ′ ( T ) = U ′ V = ν π Z ∞ dkf ( k ) k q k + m ( T ) − νT π ∂m ∂T Z ∞ dkf ( k ) k q k + m ( T ) , (28)where ν is the gluon degeneracy factor, e ′ ( T ) = U ′ V is the energy density and f ( k ) = ( e βω − − is the Bose-Einstein distribution function. Note the second term in the expression for the energy.It exists only because m = m ( T ), and it will be denoted by b ( T ) ≡ B ( T, V ) /V .Using the relation S ′ T = U ′ + P ′ V , one obtains: s ′ ( T ) = ν π T Z ∞ dkf ( k ) k k + 3 m ( T ) q k + m ( T ) − ν π ∂m ∂T Z ∞ dkf ( k ) k q k + m ( T ) . (29)If one uses the thermodynamic relation S ′ = − [ ∂ ( P ′ V ) /∂T ] V to calculate the entropydensity, with the help of P ′ given by Eq. (27), one finds exactly the same expression Eq. (29)for s ′ , confirming that the calculation is self-consistent; the additional term b ( T ) in e ′ ( T ) and s ′ ( T ) guarantees the thermodynamical consistency of the qQGP expressions. If one comparesthis extra term with the usual extra term of qQGP, the Gorenstein-Yang solution (Eq.(5) ofRef. [14]), one finds the relation b ( T ) = ∂B ( T ) /∂T . This implies that it is necessary to computean integral in T in order to get the bag constant B ( T ). In the present approach, there is not anintegral in T , what facilitates analytical calculations. Also, as m ( T ) depends on the couplingconstant, in the form of a logarithm in T , B ( T ) is harder to compute.13 .1.1 Pressure The pressure P ′ ( T ) is given by Eq. (27), and it can be rewritten as P ′ ( T ) = νβ − π Z ∞ dx x √ x + r e √ x + r − νT π I ( r ) , (30)where r ≡ mT and I is Eq. (55) of Appendix A . Eq. (30) becomes P ′ ( T ) = νπ " π T − π T m + π T m + m (cid:18) log m πT + γ E − (cid:19) + 18 ∞ X n =1 a n m n +2) T n , with a n = ( − n (2 n − ζ (2 n +1)( n +2)!2 n +1 π n . Rewriting the above integral as a function of the ideal masslessgas pressure, P = νπ T , one obtains the pressure for the massive gas of quasi-gluons: P ′ ( T ) = P " − π (cid:18) mT (cid:19) + 152 π (cid:18) mT (cid:19) + 4516 π (cid:18) mT (cid:19) (cid:18) log m πT + γ E − (cid:19) ++ 454 π ∞ X n =1 a n (cid:18) mT (cid:19) n +2) . (31)Note that the functional form of m ( T ) is unknown, i.e. statistical mechanics does not providethe function m ( T ). This problem will be treated on the next section. The energy density e ′ ( T ) is given by Eq. (28), and using the same technique as used in thecalculation of the pressure, one is able to get the expression for the energy density dependenton the temperature: e ′ ( T ) = νT π Z ∞ dx x √ x + r e √ x + r − νT π m − T ∂m ∂T ! Z ∞ dx x √ x + r e √ x + r −
1= 12 νT π I ( r ) + νT π m − T ∂m ∂T ! I ( r ) , (32)where I and I are Eqs. (54) and (55) of Appendix A . Solving these integrals one obtains:14 ′ ( T ) = e ( − π (cid:18) mT (cid:19) − π (cid:18) mT (cid:19) (cid:18) log m πT + γ E + 14 (cid:19) − π ∞ X n =1 a n (2 n + 1) (cid:18) mT (cid:19) n +2) + − T π ∂m ∂T " − π mT − π m T (cid:18) log m πT + γ E − (cid:19) − π ∞ X n =1 a n ( n + 2) (cid:18) mT (cid:19) n +1) . (33)where e = νT π is the energy density of the ideal gas. As the expression for the pressure for the massive gas of quasi-gluons has been obtained, it iseasier to compute the entropy density through the use of a Maxwell relation, and one gets: s ′ ( T ) = ∂P ′ ∂T = νπ ∂∂T " T − π (cid:18) mT (cid:19) + 152 π (cid:18) mT (cid:19) + 4516 π (cid:18) mT (cid:19) (cid:18) log m πT + γ E − (cid:19) ++ 454 π ∞ X n =1 a n (cid:18) mT (cid:19) n +2) ! . Using ∂∂T (cid:18) mT (cid:19) = ∂m∂T T − mT = 12 mT ∂m ∂T − mT ,s ′ ( T ) is written as: s ′ ( T ) = s ( − π (cid:18) mT (cid:19) + 158 π (cid:18) mT (cid:19) − π (cid:18) mT (cid:19) − π ∞ X n =1 a n n (cid:18) mT (cid:19) n +2) − T π ∂m ∂T " − π mT − π m T (cid:18) log m πT + γ E − (cid:19) − π ∞ X n =1 a n ( n + 2) (cid:18) mT (cid:19) n +1) . (34)where s = νπ T the ideal gas’ entropy density. The temperature dependence of the thermodynamics functions of a quasi-particle gas can not becompletely described using statistical mechanics only. In particular, to determine the expressionfor m ( T ) it is necessary some further information. First, we take HTL perturbation theory1510, 16] as the additional information. HTL provides a relation between m ( T ) and T that,asymptotically, is m ( T ) = N c g s T , (35)where g s is the strong coupling constant. With this input, one can compute the explicit temper-ature dependence of the thermodynamics functions. As we are interested not just in the leadingorder behavior but also in the higher order corrections, we will implement these higher ordercorrections to the thermal mass of the HTL approach [10], δm = − N c g s π T m D , (36)where m D = q N c g s T is the Debye mass. For the values of the strong coupling constantat high temperatures, g s ≪
1, expression (36) should describe correctly the next-to-leadingorder temperature dependence of the mass. However this equation runs into problems, givinga tachyon mass for the gluons, depending on the value of g s . As the aim is to construct a self-consistent model to describe the deconfined QCD phase using quasi-particles, we can not takethis next-to-leading mass as the complete quasi-gluon asymptotic mass but as an approximationto the complete mass. Analyzing this expression inside the HTL picture [10], it is seen thatEq. (36) comes exclusively from hard momenta corrections, i.e. soft momenta corrections donot enter. The complete description of the asymptotic mass term requires both soft and hardcorrections, and these will be simulated through a quadratic gap equation given by, m ( T ) = N c g s T − N c √ π g s T m ( T ) . (37)Using Eq. (37) in Eq. (31), one is able to compare the present results with lattice QCD data[2]. For the calculation of the temperature dependence of the coupling constant, we used [22]: g s ( T ) = 48 π N c log[ λT /T c + T s /T c ] , (38)with λ and T s phenomenological parameters, and N c = 3 is the number of colors that will beused hereafter. The optimal fit is achieved using λ = 5 . T s /T c = − . ν = 2( N c −
1) = 16. In Fig. 1 is plotted the thermodynamics functions Eqs. (31), (33)and (34), using the mass relation Eq. (37). The calculated curves describe quite well the latticedata [2]. For comparison, in Ref. [14], where the qQGP was based on the Gorenstein and Yangsolution, it was necessary 4 parameters to get an optimal fit of the lattice data: λ , T s /T c , theintegration constant B and the degeneracy ν . / T c / T / T / T Figure 1: Plots of the pressure, energy density and entropy density in our model, and theextrapolated lattice data [2] for the pressure (star), energy density (triangle) and entropy density(diamond), as a function of
T /T c . α = 0 Solutions
In the last section, the α = 0 solution was studied in some detail using explicit expressions.For the α = 0 cases, however, the calculation of the B term involves non trivial integrals.Nevertheless, if one is interested only in the asymptotic behavior of the gluon plasma, anexplicit calculation can also be done. In this sub-section, we will focus on this calculation.The general expression for the α = 0 solution for B is given by Eq. (11). Using the quasi-gluon dispersion relation, one can rewrite B as B = B T γα − νV T γα α π Z TT c dτ τ − γα dm ( τ ) dτ Z ∞ dk k f ( k ) q k + m ( τ ) . (39)17o analyze the high T behavior of B ( T ), we manipulate Eq. (39) in order to use the form foundin Appendix A: B = B T γα − νV T γα α π Z TT c dτ τ − γα dm ( τ ) dτ Γ(3) τ I (cid:18) mτ (cid:19) = B T γα − νV T γα α π Z TT c dτ τ − γα dm ( τ ) dτ π − π mτ + O m τ !! . (40)Integrating and using just the first two terms inside the parenthesis, one gets B = B T γα − νV α " m τ (cid:12)(cid:12)(cid:12) TT c − (cid:18) − γα (cid:19) T γα Z TT c dτ τ − γα m ++ νV απ " m τ (cid:12)(cid:12)(cid:12) TT c − (cid:18) − γα (cid:19) T γα Z TT c dτ τ − γα m + ... (41)On the other hand, for g s ≪
1, Eq. (37) is reduced to mT = g s √ − g s √ π . (42)At very high temperatures, the coupling constant, Eq. (38), decreases very slowly, implyingthat the ratio m/T is a slowly decreasing function at T /T c ≫
1. Hence, m/T can be regardedas a constant, and the integral can be easily done. Rewriting the integrand in m/T powers, onecan compute the integral of the B term at very high temperatures: B ≈ B T γα − νV α " m T − (cid:18) − γα (cid:19) T γα m T T − γα − γα + νV απ " m T − (cid:18) − γα (cid:19) T γα m T T − γα − γα = B T γα − νV α − γ/α m T + νV απ − γ/α m T, γα = 4 . (43)Note that Eq. (43) is not valid for γα = 4. For γα = 4, one has B ≈ B T − νV α (1 + 2 log T ) m T + νV απ (1 + 3 log T ) m T, γα = 4 . (44)18s the asymptotic mass is proportional to T in first order, one has to maintain the B term.To obtain the asymptotic thermodynamics functions, we write the asymptotic B in terms ofideal massless gas pressure P : b ≈ P b νπ − απ (1 + 2 log T ) m T + 152 απ (1 + 3 log T ) m T ! , γα = 4; b ≈ P b νπ T γα − − απ − γ/α m T + 452 απ − γ/α m T ! , γα = 4 , (45)where b ≡ B V . The thermodynamics functions can now be computed. For instance, with thehelp of Eqs. (5), (31) and (45) one is able to obtain the asymptotic expression for the pressurefor α = 0 and γ/α = 4: P ′ ( T ) = P " − b ανπ + 152 π m T log T − π m T log T + . . . . From Eq. (38), it is seen that the T logarithm is inversely proportional to g s . Thus, with thehelp of the asymptotic mass Eq. (35), one obtains the asymptotic pressure: P ′ ( T ) = P " − b ανπ + 3011 + 9011 √ g s π + . . . . When the coupling constant goes to zero, P ′ ( T ) should be that of an ideal massless gas. To thisaim, one has to choose b α = νπ /
33. The pressure is then P ′ ( T ) = P " √ g s π + . . . , γα = 4 . (46)The pressure for γ/α = 4 is easier to work with. Using Eqs. (5), (31) and (45) one gets P ′ ( T ) = P " − π − γα − γα (cid:18) mT (cid:19) + 152 π − γα − γα (cid:18) mT (cid:19) − αb νπ T γα − + . . . , γα = 4 . (47)In section II, the general mathematical solution for a consistent thermodynamics of massivegluons was handled. We now want to address the problem from the physics point of view: whichsolutions are physically relevant? 19he first feature that one has to keep in mind is that the gluon plasma must behave asan ideal massless gas in the limit T → ∞ . This property implies that the pressure has to beproportional to T in this limit. As it was seen, the solution for α = 0 and γ/α = 4 does nothave a problem in the T → ∞ limit. For the other solutions, one has to go back to Eq. (47)and analyze the pressure. As m/T is proportional to powers of g s , the terms in m/T do notdisagree with the ideal massless gas limit. On the other hand, the term involving b α does. Theonly way to solve this problem is to make b α term vanish, what implies that the powers of T must be negative. Hence, the general solution does not make physical sense for γ/α >
4. Asa result, the general solution obtained in section II has possible physical meaning only if thecondition γ/α ≤ weak physical condition . Noticethat solutions 1, 2 and 3 satisfy the weak physical condition. We start with solution 1 at very high temperatures. The pressure in this scenario will be denotedby P , and can be written with the help of Eq. (47) and the condition γ = 0 as: P ( T ) = P " − π (cid:18) mT (cid:19) + 158 π (cid:18) mT (cid:19) + . . . . (48)Introducing the corrected asymptotic mass, Eq. (42), in Eq. (48), one obtains P = P " − π g s + 154 π g s √ √ ! + . . . . (49)On the other hand, the perturbative QCD expression for the pressure at high temperature[21],known so far up to order g s log(1 /g s )[5], is: P QCD = P " − g s π + 154 g s π + . . . . (50)Comparing Eqs. (49) and (50), one sees that the zero and second order terms match, but the g s term does not . Even so, besides the weak physical condition being satisfied, and the matching The number inside the parenthesis which multiplies g s in the Eq. (49) is ∼ . g s order, solution 1 has anotherquality: the connection between thermodynamics and statistical mechanics can be written insuch a way which preserve the same form of standard statistical mechanics, Eq. (15) with γ = 0.One has to remember that the extra term B is usually interpreted as a modification in thezero point energy, in other words, the vacuum is modified. If the vacuum changes, and now itis T dependent, it is expected an entropy associated with this vacuum, and consequently a T dependent vacuum entropy. But solution 1 does not have an extra term associated to a vacuumentropy, which can be seen as a limitation of this particular solution.The pressure for solution 2 at extremely high temperatures, denoted by P , is: P = P " − π (cid:18) mT (cid:19) + O ( m /T ) + . . . . Notice that there is not a correction in m /T . This happens because the extra term B cancelsout the term coming from the SM pressure. If one uses the HTL asymptotic mass in the pressure P , then P = P " − g s π + 1524 g s π + . . . . (51)The pressure P does not match with pQCD at any order in g s . The term g s is about 1 . g s is 6 times smaller. However, there is one way for solution 2 tomatch the pQCD result. If one uses m = 3 g s T / m = ω = N c g s T . The effect of using the plasma frequency leaves the pressuremore convergent than P and P QCD .As it was seen in section 3 .
1, solution 3 is the only solution which has algebraic expressionsfor the thermodynamics functions. This solution, asymptotically, has a pressure P given by: P = P " − g s π + 158 g s π (3 + √
2) + . . . . (52)21omparing the asymptotic behavior of the P , calculated with the HTL asymptotic mass,with the perturbative QCD pressure, one sees that there is no match at any order, as expected.However, if instead of the HTL mass one uses m = g s T for the mass, the QCD pressure up to g s is recovered. Motivated by the HTL gap equation, one can modify the mass relation in generaland try a matching also at higher orders. One way to do this is dividing Eq. (37) by two, andredefining m √ → m . At lowest order the mass will be g s T /
2. Extending the mass equation tosecond order, the g s term for the pressure will match the perturbative QCD result well. Hence,as a bonus, solution 3 with a modified gap equation for the thermal mass, results in a matchbetween P and P QCD up to the g s order.As the internal energy and entropy are changed by a T dependent function in solution 3,the extra B term can be regarded as a zero-point energy, with an entropy associated with it.It is important to emphasize that as one has already obtained an algebraic expression for thepressure at the whole temperature range, it is possible to fit m ( T ) by QCD lattice data, andsearch for deviations of the HTL asymptotic mass near of T c .The solution γ/α = 4 has also to be analyzed. This solution is the only one which has aterm of order g s . Although it satisfies the weak physical condition, the contribution of order g s contradicts any pressure computation in the literature, either HTL, 2-Loop Φ-derivableapproximation or perturbative QCD. The absence of a term in g s can be justified by thermalfield theory, since the first-order correction to the partition function is proportional to g s [26].Therefore, we will let this solution out of the possible physical solutions. In this work we have studied the thermodynamics and the statistical mechanics of a system witha temperature dependent Hamiltonian in the canonical ensemble. Earlier works have shown thata system with a T dependent Hamiltonian is thermodynamically inconsistent if the connectionbetween thermodynamics and statistical mechanics is the same as in the T independent Hamil-tonian case. In order to have a general solution to this problem, we developed a formalism whichgives a general connection between thermodynamics and statistical mechanics, where all the22hermodynamics relations are satisfied. We have seen that it is necessary to add an extra termto the thermodynamics relations to guarantee the thermodynamics consistency, and dependingon the choice that is made for the parameters α , γ and η , the extra term B can be simpler.The general solution developed here is a generalization of the solution proposed by Goren-stein and Yang [17]. In their work, they have introduced one particular manner to maintainthe thermodynamics consistency: a correction coming from the zero point energy of the Hamil-tonian modifies the pressure and the internal energy of the system, while leaving the entropyunchanged. In the present work, it is shown that letting the Hamiltonian function to be depen-dent on the temperature of the system, represented by a function f ( T ), implies that there are,in principle, a large number of ways to render the thermodynamics of the system consistent.We emphasize that this procedure is applicable in any system with a T dependent Hamiltonian,and not only in the case of quasi-particle models.For an ideal quasi-gluon gas, we computed the pressure, the energy and the entropy, withinthe formalism developed in section 2. This calculation, named solution 3, is easier to work withas no integral in T is necessary to determine the extra term B . As the formalism does not providean expression for the thermal mass m ( T ), we used the mass calculated in the HTL approach.We achieved an optimal fit for the thermodynamics functions with two free parameters in theexpression for the strong coupling at finite T .We also studied the general asymptotic behavior of the solutions contained in α = 0 cases.The analysis of these solutions resulted in a reduced number of possible physical solutions. Themain properties of solutions 1, 2, and 3 were discussed, and it was showed that solution 1 isthe only one of the three solutions which recovers pQCD to order g s when the HTL asymptoticmass is used. In order for solutions 2 and 3 recover pQCD, it is necessary to change the gapequation for the mass. Finally, we saw that the only solutions that have physics meaning arethose where the constants α , γ and η satisfy the weak physical condition, α = 0 or γ/α < α = γ + η . An extension of the present calculation to the case ofa finite chemical potential will be presented in the near future.This work was supported by FAPESP (04/15276-2) and CNPq (307284/2006-9).23 Appendix
We need results for integrals like I n ( r ) = 1Γ( n ) Z ∞ dx x n − ( x + r ) e ( x + r ) − . (53)The relevant integrals for our calculation, following [21, 27] and generalizing their results, arethe following: I ( r ) = π − π r − r (cid:18) log r π + γ E − (cid:19) − ∞ X m =1 a m ( m + 2) r m +1) . (54) I ( r ) = π − r π π r + r (cid:18) log r π + γ E − (cid:19) + 12 ∞ X m =1 a m r m +2) . (55)where a m = ( − m (2 m − ζ (2 m +1)2 m +1 π m ( m +2)!! . If one tests the series for convergence, for instance the ratiotest, one finds that the series are convergent for r < π . References [1] A. Ali Khan et al. (CP-PACS Collaboration),Phys. Rev.
D63 , 034502 (2001).[2] F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B , 579 (2001)[arXiv:hep-lat/0012023].[3] M. Cheng et al., hep-lat/0608013.[4] J. C. Collins and M. J. Perry, Phys. Rev. Lett. , 1353 (1975).[5] K. Kajantie, M. Laine, K. Rummukainen and Y. Schroder, Phys. Rev. D , 105008 (2003)[arXiv:hep-ph/0211321].[6] P. Arnold e C.-X. Zhai, Phys. Rev. D , 1906 (1995); C.-X. Zhai e B. Kastening, Phys.Rev. D , 7232 (1995);[7] E. Braaten and R. D. Pisarski, Phys. Rev. D , 1827 (1992).248] J. O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. , 2139 (1999)[arXiv:hep-ph/9902327].[9] J. O. Andersen, E. Braaten, E. Petitgirard and M. Strickland, Phys. Rev. D , 085016(2002) [arXiv:hep-ph/0205085].[10] J. P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. , 2906 (1999)[arXiv:hep-ph/9906340]; J. P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. D , 065003(2001) [arXiv:hep-ph/0005003].[11] J. P. Blaizot, E. Iancu and A. Rebhan, arXiv:hep-ph/0303185.[12] A. Peshier, B. Kampfer, O. P. Pavlenko and G. Soff, Phys. Lett. B337 , 235 (1994).[13] P. L´evai and U. W. Heinz, Phys. Rev. C , 1879 (1998) [arXiv:hep-ph/9710463].[14] A. Peshier, B. Kampfer, O. P. Pavlenko and G. Soff, Phys. Rev. D , 2399 (1996).[15] R. A. Schneider and W. Weise, Phys. Rev. C64 , 055201 (2001).[16] A. Rebhan and P. Romatschke, Phys. Rev. D , 025022 (2003) [arXiv:hep-ph/0304294].[17] M. I. Gorenstein and S. N. Yang, Phys. Rev. D , 5206 (1995).[18] J. D. Walecka, Phys. Lett. B59 , 109 (1975).[19] V. M. Bannur, arXiv:hep-ph/0508069; V. M. Bannur, arXiv:hep-ph/0608232.[20] K. Huang,
Statistical Mechanics (John Wiley e Sons, Inc., 1963).[21] J.I. Kapusta, Nucl. Phys.
B148 , 461 (1979);
Finite Temperature Field Theory (CambridgeUniversity Press, Cambridge, MA, 1989).[22] A. Peshier, B. Kampfer and G. Soff, Phys. Rev. C , 045203 (2000)[arXiv:hep-ph/9911474].[23] S. Gupta, Phys. Rev. D , 034507 (2001) [arXiv:hep-lat/0010011].2524] R. A. Schneider, arXiv:hep-ph/0303104.[25] K. Johnson, C.B. Thorn, A. Chodos, R. L. Jaffe and V. F. Weisskopf, Phys. Rev. D , 356(1974); K. Johnson, A. Chodos, R. L. Jaffe and C.B. Thorn, Phys. Rev. D , 2599 (1974).[26] M. Le Bellac, Thermal Field Theory (Cambridge University Press, 1996).[27] L. Dolan and R. Jackiw, Phys. Rev. D9